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Michael Porter's generic strategies describe how a company can pursue competitive advantage across its chosen market scope. There are three generic strategies: lower cost, product differentiation, or focus. The focus strategy has two variants, cost focus and differentiation focus, so it is possible to see the concept in terms of four distinct strategies.
A company chooses to pursue one of two types of competitive advantage, either via lower costs than its competition or by differentiating itself along dimensions valued by customers to command a higher price. A company also chooses one of two types of scope, either focus (offering its products to selected segments of the market) or industry-wide, offering its product across many market segments. The generic strategy reflects the choices made regarding both the type of competitive advantage and the scope. The concept was described by Michael Porter in 1980.
== Concept ==
Porter wrote in 1980 that strategy targets either cost leadership, differentiation, or focus. These are known as his three generic strategies, which can be applied to any size or form of business. Porter claimed that a company must only choose one of the three or risk that the business would waste precious resources. Porter's generic strategies detail the interaction between cost minimization strategies, product differentiation strategies, and market focus strategies of firms.
Michael Porter described an industry as having multiple segments that can be targeted by a firm. The breadth of its targeting refers to the competitive scope of the business. Porter defined two types of competitive advantage: lower cost or differentiation relative to its rivals. Achieving competitive advantage results from a firm's ability to cope with the five forces better than its rivals. Porter wrote: "Achieving competitive advantage requires a firm to make a choice ... about the type of competitive advantage it seeks to attain and the scope within which it will attain it." He also wrote: "The two basic types of competitive advantage [differentiation and lower cost] combined with the scope of activities for which a firm seeks to achieve them lead to three generic strategies for achieving above average performance in an industry: cost leadership, differentiation and focus. The focus strategy has two variants, cost focus and differentiation focus."
In general:
If a firm is targeting customers in most or all segments of an industry based on offering the lowest price, it is following a cost leadership strategy;
If it targets customers in most or all segments based on attributes other than price (e.g., via higher product quality or service) to command a higher price, it is pursuing a differentiation strategy. It is attempting to differentiate itself along these dimensions favorably relative to its competition. It seeks to minimize costs in areas that do not differentiate it, to remain cost competitive; or
If it is focusing on one or a few segments, it is following a focus strategy. A firm may be attempting to offer a lower cost in that scope (cost focus) or differentiate itself in that scope (differentiation focus).
The concept of choice was a different perspective on strategy, as the 1970s paradigm was the pursuit of market share (size and scale) influenced by the experience curve. Companies that pursued the highest market share position to achieve cost advantages fit under Porter's cost leadership generic strategy, but the concept of choice regarding differentiation and focus represented a new perspective.
Porter stressed the idea that only one strategy should be adopted by a firm and failure to do so will result in the "stuck in the middle" scenario shown above. He discussed the idea that practising more than one strategy will lose the entire focus of the organization: hence clear direction of the future trajectory could not be established. The argument is based on the fundamental that differentiation will incur costs to the firm which clearly contradicts with the basis of low cost strategy and on the other hand relatively standardised products with features acceptable to many customers will not carry any differentiation hence, cost leadership and differentiation strategy will be mutually exclusive. Two focal objectives of low cost leadership and differentiation clash with each other resulting in no proper direction for a firm.
== Origin ==
Empirical research on the profit impact of marketing strategy indicated that firms with a high market share were often quite profitable, but so were many firms with low market share. The least profitable firms were those with moderate market share. This was sometimes referred to as the hole in the middle problem. Porter's explanation of this is that firms with high market share were successful because they pursued a cost leadership strategy and firms with low market share were successful because they used market segmentation to focus on a small but profitable market niche. Firms in the middle were less profitable because they did not have a strategy.
Porter suggested that combining multiple strategies is successful in only one case. Combining a market segmentation strategy with a product differentiation strategy was seen as an effective way of matching a firm's product strategy (supply side) to the characteristics of your target market segments (demand side). But combinations like cost leadership with product differentiation were seen as hard (but not impossible) to implement, due to the potential for conflict between cost minimization and the additional cost of value-added differentiation.
Since that time, empirical research has indicated companies pursuing both differentiation and low-cost strategies may be more successful than companies pursuing only one strategy.
Some commentators have made a distinction between cost leadership, that is, low cost strategies, and best cost strategies. They claim that a low cost strategy is rarely able to provide a sustainable competitive advantage. In most cases firms end up in price wars. Instead, they claim a best cost strategy is preferred. This involves providing the best value for a relatively low price.
== Cost leadership strategy ==
Cost leadership strategies involve the firm winning market share by appealing to cost-conscious or price-sensitive customers. This is achieved by having the lowest prices in the target market segment, or at least the lowest price to value ratio (price compared to what customers receive). To succeed at offering the lowest price while still achieving profitability and a high return on investment, the firm must be able to operate at a lower cost than its rivals. There are three main ways to achieve this.
The first approach is achieving a high asset utilization. In service industries, this may mean for example a restaurant that turns tables around very quickly, or an airline that turns around flights very fast. In manufacturing, it will involve production of high volumes of output. These approaches mean fixed costs are spread over a larger number of units of the product or service, resulting in a lower unit cost, i.e. the firm hopes to take advantage of economies of scale and experience curve effects. For industrial firms, mass production becomes both a strategy and an end in itself. Higher levels of output both require and result in high market share, and create an entry barrier to potential competitors, who may be unable to achieve the scale necessary to match the firms low costs and prices.
The second dimension is achieving low direct and indirect operating costs. This is achieved by offering high volumes of standardized products, offering basic no-frills products and limiting customization and personalization of service. Production costs are kept low by using fewer components, using standard components, and limiting the number of models produced to ensure larger production runs. Overheads are kept low by paying low wages, locating premises in low rent areas, establishing a cost-conscious culture, etc. Maintaining this strategy requires a continuous search for cost reductions in all aspects of the business. This will include outsourcing, controlling production costs, increasing asset capacity utilization, and minimizing other costs including distribution, R&D and advertising. The associated distribution strategy is to obtain the most extensive distribution possible. Promotional strategy often involves trying to make a virtue out of low cost product features.
The third dimension is control over the value chain encompassing all functional groups (finance, supply/procurement, marketing, inventory, information technology etc.) to ensure low costs. In a supply chain context, this could be achieved by bulk buying to enjoy quantity discounts, squeezing suppliers on price, instituting competitive bidding for contracts, working with vendors to keep inventories low using methods such as Just-in-Time purchasing or Vendor-Managed Inventory. Wal-Mart is famous for squeezing its suppliers to ensure low prices for its goods. Other procurement advantages could come from preferential access to raw materials, or backward integration. For a business which is in control of all functional groups this is suitable for cost leadership; for a business which is only in control of one functional group, this is differentiation. For example, Dell, the computer supplier, initially achieved market share by keeping inventories low and only building computers to order via applying differentiation strategies in supply/procurement chain. This will be clarified in other sections.
Cost leadership strategies are only viable for large firms with the opportunity to enjoy economies of scale and large production volumes and big market share. Small businesses can be "cost focused" not "cost leaders" if they enjoy any advantages conducive to low costs. For example, a local restaurant in a low rent location can attract price-sensitive customers if it offers a limited menu, rapid table turnover and employs staff on minimum wage. Innovation of products or processes may also enable a startup or small company to offer a cheaper product or service where incumbents' costs and prices have become too high. An example is the success of low-cost budget airlines who, despite having fewer planes than the major airlines, were able to achieve market share growth by offering cheap, no-frills services at prices much cheaper than those of the larger incumbents. At the beginning low-cost budget airlines chose "cost focused" strategies but later when the market grew, big airlines started to offer the same low-cost attributes, and so cost focus became cost leadership.
A cost leadership strategy may have the disadvantage of lower customer loyalty, as price-sensitive customers will switch once a lower-priced substitute is available. A reputation as a cost leader may also result in a reputation for low quality, which may make it difficult for a firm to rebrand itself or its products if it chooses to shift to a differentiation strategy in future.
== Differentiation strategy ==
Businesses operating this strategy differentiate their products/services in some way in order to compete successfully. Examples of the successful use of a differentiation strategy are Hero, Asian Paints, HUL, Nike athletic shoes (image and brand mark), BMW Group Automobiles, Perstorp BioProducts, Apple Computer (product's design), and Mercedes-Benz automobiles.
A differentiation strategy is appropriate where the target customer segment is not price-sensitive, the market is competitive or saturated, customers have very specific needs which are possibly under-served, and the firm has unique resources and capabilities which enable it to satisfy these needs in ways that are difficult to copy. These could include patents or other intellectual property (IP), unique technical expertise (e.g. Apple's design skills or Pixar's animation prowess), talented personnel (e.g. a sports team's star players or a brokerage firm's star traders), or innovative processes. Successful differentiation is displayed when a company accomplishes either a premium price for the product or service, increased revenue per unit, or the consumers' loyalty to purchase the company's product or service (brand loyalty). Differentiation drives profitability when the added price of the product outweighs the added expense to acquire the product or service but is ineffective when its uniqueness is easily replicated by its competitors. Successful brand management also results in perceived uniqueness even when the physical product is the same as competitors. This way, Chiquita was able to brand bananas, Starbucks could brand coffee, and Nike could brand sneakers. Fashion brands rely heavily on this form of image differentiation.
Differentiation strategy is not suitable for small companies. It is more appropriate for big companies to apply differentiation in any one or several of the functional groups (finance, purchase, marketing, inventory etc.). This point is critical. For example, GE uses its finance division to differentiate itself. A company may do so in isolation of other strategies or in conjunction with focus strategies (requires more initial investment). It provides a great advantage to use a differentiation strategy (for big companies) in conjunction with focus cost strategies or focus differentiation strategies. Coca-Cola and Royal Crown beverages are good examples of this.
Henry Mintzberg, another business writer, argues that cost leadership is really a form of differentiation, using a lower price as a form of differentiation.
=== Variants on the differentiation strategy ===
The shareholder value model holds that the timing of the use of specialized knowledge can create a differentiation advantage as long as the knowledge remains unique. This model suggests that customers buy products or services from an organization to have access to its unique knowledge. The advantage is static, rather than dynamic, because the purchase is a one-time event.
The unlimited resources model utilizes competitors by practicing a differentiation strategy. An organization with greater resources can manage risk and sustain profits more easily than one with fewer resources. This provides a short-term advantage only. If a firm lacks the capacity for continual innovation, it will not sustain its competitive position over time.
== Focus strategies ==
This dimension is not a separate strategy for big companies due to small market conditions. Big companies which chose applying differentiation strategies may also choose to apply in conjunction with focus strategies (either cost or differentiation). On the other hand, this is definitely an appropriate strategy for small companies especially for those wanting to avoid competition with big ones.
In adopting a narrow focus, the company ideally focuses on a few target markets (also called a segmentation strategy or niche strategy). These should be distinct groups with specialized needs. The choice of offering low prices or differentiated products/services should depend on the needs of the selected segment and the resources and capabilities of the firm. It is hoped that by focusing your marketing efforts on one or two narrow market segments and tailoring your marketing mix to these specialized markets, you can better meet the needs of that target market. The firm typically looks to gain a competitive advantage through product innovation and/or brand marketing rather than efficiency. A focused strategy should target market segments that are less vulnerable to substitutes or where a competition is weakest to earn above-average return on investment.
An example of an American business using a focus strategy is Southwest Airlines, which provides short-haul point-to-point flights in contrast to the hub-and-spoke model of mainstream carriers, United Airlines and American Airlines.
== Subsequent developments ==
Michael Treacy and Fred Wiersema (1993) modified Porter's three strategies in their book The Discipline of Market Leaders to describe three basic "value disciplines" which can create customer value and provide a competitive advantage: these are operational excellence, product leadership, and customer intimacy.
== Criticisms of generic strategies ==
Several commentators have questioned the use of generic strategies, claiming that they lack specificity, lack flexibility, and are limiting, and raising empirical challenges to Porter's model.
Miller questions the notion of being "caught in the middle". He claims that there is a viable middle ground between strategies. Many companies, for example, have entered a market as a niche player and gradually expanded. According to Baden-Fuller and Stopford (1992) the most successful companies are the ones that can resolve what they call "the dilemma of opposites". Furthermore, Reeves and Routledge's (2013) study of entrepreneurial spirit demonstrated this is a key factor in organisation success, differentiation and cost leadership were the least important factors.
However, contrary to the rationalisation of Porter, contemporary research has shown evidence of successful firms practising such a "hybrid strategy”. Research writings of Davis (1984 cited by Prajogo 2007, p. 74) state that firms employing the hybrid business strategy (Low cost and differentiation strategy) outperform the ones adopting one generic strategy. Sharing the same view point, Hill (1988 cited by Akan et al. 2006, p. 49) challenged Porter's concept regarding mutual exclusivity of low cost and differentiation strategy and further argued that successful combination of those two strategies will result in sustainable competitive advantage. As to Wright and other (1990 cited by Akan et al. 2006, p. 50) multiple business strategies are required to respond effectively to any environment condition. In the mid to late 1980s where the environments were relatively stable there was no requirement for flexibility in business strategies but survival in the rapidly changing, highly unpredictable present market contexts will require flexibility to face any contingency (Anderson 1997, Goldman et al. 1995, Pine 1993 cited by Radas 2005, p. 197). After eleven years Porter revised his thinking and accepted the fact that hybrid business strategy could exist (Porter cited by Prajogo 2007, p. 70) and writes in the following manner.
Although Porter had a fundamental rationalisation in his concept about the invalidity of hybrid business strategy, the highly volatile and turbulent market conditions will not permit survival of rigid business strategies since long-term establishment will depend on the agility and the quick responsiveness towards market and environmental conditions. Market and environmental turbulence will make drastic implications on the root establishment of a firm. If a firm's business strategy could not cope with the environmental and market contingencies, long-term survival becomes unrealistic. Diverging the strategy into different avenues with the view to exploit opportunities and avoid threats created by market conditions will be a pragmatic approach for a firm. Critical analysis conducted separately for cost leadership strategy and differentiation strategy identifies elementary value in both strategies in creating and sustaining a competitive advantage. Consistent and superior performance over competition could be reached with stronger foundations in the event “hybrid strategy” is adopted. Depending on the market and competitive conditions, hybrid strategy should be adjusted regarding the extent which each generic strategy (cost leadership or differentiation) should be given priority in practice.
== References ==
== Further reading ==
Critique of generic strategies and their limitations, including Porter - "Generic strategies: a substitute for thinking?"
Orcullo Jr., N. A., Fundamentals of Strategic Management
== External links ==
Data related to Porter's generic strategies at Wikidata | Wikipedia/Porter's_generic_strategies |
Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:
Superposed initial states,
Quantum entanglement of initial states,
Superposition of strategies to be used on the initial states.
This theory is based on the physics of information much like quantum computing.
== History ==
In 1969, John Clauser, Michael Horne, Abner Shimony, and Richard Holt (often referred to collectively as "CHSH") wrote an often-cited paper describing experiments which could be used to prove Bell's theorem. In one part of this paper, they describe a game where a player could have a better chance of winning by using quantum strategies than would be possible classically. While game theory was not explicitly mentioned in this paper, it is an early outline of how quantum entanglement could be used to alter a game.
In 1999, a professor in the math department at the University of California at San Diego named David A. Meyer first published Quantum Strategies which details a quantum version of the classical game theory game, matching pennies. In the quantum version, players are allowed access to quantum signals through the phenomenon of quantum entanglement.
In the same year, Jens Eisert, Martin Wilkens and Maciej Lewenstein published work entitled Quantum Games and Quantum Strategies that explored the role of quantum strategies in canonical two-player games such as the prisoner's dilemma.
Since Meyer's paper on the one hand and the
Eisert-Wilkens-Lewenstein paper on the other, a large number of publications have been published exploring quantum games and the way that quantum strategies could be used in games that have been commonly studied in classical game theory.
== Superposed initial states ==
The information transfer that occurs during a game can be viewed as a physical process.
In the simplest case of a classical game between two players with two strategies each, both the players can use a bit (a '0' or a '1') to convey their choice of strategy. A popular example of such a game is the prisoners' dilemma, where each of the convicts can either cooperate or defect: withholding knowledge or revealing that the other committed the crime. In the quantum version of the game, the bit is replaced by the qubit, which is a quantum superposition of two or more base states. In the case of a two-strategy game this can be physically implemented by the use of an entity like the electron which has a superposed spin state, with the base states being +1/2 (plus half) and −1/2 (minus half). Each of the spin states can be used to represent each of the two strategies available to the players. When a measurement is made on the electron, it collapses to one of the base states, thus conveying the strategy used by the player.
== Entangled initial states ==
The set of qubits which are initially provided to each of the players (to be used to convey their choice of strategy) may be entangled. For instance, an entangled pair of qubits implies that an operation performed on one of the qubits, affects the other qubit as well, thus altering the expected pay-offs of the game. A simple example of this is a quantum version of the Two-up coin game in which the coins are entangled.
== Superposition of strategies to be used on initial states ==
The job of a player in a game is to choose a strategy. In terms of bits this means that the player has to choose between 'flipping' the bit to its opposite state or leaving its current state untouched. When extended to the quantum domain this implies that the player can rotate the qubit to a new state, thus changing the probability amplitudes of each of the base states. Such operations on the qubits are required to be unitary transformations on the initial state of the qubit. This is different from the classical procedure which chooses the strategies with some statistical probabilities.
== Multiplayer games ==
Introducing quantum information into multiplayer games allows a new type of "equilibrium strategy" which is not found in traditional games. The entanglement of players' choices can have the effect of a contract by preventing players from profiting from other player's betrayal.
Quantum Prisoner's Dilemma
The classical Prisoner's Dilemma is a game played between two players with a choice to cooperate with or betray their opponent. Classically, the dominant strategy is to always choose betrayal. When both players choose this strategy every turn, they each ensure a suboptimal profit, but cannot lose, and the game is said to have reached a Nash equilibrium. Profit would be maximized for both players if each chose to cooperate every turn, but this is not the rational choice, thus a suboptimal solution is the dominant outcome. In the quantum Prisoner's Dilemma, both parties choosing to betray each other is still an equilibrium, however, there can also exist multiple Nash equilibriums that vary based on the entanglement of the initial states. In the case where the states are only slightly entangled, there exists a certain unitary operation for Alice so that if Bob chooses betrayal every turn, Alice will actually gain more profit than Bob and vice versa. Thus, a profitable equilibrium can be reached in 2 additional ways. The case where the initial state is most entangled shows the most change from the classical game. In this version of the game, Alice and Bob each have an operator Q that allows for a payout equal to mutual cooperation with no risk of betrayal. This is a Nash equilibrium that also happens to be Pareto optimal.
Additionally, the quantum version of the Prisoner's Dilemma differs greatly from the classical version when the game is of unknown or infinite length. Classically, the infinite Prisoner's Dilemma has no defined fixed strategy but in the quantum version it is possible to develop an equilibrium strategy.
Quantum Volunteer's Dilemma
The Volunteer's Dilemma is a well-known game in game theory that models the conflict players face when deciding whether to volunteer for a collective benefit, knowing that volunteering incurs a personal cost. One significant volunteer’s dilemma variant was introduced by Weesie and Franzen in 1998, involves cost-sharing among volunteers. In this variant of the Volunteer's Dilemma, if there is no volunteer, all players receive a payoff of 0. If there is at least one volunteer, the reward of b units is distributed to all players. In contrast, the total cost of c units incurred by volunteering is divided equally among all the volunteers. It is shown that for classical mixed strategies setting, there is a unique symmetric Nash equilibrium and the Nash equilibrium is obtained by setting the probability of volunteering for each player to be the unique root in the open interval (0,1) of the degree-n polynomial
g
n
{\displaystyle g_{n}}
given by
g
n
(
α
)
=
(
1
−
α
)
n
−
1
(
2
n
α
+
1
−
α
)
−
1.
{\displaystyle g_{n}(\alpha )=(1-\alpha )^{n-1}(2n\alpha +1-\alpha )-1.}
In 2024, a quantum variant of the classical volunteer’s dilemma is introduced with b=2 and c=1 is studied, generalizing the classical setting by allowing players to utilize quantum strategies. This is achieved by employing the Eisert–Wilkens–Lewenstein quantization framework. In this setting, the players received an entangled n-qubit state with each player controlling one qubit. The decision of each player can be viewed as determining two angles. Symmetric Nash equilibria that attain a payoff value of
2
−
1
/
n
{\displaystyle 2-1/n}
for each player is shown and each player volunteers at this Nash Equilibrium. Furthermore, these Nash Equilibrium are Pareto optimal. It is shown that the payoff function of Nash equilibrium in the quantum setting is higher than the payoff of Nash equilibrium in the classical setting.
Quantum Card Game
A classically unfair card game can be played as follows: There are two players, Alice and Bob. Alice has three cards: one has a star on both sides, one has a diamond on both sides, and one has a star on one side and a diamond on the other side. Alice places the three cards in a box and shakes it up, then Bob draws a card so that both players can only see one side of the card. If the card has the same markings on both sides, Alice wins. But if the card has different markings on each side, Bob wins. Clearly, this is an unfair game, where Alice has a probability of winning of 2/3 and Bob has a probability of winning of 1/3. Alice gives Bob one chance to "operate" on the box and then allows him to withdraw from the game if he would like, but he can only classically obtain information on one card from this operation, so the game is still unfair.
However, Alice and Bob can play a version of this game adjusted to allow for quantum strategies. If we describe the state of a card with a diamond facing up as
|
0
⟩
{\displaystyle |0\rangle }
and the state where the star is facing up as
|
1
⟩
{\displaystyle |1\rangle }
, after shaking the box up, we can describe the state of the face-up part of the cards as:
|
r
⟩
=
|
r
0
r
1
r
2
⟩
{\displaystyle |r\rangle =|r_{0}\ r_{1}\ r_{2}\rangle }
where each
r
k
{\displaystyle r_{k}}
is either 0 or 1.
Now, Bob can take advantage of his ability to operate on the box by constructing a machine as follows: First, he has a unitary matrix defined as
U
k
=
[
1
0
0
e
i
π
r
k
]
{\displaystyle U_{k}={\begin{bmatrix}1&0\\0&e^{i\pi r_{k}}\end{bmatrix}}}
. This matrix is equal to
I
{\displaystyle I}
if
r
k
{\displaystyle r_{k}}
is 0 and
Z
{\displaystyle Z}
if
r
k
{\displaystyle r_{k}}
is 1. He then creates his machine by putting this matrix between two Hadamard gates, so his machine now looks as
H
U
k
H
=
1
2
[
1
1
1
−
1
]
[
1
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i
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k
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[
1
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]
=
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i
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1
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k
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+
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i
π
r
k
]
.
{\displaystyle HU_{k}H={\frac {1}{2}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}1&0\\0&e^{i\pi r_{k}}\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1+e^{i\pi r_{k}}&1-e^{i\pi r_{k}}\\1-e^{i\pi r_{k}}&1+e^{i\pi r_{k}}\end{bmatrix}}.}
This machine operating on the state
|
0
⟩
{\displaystyle |0\rangle }
gives
H
U
k
H
|
0
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=
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.
{\displaystyle HU_{k}H|0\rangle ={\frac {1+e^{i\pi r_{k}}}{2}}|0\rangle +{\frac {1-e^{i\pi r_{k}}}{2}}|1\rangle =|r_{k}\rangle .}
So if Bob inputs
|
000
⟩
{\displaystyle |000\rangle }
to his machine, he obtains
(
H
U
k
H
⊗
H
U
k
H
⊗
H
U
k
H
)
|
000
⟩
=
|
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0
r
1
r
2
⟩
{\displaystyle (HU_{k}H\otimes HU_{k}H\otimes HU_{k}H)|000\rangle =|r_{0}\ r_{1}\ r_{2}\rangle }
and he knows the state (i.e. the mark facing up) of all three of the cards. From here, Bob can draw one card, and then choose to either withdraw, or keep playing the game. Based on the first card that he draws, he can know from his knowledge of the face-up values of the cards whether or not he has drawn a card that will give him even chances of winning going forward (in which case he can continue to play a fair game) or if he has drawn the card that will guarantee that he loses the game. In this way, he can make the game fair for himself.
This is an example of a game where a quantum strategy can make a game fair for one player when it would be unfair for them with classical strategies.
Quantum Chess
Quantum Chess was first developed by a graduate student at the University of Southern California named Chris Cantwell. His motivation to develop the game was to expose non-physicists to the world of quantum mechanics.
The game uses the same pieces as classical chess (8 pawns, 2 knights, 2 bishops, 2 rooks, 1 queen, 1 king) and is won in the same manner (by capturing the opponent's king). However, the pieces are allowed to obey laws of quantum mechanics such as superposition. By allowed the introduction of superposition, it becomes possible for pieces to occupy more than one square in an instance. The movement rules for each piece are the same as classical chess.
The biggest difference between quantum chess and classical chess is the check rule. Check is not included in quantum chess because it is possible for the king, as well as all other pieces, to occupy multiple spots on the grid at once. Another difference is the concept of movement to occupied space. Superposition also allows two occupies to share space or move through each other.
Capturing an opponent's piece is also slightly different in quantum chess than in classical chess. Quantum chess uses quantum measurement as a method of capturing. When attempting to capture an opponent's piece, a measurement is made to determine the probability of whether or not the space is occupied and if the path is blocked. If the probability is favorable, a move can be made to capture.
PQ Penny Flip Game
The PQ penny flip game involves two players: Captain Picard and Q. Q places a penny in a box, then they take turns (Q, then Picard, then Q) either flipping or not flipping the penny without revealing its state to either player. After these three moves have been made, Q wins if the penny is heads up, and Picard if the penny is face down.
The classical Nash equilibrium has both players taking a mixed strategy with each move having a 50% chance of either flipping or not flipping the penny, and Picard and Q will each win the game 50% of the time using classical strategies.
Allowing for Q to use quantum strategies, namely applying a Hadamard gate to the state of the penny places it into a superposition of face up and down, represented by the quantum state
|
ψ
1
⟩
=
1
2
(
|
0
⟩
+
|
1
⟩
)
.
{\displaystyle |\psi _{1}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle ).}
In this state, if Picard does not flip the gate, then the state remains unchanged, and flipping the penny puts it into the state
|
ψ
2
⟩
=
1
2
(
|
1
⟩
+
|
0
⟩
)
=
|
ψ
1
⟩
.
{\displaystyle |\psi _{2}\rangle ={\frac {1}{\sqrt {2}}}(|1\rangle +|0\rangle )=|\psi _{1}\rangle .}
Then, no matter Picard's move, Q can once again apply a Hadamard gate to the superposition which results in the penny being face up. In this way the quantization of Q's strategy guarantees a win against a player constrained by classical strategies.
This game is exemplary of how applying quantum strategies to classical games can shift an otherwise fair game in favor of the player using quantum strategies.
== Quantum minimax theorems ==
The concepts of a quantum player, a zero-sum quantum game and the associated expected payoff were defined by A. Boukas in 1999 (for finite games) and in 2020 by L. Accardi and A. Boukas (for infinite games) within the framework of the spectral theorem for self-adjoint operators on Hilbert spaces. Quantum versions of Von Neumann's minimax theorem were proved.
== Paradoxes ==
Quantum game theory also offers a solution to Newcomb's Paradox.
Take the two boxes offered in Newcomb's game to be coupled, as the contents of box 2 depend on if the ignorant player takes box 1. Quantum game theory enables a situation such that foreknowledge by otherwise omniscient player isn't required in order to achieve the situation. If the otherwise omniscient player operates on the state of the two boxes using a Hadamard gate, then sets up a device that operates on the state defined by the two boxes to operate again using a Hadamard gate after the ignorant player's choice. Then, no matter the pure or mixed strategy that the ignorant player uses, the ignorant player's choice will lead to its corresponding outcome as defined by the premise of the game. Because choosing a strategy for the game, then changing it to fool to otherwise omniscient player (corresponding to operating on the game state using a NOT gate) cannot give the ignorant player an additional advantage, as the two Hadamard operations ensure that the only two outcomes are those defined by the chosen strategy. In this way, the expected situation is achieved no matter the ignorant player's strategy without requiring a system knowledgeable about that player's future.
== See also ==
Quantum tic-tac-toe: not a quantum game in the sense above, but a pedagogical tool based on metaphors for quantum mechanics
Quantum pseudo-telepathy
Quantum refereed game
CHSH game
Jan Sładkowski
Jens Eisert
== References ==
== Further reading ==
Ball, Philip (18 October 1999). "Everyone wins in quantum games". Nature. doi:10.1038/news991021-3. ISSN 0028-0836. Archived from the original on 29 April 2005.
Piotrowski, E. W.; Sładkowski, J. (2003). "An Invitation to Quantum Game Theory" (PDF). International Journal of Theoretical Physics. 42 (5). Springer Nature: 1089–1099. doi:10.1023/a:1025443111388. ISSN 0020-7748. S2CID 13630647. Archived from the original (PDF) on 15 February 2012. Retrieved 17 August 2009.
Danaci, Onur; Zhang, Wenlei; Coleman, Robert; Djakam, William; Amoo, Michaela; Glasser, Ryan T.; Kirby, Brian T.; N'Gom, Moussa; Searles, Thomas A. (2023-02-28). "ManQala: Game-Inspired Strategies for Quantum State Engineering". AVS Quantum Science. 5 (3): 032002. arXiv:2302.14582. Bibcode:2023AVSQS...5c2002D. doi:10.1116/5.0148240. | Wikipedia/Quantum_game_theory |
Decision theory or the theory of rational choice is a branch of probability, economics, and analytic philosophy that uses expected utility and probability to model how individuals would behave rationally under uncertainty. It differs from the cognitive and behavioral sciences in that it is mainly prescriptive and concerned with identifying optimal decisions for a rational agent, rather than describing how people actually make decisions. Despite this, the field is important to the study of real human behavior by social scientists, as it lays the foundations to mathematically model and analyze individuals in fields such as sociology, economics, criminology, cognitive science, moral philosophy and political science.
== History ==
The roots of decision theory lie in probability theory, developed by Blaise Pascal and Pierre de Fermat in the 17th century, which was later refined by others like Christiaan Huygens. These developments provided a framework for understanding risk and uncertainty, which are central to decision-making.
In the 18th century, Daniel Bernoulli introduced the concept of "expected utility" in the context of gambling, which was later formalized by John von Neumann and Oskar Morgenstern in the 1940s. Their work on Game Theory and Expected Utility Theory helped establish a rational basis for decision-making under uncertainty.
After World War II, decision theory expanded into economics, particularly with the work of economists like Milton Friedman and others, who applied it to market behavior and consumer choice theory. This era also saw the development of Bayesian decision theory, which incorporates Bayesian probability into decision-making models.
By the late 20th century, scholars like Daniel Kahneman and Amos Tversky challenged the assumptions of rational decision-making. Their work in behavioral economics highlighted cognitive biases and heuristics that influence real-world decisions, leading to the development of prospect theory, which modified expected utility theory by accounting for psychological factors.
== Branches ==
Normative decision theory is concerned with identification of optimal decisions where optimality is often determined by considering an ideal decision maker who is able to calculate with perfect accuracy and is in some sense fully rational. The practical application of this prescriptive approach (how people ought to make decisions) is called decision analysis and is aimed at finding tools, methodologies, and software (decision support systems) to help people make better decisions.
In contrast, descriptive decision theory is concerned with describing observed behaviors often under the assumption that those making decisions are behaving under some consistent rules. These rules may, for instance, have a procedural framework (e.g. Amos Tversky's elimination by aspects model) or an axiomatic framework (e.g. stochastic transitivity axioms), reconciling the Von Neumann-Morgenstern axioms with behavioral violations of the expected utility hypothesis, or they may explicitly give a functional form for time-inconsistent utility functions (e.g. Laibson's quasi-hyperbolic discounting).
Prescriptive decision theory is concerned with predictions about behavior that positive decision theory produces to allow for further tests of the kind of decision-making that occurs in practice. In recent decades, there has also been increasing interest in "behavioral decision theory", contributing to a re-evaluation of what useful decision-making requires.
== Types of decisions ==
=== Choice under uncertainty ===
The area of choice under uncertainty represents the heart of decision theory. Known from the 17th century (Blaise Pascal invoked it in his famous wager, which is contained in his Pensées, published in 1670), the idea of expected value is that, when faced with a number of actions, each of which could give rise to more than one possible outcome with different probabilities, the rational procedure is to identify all possible outcomes, determine their values (positive or negative) and the probabilities that will result from each course of action, and multiply the two to give an "expected value", or the average expectation for an outcome; the action to be chosen should be the one that gives rise to the highest total expected value. In 1738, Daniel Bernoulli published an influential paper entitled Exposition of a New Theory on the Measurement of Risk, in which he uses the St. Petersburg paradox to show that expected value theory must be normatively wrong. He gives an example in which a Dutch merchant is trying to decide whether to insure a cargo being sent from Amsterdam to St. Petersburg in winter. In his solution, he defines a utility function and computes expected utility rather than expected financial value.
In the 20th century, interest was reignited by Abraham Wald's 1939 paper pointing out that the two central procedures of sampling-distribution-based statistical-theory, namely hypothesis testing and parameter estimation, are special cases of the general decision problem. Wald's paper renewed and synthesized many concepts of statistical theory, including loss functions, risk functions, admissible decision rules, antecedent distributions, Bayesian procedures, and minimax procedures. The phrase "decision theory" itself was used in 1950 by E. L. Lehmann.
The revival of subjective probability theory, from the work of Frank Ramsey, Bruno de Finetti, Leonard Savage and others, extended the scope of expected utility theory to situations where subjective probabilities can be used. At the time, von Neumann and Morgenstern's theory of expected utility proved that expected utility maximization followed from basic postulates about rational behavior.
The work of Maurice Allais and Daniel Ellsberg showed that human behavior has systematic and sometimes important departures from expected-utility maximization (Allais paradox and Ellsberg paradox). The prospect theory of Daniel Kahneman and Amos Tversky renewed the empirical study of economic behavior with less emphasis on rationality presuppositions. It describes a way by which people make decisions when all of the outcomes carry a risk. Kahneman and Tversky found three regularities – in actual human decision-making, "losses loom larger than gains"; people focus more on changes in their utility-states than they focus on absolute utilities; and the estimation of subjective probabilities is severely biased by anchoring.
=== Intertemporal choice ===
Intertemporal choice is concerned with the kind of choice where different actions lead to outcomes that are realized at different stages over time. It is also described as cost-benefit decision making since it involves the choices between rewards that vary according to magnitude and time of arrival. If someone received a windfall of several thousand dollars, they could spend it on an expensive holiday, giving them immediate pleasure, or they could invest it in a pension scheme, giving them an income at some time in the future. What is the optimal thing to do? The answer depends partly on factors such as the expected rates of interest and inflation, the person's life expectancy, and their confidence in the pensions industry. However even with all those factors taken into account, human behavior again deviates greatly from the predictions of prescriptive decision theory, leading to alternative models in which, for example, objective interest rates are replaced by subjective discount rates.
=== Interaction of decision makers ===
Some decisions are difficult because of the need to take into account how other people in the situation will respond to the decision that is taken. The analysis of such social decisions is often treated under decision theory, though it involves mathematical methods. In the emerging field of socio-cognitive engineering, the research is especially focused on the different types of distributed decision-making in human organizations, in normal and abnormal/emergency/crisis situations.
=== Complex decisions ===
Other areas of decision theory are concerned with decisions that are difficult simply because of their complexity, or the complexity of the organization that has to make them. Individuals making decisions are limited in resources (i.e. time and intelligence) and are therefore boundedly rational; the issue is thus, more than the deviation between real and optimal behavior, the difficulty of determining the optimal behavior in the first place. Decisions are also affected by whether options are framed together or separately; this is known as the distinction bias.
== Heuristics ==
Heuristics are procedures for making a decision without working out the consequences of every option. Heuristics decrease the amount of evaluative thinking required for decisions, focusing on some aspects of the decision while ignoring others. While quicker than step-by-step processing, heuristic thinking is also more likely to involve fallacies or inaccuracies.
One example of a common and erroneous thought process that arises through heuristic thinking is the gambler's fallacy — believing that an isolated random event is affected by previous isolated random events. For example, if flips of a fair coin give repeated tails, the coin still has the same probability (i.e., 0.5) of tails in future turns, though intuitively it might seems that heads becomes more likely. In the long run, heads and tails should occur equally often; people commit the gambler's fallacy when they use this heuristic to predict that a result of heads is "due" after a run of tails. Another example is that decision-makers may be biased towards preferring moderate alternatives to extreme ones. The compromise effect operates under a mindset that the most moderate option carries the most benefit. In an incomplete information scenario, as in most daily decisions, the moderate option will look more appealing than either extreme, independent of the context, based only on the fact that it has characteristics that can be found at either extreme.
== Alternatives ==
A highly controversial issue is whether one can replace the use of probability in decision theory with something else.
=== Probability theory ===
Advocates for the use of probability theory point to:
the work of Richard Threlkeld Cox for justification of the probability axioms,
the Dutch book paradoxes of Bruno de Finetti as illustrative of the theoretical difficulties that can arise from departures from the probability axioms, and
the complete class theorems, which show that all admissible decision rules are equivalent to the Bayesian decision rule for some utility function and some prior distribution (or for the limit of a sequence of prior distributions). Thus, for every decision rule, either the rule may be reformulated as a Bayesian procedure (or a limit of a sequence of such), or there is a rule that is sometimes better and never worse.
=== Alternatives to probability theory ===
The proponents of fuzzy logic, possibility theory, Dempster–Shafer theory, and info-gap decision theory maintain that probability is only one of many alternatives and point to many examples where non-standard alternatives have been implemented with apparent success. Notably, probabilistic decision theory can sometimes be sensitive to assumptions about the probabilities of various events, whereas non-probabilistic rules, such as minimax, are robust in that they do not make such assumptions.
=== Ludic fallacy ===
A general criticism of decision theory based on a fixed universe of possibilities is that it considers the "known unknowns", not the "unknown unknowns": it focuses on expected variations, not on unforeseen events, which some argue have outsized impact and must be considered – significant events may be "outside model". This line of argument, called the ludic fallacy, is that there are inevitable imperfections in modeling the real world by particular models, and that unquestioning reliance on models blinds one to their limits.
== See also ==
== References ==
== Further reading == | Wikipedia/Statistical_decision_theory |
Appropriation is a process by which previously unowned natural resources, particularly land, become the property of a person or group of persons.
The term is widely used in economics in this sense.
In certain cases, it proceeds under very specifically defined forms, such as driving stakes or other such markers into the land claimed, which form gave rise to the term "staking a claim."
"Squatter's rights" are another form of appropriation, but are usually asserted against land to which ownership rights of another party have been recognized.
In legal regimes recognizing such acquisition of property, the ownership of duly appropriated holdings enjoys such protections as the law provides for ownership of property in general.
Under some systems using this method of acquiring ownership of land, it is permitted to employ violence in defending the duly appropriated holding against encroachment against the ownership or usage claims, again usually according to specifically defined forms including warnings to the encroaching party, exhaustion or unavailability of duly constituted law-enforcement resources, etc.
Libertarianism and other property-rights-oriented ideologies define appropriation as requiring the "mixing" of the would-be owner's labor with the land claimed.
A prime example of such mixing is farming, although various extractive activities such as mining, and the grazing of herds are often recognized.
Personal, physical residence is often recognized after some minimum documented continuous period of time, as is built structures on the land whose ownership has not previously been recognized by the authority whose recognition is sought.
Appropriation through use can apply to resources other than the exclusive right to use of the surface of the land.
As mentioned, mineral rights are recognized under various conditions, as are riparian rights.
Appropriation can apply to inland waters within a certain distance of appropriated land, and even to the liquid water in a reservoir, lake, or stream.
Appropriation has been applied under common law to resources as disparate as radio broadcast frequencies and Internet Web site names, but many such claims have been overturned through legislated arrangements mandating other standards for the assignment of ownership rights in such things.
Appropriation as a means of acquiring property is related to the schools of thought that call for ongoing use as a condition of continued ownership, as is the case in some regimes with trademarks, but it applies to initial ownership.
== First possession ==
The "first possession" theory of property holds that ownership of something is justified simply by someone seizing it before someone else does.
This contrasts with the labor theory of property where something may become property only by applying productive labor to it, i.e. by making something out of the materials of nature.
== Real property ==
Pedis possessio is a legal phrase in common law used to describe walking on a property to establish ownership; this concept involves the establishment of first possession of land.
By walking on a property and defining its bounds, possession is established. Legal dictionaries put forth this definition.
Pedis possessio has been described as the actual possession of land within bounds set forth by the need of a mine claimant and operator to improve and work a claim for its mineral value.
Violation of set boundaries are avoided and violence prevented by the establishment of title using the concept of pedis possessio.
== Hunting results ==
In the case of Pierson v. Post, where labor theory and first possession theory were in conflict, the final verdict was that the one who caught the fox owned it.
== See also ==
== References ==
== Further reading ==
Linebaugh, Peter (1976). "Karl Marx, the theft of wood, and working class composition: A contribution to the current debate". Crime and Social Justice (6): 5--16. JSTOR 29765987.
Bensaïd, Daniel (2021). The Dispossessed: Karl Marx's Debates on Wood Theft and the Right of the Poor. U of Minnesota Press. ISBN 978-1-4529-6562-8.
Schmidtz, David (1990). "When is original appropriation required?". The Monist. 73 (4): 504--518. doi:10.5840/monist19907342. JSTOR 27903207. ProQuest 1296689122. | Wikipedia/First_possession_theory_of_property |
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.
Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy, formal semantics, and evolutionary dynamics. Its foundational appeal, together with its paradoxes, and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
== History ==
=== Early history ===
The basic notion of grouping objects has existed since at least the emergence of numbers, and the notion of treating sets as their own objects has existed since at least the Tree of Porphyry, 3rd-century AD. The simplicity and ubiquity of sets makes it hard to determine the origin of sets as now used in mathematics, however, Bernard Bolzano's Paradoxes of the Infinite (Paradoxien des Unendlichen, 1851) is generally considered the first rigorous introduction of sets to mathematics. In his work, he (among other things) expanded on Galileo's paradox, and introduced one-to-one correspondence of infinite sets, for example between the intervals
[
0
,
5
]
{\displaystyle [0,5]}
and
[
0
,
12
]
{\displaystyle [0,12]}
by the relation
5
y
=
12
x
{\displaystyle 5y=12x}
. However, he resisted saying these sets were equinumerous, and his work is generally considered to have been uninfluential in mathematics of his time.
Before mathematical set theory, basic concepts of infinity were considered to be solidly in the domain of philosophy (see: Infinity (philosophy) and Infinity § History). Since the 5th century BC, beginning with Greek philosopher Zeno of Elea in the West (and early Indian mathematicians in the East), mathematicians had struggled with the concept of infinity. With the development of calculus in the late 17th century, philosophers began to generally distinguish between actual and potential infinity, wherein mathematics was only considered in the latter. Carl Friedrich Gauss famously stated: "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics."
Development of mathematical set theory was motivated by several mathematicians. Bernhard Riemann's lecture On the Hypotheses which lie at the Foundations of Geometry (1854) proposed new ideas about topology, and about basing mathematics (especially geometry) in terms of sets or manifolds in the sense of a class (which he called Mannigfaltigkeit) now called point-set topology. The lecture was published by Richard Dedekind in 1868, along with Riemann's paper on trigonometric series (which presented the Riemann integral), The latter was a starting point a movement in real analysis for the study of “seriously” discontinuous functions. A young Georg Cantor entered into this area, which led him to the study of point-sets. Around 1871, influenced by Riemann, Dedekind began working with sets in his publications, which dealt very clearly and precisely with equivalence relations, partitions of sets, and homomorphisms. Thus, many of the usual set-theoretic procedures of twentieth-century mathematics go back to his work. However, he did not publish a formal explanation of his set theory until 1888.
=== Naive set theory ===
Set theory, as understood by modern mathematicians, is generally considered to be founded by a single paper in 1874 by Georg Cantor titled On a Property of the Collection of All Real Algebraic Numbers. In his paper, he developed the notion of cardinality, comparing the sizes of two sets by setting them in one-to-one correspondence. His "revolutionary discovery" was that the set of all real numbers is uncountable, that is, one cannot put all real numbers in a list. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument.
Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem. Cantor developed a theory of transfinite numbers, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter
ℵ
{\displaystyle \aleph }
(ℵ, aleph) with a natural number subscript; for the ordinals he employed the Greek letter
ω
{\displaystyle \omega }
(ω, omega).
Set theory was beginning to become an essential ingredient of the new “modern” approach to mathematics. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections (see: Controversy over Cantor's theory). Dedekind's algebraic style only began to find followers in the 1890s
Despite the controversy, Cantor's set theory gained remarkable ground around the turn of the 20th century with the work of several notable mathematicians and philosophers. Richard Dedekind, around the same time, began working with sets in his publications, and famously constructing the real numbers using Dedekind cuts. He also worked with Giuseppe Peano in developing the Peano axioms, which formalized natural-number arithmetic, using set-theoretic ideas, which also introduced the epsilon symbol for set membership. Possibly most prominently, Gottlob Frege began to develop his Foundations of Arithmetic.
In his work, Frege tries to ground all mathematics in terms of logical axioms using Cantor's cardinality. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept horse in the barn. Frege attempted to explain our grasp of numbers through cardinality ('the number of...', or
N
x
:
F
x
{\displaystyle Nx:Fx}
), relying on Hume's principle.
However, Frege's work was short-lived, as it was found by Bertrand Russell that his axioms lead to a contradiction. Specifically, Frege's Basic Law V (now known as the axiom schema of unrestricted comprehension). According to Basic Law V, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. The contradiction, called Russell's paradox, is shown as follows:
Let R be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols:
Let
R
=
{
x
∣
x
∉
x
}
, then
R
∈
R
⟺
R
∉
R
{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}
This came around a time of several paradoxes or counter-intuitive results. For example, that the parallel postulate cannot be proved, the existence of mathematical objects that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with Peano arithmetic. The result was a foundational crisis of mathematics.
== Basic concepts and notation ==
Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, the notation o ∈ A is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets.
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined, variously denoted
A
⊂
B
{\displaystyle A\subset B}
,
A
⊊
B
{\displaystyle A\subsetneq B}
, or
A
⫋
B
{\displaystyle A\subsetneqq B}
(note however that the notation
A
⊂
B
{\displaystyle A\subset B}
is sometimes used synonymously with
A
⊆
B
{\displaystyle A\subseteq B}
; that is, allowing the possibility that A and B are equal). We call A a proper subset of B if and only if A is a subset of B, but A is not equal to B. Also, 1, 2, and 3 are members (elements) of the set {1, 2, 3}, but are not subsets of it; and in turn, the subsets, such as {1}, are not members of the set {1, 2, 3}. More complicated relations can exist; for example, the set {1} is both a member and a proper subset of the set {1, {1}}.
Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The following is a partial list of them:
Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. For example, the union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.
Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. For example, the intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3}.
Set difference of U and A, denoted U ∖ A, is the set of all members of U that are not members of A. The set difference {1, 2, 3} ∖ {2, 3, 4} is {1}, while conversely, the set difference {2, 3, 4} ∖ {1, 2, 3} is {4}. When A is a subset of U, the set difference U ∖ A is also called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U ∖ A, particularly if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1, 2, 3} and {2, 3, 4}, the symmetric difference set is {1, 4}. It is the set difference of the union and the intersection, (A ∪ B) ∖ (A ∩ B) or (A ∖ B) ∪ (B ∖ A).
Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a, b), where a is a member of A and b is a member of B. For example, the Cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.
Some basic sets of central importance are the set of natural numbers, the set of real numbers and the empty set – the unique set containing no elements. The empty set is also occasionally called the null set, though this name is ambiguous and can lead to several interpretations. The empty set can be denoted with empty braces "
{
}
{\displaystyle \{\}}
" or the symbol "
∅
{\displaystyle \varnothing }
" or "
∅
{\displaystyle \emptyset }
".
The power set of a set A, denoted
P
(
A
)
{\displaystyle {\mathcal {P}}(A)}
, is the set whose members are all of the possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} }. Notably,
P
(
A
)
{\displaystyle {\mathcal {P}}(A)}
contains both A and the empty set.
== Ontology ==
A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number
α
{\displaystyle \alpha }
, known as its rank. The rank of a pure set
X
{\displaystyle X}
is defined to be the least ordinal that is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set containing only the empty set is assigned rank 1. For each ordinal
α
{\displaystyle \alpha }
, the set
V
α
{\displaystyle V_{\alpha }}
is defined to consist of all pure sets with rank less than
α
{\displaystyle \alpha }
. The entire von Neumann universe is denoted
V
{\displaystyle V}
.
== Formalized set theory ==
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.
The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:
Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Fragments of ZFC include:
Zermelo set theory, which replaces the axiom schema of replacement with that of separation;
General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
Kripke–Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement.
Sets and proper classes. These include Von Neumann–Bernays–Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse–Kelley set theory and Tarski–Grothendieck set theory, both of which are stronger than ZFC.
The above systems can be modified to allow urelements, objects that can be members of sets but that are not themselves sets and do not have any members.
The New Foundations systems of NFU (allowing urelements) and NF (lacking them), associate with Willard Van Orman Quine, are not based on a cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness, Thomas Forster has argued that it does reflect an iterative conception of set.
Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic. Yet other systems accept classical logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject.
An enrichment of ZFC called internal set theory was proposed by Edward Nelson in 1977.
== Applications ==
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, vector spaces, and relational algebras can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic. For example, properties of the natural and real numbers can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms.
Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes human-written, computer-verified derivations of more than 12,000 theorems starting from ZFC set theory, first-order logic and propositional logic.
== Areas of study ==
Set theory is a major area of research in mathematics with many interrelated subfields:
=== Combinatorial set theory ===
Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and the study of extensions of Ramsey's theorem such as the Erdős–Rado theorem.
=== Descriptive set theory ===
Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.
The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.
A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics.
=== Fuzzy set theory ===
In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.
=== Inner model theory ===
An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel.
One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model V of ZF satisfies the continuum hypothesis or the axiom of choice, the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent.
The study of inner models is common in the study of determinacy and large cardinals, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).
=== Large cardinals ===
A large cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo–Fraenkel set theory.
=== Determinacy ===
Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure.
=== Forcing ===
Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models.
=== Cardinal invariants ===
A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.
=== Set-theoretic topology ===
Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.
== Controversy ==
From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop's influential book Foundations of Constructive Analysis.
A different objection put forth by Henri Poincaré is that defining sets using the axiom schemas of specification and replacement, as well as the axiom of power set, introduces impredicativity, a type of circularity, into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]".
Ludwig Wittgenstein condemned set theory philosophically for its connotations of mathematical platonism. He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein identified mathematics with algorithmic human deduction; the need for a secure foundation for mathematics seemed, to him, nonsensical. Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical constructivism and finitism. Meta-mathematical statements – which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory – are not mathematics. Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in Remarks on the Foundations of Mathematics: Wittgenstein attempted to refute Gödel's incompleteness theorems after having only read the abstract. As reviewers Kreisel, Bernays, Dummett, and Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as Crispin Wright begun to rehabilitate Wittgenstein's arguments.
Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory. Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone spaces.
An active area of research is the univalent foundations and related to it homotopy type theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types. Principles such as the axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results.
== Mathematical education ==
As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing the basics of naive set theory early in mathematics education.
In the US in the 1960s, the New Math experiment aimed to teach basic set theory, among other abstract concepts, to primary school students but was met with much criticism. The math syllabus in European schools followed this trend and currently includes the subject at different levels in all grades. Venn diagrams are widely employed to explain basic set-theoretic relationships to primary school students (even though John Venn originally devised them as part of a procedure to assess the validity of inferences in term logic).
Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition) of sets (e.g. "months starting with the letter A"), which may be useful when learning computer programming, since Boolean logic is used in various programming languages. Likewise, sets and other collection-like objects, such as multisets and lists, are common datatypes in computer science and programming.
In addition to that, certain sets are commonly used in mathematical teaching, such as the sets
N
{\displaystyle \mathbb {N} }
of natural numbers,
Z
{\displaystyle \mathbb {Z} }
of integers,
R
{\displaystyle \mathbb {R} }
of real numbers, etc.). These are commonly used when defining a mathematical function as a relation from one set (the domain) to another set (the range).
== See also ==
Glossary of set theory
Class (set theory)
List of set theory topics
Relational model – borrows from set theory
Venn diagram
Elementary Theory of the Category of Sets
Structural set theory
== Notes ==
== Citations ==
== References ==
Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory, Undergraduate Texts in Mathematics (2nd ed.), Springer Verlag, doi:10.1007/978-1-4612-0903-4, ISBN 0-387-94094-4
Ferreirós, Jose (2001), Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Berlin: Springer, ISBN 978-3-7643-5749-8
Monk, J. Donald (1969), Introduction to Set Theory, McGraw-Hill Book Company, ISBN 978-0-898-74006-6
Potter, Michael (2004), Set Theory and Its Philosophy: A Critical Introduction, Oxford University Press, ISBN 978-0-191-55643-2
Smullyan, Raymond M.; Fitting, Melvin (2010), Set Theory and the Continuum Problem, Dover Publications, ISBN 978-0-486-47484-7
Tiles, Mary (2004), The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise, Dover Publications, ISBN 978-0-486-43520-6
Dauben, Joseph W. (1977), "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite", Journal of the History of Ideas, 38 (1): 85–108, doi:10.2307/2708842, JSTOR 2708842
Dauben, Joseph W. (1979), [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite, Boston: Harvard University Press, ISBN 978-0-691-02447-9
== External links ==
Daniel Cunningham, Set Theory article in the Internet Encyclopedia of Philosophy.
Jose Ferreiros, "The Early Development of Set Theory" article in the [Stanford Encyclopedia of Philosophy].
Foreman, Matthew, Akihiro Kanamori, eds. Handbook of Set Theory. 3 vols., 2010. Each chapter surveys some aspect of contemporary research in set theory. Does not cover established elementary set theory, on which see Devlin (1993).
"Axiomatic set theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
"Set theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Schoenflies, Arthur (1898). Mengenlehre in Klein's encyclopedia.
Online books, and library resources in your library and in other libraries about set theory
Rudin, Walter B. (April 6, 1990), "Set Theory: An Offspring of Analysis", Marden Lecture in Mathematics, University of Wisconsin-Milwaukee, archived from the original on 2021-10-31 – via YouTube | Wikipedia/Axiomatic_set_theory |
In game theory and in particular the study of Blotto games and operational research, the Gibbs lemma is a result that is useful in maximization problems. It is named for Josiah Willard Gibbs.
Consider
ϕ
=
∑
i
=
1
n
f
i
(
x
i
)
{\displaystyle \phi =\sum _{i=1}^{n}f_{i}(x_{i})}
. Suppose
ϕ
{\displaystyle \phi }
is maximized, subject to
∑
x
i
=
X
{\displaystyle \sum x_{i}=X}
and
x
i
≥
0
{\displaystyle x_{i}\geq 0}
, at
x
0
=
(
x
1
0
,
…
,
x
n
0
)
{\displaystyle x^{0}=(x_{1}^{0},\ldots ,x_{n}^{0})}
. If the
f
i
{\displaystyle f_{i}}
are differentiable, then the Gibbs lemma states that there exists a
λ
{\displaystyle \lambda }
such that
f
i
′
(
x
i
0
)
=
λ
if
x
i
0
>
0
≤
λ
if
x
i
0
=
0.
{\displaystyle {\begin{aligned}f'_{i}(x_{i}^{0})&=\lambda {\mbox{ if }}x_{i}^{0}>0\\&\leq \lambda {\mbox{ if }}x_{i}^{0}=0.\end{aligned}}}
== Notes ==
== References == | Wikipedia/Gibbs_lemma |
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games (Friedman 1971). The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.
The Folk Theorem suggests that if the players are patient enough and far-sighted (i.e. if the discount factor
δ
→
1
{\displaystyle \delta \to 1}
), then repeated interaction can result in virtually any average payoff in an SPE equilibrium. "Virtually any" is here technically defined as "feasible" and "individually rational".
== Setup and definitions ==
We start with a basic game, also known as the stage game, which is an n-player game. In this game, each player has finitely many actions to choose from, and they make their choices simultaneously and without knowledge of the other player's choices. The collective choices of the players leads to a payoff profile, i.e. to a payoff for each of the players. The mapping from collective choices to payoff profiles is known to the players, and each player aims to maximize their payoff. If the collective choice is denoted by x, the payoff that player i receives, also known as player i's utility, will be denoted by
u
i
(
x
)
{\displaystyle u_{i}(x)}
.
We then consider a repetition of this stage game, finitely or infinitely many times. In each repetition, each player chooses one of their stage game options, and when making that choice, they may take into account the choices of the other players in the prior iterations. In this repeated game, a strategy for one of the players is a deterministic rule that specifies the player's choice in each iteration of the stage game, based on all other player's choices in the prior iterations. A choice of strategy for each of the players is a strategy profile, and it leads to a payout profile for the repeated game. There are a number of different ways such a strategy profile can be translated into a payout profile, outlined below.
Any Nash equilibrium payoff profile of a repeated game must satisfy two properties:
Individual rationality: the payoff must weakly dominate the minmax payoff profile of the constituent stage game. That is, the equilibrium payoff of each player must be at least as large as the minmax payoff of that player. This is because a player achieving less than their minmax payoff always has incentive to deviate by simply playing their minmax strategy at every history.
Feasibility: the payoff must be a convex combination of possible payoff profiles of the stage game. This is because the payoff in a repeated game is just a weighted average of payoffs in the basic games.
Folk theorems are partially converse claims: they say that, under certain conditions (which are different in each folk theorem), every payoff profile that is both individually rational and feasible can be realized as a Nash equilibrium payoff profile of the repeated game.
There are various folk theorems; some relate to finitely-repeated games while others relate to infinitely-repeated games.
== Infinitely-repeated games without discounting ==
In the undiscounted model, the players are patient. They do not differentiate between utilities in different time periods. Hence, their utility in the repeated game is represented by the sum of utilities in the basic games.
When the game is infinite, a common model for the utility in the infinitely-repeated game is the limit inferior of mean utility: If the game results in a path of outcomes
x
t
{\displaystyle x_{t}}
, where
x
t
{\displaystyle x_{t}}
denotes the collective choices of the players at iteration t (t=0,1,2,...), player i's utility is defined as
U
i
=
lim inf
T
→
∞
1
T
∑
t
=
0
T
u
i
(
x
t
)
,
{\displaystyle U_{i}=\liminf _{T\to \infty }{\frac {1}{T}}\sum _{t=0}^{T}u_{i}(x_{t}),}
where
u
i
{\displaystyle u_{i}}
is the basic-game utility function of player i.
An infinitely-repeated game without discounting is often called a "supergame".
The folk theorem in this case is very simple and contains no pre-conditions: every individually rational and feasible payoff profile in the basic game is a Nash equilibrium payoff profile in the repeated game.
The proof employs what is called a grim or grim trigger strategy. All players start by playing the prescribed action and continue to do so until someone deviates. If player i deviates, all other players switch to picking the action which minmaxes player i forever after. The one-stage gain from deviation contributes 0 to the total utility of player i. The utility of a deviating player cannot be higher than his minmax payoff. Hence all players stay on the intended path and this is indeed a Nash equilibrium.
=== Subgame perfection ===
The above Nash equilibrium is not always subgame perfect. If punishment is costly for the punishers, the threat of punishment is not credible.
A subgame perfect equilibrium requires a slightly more complicated strategy.: 146–149 The punishment should not last forever; it should last only a finite time which is sufficient to wipe out the gains from deviation. After that, the other players should return to the equilibrium path.
The limit-of-means criterion ensures that any finite-time punishment has no effect on the final outcome. Hence, limited-time punishment is a subgame-perfect equilibrium.
Coalition subgame-perfect equilibria: An equilibrium is called a coalition Nash equilibrium if no coalition can gain from deviating. It is called a coalition subgame-perfect equilibrium if no coalition can gain from deviating after any history. With the limit-of-means criterion, a payoff profile is attainable in coalition-Nash-equilibrium or in coalition-subgame-perfect-equilibrium, if-and-only-if it is Pareto efficient and weakly-coalition-individually-rational.
=== Overtaking ===
Some authors claim that the limit-of-means criterion is unrealistic, because it implies that utilities in any finite time-span contribute 0 to the total utility. However, if the utilities in any finite time-span contribute a positive value, and the value is undiscounted, then it is impossible to attribute a finite numeric utility to an infinite outcome sequence. A possible solution to this problem is that, instead of defining a numeric utility for each infinite outcome sequence, we just define the preference relation between two infinite sequences. We say that agent
i
{\displaystyle i}
(strictly) prefers the sequence of outcomes
y
t
{\displaystyle y_{t}}
over the sequence
x
t
{\displaystyle x_{t}}
, if:: 139
lim inf
T
→
∞
∑
t
=
0
T
(
u
i
(
y
t
)
−
u
i
(
x
t
)
)
>
0
{\displaystyle \liminf _{T\to \infty }\sum _{t=0}^{T}(u_{i}(y_{t})-u_{i}(x_{t}))>0}
For example, consider the sequences
u
i
(
x
)
=
(
0
,
0
,
0
,
0
,
…
)
{\displaystyle u_{i}(x)=(0,0,0,0,\ldots )}
and
u
i
(
y
)
=
(
−
1
,
2
,
0
,
0
,
…
)
{\displaystyle u_{i}(y)=(-1,2,0,0,\ldots )}
. According to the limit-of-means criterion, they provide the same utility to player i, but according to the overtaking criterion,
y
{\displaystyle y}
is better than
x
{\displaystyle x}
for player i. See overtaking criterion for more information.
The folk theorems with the overtaking criterion are slightly weaker than with the limit-of-means criterion. Only outcomes that are strictly individually rational, can be attained in Nash equilibrium. This is because, if an agent deviates, he gains in the short run, and this gain can be wiped out only if the punishment gives the deviator strictly less utility than the agreement path. The following folk theorems are known for the overtaking criterion:
Strict stationary equilibria: A Nash equilibrium is called strict if each player strictly prefers the infinite sequence of outcomes attained in equilibrium, over any other sequence he can deviate to. A Nash equilibrium is called stationary if the outcome is the same in each time-period. An outcome is attainable in strict-stationary-equilibrium if-and-only-if for every player the outcome is strictly better than the player's minimax outcome.
Strict stationary subgame-perfect equilibria: An outcome is attainable in strict-stationary-subgame-perfect-equilibrium, if for every player the outcome is strictly better than the player's minimax outcome (note that this is not an "if-and-only-if" result). To achieve subgame-perfect equilibrium with the overtaking criterion, it is required to punish not only the player that deviates from the agreement path, but also every player that does not cooperate in punishing the deviant.: 149–150
The "stationary equilibrium" concept can be generalized to a "periodic equilibrium", in which a finite number of outcomes is repeated periodically, and the payoff in a period is the arithmetic mean of the payoffs in the outcomes. That mean payoff should be strictly above the minimax payoff.
Strict stationary coalition equilibria: With the overtaking criterion, if an outcome is attainable in coalition-Nash-equilibrium, then it is Pareto efficient and weakly-coalition-individually-rational. On the other hand, if it is Pareto efficient and strongly-coalition-individually-rational it can be attained in strict-stationary-coalition-equilibrium.
== Infinitely-repeated games with discounting ==
Assume that the payoff of a player in an infinitely repeated game is given by the average discounted criterion with discount factor 0 < δ < 1:
U
i
=
(
1
−
δ
)
∑
t
≥
0
δ
t
u
i
(
x
t
)
,
{\displaystyle U_{i}=(1-\delta )\sum _{t\geq 0}\delta ^{t}u_{i}(x_{t}),}
The discount factor indicates how patient the players are. The factor
(
1
−
δ
)
{\displaystyle (1-\delta )}
is introduced so that the payoff remain bounded when
δ
→
1
{\displaystyle \delta \rightarrow 1}
.
The folk theorem in this case requires that the payoff profile in the repeated game strictly dominates the minmax payoff profile (i.e., each player receives strictly more than the minmax payoff).
Let a be a strategy profile of the stage game with payoff profile u which strictly dominates the minmax payoff profile. One can define a Nash equilibrium of the game with u as resulting payoff profile as follows:
1. All players start by playing a and continue to play a if no deviation occurs.
2. If any one player, say player i, deviated, play the strategy profile m which minmaxes i forever after.
3. Ignore multilateral deviations.
If player i gets ε more than his minmax payoff each stage by following 1, then the potential loss from punishment is
1
1
−
δ
ε
.
{\displaystyle {\frac {1}{1-\delta }}\varepsilon .}
If δ is close to 1, this outweighs any finite one-stage gain, making the strategy a Nash equilibrium.
An alternative statement of this folk theorem allows the equilibrium payoff profile u to be any individually rational feasible payoff profile; it only requires there exist an individually rational feasible payoff profile that strictly dominates the minmax payoff profile. Then, the folk theorem guarantees that it is possible to approach u in equilibrium to any desired precision (for every ε there exists a Nash equilibrium where the payoff profile is a distance ε away from u).
=== Subgame perfection ===
Attaining a subgame perfect equilibrium in discounted games is more difficult than in undiscounted games. The cost of punishment does not vanish (as with the limit-of-means criterion).
It is not always possible to punish the non-punishers endlessly (as with the overtaking criterion) since the discount factor makes punishments far away in the future irrelevant for the present. Hence, a different approach is needed: the punishers should be rewarded.
This requires an additional assumption, that the set of feasible payoff profiles is full dimensional and the min-max profile lies in its interior. The strategy is as follows.
1. All players start by playing a and continue to play a if no deviation occurs.
2. If any one player, say player i, deviated, play the strategy profile m which minmaxes i for N periods. (Choose N and δ large enough so that no player has incentive to deviate from phase 1.)
3. If no players deviated from phase 2, all player j ≠ i gets rewarded ε above j's min-max forever after, while player i continues receiving his min-max. (Full-dimensionality and the interior assumption is needed here.)
4. If player j deviated from phase 2, all players restart phase 2 with j as target.
5. Ignore multilateral deviations.
Player j ≠ i now has no incentive to deviate from the punishment phase 2. This proves the subgame perfect folk theorem.
== Finitely-repeated games without discount ==
Assume that the payoff of player i in a game that is repeated T times is given by a simple arithmetic mean:
U
i
=
1
T
∑
t
=
0
T
u
i
(
x
t
)
{\displaystyle U_{i}={\frac {1}{T}}\sum _{t=0}^{T}u_{i}(x_{t})}
A folk theorem for this case has the following additional requirement:
In the basic game, for every player i, there is a Nash-equilibrium
E
i
{\displaystyle E_{i}}
that is strictly better, for i, than his minmax payoff.
This requirement is stronger than the requirement for discounted infinite games, which is in turn stronger than the requirement for undiscounted infinite games.
This requirement is needed because of the last step. In the last step, the only stable outcome is a Nash-equilibrium in the basic game. Suppose a player i gains nothing from the Nash equilibrium (since it gives him only his minmax payoff). Then, there is no way to punish that player.
On the other hand, if for every player there is a basic equilibrium which is strictly better than minmax, a repeated-game equilibrium can be constructed in two phases:
In the first phase, the players alternate strategies in the required frequencies to approximate the desired payoff profile.
In the last phase, the players play the preferred equilibrium of each of the players in turn.
In the last phase, no player deviates since the actions are already a basic-game equilibrium. If an agent deviates in the first phase, he can be punished by minmaxing him in the last phase. If the game is sufficiently long, the effect of the last phase is negligible, so the equilibrium payoff approaches the desired profile.
== Applications ==
Folk theorems can be applied to a diverse number of fields. For example:
Anthropology: in a community where all behavior is well known, and where members of the community know that they will continue to have to deal with each other, then any pattern of behavior (traditions, taboos, etc.) may be sustained by social norms so long as the individuals of the community are better off remaining in the community than they would be leaving the community (the minimax condition).
International politics: agreements between countries cannot be effectively enforced. They are kept, however, because relations between countries are long-term and countries can use "minimax strategies" against each other. This possibility often depends on the discount factor of the relevant countries. If a country is very impatient (pays little attention to future outcomes), then it may be difficult to punish it (or punish it in a credible way).
On the other hand, MIT economist Franklin Fisher has noted that the folk theorem is not a positive theory. In considering, for instance, oligopoly behavior, the folk theorem does not tell the economist what firms will do, but rather that cost and demand functions are not sufficient for a general theory of oligopoly, and the economists must include the context within which oligopolies operate in their theory.
In 2007, Borgs et al. proved that, despite the folk theorem, in the general case computing the Nash equilibria for repeated games is not easier than computing the Nash equilibria for one-shot finite games, a problem which lies in the PPAD complexity class. The practical consequence of this is that no efficient (polynomial-time) algorithm is known that computes the strategies required by folk theorems in the general case.
== Summary of folk theorems ==
The following table compares various folk theorems in several aspects:
Horizon – whether the stage game is repeated finitely or infinitely many times.
Utilities – how the utility of a player in the repeated game is determined from the player's utilities in the stage game iterations.
Conditions on G (the stage game) – whether there are any technical conditions that should hold in the one-shot game in order for the theorem to work.
Conditions on x (the target payoff vector of the repeated game) – whether the theorem works for any individually rational and feasible payoff vector, or only on a subset of these vectors.
Equilibrium type – if all conditions are met, what kind of equilibrium is guaranteed by the theorem – Nash or Subgame-perfect?
Punishment type – what kind of punishment strategy is used to deter players from deviating?
== Folk theorems in other settings ==
In allusion to the folk theorems for repeated games, some authors have used the term "folk theorem" to refer to results on the set of possible equilibria or equilibrium payoffs in other settings, especially if the results are similar in what equilibrium payoffs they allow. For instance, Tennenholtz proves a "folk theorem" for program equilibrium. Many other folk theorems have been proved in settings with commitment.
== Notes ==
== References ==
Friedman, J. (1971). "A non-cooperative equilibrium for supergames". Review of Economic Studies. 38 (1): 1–12. doi:10.2307/2296617. JSTOR 2296617.
Ichiishi, Tatsuro (1997). Microeconomic Theory. Oxford: Blackwell. pp. 263–269. ISBN 1-57718-037-2.
Mas-Colell, A.; Whinston, M.; Green, J. (1995). Microeconomic Theory. New York: Oxford University Press. ISBN 0-19-507340-1.
Ratliff, J. (1996). "A Folk Theorem Sampler" (PDF). A set of introductory notes to the Folk Theorem. | Wikipedia/Folk_theorem_(game_theory) |
Rationalizability is a solution concept in game theory. It is the most permissive possible solution concept that still requires both players to be at least somewhat rational and know the other players are also somewhat rational, i.e. that they do not play dominated strategies. A strategy is rationalizable if there exists some possible set of beliefs both players could have about each other's actions, that would still result in the strategy being played.
Rationalizability is a broader concept than a Nash equilibrium. Both require players to respond optimally to some belief about their opponents' actions, but Nash equilibrium requires these beliefs to be correct, while rationalizability does not. Rationalizability was first defined, independently, by Bernheim (1984) and Pearce (1984).
== Definition ==
Starting with a normal-form game, the rationalizable set of actions can be computed as follows:
Start with the full action set for each player.
Remove all dominated strategies, i.e. strategies that "never make sense" (are never a best reply to any belief about the opponents' actions). The motivation for this step is no rational player would ever choose such actions.
Remove all actions which are never a best reply to any belief about the opponents' remaining actions—this second step is justified because each player knows that the other players are rational.
Continue the process until no further actions can be eliminated.
In a game with finitely many actions, this process always terminates and leaves a non-empty set of actions for each player. These are the rationalizable actions.
== Iterated elimination of strictly dominated strategies (IESDS) ==
The iterated elimination (or deletion, or removal) of dominated strategies (also denominated as IESDS, or IDSDS, or IRSDS) is one common technique for solving games that involves iteratively removing dominated strategies. In the first step, at most one dominated strategy is removed from the strategy space of each of the players since no rational player would ever play these strategies. This results in a new, smaller game. Some strategies—that were not dominated before—may be dominated in the smaller game. The first step is repeated, creating a new even smaller game, and so on. The process stops when no dominated strategy is found for any player. This process is valid since it is assumed that rationality among players is common knowledge, that is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on ad infinitum (see Aumann, 1976).
There are two versions of this process. One version involves only eliminating strictly dominated strategies. If, after completing this process, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium. Moreover, iterated elimination of strictly dominated strategies is path independent. That is, if at any point in the process there are multiple strictly dominated strategies, then it doesn't matter for the end result which strategies we remove first.
Strict Dominance Deletion Step-by-Step Example:
C is strictly dominated by A for Player 1. Therefore, Player 1 will never play strategy C. Player 2 knows this. (see IESDS Figure 1)
Of the remaining strategies (see IESDS Figure 2), Z is strictly dominated by Y and X for Player 2. Therefore, Player 2 will never play strategy Z. Player 1 knows this.
Of the remaining strategies (see IESDS Figure 3), B is strictly dominated by A for Player 1. Therefore, Player 1 will never play B. Player 2 knows this.
Of the remaining strategies (see IESDS Figure 4), Y is strictly dominated by X for Player 2. Therefore, Player 2 will never play Y. Player 1 knows this.
Only one rationalizable strategy is left {A,X} which results in a payoff of (10,4). This is the single Nash Equilibrium for this game.
Another version involves eliminating both strictly and weakly dominated strategies. If, at the end of the process, there is a single strategy for each player, this strategy set is also a Nash equilibrium. However, unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria. As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium. (In some games, if we remove weakly dominated strategies in a different order, we may end up with a different Nash equilibrium.)
Weak Dominance Deletion Step-by-Step Example:
O is strictly dominated by N for Player 1. Therefore, Player 1 will never play strategy O. Player 2 knows this. (see IESDS Figure 5)
U is weakly dominated by T for Player 2. If Player 2 chooses T, then the final equilibrium is (N,T)
O is strictly dominated by N for Player 1. Therefore, Player 1 will never play strategy O. Player 2 knows this. (see IESDS Figure 6)
T is weakly dominated by U for Player 2. If Player 2 chooses U, then the final equilibrium is (N,U)
In any case, if by iterated elimination of dominated strategies there is only one strategy left for each player, the game is called a dominance-solvable game.
== Iterated elimination by mixed strategy ==
There are instances when there is no pure strategy that dominates another pure strategy, but a mixture of two or more pure strategies can dominate another strategy. This is called Strictly Dominant Mixed Strategies. Some authors allow for elimination of strategies dominated by a mixed strategy in this way.
Example 1:
In this scenario, for player 1, there is no pure strategy that dominates another pure strategy. Let's define the probability of player 1 playing up as p, and let p = 1/2. We can set a mixed strategy where player 1 plays up and down with probabilities (1/2,1/2). When player 2 plays left, then the payoff for player 1 playing the mixed strategy of up and down is 1, when player 2 plays right, the payoff for player 1 playing the mixed strategy is 0.5. Thus regardless of whether player 2 chooses left or right, player 1 gets more from playing this mixed strategy between up and down than if the player were to play the middle strategy. In this case, we should eliminate the middle strategy for player 1 since it's been dominated by the mixed strategy of playing up and down with probability (1/2,1/2).
Example 2:
We can demonstrate the same methods on a more complex game and solve for the rational strategies. In this scenario, the blue coloring represents the dominating numbers in the particular strategy.
Step-by-step solving:
For Player 2, X is dominated by the mixed strategy 1/2Y and 1/2Z.
The expected payoff for playing strategy 1/2Y + 1/2Z must be greater than the expected payoff for playing pure strategy X, assigning 1/2 and 1/2 as tester values. The argument for mixed strategy dominance can be made if there is at least one mixed strategy that allows for dominance.
Testing with 1/2 and 1/2 gets the following:
Expected average payoff of 1/2 Strategy Y: 1/2(4+0+4) = 4
Expected average payoff of 1/2 Strategy Z: 1/2(0+5+5) = 5
Expected average payoff of pure strategy X: (1+1+3) = 5
Set up the inequality to determine whether the mixed strategy will dominate the pure strategy based on expected payoffs.
u1/2Y + u1/2Z ⩼ uX
4 + 5 > 5
Mixed strategy 1/2Y and 1/2Z will dominate pure strategy X for Player 2, and thus X can be eliminated from the rationalizable strategies for P2.
For Player 1, U is dominated by the pure strategy D.
For player 2, Y is dominated by the pure strategy Z.
This leaves M dominating D for Player 1.
The only rationalizable strategy for Players 1 and 2 is then (M,Z) or (3,5).
== Constraints on beliefs ==
Consider a simple coordination game (the payoff matrix is to the right). The row player can play a if he can reasonably believe that the column player could play A, since a is a best response to A. He can reasonably believe that the column player can play A if it is reasonable for the column player to believe that the row player could play a. She can believe that he will play a if it is reasonable for her to believe that he could play a, etc.
This provides an infinite chain of consistent beliefs that result in the players playing (a, A). This makes (a, A) a rationalizable pair of actions. A similar process can be repeated for (b, B).
As an example where not all strategies are rationalizable, consider a prisoner's dilemma pictured to the left. Row player would never play c, since c is not a best response to any strategy by the column player. For this reason, c is not rationalizable.
Conversely, for two-player games, the set of all rationalizable strategies can be found by iterated elimination of strictly dominated strategies. For this method to hold however, one also needs to consider strict domination by mixed strategies. Consider the game on the right with payoffs of the column player omitted for simplicity. Notice that "b" is not strictly dominated by either "t" or "m" in the pure strategy sense, but it is still dominated by a strategy that would mix "t" and "m" with probability of each equal to 1/2. This is due to the fact that given any belief about the action of the column player, the mixed strategy will always yield higher expected payoff. This implies that "b" is not rationalizable.
Moreover, "b" is not a best response to either "L" or "R" or any mix of the two. This is because an action that is not rationalizable can never be a best response to any opponent's strategy (pure or mixed). This would imply another version of the previous method of finding rationalizable strategies as those that survive the iterated elimination of strategies that are never a best response (in pure or mixed sense).
In games with more than two players, however, there may be strategies that are not strictly dominated, but which can never be the best response. By the iterated elimination of all such strategies one can find the rationalizable strategies for a multiplayer game.
== Rationalizability and Nash equilibria ==
It can be easily proved that a Nash equilibrium is a rationalizable equilibrium; however, the converse is not true. Some rationalizable equilibria are not Nash equilibria. This makes the rationalizability concept a generalization of Nash equilibrium concept.
As an example, consider the game matching pennies pictured to the right. In this game the only Nash equilibrium is row playing h and t with equal probability and column playing H and T with equal probability. However, all pure strategies in this game are rationalizable.
Consider the following reasoning: row can play h if it is reasonable for her to believe that column will play H. Column can play H if its reasonable for him to believe that row will play t. Row can play t if it is reasonable for her to believe that column will play T. Column can play T if it is reasonable for him to believe that row will play h (beginning the cycle again). This provides an infinite set of consistent beliefs that results in row playing h. A similar argument can be given for row playing t, and for column playing either H or T.
== See also ==
Self-confirming equilibrium
Strategic dominance
== Footnotes ==
== References ==
Bernheim, D. (1984) Rationalizable Strategic Behavior. Econometrica 52: 1007–1028.
Fudenberg, Drew and Jean Tirole (1993) Game Theory. Cambridge: MIT Press.
Pearce, D. (1984) Rationalizable Strategic Behavior and the Problem of Perfection. Econometrica 52: 1029–1050.
Ratcliff, J. (1992–1997) lecture notes on game theory, §2.2: "Iterated Dominance and Rationalizability" | Wikipedia/Rationalizable_strategy |
In game theory, perfect recall is a property of players within extensive-form games, introduced by Harold W. Kuhn in 1953. it describes a player's ability to remember their past actions and the information they possessed at previous decision points. For example, in a simplified card game where a player makes multiple betting rounds, perfect recall means they remember their own previous bets and the cards they've seen. Essentially, it indicates that a player does not "forget" relevant information acquired during the game.
It is important to distinguish perfect recall from perfect information. While perfect information means all players know all previous actions of all players, perfect recall means a player remembers their own past actions and knowledge.
== Significance ==
Perfect recall is crucial for the consistency of rational decision-making in sequential games. If a player forgets past information, their current decisions may contradict their earlier intentions. The concept plays a key role in the relationship between mixed and behavioral strategies. In games where players have perfect recall, these two types of strategies are essentially equivalent, meaning that any outcome that can be achieved with a mixed strategy can also be achieved with a behavioral strategy, and vice versa. This equivalence, notably formalized in Kuhn's theorem, simplifies the analysis of such games. It is a core component of how game theorists analyze extensive-form games.
The formal definition of perfect recall involves the concept of information sets in extensive-form games. It ensures that if a player reaches a certain information set, the player's past actions and information are consistent with all the nodes within that information set. Games with players possessing perfect recall are often easier to analyze than those where players do not. Conversely, a lack of perfect recall by a player can lead to situations where that player is unable to execute planned strategies, affecting game outcomes.
== See also ==
Kuhn's theorem
== References == | Wikipedia/Perfect_recall_(game_theory) |
In game theory, a max-dominated strategy is a strategy that is never a best response to any possible strategy profile of the other players. This means there is no situation in which the strategy is optimal to play, even if it is not strictly worse than another strategy in every case.
The concept generalizes the notion of a strictly dominated strategy, which is a strategy that always yields a lower payoff than some other strategy, no matter what the other players do. Every strictly dominated strategy is max-dominated, but not every max-dominated strategy is strictly dominated. For example, suppose strategy A gives the same payoff as another strategy B against some opponent choices, but never gives a higher payoff than B—and is strictly worse in some cases. In this case, A is never a best response, so it is max-dominated, even though it is not strictly dominated.
== Definition ==
=== Max-dominated strategies ===
A strategy
s
i
∈
S
i
{\displaystyle s_{i}\in S_{i}}
of player
i
{\displaystyle i}
is max-dominated if for every strategy profile of the other players
s
−
i
∈
S
−
i
{\displaystyle s_{-i}\in S_{-i}}
there is a strategy
s
i
′
∈
S
i
{\displaystyle s_{i}^{\prime }\in S_{i}}
such that
u
i
(
s
i
′
,
s
−
i
)
>
u
i
(
s
i
,
s
−
i
)
{\displaystyle u_{i}(s_{i}^{\prime },s_{-i})>u_{i}(s_{i},s_{-i})}
. This definition means that
s
i
{\displaystyle s_{i}}
is not a best response to any strategy profile
s
−
i
{\displaystyle s_{-i}}
, since for every such strategy profile there is another strategy
s
i
′
{\displaystyle s_{i}^{\prime }}
which gives higher utility than
s
i
{\displaystyle s_{i}}
for player
i
{\displaystyle i}
.
If a strategy
s
i
∈
S
i
{\displaystyle s_{i}\in S_{i}}
is strictly dominated by strategy
s
i
′
∈
S
i
{\displaystyle s_{i}^{\prime }\in S_{i}}
then it is also max-dominated, since for every strategy profile of the other players
s
−
i
∈
S
−
i
{\displaystyle s_{-i}\in S_{-i}}
,
s
i
′
{\displaystyle s_{i}^{\prime }}
is the strategy for which
u
i
(
s
i
′
,
s
−
i
)
>
u
i
(
s
i
,
s
−
i
)
{\displaystyle u_{i}(s_{i}^{\prime },s_{-i})>u_{i}(s_{i},s_{-i})}
.
Even if
s
i
{\displaystyle s_{i}}
is strictly dominated by a mixed strategy it is also max-dominated.
=== Weakly max-dominated strategies ===
A strategy
s
i
∈
S
i
{\displaystyle s_{i}\in S_{i}}
of player
i
{\displaystyle i}
is weakly max-dominated if for every strategy profile of the other players
s
−
i
∈
S
−
i
{\displaystyle s_{-i}\in S_{-i}}
there is a strategy
s
i
′
∈
S
i
{\displaystyle s_{i}^{\prime }\in S_{i}}
such that
u
i
(
s
i
′
,
s
−
i
)
≥
u
i
(
s
i
,
s
−
i
)
{\displaystyle u_{i}(s_{i}^{\prime },s_{-i})\geq u_{i}(s_{i},s_{-i})}
. This definition means that
s
i
{\displaystyle s_{i}}
is either not a best response or not the only best response to any strategy profile
s
−
i
{\displaystyle s_{-i}}
, since for every such strategy profile there is another strategy
s
i
′
{\displaystyle s_{i}^{\prime }}
which gives at least the same utility as
s
i
{\displaystyle s_{i}}
for player
i
{\displaystyle i}
.
If a strategy
s
i
∈
S
i
{\displaystyle s_{i}\in S_{i}}
is weakly dominated by strategy
s
i
′
∈
S
i
{\displaystyle s_{i}^{\prime }\in S_{i}}
then it is also weakly max-dominated, since for every strategy profile of the other players
s
−
i
∈
S
−
i
{\displaystyle s_{-i}\in S_{-i}}
,
s
i
′
{\displaystyle s_{i}^{\prime }}
is the strategy for which
u
i
(
s
i
′
,
s
−
i
)
≥
u
i
(
s
i
,
s
−
i
)
{\displaystyle u_{i}(s_{i}^{\prime },s_{-i})\geq u_{i}(s_{i},s_{-i})}
.
Even if
s
i
{\displaystyle s_{i}}
is weakly dominated by a mixed strategy it is also weakly max-dominated.
== Max-solvable games ==
=== Definition ===
A game
G
{\displaystyle G}
is said to be max-solvable if by iterated elimination of max-dominated strategies only one strategy profile is left at the end.
More formally we say that
G
{\displaystyle G}
is max-solvable if there exists a sequence of games
G
0
,
.
.
.
,
G
r
{\displaystyle G_{0},...,G_{r}}
such that:
G
0
=
G
{\displaystyle G_{0}=G}
G
k
+
1
{\displaystyle G_{k+1}}
is obtained by removing a single max-dominated strategy from the strategy space of a single player in
G
k
{\displaystyle G_{k}}
.
There is only one strategy profile left in
G
r
{\displaystyle G_{r}}
.
Obviously every max-solvable game has a unique pure Nash equilibrium which is the strategy profile left in
G
r
{\displaystyle G_{r}}
.
As in the previous part one can define respectively the notion of weakly max-solvable games, which are games for which a game with a single strategy profile can be reached by eliminating weakly max-dominated strategies. The main difference would be that weakly max-dominated games may have more than one pure Nash equilibrium, and that the order of elimination might result in different Nash equilibria.
=== Example ===
The prisoner's dilemma is an example of a max-solvable game (as it is also dominance solvable). The strategy cooperate is max-dominated by the strategy defect for both players, since playing defect always gives the player a higher utility, no matter what the other player plays. To see this note that if the row player plays cooperate then the column player would prefer playing defect and go free than playing cooperate and serving one year in jail. If the row player plays defect then the column player would prefer playing defect and serve three years in jail rather than playing cooperate and serving five years in jail.
=== Max-solvable games and best-reply dynamics ===
In any max-solvable game, best-reply dynamics ultimately leads to the unique pure Nash equilibrium of the game. In order to see this, all we need to do is notice that if
s
1
,
s
2
,
s
3
,
.
.
.
,
s
k
{\displaystyle s_{1},s_{2},s_{3},...,s_{k}}
is an elimination sequence of the game (meaning that first
s
1
{\displaystyle s_{1}}
is eliminated from the strategy space of some player since it is max-dominated, then
s
2
{\displaystyle s_{2}}
is eliminated, and so on), then in the best-response dynamics
s
1
{\displaystyle s_{1}}
will be never played by its player after one iteration of best responses,
s
2
{\displaystyle s_{2}}
will never be played by its player after two iterations of best responses and so on. The reason for this is that
s
1
{\displaystyle s_{1}}
is not a best response to any strategy profile of the other players
s
−
i
{\displaystyle s_{-i}}
so after one iteration of best responses its player must have chosen a different strategy. Since we understand that we will never return to
s
1
{\displaystyle s_{1}}
in any iteration of the best responses, we can treat the game after one iteration of best responses as if
s
1
{\displaystyle s_{1}}
has been eliminated from the game, and complete the proof by induction.
It may come by surprise then that weakly max-solvable games do not necessarily converge to a pure Nash equilibrium when using the best-reply dynamics, as can be seen in the game on the right. If the game starts of the bottom left cell of the matrix, then the following best replay dynamics is possible: the row player moves one row up to the center row, the column player moves to the right column, the row player moves back to the bottom row, the column player moves back to the left column and so on. This obviously never converges to the unique pure Nash equilibrium of the game (which is the upper left cell in the payoff matrix).
== See also ==
Dominance (game theory)
== External links and references ==
Nisan, Noam; Schapira, Michael; Zohar, Aviv (2009), Asynchronous best reply dynamics, Berlin: Springer-Verlag, archived from the original on 2003-04-17. Asynchronous best-reply dynamics. [1]. | Wikipedia/Max-dominated_strategy |
Game theory is the study of participants' behavior in strategic situations.
Game theory may also refer to:
Game Theory (web show), an ongoing web series created and formerly hosted by Matthew Patrick
Combinatorial game theory, the study of move combinations in games like nim, chess, and go
Game Theory (band), a 1980s American rock band
Game Theory (album), a 2006 album by hip-hop band The Roots
Political Game Theory, a book by Nolan McCarty with Adam Meirowitz
Role-playing game theory
== See also ==
Game studies, the discipline of studying games in the context of entertainment and education
Game design, the process of designing the rules and content of a game | Wikipedia/Game_theory_(disambiguation) |
In game theory, a strategy A dominates another strategy B if A will always produce a better result than B, regardless of how any other player plays. Some very simple games (called straightforward games) can be solved using dominance.
== Terminology ==
A player can compare two strategies, A and B, to determine which one is better. The result of the comparison is one of:
B strictly dominates (>) A: choosing B always gives a better outcome than choosing A, no matter what the other players do.
B weakly dominates (≥) A: choosing B always gives at least as good an outcome as choosing A, no matter what the other players do, and there is at least one set of opponents' actions for which B gives a better outcome than A. (Notice that if B strictly dominates A, then B weakly dominates A. Therefore, we can say "B dominates A" to mean "B weakly dominates A".)
B is weakly dominated by A: there is at least one set of opponents' actions for which B gives a worse outcome than A, while all other sets of opponents' actions give B the same payoff as A. (Strategy A weakly dominates B).
B is strictly dominated by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. (Strategy A strictly dominates B).
Neither A nor B dominates the other: B and A are not equivalent, and B neither dominates, nor is dominated by, A. Choosing A is better in some cases, while choosing B is better in other cases, depending on exactly how the opponent chooses to play. For example, B is "throw rock" while A is "throw scissors" in Rock, Paper, Scissors.
This notion can be generalized beyond the comparison of two strategies.
Strategy B is strictly dominant if strategy B strictly dominates every other possible strategy.
Strategy B is weakly dominant if strategy B weakly dominates every other possible strategy.
Strategy B is strictly dominated if some other strategy exists that strictly dominates B.
Strategy B is weakly dominated if some other strategy exists that weakly dominates B.
Strategy: A complete contingent plan for a player in the game. A complete contingent plan is a full specification of a player's behavior, describing each action a player would take at every possible decision point. Because information sets represent points in a game where a player must make a decision, a player's strategy describes what that player will do at each information set.
Rationality: The assumption that each player acts in a way that is designed to bring about what he or she most prefers given probabilities of various outcomes; von Neumann and Morgenstern showed that if these preferences satisfy certain conditions, this is mathematically equivalent to maximizing a payoff. A straightforward example of maximizing payoff is that of monetary gain, but for the purpose of a game theory analysis, this payoff can take any desired outcome—cash reward, minimization of exertion or discomfort, or promoting justice can all be modeled as amassing an overall “utility” for the player. The assumption of rationality states that players will always act in the way that best satisfies their ordering from best to worst of various possible outcomes.
Common Knowledge: The assumption that each player has knowledge of the game, knows the rules and payoffs associated with each course of action, and realizes that every other player has this same level of understanding. This is the premise that allows a player to make a value judgment on the actions of another player, backed by the assumption of rationality, into consideration when selecting an action.
== Dominance and Nash equilibria ==
If a strictly dominant strategy exists for one player in a game, that player will play that strategy in each of the game's Nash equilibria. If both players have a strictly dominant strategy, the game has only one unique Nash equilibrium, referred to as a "dominant strategy equilibrium". However, that Nash equilibrium is not necessarily "efficient", meaning that there may be non-equilibrium outcomes of the game that would be better for both players. The classic game used to illustrate this is the Prisoner's Dilemma.
Strictly dominated strategies cannot be a part of a Nash equilibrium, and as such, it is irrational for any player to play them. On the other hand, weakly dominated strategies may be part of Nash equilibria. For instance, consider the payoff matrix pictured at the right.
Strategy C weakly dominates strategy D. Consider playing C: If one's opponent plays C, one gets 1; if one's opponent plays D, one gets 0. Compare this to D, where one gets 0 regardless. Since in one case, one does better by playing C instead of D and never does worse, C weakly dominates D. Despite this,
(
D
,
D
)
{\displaystyle (D,D)}
is a Nash equilibrium. Suppose both players choose D. Neither player will do any better by unilaterally deviating—if a player switches to playing C, they will still get 0. This satisfies the requirements of a Nash equilibrium. Suppose both players choose C. Neither player will do better by unilaterally deviating—if a player switches to playing D, they will get 0. This also satisfies the requirements of a Nash equilibrium.
== Iterated elimination of strictly dominated strategies ==
The iterated elimination (or deletion, or removal) of dominated strategies (also denominated as IESDS, or IDSDS, or IRSDS) is one common technique for solving games that involves iteratively removing dominated strategies. In the first step, all dominated strategies are removed from the strategy space of each of the players, since no rational player would ever play these strategies. This results in a new, smaller game. Some strategies—that were not dominated before—may be dominated in the smaller game. The first step is repeated, creating a new even smaller game, and so on.
This process is valid since it is assumed that rationality among players is common knowledge, that is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on ad infinitum (see Aumann, 1976).
== See also ==
Max-dominated strategy
Risk dominance
Winning strategy
== References ==
Fudenberg, Drew; Tirole, Jean (1993). Game Theory. MIT Press.
Gibbons, Robert (1992). Game Theory for Applied Economists. Princeton University Press. ISBN 0-691-00395-5.
Gintis, Herbert (2000). Game Theory Evolving. Princeton University Press. ISBN 0-691-00943-0.
Leyton-Brown, Kevin; Shoham, Yoav (2008). Essentials of Game Theory: A Concise, Multidisciplinary Introduction. San Rafael, CA: Morgan & Claypool Publishers. ISBN 978-1-59829-593-1.. An 88-page mathematical introduction; see Section 3.3. Free online at many universities.
Rapoport, A. (1966). Two-Person Game Theory: The Essential Ideas. University of Michigan Press.
Jim Ratliff's Game Theory Course: Strategic Dominance
Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7. A comprehensive reference from a computational perspective; see Sections 3.4.3, 4.5. Downloadable free online.
"Strict Dominance in Mixed Strategies – Game Theory 101". gametheory101.com. Retrieved 2021-12-17.
Watson Joel. Strategy : An Introduction to Game Theory. Third ed. W.W. Norton & Company 2013.
This article incorporates material from Dominant strategy on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. | Wikipedia/Iterated_elimination_of_dominated_strategies |
In game theory, a move, action, or play is any one of the options which a player can choose in a setting where the optimal outcome depends not only on their own actions but on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship.
The term strategy is typically used to mean a complete algorithm for playing a game, telling a player what to do for every possible situation. A player's strategy determines the action the player will take at any stage of the game. However, the idea of a strategy is often confused or conflated with that of a move or action, because of the correspondence between moves and pure strategies in most games: for any move X, "always play move X" is an example of a valid strategy, and as a result every move can also be considered to be a strategy. Other authors treat strategies as being a different type of thing from actions, and therefore distinct.
It is helpful to think about a "strategy" as a list of directions, and a "move" as a single turn on the list of directions itself. This strategy is based on the payoff or outcome of each action. The goal of each agent is to consider their payoff based on a competitors action. For example, competitor A can assume competitor B enters the market. From there, Competitor A compares the payoffs they receive by entering and not entering. The next step is to assume Competitor B does not enter and then consider which payoff is better based on if Competitor A chooses to enter or not enter. This technique can identify dominant strategies where a player can identify an action that they can take no matter what the competitor does to try to maximize the payoff.
A strategy profile (sometimes called a strategy combination) is a set of strategies for all players which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.
== Strategy set ==
A player's strategy set defines what strategies are available for them to play.
A player has a finite strategy set if they have a number of discrete strategies available to them. For instance, a game of rock paper scissors comprises a single move by each player—and each player's move is made without knowledge of the other's, not as a response—so each player has the finite strategy set {rock paper scissors}.
A strategy set is infinite otherwise. For instance the cake cutting game has a bounded continuum of strategies in the strategy set {Cut anywhere between zero percent and 100 percent of the cake}.
In a dynamic game, games that are played over a series of time, the strategy set consists of the possible rules a player could give to a robot or agent on how to play the game. For instance, in the ultimatum game, the strategy set for the second player would consist of every possible rule for which offers to accept and which to reject.
In a Bayesian game, or games in which players have incomplete information about one another, the strategy set is similar to that in a dynamic game. It consists of rules for what action to take for any possible private information.
=== Choosing a strategy set ===
In applied game theory, the definition of the strategy sets is an important part of the art of making a game simultaneously solvable and meaningful. The game theorist can use knowledge of the overall problem, that is the friction between two or more players, to limit the strategy spaces, and ease the solution.
For instance, strictly speaking in the Ultimatum game a player can have strategies such as: Reject offers of ($1, $3, $5, ..., $19), accept offers of ($0, $2, $4, ..., $20). Including all such strategies makes for a very large strategy space and a somewhat difficult problem. A game theorist might instead believe they can limit the strategy set to: {Reject any offer ≤ x, accept any offer > x; for x in ($0, $1, $2, ..., $20)}.
== Pure and mixed strategies ==
A pure strategy provides a complete and deterministic plan for how a player will act in every possible situation in a game. It specifies exactly what action the player will take at each decision point, given any information they may have. A player's strategy set consists of all the pure strategies available to them.
A mixed strategy is a probability distribution over the set of pure strategies. Rather than committing to a single course of action, the player randomizes among pure strategies according to specified probabilities. Mixed strategies are particularly useful in games where no pure strategy constitutes a best response, allowing players to avoid being predictable. Since the outcomes depend on probabilities, we refer to the resulting payoffs as expected payoffs.
A pure strategy can be viewed as a special case of a mixed strategy—one in which a single pure strategy is chosen with probability 1, and all others with probability 0.
A totally mixed strategy is a mixed strategy in which every pure strategy in the player's strategy set is assigned a strictly positive probability—that is, no pure strategy is excluded or played with zero probability. This means the player randomizes across all of their options, never fully ruling any one out. Totally mixed strategies are important in some advanced game theory concepts like trembling hand perfect equilibrium, where the idea is to model players as occasionally making small mistakes. In that context, assigning positive probability to every strategy—even suboptimal ones—helps capture how players might still end up choosing them due to small "trembles" in decision-making.
== Mixed strategy ==
=== Illustration ===
In a soccer penalty kick, the kicker must choose whether to kick to the right or left side of the goal, and simultaneously the goalie must decide which way to block it. Also, the kicker has a direction they are best at shooting, which is left if they are right-footed. The matrix for the soccer game illustrates this situation, a simplified form of the game studied by Chiappori, Levitt, and Groseclose (2002). It assumes that if the goalie guesses correctly, the kick is blocked, which is set to the base payoff of 0 for both players. If the goalie guesses wrong, the kick is more likely to go in if it is to the left (payoffs of +2 for the kicker and -2 for the goalie) than if it is to the right (the lower payoff of +1 to kicker and -1 to goalie).
This game has no pure-strategy equilibrium, because one player or the other would deviate from any profile of strategies—for example, (Left, Left) is not an equilibrium because the Kicker would deviate to Right and increase his payoff from 0 to 1.
The kicker's mixed-strategy equilibrium is found from the fact that they will deviate from randomizing unless their payoffs from Left Kick and Right Kick are exactly equal. If the goalie leans left with probability g, the kicker's expected payoff from Kick Left is g(0) + (1-g)(2), and from Kick Right is g(1) + (1-g)(0). Equating these yields g= 2/3. Similarly, the goalie is willing to randomize only if the kicker chooses mixed strategy probability k such that Lean Left's payoff of k(0) + (1-k)(-1) equals Lean Right's payoff of k(-2) + (1-k)(0), so k = 1/3. Thus, the mixed-strategy equilibrium is (Prob(Kick Left) = 1/3, Prob(Lean Left) = 2/3).
In equilibrium, the kicker kicks to their best side only 1/3 of the time. That is because the goalie is guarding that side more. Also, in equilibrium, the kicker is indifferent which way they kick, but for it to be an equilibrium they must choose exactly 1/3 probability.
Chiappori, Levitt, and Groseclose try to measure how important it is for the kicker to kick to their favored side, add center kicks, etc., and look at how professional players actually behave. They find that they do randomize, and that kickers kick to their favored side 45% of the time and goalies lean to that side 57% of the time. Their article is well-known as an example of how people in real life use mixed strategies.
=== Significance ===
In his famous paper, John Forbes Nash proved that there is an equilibrium for every finite game. One can divide Nash equilibria into two types. Pure strategy Nash equilibria are Nash equilibria where all players are playing pure strategies. Mixed strategy Nash equilibria are equilibria where at least one player is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies, see Matching pennies. However, many games do have pure strategy Nash equilibria (e.g. the Coordination game, the Prisoner's dilemma, the Stag hunt). Further, games can have both pure strategy and mixed strategy equilibria. An easy example is the pure coordination game, where in addition to the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both players play either strategy with probability 1/2.
=== Interpretations of mixed strategies ===
During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic", since they are weak Nash equilibria, and a player is indifferent about whether to follow their equilibrium strategy probability or deviate to some other probability.
Game theorist Ariel Rubinstein describes alternative ways of understanding the concept. The first, due to Harsanyi (1973), is called purification, and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors.
A second interpretation imagines the game players standing for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents.
Later, Aumann and Brandenburger (1995), re-interpreted Nash equilibrium as an equilibrium in beliefs, rather than actions. For instance, in rock paper scissors an equilibrium in beliefs would have each player believing the other was equally likely to play each strategy. This interpretation weakens the descriptive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to actually play a pure strategy of Rock in each play of the game, even though over time the probabilities are those of the mixed strategy.
== Behavior strategy ==
While a mixed strategy assigns a probability distribution over pure strategies, a behavior strategy (or behavioral strategy) assigns at each information set a probability distribution over the set of possible actions. While the two concepts are very closely related in the context of normal form games, they have very different implications for extensive form games. Roughly, a mixed strategy randomly chooses a deterministic path through the game tree, while a behavior strategy can be seen as a stochastic path.
The relationship between mixed and behavior strategies is the subject of Kuhn's theorem, a behavioral outlook on traditional game-theoretic hypotheses. The result establishes that in any finite extensive-form game with perfect recall, for any player and any mixed strategy, there exists a behavior strategy that, against all profiles of strategies (of other players), induces the same distribution over terminal nodes as the mixed strategy does. The converse is also true.
A famous example of why perfect recall is required for the equivalence is given by Piccione and Rubinstein (1997) with their Absent-Minded Driver game.
=== Outcome equivalence ===
Outcome equivalence combines the mixed and behavioral strategy of Player i in relation to the pure strategy of Player i’s opponent. Outcome equivalence is defined as the situation in which, for any mixed and behavioral strategy that Player i takes, in response to any pure strategy that Player I’s opponent plays, the outcome distribution of the mixed and behavioral strategy must be equal. This equivalence can be described by the following formula: (Q^(U(i), S(-i)))(z) = (Q^(β(i), S(-i)))(z), where U(i) describes Player i's mixed strategy, β(i) describes Player i's behavioral strategy, and S(-i) is the opponent's strategy.
=== Strategy with perfect recall ===
Perfect recall is defined as the ability of every player in game to remember and recall all past actions within the game. Perfect recall is required for equivalence as, in finite games with imperfect recall, there will be existing mixed strategies of Player I in which there is no equivalent behavior strategy. This is fully described in the Absent-Minded Driver game formulated by Piccione and Rubinstein. In short, this game is based on the decision-making of a driver with imperfect recall, who needs to take the second exit off the highway to reach home but does not remember which intersection they are at when they reach it. Figure [2] describes this game.
Without perfect information (i.e. imperfect information), players make a choice at each decision node without knowledge of the decisions that have preceded it. Therefore, a player’s mixed strategy can produce outcomes that their behavioral strategy cannot, and vice versa. This is demonstrated in the Absent-minded Driver game. With perfect recall and information, the driver has a single pure strategy, which is [continue, exit], as the driver is aware of what intersection (or decision node) they are at when they arrive to it. On the other hand, looking at the planning-optimal stage only, the maximum payoff is achieved by continuing at both intersections, maximized at p=2/3 (reference). This simple one player game demonstrates the importance of perfect recall for outcome equivalence, and its impact on normal and extended form games.
== See also ==
Nash equilibrium
Haven (graph theory)
Evolutionarily stable strategy
== References == | Wikipedia/Mixed_strategy |
In game theory, the traveler's dilemma (sometimes abbreviated TD) is a non-zero-sum game in which each player proposes a payoff. The lower of the two proposals wins; the lowball player receives the lowball payoff plus a small bonus, and the highball player receives the same lowball payoff, minus a small penalty. Surprisingly, the Nash equilibrium is for both players to aggressively lowball. The traveler's dilemma is notable in that naive play appears to outperform the Nash equilibrium; this apparent paradox also appears in the centipede game and the finitely-iterated prisoner's dilemma.
== Formulation ==
The original game scenario was formulated in 1994 by Kaushik Basu and goes as follows:
"An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical antiques. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of $100 per suitcase—he is unable to find out directly the price of the antiques."
"To determine an honest appraised value of the antiques, the manager separates both travelers so they can't confer, and asks them to write down the amount of their value at no less than $2 and no larger than $100. He also tells them that if both write down the same number, he will treat that number as the true dollar value of both suitcases and reimburse both travelers that amount. However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: $2 extra will be paid to the traveler who wrote down the lower value and a $2 deduction will be taken from the person who wrote down the higher amount. The challenge is: what strategy should both travelers follow to decide the value they should write down?"
The two players attempt to maximize their own payoff, without any concern for the other player's payoff.
== Analysis ==
One might expect a traveler's optimum choice to be $100; that is, the traveler values the antiques at the airline manager's maximum allowed price. Remarkably, and, to many, counter-intuitively, the Nash equilibrium solution is in fact just $2; that is, the traveler values the antiques at the airline manager's minimum allowed price.
For an understanding of why $2 is the Nash equilibrium consider the following proof:
Alice, having lost her antiques, is asked their value. Alice's first thought is to quote $100, the maximum permissible value.
On reflection, though, she realizes that her fellow traveler, Bob, might also quote $100. And so Alice changes her mind, and decides to quote $99, which, if Bob quotes $100, will pay $101.
But Bob, being in an identical position to Alice, might also think of quoting $99. And so Alice changes her mind, and decides to quote $98, which, if Bob quotes $99, will pay $100. This is greater than the $99 Alice would receive if both she and Bob quoted $99.
This cycle of thought continues, until Alice finally decides to quote just $2—the minimum permissible price.
Another proof goes as follows:
If Alice only wants to maximize her own payoff, choosing $99 trumps choosing $100. If Bob chooses any dollar value 2–98 inclusive, $99 and $100 give equal payoffs; if Bob chooses $99 or $100, choosing $99 nets Alice an extra dollar.
A similar line of reasoning shows that choosing $98 is always better for Alice than choosing $99. The only situation where choosing $99 would give a higher payoff than choosing $98 is if Bob chooses $100—but if Bob is only seeking to maximize his own profit, he will always choose $99 instead of $100.
This line of reasoning can be applied to all of Alice's whole-dollar options until she finally reaches $2, the lowest price.
== Experimental results ==
The ($2, $2) outcome in this instance is the Nash equilibrium of the game. By definition this means that if your opponent chooses this Nash equilibrium value then your best choice is that Nash equilibrium value of $2. This will not be the optimum choice if there is a chance of your opponent choosing a higher value than $2. When the game is played experimentally, most participants select a value higher than the Nash equilibrium and closer to $100 (corresponding to the Pareto optimal solution). More precisely, the Nash equilibrium strategy solution proved to be a bad predictor of people's behavior in a traveler's dilemma with small bonus/malus and a rather good predictor if the bonus/malus parameter was big.
Furthermore, the travelers are rewarded by deviating strongly from the Nash equilibrium in the game and obtain much higher rewards than would be realized with the purely rational strategy. These experiments (and others, such as focal points) show that the majority of people do not use purely rational strategies, but the strategies they do use are demonstrably optimal. This paradox could reduce the value of pure game theory analysis, but could also point to the benefit of an expanded reasoning that understands how it can be quite rational to make non-rational choices, at least in the context of games that have players that can be counted on to not play "rationally." For instance, Capraro has proposed a model where humans do not act a priori as single agents but they forecast how the game would be played if they formed coalitions and then they act so as to maximize the forecast. His model fits the experimental data on the Traveler's dilemma and similar games quite well. Recently, the traveler's dilemma was tested with decision undertaken in groups rather than individually, in order to test the assumption that groups decisions are more rational, delivering the message that, usually, two heads are better than one. Experimental findings show that groups are always more rational – i.e. their claims are closer to the Nash equilibrium - and more sensitive to the size of the bonus/malus.
Some players appear to pursue a Bayesian Nash equilibrium.
== Similar games ==
The traveler's dilemma can be framed as a finitely repeated prisoner's dilemma. Similar paradoxes are attributed to the centipede game and to the p-beauty contest game (or more specifically, "Guess 2/3 of the average"). One variation of the original traveler's dilemma in which both travelers are offered only two integer choices, $2 or $3, is identical mathematically to the standard non-iterated Prisoner's dilemma and thus the traveler's dilemma can be viewed as an extension of prisoner's dilemma. (The minimum guaranteed payout is $1, and each dollar beyond that may be considered equivalent to a year removed from a three-year prison sentence.) These games tend to involve deep iterative deletion of dominated strategies in order to demonstrate the Nash equilibrium, and tend to lead to experimental results that deviate markedly from classical game-theoretical predictions.
== Payoff matrix ==
The canonical payoff matrix is shown below (if only integer inputs are taken into account):
Denoting by
S
=
{
2
,
.
.
.
,
100
}
{\displaystyle S=\{2,...,100\}}
the set of strategies available to both players and by
F
:
S
×
S
→
R
{\displaystyle F:S\times S\rightarrow \mathbb {R} }
the payoff function of one of them we can write
F
(
x
,
y
)
=
min
(
x
,
y
)
+
2
⋅
sgn
(
y
−
x
)
{\displaystyle F(x,y)=\min(x,y)+2\cdot \operatorname {sgn}(y-x)}
(Note that the other player receives
F
(
y
,
x
)
{\displaystyle F(y,x)}
since the game is quantitatively symmetric).
== References == | Wikipedia/Traveler's_dilemma |
In game theory, a non-cooperative game is a game in which there are no external rules or binding agreements that enforce the cooperation of the players. A non-cooperative game is typically used to model a competitive environment. This is stated in various accounts most prominent being John Nash's 1951 paper in the journal Annals of Mathematics.
Counterintuitively, non-cooperative game models can be used to model cooperation as well, and vice versa, cooperative game theory can be used to model competition. Some examples of this would be the use of non-cooperative game models in determining the stability and sustainability of cartels and coalitions.
== The difference between cooperative and non-cooperative game theory ==
According to Nash, the difference between cooperative game theory and non-cooperative game theory is that “(cooperative game) theory is based on an analysis of the interrelationships of the various coalitions which can be formed by the players of the game. Our (non-cooperative game) theory, in contradistinction, is based on the absence of coalitions in that it is assumed that each participant acts independently, without collaboration or communication with any of the others.”
Non-cooperative game theory models different situations in which agents are unable to reach a resolution to a conflict that enforces some action on one another. This form of game theory pays close attention to the individuals involved and their rational decision making. There are winners and losers in each case, and yet agents may end up in Pareto-inferior outcomes, where every agent is worse off and there is a potential outcome for every agent to be better off. Agents will have the ability to predict what their opponents will do. Cooperative game theory models situations in which a binding agreement is possible. In other words, the cooperative game theory implies that agents cooperate to achieve a common goal and they are not necessarily referred to as a team because the correct term is the coalition. Each agent has its skills or contributions that provide strength to the coalition.
Further, it has been supposed that non-cooperative game theory is purported to analyse the effect of independent decisions on society as a whole. In comparison, cooperative game theory focuses only on the effects of participants in a certain coalition, when the coalition attempts to improve the collective welfare.
Many results or solutions proposed by the agents involved in Game Theory are important in understanding the rivalry between these agents under a set of conditions that are strategic.
== Elements of a non-cooperative game ==
To specify a non-cooperative game completely, one must specify
The number of players,
The actions available to each player at any given state of the game,
The function that each player is attempting to maximize,
The time ordering of actions (if needed),
How information is acquired by the players.
Whether there is any randomness in the game.
The following assumptions are commonly made:
Perfect recall: each player remembers their decisions and known information.
Self-interest: each player does not consider the effect of actions on the others but only on their own.
Rational: each player is interested to maximise their utility or payoff.
Complete information: each player knows the preferences and strategies of the other players.
Each player has the same understanding of how the game is.
== Examples ==
Strategic games are a form of non-cooperative game, where only the available strategies and combinations of options are listed to produce outcomes.
=== Rock paper scissors ===
In the game of rock-paper-scissors, if Player 1 decides to play "rock", it is in Player 2's interest to play "paper"; if Player 2 chooses to play "paper", it is in Player 1's interest to play "scissors"; and if Player 1 plays "scissors", Player 2 will, in their own interests, play "rock".
=== Prisoner's Dilemma ===
The Prisoner's Dilemma game is another well-known example of a non-cooperative game. The game involves two players, or defendants, who are kept in separate rooms and thus are unable to communicate. Players must decide, by themselves in isolation, whether to cooperate with the other player or to betray them and confess to law authorities. As shown in the diagram, both players will receive a higher payoff in the form of a lower jail sentence if they both remain silent. If both confess, they receive a lower payoff in the form of a higher jail sentence. If one player confesses and the other remain silent and cooperates, the confessor will receive a higher payoff, while the silent player will receive a lower payoff than if both players cooperated with each other.
The Nash equilibrium therefore lies where players both betray each other, in the players protecting oneself from being punished more.
=== The battle of the sexes ===
The game involves two players, boy and girl, deciding either going to a football game or going to an opera for their date, which respectively represent boy's and girl's preferred activity (i.e. boy prefers football game and girl prefers opera). This example is a two-person non-cooperative non-zero sum (TNNC) game with opposite payoffs or conflicting preferences. Because there are two Nash equilibria, this case is a pure coordination problem with no possibility of refinement or selection. Thus, the two players will try to maximise their own payoff or to sacrifice for the other and yet these strategies without coordination will lead to two outcomes with even worse payoffs for both if they disagree on what to do on their date.
=== Matching pennies game ===
This game is a two-person zero-sum game. In order to play this game, both players will each need to be given a fair two-sided penny. To start the game, both player will each choose to either flip their penny to heads or tails. This action is to be done in secrecy and there should be no attempt at investigating the choice of the other player. After both players have confirmed their decisions, they will simultaneously reveal their choices. This concludes the actions taken by the players to determine the outcome.
The win condition for this game is different for both players. For simplicity in explanation, lets denote the players as Player 1 and Player 2. In order for Player 1 to win, the faces of the pennies must match (This means they must both be heads or tails). In order for Player 2 to win, the faces of the pennies must be different (This means that they must be in a combination of heads and tails).
The payoff/prize of this game is receiving the loser's penny in addition to your own.
Therefore the payoff matrix will look like this:
Looking at this matrix, we can conclude a few basic observations.
For all scenarios, there will be a loser and a winner.
This is a zero sum game where the pay out to the winner is equal to the loss of the loser.
There is no Pure Strategy Nash Equilibrium.
== Analysis ==
Non-cooperative games are generally analysed through the non-cooperative game theory framework, which attempts to predict players' individual strategies and payoffs and in order to find the Nash equilibria. This framework often requires a detailed knowledge in the possible actions and the levels of information of each player. It is opposed to cooperative game theory, which focuses on predicting which groups of players ("coalitions") will form, the joint actions that these groups will take, and the resulting collective payoffs that arise. Cooperative game theory does not analyse the strategic bargaining that occurs within each coalition and affects the distribution of the collective payoff between the members. Further in contrast to cooperative game theory, it is assumed that players involved have prior knowledge of their game in which they are involved, due to built in commitments.
Non-cooperative game theory provides a low-level approach as it models all the procedural details of the game, whereas cooperative game theory only describes the structure, strategies and payoffs of coalitions. Therefore, cooperative game theory is referred to as coalitional, and non-cooperative game theory is procedural. Non-cooperative game theory is in this sense more inclusive than cooperative game theory.
It is also more general, as cooperative games can be analysed using the terms of non-cooperative game theory where arbitration is available to enforce an agreement, that agreement falls outside the scope of non-cooperative theory: but it may be possible to state sufficient assumptions to encompass all the possible strategies players may adopt, in relation to arbitration. This will bring the agreement within the scope of non-cooperative theory. Alternatively, it may be possible to describe the arbitrator as a party to the agreement and model the relevant processes and payoffs suitably.
Accordingly, it would be desirable to have all games expressed under a non-cooperative framework. But in many instances insufficient information is available to accurately model the formal procedures available to the players during the strategic bargaining process; or the resulting model would be of too high complexity to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make any assumption about bargaining powers.
Additionally, we must also look at the limitations that the non-cooperative model may have. We can have a clearer picture when looking at the list of assumptions stated above. As already mentioned, there are many scenarios where perfect symmetry of information is not possible which therefore results in the decision making process to be flawed.
Secondly, the assumption of self-interest and rationality could be argued. Arguments are made that being rational can result in the assumption of self-interest being invalidated and vice versa. One such example could be the reduction in profits and revenue in attempts to drive out competitors for a higher market share. This thus does not follow both of the assumptions as the player is concerned with the downfall of their opponent more than the maximisation of their profits. There is the argument to be made that although mathematically sound and feasible, it is not necessarily the best method of looking at real life economical problems that are more complex in nature.
== Solutions ==
Solutions in non-cooperative games are similar to all other games in game theory, but without the ones involved binding agreements enforced by the external authority. The solutions are normally based on the concept of Nash equilibrium, and these solutions are reached by using methods listed in Solution concept. Most solutions used in non-cooperative game are refinements developed from Nash equilibrium, including the minimax mixed-strategy proved by John von Neumann.
== See also ==
Mutual assured destruction – Doctrine of military strategy
Grim trigger – Trigger strategy
Intra-household bargaining – negotiations between members of a household to reach decisionsPages displaying wikidata descriptions as a fallback
Proper equilibrium – in game theory, a refinement of the Nash equilibriumPages displaying wikidata descriptions as a fallback
Tit for tat – English saying meaning "equivalent retaliation"
Trembling hand perfect equilibrium – Variant of Nash equilibrium in game theory
Trigger strategy – Class of strategies employed in a repeated non-cooperative game
War of attrition (game) – Game theory model of aggression
Game theory – Mathematical models of strategic interactions
== References ==
== External links ==
A brief introduction to non-cooperative game theory | Wikipedia/Non-cooperative_game_theory |
A business model describes how a business organization creates, delivers, and captures value, in economic, social, cultural or other contexts. The model describes the specific way in which the business conducts itself, spends, and earns money in a way that generates profit. The process of business model construction and modification is also called business model innovation and forms a part of business strategy.
In theory and practice, the term business model is used for a broad range of informal and formal descriptions to represent core aspects of an organization or business, including purpose, business process, target customers, offerings, strategies, infrastructure, organizational structures, profit structures, sourcing, trading practices, and operational processes and policies including culture.
== Context ==
The literature has provided very diverse interpretations and definitions of a business model. A systematic review and analysis of manager responses to a survey defines business models as the design of organizational structures to enact a commercial opportunity. Further extensions to this design logic emphasize the use of narrative or coherence in business model descriptions as mechanisms by which entrepreneurs create extraordinarily successful growth firms.
Business models are used to describe and classify businesses, especially in an entrepreneurial setting, but they are also used by managers inside companies to explore possibilities for future development. Well-known business models can operate as "recipes" for creative managers. Business models are also referred to in some instances within the context of accounting for purposes of public reporting.
== History ==
According to the Oxford English Dictionary, the term "business model", a compound of business and model, was first used in 1832 in the sense of "a plan for the operation of a business".
Over the years, business models have become much more sophisticated. The bait and hook business model (also referred to as the "razor and blades business model" or the "tied products business model") was introduced in the early 20th century. This involves offering a basic product at a very low cost, often at a loss (the "bait"), then charging compensatory recurring amounts for refills or associated products or services (the "hook"). Examples include: razor (bait) and blades (hook); cell phones (bait) and air time (hook); computer printers (bait) and ink cartridge refills (hook); and cameras (bait) and prints (hook). A variant of this model was employed by Adobe, a software developer that gave away its document reader free of charge but charged several hundred dollars for its document writer.
In the 1950s, new business models came from McDonald's Restaurants and Toyota. In the 1960s, the innovators were Wal-Mart and Hypermarkets. The 1970s saw new business models from FedEx and Toys R Us; the 1980s from Blockbuster, Home Depot, Intel, and Dell Computer; the 1990s from Southwest Airlines, Netflix, eBay, Amazon.com, and Starbucks.
Today, the type of business models might depend on how technology is used. For example, entrepreneurs on the internet have also created new models that depend entirely on existing or emergent technology. Using technology, businesses can reach a large number of customers with minimal costs. In addition, the rise of outsourcing and globalization has meant that business models must also account for strategic sourcing, complex supply chains and moves to collaborative, relational contracting structures.
== Theoretical and empirical insights ==
=== Design logic and narrative coherence ===
Design logic views the business model as an outcome of creating new organizational structures or changing existing structures to pursue a new opportunity. Gerry George and Adam Bock (2011) conducted a comprehensive literature review and surveyed managers to understand how they perceived the components of a business model. In that analysis these authors show that there is a design logic behind how entrepreneurs and managers perceive and explain their business model. In further extensions to the design logic, George and Bock (2012) use case studies and the IBM survey data on business models in large companies, to describe how CEOs and entrepreneurs create narratives or stories in a coherent manner to move the business from one opportunity to another. They also show that when the narrative is incoherent or the components of the story are misaligned, that these businesses tend to fail. They recommend ways in which the entrepreneur or CEO can create strong narratives for change.
=== Complementarities between partnering firms ===
Berglund and Sandström (2013) argued that business models should be understood from an open systems perspective as opposed to being a firm-internal concern. Since innovating firms do not have executive control over their surrounding network, business model innovation tends to require soft power tactics with the goal of aligning heterogeneous interests. In a study of collaborative research and external sourcing of technology, Hummel et al. (2010) similarly found that in deciding on business partners, it is important to make sure that both parties' business models are complementary. For example, they found that it was important to identify the value drivers of potential partners by analyzing their business models, and that it is beneficial to find partner firms that understand key aspects of one's own firm's business model.
The University of Tennessee conducted research into highly collaborative business relationships. Researchers codified their research into a sourcing business model known as Vested Outsourcing, a hybrid sourcing business model in which buyers and suppliers in an outsourcing or business relationship focus on shared values and goals to create an arrangement that is highly collaborative and mutually beneficial to each.
== Categorization ==
From about 2012, some research and experimentation has theorized about a so-called "liquid business model".
=== Shift from pipes to platforms ===
Sangeet Paul Choudary distinguishes between two broad families of business models in an article in Wired magazine. Choudary contrasts pipes (linear business models) with platforms (networked business models). In the case of pipes, firms create goods and services, push them out and sell them to customers. Value is produced upstream and consumed downstream. There is a linear flow, much like water flowing through a pipe. Unlike pipes, platforms do not just create and push stuff out. They allow users to create and consume value.
Alex Moazed, founder and CEO of Applico, defines a platform as a business model that creates value by facilitating exchanges between two or more interdependent groups, usually consumers and producers, of a given value. As a result of digital transformation, it is the predominant business model of the 21st century.
In an op-ed on MarketWatch, Choudary, Van Alstyne and Parker further explain how business models are moving from pipes to platforms, leading to disruption of entire industries.
=== Platform ===
There are three elements to a successful platform business model. The toolbox creates connection by making it easy for others to plug into the platform. This infrastructure enables interactions between participants. The magnet creates pull that attracts participants to the platform. For transaction platforms, both producers and consumers must be present to achieve critical mass. The matchmaker fosters the flow of value by making connections between producers and consumers. Data is at the heart of successful matchmaking, and distinguishes platforms from other business models.
Chen (2009) stated that the business model has to take into account the capabilities of Web 2.0, such as collective intelligence, network effects, user-generated content, and the possibility of self-improving systems. He suggested that the service industry such as the airline, traffic, transportation, hotel, restaurant, information and communications technology and online gaming industries will be able to benefit in adopting business models that take into account the characteristics of Web 2.0. He also emphasized that Business Model 2.0 has to take into account not just the technology effect of Web 2.0 but also the networking effect. He gave the example of the success story of Amazon in making huge revenues each year by developing an open platform that supports a community of companies that re-use Amazon's on-demand commerce services.
==== Impacts of platform business models ====
Jose van Dijck (2013) identifies three main ways that media platforms choose to monetize, which mark a change from traditional business models. One is the subscription model, in which platforms charge users a small monthly fee in exchange for services. She notes that the model was ill-suited for those "accustomed to free content and services", leading to a variant, the freemium model. A second method is via advertising. Arguing that traditional advertising is no longer appealing to people used to "user-generated content and social networking", she states that companies now turn to strategies of customization and personalization in targeted advertising. Eric K. Clemons (2009) asserts that consumers no longer trust most commercial messages; Van Dijck argues platforms are able to circumvent the issue through personal recommendations from friends or influencers on social media platforms, which can serve as a more subtle form of advertisement. Finally, a third common business model is monetization of data and metadata generated from the use of platforms.
== Applications ==
Malone et al. found that some business models, as defined by them, indeed performed better than others in a dataset consisting of the largest U.S. firms, in the period 1998 through 2002, while they did not prove whether the existence of a business model mattered.
In the healthcare space, and in particular in companies that leverage the power of Artificial Intelligence, the design of business models is particularly challenging as there are a multitude of value creation mechanisms and a multitude of possible stakeholders. An emerging categorization has identified seven archetypes.
The concept of a business model has been incorporated into certain accounting standards. For example, the International Accounting Standards Board (IASB) utilizes an "entity's business model for managing the financial assets" as a criterion for determining whether such assets should be measured at amortized cost or at fair value in its International Financial Reporting Standard, IFRS 9. In their 2013 proposal for accounting for financial instruments, the Financial Accounting Standards Board also proposed a similar use of business model for classifying financial instruments. The concept of business model has also been introduced into the accounting of deferred taxes under International Financial Reporting Standards with 2010 amendments to IAS 12 addressing deferred taxes related to investment property.
Both IASB and FASB have proposed using the concept of business model in the context of reporting a lessor's lease income and lease expense within their joint project on accounting for leases. In its 2016 lease accounting model, IFRS 16, the IASB chose not to include a criterion of "stand alone utility" in its lease definition because "entities might reach different conclusions for contracts that contain the same rights of use, depending on differences between customers' resources or suppliers' business models." The concept has also been proposed as an approach for determining the measurement and classification when accounting for insurance contracts. As a result of the increasing prominence the concept of business model has received in the context of financial reporting, the European Financial Reporting Advisory Group (EFRAG), which advises the European Union on endorsement of financial reporting standards, commenced a project on the "Role of the Business Model in Financial Reporting" in 2011.
== Design ==
Business model design generally refers to the activity of designing a company's business model. It is part of the business development and business strategy process and involves design methods. Massa and Tucci (2014) highlighted the difference between crafting a new business model when none is in place, as it is often the case with academic spinoffs and high technology entrepreneurship, and changing an existing business model, such as when the tooling company Hilti shifted from selling its tools to a leasing model. They suggested that the differences are so profound (for example, lack of resource in the former case and inertia and conflicts with existing configurations and organisational structures in the latter) that it could be worthwhile to adopt different terms for the two. They suggest business model design to refer to the process of crafting a business model when none is in place and business model reconfiguration for the process of changing an existing business model, also highlighting that the two processes are not mutually exclusive, meaning reconfiguration may involve steps which parallel those of designing a business model.
=== Economic consideration ===
Al-Debei and Avison (2010) consider value finance as one of the main dimensions of business modelling which depicts information related to costing, pricing methods, and revenue structure. Stewart and Zhao (2000) defined the business model as "a statement of how a firm will make money and sustain its profit stream over time."
=== Component consideration ===
Osterwalder et al. (2005) consider the Business Model as the blueprint of how a company does business. Slywotzky (1996) regards the business model as "the totality of how a company selects its customers, defines and differentiates it offerings, defines the tasks it will perform itself and those it will outsource, configures its resources, goes to market, creates utility for customers and captures profits."
=== Strategic outcome ===
Mayo and Brown (1999) considered the business model as "the design of key interdependent systems that create and sustain a competitive business." Casadesus-Masanell and Ricart (2011) explain a business model as a set of "choices (policy, assets and governance)" and "consequences (flexible and rigid)" and underline the importance of considering "how it interacts with models of other players in the industry" instead of thinking of it in isolation.
== Definitions of design or development ==
Zott and Amit (2009) consider business model design from the perspectives of design themes and design content. Design themes refer to the system's dominant value creation drivers and design content examines in greater detail the activities to be performed, the linking and sequencing of the activities and who will perform the activities.
=== Design themes emphasis ===
Developing a framework for business model development with an emphasis on design themes, Lim (2010) proposed the environment-strategy-structure-operations (ESSO) business model development which takes into consideration the alignment of the organization's strategy with the organization's structure, operations, and the environmental factors in achieving competitive advantage in varying combination of cost, quality, time, flexibility, innovation and affective.
=== Design content emphasis ===
Business model design includes the modeling and description of a company's:
value propositions
target customer segments
distribution channels
customer relationships
value configurations
core capabilities
commercial network
partner network
cost structure
revenue model
A business model design template can facilitate the process of designing and describing a company's business model. In a paper published in 2017, Johnson demonstrated how matrix methods may usefully be deployed to characterise the architecture of resources, costs, and revenues that a business uses to create and deliver value to customers which defines its business model. Systematisation of this technique (Johnson settles on a business genomic code of seven matrix elements of a business model) would support a taxonomical approach to empirical studies of business models in the same way that Linnaeus' taxonomy revolutionised biology.
Daas et al. (2012) developed a decision support system (DSS) for business model design. In their study a decision support system (DSS) is developed to help SaaS in this process, based on a design approach consisting of a design process that is guided by various design methods.
== Examples ==
In the early history of business models it was very typical to define business model types such as bricks-and-mortar or e-broker. However, these types usually describe only one aspect of the business (most often the revenue model). Therefore, more recent literature on business models concentrate on describing a business model as a whole, instead of only the most visible aspects.
The following examples provide an overview for various business model types that have been in discussion since the invention of term business model:
Bricks and clicks business model
Business model by which a company integrates both offline (bricks) and online (clicks) presences. One example of the bricks-and-clicks model is when a chain of stores allows the user to order products online, but lets them pick up their order at a local store.
Collective business models
Business system, organization or association typically composed of relatively large numbers of businesses, tradespersons or professionals in the same or related fields of endeavor, which pools resources, shares information or provides other benefits for their members. For example, a science park or high-tech campus provides shared resources (e.g. cleanrooms and other lab facilities) to the firms located on its premises, and in addition seeks to create an innovation community among these firms and their employees.
Cutting out the middleman model
The removal of intermediaries in a supply chain: "cutting out the middleman". Instead of going through traditional distribution channels, which had some type of intermediate (such as a distributor, wholesaler, broker, or agent), companies may now deal with every customer directly, for example via the Internet.
Direct sales model
Direct selling is marketing and selling products to consumers directly, away from a fixed retail location. Sales are typically made through party plan, one-to-one demonstrations, and other personal contact arrangements. A text book definition is: "The direct personal presentation, demonstration, and sale of products and services to consumers, usually in their homes or at their jobs."
Distribution business models, various
Fee in, free out
Business model which works by charging the first client a fee for a service, while offering that service free of charge to subsequent clients.
Franchise
Franchising is the practice of using another firm's successful business model. For the franchisor, the franchise is an alternative to building 'chain stores' to distribute goods and avoid investment and liability over a chain. The franchisor's success is the success of the franchisees. The franchisee is said to have a greater incentive than a direct employee because he or she has a direct stake in the business.
Sourcing business model
Sourcing Business Models are a systems-based approach to structuring supplier relationships. A sourcing business model is a type of business model that is applied to business relationships where more than one party needs to work with another party to be successful. There are seven sourcing business models that range from the transactional to investment-based. The seven models are: Basic Provider, Approved Provider, Preferred Provider, Performance-Based/Managed Services Model, Vested outsourcing Business Model, Shared Services Model, and Equity Partnership Model. Sourcing business models are targeted for procurement professionals who seek a modern approach to achieve the best fit between buyers and suppliers. Sourcing business model theory is based on a collaborative research effort by the University of Tennessee (UT), the Sourcing Industry Group (SIG), the Center for Outsourcing Research and Education (CORE), and the International Association for Contracts and Commercial Management (IACCM). This research formed the basis for the 2016 book, Strategic Sourcing in the New Economy: Harnessing the Potential of Sourcing Business Models in Modern Procurement.
Freemium business model
Business model that works by offering basic Web services, or a basic downloadable digital product, for free, while charging a premium for advanced or special features.
Pay what you can (PWYC)
A non-profit or for-profit business model which does not depend on set prices for its goods, but instead asks customers to pay what they feel the product or service is worth to them. It is often used as a promotional tactic, but can also be the regular method of doing business. It is a variation on the gift economy and cross-subsidization, in that it depends on reciprocity and trust to succeed.: "Pay what you want" (PWYW) is sometimes used synonymously, but "pay what you can" is often more oriented to charity or socially oriented uses, based more on ability to pay, while "pay what you want" is often more broadly oriented to perceived value in combination with willingness and ability to pay.
Value-added reseller model
Value Added Reseller is a model where a business makes something which is resold by other businesses but with modifications which add value to the original product or service. These modifications or additions are mostly industry specific in nature and are essential for the distribution. Businesses going for a VAR model have to develop a VAR network. It is one of the latest collaborative business models which can help in faster development cycles and is adopted by many Technology companies especially software.
Other examples of business models are:
Auction business model
All-in-one business model
Chemical leasing
Low-cost carrier business model
Loyalty business models
Monopolistic business model
Multi-level marketing business model
Network effects business model
Online auction business model
Online content business model
Premium business model
Professional open-source model
Pyramid scheme business model
Razor and blades model
Servitization of products business model
Subscription business model
Network Orchestrators Companies
Virtual business model
== Frameworks ==
Technology centric communities have defined "frameworks" for business modeling. These frameworks attempt to define a rigorous approach to defining business value streams. It is not clear, however, to what extent such frameworks are actually important for business planning. Business model frameworks represent the core aspect of any company; they involve "the totality of how a company selects its customers defines and differentiates its offerings, defines the tasks it will perform itself and those it will outsource, configures its resource, goes to market, creates utility for customers, and captures profits". A business framework involves internal factors (market analysis; products/services promotion; development of trust; social influence and knowledge sharing) and external factors (competitors and technological aspects).
Business reference model
Business reference model is a reference model, concentrating on the architectural aspects of the core business of an enterprise, service organization or government agency.
Component business model
Technique developed by IBM to model and analyze an enterprise. It is a logical representation or map of business components or "building blocks" and can be depicted on a single page. It can be used to analyze the alignment of enterprise strategy with the organization's capabilities and investments, identify redundant or overlapping business capabilities, etc.
Industrialization of services business model
Business model used in strategic management and services marketing that treats service provision as an industrial process, subject to industrial optimization procedures
Business Model Canvas
Developed by A. Osterwalder, Yves Pigneur, Alan Smith, and 470 practitioners from 45 countries, the business model canvas is one of the most used frameworks for describing the elements of business models.
OGSM
The OGSM is developed by Marc van Eck and Ellen van Zanten of Business Openers into the 'Business plan on 1 page'. Translated in several languages all over the world. #1 Management book in The Netherlands in 2015. The foundation of Business plan on 1 page is the OGSM. Objectives, Goals, Strategies and Measures (dashboard and actions).
== Related concepts ==
The process of business model design is part of business strategy. Business model design and innovation refer to the way a firm (or a network of firms) defines its business logic at the strategic level.
In contrast, firms implement their business model at the operational level, through their business operations. This refers to their process-level activities, capabilities, functions and infrastructure (for example, their business processes and business process modeling), their organizational structures (e.g. organograms, workflows, human resources) and systems (e.g. information technology architecture, production lines).
The brand is a consequence of the business model and has a symbiotic relationship with it, because the business model determines the brand promise, and the brand equity becomes a feature of the model. Managing this is a task of integrated marketing.
The standard terminology and examples of business models do not apply to most nonprofit organizations, since their sources of income are generally not the same as the beneficiaries. The term 'funding model' is generally used instead.
The model is defined by the organization's vision, mission, and values, as well as sets of boundaries for the organization—what products or services it will deliver, what customers or markets it will target, and what supply and delivery channels it will use. Mission and vision together make part of the overall business purpose. While the business model includes high-level strategies and tactical direction for how the organization will implement the model, it also includes the annual goals that set the specific steps the organization intends to undertake in the next year and the measures for their expected accomplishment. Each of these is likely to be part of internal documentation that is available to the internal auditor.
== Business model innovation ==
When an organisation creates a new business model, the process is called business model innovation. There is a range of reviews on the topic, The concept facilitates the analysis and planning of transformations from one business model to another. Frequent and successful business model innovation can increase an organisation's resilience to changes in its environment and if an organisation has the capability to do this, it can become a competitive advantage. Although business model innovation promises financial returns, periods of radical business model innovation can reduce the person-organization fit and thus lead to a greater fluctuation in the workforce.
== Business Model Adaptation ==
As a specific instance of Business Model Dynamics, a research strand derived from the evolving changes in business models, BMA identifies an update of the current business model to changes derived from the context. BMA can be innovative or not, depending on the degree of novelty of the changes implemented. As a consequence of the new context, several business model elements are promoted to answer those challenges, pivoting the business model towards new models. Companies adapt their business model when someone or something such as COVID-19 has disrupted the market. BMA could fit any organization, but incumbents are more motivated to adapt their current BM than to change it radically or create a new one.
== See also ==
== References ==
== Further reading ==
A. Afuah and C. Tucci, Internet Business Models and Strategies, Boston, McGraw Hill, 2003.
T. Burkhart, J. Krumeich, D. Werth, and P. Loos, Analyzing the Business Model Concept — A Comprehensive Classification of Literature, Proceedings of the International Conference on Information Systems (ICIS 2011). Paper 12. http://aisel.aisnet.org/icis2011/proceedings/generaltopics/12
H. Chesbrough and R. S. Rosenbloom, The Role of the Business Model in capturing value from Innovation: Evidence from XEROX Corporation's Technology Spinoff Companies., Boston, Massachusetts, Harvard Business School, 2002.
Marc Fetscherin and Gerhard Knolmayer, Focus Theme Articles: Business Models for Content Delivery: An Empirical Analysis of the Newspaper and Magazine Industry, International Journal on Media Management, Volume 6, Issue 1 & 2 September 2004, pages 4 – 11, September 2004.
George, G., Bock, AJ. Models of opportunity: How entrepreneurs design firms to achieve the unexpected. Cambridge University Press, 2012, ISBN 978-0-521-17084-0.
J. Gordijn, Value-based Requirements Engineering – Exploring Innovative e-Commerce Ideas, Amsterdam, Vrije Universiteit, 2002.
G. Hamel, Leading the revolution., Boston, Harvard Business School Press, 2000.
J. Linder and S. Cantrell, Changing Business Models: Surveying the Landscape, Accenture Institute for Strategic Change, 2000.
Lindgren, P. and Jørgensen, R., M.S. Li, Y. Taran, K.F. Saghaug, "Towards a new generation of business model innovation model", presented at the 12th International CINet Conference: Practicing innovation in times of discontinuity, Aarhus, Denmark, 10–13 September 2011
Long Range Planning, vol 43 April 2010, "Special Issue on Business Models," includes 19 pieces by leading scholars on the nature of business models
S. Muegge. Business Model Discovery by Technology Entrepreneurs Archived 2021-12-31 at the Wayback Machine. Technology Innovation Management Review Archived 2021-03-10 at the Wayback Machine, April 2012, pp. 5–16.
S. Muegge, C. Haw, and Sir T. Matthews, Business Models for Entrepreneurs and Startups, Best of TIM Review, Book 2, Talent First Network, 2013.
Alex Osterwalder et al. Business Model Generation, Co-authored with Yves Pigneur, Alan Smith, and 470 practitioners from 45 countries, self-published, 2009
O. Peterovic and C. Kittl et al., Developing Business Models for eBusiness., International Conference on Electronic Commerce 2001, 2001.
Alt, Rainer; Zimmermann, Hans-Dieter: Introduction to Special Section – Business Models. In: Electronic Markets Anniversary Edition, Vol. 11 (2001), No. 1. link
Santiago Restrepo Barrera, Business model tool, Business life model, Colombia 2012, http://www.imaginatunegocio.com/#!business-life-model/c1o75 (Spanish)
Paul Timmers. Business Models for Electronic Markets, Electronic Markets, Vol 8 (1998) No 2, pp. 3 – 8.
Peter Weill and M. R. Vitale, Place to space: Migrating to eBusiness Models., Boston, Harvard Business School Press, 2001.
C. Zott, R. Amit, & L.Massa. 'The Business Model: Theoretical Roots, Recent Developments, and Future Research', WP-862, IESE, June, 2010 – revised September 2010 (PDF)
Magretta, J. (2002). Why Business Models Matter, Harvard Business Review, May: 86–92.
Govindarajan, V. and Trimble, C. (2011). The CEO's role in business model reinvention. Harvard Business Review, January–February: 108–114.
van Zyl, Jay. (2011). Built to Thrive: using innovation to make your mark in a connected world. Chapter 7 Towards a universal service delivery platform. San Francisco.
== External links ==
Media related to Business models at Wikimedia Commons
Sustaining Digital Resources: An on-the-ground view of projects today, Ithaka, November 2009. Overview of the models being deployed and analysis on the effects of income generation and cost management. | Wikipedia/Business_model |
Strategy (from Greek στρατηγία stratēgia, "troop leadership; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the sense of the "art of the general", which included several subsets of skills including military tactics, siegecraft, logistics etc., the term came into use in the 6th century C.E. in Eastern Roman terminology, and was translated into Western vernacular languages only in the 18th century. From then until the 20th century, the word "strategy" came to denote "a comprehensive way to try to pursue political ends, including the threat or actual use of force, in a dialectic of wills" in a military conflict, in which both adversaries interact.
Strategy is important because the resources available to achieve goals are usually limited. Strategy generally involves setting goals and priorities, determining actions to achieve the goals, and mobilizing resources to execute the actions. A strategy describes how the ends (goals) will be achieved by the means (resources). Strategy can be intended or can emerge as a pattern of activity as the organization adapts to its environment or competes. It involves activities such as strategic planning and strategic thinking.
Henry Mintzberg from McGill University defined strategy as a pattern in a stream of decisions to contrast with a view of strategy as planning,. while Max McKeown (2011) argues that "strategy is about shaping the future" and is the human attempt to get to "desirable ends with available means". Vladimir Kvint defines strategy as "a system of finding, formulating, and developing a doctrine that will ensure long-term success if followed faithfully."
== Military theory ==
Subordinating the political point of view to the military would be absurd, for it is policy that has created war...Policy is the guiding intelligence, and war only the instrument, not vice-versa.
In military theory, strategy is "the utilization during both peace and war, of all of the nation's forces, through large scale, long-range planning and development, to ensure security and victory" (Random House Dictionary).
The father of Western modern strategic study, Carl von Clausewitz, defined military strategy as "the employment of battles to gain the end of war." B. H. Liddell Hart's definition put less emphasis on battles, defining strategy as "the art of distributing and applying military means to fulfill the ends of policy". Hence, both gave the pre-eminence to political aims over military goals. U.S. Naval War College instructor Andrew Wilson defined strategy as the "process by which political purpose is translated into military action." Lawrence Freedman defined strategy as the "art of creating power."
Eastern military philosophy dates back much further, with examples such as The Art of War by Sun Tzu dated around 500 B.C.
=== Counterterrorism Strategy ===
Because counterterrorism involves the synchronized efforts of numerous competing bureaucratic entities, national governments frequently create overarching counterterrorism strategies at the national level. A national counterterrorism strategy is a government's plan to use the instruments of national power to neutralize terrorists, their organizations, and their networks in order to render them incapable of using violence to instill fear and to coerce the government or its citizens to react in accordance with the terrorists' goals. The United States has had several such strategies in the past, including the United States National Strategy for Counterterrorism (2018); the Obama-era National Strategy for Counterterrorism (2011); and the National Strategy for Combatting Terrorism (2003). There have also been a number of ancillary or supporting plans, such as the 2014 Strategy to Counter the Islamic State of Iraq and the Levant, and the 2016 Strategic Implementation Plan for Empowering Local Partners to Prevent Violent Extremism in the United States. Similarly, the United Kingdom's counterterrorism strategy, CONTEST, seeks "to reduce the risk to the UK and its citizens and interests overseas from terrorism so that people can go about their lives freely and with confidence."
== Management theory ==
The essence of formulating competitive strategy is relating a company to its environment.
Modern business strategy emerged as a field of study and practice in the 1960s; prior to that time, the words "strategy" and "competition" rarely appeared in the most prominent management literature.
Alfred Chandler wrote in 1962 that: "Strategy is the determination of the basic long-term goals of an enterprise, and the adoption of courses of action and the allocation of resources necessary for carrying out these goals." Michael Porter defined strategy in 1980 as the "...broad formula for how a business is going to compete, what its goals should be, and what policies will be needed to carry out those goals" and the "...combination of the ends (goals) for which the firm is striving and the means (policies) by which it is seeking to get there."
=== Definition ===
Henry Mintzberg described five definitions of strategy in 1998:
Strategy as plan – a directed course of action to achieve an intended set of goals; similar to the strategic planning concept;
Strategy as pattern – a consistent pattern of past behavior, with a strategy realized over time rather than planned or intended. Where the realized pattern was different from the intent, he referred to the strategy as emergent;
Strategy as position – locating brands, products, or companies within the market, based on the conceptual framework of consumers or other stakeholders; a strategy determined primarily by factors outside the firm;
Strategy as ploy – a specific maneuver intended to outwit a competitor; and
Strategy as perspective – executing strategy based on a "theory of the business" or natural extension of the mindset or ideological perspective of the organization.
Complexity theorists define strategy as the unfolding of the internal and external aspects of the organization that results in actions in a socio-economic context.
=== Strategic Problem ===
In 1998, Crouch defined the strategic problem as maintaining flexible relationships that can range from intense competition to harmonious cooperation among different players in a dynamic market. While Crouch was open to the idea of cooperation between players, his approach still emphasized that strategy is shaped by market conditions and organizational structure. This view aligns with the definitions of strategy proposed by Porter and Mintzberg.
In contrast, Burnett regards strategy as a plan formulated through methodology in which the strategic problem encompasses six tasks: goal formulation, environmental analysis, strategy formulation, strategy evaluation, strategy implementation, and strategy control.
The literature identifies two main sources for defining a strategic problem. The first is related to environmental factors, and the second focuses on the organizational context (Mukherji and Hurtado, 2001). These two sources summarize three dimensions originally proposed by Ansoff and Hayes (1981). According to them, a strategic problem arises from analysis of internal and external issues, the processes to solve them, and the variables involved.
In Terra and Passador's view, organizations and the systems around them are tightly connected, so they rely on each other to survive. This means a strategy should balance being proactive and reactive. This involves recognizing the organization’s impact on the environment and acting to minimize harm while adapting to new demands. The strategy should also align internal and external aspects of the organization and include all related entities. This helps build a complex socio-economic system where the organization is part of a sustainable ecosystem.
=== Complexity theory ===
Complexity science, as articulated by R. D. Stacey, represents a conceptual framework capable of harmonizing emergent and deliberate strategies. Within complexity approaches the term "strategy" is intricately linked to action. Complexity theorists view programs merely as predetermined sequences effective in highly ordered and less chaotic environments. Conversely, strategy emerges from a simultaneous examination of determined conditions (order) and uncertainties (disorder) that drive action. Complexity theory posits that strategy involves execution, encompasses control and emergence, scrutinizes both internal and external organizational aspects, and can take the form of maneuvers or any other act or process.
The works of Stacey stand as pioneering efforts in applying complexity principles to the field of strategy. This author applied self-organization and chaos principles to describe strategy, organizational change dynamics, and learning. Their propositions advocate for strategy approached through choices and the evolutionary process of competitive selection. In this context, corrections of anomalies occur through actions involving negative feedback, while innovation and continuous change stem from actions guided by positive feedback.
Dynamically, complexity in strategic management can be elucidated through the model of "Symbiotic Dynamics" by Terra and Passador. This model conceives the social organization of production as an interplay between two distinct systems existing in a symbiotic relationship while interconnected with the external environment. The organization's social network acts as a self-referential entity controlling the organization's life, while its technical structure resembles a purposeful "machine" supplying the social system by processing resources. These intertwined structures exchange disturbances and residues while interacting with the external world through their openness. Essentially, as the organization produces itself, it also hetero-produces, surviving through energy and resource flows across its subsystems.
This dynamic has strategic implications, governing organizational dynamics through a set of attraction basins establishing operational and regenerative capabilities. Hence, one of the primary roles of strategists is to identify "human attractors" and assess their impacts on organizational dynamics. According to the theory of Symbiotic Dynamics, both leaders and the technical system can act as attractors, directly influencing organizational dynamics and responses to external disruptions. Terra and Passador further assert that while producing, organizations contribute to environmental entropy, potentially leading to abrupt ruptures and collapses within their subsystems, even within the organizations themselves. Given this issue, the authors conclude that organizations intervening to maintain the environment's stability within suitable parameters for survival tend to exhibit greater longevity.
The theory of Symbiotic Dynamics posits that organizations must acknowledge their impact on the external environment (markets, society, and the environment) and act systematically to reduce their degradation while adapting to the demands arising from these interactions. To achieve this, organizations need to incorporate all interconnected systems into their decision-making processes, enabling the envisioning of complex socio-economic systems where they integrate in a stable and sustainable manner. This blend of proactivity and reactivity is fundamental to ensuring the survival of the organization itself.
=== Components ===
Professor Richard P. Rumelt described strategy as a type of problem solving in 2011. He wrote that good strategy has an underlying structure he called a kernel. The kernel has three parts: 1) A diagnosis that defines or explains the nature of the challenge; 2) A guiding policy for dealing with the challenge; and 3) Coherent actions designed to carry out the guiding policy.
President Kennedy illustrated these three elements of strategy in his Cuban Missile Crisis Address to the Nation of 22 October 1962:
Diagnosis: "This Government, as promised, has maintained the closest surveillance of the Soviet military buildup on the island of Cuba. Within the past week, unmistakable evidence has established the fact that a series of offensive missile sites are now in preparation on that imprisoned island. The purpose of these bases can be none other than to provide a nuclear strike capability against the Western Hemisphere."
Guiding Policy: "Our unswerving objective, therefore, must be to prevent the use of these missiles against this or any other country, and to secure their withdrawal or elimination from the Western Hemisphere."
Action Plans: First among seven numbered steps was the following: "To halt this offensive buildup a strict quarantine on all offensive military equipment under shipment to Cuba is being initiated. All ships of any kind bound for Cuba from whatever nation or port will, if found to contain cargoes of offensive weapons, be turned back."
Rumelt wrote in 2011 that three important aspects of strategy include "premeditation, the anticipation of others' behavior, and the purposeful design of coordinated actions." He described strategy as solving a design problem, with trade-offs among various elements that must be arranged, adjusted and coordinated, rather than a plan or choice.
=== Formulation and implementation ===
Strategy typically involves two major processes: formulation and implementation. Formulation involves analyzing the environment or situation, making a diagnosis, and developing guiding policies. It includes such activities as strategic planning and strategic thinking. Implementation refers to the action plans taken to achieve the goals established by the guiding policy.
Bruce Henderson wrote in 1981 that: "Strategy depends upon the ability to foresee future consequences of present initiatives." He wrote that the basic requirements for strategy development include, among other factors: 1) extensive knowledge about the environment, market and competitors;
2) ability to examine this knowledge as an interactive dynamic system; and
3) the imagination and logic to choose between specific alternatives. Henderson wrote that strategy was valuable because of: "finite resources, uncertainty about an adversary's capability and intentions; the irreversible commitment of resources; necessity of coordinating action over time and distance; uncertainty about control of the initiative; and the nature of adversaries' mutual perceptions of each other."
== Game theory ==
In game theory, a player's strategy is any of the options that the player would choose in a specific setting. Any optimal outcomes they receive depend not only on their actions but also, the actions of other players.
== See also ==
Concept Driven Strategy
Consultant
Odds algorithm (Odds strategy)
Sports strategy
Strategy game
Strategic management
Strategy pattern
Strategic planning
Strategic voting
Strategist
Strategy Markup Language
Tactic (method)
Time management
U.S. Army Strategist
== Further reading ==
Burgelman, James. Strategy is Destiny (2002): Strategy Is Destiny: How Strategy-Making Shapes a Company's Future
Freedman, Lawrence. Strategy: A History (2013): Strategy: A History 1st Edition
Heuser, Beatrice. The Evolution of Strategy (2010): The Evolution of Strategy: Thinking War from Antiquity to the Present
Kvint, Vladimir. Strategy for the Global Market: Theory and Practical Applications (2016): Excerpt from Google Books
== References ==
== External links == | Wikipedia/Strategy |
In game theory, the unscrupulous diner's dilemma (or just diner's dilemma) is an n-player prisoner's dilemma. The situation imagined is that several people go out to eat, and before ordering, they agree to split the cost equally between them. Each diner must now choose whether to order the costly or cheap dish. It is presupposed that the costlier dish is better than the cheaper, but not by enough to warrant paying the difference when eating alone. Each diner reasons that, by ordering the costlier dish, the extra cost to their own bill will be small, and thus the better dinner is worth the money. However, all diners having reasoned thus, they each end up paying for the costlier dish, which by assumption, is worse than had they each ordered the cheaper.
== Formal definition and equilibrium analysis ==
Let a represent the joy of eating the expensive meal, b the joy of eating the cheap meal, k is the cost of the expensive meal, l the cost of the cheap meal, and n the number of players. From the description above we have the following ordering
k
−
l
>
a
−
b
{\displaystyle k-l>a-b}
. Also, in order to make the game sufficiently similar to the Prisoner's dilemma we presume that one would prefer to order the expensive meal given others will help defray the cost,
a
−
1
n
k
>
b
−
1
n
l
{\displaystyle a-{\frac {1}{n}}k>b-{\frac {1}{n}}l}
Consider an arbitrary set of strategies by a player's opponent. Let the total cost of the other players' meals be x. The cost of ordering the cheap meal is
1
n
x
+
1
n
l
{\displaystyle {\frac {1}{n}}x+{\frac {1}{n}}l}
and the cost of ordering the expensive meal is
1
n
x
+
1
n
k
{\displaystyle {\frac {1}{n}}x+{\frac {1}{n}}k}
. So the utilities for each meal are
a
−
1
n
x
−
1
n
k
{\displaystyle a-{\frac {1}{n}}x-{\frac {1}{n}}k}
for the expensive meal and
b
−
1
n
x
−
1
n
l
{\displaystyle b-{\frac {1}{n}}x-{\frac {1}{n}}l}
for the cheaper meal. By assumption, the utility of ordering the expensive meal is higher. Remember that the choice of opponents' strategies was arbitrary and that the situation is symmetric. This proves that the expensive meal is strictly dominant and thus the unique Nash equilibrium.
If everyone orders the expensive meal all of the diners pay k and the utility of every player is
a
−
k
{\displaystyle a-k}
. On the other hand, if all the individuals had ordered the cheap meal, the utility of every player would have been
b
−
l
{\displaystyle b-l}
. Since by assumption
b
−
l
>
a
−
k
{\displaystyle b-l>a-k}
, everyone would be better off. This demonstrates the similarity between the diner's dilemma and the prisoner's dilemma. Like the prisoner's dilemma, everyone is worse off by playing the unique equilibrium than they would have been if they collectively pursued another strategy.
== Experimental evidence ==
Uri Gneezy, Ernan Haruvy, and Hadas Yafe (2004) tested these results in a field experiment. Groups of six diners faced different billing arrangements. In one arrangement the diners pay individually, in the second they split the bill evenly between themselves and in the third the meal is paid entirely by the experimenter. As predicted, the consumption is the smallest when the payment is individually made, the largest when the meal is free and in-between for the even split. In a fourth arrangement, each participant pays only one sixth of their individual meal and the experimenter pay the rest, to account for possible unselfishness and social considerations. There was no difference between the amount consumed by these groups and those splitting the total cost of the meal equally. As the private cost of increased consumption is the same for both treatments but splitting the cost imposes a burden on other group members, this indicates that participants did not take the welfare of others into account when making their choices. This contrasts to a large number of laboratory experiments where subjects face analytically similar choices but the context is more abstract.
== See also ==
Tragedy of the commons
Free-rider problem
Abilene paradox
== References ==
== External links ==
If You're Paying, I'll Have Top Sirloin by Russell Roberts | Wikipedia/Unscrupulous_diner's_dilemma |
Theory of Games and Economic Behavior, published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinary research field of game theory. In the introduction of its 60th anniversary commemorative edition from the Princeton University Press, the book is described as "the classic work upon which modern-day game theory is based."
== Overview ==
The book is based partly on earlier research by von Neumann, published in 1928 under the German title "Zur Theorie der Gesellschaftsspiele" ("On the Theory of Board Games").
The derivation of expected utility from its axioms appeared in an appendix to the Second Edition (1947). Von Neumann and Morgenstern used objective probabilities, supposing that all the agents had the same probability distribution, as a convenience. However, Neumann and Morgenstern mentioned that a theory of subjective probability could be provided, and this task was completed by Jimmie Savage in 1954 and Johann Pfanzagl in 1967. Savage extended von Neumann and Morgenstern's axioms of rational preferences to endogenize probability and make it subjective. He then used Bayes' theorem to update these subject probabilities in light of new information, thus linking rational choice and inference.
== Reception ==
Herbert A. Simon praised the book.
== See also ==
Pfanzagl, J (1967). "Subjective Probability Derived from the Morgenstern-von Neumann Utility Theory". In Martin Shubik (ed.). Essays in Mathematical Economics In Honor of Oskar Morgenstern. Princeton University Press. pp. 237–251.
Pfanzagl, J. in cooperation with V. Baumann and H. Huber (1968). "Events, Utility and Subjective Probability". Theory of Measurement. Wiley. pp. 195–220.
Morgenstern, Oskar (1976). "Some Reflections on Utility". In Andrew Schotter (ed.). Selected Economic Writings of Oskar Morgenstern. New York University Press. pp. 65–70.
Morgenstern Oskar (1976). "The Collaboration Between Oskar Morgenstern and John von Neumann on the Theory of Games". Journal of Economic Literature. 14 (3): 805–816. JSTOR 2722628.
Commemorative edition of the book Theory of Games and Economic Behavior
Copeland A. H. (1945). "Review of 'The Theory of Games and Economic Behavior". Bulletin of the American Mathematical Society. 51: 498–504. doi:10.1090/s0002-9904-1945-08391-8.
Hurwicz Leonid (1945). "The Theory of Economic Behavior". American Economic Review. 35 (5): 909–925. JSTOR 1812602.
Kaysen Carl (1946). "A Revolution in Economic Theory?". Review of Economic Studies. 14 (1): 1–15. doi:10.2307/2295753. JSTOR 2295753.
Marschak Jacob (1946). "Neumann's and Morgenstern's New Approach to Static Economics" (PDF). Journal of Political Economy. 54 (2): 97–115. doi:10.1086/256327. S2CID 154536775.
Stone Richard (1948). "The Theory of Games". Economic Journal. 58 (230): 185–201. doi:10.2307/2225934. JSTOR 2225934.
== References ==
== External links ==
Theory of Games and Economic Behavior, full text at archive.org (public domain) | Wikipedia/Theory_of_Games_and_Economic_Behavior |
In game theory, an information set is the basis for decision making in a game, which includes the actions available to players and the potential outcomes of each action. It consists of a collection of decision nodes that a player cannot distinguish between when making a move, due to incomplete information about previous actions or the current state of the game. In other words, when a player's turn comes, they may be uncertain about which exact node in the game tree they are currently at, and the information set represents all the possibilities they must consider. Information sets are a fundamental concept particularly important in games with imperfect information.
In games with perfect information (such as chess or Go), every information set contains exactly one decision node, as each player can observe all previous moves and knows the exact game state. However, in games with imperfect information—such as most card games like poker or bridge—information sets may contain multiple nodes, reflecting the player's uncertainty about the true state of the game. This uncertainty fundamentally changes how players must reason about optimal strategies.
The concept of information set was introduced by John von Neumann, motivated by his study of poker, and is now essential to the analysis of sequential games and the development of solution concepts such as subgame perfect equilibrium and perfect Bayesian equilibrium.
== In extensive form games ==
Information sets are primarily used in extensive form representations of games and are typically depicted in game trees. A game tree shows all possible paths from the start of a game to its various endings, with branches representing the choices available to players at each decision point.
For games with imperfect information, the challenge lies in representing situations where a player cannot determine their exact position in the game. For example, in a card game, a player knows their own cards but not their opponent's cards, creating uncertainty about the true game state. This uncertainty is modeled using information sets.
Information sets are typically represented in game trees using dotted lines connecting indistinguishable nodes, ovals encompassing multiple nodes, or similar notations indicating that a player cannot tell which of several positions they are actually in. This visual representation helps analyze how uncertainty affects optimal play.
=== Formal definition ===
An information set in an extensive form game must satisfy the following properties:
Every node in the information set belongs to the same player.
The player cannot distinguish between any nodes within the same information set based on their available information.
All nodes in the same information set must have identical available actions.
No node in an information set can be an ancestor of another node in the same set (this would create a logical impossibility in the game timeline).
=== Strategic implications ===
The structure of information sets profoundly affects strategic reasoning. When a player faces an information set with multiple nodes, they must formulate strategies that are optimal across all possible game states represented by that information set.
This leads to several important game-theoretic concepts:
Mixed strategies often become necessary when facing uncertainty, as pure strategies might be exploitable by opponents who can predict them.
Bayesian updating occurs as players update their beliefs about which node in an information set they are at based on observed actions.
Signaling and information revelation become strategic considerations, as players may take actions specifically to reveal or conceal information.
=== Dynamic games and backward induction ===
In games with multiple information sets, the strategic interaction becomes dynamic rather than static. Players must reason not just about current decisions but about future information sets that might arise.
The standard solution technique for such games is backward induction, where players reason from the end of the game toward the beginning. For example, when player A chooses first, player B will make the best decision for themselves based on A's choice and their own information set at that time. Player A, anticipating this reaction, makes their initial choice to maximize their own payoff.
This sequential reasoning process is complicated in games with imperfect information, requiring more sophisticated solution concepts like sequential equilibrium that account for beliefs about which node in an information set a player is actually at.
== Example ==
At the right are two versions of the battle of the sexes game, shown in extensive form. Below, the normal form for both of these games is shown as well.
The first game is simply sequential―when player 2 makes a choice, both parties are already aware of whether player 1 has chosen O(pera) or F(ootball).
The second game is also sequential, but the dotted line shows player 2's information set. This is the common way to show that when player 2 moves, he or she is not aware of what player 1 did.
This difference also leads to different predictions for the two games. In the first game, player 1 has the upper hand. They know that they can choose O(pera) safely because once player 2 knows that player 1 has chosen opera, player 2 would rather go along for o(pera) and get 2 than choose f(ootball) and get 0. Formally, that's applying subgame perfection to solve the game.
In the second game, player 2 can't observe what player 1 did, so it might as well be a simultaneous game. So subgame perfection doesn't get us anything that Nash equilibrium can't get us, and we have the standard 3 possible equilibria:
Both choose opera
both choose football
or both use a mixed strategy, with player 1 choosing O(pera) 3/5 of the time and choosing football 2/5 of the time, and player 2 choosing f(ootball) 3/5 of the time and opera 2/5 of the time
== See also ==
Self-confirming equilibrium
== References ==
== Further reading ==
Binmore, Ken (2007). Game Theory: A very short introduction. Oxford University Press. pp. 88–89. ISBN 978-0-19-921846-2. | Wikipedia/Information_set_(game_theory) |
Mechanism design (sometimes implementation theory or institution design) is a branch of economics and game theory. It studies how to construct rules—called mechanisms or institutions—that produce good outcomes according to some predefined metric, even when the designer does not know the players' true preferences or what information they have. Mechanism design thus focuses on the study of solution concepts for a class of private-information games.
Mechanism design has broad applications, including traditional domains of economics such as market design, but also political science (through voting theory). It is a foundational component in the operation of the internet, being used in networked systems (such as inter-domain routing), e-commerce, and advertisement auctions by Facebook and Google.
Because it starts with the end of the game (a particular result), then works backwards to find a game that implements it, it is sometimes described as reverse game theory. Leonid Hurwicz explains that "in a design problem, the goal function is the main given, while the mechanism is the unknown. Therefore, the design problem is the inverse of traditional economic theory, which is typically devoted to the analysis of the performance of a given mechanism."
The 2007 Nobel Memorial Prize in Economic Sciences was awarded to Leonid Hurwicz, Eric Maskin, and Roger Myerson "for having laid the foundations of mechanism design theory." The related works of William Vickrey that established the field earned him the 1996 Nobel prize.
== Description ==
One person, called the "principal", would like to condition his behavior on information privately known to the players of a game. For example, the principal would like to know the true quality of a used car a salesman is pitching. He cannot learn anything simply by asking the salesman, because it is in the salesman's interest to distort the truth. However, in mechanism design, the principal does have one advantage: He may design a game whose rules influence others to act the way he would like.
Without mechanism design theory, the principal's problem would be difficult to solve. He would have to consider all the possible games and choose the one that best influences other players' tactics. In addition, the principal would have to draw conclusions from agents who may lie to him. Thanks to the revelation principle, the principal only needs to consider games in which agents truthfully report their private information.
== Foundations ==
=== Mechanism ===
A game of mechanism design is a game of private information in which one of the agents, called the principal, chooses the payoff structure. Following Harsanyi (1967), the agents receive secret "messages" from nature containing information relevant to payoffs. For example, a message may contain information about their preferences or the quality of a good for sale. We call this information the agent's "type" (usually noted
θ
{\displaystyle \theta }
and accordingly the space of types
Θ
{\displaystyle \Theta }
). Agents then report a type to the principal (usually noted with a hat
θ
^
{\displaystyle {\hat {\theta }}}
) that can be a strategic lie. After the report, the principal and the agents are paid according to the payoff structure the principal chose.
The timing of the game is:
The principal commits to a mechanism
y
(
)
{\displaystyle y()}
that grants an outcome
y
{\displaystyle y}
as a function of reported type
The agents report, possibly dishonestly, a type profile
θ
^
{\displaystyle {\hat {\theta }}}
The mechanism is executed (agents receive outcome
y
(
θ
^
)
{\displaystyle y({\hat {\theta }})}
)
In order to understand who gets what, it is common to divide the outcome
y
{\displaystyle y}
into a goods allocation and a money transfer,
y
(
θ
)
=
{
x
(
θ
)
,
t
(
θ
)
}
,
x
∈
X
,
t
∈
T
{\displaystyle y(\theta )=\{x(\theta ),t(\theta )\},\ x\in X,t\in T}
where
x
{\displaystyle x}
stands for an allocation of goods rendered or received as a function of type, and
t
{\displaystyle t}
stands for a monetary transfer as a function of type.
As a benchmark the designer often defines what should happen under full information. Define a social choice function
f
(
θ
)
{\displaystyle f(\theta )}
mapping the (true) type profile directly to the allocation of goods received or rendered,
f
(
θ
)
:
Θ
→
Y
{\displaystyle f(\theta ):\Theta \rightarrow Y}
In contrast a mechanism maps the reported type profile to an outcome (again, both a goods allocation
x
{\displaystyle x}
and a money transfer
t
{\displaystyle t}
)
y
(
θ
^
)
:
Θ
→
Y
{\displaystyle y({\hat {\theta }}):\Theta \rightarrow Y}
=== Revelation principle ===
A proposed mechanism constitutes a Bayesian game (a game of private information), and if it is well-behaved the game has a Bayesian Nash equilibrium. At equilibrium agents choose their reports strategically as a function of type
θ
^
(
θ
)
{\displaystyle {\hat {\theta }}(\theta )}
It is difficult to solve for Bayesian equilibria in such a setting because it involves solving for agents' best-response strategies and for the best inference from a possible strategic lie. Thanks to a sweeping result called the revelation principle, no matter the mechanism a designer can confine attention to equilibria in which agents truthfully report type. The revelation principle states: "To every Bayesian Nash equilibrium there corresponds a Bayesian game with the same equilibrium outcome but in which players truthfully report type."
This is extremely useful. The principle allows one to solve for a Bayesian equilibrium by assuming all players truthfully report type (subject to an incentive compatibility constraint). In one blow it eliminates the need to consider either strategic behavior or lying.
Its proof is quite direct. Assume a Bayesian game in which the agent's strategy and payoff are functions of its type and what others do,
u
i
(
s
i
(
θ
i
)
,
s
−
i
(
θ
−
i
)
,
θ
i
)
{\displaystyle u_{i}\left(s_{i}(\theta _{i}),s_{-i}(\theta _{-i}),\theta _{i}\right)}
. By definition agent i's equilibrium strategy
s
(
θ
i
)
{\displaystyle s(\theta _{i})}
is Nash in expected utility:
s
i
(
θ
i
)
∈
arg
max
s
i
′
∈
S
i
∑
θ
−
i
p
(
θ
−
i
∣
θ
i
)
u
i
(
s
i
′
,
s
−
i
(
θ
−
i
)
,
θ
i
)
{\displaystyle s_{i}(\theta _{i})\in \arg \max _{s'_{i}\in S_{i}}\sum _{\theta _{-i}}\ p(\theta _{-i}\mid \theta _{i})\ u_{i}\left(s'_{i},s_{-i}(\theta _{-i}),\theta _{i}\right)}
Simply define a mechanism that would induce agents to choose the same equilibrium. The easiest one to define is for the mechanism to commit to playing the agents' equilibrium strategies for them.
y
(
θ
^
)
:
Θ
→
S
(
Θ
)
→
Y
{\displaystyle y({\hat {\theta }}):\Theta \rightarrow S(\Theta )\rightarrow Y}
Under such a mechanism the agents of course find it optimal to reveal type since the mechanism plays the strategies they found optimal anyway. Formally, choose
y
(
θ
)
{\displaystyle y(\theta )}
such that
θ
i
∈
arg
max
θ
i
′
∈
Θ
∑
θ
−
i
p
(
θ
−
i
∣
θ
i
)
u
i
(
y
(
θ
i
′
,
θ
−
i
)
,
θ
i
)
=
∑
θ
−
i
p
(
θ
−
i
∣
θ
i
)
u
i
(
s
i
(
θ
)
,
s
−
i
(
θ
−
i
)
,
θ
i
)
{\displaystyle {\begin{aligned}\theta _{i}\in {}&\arg \max _{\theta '_{i}\in \Theta }\sum _{\theta _{-i}}\ p(\theta _{-i}\mid \theta _{i})\ u_{i}\left(y(\theta '_{i},\theta _{-i}),\theta _{i}\right)\\[5pt]&=\sum _{\theta _{-i}}\ p(\theta _{-i}\mid \theta _{i})\ u_{i}\left(s_{i}(\theta ),s_{-i}(\theta _{-i}),\theta _{i}\right)\end{aligned}}}
=== Implementability ===
The designer of a mechanism generally hopes either
to design a mechanism
y
(
)
{\displaystyle y()}
that "implements" a social choice function
to find the mechanism
y
(
)
{\displaystyle y()}
that maximizes some value criterion (e.g. profit)
To implement a social choice function
f
(
θ
)
{\displaystyle f(\theta )}
is to find some transfer function
t
(
θ
)
{\displaystyle t(\theta )}
that motivates agents to pick
f
(
θ
)
{\displaystyle f(\theta )}
. Formally, if the equilibrium strategy profile under the mechanism maps to the same goods allocation as a social choice function,
f
(
θ
)
=
x
(
θ
^
(
θ
)
)
{\displaystyle f(\theta )=x\left({\hat {\theta }}(\theta )\right)}
we say the mechanism implements the social choice function.
Thanks to the revelation principle, the designer can usually find a transfer function
t
(
θ
)
{\displaystyle t(\theta )}
to implement a social choice by solving an associated truthtelling game. If agents find it optimal to truthfully report type,
θ
^
(
θ
)
=
θ
{\displaystyle {\hat {\theta }}(\theta )=\theta }
we say such a mechanism is truthfully implementable. The task is then to solve for a truthfully implementable
t
(
θ
)
{\displaystyle t(\theta )}
and impute this transfer function to the original game. An allocation
x
(
θ
)
{\displaystyle x(\theta )}
is truthfully implementable if there exists a transfer function
t
(
θ
)
{\displaystyle t(\theta )}
such that
u
(
x
(
θ
)
,
t
(
θ
)
,
θ
)
≥
u
(
x
(
θ
^
)
,
t
(
θ
^
)
,
θ
)
∀
θ
,
θ
^
∈
Θ
{\displaystyle u(x(\theta ),t(\theta ),\theta )\geq u(x({\hat {\theta }}),t({\hat {\theta }}),\theta )\ \forall \theta ,{\hat {\theta }}\in \Theta }
which is also called the incentive compatibility (IC) constraint.
In applications, the IC condition is the key to describing the shape of
t
(
θ
)
{\displaystyle t(\theta )}
in any useful way. Under certain conditions it can even isolate the transfer function analytically. Additionally, a participation (individual rationality) constraint is sometimes added if agents have the option of not playing.
==== Necessity ====
Consider a setting in which all agents have a type-contingent utility function
u
(
x
,
t
,
θ
)
{\displaystyle u(x,t,\theta )}
. Consider also a goods allocation
x
(
θ
)
{\displaystyle x(\theta )}
that is vector-valued and size
k
{\displaystyle k}
(which permits
k
{\displaystyle k}
number of goods) and assume it is piecewise continuous with respect to its arguments.
The function
x
(
θ
)
{\displaystyle x(\theta )}
is implementable only if
∑
k
=
1
n
∂
∂
θ
(
∂
u
/
∂
x
k
|
∂
u
/
∂
t
|
)
∂
x
∂
θ
≥
0
{\displaystyle \sum _{k=1}^{n}{\frac {\partial }{\partial \theta }}\left({\frac {\partial u/\partial x_{k}}{\left|\partial u/\partial t\right|}}\right){\frac {\partial x}{\partial \theta }}\geq 0}
whenever
x
=
x
(
θ
)
{\displaystyle x=x(\theta )}
and
t
=
t
(
θ
)
{\displaystyle t=t(\theta )}
and x is continuous at
θ
{\displaystyle \theta }
. This is a necessary condition and is derived from the first- and second-order conditions of the agent's optimization problem assuming truth-telling.
Its meaning can be understood in two pieces. The first piece says the agent's marginal rate of substitution (MRS) increases as a function of the type,
∂
∂
θ
(
∂
u
/
∂
x
k
|
∂
u
/
∂
t
|
)
=
∂
∂
θ
M
R
S
x
,
t
{\displaystyle {\frac {\partial }{\partial \theta }}\left({\frac {\partial u/\partial x_{k}}{\left|\partial u/\partial t\right|}}\right)={\frac {\partial }{\partial \theta }}\mathrm {MRS} _{x,t}}
In short, agents will not tell the truth if the mechanism does not offer higher agent types a better deal. Otherwise, higher types facing any mechanism that punishes high types for reporting will lie and declare they are lower types, violating the truthtelling incentive-compatibility constraint. The second piece is a monotonicity condition waiting to happen,
∂
x
∂
θ
{\displaystyle {\frac {\partial x}{\partial \theta }}}
which, to be positive, means higher types must be given more of the good.
There is potential for the two pieces to interact. If for some type range the contract offered less quantity to higher types
∂
x
/
∂
θ
<
0
{\displaystyle \partial x/\partial \theta <0}
, it is possible the mechanism could compensate by giving higher types a discount. But such a contract already exists for low-type agents, so this solution is pathological. Such a solution sometimes occurs in the process of solving for a mechanism. In these cases it must be "ironed". In a multiple-good environment it is also possible for the designer to reward the agent with more of one good to substitute for less of another (e.g. butter for margarine). Multiple-good mechanisms are an area of continuing research in mechanism design.
==== Sufficiency ====
Mechanism design papers usually make two assumptions to ensure implementability:
∂
∂
θ
∂
u
/
∂
x
k
|
∂
u
/
∂
t
|
>
0
∀
k
{\displaystyle {\frac {\partial }{\partial \theta }}{\frac {\partial u/\partial x_{k}}{\left|\partial u/\partial t\right|}}>0\ \forall k}
This is known by several names: the single-crossing condition, the sorting condition and the Spence–Mirrlees condition. It means the utility function is of such a shape that the agent's MRS is increasing in type.
∃
K
0
,
K
1
such that
|
∂
u
/
∂
x
k
∂
u
/
∂
t
|
≤
K
0
+
K
1
|
t
|
{\displaystyle \exists K_{0},K_{1}{\text{ such that }}\left|{\frac {\partial u/\partial x_{k}}{\partial u/\partial t}}\right|\leq K_{0}+K_{1}|t|}
This is a technical condition bounding the rate of growth of the MRS.
These assumptions are sufficient to provide that any monotonic
x
(
θ
)
{\displaystyle x(\theta )}
is implementable (a
t
(
θ
)
{\displaystyle t(\theta )}
exists that can implement it). In addition, in the single-good setting the single-crossing condition is sufficient to provide that only a monotonic
x
(
θ
)
{\displaystyle x(\theta )}
is implementable, so the designer can confine his search to a monotonic
x
(
θ
)
{\displaystyle x(\theta )}
.
== Highlighted results ==
=== Revenue equivalence theorem ===
Vickrey (1961) gives a celebrated result that any member of a large class of auctions assures the seller of the same expected revenue and that the expected revenue is the best the seller can do. This is the case if
The buyers have identical valuation functions (which may be a function of type)
The buyers' types are independently distributed
The buyers types are drawn from a continuous distribution
The type distribution bears the monotone hazard rate property
The mechanism sells the good to the buyer with the highest valuation
The last condition is crucial to the theorem. An implication is that for the seller to achieve higher revenue he must take a chance on giving the item to an agent with a lower valuation. Usually this means he must risk not selling the item at all.
=== Vickrey–Clarke–Groves mechanisms ===
The Vickrey (1961) auction model was later expanded by Clarke (1971) and Groves to treat a public choice problem in which a public project's cost is borne by all agents, e.g. whether to build a municipal bridge. The resulting "Vickrey–Clarke–Groves" mechanism can motivate agents to choose the socially efficient allocation of the public good even if agents have privately known valuations. In other words, it can solve the "tragedy of the commons"—under certain conditions, in particular quasilinear utility or if budget balance is not required.
Consider a setting in which
I
{\displaystyle I}
number of agents have quasilinear utility with private valuations
v
(
x
,
t
,
θ
)
{\displaystyle v(x,t,\theta )}
where the currency
t
{\displaystyle t}
is valued linearly. The VCG designer designs an incentive compatible (hence truthfully implementable) mechanism to obtain the true type profile, from which the designer implements the socially optimal allocation
x
I
∗
(
θ
)
∈
argmax
x
∈
X
∑
i
∈
I
v
(
x
,
θ
i
)
{\displaystyle x_{I}^{*}(\theta )\in {\underset {x\in X}{\operatorname {argmax} }}\sum _{i\in I}v(x,\theta _{i})}
The cleverness of the VCG mechanism is the way it motivates truthful revelation. It eliminates incentives to misreport by penalizing any agent by the cost of the distortion he causes. Among the reports the agent may make, the VCG mechanism permits a "null" report saying he is indifferent to the public good and cares only about the money transfer. This effectively removes the agent from the game. If an agent does choose to report a type, the VCG mechanism charges the agent a fee if his report is pivotal, that is if his report changes the optimal allocation x so as to harm other agents. The payment is calculated
t
i
(
θ
^
)
=
∑
j
∈
I
−
i
v
j
(
x
I
−
i
∗
(
θ
I
−
i
)
,
θ
j
)
−
∑
j
∈
I
−
i
v
j
(
x
I
∗
(
θ
^
i
,
θ
I
)
,
θ
j
)
{\displaystyle t_{i}({\hat {\theta }})=\sum _{j\in I-i}v_{j}(x_{I-i}^{*}(\theta _{I-i}),\theta _{j})-\sum _{j\in I-i}v_{j}(x_{I}^{*}({\hat {\theta }}_{i},\theta _{I}),\theta _{j})}
which sums the distortion in the utilities of the other agents (and not his own) caused by one agent reporting.
=== Gibbard–Satterthwaite theorem ===
Gibbard (1973) and Satterthwaite (1975) give an impossibility result similar in spirit to Arrow's impossibility theorem. For a very general class of games, only "dictatorial" social choice functions can be implemented.
A social choice function
f
(
⋅
)
{\displaystyle f(\cdot )}
is dictatorial if one agent always receives his most-favored goods allocation,
for
f
(
Θ
)
,
∃
i
∈
I
such that
u
i
(
x
,
θ
i
)
≥
u
i
(
x
′
,
θ
i
)
∀
x
′
∈
X
{\displaystyle {\text{for }}f(\Theta ){\text{, }}\exists i\in I{\text{ such that }}u_{i}(x,\theta _{i})\geq u_{i}(x',\theta _{i})\ \forall x'\in X}
The theorem states that under general conditions any truthfully implementable social choice function must be dictatorial if,
X is finite and contains at least three elements
Preferences are rational
f
(
Θ
)
=
X
{\displaystyle f(\Theta )=X}
=== Myerson–Satterthwaite theorem ===
Myerson and Satterthwaite (1983) show there is no efficient way for two parties to trade a good when they each have secret and probabilistically varying valuations for it, without the risk of forcing one party to trade at a loss. It is among the most remarkable negative results in economics—a kind of negative mirror to the fundamental theorems of welfare economics.
=== Shapley value ===
Phillips and Marden (2018) proved that for cost-sharing games with concave cost functions, the optimal cost-sharing rule that firstly optimizes the worst-case inefficiencies in a game (the price of anarchy), and then secondly optimizes the best-case outcomes (the price of stability), is precisely the Shapley value cost-sharing rule. A symmetrical statement is similarly valid for utility-sharing games with convex utility functions.
=== Price discrimination ===
Mirrlees (1971) introduces a setting in which the transfer function t() is easy to solve for. Due to its relevance and tractability it is a common setting in the literature. Consider a single-good, single-agent setting in which the agent has quasilinear utility with an unknown type parameter
θ
{\displaystyle \theta }
u
(
x
,
t
,
θ
)
=
V
(
x
,
θ
)
−
t
{\displaystyle u(x,t,\theta )=V(x,\theta )-t}
and in which the principal has a prior CDF over the agent's type
P
(
θ
)
{\displaystyle P(\theta )}
. The principal can produce goods at a convex marginal cost c(x) and wants to maximize the expected profit from the transaction
max
x
(
θ
)
,
t
(
θ
)
E
θ
[
t
(
θ
)
−
c
(
x
(
θ
)
)
]
{\displaystyle \max _{x(\theta ),t(\theta )}\mathbb {E} _{\theta }\left[t(\theta )-c\left(x(\theta )\right)\right]}
subject to IC and IR conditions
u
(
x
(
θ
)
,
t
(
θ
)
,
θ
)
≥
u
(
x
(
θ
′
)
,
t
(
θ
′
)
,
θ
)
∀
θ
,
θ
′
{\displaystyle u(x(\theta ),t(\theta ),\theta )\geq u(x(\theta '),t(\theta '),\theta )\ \forall \theta ,\theta '}
u
(
x
(
θ
)
,
t
(
θ
)
,
θ
)
≥
u
_
(
θ
)
∀
θ
{\displaystyle u(x(\theta ),t(\theta ),\theta )\geq {\underline {u}}(\theta )\ \forall \theta }
The principal here is a monopolist trying to set a profit-maximizing price scheme in which it cannot identify the type of the customer. A common example is an airline setting fares for business, leisure and student travelers. Due to the IR condition it has to give every type a good enough deal to induce participation. Due to the IC condition it has to give every type a good enough deal that the type prefers its deal to that of any other.
A trick given by Mirrlees (1971) is to use the envelope theorem to eliminate the transfer function from the expectation to be maximized,
let
U
(
θ
)
=
max
θ
′
u
(
x
(
θ
′
)
,
t
(
θ
′
)
,
θ
)
{\displaystyle {\text{let }}U(\theta )=\max _{\theta '}u\left(x(\theta '),t(\theta '),\theta \right)}
d
U
d
θ
=
∂
u
∂
θ
=
∂
V
∂
θ
{\displaystyle {\frac {dU}{d\theta }}={\frac {\partial u}{\partial \theta }}={\frac {\partial V}{\partial \theta }}}
Integrating,
U
(
θ
)
=
u
_
(
θ
0
)
+
∫
θ
0
θ
∂
V
∂
θ
~
d
θ
~
{\displaystyle U(\theta )={\underline {u}}(\theta _{0})+\int _{\theta _{0}}^{\theta }{\frac {\partial V}{\partial {\tilde {\theta }}}}d{\tilde {\theta }}}
where
θ
0
{\displaystyle \theta _{0}}
is some index type. Replacing the incentive-compatible
t
(
θ
)
=
V
(
x
(
θ
)
,
θ
)
−
U
(
θ
)
{\displaystyle t(\theta )=V(x(\theta ),\theta )-U(\theta )}
in the maximand,
E
θ
[
V
(
x
(
θ
)
,
θ
)
−
u
_
(
θ
0
)
−
∫
θ
0
θ
∂
V
∂
θ
~
d
θ
~
−
c
(
x
(
θ
)
)
]
=
E
θ
[
V
(
x
(
θ
)
,
θ
)
−
u
_
(
θ
0
)
−
1
−
P
(
θ
)
p
(
θ
)
∂
V
∂
θ
−
c
(
x
(
θ
)
)
]
{\displaystyle {\begin{aligned}&\mathbb {E} _{\theta }\left[V(x(\theta ),\theta )-{\underline {u}}(\theta _{0})-\int _{\theta _{0}}^{\theta }{\frac {\partial V}{\partial {\tilde {\theta }}}}d{\tilde {\theta }}-c\left(x(\theta )\right)\right]\\&{}=\mathbb {E} _{\theta }\left[V(x(\theta ),\theta )-{\underline {u}}(\theta _{0})-{\frac {1-P(\theta )}{p(\theta )}}{\frac {\partial V}{\partial \theta }}-c\left(x(\theta )\right)\right]\end{aligned}}}
after an integration by parts. This function can be maximized pointwise.
Because
U
(
θ
)
{\displaystyle U(\theta )}
is incentive-compatible already the designer can drop the IC constraint. If the utility function satisfies the Spence–Mirrlees condition then a monotonic
x
(
θ
)
{\displaystyle x(\theta )}
function exists. The IR constraint can be checked at equilibrium and the fee schedule raised or lowered accordingly. Additionally, note the presence of a hazard rate in the expression. If the type distribution bears the monotone hazard ratio property, the FOC is sufficient to solve for t(). If not, then it is necessary to check whether the monotonicity constraint (see sufficiency, above) is satisfied everywhere along the allocation and fee schedules. If not, then the designer must use Myerson ironing.
=== Myerson ironing ===
In some applications the designer may solve the first-order conditions for the price and allocation schedules yet find they are not monotonic. For example, in the quasilinear setting this often happens when the hazard ratio is itself not monotone. By the Spence–Mirrlees condition the optimal price and allocation schedules must be monotonic, so the designer must eliminate any interval over which the schedule changes direction by flattening it.
Intuitively, what is going on is the designer finds it optimal to bunch certain types together and give them the same contract. Normally the designer motivates higher types to distinguish themselves by giving them a better deal. If there are insufficiently few higher types on the margin the designer does not find it worthwhile to grant lower types a concession (called their information rent) in order to charge higher types a type-specific contract.
Consider a monopolist principal selling to agents with quasilinear utility, the example above. Suppose the allocation schedule
x
(
θ
)
{\displaystyle x(\theta )}
satisfying the first-order conditions has a single interior peak at
θ
1
{\displaystyle \theta _{1}}
and a single interior trough at
θ
2
>
θ
1
{\displaystyle \theta _{2}>\theta _{1}}
, illustrated at right.
Following Myerson (1981) flatten it by choosing
x
{\displaystyle x}
satisfying
∫
ϕ
2
(
x
)
ϕ
1
(
x
)
(
∂
V
∂
x
(
x
,
θ
)
−
1
−
P
(
θ
)
p
(
θ
)
∂
2
V
∂
θ
∂
x
(
x
,
θ
)
−
∂
c
∂
x
(
x
)
)
d
θ
=
0
{\displaystyle \int _{\phi _{2}(x)}^{\phi _{1}(x)}\left({\frac {\partial V}{\partial x}}(x,\theta )-{\frac {1-P(\theta )}{p(\theta )}}{\frac {\partial ^{2}V}{\partial \theta \,\partial x}}(x,\theta )-{\frac {\partial c}{\partial x}}(x)\right)d\theta =0}
where
ϕ
1
(
x
)
{\displaystyle \phi _{1}(x)}
is the inverse function of x mapping to
θ
≤
θ
1
{\displaystyle \theta \leq \theta _{1}}
and
ϕ
2
(
x
)
{\displaystyle \phi _{2}(x)}
is the inverse function of x mapping to
θ
≥
θ
2
{\displaystyle \theta \geq \theta _{2}}
. That is,
ϕ
1
{\displaystyle \phi _{1}}
returns a
θ
{\displaystyle \theta }
before the interior peak and
ϕ
2
{\displaystyle \phi _{2}}
returns a
θ
{\displaystyle \theta }
after the interior trough.
If the nonmonotonic region of
x
(
θ
)
{\displaystyle x(\theta )}
borders the edge of the type space, simply set the appropriate
ϕ
(
x
)
{\displaystyle \phi (x)}
function (or both) to the boundary type. If there are multiple regions, see a textbook for an iterative procedure; it may be that more than one troughs should be ironed together.
==== Proof ====
The proof uses the theory of optimal control. It considers the set of intervals
[
θ
_
,
θ
¯
]
{\displaystyle \left[{\underline {\theta }},{\overline {\theta }}\right]}
in the nonmonotonic region of
x
(
θ
)
{\displaystyle x(\theta )}
over which it might flatten the schedule. It then writes a Hamiltonian to obtain necessary conditions for a
x
(
θ
)
{\displaystyle x(\theta )}
within the intervals
that does satisfy monotonicity
for which the monotonicity constraint is not binding on the boundaries of the interval
Condition two ensures that the
x
(
θ
)
{\displaystyle x(\theta )}
satisfying the optimal control problem reconnects to the schedule in the original problem at the interval boundaries (no jumps). Any
x
(
θ
)
{\displaystyle x(\theta )}
satisfying the necessary conditions must be flat because it must be monotonic and yet reconnect at the boundaries.
As before maximize the principal's expected payoff, but this time subject to the monotonicity constraint
∂
x
∂
θ
≥
0
{\displaystyle {\frac {\partial x}{\partial \theta }}\geq 0}
and use a Hamiltonian to do it, with shadow price
ν
(
θ
)
{\displaystyle \nu (\theta )}
H
=
(
V
(
x
,
θ
)
−
u
_
(
θ
0
)
−
1
−
P
(
θ
)
p
(
θ
)
∂
V
∂
θ
(
x
,
θ
)
−
c
(
x
)
)
p
(
θ
)
+
ν
(
θ
)
∂
x
∂
θ
{\displaystyle H=\left(V(x,\theta )-{\underline {u}}(\theta _{0})-{\frac {1-P(\theta )}{p(\theta )}}{\frac {\partial V}{\partial \theta }}(x,\theta )-c(x)\right)p(\theta )+\nu (\theta ){\frac {\partial x}{\partial \theta }}}
where
x
{\displaystyle x}
is a state variable and
∂
x
/
∂
θ
{\displaystyle \partial x/\partial \theta }
the control. As usual in optimal control the costate evolution equation must satisfy
∂
ν
∂
θ
=
−
∂
H
∂
x
=
−
(
∂
V
∂
x
(
x
,
θ
)
−
1
−
P
(
θ
)
p
(
θ
)
∂
2
V
∂
θ
∂
x
(
x
,
θ
)
−
∂
c
∂
x
(
x
)
)
p
(
θ
)
{\displaystyle {\frac {\partial \nu }{\partial \theta }}=-{\frac {\partial H}{\partial x}}=-\left({\frac {\partial V}{\partial x}}(x,\theta )-{\frac {1-P(\theta )}{p(\theta )}}{\frac {\partial ^{2}V}{\partial \theta \,\partial x}}(x,\theta )-{\frac {\partial c}{\partial x}}(x)\right)p(\theta )}
Taking advantage of condition 2, note the monotonicity constraint is not binding at the boundaries of the
θ
{\displaystyle \theta }
interval,
ν
(
θ
_
)
=
ν
(
θ
¯
)
=
0
{\displaystyle \nu ({\underline {\theta }})=\nu ({\overline {\theta }})=0}
meaning the costate variable condition can be integrated and also equals 0
∫
θ
_
θ
¯
(
∂
V
∂
x
(
x
,
θ
)
−
1
−
P
(
θ
)
p
(
θ
)
∂
2
V
∂
θ
∂
x
(
x
,
θ
)
−
∂
c
∂
x
(
x
)
)
p
(
θ
)
d
θ
=
0
{\displaystyle \int _{\underline {\theta }}^{\overline {\theta }}\left({\frac {\partial V}{\partial x}}(x,\theta )-{\frac {1-P(\theta )}{p(\theta )}}{\frac {\partial ^{2}V}{\partial \theta \,\partial x}}(x,\theta )-{\frac {\partial c}{\partial x}}(x)\right)p(\theta )\,d\theta =0}
The average distortion of the principal's surplus must be 0. To flatten the schedule, find an
x
{\displaystyle x}
such that its inverse image maps to a
θ
{\displaystyle \theta }
interval satisfying the condition above.
== See also ==
== Notes ==
== References ==
Clarke, Edward H. (1971). "Multipart Pricing of Public Goods" (PDF). Public Choice. 11 (1): 17–33. doi:10.1007/BF01726210. JSTOR 30022651. S2CID 154860771. Archived from the original (PDF) on 2017-05-10. Retrieved 2016-08-12.
Gibbard, Allan (1973). "Manipulation of voting schemes: A general result" (PDF). Econometrica. 41 (4): 587–601. doi:10.2307/1914083. JSTOR 1914083. Archived from the original (PDF) on 2016-08-23. Retrieved 2016-08-12.
Groves, Theodore (1973). "Incentives in Teams" (PDF). Econometrica. 41 (4): 617–631. doi:10.2307/1914085. JSTOR 1914085.
Harsanyi, John C. (1967). "Games with incomplete information played by "Bayesian" players, I-III. part I. The Basic Model". Management Science. 14 (3): 159–182. doi:10.1287/mnsc.14.3.159. JSTOR 2628393.
Mirrlees, J. A. (1971). "An Exploration in the Theory of Optimum Income Taxation" (PDF). Review of Economic Studies. 38 (2): 175–208. doi:10.2307/2296779. JSTOR 2296779. Archived from the original (PDF) on 2017-05-10. Retrieved 2016-08-12.
Myerson, Roger B.; Satterthwaite, Mark A. (1983). "Efficient Mechanisms for Bilateral Trading" (PDF). Journal of Economic Theory. 29 (2): 265–281. doi:10.1016/0022-0531(83)90048-0. hdl:10419/220829. Archived from the original (PDF) on 2017-05-10. Retrieved 2016-08-12.
Satterthwaite, Mark Allen (1975). "Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions". Journal of Economic Theory. 10 (2): 187–217. CiteSeerX 10.1.1.471.9842. doi:10.1016/0022-0531(75)90050-2.
Vickrey, William (1961). "Counterspeculation, Auctions, and Competitive Sealed Tenders" (PDF). The Journal of Finance. 16 (1): 8–37. doi:10.1111/j.1540-6261.1961.tb02789.x.
== Further reading ==
Chapter 7 of Fudenberg, Drew; Tirole, Jean (1991), Game Theory, Boston: MIT Press, ISBN 978-0-262-06141-4. A standard text for graduate game theory.
Chapter 23 of Mas-Colell; Whinston; Green (1995), Microeconomic Theory, Oxford: Oxford University Press, ISBN 978-0-19-507340-9. A standard text for graduate microeconomics.
Milgrom, Paul (2004), Putting Auction Theory to Work, New York: Cambridge University Press, ISBN 978-0-521-55184-7. Applications of mechanism design principles in the context of auctions.
Noam Nisan. A Google tech talk on mechanism design.
Legros, Patrick; Cantillon, Estelle (2007). "What is mechanism design and why does it matter for policy-making?". Centre for Economic Policy Research.
Roger B. Myerson (2008), "Mechanism Design", The New Palgrave Dictionary of Economics Online, Abstract.
Diamantaras, Dimitrios (2009), A Toolbox for Economic Design, New York: Palgrave Macmillan, ISBN 978-0-230-61060-6. A graduate text specifically focused on mechanism design.
== External links ==
Eric Maskin "Nobel Prize Lecture" delivered on 8 December 2007 at Aula Magna, Stockholm University. | Wikipedia/Mechanism_design |
In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.
Many solution concepts, for many games, will result in more than one solution. This puts any one of the solutions in doubt, so a game theorist may apply a refinement to narrow down the solutions. Each successive solution concept presented in the following improves on its predecessor by eliminating implausible equilibria in richer games.
== Formal definition ==
Let
Γ
{\displaystyle \Gamma }
be the class of all games and, for each game
G
∈
Γ
{\displaystyle G\in \Gamma }
, let
S
G
{\displaystyle S_{G}}
be the set of strategy profiles of
G
{\displaystyle G}
. A solution concept is an element of the direct product
Π
G
∈
Γ
2
S
G
;
{\displaystyle \Pi _{G\in \Gamma }2^{S_{G}};}
i.e., a function
F
:
Γ
→
⋃
G
∈
Γ
2
S
G
{\displaystyle F:\Gamma \rightarrow \bigcup \nolimits _{G\in \Gamma }2^{S_{G}}}
such that
F
(
G
)
⊆
S
G
{\displaystyle F(G)\subseteq S_{G}}
for all
G
∈
Γ
.
{\displaystyle G\in \Gamma .}
== Rationalizability and iterated dominance ==
In this solution concept, players are assumed to be rational and so strictly dominated strategies are eliminated from the set of strategies that might feasibly be played. A strategy is strictly dominated when there is some other strategy available to the player that always has a higher payoff, regardless of the strategies that the other players choose. (Strictly dominated strategies are also important in minimax game-tree search.) For example, in the (single period) prisoners' dilemma (shown below), cooperate is strictly dominated by defect for both players because either player is always better off playing defect, regardless of what his opponent does.
== Nash equilibrium ==
A Nash equilibrium is a strategy profile (a strategy profile specifies a strategy for every player, e.g. in the above prisoners' dilemma game (cooperate, defect) specifies that prisoner 1 plays cooperate and prisoner 2 plays defect) in which every strategy played by every agent (agent i) is a best response to every other strategy played by all the other opponents (agents j for every j≠i) . A strategy by a player is a best response to another player's strategy if there is no other strategy that could be played that would yield a higher pay-off in any situation in which the other player's strategy is played.
== Backward induction ==
In some games, there are multiple Nash equilibria, but not all of them are realistic. In dynamic games, backward induction can be used to eliminate unrealistic Nash equilibria. Backward induction assumes that players are rational and will make the best decisions based on their future expectations. This eliminates noncredible threats, which are threats that a player would not carry out if they were ever called upon to do so.
For example, consider a dynamic game with an incumbent firm and a potential entrant to the industry. The incumbent has a monopoly and wants to maintain its market share. If the entrant enters, the incumbent can either fight or accommodate the entrant. If the incumbent accommodates, the entrant will enter and gain profit. If the incumbent fights, it will lower its prices, run the entrant out of business (incurring exit costs), and damage its own profits.
The best response for the incumbent if the entrant enters is to accommodate, and the best response for the entrant if the incumbent accommodates is to enter. This results in a Nash equilibrium. However, if the incumbent chooses to fight, the best response for the entrant is to not enter. If the entrant does not enter, it does not matter what the incumbent chooses to do. Hence, fight can be considered a best response for the incumbent if the entrant does not enter, resulting in another Nash equilibrium.
However, this second Nash equilibrium can be eliminated by backward induction because it relies on a noncredible threat from the incumbent. By the time the incumbent reaches the decision node where it can choose to fight, it would be irrational to do so because the entrant has already entered. Therefore, backward induction eliminates this unrealistic Nash equilibrium.
See also:
Monetary policy theory
Stackelberg competition
== Subgame perfect Nash equilibrium ==
A generalization of backward induction is subgame perfection. Backward induction assumes that all future play will be rational. In subgame perfect equilibria, play in every subgame is rational (specifically a Nash equilibrium). Backward induction can only be used in terminating (finite) games of definite length and cannot be applied to games with imperfect information. In these cases, subgame perfection can be used. The eliminated Nash equilibrium described above is subgame imperfect because it is not a Nash equilibrium of the subgame that starts at the node reached once the entrant has entered.
== Perfect Bayesian equilibrium ==
Sometimes subgame perfection does not impose a large enough restriction on unreasonable outcomes. For example, since subgames cannot cut through information sets, a game of imperfect information may have only one subgame – itself – and hence subgame perfection cannot be used to eliminate any Nash equilibria. A perfect Bayesian equilibrium (PBE) is a specification of players' strategies and beliefs about which node in the information set has been reached by the play of the game. A belief about a decision node is the probability that a particular player thinks that node is or will be in play (on the equilibrium path). In particular, the intuition of PBE is that it specifies player strategies that are rational given the player beliefs it specifies and the beliefs it specifies are consistent with the strategies it specifies.
In a Bayesian game a strategy determines what a player plays at every information set controlled by that player. The requirement that beliefs are consistent with strategies is something not specified by subgame perfection. Hence, PBE is a consistency condition on players' beliefs. Just as in a Nash equilibrium no player's strategy is strictly dominated, in a PBE, for any information set no player's strategy is strictly dominated beginning at that information set. That is, for every belief that the player could hold at that information set there is no strategy that yields a greater expected payoff for that player. Unlike the above solution concepts, no player's strategy is strictly dominated beginning at any information set even if it is off the equilibrium path. Thus in PBE, players cannot threaten to play strategies that are strictly dominated beginning at any information set off the equilibrium path.
The Bayesian in the name of this solution concept alludes to the fact that players update their beliefs according to Bayes' theorem. They calculate probabilities given what has already taken place in the game.
== Forward induction ==
Forward induction is so called because just as backward induction assumes future play will be rational, forward induction assumes past play was rational. Where a player does not know what type another player is (i.e. there is imperfect and asymmetric information), that player may form a belief of what type that player is by observing that player's past actions. Hence the belief formed by that player of what the probability of the opponent being a certain type is based on the past play of that opponent being rational. A player may elect to signal his type through his actions.
Kohlberg and Mertens (1986) introduced the solution concept of Stable equilibrium, a refinement that satisfies forward induction. A counter-example was found where such a stable equilibrium did not satisfy backward induction. To resolve the problem Jean-François Mertens introduced what game theorists now call Mertens-stable equilibrium concept, probably the first solution concept satisfying both forward and backward induction.
Forward induction yields a unique solution for the burning money game.
== See also ==
Extensive form game
Trembling hand equilibrium
"The Intuitive Criterion" (Cho & Kreps 1987)
== References ==
Cho, I-K.; Kreps, D. M. (1987). "Signaling Games and Stable Equilibria". Quarterly Journal of Economics. 102 (2): 179–221. CiteSeerX 10.1.1.407.5013. doi:10.2307/1885060. JSTOR 1885060. S2CID 154404556.
Fudenberg, Drew; Tirole, Jean (1991). Game Theory. Cambridge, Massachusetts: MIT Press. ISBN 9780262061414. Book preview.
Harsanyi, J. (1973) Oddness of the number of equilibrium points: a new proof. International Journal of Game Theory 2:235–250.
Govindan, Srihari & Robert Wilson, 2008. "Refinements of Nash Equilibrium," The New Palgrave Dictionary of Economics, 2nd Edition.[1]
Hines, W. G. S. (1987) Evolutionary stable strategies: a review of basic theory. Theoretical Population Biology 31:195–272.
Kohlberg, Elon & Jean-François Mertens, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-37, September.
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Behavioral game theory seeks to examine how people's strategic decision-making behavior is shaped by social preferences, social utility and other psychological factors. Behavioral game theory analyzes interactive strategic decisions and behavior using the methods of game theory, experimental economics, and experimental psychology. Experiments include testing deviations from typical simplifications of economic theory such as the independence axiom and neglect of altruism, fairness, and framing effects. As a research program, the subject is a development of the last three decades.
Traditional game theory is a critical principle of economic theory, and assumes that people's strategic decisions are shaped by rationality, selfishness and utility maximisation. It focuses on the mathematical structure of equilibria, and tends to use basic rational choice theory and utility maximization as the primary principles within economic models. At the same time rational choice theory is an ideal model that assumes that individuals will actively choose the option with the greatest benefit. The fact is that consumers have different preferences and rational choice theory is not accurate in its assumptions about consumer behavior. In contrast to traditional game theory, behavioral game theory examines how actual human behavior tends to deviate from standard predictions and models. In order to more accurately understand these deviations and determine the factors and conditions involved in strategic decision making, behavioral game theory aims to create new models that incorporate psychological principles. Studies of behavioral game theory demonstrate that choices are not always rational and do not always represent the utility maximizing choice.
Behavioral game theory largely utilizes empirical and theoretical research to understand human behavior. It also uses laboratory and field experiments, as well as modeling – both theoretical and computational. Recently, methods from machine learning have been applied in work at the intersection of economics, psychology, and computer science to improve both prediction and understanding of behavior in games.
== History ==
Behavioral game theory began with the work of Allais in 1953 and Ellsberg in 1961. They discovered the Allais paradox and the Ellsberg paradox, respectively. Both paradoxes show that choices made by participants in a game do not reflect the benefit they expect to receive from making those choices. In 1956, the work of Vernon Smith showed that economic markets could be examined experimentally rather than only theoretically, and reinforced the importance of rationality and self-interest within economic models. According to rational choice theory, consumers' behavior depends on three reasons. The first reason is that the degree of emotional pleasure consumers derive from their purchases depends on their preferences. The second reason is that consumers do not have enough choices. The third reason is that consumers derive greater pleasure from a limited number of choices. Later in the 1970s, economists Tversky and Kahneman, as well as several other co-workers, conducted experiments that discovered variations of traditional decision-making models such as regret theory, prospect theory, and hyperbolic discounting. These discoveries showed that decision makers consider many factors when making choices. For example, a person may seek to minimize the amount of regret they will feel after making a decision and weigh their options based on the amount of regret they anticipate from each. Due to the fact that these theories were not previously examined by traditional economic theory, factors such as regret along with many others fueled further research on the subject of social preferences and decision making.
Beginning in the 1980s experimenters started examining the conditions that cause divergence from rational choice. Ultimatum and bargaining games examined the effect of emotions on predictions of opponent behavior. One of the most well known examples of an ultimatum game is the television show Deal or No Deal in which participants must make decisions to sell or continue playing based on monetary ultimatums given to them by "the banker." These games also explored the effect of trust on decision-making outcomes and utility maximizing behavior. Common resource games were used to experimentally test how cooperation and social desirability affect subject's choices. A real-life example of a common resource game might be a party guest's decision to take from a food platter. The guest's decisions be affected by how hungry they are, how much of the shared resource (the food) is left and if the guest believes others would judge them for taking more. Experimenters believed that any behavior that did not maximize utility as the result of participant's flawed reasoning. By the turn of the century economists and psychologists expanded this research. Models based on the rational choice theory were adapted to reflect decision maker preferences and attempt to rationalize choices that did not maximize utility.
== Comparison to traditional game theory ==
There are various distinctions between traditional game theory and behavioral game theory. Traditional game theory uses theoretical and mathematical models to determine the most beneficial choice of all players in a game. Game theory uses rational choice theory to predict people's decisions in conditions of uncertainty. It understands strategic behavior to be influenced by utility-maximising preferences, as well as player's assumed knowledge of their opponents and material constraints. It also allows for players to predict their opponents' strategies. Also consumers' decisions are affected by psychological issues, and inattentional blindness is important in influencing the outcome of decisions. This is due to the fact that when consumers' attention is focused on one thing, they ignore other choices. Inattentional blindness believes that human attention and cognition are limited, which explains why consumers will make choices based on their personal preferences. Traditional game theory is a primarily normative theory as it seeks to pinpoint the decision that rational players should choose, but does not attempt to explain why that decision was made. Rationality is a primary assumption of game theory, so there are not explanations for different forms of rational decisions or irrational decisions.
In contrast to traditional game theory, behavioral game theory uses empirical models to explain how social preferences, such as ideals of fairness, efficiency or equity, influence human decisions and strategic reasoning. Behavioral game theory attempts to explain factors that influence real-world decisions. These factors are not explored in the area of traditional game theory, but can be postulated and observed using empirical data. Findings from behavioral game theory will tend to have higher external validity and can be better applied to real world decision-making behavior. Behavioral game theory is a primarily positive theory rather than a normative theory. A positive theory seeks to describe phenomena rather than prescribe a correct action. Positive theories must be testable and can be proven true or false. A normative theory is subjective and based on opinions. Because of this, normative theories cannot be proven true or false. Behavioral game theory attempts to explain decision making using experimental data. The theory allows for rational and irrational decisions because both are examined using real-life experiments in the form of simple games. Simple games are often used in behavioral game theory research as a way of analyzing unexplored phenomena, such as social preferences and social utility, that are not explored in traditional game theory.
== Examining social utility and preferences through games ==
Simple games are regularly utilized in behavioral game theory experiments in order to examine player's social utility. The simplicity of these games means that players do not face intellectual challenges, and player's choices are not impacted or altered by the game itself. This makes the games extremely useful in understanding social preferences. Games often include monetary rewards to easily calculate how players will act if their choices are driven by monetary incentives and payoffs. Player's actions are often shaped by the social utility function, whereby their choices are shaped by the benefits that both they and their opponent would receive. Traditional game theory would expect rational players to attempt to maximise their monetary rewards. If these calculations were wrong, however, and if players choose not to maximise their utility, then players would be exhibiting a social preference for a particular action. Behavioral game theory explains how players often deviate from traditional norms, and quite regularly consider factors such as social welfare when making their strategic decisions. For example, players are known to sacrifice high monetary rewards in order to maintain fairness within the game.
Different games demonstrate different social preferences. For example, the ultimatum game is known to demonstrate negative reciprocity. The premise of the ultimatum game is that Player 1 is given a certain amount of money, and is then forced to offer a certain amount to Player 2. Player 2 can then choose to either accept or reject Player 1's offer. If Player 2 accepts the offer, then both players are able to enjoy the amount offered. If Player 2 rejects the offer, then neither player is able to receive the money. Results from ultimatum game experiments demonstrate that players value being treated fairly and do not react well when one player is attempting to receive better payoffs than the other. Studies show that people are more likely sacrifice all monetary rewards if they are offered less than 20 percent of the original amount. This represents negative reciprocity preferences, as players would rather sacrifice their payoff in order to punish their opponent for their unkind behavior. However, being scared of having their offers rejected, people often give Player 2 around 40-50 percent of the original amount.
Another example of a social preference is positive reciprocity, which is displayed in the gift exchange game. The gift exchange game involves Player 1 either keeping set amount of money, or offering an even larger amount to Player 2. Player 2 is then able to decide how they wish to divide the money between the two of them. In this game, Player 1 trusts that Player 2 will return a certain amount of money to them. Findings from this game often show that if Player 2 is offered a generous amount of money from Player 1, then they are more likely to return the favour and give Player 1 back an equally generous amount. This demonstrates how players appreciate being treated kindly, and are more likely to treat their opponent kindly in return. The concept of positive reciprocity can be seen in real-life examples, such as the workplace. If an employer offers a large wage to their employees, then the employees often pay back the favour by working harder.
Altruism is another social preference seen in the dictator game. This game is similar to the ultimatum and gift exchange games. In this game, however, Player 1 is given an amount of money, and can then offer however much they would like to Player 2. Unlike the ultimatum game, Player 2 cannot reject the amount they have been offered. As a result, people are more likely to reduce the amount of money they offer to Player 2. Despite this, results show that people still offer Player 2 a sum of around 20-30 percent of the original amount. The dictator game shows how people are willing to share their rewards with people, despite not being forced to.
The prisoner's dilemma game is effective in examining the social preference of cooperation. The logic behind the prisoner's dilemma is that every players rational choice is to defect, rather than cooperate. As it is in each player's best interest to defect, both players would rationally choose to defect. This results in a worse payoff for both players. The ultimatum game requires two players to agree on the allocation of money, yet what is reflected by the game is that humans are more concerned with whether the allocation is fair than whether the benefits are maximized. This behavior also illustrates that behavioral game theory is more well thought out than traditional game theory. However, in an attempt to reach a fair equilibrium for both players, results from the prisoner's dilemma game show that people cooperate much more than traditionally thought. When one player decides to cooperate, then the other players are more likely to cooperate too. This goes against the traditional beliefs that people only make decisions that maximise their utility.
== Examples of games used in behavioral game theory research ==
Signaling game
Dictator Game
Ultimatum Game
Keynesian beauty contest
Normal form game
Cooperative game
Gift-exchange game
Prisoner's Dilemma
Zero-sum Games
== Factors that affect rationality in games ==
=== Learning ===
Learning models are a way of explaining and predicting strategic decisions in behavioral game theory. More specifically, they aim to explain how player's choices may change when given the chance to learn about their opponents or the game. There are three different types of learning models. The first is reinforcement learning. Reinforcement learning suggests that if a player received a high reward from choosing a certain behavior or strategy, then that player would be more inclined to use the same strategy again. If a particular strategy has not been used before however, then the strategy would not appear to be more or less appealing to the player. Another learning model is belief learning. Belief learning assumes that players often remember their opponents previous strategies in games, and will henceforth change their own strategies based on their opponents past behavior. Lastly, experience weighted attraction learning uses a mixture of belief learning and reinforcement learning in its model. This model accounts for the strategies and payoffs that have been played and unplayed. The experience weighted attraction learning framework posits that people learn from past experiences as well as by questioning what they could've done differently. Furthermore, it also believes that players evaluate their past rewards half as much as their actual rewards.
=== Beliefs ===
Beliefs about other people in a decision-making game are expected to influence ones ability to make rational choices. However, beliefs of others can also cause experimental results to deviate from equilibrium, utility-maximizing decisions. In an experiment by Costa-Gomez (2008) participants were questioned about their first order beliefs about their opponent's actions prior to completing a series of normal-form games with other participants. Participants complied with Nash Equilibrium only 35% of the time. Further, participants only stated beliefs that their opponents would comply with traditional game theory equilibrium 15% of the time. This means participants believed their opponents would be less rational than they really were. The results of this study show that participants do not choose the utility-maximizing action and they expect their opponents to do the same. Also, the results show that participants did not choose the utility-maximizing action that corresponded to their beliefs about their opponent's action. While participants may have believed their opponent was more likely to make a certain decision, they still made decisions as if their opponent was choosing randomly. Another study that examined participants from the TV show Deal or No Deal found divergence from rational choice. Participants were more likely to base their decisions on previous outcomes when progressing through the game. Risk aversion decreased when participants' expectations were not met within the game. For example, a subject that experienced a string of positive outcomes was less likely to accept the deal and end the game. The same was true for a subject that experienced primarily negative outcomes early in the game.
=== Social cooperation ===
Social behavior and cooperation with other participants are two factors that are not modeled in traditional game theory, but are often seen in an experimental setting. The evolution of social norms has been neglected in decision-making models, but these norms influence the ways in which real people interact with one another and make choices. One tendency is for a person to be a strong reciprocator. This type of person enters a game with the predisposition to cooperate with other players. They will increase their cooperation levels in response to cooperation from other players and decrease their cooperation levels, even at their own expense, to punish players who do not cooperate. This is not payoff-maximizing behavior, as a strong reciprocator is willing to reduce their payoff in order to encourage cooperation from others. Rational choice theory has limitations in interactive decision making, and it is also difficult to accurately predict human behavior in social cooperation. Behavioral games not only require players to make rational choices, but also require players to be able to predict the decisions of other players in their interactions. In game experiments, rational choice conflicts with individual decision making, and individual behavior may be able to achieve greater gains than rational choice. Rational choice theory has limitations and uncertainties for social interaction decisions, so the predicted results are not the same as the experimental results.
Dufwenberg and Kirchsteiger (2004) developed a model based on reciprocity called the sequential reciprocity equilibrium. This model adapts traditional game theory logic to the idea that players reciprocate actions in order to cooperate. The model had been used to more accurately predict experimental outcomes of classic games such as the prisoner's dilemma and the centipede game. Rabin (1993) also created a fairness equilibrium that measures altruism's effect on choices. He found that when a player is altruistic to another player the second player is more likely to reciprocate that altruism. This is due to the idea of fairness. Fairness equilibriums take the form of mutual maximum, where both players choose an outcome that benefits both of them the most, or mutual minimum, where both players choose an outcome that hurts both of them the most. These equilibriums are also Nash equilibriums, but they incorporate the willingness of participants to cooperate and play fair.
=== Incentives, consequences, and deception ===
The role of incentives and consequences in decision-making is interesting to behavioral game theorists because it affects rational behavior. Post (2008) analyzed Deal or no Deal contestant behavior in order to reach conclusions about decision-making when stakes are high. Studying the contestant's choices formed the conclusion that, in a sequential game with high stakes, decisions were based on previous outcomes rather than rationality. Players who face a succession of good outcomes, in this case they eliminate the low-value cases from play, or players who face a succession of poor outcomes become less risk averse. This means that players who are having exceptionally good or exceptionally bad outcomes are more likely to gamble and continue playing than average players. The lucky or unlucky players were willing to reject offers of over one hundred percent of the expected value of their case in order to continue playing. This shows a shift from risk avoiding behavior to risk seeking behavior. This study highlights behavioral biases that are not accounted for by traditional game theory. Riskier behavior in unlucky contestants can be attributed to the break-even effect, which states that gamblers will continue to make risky decisions in order to win back money. On the other hand, riskier behavior in lucky contestants can be explained by the house-money effect, which states that winning gamblers are more likely to make risky decisions because they perceive that they are not gambling with their own money. This analysis shows that incentives influence rational choice, especially when players make a series of decisions.
Incentives and consequences also play a large role in deception in games. Gneezy (2005) studied deception using a cheap talk sender-receiver game. In this type of game player one receives information about the payouts of option A and option B. Then, player one gives a recommendation to player two about which option to take. Player one can choose to deceive player two, and player two can choose to reject player one's advice. Gneezy found that participants were more sensitive to their gain from lying than to their opponent's loss. He also found that participants were not wholly selfish and cared about how much their opponents lost from their deception, but this effect diminished as their own payout increased. These findings show that decision makers examine both incentives to lie and consequences of lying in order to decide whether or not to lie. In general people are averse to lying, but given the right incentives they tend to ignore consequences. Wang (2009) also used a cheap talk game to study deception in participants with an incentive to deceive. Using eye tracking he found that participants who received information about payoffs focused on their own payoff twice as often as their opponents. This suggests minimal strategic thinking. Further, participants' pupils dilated when they sent a deceiving, and they dilated more when telling a bigger lie. Through these physical cues Wang concluded that deception is cognitively difficult. These findings show that factors such as incentives, consequences, and deception can create irrational decisions and affect the way games unfold.
A consequence of the game theory is its lack of use of empirical data to predict outcomes. "game theory will be no substitute for an empirically grounded behavioral theory when we want to predict what people will actually do in a competitive situation" Predicting rational behavior is possible with game theory but it can be improved if the theory is open to adjustment. The predicted result of the game can be improved and long-lasting if the discipline expands its knowledge of behavioral theory. How people act, think, and feel affect their decisions to play a role in this theory.,. Ken Binmore makes an excellent point that when empirically sound data is present, game theory should not hold the final decision outcome. That this is good for trying to understand if the rational decision being made is due to game theory or is just a consistent behavioral decision being made. The field of economics should try to take every step in improving empirical information in that there is little reliance on just a theory. Businesses value game theory, and the economic discipline must improve the strength of game theory by trying to establish an empirical database. Society will be able to advance its knowledge of behavioral game theory just by expanding the economic discipline of data. Alvin E Roth states, "if we do not take steps in the direction of adding a solid empirical base to game theory, but instead continue to rely on game theory primarily for conceptual insights, then it is likely that long before a hundred-year game theory will have experienced sharply diminishing return"
=== Group decisions ===
Behavioral game theory considers the effects of groups on rationality. In the real world many decisions are made by teams, yet traditional game theory uses an individual as a decision maker. Milton Friedman argues that usually people ignore individual behavior and focus more on group behavior, so group behavior is often perceived as more rational. This created a need to model group decision-making behavior. Bornstein and Yaniv (1998) examined the difference in rationality between groups and individuals in an ultimatum game. In this game player one (or group one) decides what percentage of a payout to give to player two (or group two) and then player two decides whether to accept or reject this offer. Participants in the group condition were put in groups of three and allowed to deliberate on their decisions. Perfect rationality in this game would be player one offering player two none of the payout, but that is almost never the case in observed offers. Bornstein and Yaniv found that groups were less generous, willing to give up a smaller portion of the payoff, in the player one condition and more accepting of low offers in the player two condition than individuals. These results suggest that groups are more rational than individuals.
Kocher and Sutter (2005) used a beauty contest game to study and compare individual and group behavior. A beauty contest game is one where all participants choose a number between zero and one hundred. The winner is the participant who chooses a number closest to two thirds of the average number. In the first round the rational choice would be thirty-three, as it is two thirds of the average number, fifty. Given an infinite number of rounds all participants should choose zero according to game theory. Kocher and Sutter found that groups did not perform more rationally than individuals in the first round of the game. However, groups performed more rationally than individuals in subsequent rounds. This shows that groups are able to learn the game and adapt their strategy faster than individuals.
== See also ==
Behavioral economics
Experimental economics
Game theory
== References == | Wikipedia/Behavioural_game_theory |
In the field of management, strategic management involves the formulation and implementation of the major goals and initiatives taken by an organization's managers on behalf of stakeholders, based on consideration of resources and an assessment of the internal and external environments in which the organization operates. Strategic management provides overall direction to an enterprise and involves specifying the organization's objectives, developing policies and plans to achieve those objectives, and then allocating resources to implement the plans. Academics and practicing managers have developed numerous models and frameworks to assist in strategic decision-making in the context of complex environments and competitive dynamics. Strategic management is not static in nature; the models can include a feedback loop to monitor execution and to inform the next round of planning.
Michael Porter identifies three principles underlying strategy:
creating a "unique and valuable [market] position"
making trade-offs by choosing "what not to do"
creating "fit" by aligning company activities with one another to support the chosen strategy.
Corporate strategy involves answering a key question from a portfolio perspective: "What business should we be in?" Business strategy involves answering the question: "How shall we compete in this business?" Alternatively, corporate strategy may be thought of as the strategic management of a corporation (a particular legal structure of a business), and business strategy as the strategic management of a business.
Management theory and practice often make a distinction between strategic management and operational management, where operational management is concerned primarily with improving efficiency and controlling costs within the boundaries set by the organization's strategy.
== Definitions ==
In 1988, Henry Mintzberg described the many different definitions and perspectives on strategy reflected in both academic research and in practice. He examined the strategic process and concluded it was much more fluid and unpredictable than people had thought. Because of this, he could not point to one process that could be called strategic planning. Instead Mintzberg concludes that there are five types of strategies:
Strategy as plan – a directed course of action to achieve an intended set of goals; similar to the strategic planning concept;
Strategy as pattern – a consistent pattern of past behavior, with a strategy realized over time rather than planned or intended. Where the realized pattern was different from the intent, he referred to the strategy as emergent;
Strategy as position – locating brands, products, or companies within the market, based on the conceptual framework of consumers or other stakeholders; a strategy determined primarily by factors outside the firm;
Strategy as ploy – a specific maneuver intended to outwit a competitor; and
Strategy as perspective – executing strategy based on a "theory of the business" or natural extension of the mindset or ideological perspective of the organization.
In 1998, Mintzberg developed these five types of management strategy into 10 "schools of thought" and grouped them into three categories. The first group is normative. It consists of the schools of informal design and conception, the formal planning, and analytical positioning. The second group, consisting of six schools, is more concerned with how strategic management is actually done, rather than prescribing optimal plans or positions. The six schools are entrepreneurial, visionary, cognitive, learning/adaptive/emergent, negotiation, corporate culture and business environment. The third and final group consists of one school, the configuration or transformation school, a hybrid of the other schools organized into stages, organizational life cycles, or "episodes".
Michael Porter defined strategy in 1980 as the "...broad formula for how a business is going to compete, what its goals should be, and what policies will be needed to carry out those goals" and the "...combination of the ends (goals) for which the firm is striving and the means (policies) by which it is seeking to get there." He continued that: "The essence of formulating competitive strategy is relating a company to its environment."
Some complexity theorists define strategy as the unfolding of the internal and external aspects of the organization that results in actions in a socio-economic context.
Michael D. Watkins (2007) argued that strategic management operates as a critical bridge between an organization's mission, vision, and execution. He asserted that if the mission statement and goals answer the 'what' question, and if the vision statement answers the 'why' questions, then strategy provides answers to the 'how' question of business management. In other words, strategy encompasses the methods, frameworks, and decision-making processes that enable a company to translate its aspirations into concrete actions and competitive success.
== Application ==
Strategy is defined as "the determination of the basic long-term goals of an enterprise, and the adoption of courses of action and the allocation of resources necessary for carrying out these goals". Strategies are established to set direction, focus effort, define or clarify the organization, and provide consistency or guidance in response to the environment.
Strategic management involves the related concepts of strategic planning and strategic thinking. Strategic planning is analytical in nature and refers to formalized procedures to produce the data and analyses used as inputs for strategic thinking, which synthesizes the data resulting in the strategy. Strategic planning may also refer to control mechanisms used to implement the strategy once it is determined. In other words, strategic planning happens around the strategic thinking or strategy making activity.
Strategic management is often described as involving two major processes: formulation and implementation of strategy. While described sequentially below, in practice the two processes are iterative and each provides input for the other.
=== Formulation ===
Formulation of strategy involves analyzing the environment in which the organization operates, then making a series of strategic decisions about how the organization will fulfill its mission. Formulation ends with a series of goals or objectives and measures for the organization to pursue. Environmental analysis includes the:
Remote external environment, including the political, economic, social, technological, legal and environmental landscape (PESTLE);
Industry environment, such as the competitive behavior of rival organizations, the bargaining power of buyers/customers and suppliers, threats from new entrants to the industry, and the ability of buyers to substitute products (Porter's 5 forces); and
Internal environment, regarding the strengths and weaknesses of the organization's resources (i.e., its people, processes and IT systems).
Strategic decisions are based on insight from the environmental assessment and are responses to strategic questions about how the organization will compete, such as:
What is the organization's business?
Who is the target customer for the organization's products and services?
Where are the customers and how do they buy? What is considered "value" to the customer?
Which businesses, products and services should be included or excluded from the portfolio of offerings?
What is the geographic scope of the business?
What differentiates the company from its competitors in the eyes of customers and other stakeholders?
Which resources, skills and capabilities should be developed within the firm?
What are the important opportunities and risks for the organization?
How can the firm grow, through both its base business and new business?
How can the firm generate more value for investors?
The answers to these and many other strategic questions result in the organization's strategy and a series of specific short-term and long-term goals or objectives and related measures.
=== Implementation ===
The second major process of strategic management is implementation, which involves decisions regarding how the organization's resources (i.e., people, process and IT systems) will be aligned and mobilized towards the objectives. Implementation results in how the organization's resources are structured (such as by product or service or geography), leadership arrangements, communication, incentives, and monitoring mechanisms to track progress towards objectives, among others.
Running the day-to-day operations of the business is often referred to as "operations management" or specific terms for key departments or functions, such as "logistics management" or "marketing management", which take over once strategic management decisions are implemented.
== Historical development ==
=== Origins ===
The strategic management discipline originated in the 1950s and 1960s. Among the numerous early contributors, the most influential were Peter Drucker, Philip Selznick, Alfred Chandler, Igor Ansoff, and Bruce Henderson. The discipline draws from earlier thinking and texts on 'strategy' dating back thousands of years. Prior to 1960, the term "strategy" was primarily used regarding war and politics, not business. Many companies built strategic planning functions to develop and execute the formulation and implementation processes during the 1960s.
Peter Drucker was a prolific management theorist and author of dozens of management books, with a career spanning five decades. He addressed fundamental strategic questions in a 1954 book The Practice of Management writing: "... the first responsibility of top management is to ask the question 'what is our business?' and to make sure it is carefully studied and correctly answered." He wrote that the answer was determined by the customer. He recommended eight areas where objectives should be set, such as market standing, innovation, productivity, physical and financial resources, worker performance and attitude, profitability, manager performance and development, and public responsibility.
In 1957, Philip Selznick initially used the term "distinctive competence" in referring to how the Navy was attempting to differentiate itself from the other services. He also formalized the idea of matching the organization's internal factors with external environmental circumstances. This core idea was developed further by Kenneth R. Andrews in 1963 into what we now call SWOT analysis, in which the strengths and weaknesses of the firm are assessed in light of the opportunities and threats in the business environment.
Alfred Chandler recognized the importance of coordinating management activity under an all-encompassing strategy. Interactions between functions were typically handled by managers who relayed information back and forth between departments. Chandler stressed the importance of taking a long-term perspective when looking to the future. In his 1962 ground breaking work Strategy and Structure, Chandler showed that a long-term coordinated strategy was necessary to give a company structure, direction and focus. He says it concisely, "Structure follows Strategy." Chandler wrote that: "Strategy is the determination of the basic long-term goals of an enterprise, and the adoption of courses of action and the allocation of resources necessary for carrying out these goals."
Igor Ansoff built on Chandler's work by adding concepts and inventing a vocabulary. He developed a grid that compared strategies for market penetration, product development, market development and horizontal and vertical integration and diversification. He felt that management could use the grid to systematically prepare for the future. In his 1965 classic Corporate Strategy, he developed gap analysis to clarify the gap between the current reality and the goals and to develop what he called "gap reducing actions". Ansoff wrote that strategic management had three parts: strategic planning; the skill of a firm in converting its plans into reality; and the skill of a firm in managing its own internal resistance to change.
Bruce Henderson, founder of the Boston Consulting Group, wrote about the concept of the experience curve in 1968, following initial work begun in 1965. The experience curve refers to a hypothesis that unit production costs decline by 20–30% every time cumulative production doubles. This supported the argument for achieving higher market share and economies of scale.
Porter wrote in 1980 that companies have to make choices about their scope and the type of competitive advantage they seek to achieve, whether lower cost or differentiation. The idea of strategy targeting particular industries and customers (i.e., competitive positions) with a differentiated offering was a departure from the experience-curve influenced strategy paradigm, which was focused on larger scale and lower cost. Porter revised the strategy paradigm again in 1985, writing that superior performance of the processes and activities performed by organizations as part of their value chain is the foundation of competitive advantage, thereby outlining a process view of strategy.
=== Change in focus from production to marketing ===
The direction of strategic research also paralleled a major paradigm shift in how companies competed, specifically a shift from the production focus to market focus. The prevailing concept in strategy up to the 1950s was to create a product of high technical quality. If you created a product that worked well and was durable, it was assumed you would have no difficulty profiting. This was called the production orientation. Henry Ford famously said of the Model T car: "Any customer can have a car painted any color that he wants, so long as it is black."
Management theorist Peter F Drucker wrote in 1954 that it was the customer who defined what business the organization was in. In 1960 Theodore Levitt argued that instead of producing products then trying to sell them to the customer, businesses should start with the customer, find out what they wanted, and then produce it for them. The fallacy of the production orientation was also referred to as marketing myopia in an article of the same name by Levitt.
Over time, the customer became the driving force behind all strategic business decisions. This marketing concept, in the decades since its introduction, has been reformulated and repackaged under names including market orientation, customer orientation, customer intimacy, customer focus, customer-driven and market focus.
=== Nature of strategy ===
In 1985, Ellen Earle Chaffee summarized what she thought were the main elements of strategic management theory where consensus generally existed as of the 1970s, writing that strategic management:
involves adapting the organization to its business environment;
is fluid and complex. Change creates novel combinations of circumstances requiring unstructured non-repetitive responses;
affects the entire organization by providing direction;
involves both strategy formulation processes and also implementation of the content of the strategy;
may be planned (intended) or unplanned (emergent); these may differ from each other and also from the realized strategy which results from them (Chaffee, p. 89)
is done at several levels: overall corporate-level strategy, and individual business-level strategies; and
involves both conceptual and analytical thought processes.
Chaffee further wrote that research up to that point covered three models of strategy, which were not mutually exclusive:
Linear strategy: a planned determination of goals, initiatives, and allocation of resources, along the lines of the Chandler definition above. This is most consistent with strategic planning approaches and may have a long planning horizon. The strategist "deals with" the environment but it is not the central concern.
Adaptive strategy: in this model, the organization's goals and activities are primarily concerned with adaptation to its environment, analogous to a biological organism. The need for continuous adaption reduces or eliminates the planning window. There is more focus on means (resource mobilization to address the environment) rather than ends (goals). Strategy is less centralized than in the linear model.
Interpretive strategy: as a less developed model than the linear and adaptive models, dating from the 1970s, interpretive strategy is concerned with "orienting metaphors constructed for the purpose of conceptualizing and guiding individual attitudes or organizational participants". The aim of interpretive strategy is legitimacy or credibility in the mind of stakeholders. It places emphasis on symbols and language to influence the minds of customers, rather than the physical product of the organization.
J I Moore identifies four related levels at which strategies can be devised: enterprise, corporate, business and functional Levels. The functional level applies to specific functional areas within an organisation such as its finance department, HR team or IT section.
In 2004, George Stalk, a Boston Consulting Group writer, distinguished between two extremes of business strategy using baseball metaphors:
Softball: relying on weak competitive tactics which appear to be "strategic" but in fact "do little more than keep the company in the game for the short term";
Hardball: engaging with tough competitive strategies, "relentlessly" aiming for success.
== Concepts and frameworks ==
The progress of strategy since 1960 can be charted by a variety of frameworks and concepts introduced by management consultants and academics. These reflect an increased focus on cost, competition and customers. These "3 Cs" were illuminated by much more robust empirical analysis at ever-more granular levels of detail, as industries and organizations were disaggregated into business units, activities, processes, and individuals in a search for sources of competitive advantage.
=== SWOT analysis ===
By the 1960s, the capstone business policy course at the Harvard Business School included the concept of matching the distinctive competence of a company (its internal strengths and weaknesses) with its environment (external opportunities and threats) in the context of its objectives. This framework came to be known by the acronym SWOT and was "a major step forward in bringing explicitly competitive thinking to bear on questions of strategy". Kenneth R. Andrews helped popularize the framework via a 1963 conference and it remains commonly used in practice.
=== Experience curve ===
The experience curve was developed by the Boston Consulting Group in 1966. It reflects a hypothesis that total per unit costs decline systematically by as much as 15–25% every time cumulative production (i.e., "experience") doubles. It has been empirically confirmed by some firms at various points in their history. Costs decline due to a variety of factors, such as the learning curve, substitution of labor for capital (automation), and technological sophistication. Author Walter Kiechel wrote that it reflected several insights, including:
A company can always improve its cost structure;
Competitors have varying cost positions based on their experience;
Firms could achieve lower costs through higher market share, attaining a competitive advantage; and
An increased focus on empirical analysis of costs and processes, a concept which author Kiechel refers to as "Greater Taylorism".
Kiechel wrote in 2010: "The experience curve was, simply, the most important concept in launching the strategy revolution...with the experience curve, the strategy revolution began to insinuate an acute awareness of competition into the corporate consciousness." Prior to the 1960s, the word competition rarely appeared in the most prominent management literature; U.S. companies then faced considerably less competition and did not focus on performance relative to peers. Further, the experience curve provided a basis for the retail sale of business ideas, helping drive the management consulting
industry.
=== Importance-performance matrix ===
Completion of an importance-performance matrix forms "a crucial stage in the formulation of operations strategy", and may be considered a "simple, yet useful, method for simultaneously considering both the importance and performance dimensions when evaluating or defining strategy". Notes on this subject from the Department of Engineering at the University of Cambridge suggest that a binary matrix may be used "but may be found too crude", and nine point scales on both the importance and performance axes are recommended. An importance scale could be labelled from "the main thrust of competitiveness" to "never considered by customers and never likely to do so", and performance can be segmented into "better than", "the same as", and "worse than" the company's competitors. The highest urgency would than be directed to the most important areas where performance is poorer than competitors.
The technique is also used in relation to marketing, where the variable "importance" is related to buyers' perception of important attributes of a product: for attributes which might be considered important to buyers, both their perceived importance and their performance are assessed.
=== Corporate strategy and portfolio theory ===
The concept of the corporation as a portfolio of business units, with each plotted graphically based on its market share (a measure of its competitive position relative to its peers) and industry growth rate (a measure of industry attractiveness), was summarized in the growth–share matrix developed by the Boston Consulting Group around 1970. By 1979, one study estimated that 45% of the Fortune 500 companies were using some variation of the matrix in their strategic planning. This framework helped companies decide where to invest their resources (i.e., in their high market share, high growth businesses) and which businesses to divest (i.e., low market share, low growth businesses.) The growth-share matrix was followed by G.E. multi factoral model, developed by General Electric.
Companies continued to diversify as conglomerates until the 1980s, when deregulation and a less restrictive antitrust environment led to the view that a portfolio of operating divisions in different industries was worth more as many independent companies, leading to the breakup of many conglomerates. While the popularity of portfolio theory has waxed and waned, the key dimensions considered (industry attractiveness and competitive position) remain central to strategy.
In response to the evident problems of "over diversification", C. K. Prahalad and Gary Hamel suggested that companies should build portfolios of businesses around shared technical or operating competencies, and should develop structures and processes to enhance their core competencies.
Michael Porter also addressed the issue of the appropriate level of diversification. In 1987, he argued that corporate strategy involves two questions: 1) What business should the corporation be in? and 2) How should the corporate office manage its business units? He mentioned four concepts of corporate strategy each of which suggest a certain type of portfolio and a certain role for the corporate office; the latter three can be used together:
Portfolio theory: A strategy based primarily on diversification through acquisition. The corporation shifts resources among the units and monitors the performance of each business unit and its leaders. Each unit generally runs autonomously, with limited interference from the corporate center provided goals are met.
Restructuring: The corporate office acquires then actively intervenes in a business where it detects potential, often by replacing management and implementing a new business strategy.
Transferring skills: Important managerial skills and organizational capability are essentially spread to multiple businesses. The skills must be necessary to competitive advantage.
Sharing activities: Ability of the combined corporation to leverage centralized functions, such as sales, finance, etc. thereby reducing costs.
Building on Porter's ideas, Michael Goold, Andrew Campbell and Marcus Alexander developed the concept of "parenting advantage" to be applied at the corporate level, as a parallel to the concept of "competitive advantage" applied at the business level. Parent companies, they argued, should aim to "add more value" to their portfolio of businesses than rivals. If they succeed, they have a parenting advantage. The right level of diversification depends, therefore, on the ability of the parent company to add value in comparison to others. Different parent companies with different skills should expect to have different portfolios. See Corporate Level Strategy 1995 and Strategy for the Corporate Level 2014
=== Competitive advantage ===
In 1980, Porter defined the two types of competitive advantage an organization can achieve relative to its rivals: lower cost or differentiation. This advantage derives from attribute(s) that allow an organization to outperform its competition, such as superior market position, skills, or resources. In Porter's view, strategic management should be concerned with building and sustaining competitive advantage.
=== Industry structure and profitability ===
Porter developed a framework for analyzing the profitability of industries and how those profits are divided among the participants in 1980. In five forces analysis he identified the forces that shape the industry structure or environment. The framework involves the bargaining power of buyers and suppliers, the threat of new entrants, the availability of substitute products, and the competitive rivalry of firms in the industry. These forces affect the organization's ability to raise its prices as well as the costs of inputs (such as raw materials) for its processes.
The five forces framework helps describe how a firm can use these forces to obtain a sustainable competitive advantage, either lower cost or differentiation. Companies can maximize their profitability by competing in industries with favorable structure. Competitors can take steps to grow the overall profitability of the industry, or to take profit away from other parts of the industry structure. Porter modified Chandler's dictum about structure following strategy by introducing a second level of structure: while organizational structure follows strategy, it in turn follows industry structure.
=== Generic competitive strategies ===
Porter wrote in 1980 that strategy target either cost leadership, differentiation, or focus. These are known as Porter's three generic strategies and can be applied to any size or form of business. Porter claimed that a company must only choose one of the three or risk that the business would waste precious resources. Porter's generic strategies detail the interaction between cost minimization strategies, product differentiation strategies, and market focus strategies.
Porter described an industry as having multiple segments that can be targeted by a firm. The breadth of its targeting refers to the competitive scope of the business. Porter defined two types of competitive advantage: lower cost or differentiation relative to its rivals. Achieving competitive advantage results from a firm's ability to cope with the five forces better than its rivals. Porter wrote: "[A]chieving competitive advantage requires a firm to make a choice...about the type of competitive advantage it seeks to attain and the scope within which it will attain it." He also wrote: "The two basic types of competitive advantage [differentiation and lower cost] combined with the scope of activities for which a firm seeks to achieve them lead to three generic strategies for achieving above average performance in an industry: cost leadership, differentiation and focus. The focus strategy has two variants, cost focus and differentiation focus."
The concept of choice was a different perspective on strategy, as the 1970s paradigm was the pursuit of market share (size and scale) influenced by the experience curve. Companies that pursued the highest market share position to achieve cost advantages fit under Porter's cost leadership generic strategy, but the concept of choice regarding differentiation and focus represented a new perspective.
=== Value chain ===
Porter's 1985 description of the value chain refers to the chain of activities (processes or collections of processes) that an organization performs in order to deliver a valuable product or service for the market. These include functions such as inbound logistics, operations, outbound logistics, marketing and sales, and service, supported by systems and technology infrastructure. By aligning the various activities in its value chain with the organization's strategy in a coherent way, a firm can achieve a competitive advantage. Porter also wrote that strategy is an internally consistent configuration of activities that differentiates a firm from its rivals. A robust competitive position cumulates from many activities which should fit coherently together.
Porter wrote in 1985: "Competitive advantage cannot be understood by looking at a firm as a whole. It stems from the many discrete activities a firm performs in designing, producing, marketing, delivering and supporting its product. Each of these activities can contribute to a firm's relative cost position and create a basis for differentiation...the value chain disaggregates a firm into its strategically relevant activities in order to understand the behavior of costs and the existing and potential sources of differentiation."
=== Interorganizational relationships ===
Interorganizational relationships allow independent organizations to get access to resources or to enter new markets. Interorganizational relationships represent a critical lever of competitive advantage.
The field of strategic management has paid much attention to the different forms of relationships between organizations ranging from strategic alliances to buyer-supplier relationships, joint ventures, networks, R&D consortia, licensing, and franchising.
On the one hand, scholars drawing on organizational economics (e.g., transaction costs theory) have argued that firms use interorganizational relationships when they are the most efficient form comparatively to other forms of organization such as operating on its own or using the market. On the other hand, scholars drawing on organizational theory (e.g., resource dependence theory) suggest that firms tend to partner with others when such relationships allow them to improve their status, power, reputation, or legitimacy.
A key component to the strategic management of inter-organizational relationships relates to the choice of governance mechanisms. While early research focused on the choice between equity and non equity forms, recent scholarship studies the nature of the contractual and relational arrangements between organizations.
Researchers have also noted, although to a lesser extent, the dark side of interorganizational relationships, such as conflict, disputes, opportunism and unethical behaviors. Relational or collaborative risk can be defined as the uncertainty about whether potentially significant and/or disappointing outcomes of collaborative activities will be realized. Companies can assess, monitor and manage collaborative risks. Empirical studies show that managers assess risks as lower when they external partners, higher if they are satisfied with their own performance, and lower when their business environment is turbulent.
=== Core competence ===
Gary Hamel and C. K. Prahalad described the idea of core competency in 1990, the idea that each organization has some capability in which it excels and that the business should focus on opportunities in that area, letting others go or outsourcing them. Further, core competency is difficult to duplicate, as it involves the skills and coordination of people across a variety of functional areas or processes used to deliver value to customers. By outsourcing, companies expanded the concept of the value chain, with some elements within the entity and others without. Core competency is part of a branch of strategy called the resource-based view of the firm, which postulates that if activities are strategic as indicated by the value chain, then the organization's capabilities and ability to learn or adapt are also strategic.
=== Theory of the business ===
According to Peter Drucker, business theory refers to the key points and strategies of a company, which are divided into three parts:
1. The external environment (society, technology, customers, and competition).
2. The goal of an organization.
3. Guidelines essential to achieving the mission.
This business theory has four differentiations:
1. Hypotheses maintain that mission and guidelines must be reality focused.
2. Thoughts must have agreement.
3. The business theory must be notable and interpreted by the members of the organization.
4. Business theory must be continuously analyzed.
Companies have difficulties when the assumptions of such a theory do not align with reality, Peter Drucker took as an example large retail premises, his goal was that people who wanted to buy in large commercial premises do so, but many consumers rejected commercial premises and preferred retailers (which focus on one or two categories of products and own their own premises) time was essential in shopping instead of profits .
This theory is classified as an assumption and a discipline, which focused on the elaboration of systematic diagnoses, monitoring and testing of the guidelines that make up the business theory in order to maintain competition.
== Strategic thinking ==
Strategic thinking involves the generation and application of unique business insights to opportunities intended to create competitive advantage for a firm or organization. It involves challenging the assumptions underlying the organization's strategy and value proposition. Mintzberg wrote in 1994 that it is more about synthesis (i.e., "connecting the dots") than analysis (i.e., "finding the dots"). It is about "capturing what the manager learns from all sources (both the soft insights from his or her personal experiences and the experiences of others throughout the organization and the hard data from market research and the like) and then synthesizing that learning into a vision of the direction that the business should pursue." Mintzberg argued that strategic thinking is the critical part of formulating strategy, more so than strategic planning exercises.
General Andre Beaufre wrote in 1963 that strategic thinking "is a mental process, at once abstract and rational, which must be capable of synthesizing both psychological and material data. The strategist must have a great capacity for both analysis and synthesis; analysis is necessary to assemble the data on which he makes his diagnosis, synthesis in order to produce from these data the diagnosis itself--and the diagnosis in fact amounts to a choice between alternative courses of action."
Will Mulcaster argued that while much research and creative thought has been devoted to generating alternative strategies, too little work has been done on what influences the quality of strategic decision making and the effectiveness with which strategies are implemented. For instance, in retrospect it can be seen that the 2008 financial crisis could have been avoided if the banks had paid more attention to the risks associated with their investments, but how should banks change the way they make decisions to improve the quality of their decisions in the future? Mulcaster's Managing Forces framework addresses this issue by identifying 11 forces that should be incorporated into the processes of decision making and strategic implementation. The 11 forces are: Time; Opposing forces; Politics; Perception; Holistic effects; Adding value; Incentives; Learning capabilities; Opportunity cost; Risk and Style.
Classic strategy thinking, and vision have some limitations in a turbulent environment and uncertainty. The limitations relate to the heterogeneity and future-oriented goals and possession of cognitive capabilities in classic definition. Strategy should not be seen only from the top managerial hierarchy visions. The newer micro foundation framework suggests that people from different managerial levels are needed to work and interact dynamically to result in the knowledge strategy.
== Strategic planning ==
Strategic planning is a means of administering the formulation and implementation of strategy. Strategic planning is analytical in nature and refers to formalized procedures to produce the data and analyses used as inputs for strategic thinking, which synthesizes the data resulting in the strategy. Strategic planning may also refer to control mechanisms used to implement the strategy once it is determined. In other words, strategic planning happens around the strategy formation process.
=== Environmental analysis ===
Porter wrote in 1980 that formulation of competitive strategy includes consideration of four key elements:
Company strengths and weaknesses;
Personal values of the key implementers (i.e., management and the board)
Industry opportunities and threats; and
Broader societal expectations.
The first two elements relate to factors internal to the company (i.e., the internal environment), while the latter two relate to factors external to the company (i.e., the external environment).
There are many analytical frameworks which attempt to organize the strategic planning process. Examples of frameworks that address the four elements described above include:
External environment: PEST analysis or STEEP analysis is a framework used to examine the remote external environmental factors that can affect the organization, such as political, economic, social/demographic, and technological. Common variations include SLEPT, PESTLE, STEEPLE, and STEER analysis, each of which incorporates slightly different emphases.
Industry environment: The Porter Five Forces Analysis framework helps to determine the competitive rivalry and therefore attractiveness of a market. It is used to help determine the portfolio of offerings the organization will provide and in which markets.
Relationship of internal and external environment: SWOT analysis is one of the most basic and widely used frameworks, which examines both internal elements of the organization—Strengths and Weaknesses—and external elements—Opportunities and Threats. It helps examine the organization's resources in the context of its environment.
=== Scenario planning ===
A number of strategists use scenario planning techniques to deal with change. The way Peter Schwartz put it in 1991 is that strategic outcomes cannot be known in advance so the sources of competitive advantage cannot be predetermined. The fast changing business environment is too uncertain for us to find sustainable value in formulas of excellence or competitive advantage. Instead, scenario planning is a technique in which multiple outcomes can be developed, their implications assessed, and their likeliness of occurrence evaluated. According to Pierre Wack, scenario planning is about insight, complexity, and subtlety, not about formal analysis and numbers.
Some business planners are starting to use a complexity theory approach to strategy. Complexity can be thought of as chaos with a dash of order. Chaos theory deals with turbulent systems that rapidly become disordered. Complexity is not quite so unpredictable. It involves multiple agents interacting in such a way that a glimpse of structure may appear.
=== Measuring and controlling implementation ===
Once the strategy is determined, various goals and measures may be established to chart a course for the organization, measure performance and control implementation of the strategy. Tools such as the balanced scorecard and strategy maps help crystallize the strategy, by relating key measures of success and performance to the strategy. These tools measure financial, marketing, production, organizational development, and innovation measures to achieve a 'balanced' perspective. Advances in information technology and data availability enable the gathering of more information about performance, allowing managers to take a much more analytical view of their business than before.
Strategy may also be organized as a series of "initiatives" or "programs", each of which comprises one or more projects. Various monitoring and feedback mechanisms may also be established, such as regular meetings between divisional and corporate management to control implementation.
=== Evaluation ===
A key component to strategic management which is often overlooked when planning is evaluation. Evaluation may involve looking at what was done (implementation) and what happened as a result, or it may involve evaluating options to see what potential different options may open up, in order to decide on planned actions.
There are many ways to evaluate whether or not strategic priorities and plans have been achieved, one such method is Robert Stake's Responsive Evaluation. Responsive evaluation provides a naturalistic and humanistic approach to program evaluation. In expanding beyond the goal-oriented or pre-ordinate evaluation design, responsive evaluation takes into consideration the program's background (history), conditions, and transactions among stakeholders. It is largely emergent, the design unfolds as contact is made with stakeholders.
== Limitations ==
While strategies are established to set direction, focus effort, define or clarify the organization, and provide consistency or guidance in response to the environment, these very elements also mean that certain signals are excluded from consideration or de-emphasized. Mintzberg wrote in 1987: "Strategy is a categorizing scheme by which incoming stimuli can be ordered and dispatched." Since a strategy orients the organization in a particular manner or direction, that direction may not effectively match the environment, initially (if a bad strategy) or over time as circumstances change. As such, Mintzberg continued, "Strategy [once established] is a force that resists change, not encourages it."
Therefore, a critique of strategic management is that it can overly constrain managerial discretion in a dynamic environment. "How can individuals, organizations and societies cope as well as possible with ... issues too complex to be fully understood, given the fact that actions initiated on the basis of inadequate understanding may lead to significant regret?" Some theorists insist on an iterative approach, considering in turn objectives, implementation and resources. I.e., a "...repetitive learning cycle [rather than] a linear progression towards a clearly defined final destination." Strategies must be able to adjust during implementation because "humans rarely can proceed satisfactorily except by learning from experience; and modest probes, serially modified on the basis of feedback, usually are the best method for such learning."
In 2000, Gary Hamel coined the term strategic convergence to explain the limited scope of the strategies being used by rivals in greatly differing circumstances. He lamented that successful strategies are imitated by firms that do not understand that for a strategy to work, it must account for the specifics of each situation.
Woodhouse and Collingridge claim that the essence of being "strategic" lies in a capacity for "intelligent trial-and error" rather than strict adherence to finely honed strategic plans. Strategy should be seen as laying out the general path rather than precise steps. Means are as likely to determine ends as ends are to determine means. The objectives that an organization might wish to pursue are limited by the range of feasible approaches to implementation. (There will usually be only a small number of approaches that will not only be technically and administratively possible, but also satisfactory to the full range of organizational stakeholders.) In turn, the range of feasible implementation approaches is determined by the availability of resources.
== Strategic themes ==
Various strategic approaches used across industries (themes) have arisen over the years. These include the shift from product-driven demand to customer- or marketing-driven demand (described above), the increased use of self-service approaches to lower cost, changes in the value chain or corporate structure due to globalization (e.g., off-shoring of production and assembly), and the internet.
=== Self-service ===
One theme in strategic competition has been the trend towards self-service, often enabled by technology, where the customer takes on a role previously performed by a worker to lower costs for the firm and perhaps prices. Examples include:
Automated teller machine (ATM) to obtain cash rather via a bank teller;
Self-service at the gas pump rather than with help from an attendant;
Retail internet orders input by the customer rather than a retail clerk, such as online book sales;
Mass-produced ready-to-assemble furniture transported by the customer;
Self-checkout at the grocery store; and
Online banking and bill payment.
=== Globalization and the virtual firm ===
One definition of globalization refers to the integration of economies due to technology and supply chain process innovation. Companies are no longer required to be vertically integrated (i.e., designing, producing, assembling, and selling their products). In other words, the value chain for a company's product may no longer be entirely within one firm; several entities comprising a virtual firm may exist to fulfill the customer requirement. For example, some companies have chosen to outsource production to third parties, retaining only design and sales functions inside their organization.
=== Internet and information availability ===
The internet has dramatically empowered consumers and enabled buyers and sellers to come together with drastically reduced transaction and intermediary costs, creating much more robust marketplaces for the purchase and sale of goods and services. The Internet has enabled many Internet-based entrepreneurs to tap serendipity as a strategic advantage and thrive. Examples include online auction sites, internet dating services, and internet book sellers. In many industries, the internet has dramatically altered the competitive landscape. Services that used to be provided within one entity (e.g., a car dealership providing financing and pricing information) are now provided by third parties. Further, compared to traditional media like television, the internet has caused a major shift in viewing habits through on demand content which has led to an increasingly fragmented audience.
Author Phillip Evans said in 2013 that networks are challenging traditional hierarchies. Value chains may also be breaking up ("deconstructing") where information aspects can be separated from functional activity. Data that is readily available for free or very low cost makes it harder for information-based, vertically integrated businesses to remain intact. Evans said: "The basic story here is that what used to be vertically integrated, oligopolistic competition among essentially similar kinds of competitors is evolving, by one means or another, from a vertical structure to a horizontal one. Why is that happening? It's happening because transaction costs are plummeting and because scale is polarizing. The plummeting of transaction costs weakens the glue that holds value chains together, and allows them to separate." He used Wikipedia as an example of a network that has challenged the traditional encyclopedia business model. Evans predicts the emergence of a new form of industrial organization called a "stack", analogous to a technology stack, in which competitors rely on a common platform of inputs (services or information), essentially layering the remaining competing parts of their value chains on top of this common platform.
=== Sustainability ===
In the recent decade, sustainability—or ability to successfully sustain a company in a context of rapidly changing environmental, social, health, and economic circumstances—has emerged as crucial aspect of any strategy development. Research focusing on sustainability in commercial strategies has led to emergence of the concept of "embedded sustainability" – defined by its authors Chris Laszlo and Nadya Zhexembayeva as "incorporation of environmental, health, and social value into the core business with no trade-off in price or quality—in other words, with no social or green premium." Their research showed that embedded sustainability offers at least seven distinct opportunities for business value and competitive advantage creation: a) better risk management, b) increased efficiency through reduced waste and resource use, c) better product differentiation, d) new market entrances, e) enhanced brand and reputation, f) greater opportunity to influence industry standards, and g) greater opportunity for radical innovation. Research further suggested that innovation driven by resource depletion can result in fundamental competitive advantages for a company's products and services, as well as the company strategy as a whole, when right principles of innovation are applied. Asset managers who committed to integrating embedded sustainability factors in their capital allocation decisions created a stronger return on investment than managers that did not strategically integrate sustainability into their similar business model.
To achieve genuine sustainability and these associated benefits, corporations have historically relied on a variety of mechanisms that can be integrated into their management strategy. Timothy Galpin in his chapter of “Business Strategies for Sustainability: A Research Anthology” discusses four “Internal Strategic Management Components” to build sustainability. They are as follows:
Mission: Defines the purpose and priorities of the organization, ultimately providing critical signals to organizational stakeholders regarding the aims of the firm.
Values: Refers to the expectations of internal stakeholders, and communicates the organisation’s belief system to various external stakeholders
Goals: Provides a roadmap of the firm’s organisational activity and a basis for which to measure progress and performance.
Capabilities and resources: The development of patterns of activity and investment decisions that facilitate sustainable business practices.
To fully utilise these strategic management components, a firm’s mission, values, goals, resources, and capabilities need to be functioning in alignment with one another. This develops consistency across management and employee behaviour. Research has indicated that this alignment has led to improved firm performance.
Following the embedding of sustainability in a firm’s strategic management plan, to fully reap the benefits the agenda must be communicated effectively to internal and external stakeholders. Doing so satisfies stakeholder theory, whereby the firm maintains ‘trustful and mutually respectful relationships with the various stakeholders’. In the past, this has consisted of advertising and disclosing sustainability information and reports. Firms are available to promote their superior sustainability performance and ultimately possess higher market valuations in comparison to firms that do not provide sustainability reporting.
The amalgamation and alignment of these key internal strategic management components, in conjunction with thorough communication of the firm’s sustainability agenda, is required to achieve these associated benefits and is the reason many firms are pursuing such tactics more frequently.
== Strategy as learning ==
=== Learning organization ===
In 1990, Peter Senge, who had collaborated with Arie de Geus at Dutch Shell, popularized de Geus' notion of the "learning organization". The theory is that gathering and analyzing information is a necessary requirement for business success in the information age. To do this, Senge claimed that an organization would need to be structured such that:
People can continuously expand their capacity to learn and be productive.
New patterns of thinking are nurtured.
Collective aspirations are encouraged.
People are encouraged to see the "whole picture" together.
Senge identified five disciplines of a learning organization. They are:
Personal responsibility, self-reliance, and mastery – We accept that we are the masters of our own destiny. We make decisions and live with the consequences of them. When a problem needs to be fixed, or an opportunity exploited, we take the initiative to learn the required skills to get it done.
Mental models – We need to explore our personal mental models to understand the subtle effect they have on our behaviour.
Shared vision – The vision of where we want to be in the future is discussed and communicated to all. It provides guidance and energy for the journey ahead.
Team learning – We learn together in teams. This involves a shift from "a spirit of advocacy to a spirit of enquiry".
Systems thinking – We look at the whole rather than the parts. This is what Senge calls the "Fifth discipline". It is the glue that integrates the other four into a coherent strategy. For an alternative approach to the "learning organization", see Garratt, B. (1987).
Geoffrey Moore (1991) and R. Frank and P. Cook also detected a shift in the nature of competition. Markets driven by technical standards or by "network effects" can give the dominant firm a near-monopoly. The same is true of networked industries in which interoperability requires compatibility between users. Examples include Internet Explorer's and Amazon's early dominance of their respective industries. IE's later decline shows that such dominance may be only temporary.
Moore showed how firms could attain this enviable position by using E.M. Rogers' five stage adoption process and focusing on one group of customers at a time, using each group as a base for reaching the next group. The most difficult step is making the transition between introduction and mass acceptance. (See Crossing the Chasm). If successful a firm can create a bandwagon effect in which the momentum builds and its product becomes a de facto standard.
=== Integrated view to learning ===
Bolisani & Bratianu (2017) have defined knowledge strategy as an integration of rational thinking and dynamic learning. Rational planning contains a three-step process where the first step is to collect information, the second step is to analyze the information and the third step is to formulate goals and plans based on information. Emergent planning also contains three steps to the opposite direction starting from practical experience, what is analyzed in the second step, and then formulated to a strategy in the third step. These two approaches are combined to the “integrated view” with the Bolisani and Bratianu research implications. To start the planning process for knowledge and KM strategy creation, company can prepare a preliminary plan with the basis of rational analysis from internal or external environments. While creating rational and predictive plans, company can similarly utilize practical adapted knowledge for example learning from the ground. The idea behind the integrated view is to combine the general visions of knowledge strategy with both the current practical understanding and future ideas. This model will move the decision-making process in a more interactive and co-creative direction.
== Strategy as adapting to change ==
In 1969, Peter Drucker coined the phrase Age of Discontinuity to describe the way change disrupts lives. In an age of continuity attempts to predict the future by extrapolating from the past can be accurate. But according to Drucker, we are now in an age of discontinuity and extrapolating is ineffective. He identifies four sources of discontinuity: new technologies, globalization, cultural pluralism and knowledge capital.
In 1970, Alvin Toffler in Future Shock described a trend towards accelerating rates of change. He illustrated how social and technical phenomena had shorter lifespans with each generation, and he questioned society's ability to cope with the resulting turmoil and accompanying anxiety. In past eras periods of change were always punctuated with times of stability. This allowed society to assimilate the change before the next change arrived. But these periods of stability had all but disappeared by the late 20th century. In 1980 in The Third Wave, Toffler characterized this shift to relentless change as the defining feature of the third phase of civilization (the first two phases being the agricultural and industrial waves).
In 1978, Derek F. Abell (Abell, D. 1978) described "strategic windows" and stressed the importance of the timing (both entrance and exit) of any given strategy. This led some strategic planners to build planned obsolescence into their strategies.
In 1983, Noel Tichy wrote that because we are all beings of habit we tend to repeat what we are comfortable with. He wrote that this is a trap that constrains our creativity, prevents us from exploring new ideas, and hampers our dealing with the full complexity of new issues. He developed a systematic method of dealing with change that involved looking at any new issue from three angles: technical and production, political and resource allocation, and corporate culture.
In 1989, Charles Handy identified two types of change. "Strategic drift" is a gradual change that occurs so subtly that it is not noticed until it is too late. By contrast, "transformational change" is sudden and radical. It is typically caused by discontinuities (or exogenous shocks) in the business environment. The point where a new trend is initiated is called a "strategic inflection point" by Andy Grove. Inflection points can be subtle or radical.
In 1990, Richard Pascale wrote that relentless change requires that businesses continuously reinvent themselves. His famous maxim is "Nothing fails like success" by which he means that what was a strength yesterday becomes the root of weakness today, We tend to depend on what worked yesterday and refuse to let go of what worked so well for us in the past. Prevailing strategies become self-confirming. To avoid this trap, businesses must stimulate a spirit of inquiry and healthy debate. They must encourage a creative process of self-renewal based on constructive conflict.
In 1996, Adrian Slywotzky showed how changes in the business environment are reflected in value migrations between industries, between companies, and within companies. He claimed that recognizing the patterns behind these value migrations is necessary if we wish to understand the world of chaotic change. In "Profit Patterns" (1999) he described businesses as being in a state of strategic anticipation as they try to spot emerging patterns. Slywotsky and his team identified 30 patterns that have transformed industry after industry.
In 1997, Clayton Christensen (1997) took the position that great companies can fail precisely because they do everything right since the capabilities of the organization also define its disabilities. Christensen's thesis is that outstanding companies lose their market leadership when confronted with disruptive technology. He called the approach to discovering the emerging markets for disruptive technologies agnostic marketing, i.e., marketing under the implicit assumption that no one – not the company, not the customers – can know how or in what quantities a disruptive product can or will be used without the experience of using it.
In 1999, Constantinos Markides reexamined the nature of strategic planning. He described strategy formation and implementation as an ongoing, never-ending, integrated process requiring continuous reassessment and reformation. Strategic management is planned and emergent, dynamic and interactive.
J. Moncrieff (1999) stressed strategy dynamics. He claimed that strategy is partially deliberate and partially unplanned. The unplanned element comes from emergent strategies that result from the emergence of opportunities and threats in the environment and from "strategies in action" (ad hoc actions across the organization).
David Teece pioneered research on resource-based strategic management and the dynamic capabilities perspective, defined as "the ability to integrate, build, and reconfigure internal and external competencies to address rapidly changing environments". His 1997 paper (with Gary Pisano and Amy Shuen) "Dynamic Capabilities and Strategic Management" was the most cited paper in economics and business for the period from 1995 to 2005.
In 2000, Gary Hamel discussed strategic decay, the notion that the value of every strategy, no matter how brilliant, decays over time.
== Strategy as operational excellence ==
=== Quality ===
A large group of theorists felt the area where western business was most lacking was product quality. W. Edwards Deming, Joseph M. Juran, Andrew Thomas Kearney, Philip Crosby and Armand V. Feigenbaum suggested quality improvement techniques such total quality management (TQM), continuous improvement (kaizen), lean manufacturing, Six Sigma, and return on quality (ROQ).
Contrarily, James Heskett (1988), Earl Sasser (1995), William Davidow, Len Schlesinger, A. Paraurgman (1988), Len Berry, Jane Kingman-Brundage, Christopher Hart, and Christopher Lovelock (1994), felt that poor customer service was the problem. They gave us fishbone diagramming, service charting, Total Customer Service (TCS), the service profit chain, service gaps analysis, the service encounter, strategic service vision, service mapping, and service teams. Their underlying assumption was that there is no better source of competitive advantage than a continuous stream of delighted customers.
Process management uses some of the techniques from product quality management and some of the techniques from customer service management. It looks at an activity as a sequential process. The objective is to find inefficiencies and make the process more effective. Although the procedures have a long history, dating back to Taylorism, the scope of their applicability has been greatly widened, leaving no aspect of the firm free from potential process improvements. Because of the broad applicability of process management techniques, they can be used as a basis for competitive advantage.
Carl Sewell, Frederick F. Reichheld, Christian Grönroos, and Earl Sasser observed that businesses were spending more on customer acquisition than on retention. They showed how a competitive advantage could be found in ensuring that customers returned again and again. Reicheld broadened the concept to include loyalty from employees, suppliers, distributors and shareholders. They developed techniques for estimating customer lifetime value (CLV) for assessing long-term relationships. The concepts begat attempts to recast selling and marketing into a long term endeavor that created a sustained relationship (called relationship selling, relationship marketing, and customer relationship management). Customer relationship management (CRM) software became integral to many firms.
=== Reengineering ===
Michael Hammer and James Champy felt that these resources needed to be restructured. In a process that they labeled reengineering, firm's reorganized their assets around whole processes rather than tasks. In this way a team of people saw a project through, from inception to completion. This avoided functional silos where isolated departments seldom talked to each other. It also eliminated waste due to functional overlap and interdepartmental communications.
In 1989 Richard Lester and the researchers at the MIT Industrial Performance Center identified seven best practices and concluded that firms must accelerate the shift away from the mass production of low cost standardized products. The seven areas of best practice were:
Simultaneous continuous improvement in cost, quality, service, and product innovation
Breaking down organizational barriers between departments
Eliminating layers of management creating flatter organizational hierarchies.
Closer relationships with customers and suppliers
Intelligent use of new technology
Global focus
Improving human resource skills
The search for best practices is also called benchmarking. This involves determining where you need to improve, finding an organization that is exceptional in this area, then studying the company and applying its best practices in your firm.
== Other perspectives on strategy ==
=== Strategy as problem solving ===
Professor Richard P. Rumelt described strategy as a type of problem solving in 2011. He wrote that good strategy has an underlying structure called a kernel. The kernel has three parts: 1) A diagnosis that defines or explains the nature of the challenge; 2) A guiding policy for dealing with the challenge; and 3) Coherent actions designed to carry out the guiding policy.
President Kennedy outlined these three elements of strategy in his Cuban Missile Crisis Address to the Nation of 22 October 1962:
Diagnosis: "This Government, as promised, has maintained the closest surveillance of the Soviet military buildup on the island of Cuba. Within the past week, unmistakable evidence has established the fact that a series of offensive missile sites is now in preparation on that imprisoned island. The purpose of these bases can be none other than to provide a nuclear strike capability against the Western Hemisphere."
Guiding Policy: "Our unswerving objective, therefore, must be to prevent the use of these missiles against this or any other country, and to secure their withdrawal or elimination from the Western Hemisphere."
Action Plans: First among seven numbered steps was the following: "To halt this offensive buildup a strict quarantine on all offensive military equipment under shipment to Cuba is being initiated. All ships of any kind bound for Cuba from whatever nation or port will, if found to contain cargoes of offensive weapons, be turned back."
Active strategic management required active information gathering and active problem solving. In the early days of Hewlett-Packard (HP), Dave Packard and Bill Hewlett devised an active management style that they called management by walking around (MBWA). Senior HP managers were seldom at their desks. They spent most of their days visiting employees, customers, and suppliers. This direct contact with key people provided them with a solid grounding from which viable strategies could be crafted. Management consultants Tom Peters and Robert H. Waterman had used the term in their 1982 book In Search of Excellence: Lessons From America's Best-Run Companies. Some Japanese managers employ a similar system, which originated at Honda, and is sometimes called the 3 G's (Genba, Genbutsu, and Genjitsu, which translate into "actual place", "actual thing", and "actual situation").
=== Creative vs analytic approaches ===
In 2010, IBM released a study summarizing three conclusions of 1500 CEOs around the world: 1) complexity is escalating, 2) enterprises are not equipped to cope with this complexity, and 3) creativity is now the single most important leadership competency. IBM said that it is needed in all aspects of leadership, including strategic thinking and planning.
Similarly, McKeown argued that over-reliance on any particular approach to strategy is dangerous and that multiple methods can be used to combine the creativity and analytics to create an "approach to shaping the future", that is difficult to copy.
=== Non-strategic management ===
A 1938 treatise by Chester Barnard, based on his own experience as a business executive, described the process as informal, intuitive, non-routinized and involving primarily oral, 2-way communications. Bernard says "The process is the sensing of the organization as a whole and the total situation relevant to it. It transcends the capacity of merely intellectual methods, and the techniques of discriminating the factors of the situation. The terms pertinent to it are "feeling", "judgement", "sense", "proportion", "balance", "appropriateness". It is a matter of art rather than science."
In 1973, Mintzberg found that senior managers typically deal with unpredictable situations so they strategize in ad hoc, flexible, dynamic, and implicit ways. He wrote, "The job breeds adaptive information-manipulators who prefer the live concrete situation. The manager works in an environment of stimulus-response, and he develops in his work a clear preference for live action."
In 1982, John Kotter studied the daily activities of 15 executives and concluded that they spent most of their time developing and working a network of relationships that provided general insights and specific details for strategic decisions. They tended to use "mental road maps" rather than systematic planning techniques.
Daniel Isenberg's 1984 study of senior managers found that their decisions were highly intuitive. Executives often sensed what they were going to do before they could explain why. He claimed in 1986 that one of the reasons for this is the complexity of strategic decisions and the resultant information uncertainty.
Zuboff claimed that information technology was widening the divide between senior managers (who typically make strategic decisions) and operational level managers (who typically make routine decisions). She alleged that prior to the widespread use of computer systems, managers, even at the most senior level, engaged in both strategic decisions and routine administration, but as computers facilitated (She called it "deskilled") routine processes, these activities were moved further down the hierarchy, leaving senior management free for strategic decision making.
In 1977, Abraham Zaleznik distinguished leaders from managers. He described leaders as visionaries who inspire, while managers care about process. He claimed that the rise of managers was the main cause of the decline of American business in the 1970s and 1980s. Lack of leadership is most damaging at the level of strategic management where it can paralyze an entire organization.
According to Corner, Kinichi, and Keats, strategic decision making in organizations occurs at two levels: individual and aggregate. They developed a model of parallel strategic decision making. The model identifies two parallel processes that involve getting attention, encoding information, storage and retrieval of information, strategic choice, strategic outcome and feedback. The individual and organizational processes interact at each stage. For instance, competition-oriented objectives are based on the knowledge of competing firms, such as their market share.
=== Strategy as marketing ===
The 1980s also saw the widespread acceptance of positioning theory. Although the theory originated with Jack Trout in 1969, it didn't gain wide acceptance until Al Ries and Jack Trout wrote their classic book Positioning: The Battle For Your Mind (1979). The basic premise is that a strategy should not be judged by internal company factors but by the way customers see it relative to the competition. Crafting and implementing a strategy involves creating a position in the mind of the collective consumer. Several techniques enabled the practical use of positioning theory. Perceptual mapping for example, creates visual displays of the relationships between positions. Multidimensional scaling, discriminant analysis, factor analysis and conjoint analysis are mathematical techniques used to determine the most relevant characteristics (called dimensions or factors) upon which positions should be based. Preference regression can be used to determine vectors of ideal positions and cluster analysis can identify clusters of positions.
In 1992 Jay Barney saw strategy as assembling the optimum mix of resources, including human, technology and suppliers, and then configuring them in unique and sustainable ways.
James Gilmore and Joseph Pine found competitive advantage in mass customization. Flexible manufacturing techniques allowed businesses to individualize products for each customer without losing economies of scale. This effectively turned the product into a service. They also realized that if a service is mass-customized by creating a "performance" for each individual client, that service would be transformed into an "experience". Their book, The Experience Economy, along with the work of Bernd Schmitt, convinced many to see service provision as a form of theatre. This school of thought is sometimes referred to as customer experience management (CEM).
=== Information- and technology-driven strategy ===
Many industries with a high information component are being transformed. For example, Encarta demolished Encyclopædia Britannica (whose sales have plummeted 80% since their peak of $650 million in 1990) before it was, in turn, eclipsed by collaborative encyclopedias like Wikipedia. The music industry was similarly disrupted. The technology sector has provided some strategies directly. For example, from the software development industry agile software development provides a model for shared development processes.
Peter Drucker conceived of the "knowledge worker" in the 1950s. He described how fewer workers would do physical labor, and more would apply their minds. In 1984, John Naisbitt theorized that the future would be driven largely by information: companies that managed information well could obtain an advantage, however the profitability of what he called "information float" (information that the company had and others desired) would disappear as inexpensive computers made information more accessible.
Daniel Bell (1985) examined the sociological consequences of information technology, while Gloria Schuck and Shoshana Zuboff looked at psychological factors. Zuboff distinguished between "automating technologies" and "informating technologies". She studied the effect that both had on workers, managers and organizational structures. She largely confirmed Drucker's predictions about the importance of flexible decentralized structure, work teams, knowledge sharing and the knowledge worker's central role. Zuboff also detected a new basis for managerial authority, based on knowledge (also predicted by Drucker) which she called "participative management".
=== Regulatory strategy ===
An organisation's regulatory strategy accounts for how the organisation will respond to its regulatory bodies and standards as a feature of its operating environment, for example for businesses in the financial services, health care or energy industries. Beardsley et al., for example, refer to companies who are fatalistic or confrontational in their approach to being regulated. They recommend instead that the regulatory aspects of the business environment need to be integrated into the wider aspects of strategic planning and a coordinated approach taken in dialogue with regulators.
The term "regulatory strategy" is also used by regulators and legislators to define the aims and processes through which they will undertake their regulatory functions.
=== Maturity of planning process ===
McKinsey & Company developed a capability maturity model in the 1970s to describe the sophistication of planning processes, with strategic management ranked the highest. The four stages include:
Financial planning, which is primarily about annual budgets and a functional focus, with limited regard for the environment;
Forecast-based planning, which includes multi-year budgets and more robust capital allocation across business units;
Externally oriented planning, where a thorough situation analysis and competitive assessment is performed;
Strategic management, where widespread strategic thinking occurs and a well-defined strategic framework is used.
=== PIMS study ===
The long-term PIMS study, started in the 1960s and lasting for 19 years, attempted to understand the Profit Impact of Marketing Strategies (PIMS), particularly the effect of market share. The initial conclusion of the study was unambiguous: the greater a company's market share, the greater their rate of profit. Market share provides economies of scale. It also provides experience curve advantages. The combined effect is increased profits.
The benefits of high market share naturally led to an interest in growth strategies. The relative advantages of horizontal integration, vertical integration, diversification, franchises, mergers and acquisitions, joint ventures and organic growth were discussed. Other research indicated that a low market share strategy could still be very profitable. Schumacher (1973), Woo and Cooper (1982), Levenson (1984), and later Traverso (2002) showed how smaller niche players obtained very high returns.
== Other influences on business strategy ==
=== Military strategy ===
In the 1980s business strategists realized that there was a vast knowledge base stretching back thousands of years that they had barely examined. They turned to military strategy for guidance. Military strategy books such as The Art of War by Sun Tzu, On War by von Clausewitz, and The Red Book by Mao Zedong became business classics. From Sun Tzu, they learned the tactical side of military strategy and specific tactical prescriptions. From von Clausewitz, they learned the dynamic and unpredictable nature of military action. From Mao, they learned the principles of guerrilla warfare. Important marketing warfare books include Business War Games by Barrie James, Marketing Warfare by Al Ries and Jack Trout and Leadership Secrets of Attila the Hun by Wess Roberts. The marketing warfare literature also examined leadership and motivation, intelligence gathering, types of marketing weapons, logistics and communications.
By the twenty-first century marketing warfare strategies had gone out of favour in favor of non-confrontational approaches. In 1989, Dudley Lynch and Paul L. Kordis published Strategy of the Dolphin: Scoring a Win in a Chaotic World. "The Strategy of the Dolphin" was developed to give guidance as to when to use aggressive strategies and when to use passive strategies. A variety of aggressive strategies were developed.
In 1993, J. Moore used a similar metaphor. Instead of using military terms, he created an ecological theory of predators and prey(see ecological model of competition), a sort of Darwinian management strategy in which market interactions mimic long term ecological stability.
Author Phillip Evans said in 2014 that "Henderson's central idea was what you might call the Napoleonic idea of concentrating mass against weakness, of overwhelming the enemy. What Henderson recognized was that, in the business world, there are many phenomena which are characterized by what economists would call increasing returns—scale, experience. The more you do of something, disproportionately the better you get. And therefore he found a logic for investing in such kinds of overwhelming mass in order to achieve competitive advantage. And that was the first introduction of essentially a military concept of strategy into the business world. ... It was on those two ideas, Henderson's idea of increasing returns to scale and experience, and Porter's idea of the value chain, encompassing heterogenous elements, that the whole edifice of business strategy was subsequently erected."
== Traits of successful companies ==
Like Peters and Waterman a decade earlier, James Collins and Jerry Porras spent years conducting empirical research on what makes great companies. Six years of research uncovered a key underlying principle behind the 19 successful companies that they studied: They all encourage and preserve a core ideology that nurtures the company. Even though strategy and tactics change daily, the companies, nevertheless, were able to maintain a core set of values. These core values encourage employees to build an organization that lasts. In Built To Last (1994) they claim that short term profit goals, cost cutting, and restructuring will not stimulate dedicated employees to build a great company that will endure. In 2000 Collins coined the term "built to flip" to describe the prevailing business attitudes in Silicon Valley. It describes a business culture where technological change inhibits a long term focus. He also popularized the concept of the BHAG (Big Hairy Audacious Goal).
Arie de Geus (1997) undertook a similar study and obtained similar results. He identified four key traits of companies that had prospered for 50 years or more. They are:
Sensitivity to the business environment – the ability to learn and adjust
Cohesion and identity – the ability to build a community with personality, vision, and purpose
Tolerance and decentralization – the ability to build relationships
Conservative financing
A company with these key characteristics he called a living company because it is able to perpetuate itself. If a company emphasizes knowledge rather than finance, and sees itself as an ongoing community of human beings, it has the potential to become great and endure for decades. Such an organization is an organic entity capable of learning (he called it a "learning organization") and capable of creating its own processes, goals, and persona.
Will Mulcaster suggests that firms engage in a dialogue that centres around these questions:
Will the proposed competitive advantage create Perceived Differential Value?"
Will the proposed competitive advantage create something that is different from the competition?"
Will the difference add value in the eyes of potential customers?" – This question will entail a discussion of the combined effects of price, product features and consumer perceptions.
Will the product add value for the firm?" – Answering this question will require an examination of cost effectiveness and the pricing strategy.
== See also ==
== References ==
=== Further reading ===
Cameron, Bobby Thomas. (2014). Using responsive evaluation in Strategic Management.Strategic Leadership Review 4 (2), 22–27.
David Besanko, David Dranove, Scott Schaefer, and Mark Shanley (2012) Economics of Strategy, John Wiley & Sons, ISBN 978-1118273630
Edwards, Janice et al. Mastering Strategic Management- 1st Canadian Edition. BC Open Textbooks, 2014.
Kemp, Roger L. "Strategic Planning for Local Government: A Handbook for Officials and Citizens," McFarland and Co., Inc., Jefferson, NC, USA, and London, England, UK, 2008 (ISBN 978-0-7864-3873-0)
Kvint, Vladimir (2009) The Global Emerging Market: Strategic Management and Economics Excerpt from Google Books
Pankaj Ghemawhat - Harvard Strategy Professor: Competition and Business Strategy in Historical Perspective Social Science History Network-Spring 2002
== External links ==
Media related to Strategic management at Wikimedia Commons
Institute for Strategy and Competitiveness at Harvard Business School – recent publications
The Journal of Business Strategies – online library | Wikipedia/Strategic_management |
Prim–Read theory, or Prim–Read defense, was an important development in game theory that led to radical changes in the United States' views on the value of anti-ballistic missile (ABM) systems. The theory assigns a certain cost to deploying defensive missiles and suggests a way to maximize their value in terms of the amount of damage they could reduce. By comparing the cost of various deployments, one can determine the relative amount of money needed to provide a defense against a certain number of ICBMs.
The theory was first introduced in the late 1950s and might have been lost to history had it not been picked up during the debate on the Nike-X ABM. Nike-X called for the deployment of a heavy defensive system around major US cities with the intent of seriously blunting the effect of any Soviet strike. A number of operations researchers, notably US Air Force General Glenn Kent, used Prim–Read to conclusively demonstrate that the cost of reducing damage back to a given level was always more than the cost of causing additional damage by building more ICBMs.
The outcome of these studies suggested that any US deployment of an ABM system would result in the USSR building a small number of additional missiles to defeat it. Assuming the Soviets would come to the same conclusion, Robert McNamara became highly critical of any large-scale ABM system, and began efforts that would ultimately lead to the ABM treaty in 1972. The underlying concept became known as the cost-exchange ratio.
== History ==
=== Nike Zeus ===
The US Army began studying the anti-ballistic missile in a serious fashion in 1955. Working with Bell Labs, who had delivered the successful Nike and Nike B surface-to-air missiles (SAM), they began by considering what was essentially a direct update of the Nike concepts to the ABM mission. Bell returned a report suggesting that minor upgrades to the Hercules missile, along with much more powerful radars and computers, would do the trick. This was initially known as Nike II, but renamed Nike Zeus in 1956.
Early in the Zeus effort the US Air Force attempted to derail the project by pointing out that if Zeus cost the same as an ICBM, and the Soviets were building them as quickly as Nikita Khrushchev claimed, then they could simply build a few more to "soak up" any Zeus' the Army deployed. But in fact, it seemed the ICBMs were actually cheaper than Zeus, perhaps significantly, which meant the US would lose the resulting arms race. This basic concept became known as the cost exchange ratio.
President Eisenhower's Secretary of Defense Neil McElroy identified the Air Force complaints as an example of sour grapes, having lost funding for their own ABM efforts, Project Wizard, in favor of Zeus. But the math appeared to be correct, so he asked for a second opinion from the President's Science Advisory Committee (PSAC). They largely agreed with the Air Force's take, and then added several additional concerns of their own.
By the late 1950s, several new problems became evident. One was that the newly discovered nuclear blackout effect would allow an enemy to blanket an area hundreds of miles wide with a radar-opaque layer for the cost of one warhead. This would render Zeus blind to anything above the layer; following warheads would not become visible until too close to the base to attack. Another issue was the addition of decoys to the ICBMs, which presented radar targets that looked the same as the warhead. These cleared away due to drag as they reentered the atmosphere, but once again, this occurred at too low an altitude to attack.
=== Nike-X ===
At the suggestion of ARPA, the Army responded by redesigning the system as Nike-X. Nike-X used a short-range but extremely high-speed interceptor known as Sprint that was optimized for interceptions under 60 kilometres (200,000 ft) and combined that with an extremely high-speed radar and computer system. The plan was to wait until the warhead cleared any blackout and the decoys were slowing, allowing the radar to pick out the warhead and attack it with the Sprint. The entire engagement would last only a few seconds.
The Army produced a study that considered a real deployment scenario and then estimated the number of lives it would save. They started by assuming that the Soviets would want to launch two warheads at every target, to ensure at least one would go off. In order to confuse the defense, they would add nine credible decoys to each ICBM. This would present each base with 20 radar targets in total. For the same redundancy reasons, they would launch two Sprint missiles at each one, so a total of 40 Sprints would be needed to protect every target. Given the relative costs of the Sprint and an ICBM, the Army demonstrated that the Sprint system would save a considerable number of civilian lives for less than the cost of an ICBM.
=== Last-move problem ===
When this was presented as a part of a PSAC study of the Nike-X system, one member of the group immediately noted a problem. Air Force Brigadier General Glenn Kent had been taught to always consider who had the last move in any plan, and in this case, he concluded that the Soviets had that advantage. Facing a Nike-X deployment, they could change their ICBM targets without the US having any idea what those were.
For instance, one response would be to ignore the defended targets entirely, and use the missiles to attack the next cities on their target list. Since those targets would be smaller, they could be assigned one missile each. Although this would increase the number of targets that were not destroyed due to failures, the total number of targets hit would be greater.
Another solution would be to ignore targets further down the list and reassign those warheads to ones further up the list, ensuring the defenses at those targets would be overwhelmed through sheer numbers. Although the targets further down the list would no longer be attacked, they had smaller populations so their value was less.
In either case, the attacker could once again cause enormous damage without spending a single extra dollar on the attack. Worse, the US has no idea which strategy the Soviets picked, and therefore have no idea how to respond. The question, then, was how does one plan a defensive layout when there is no clear answer what the enemy's response will be?
When Kent pointed this issue out to Director of Defense Research and Engineering (DDR&E) Harold Brown, Brown immediately grasped the problem and recalled the Army group to explain why their analysis was essentially useless. He then tasked Kent with coming up with a way to analyze the problem that would not be dependent on knowing the Soviet attack allocations.
=== Prim–Read ===
Kent learned that two researchers at Bell Labs had considered this exact issue in a 1957 paper. Robert Prim and Thornton Read solved the problem by developing a simple mathematical formula that maximized the damage reduction in terms of any given expenditure on the defense. Prim visited Kent at the Pentagon to explain the idea, which was extremely simple in conceptual terms.
The basic idea was a reflection of the targeting priorities the Soviets would use. Against "soft targets" like cities, a single warhead will effectively destroy it, so launching additional warheads at the same target will not cause a corresponding doubling of damage inflicted. However, the missiles have a certain probability of successfully reaching the target and detonating, the probability of kill, or Pk. If the Pk is 50%, for instance, the Soviets will want to launch more than one ICBM at a target to increase the chances of destroying it. Two warheads improve this to 75%, and three to 87.5%, but in that case, if the first one does work the following two are wasted. They have to balance the desire to guarantee destruction of certain targets with the knowledge that other targets would then be skipped entirely.
The Prim–Read concept used the same basic logic but applied it to the chance of successfully destroying an enemy missile. For instance, if a city is expecting to be attacked by two warheads, then its chance of being destroyed is 75%. Assigning a single interceptor to defend that city means one of the two warheads will be shot down 50% of the time. This means the chance of not getting hit is now 50%, a 25% improvement. Critically, adding a second interceptor means a 50% of hitting either, a 75% chance of hitting both. The chance you hit the one that would go off is 50-50, so now the chance the target does not get hit is 62.5%. Thus adding the second interceptor only improves the survival rate by 12.5%.
The key point here is that instead of applying the second interceptor to improve the survival rate of that target 12.5%, it might be better to instead put that interceptor over some other target that formerly had no protection, and improve its survival rate by 50%. Of course, this requires one to put a value of some sort on the targets so one can calculate if 50% of one target is worth more than 12.5% of another.
Consider a real-world example in which New York is considered to have twice the "value" of Los Angeles. In this case, a naive arrangement would be to assign twice as many interceptors to New York. However, due to the Pk considerations, this does not provide twice the defensive capability, but a fractional addition. In the case of large numbers of interceptors and enemy warheads, additional missiles may provide only a tiny benefit. In contrast, assigning those missiles to Los Angeles may dramatically improve its survival if it otherwise had only a few. Improving Los Angeles' survival by 25% is likely "better" than improving New York's by 12.5%.
The paper goes on to explain how to arrange the overall deployment. Each target is assigned a worth, W, and the price of the defense assigned to protect it is P. The ratio of W to P is λ. If one were to assign a single missile to all potential targets, then the list of resulting λ values would mirror the W values.
If λ is less than 1, that means the cost of defense is more than the worth of the target. In this case, that target's missile is much better off being assigned to another target, the one with the highest λ. When that happens, the λ of that target drops because more P is being spent on it. As a result, another target becomes the highest on the list of λ. One then continues this process of reassigning missiles until the resulting list of targets that are protected have the same λ, or as close to that as possible. λ, in effect, represents the damage percent you are willing to accept.
One can make real-world calculations by selecting the population of the urban area to be a proxy for W. In this case New York has the highest W and initial λ, and it is naturally assigned the largest number of interceptors. One might be inclined to move a missile from Los Angeles to New York to offer higher protection, but the brilliance of Prim–Read is that demonstrates that while doing so would improve New York's survival rate a tiny bit, it would lower Los Angeles' even more.
One outcome of the Prim–Read deployment is that it is based entirely on the number of ABMs constructed and the total worth of the targets they protect. It does not matter what the Soviet response to the deployment is; if they choose to reduce the number of missiles assigned to one target to ensure they penetrate the defenses of another, that will always increase the overall survival rate of the defenders. It is possible for the Soviets to overwhelm the entire system, but even in that case the Prim–Read deployment will reduce whatever damage will be caused by the maximal amount possible.
=== Prim–Read becomes notorious ===
With Prim–Read, one can construct a mathematically maximal defense for any given expenditure. Because that defense is probabilistic, it means that it assumes some damage even when the defense is overwhelming, and at the same time it means there will be some reduction in damage even if the attack is overwhelming. The question then becomes whether or not the amount of damage reduction desired can be achieved for a reasonable total expenditure, given various estimates of the Soviet fleet.
Kent began developing Prim–Read deployments of various numbers of ABMs to determine their effectiveness against various numbers of ICBMs. The results were clear. Limited amounts of protection could be offered with small expenditures even if the Soviets built huge numbers of ICBMs; by pure chance, some of the targets would not be hit and ABMs would improve those numbers. The opposite was also true; if the US built an enormous fleet of ABMs, some enemy warheads would still hit their targets purely by chance.
If one wanted to save 90% of the US population, one required huge numbers of ABMs, and the relative cost of the defense compared to the offense was about 1.7 times. In other words, if the Soviets spent $10 billion producing ICBMs in a given year, the US would have to spend $17 billion on ABMs. However, when they found the official exchange rate between the US Dollar and Ruble was a fiction, and the actual value was very different, the ratio inflated to 6-to-1. In this sort of regime, the USSR could easily afford to build enough missiles to overwhelm any defense the US could afford.
Kent presented his results to Brown, who began to have serious questions about any sort of active defense. While this had no immediate effect on Nike-X planning, this was all taking place while another group was forming to consider the entire issue of the nuclear age under the direction of Frank Trinkl, part of Alain Enthoven's group at RAND. Kent was put into the group and noted that of the twenty items they had been tasked to consider, eight of those were purely defensive and he suggested grouping them together under the topic of damage limitation. Trinkl disagreed, and when Kent continued to pester him about it, Trinkl fired him from the group.
Brown then tasked Kent with going ahead and considering the eight issues in his own report, and this time assigned members of the Joint Chiefs of Staff to help. The report, on the topic of "damage limitation", immediately caught the eye of Robert McNamara who "bought it lock, stock, and barrel." McNamara put his feelings on the matter succinctly, stating to Kent that "At 70 percent surviving, you say 70 percent surviving, General, that sounds pretty good. Do you know what our detractors will say? 'Only 60 million dead.'"
From that point on, McNamara was against any sort of large scale Nike-X deployment, and the system was ultimately canceled. The basic concept, which became known as the cost-exchange ratio, ultimately ended any large-scale ABM deployment in the United States, and led directly to the 1972 ABM Treaty. This did not end well for Kent, who was blamed for this situation, with one detractor stating "There’s the man that was the genesis of the ABM Treaty, the worst of our greatest strategic disasters, the ABM Treaty of 1972."
== References ==
=== Citations ===
=== Bibliography ===
Baucom, Donald (1992). The Origins of SDI, 1944–1983. Lawrence, Kansas: University Press of Kansas. ISBN 978-0-7006-0531-6. OCLC 25317621.
Bexfield, Jim (17 February 2012). "Glenn A. Kent Interview" (PDF). Military Operations Research Society (MORS) Oral History Interview.
Kaplan, Fred (2008). Daydream Believers: How a Few Grand Ideas Wrecked American Power. John Wiley & Sons. ISBN 9780470121184.
Kent, Glenn (2008). Thinking about America's Defense: An Analytical Memoir. Rand Corporation. ISBN 9780833044525.
=== Further reading ===
Aumann, Robert; Hart, Sergiu (1994). Handbook of Game Theory with Economic Applications, Volume 2. Elsevier. ISBN 9780444894274. | Wikipedia/Prim–Read_theory |
The Hamilton-Jacobi-Bellman (HJB) equation is a nonlinear partial differential equation that provides necessary and sufficient conditions for optimality of a control with respect to a loss function. Its solution is the value function of the optimal control problem which, once known, can be used to obtain the optimal control by taking the maximizer (or minimizer) of the Hamiltonian involved in the HJB equation.
The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The connection to the Hamilton–Jacobi equation from classical physics was first drawn by Rudolf Kálmán. In discrete-time problems, the analogous difference equation is usually referred to as the Bellman equation.
While classical variational problems, such as the brachistochrone problem, can be solved using the Hamilton–Jacobi–Bellman equation, the method can be applied to a broader spectrum of problems. Further it can be generalized to stochastic systems, in which case the HJB equation is a second-order elliptic partial differential equation. A major drawback, however, is that the HJB equation admits classical solutions only for a sufficiently smooth value function, which is not guaranteed in most situations. Instead, the notion of a viscosity solution is required, in which conventional derivatives are replaced by (set-valued) subderivatives.
== Optimal Control Problems ==
Consider the following problem in deterministic optimal control over the time period
[
0
,
T
]
{\displaystyle [0,T]}
:
V
(
x
(
0
)
,
0
)
=
min
u
{
∫
0
T
C
[
x
(
t
)
,
u
(
t
)
]
d
t
+
D
[
x
(
T
)
]
}
{\displaystyle V(x(0),0)=\min _{u}\left\{\int _{0}^{T}C[x(t),u(t)]\,dt+D[x(T)]\right\}}
where
C
[
⋅
]
{\displaystyle C[\cdot ]}
is the scalar cost rate function and
D
[
⋅
]
{\displaystyle D[\cdot ]}
is a function that gives the bequest value at the final state,
x
(
t
)
{\displaystyle x(t)}
is the system state vector,
x
(
0
)
{\displaystyle x(0)}
is assumed given, and
u
(
t
)
{\displaystyle u(t)}
for
0
≤
t
≤
T
{\displaystyle 0\leq t\leq T}
is the control vector that we are trying to find. Thus,
V
(
x
,
t
)
{\displaystyle V(x,t)}
is the value function.
The system must also be subject to
x
˙
(
t
)
=
F
[
x
(
t
)
,
u
(
t
)
]
{\displaystyle {\dot {x}}(t)=F[x(t),u(t)]\,}
where
F
[
⋅
]
{\displaystyle F[\cdot ]}
gives the vector determining physical evolution of the state vector over time.
== The Partial Differential Equation ==
For this simple system, the Hamilton–Jacobi–Bellman partial differential equation is
∂
V
(
x
,
t
)
∂
t
+
min
u
{
∂
V
(
x
,
t
)
∂
x
⋅
F
(
x
,
u
)
+
C
(
x
,
u
)
}
=
0
{\displaystyle {\frac {\partial V(x,t)}{\partial t}}+\min _{u}\left\{{\frac {\partial V(x,t)}{\partial x}}\cdot F(x,u)+C(x,u)\right\}=0}
subject to the terminal condition
V
(
x
,
T
)
=
D
(
x
)
,
{\displaystyle V(x,T)=D(x),\,}
As before, the unknown scalar function
V
(
x
,
t
)
{\displaystyle V(x,t)}
in the above partial differential equation is the Bellman value function, which represents the cost incurred from starting in state
x
{\displaystyle x}
at time
t
{\displaystyle t}
and controlling the system optimally from then until time
T
{\displaystyle T}
.
== Deriving the Equation ==
Intuitively, the HJB equation can be derived as follows. If
V
(
x
(
t
)
,
t
)
{\displaystyle V(x(t),t)}
is the optimal cost-to-go function (also called the 'value function'), then by Richard Bellman's principle of optimality, going from time t to t + dt, we have
V
(
x
(
t
)
,
t
)
=
min
u
{
V
(
x
(
t
+
d
t
)
,
t
+
d
t
)
+
∫
t
t
+
d
t
C
(
x
(
s
)
,
u
(
s
)
)
d
s
}
.
{\displaystyle V(x(t),t)=\min _{u}\left\{V(x(t+dt),t+dt)+\int _{t}^{t+dt}C(x(s),u(s))\,ds\right\}.}
Note that the Taylor expansion of the first term on the right-hand side is
V
(
x
(
t
+
d
t
)
,
t
+
d
t
)
=
V
(
x
(
t
)
,
t
)
+
∂
V
(
x
,
t
)
∂
t
d
t
+
∂
V
(
x
,
t
)
∂
x
⋅
x
˙
(
t
)
d
t
+
o
(
d
t
)
,
{\displaystyle V(x(t+dt),t+dt)=V(x(t),t)+{\frac {\partial V(x,t)}{\partial t}}\,dt+{\frac {\partial V(x,t)}{\partial x}}\cdot {\dot {x}}(t)\,dt+{\mathcal {o}}(dt),}
where
o
(
d
t
)
{\displaystyle {\mathcal {o}}(dt)}
denotes the terms in the Taylor expansion of higher order than one in little-o notation. Then if we subtract
V
(
x
(
t
)
,
t
)
{\displaystyle V(x(t),t)}
from both sides, divide by dt, and take the limit as dt approaches zero, we obtain the HJB equation defined above.
== Solving the Equation ==
The HJB equation is usually solved backwards in time, starting from
t
=
T
{\displaystyle t=T}
and ending at
t
=
0
{\displaystyle t=0}
.
When solved over the whole of state space and
V
(
x
)
{\displaystyle V(x)}
is continuously differentiable, the HJB equation is a necessary and sufficient condition for an optimum when the terminal state is unconstrained. If we can solve for
V
{\displaystyle V}
then we can find from it a control
u
{\displaystyle u}
that achieves the minimum cost.
In general case, the HJB equation does not have a classical (smooth) solution. Several notions of generalized solutions have been developed to cover such situations, including viscosity solution (Pierre-Louis Lions and Michael Crandall), minimax solution (Andrei Izmailovich Subbotin), and others.
Approximate dynamic programming has been introduced by D. P. Bertsekas and J. N. Tsitsiklis with the use of artificial neural networks (multilayer perceptrons) for approximating the Bellman function in general. This is an effective mitigation strategy for reducing the impact of dimensionality by replacing the memorization of the complete function mapping for the whole space domain with the memorization of the sole neural network parameters. In particular, for continuous-time systems, an approximate dynamic programming approach that combines both policy iterations with neural networks was introduced. In discrete-time, an approach to solve the HJB equation combining value iterations and neural networks was introduced.
Alternatively, it has been shown that sum-of-squares optimization can yield an approximate polynomial solution to the Hamilton–Jacobi–Bellman equation arbitrarily well with respect to the
L
1
{\displaystyle L^{1}}
norm.
== Extension to Stochastic Problems ==
The idea of solving a control problem by applying Bellman's principle of optimality and then working out backwards in time an optimizing strategy can be generalized to stochastic control problems. Consider similar as above
min
u
E
{
∫
0
T
C
(
t
,
X
t
,
u
t
)
d
t
+
D
(
X
T
)
}
{\displaystyle \min _{u}\mathbb {E} \left\{\int _{0}^{T}C(t,X_{t},u_{t})\,dt+D(X_{T})\right\}}
now with
(
X
t
)
t
∈
[
0
,
T
]
{\displaystyle (X_{t})_{t\in [0,T]}\,\!}
the stochastic process to optimize and
(
u
t
)
t
∈
[
0
,
T
]
{\displaystyle (u_{t})_{t\in [0,T]}\,\!}
the steering. By first using Bellman and then expanding
V
(
X
t
,
t
)
{\displaystyle V(X_{t},t)}
with Itô's rule, one finds the stochastic HJB equation
min
u
{
A
V
(
x
,
t
)
+
C
(
t
,
x
,
u
)
}
=
0
,
{\displaystyle \min _{u}\left\{{\mathcal {A}}V(x,t)+C(t,x,u)\right\}=0,}
where
A
{\displaystyle {\mathcal {A}}}
represents the stochastic differentiation operator, and subject to the terminal condition
V
(
x
,
T
)
=
D
(
x
)
.
{\displaystyle V(x,T)=D(x)\,\!.}
Note that the randomness has disappeared. In this case a solution
V
{\displaystyle V\,\!}
of the latter does not necessarily solve the primal problem, it is a candidate only and a further verifying argument is required. This technique is widely used in Financial Mathematics to determine optimal investment strategies in the market (see for example Merton's portfolio problem).
=== Application to LQG-Control ===
As an example, we can look at a system with linear stochastic dynamics and quadratic cost. If the system dynamics is given by
d
x
t
=
(
a
x
t
+
b
u
t
)
d
t
+
σ
d
w
t
,
{\displaystyle dx_{t}=(ax_{t}+bu_{t})dt+\sigma dw_{t},}
and the cost accumulates at rate
C
(
x
t
,
u
t
)
=
r
(
t
)
u
t
2
/
2
+
q
(
t
)
x
t
2
/
2
{\displaystyle C(x_{t},u_{t})=r(t)u_{t}^{2}/2+q(t)x_{t}^{2}/2}
, the HJB equation is given by
−
∂
V
(
x
,
t
)
∂
t
=
1
2
q
(
t
)
x
2
+
∂
V
(
x
,
t
)
∂
x
a
x
−
b
2
2
r
(
t
)
(
∂
V
(
x
,
t
)
∂
x
)
2
+
σ
2
2
∂
2
V
(
x
,
t
)
∂
x
2
.
{\displaystyle -{\frac {\partial V(x,t)}{\partial t}}={\frac {1}{2}}q(t)x^{2}+{\frac {\partial V(x,t)}{\partial x}}ax-{\frac {b^{2}}{2r(t)}}\left({\frac {\partial V(x,t)}{\partial x}}\right)^{2}+{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}V(x,t)}{\partial x^{2}}}.}
with optimal action given by
u
t
=
−
b
r
(
t
)
∂
V
(
x
,
t
)
∂
x
{\displaystyle u_{t}=-{\frac {b}{r(t)}}{\frac {\partial V(x,t)}{\partial x}}}
Assuming a quadratic form for the value function, we obtain the usual Riccati equation for the Hessian of the value function as is usual for Linear-quadratic-Gaussian control.
== See also ==
Bellman equation, discrete-time counterpart of the Hamilton–Jacobi–Bellman equation.
Pontryagin's maximum principle, necessary but not sufficient condition for optimum, by maximizing a Hamiltonian, but this has the advantage over HJB of only needing to be satisfied over the single trajectory being considered.
== References ==
== Further reading ==
Bertsekas, Dimitri P. (2005). Dynamic Programming and Optimal Control. Athena Scientific.
Pham, Huyên (2009). "The Classical PDE Approach to Dynamic Programming". Continuous-time Stochastic Control and Optimization with Financial Applications. Springer. pp. 37–60. ISBN 978-3-540-89499-5.
Stengel, Robert F. (1994). "Conditions for Optimality". Optimal Control and Estimation. New York: Dover. pp. 201–222. ISBN 0-486-68200-5. | Wikipedia/Hamilton–Jacobi–Bellman_equation |
Combinatorial Games: Tic-Tac-Toe Theory is a monograph on the mathematics of tic-tac-toe and other positional games, written by József Beck. It was published in 2008 by the Cambridge University Press as volume 114 of their Encyclopedia of Mathematics and its Applications book series (ISBN 978-0-521-46100-9).
== Topics ==
A positional game is a game in which players alternate in taking possession of a given set of elements, with the goal of forming a winning configuration of elements; for instance, in tic-tac-toe and gomoku, the elements are the squares of a grid, and the winning configurations are lines of squares. These examples are symmetric: both players have the same winning configurations. However, positional games also include other possibilities such as the maker-breaker games in which one player (the "maker") tries to form a winning configuration and the other (the "breaker") tries to put off that outcome indefinitely or until the end of the game. In symmetric positional games one can use a strategy-stealing argument to prove that the first player has an advantage, but realizing this advantage by a constructive strategy can be very difficult.
According to the Hales–Jewett theorem, in tic-tac-toe-like games involving forming lines on a grid or higher-dimensional lattice, grids that are small relative to their dimension cannot lead to a drawn game: once the whole grid is partitioned between the two players, one of them will necessarily have a line. One of the main results of the book is that somewhat larger grids lead to a "weak win", a game in which one player can always force the formation of a line (not necessarily before the other player does), but that grid sizes beyond a certain threshold lead to a "strong draw", a game in which both players can prevent the other from forming a line. Moreover, the threshold between a weak win and a strong draw can often be determined precisely. The proof of this result uses a combination of the probabilistic method, to prove the existence of strategies for achieving the desired outcome, and derandomization, to make those strategies explicit.
The book is long (732 pages), organized into 49 chapters and four sections. Part A looks at the distinction between weak wins (the player can force the existence of a winning configuration) and strong wins (the winning configuration can be forced to exist before the other player gets a win). It shows that, for maker-breaker games over the points on the plane in which the players attempt to create a congruent copy of some finite point set, the maker always has a weak win, but to do so must sometimes allow the breaker to form a winning configuration earlier. It also includes an extensive analysis of tic-tac-toe-like symmetric line-forming games, and discusses the Erdős–Selfridge theorem according to which sparse-enough sets of winning configurations lead to drawn maker-breaker games. Part B of the book discusses the potential-based method by which the Erdős–Selfridge theorem was proven, and extends it to additional examples, including some in which the maker wins. Part C covers more advanced techniques of determining the outcome of a positional game, and introduces more complex games of this type, including picker-chooser games in which one player picks two unchosen elements and the other player chooses which one to give to each player. Part D includes the decomposition of games and the use of techniques from Ramsey theory to prove theorems about games. A collection of open problems in this area is provided at the end of the book.
== Audience and reception ==
This is a monograph, aimed at researchers in this area rather than at a popular audience. Reviewer William Gasarch writes that, although this work assumes little background knowledge of its readers, beyond low-level combinatorics and probability, "the material is still difficult". Similarly, reviewer Kyle Burke complains that "many definitions and explanations are awkwardly 'math heavy'; undefined terms from advanced mathematics abound in small examples, where simpler descriptions would suffice".
Much of the book concerns new research rather than merely summarizing what was previously known. Reviewer Ales Pultr calls this book "a most thorough and useful treatment of the subject (so far insufficiently presented in the literature), with an enormous store of results, links with other theories, and interesting open problems". Gasarch agrees: "Once you get through it you will have learned a great deal of mathematics." A pseudonymous reviewer for the European Mathematical Society adds that the book could be "a milestone in the development of combinatorial game theory".
== References == | Wikipedia/Combinatorial_Games:_Tic-Tac-Toe_Theory |
Cognitive science is the interdisciplinary, scientific study of the mind and its processes. It examines the nature, the tasks, and the functions of cognition (in a broad sense). Mental faculties of concern to cognitive scientists include perception, memory, attention, reasoning, language, and emotion. To understand these faculties, cognitive scientists borrow from fields such as psychology, economics, artificial intelligence, neuroscience, linguistics, and anthropology. The typical analysis of cognitive science spans many levels of organization, from learning and decision-making to logic and planning; from neural circuitry to modular brain organization. One of the fundamental concepts of cognitive science is that "thinking can best be understood in terms of representational structures in the mind and computational procedures that operate on those structures."
== History ==
The cognitive sciences began as an intellectual movement in the 1950s, called the cognitive revolution. Cognitive science has a prehistory traceable back to ancient Greek philosophical texts (see Plato's Meno and Aristotle's De Anima); Modern philosophers such as Descartes, David Hume, Immanuel Kant, Benedict de Spinoza, Nicolas Malebranche, Pierre Cabanis, Leibniz and John Locke, rejected scholasticism while mostly having never read Aristotle, and they were working with an entirely different set of tools and core concepts than those of the cognitive scientist.
The modern culture of cognitive science can be traced back to the early cyberneticists in the 1930s and 1940s, such as Warren McCulloch and Walter Pitts, who sought to understand the organizing principles of the mind. McCulloch and Pitts developed the first variants of what are now known as artificial neural networks, models of computation inspired by the structure of biological neural networks.
Another precursor was the early development of the theory of computation and the digital computer in the 1940s and 1950s. Kurt Gödel, Alonzo Church, Alan Turing, and John von Neumann were instrumental in these developments. The modern computer, or Von Neumann machine, would play a central role in cognitive science, both as a metaphor for the mind, and as a tool for investigation.
The first instance of cognitive science experiments being done at an academic institution took place at MIT Sloan School of Management, established by J.C.R. Licklider working within the psychology department and conducting experiments using computer memory as models for human cognition. In 1959, Noam Chomsky published a scathing review of B. F. Skinner's book Verbal Behavior. At the time, Skinner's behaviorist paradigm dominated the field of psychology within the United States. Most psychologists focused on functional relations between stimulus and response, without positing internal representations. Chomsky argued that in order to explain language, we needed a theory like generative grammar, which not only attributed internal representations but characterized their underlying order.
The term cognitive science was coined by Christopher Longuet-Higgins in his 1973 commentary on the Lighthill report, which concerned the then-current state of artificial intelligence research. In the same decade, the journal Cognitive Science and the Cognitive Science Society were founded. The founding meeting of the Cognitive Science Society was held at the University of California, San Diego in 1979, which resulted in cognitive science becoming an internationally visible enterprise. In 1972, Hampshire College started the first undergraduate education program in Cognitive Science, led by Neil Stillings. In 1982, with assistance from Professor Stillings, Vassar College became the first institution in the world to grant an undergraduate degree in Cognitive Science. In 1986, the first Cognitive Science Department in the world was founded at the University of California, San Diego.
In the 1970s and early 1980s, as access to computers increased, artificial intelligence research expanded. Researchers such as Marvin Minsky would write computer programs in languages such as LISP to attempt to formally characterize the steps that human beings went through, for instance, in making decisions and solving problems, in the hope of better understanding human thought, and also in the hope of creating artificial minds. This approach is known as "symbolic AI".
Eventually the limits of the symbolic AI research program became apparent. For instance, it seemed to be unrealistic to comprehensively list human knowledge in a form usable by a symbolic computer program. The late 80s and 90s saw the rise of neural networks and connectionism as a research paradigm. Under this point of view, often attributed to James McClelland and David Rumelhart, the mind could be characterized as a set of complex associations, represented as a layered network. Critics argue that there are some phenomena which are better captured by symbolic models, and that connectionist models are often so complex as to have little explanatory power. Recently symbolic and connectionist models have been combined, making it possible to take advantage of both forms of explanation. While both connectionism and symbolic approaches have proven useful for testing various hypotheses and exploring approaches to understanding aspects of cognition and lower level brain functions, neither are biologically realistic and therefore, both suffer from a lack of neuroscientific plausibility. Connectionism has proven useful for exploring computationally how cognition emerges in development and occurs in the human brain, and has provided alternatives to strictly domain-specific / domain general approaches. For example, scientists such as Jeff Elman, Liz Bates, and Annette Karmiloff-Smith have posited that networks in the brain emerge from the dynamic interaction between them and environmental input.
Recent developments in quantum computation, including the ability to run quantum circuits on quantum computers such as IBM Quantum Platform, has accelerated work using elements from quantum mechanics in cognitive models.
== Principles ==
=== Levels of analysis ===
A central tenet of cognitive science is that a complete understanding of the mind/brain cannot be attained by studying only a single level. An example would be the problem of remembering a phone number and recalling it later. One approach to understanding this process would be to study behavior through direct observation, or naturalistic observation. A person could be presented with a phone number and be asked to recall it after some delay of time; then the accuracy of the response could be measured. Another approach to measure cognitive ability would be to study the firings of individual neurons while a person is trying to remember the phone number. Neither of these experiments on its own would fully explain how the process of remembering a phone number works. Even if the technology to map out every neuron in the brain in real-time were available and it were known when each neuron fired it would still be impossible to know how a particular firing of neurons translates into the observed behavior. Thus an understanding of how these two levels relate to each other is imperative. Francisco Varela, in The Embodied Mind: Cognitive Science and Human Experience, argues that "the new sciences of the mind need to enlarge their horizon to encompass both lived human experience and the possibilities for transformation inherent in human experience". On the classic cognitivist view, this can be provided by a functional level account of the process. Studying a particular phenomenon from multiple levels creates a better understanding of the processes that occur in the brain to give rise to a particular behavior.
Marr gave a famous description of three levels of analysis:
The computational theory, specifying the goals of the computation;
Representation and algorithms, giving a representation of the inputs and outputs and the algorithms which transform one into the other; and
The hardware implementation, or how algorithm and representation may be physically realized.
=== Interdisciplinary nature ===
Cognitive science is an interdisciplinary field with contributors from various fields, including psychology, neuroscience, linguistics, philosophy of mind, computer science, anthropology and biology. Cognitive scientists work collectively in hope of understanding the mind and its interactions with the surrounding world much like other sciences do. The field regards itself as compatible with the physical sciences and uses the scientific method as well as simulation or modeling, often comparing the output of models with aspects of human cognition. Similarly to the field of psychology, there is some doubt whether there is a unified cognitive science, which have led some researchers to prefer 'cognitive sciences' in plural.
Many, but not all, who consider themselves cognitive scientists hold a functionalist view of the mind—the view that mental states and processes should be explained by their function – what they do. According to the multiple realizability account of functionalism, even non-human systems such as robots and computers can be ascribed as having cognition.
=== Cognitive science, the term ===
The term "cognitive" in "cognitive science" is used for "any kind of mental operation or structure that can be studied in precise terms" (Lakoff and Johnson, 1999). This conceptualization is very broad, and should not be confused with how "cognitive" is used in some traditions of analytic philosophy, where "cognitive" has to do only with formal rules and truth-conditional semantics.
The earliest entries for the word "cognitive" in the OED take it to mean roughly "pertaining to the action or process of knowing". The first entry, from 1586, shows the word was at one time used in the context of discussions of Platonic theories of knowledge. Most in cognitive science, however, presumably do not believe their field is the study of anything as certain as the knowledge sought by Plato.
== Scope ==
Cognitive science is a large field, and covers a wide array of topics on cognition. However, it should be recognized that cognitive science has not always been equally concerned with every topic that might bear relevance to the nature and operation of minds. Classical cognitivists have largely de-emphasized or avoided social and cultural factors, embodiment, emotion, consciousness, animal cognition, and comparative and evolutionary psychologies. However, with the decline of behaviorism, internal states such as affects and emotions, as well as awareness and covert attention became approachable again. For example, situated and embodied cognition theories take into account the current state of the environment as well as the role of the body in cognition. With the newfound emphasis on information processing, observable behavior was no longer the hallmark of psychological theory, but the modeling or recording of mental states.
Below are some of the main topics that cognitive science is concerned with; see List of cognitive science topics for a more exhaustive list.
=== Artificial intelligence ===
Artificial intelligence (AI) involves the study of cognitive phenomena in machines. One of the practical goals of AI is to implement aspects of human intelligence in computers. Computers are also widely used as a tool with which to study cognitive phenomena. Computational modeling uses simulations to study how human intelligence may be structured. (See § Computational modeling.)
There is some debate in the field as to whether the mind is best viewed as a huge array of small but individually feeble elements (i.e. neurons), or as a collection of higher-level structures such as symbols, schemes, plans, and rules. The former view uses connectionism to study the mind, whereas the latter emphasizes symbolic artificial intelligence. One way to view the issue is whether it is possible to accurately simulate a human brain on a computer without accurately simulating the neurons that make up the human brain.
=== Attention ===
Attention is the selection of important information. The human mind is bombarded with millions of stimuli and it must have a way of deciding which of this information to process. Attention is sometimes seen as a spotlight, meaning one can only shine the light on a particular set of information. Experiments that support this metaphor include the dichotic listening task (Cherry, 1957) and studies of inattentional blindness (Mack and Rock, 1998). In the dichotic listening task, subjects are bombarded with two different messages, one in each ear, and told to focus on only one of the messages. At the end of the experiment, when asked about the content of the unattended message, subjects cannot report it.
The psychological construct of attention is sometimes confused with the concept of intentionality due to some degree of semantic ambiguity in their definitions. At the beginning of experimental research on attention, Wilhelm Wundt defined this term as "that psychical process, which is operative in the clear perception of the narrow region of the content of consciousness." His experiments showed the limits of attention in space and time, which were 3-6 letters during an exposition of 1/10 s. Because this notion develops within the framework of the original meaning during a hundred years of research, the definition of attention would reflect the sense when it accounts for the main features initially attributed to this term – it is a process of controlling thought that continues over time. While intentionality is the power of minds to be about something, attention is the concentration of awareness on some phenomenon during a period of time, which is necessary to elevate the clear perception of the narrow region of the content of consciousness and which is feasible to control this focus in mind.
The significance of knowledge about the scope of attention for studying cognition is that it defines the intellectual functions of cognition such as apprehension, judgment, reasoning, and working memory. The development of attention scope increases the set of faculties responsible for the mind relies on how it perceives, remembers, considers, and evaluates in making decisions. The ground of this statement is that the more details (associated with an event) the mind may grasp for their comparison, association, and categorization, the closer apprehension, judgment, and reasoning of the event are in accord with reality. According to Latvian professor Sandra Mihailova and professor Igor Val Danilov, the more elements of the phenomenon (or phenomena ) the mind can keep in the scope of attention simultaneously, the more significant number of reasonable combinations within that event it can achieve, enhancing the probability of better understanding features and particularity of the phenomenon (phenomena). For example, three items in the focal point of consciousness yield six possible combinations (3 factorial) and four items – 24 (4 factorial) combinations. The number of reasonable combinations becomes significant in the case of a focal point with six items with 720 possible combinations (6 factorial).
=== Bodily processes related to cognition ===
Embodied cognition approaches to cognitive science emphasize the role of body and environment in cognition. This includes both neural and extra-neural bodily processes, and factors that range from affective and emotional processes, to posture, motor control, proprioception, and kinaesthesis, to autonomic processes that involve heartbeat and respiration, to the role of the enteric gut microbiome. It also includes accounts of how the body engages with or is coupled to social and physical environments. 4E (embodied, embedded, extended and enactive) cognition includes a broad range of views about brain-body-environment interaction, from causal embeddedness to stronger claims about how the mind extends to include tools and instruments, as well as the role of social interactions, action-oriented processes, and affordances. 4E theories range from those closer to classic cognitivism (so-called "weak" embodied cognition) to stronger extended and enactive versions that are sometimes referred to as radical embodied cognitive science.
A hypothesis of pre-perceptual multimodal integration supports embodied cognition approaches and converges two competing naturalist and constructivist viewpoints about cognition and the development of emotions. According to this hypothesis supported by empirical data, cognition and emotion development are initiated by the association of affective cues with stimuli responsible for triggering the neuronal pathways of simple reflexes. This pre-perceptual multimodal integration can succeed owing to neuronal coherence in mother-child dyads beginning from pregnancy. These cognitive-reflex and emotion-reflex stimuli conjunctions further form simple innate neuronal assemblies, shaping the cognitive and emotional neuronal patterns in statistical learning that are continuously connected with the neuronal pathways of reflexes.
=== Knowledge and processing of language ===
The ability to learn and understand language is an extremely complex process. Language is acquired within the first few years of life, and all humans under normal circumstances are able to acquire language proficiently. A major driving force in the theoretical linguistic field is discovering the nature that language must have in the abstract in order to be learned in such a fashion. Some of the driving research questions in studying how the brain itself processes language include: (1) To what extent is linguistic knowledge innate or learned?, (2) Why is it more difficult for adults to acquire a second-language than it is for infants to acquire their first-language?, and (3) How are humans able to understand novel sentences?
The study of language processing ranges from the investigation of the sound patterns of speech to the meaning of words and whole sentences. Linguistics often divides language processing into orthography, phonetics, phonology, morphology, syntax, semantics, and pragmatics. Many aspects of language can be studied from each of these components and from their interaction.
The study of language processing in cognitive science is closely tied to the field of linguistics. Linguistics was traditionally studied as a part of the humanities, including studies of history, art and literature. In the last fifty years or so, more and more researchers have studied knowledge and use of language as a cognitive phenomenon, the main problems being how knowledge of language can be acquired and used, and what precisely it consists of. Linguists have found that, while humans form sentences in ways apparently governed by very complex systems, they are remarkably unaware of the rules that govern their own speech. Thus linguists must resort to indirect methods to determine what those rules might be, if indeed rules as such exist. In any event, if speech is indeed governed by rules, they appear to be opaque to any conscious consideration.
=== Learning and development ===
Learning and development are the processes by which we acquire knowledge and information over time. Infants are born with little or no knowledge (depending on how knowledge is defined), yet they rapidly acquire the ability to use language, walk, and recognize people and objects. Research in learning and development aims to explain the mechanisms by which these processes might take place.
A major question in the study of cognitive development is the extent to which certain abilities are innate or learned. This is often framed in terms of the nature and nurture debate. The nativist view emphasizes that certain features are innate to an organism and are determined by its genetic endowment. The empiricist view, on the other hand, emphasizes that certain abilities are learned from the environment. Although clearly both genetic and environmental input is needed for a child to develop normally, considerable debate remains about how genetic information might guide cognitive development. In the area of language acquisition, for example, some (such as Steven Pinker) have argued that specific information containing universal grammatical rules must be contained in the genes, whereas others (such as Jeffrey Elman and colleagues in Rethinking Innateness) have argued that Pinker's claims are biologically unrealistic. They argue that genes determine the architecture of a learning system, but that specific "facts" about how grammar works can only be learned as a result of experience.
=== Memory ===
Memory allows us to store information for later retrieval. Memory is often thought of as consisting of both a long-term and short-term store. Long-term memory allows us to store information over prolonged periods (days, weeks, years). We do not yet know the practical limit of long-term memory capacity. Short-term memory allows us to store information over short time scales (seconds or minutes).
Memory is also often grouped into declarative and procedural forms. Declarative memory—grouped into subsets of semantic and episodic forms of memory—refers to our memory for facts and specific knowledge, specific meanings, and specific experiences (e.g. "Are apples food?", or "What did I eat for breakfast four days ago?"). Procedural memory allows us to remember actions and motor sequences (e.g. how to ride a bicycle) and is often dubbed implicit knowledge or memory .
Cognitive scientists study memory just as psychologists do, but tend to focus more on how memory bears on cognitive processes, and the interrelationship between cognition and memory. One example of this could be, what mental processes does a person go through to retrieve a long-lost memory? Or, what differentiates between the cognitive process of recognition (seeing hints of something before remembering it, or memory in context) and recall (retrieving a memory, as in "fill-in-the-blank")?
=== Perception and action ===
Perception is the ability to take in information via the senses, and process it in some way. Vision and hearing are two dominant senses that allow us to perceive the environment. Some questions in the study of visual perception, for example, include: (1) How are we able to recognize objects?, (2) Why do we perceive a continuous visual environment, even though we only see small bits of it at any one time? One tool for studying visual perception is by looking at how people process optical illusions. The image on the right of a Necker cube is an example of a bistable percept, that is, the cube can be interpreted as being oriented in two different directions.
The study of haptic (tactile), olfactory, and gustatory stimuli also fall into the domain of perception.
Action is taken to refer to the output of a system. In humans, this is accomplished through motor responses. Spatial planning and movement, speech production, and complex motor movements are all aspects of action.
=== Consciousness ===
== Research methods ==
Many different methodologies are used to study cognitive science. As the field is highly interdisciplinary, research often cuts across multiple areas of study, drawing on research methods from psychology, neuroscience, computer science and systems theory.
=== Behavioral experiments ===
In order to have a description of what constitutes intelligent behavior, one must study behavior itself. This type of research is closely tied to that in cognitive psychology and psychophysics. By measuring behavioral responses to different stimuli, one can understand something about how those stimuli are processed. Lewandowski & Strohmetz (2009) reviewed a collection of innovative uses of behavioral measurement in psychology including behavioral traces, behavioral observations, and behavioral choice. Behavioral traces are pieces of evidence that indicate behavior occurred, but the actor is not present (e.g., litter in a parking lot or readings on an electric meter). Behavioral observations involve the direct witnessing of the actor engaging in the behavior (e.g., watching how close a person sits next to another person). Behavioral choices are when a person selects between two or more options (e.g., voting behavior, choice of a punishment for another participant).
Reaction time. The time between the presentation of a stimulus and an appropriate response can indicate differences between two cognitive processes, and can indicate some things about their nature. For example, if in a search task the reaction times vary proportionally with the number of elements, then it is evident that this cognitive process of searching involves serial instead of parallel processing.
Psychophysical responses. Psychophysical experiments are an old psychological technique, which has been adopted by cognitive psychology. They typically involve making judgments of some physical property, e.g. the loudness of a sound. Correlation of subjective scales between individuals can show cognitive or sensory biases as compared to actual physical measurements. Some examples include:
sameness judgments for colors, tones, textures, etc.
threshold differences for colors, tones, textures, etc.
Eye tracking. This methodology is used to study a variety of cognitive processes, most notably visual perception and language processing. The fixation point of the eyes is linked to an individual's focus of attention. Thus, by monitoring eye movements, we can study what information is being processed at a given time. Eye tracking allows us to study cognitive processes on extremely short time scales. Eye movements reflect online decision making during a task, and they provide us with some insight into the ways in which those decisions may be processed.
=== Brain imaging ===
Brain imaging involves analyzing activity within the brain while performing various tasks. This allows us to link behavior and brain function to help understand how information is processed. Different types of imaging techniques vary in their temporal (time-based) and spatial (location-based) resolution. Brain imaging is often used in cognitive neuroscience.
Single-photon emission computed tomography and positron emission tomography. SPECT and PET use radioactive isotopes, which are injected into the subject's bloodstream and taken up by the brain. By observing which areas of the brain take up the radioactive isotope, we can see which areas of the brain are more active than other areas. PET has similar spatial resolution to fMRI, but it has extremely poor temporal resolution.
Electroencephalography. EEG measures the electrical fields generated by large populations of neurons in the cortex by placing a series of electrodes on the scalp of the subject. This technique has an extremely high temporal resolution, but a relatively poor spatial resolution.
Functional magnetic resonance imaging. fMRI measures the relative amount of oxygenated blood flowing to different parts of the brain. More oxygenated blood in a particular region is assumed to correlate with an increase in neural activity in that part of the brain. This allows us to localize particular functions within different brain regions. fMRI has moderate spatial and temporal resolution.
Optical imaging. This technique uses infrared transmitters and receivers to measure the amount of light reflectance by blood near different areas of the brain. Since oxygenated and deoxygenated blood reflects light by different amounts, we can study which areas are more active (i.e., those that have more oxygenated blood). Optical imaging has moderate temporal resolution, but poor spatial resolution. It also has the advantage that it is extremely safe and can be used to study infants' brains.
Magnetoencephalography. MEG measures magnetic fields resulting from cortical activity. It is similar to EEG, except that it has improved spatial resolution since the magnetic fields it measures are not as blurred or attenuated by the scalp, meninges and so forth as the electrical activity measured in EEG is. MEG uses SQUID sensors to detect tiny magnetic fields.
=== Computational modeling ===
Computational models require a mathematically and logically formal representation of a problem. Computer models are used in the simulation and experimental verification of different specific and general properties of intelligence. Computational modeling can help us understand the functional organization of a particular cognitive phenomenon.
Approaches to cognitive modeling can be categorized as: (1) symbolic, on abstract mental functions of an intelligent mind by means of symbols; (2) subsymbolic, on the neural and associative properties of the human brain; and (3) across the symbolic–subsymbolic border, including hybrid.
Symbolic modeling evolved from the computer science paradigms using the technologies of knowledge-based systems, as well as a philosophical perspective (e.g. "Good Old-Fashioned Artificial Intelligence" (GOFAI)). They were developed by the first cognitive researchers and later used in information engineering for expert systems. Since the early 1990s it was generalized in systemics for the investigation of functional human-like intelligence models, such as personoids, and, in parallel, developed as the SOAR environment. Recently, especially in the context of cognitive decision-making, symbolic cognitive modeling has been extended to the socio-cognitive approach, including social and organizational cognition, interrelated with a sub-symbolic non-conscious layer.
Subsymbolic modeling includes connectionist/neural network models. Connectionism relies on the idea that the mind/brain is composed of simple nodes and its problem-solving capacity derives from the connections between them. Neural nets are textbook implementations of this approach. Some critics of this approach feel that while these models approach biological reality as a representation of how the system works, these models lack explanatory powers because, even in systems endowed with simple connection rules, the emerging high complexity makes them less interpretable at the connection-level than they apparently are at the macroscopic level.
Other approaches gaining in popularity include (1) dynamical systems theory, (2) mapping symbolic models onto connectionist models (Neural-symbolic integration or hybrid intelligent systems), and (3) and Bayesian models, which are often drawn from machine learning.
All the above approaches tend either to be generalized to the form of integrated computational models of a synthetic/abstract intelligence (i.e. cognitive architecture) in order to be applied to the explanation and improvement of individual and social/organizational decision-making and reasoning or to focus on single simulative programs (or microtheories/"middle-range" theories) modelling specific cognitive faculties (e.g. vision, language, categorization etc.).
=== Neurobiological methods ===
Research methods borrowed directly from neuroscience and neuropsychology can also help us to understand aspects of intelligence. These methods allow us to understand how intelligent behavior is implemented in a physical system.
Single-unit recording
Direct brain stimulation
Animal models
Postmortem studies
== Key findings ==
Cognitive science has given rise to models of human cognitive bias and risk perception, and has been influential in the development of behavioral finance, part of economics. It has also given rise to a new theory of the philosophy of mathematics (related to denotational mathematics), and many theories of artificial intelligence, persuasion and coercion. It has made its presence known in the philosophy of language and epistemology as well as constituting a substantial wing of modern linguistics. Fields of cognitive science have been influential in understanding the brain's particular functional systems (and functional deficits) ranging from speech production to auditory processing and visual perception. It has made progress in understanding how damage to particular areas of the brain affect cognition, and it has helped to uncover the root causes and results of specific dysfunction, such as dyslexia, anopsia, and hemispatial neglect.
== Notable researchers ==
Some of the more recognized names in cognitive science are usually either the most controversial or the most cited. Within philosophy, some familiar names include Daniel Dennett, who writes from a computational systems perspective, John Searle, known for his controversial Chinese room argument, and Jerry Fodor, who advocates functionalism.
Others include David Chalmers, who advocates Dualism and is also known for articulating the hard problem of consciousness, and Douglas Hofstadter, famous for writing Gödel, Escher, Bach, which questions the nature of words and thought.
In the realm of linguistics, Noam Chomsky and George Lakoff have been influential (both have also become notable as political commentators). In artificial intelligence, Marvin Minsky, Herbert A. Simon, and Allen Newell are prominent.
Popular names in the discipline of psychology include George A. Miller, James McClelland, Philip Johnson-Laird, Lawrence Barsalou, Vittorio Guidano, Howard Gardner and Steven Pinker. Anthropologists Dan Sperber, Edwin Hutchins, Bradd Shore, James Wertsch and Scott Atran, have been involved in collaborative projects with cognitive and social psychologists, political scientists and evolutionary biologists in attempts to develop general theories of culture formation, religion, and political association.
Computational theories (with models and simulations) have also been developed, by David Rumelhart, James McClelland and Philip Johnson-Laird.
== Epistemics ==
Epistemics is a term coined in 1969 by the University of Edinburgh with the foundation of its School of Epistemics. Epistemics is to be distinguished from epistemology in that epistemology is the philosophical theory of knowledge, whereas epistemics signifies the scientific study of knowledge.
Christopher Longuet-Higgins has defined it as "the construction of formal models of the processes (perceptual, intellectual, and linguistic) by which knowledge and understanding are achieved and communicated."
In his 1978 essay "Epistemics: The Regulative Theory of Cognition", Alvin I. Goldman claims to have coined the term "epistemics" to describe a reorientation of epistemology. Goldman maintains that his epistemics is continuous with traditional epistemology and the new term is only to avoid opposition. Epistemics, in Goldman's version, differs only slightly from traditional epistemology in its alliance with the psychology of cognition; epistemics stresses the detailed study of mental processes and information-processing mechanisms that lead to knowledge or beliefs.
In the mid-1980s, the School of Epistemics was renamed as The Centre for Cognitive Science (CCS). In 1998, CCS was incorporated into the University of Edinburgh's School of Informatics.
== Binding problem in cognitive science ==
One of the core aims of cognitive science is to achieve an integrated theory of cognition. This requires integrative mechanisms explaining how the information processing that occurs simultaneously in spatially segregated (sub-)cortical areas in the brain is coordinated and bound together to give rise to coherent perceptual and symbolic representations. One approach is to solve this "Binding problem" (that is, the problem of dynamically representing conjunctions of informational elements, from the most basic perceptual representations ("feature binding") to the most complex cognitive representations, like symbol structures ("variable binding")), by means of integrative synchronization mechanisms. In other words, one of the coordinating mechanisms appears to be the temporal (phase) synchronization of neural activity based on dynamical self-organizing processes in neural networks, described by the Binding-by-synchrony (BBS) Hypothesis from neurophysiology. Connectionist cognitive neuroarchitectures have been developed that use integrative synchronization mechanisms to solve this binding problem in perceptual cognition and in language cognition. In perceptual cognition the problem is to explain how elementary object properties and object relations, like the object color or the object form, can be dynamically bound together or can be integrated to a representation of this perceptual object by means of a synchronization mechanism ("feature binding", "feature linking"). In language cognition the problem is to explain how semantic concepts and syntactic roles can be dynamically bound together or can be integrated to complex cognitive representations like systematic and compositional symbol structures and propositions by means of a synchronization mechanism ("variable binding") (see also the "Symbolism vs. connectionism debate" in connectionism).
However, despite significant advances in understanding the integrated theory of cognition (specifically the Binding problem), the debate on this issue of beginning cognition is still in progress. From the different perspectives noted above, this problem can be reduced to the issue of how organisms at the simple reflexes stage of development overcome the threshold of the environmental chaos of sensory stimuli: electromagnetic waves, chemical interactions, and pressure fluctuations. The so-called Primary Data Entry (PDE) thesis poses doubts about the ability of such an organism to overcome this cue threshold on its own. In terms of mathematical tools, the PDE thesis underlines the insuperable high threshold of the cacophony of environmental stimuli (the stimuli noise) for young organisms at the onset of life. It argues that the temporal (phase) synchronization of neural activity based on dynamical self-organizing processes in neural networks, any dynamical bound together or integration to a representation of the perceptual object by means of a synchronization mechanism can not help organisms in distinguishing relevant cue (informative stimulus) for overcome this noise threshold.
== See also ==
Outlines
Outline of human intelligence – topic tree presenting the traits, capacities, models, and research fields of human intelligence, and more.
Outline of thought – topic tree that identifies many types of thoughts, types of thinking, aspects of thought, related fields, and more.
== References ==
== External links ==
Media related to Cognitive science at Wikimedia Commons
Quotations related to Cognitive science at Wikiquote
Learning materials related to Cognitive science at Wikiversity
"Cognitive Science" on the Stanford Encyclopedia of Philosophy
Cognitive Science Society
Cognitive Science Movie Index: A broad list of movies showcasing themes in the Cognitive Sciences Archived 4 September 2015 at the Wayback Machine
List of leading thinkers in cognitive science | Wikipedia/Cognitive_science |
Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. The use of the term "mean field" is inspired by mean-field theory in physics, which considers the behavior of systems of large numbers of particles where individual particles have negligible impacts upon the system. In other words, each agent acts according to his minimization or maximization problem taking into account other agents’ decisions and because their population is large we can assume the number of agents goes to infinity and a representative agent exists.
In traditional game theory, the subject of study is usually a game with two players and discrete time space, and extends the results to more complex situations by induction. However, for games in continuous time with continuous states (differential games or stochastic differential games) this strategy cannot be used because of the complexity that the dynamic interactions generate. On the other hand with MFGs we can handle large numbers of players through the mean representative agent and at the same time describe complex state dynamics.
This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal, in the engineering literature by Minyi Huang, Roland Malhame, and Peter E. Caines and independently and around the same time by mathematicians Jean-Michel Lasry and Pierre-Louis Lions.
In continuous time a mean-field game is typically composed of a Hamilton–Jacobi–Bellman equation that describes the optimal control problem of an individual and a Fokker–Planck equation that describes the dynamics of the aggregate distribution of agents. Under fairly general assumptions it can be proved that a class of mean-field games is the limit as
N
→
∞
{\displaystyle N\to \infty }
of an N-player Nash equilibrium.
A related concept to that of mean-field games is "mean-field-type control". In this case, a social planner controls the distribution of states and chooses a control strategy. The solution to a mean-field-type control problem can typically be expressed as a dual adjoint Hamilton–Jacobi–Bellman equation coupled with Kolmogorov equation. Mean-field-type game theory is the multi-agent generalization of the single-agent mean-field-type control.
== General Form of a Mean-field Game ==
The following system of equations can be used to model a typical Mean-field game:
{
−
∂
t
u
−
ν
Δ
u
+
H
(
x
,
m
,
D
u
)
=
0
(
1
)
∂
t
m
−
ν
Δ
m
−
div
(
D
p
H
(
x
,
m
,
D
u
)
m
)
=
0
(
2
)
m
(
0
)
=
m
0
(
3
)
u
(
x
,
T
)
=
G
(
x
,
m
(
T
)
)
(
4
)
{\displaystyle {\begin{cases}-\partial _{t}u-\nu \Delta u+H(x,m,Du)=0&(1)\\\partial _{t}m-\nu \Delta m-\operatorname {div} (D_{p}H(x,m,Du)m)=0&(2)\\m(0)=m_{0}&(3)\\u(x,T)=G(x,m(T))&(4)\end{cases}}}
The basic dynamics of this set of Equations can be explained by an average agent's optimal control problem. In a mean-field game, an average agent can control their movement
α
{\displaystyle \alpha }
to influence the population's overall location by:
d
X
t
=
α
t
d
t
+
2
ν
d
B
t
{\displaystyle dX_{t}=\alpha _{t}dt+{\sqrt {2\nu }}dB_{t}}
where
ν
{\displaystyle \nu }
is a parameter and
B
t
{\displaystyle B_{t}}
is a standard Brownian motion. By controlling their movement, the agent aims to minimize their overall expected cost
C
{\displaystyle C}
throughout the time period
[
0
,
T
]
{\displaystyle [0,T]}
:
C
=
E
[
∫
0
T
L
(
X
s
,
α
s
,
m
(
s
)
)
d
s
+
G
(
X
T
,
m
(
T
)
)
]
{\displaystyle C=\mathbb {E} \left[\int _{0}^{T}L(X_{s},\alpha _{s},m(s))ds+G(X_{T},m(T))\right]}
where
L
(
X
s
,
α
s
,
m
(
s
)
)
{\displaystyle L(X_{s},\alpha _{s},m(s))}
is the running cost at time
s
{\displaystyle s}
and
G
(
X
T
,
m
(
T
)
)
{\displaystyle G(X_{T},m(T))}
is the terminal cost at time
T
{\displaystyle T}
. By this definition, at time
t
{\displaystyle t}
and position
x
{\displaystyle x}
, the value function
u
(
t
,
x
)
{\displaystyle u(t,x)}
can be determined as:
u
(
t
,
x
)
=
inf
α
E
[
∫
t
T
L
(
X
s
,
α
s
,
m
(
s
)
)
d
s
+
G
(
X
T
,
m
(
T
)
)
]
{\displaystyle u(t,x)=\inf _{\alpha }\mathbb {E} \left[\int _{t}^{T}L(X_{s},\alpha _{s},m(s))ds+G(X_{T},m(T))\right]}
Given the definition of the value function
u
(
t
,
x
)
{\displaystyle u(t,x)}
, it can be tracked by the Hamilton-Jacobi equation (1). The optimal action of the average players
α
∗
(
x
,
t
)
{\displaystyle \alpha ^{*}(x,t)}
can be determined as
α
∗
(
x
,
t
)
=
D
p
H
(
x
,
m
,
D
u
)
{\displaystyle \alpha ^{*}(x,t)=D_{p}H(x,m,Du)}
. As all agents are relatively small and cannot single-handedly change the dynamics of the population, they will individually adapt the optimal control and the population would move in that way. This is similar to a Nash Equilibrium, in which all agents act in response to a specific set of others' strategies. The optimal control solution then leads to the Kolmogorov-Fokker-Planck equation (2).
== Finite State Games ==
A prominent category of mean field is games with a finite number of states and a finite number of actions per player. For those games, the analog of the Hamilton-Jacobi-Bellman equation is the Bellman equation, and the discrete version of the Fokker-Planck equation is the Kolmogorov equation. Specifically, for discrete-time models, the players' strategy is the Kolmogorov equation's probability matrix. In continuous time models, players have the ability to control the transition rate matrix.
A discrete mean field game can be defined by a tuple
G
=
(
E
,
A
,
{
Q
a
}
,
m
0
,
{
c
a
}
,
β
)
{\displaystyle {\mathcal {G}}=({\mathcal {E}},{\mathcal {A}},\{Q_{a}\},{\bf {m}}_{0},\{c_{a}\},\beta )}
, where
E
{\displaystyle {\mathcal {E}}}
is the state space,
A
{\displaystyle {\mathcal {A}}}
the action set,
Q
a
{\displaystyle Q_{a}}
the transition rate matrices,
m
0
{\displaystyle {\bf {m}}_{0}}
the initial state,
{
c
a
}
{\displaystyle \{c_{a}\}}
the cost functions and
β
{\displaystyle \beta }
∈
R
{\displaystyle \in \mathbb {R} }
a discount factor. Furthermore, a mixed strategy is a measurable function
π
:
E
×
R
+
→
P
(
A
)
{\displaystyle \pi :\mathbb {E} \times \mathbb {R} ^{+}{\xrightarrow[{}]{}}{\mathcal {P(A)}}}
, that associates to each state
i
∈
E
{\displaystyle i\in {\mathcal {E}}}
and each time
t
≥
0
{\displaystyle t\geq 0}
a probability measure
π
i
(
t
)
∈
P
(
A
)
{\displaystyle \pi _{i}(t)\in {\mathcal {P(A)}}}
on the set of possible actions. Thus
π
i
,
a
(
t
)
{\displaystyle \pi _{i,a}(t)}
is the probability that, at time
t
{\displaystyle t}
a player in state
i
{\displaystyle i}
takes action
a
{\displaystyle a}
, under strategy
π
{\displaystyle \pi }
. Additionally, rate matrices
{
Q
a
(
m
π
(
t
)
)
}
a
∈
A
{\displaystyle \{Q_{a}({\bf {m}}^{\pi }(t))\}_{a\in {\mathcal {A}}}}
define the evolution over the time of population distribution, where
m
π
(
t
)
∈
P
(
E
)
{\displaystyle {\bf {m}}^{\pi }(t)\in {\mathcal {P({\mathcal {E}})}}}
is the population distribution at time
t
{\displaystyle t}
.
== Linear-quadratic Gaussian game problem ==
From Caines (2009), a relatively simple model of large-scale games is the linear-quadratic Gaussian model. The individual agent's dynamics are modeled as a stochastic differential equation
d
X
i
=
(
a
i
X
i
+
b
i
u
i
)
d
t
+
σ
i
d
W
i
,
i
=
1
,
…
,
N
,
{\displaystyle dX_{i}=(a_{i}X_{i}+b_{i}u_{i})\,dt+\sigma _{i}\,dW_{i},\quad i=1,\dots ,N,}
where
X
i
{\displaystyle X_{i}}
is the state of the
i
{\displaystyle i}
-th agent,
u
i
{\displaystyle u_{i}}
is the control of the
i
{\displaystyle i}
-th agent, and
W
i
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{\displaystyle J_{i}(u_{i},\nu )=\mathbb {E} \left\{\int _{0}^{\infty }e^{-\rho t}\left[(X_{i}-\nu )^{2}+ru_{i}^{2}\right]\,dt\right\},\quad \nu =\Phi \left({\frac {1}{N}}\sum _{k\neq i}^{N}X_{k}+\eta \right).}
The coupling between agents occurs in the cost function.
== General and Applied Use ==
The paradigm of Mean Field Games has become a major connection between distributed decision-making and stochastic modeling. Starting out in the stochastic control literature, it is gaining rapid adoption across a range of applications, including:
a. Financial market
Carmona reviews applications in financial engineering and economics that can be cast and tackled within the framework of the MFG paradigm. Carmona argues that models in macroeconomics, contract theory, finance, …, greatly benefit from the switch to continuous time from the more traditional discrete-time models. He considers only continuous time models in his review chapter, including systemic risk, price impact, optimal execution, models for bank runs, high-frequency trading, and cryptocurrencies.
b. Crowd motions
MFG assumes that individuals are smart players which try to optimize their strategy and path with respect to certain costs (equilibrium with rational expectations approach). MFG models are useful to describe the anticipation phenomenon: the forward part describes the crowd evolution while the backward gives the process of how the anticipations are built. Additionally, compared to multi-agent microscopic model computations, MFG only requires lower computational costs for the macroscopic simulations. Some researchers have turned to MFG in order to model the interaction between populations and study the decision-making process of intelligent agents, including aversion and congestion behavior between two groups of pedestrians, departure time choice of morning commuters, and decision-making processes for autonomous vehicle.
c. Control and mitigation of Epidemics
Since the epidemic has affected society and individuals significantly, MFG and mean-field controls (MFCs) provide a perspective to study and understand the underlying population dynamics, especially in the context of the Covid-19 pandemic response. MFG has been used to extend the SIR-type dynamics with spatial effects or allowing for individuals to choose their behaviors and control their contributions to the spread of the disease. MFC is applied to design the optimal strategy to control the virus spreading within a spatial domain, control individuals’ decisions to limit their social interactions, and support the government’s nonpharmaceutical interventions.
== See also ==
== References ==
== External links ==
Mean Field Stochastic Control (Slides), 2009 IEEE Control Systems Society Bode Prize Lecture by Peter E. Caines
Caines, Peter E. (2013). "Mean Field Games". Encyclopedia of Systems and Control. pp. 1–6. doi:10.1007/978-1-4471-5102-9_30-1. ISBN 978-1-4471-5102-9. S2CID 33954904.
Notes on Mean Field Games, from Pierre-Louis Lions' lectures at Collège de France
(in French) Video lectures by Pierre-Louis Lions
Mean field games and applications by Olivier Guéant, Jean-Michel Lasry, and Pierre-Louis Lions | Wikipedia/Mean-field_game_theory |
In game theory, Zermelo's theorem is a theorem about finite two-person games of perfect information in which the players move alternately and in which chance does not affect the decision making process. It says that if the game cannot end in a draw, then one of the two players must have a winning strategy (i.e. can force a win). An alternate statement is that for a game meeting all of these conditions except the condition that a draw is now possible, then either the first-player can force a win, or the second-player can force a win, or both players can at least force a draw.
The theorem is named after Ernst Zermelo, a German mathematician and logician, who proved the theorem for the example game of chess in 1913.
== Example ==
Zermelo's theorem can be applied to all finite-stage two-player games with complete information and alternating moves. The game must satisfy the following criteria: there are two players in the game; the game is of perfect information; the board game is finite; the two players can take alternate turns; and there is no chance element present. Zermelo has stated that there are many games of this type; however his theorem has been applied mostly to the game chess.
When applied to chess, Zermelo's theorem states "either White can force a win, or Black can force a win, or both sides can force at least a draw".
Zermelo's algorithm is a cornerstone algorithm in game-theory; however, it can also be applied in areas outside of finite games.
Apart from chess, Zermelo's theorem is applied across all areas of computer science. In particular, it is applied in model checking and value interaction.
== Conclusions of Zermelo's theorem ==
Zermelo's work shows that in two-person zero-sum games with perfect information, if a player is in a winning position, then that player can always force a win no matter what strategy the other player may employ. Furthermore, and as a consequence, if a player is in a winning position, it will never require more moves than there are positions in the game (with a position defined as position of pieces as well as the player next to move).
== Publication history ==
In 1912, during the Fifth International Congress of Mathematicians in Cambridge, Ernst Zermelo gave two talks. The first one covered axiomatic and genetic methods in the foundation of mathematical disciplines, and the second speech was on the game of chess. The second speech prompted Zermelo to write a paper on game theory. Being an avid chess player, Zermelo was concerned with application of set theory to the game of chess. Zermelo's original paper describing the theorem,
Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, was published in German in 1913. It can be considered as the first known paper on game theory. Ulrich Schwalbe and Paul Walker translated Zermelo's paper into English in 1997 and published the translation in the appendix to Zermelo and the Early History of Game Theory.
== Details ==
Zermelo considers the class of two-person games without chance, where players have strictly opposing interests and where only a finite number of positions are possible. Although in the game only finitely many positions are possible, Zermelo allows infinite sequences of moves since he does not consider stopping rules. Thus, he allows for the possibility of infinite games. Then he addresses two problems:
What does it mean for a player to be in a 'winning' position and is it possible to define this in an objective mathematical manner?
If the player is in a winning position, can the number of moves needed to force the win be determined?
To answer the first question, Zermelo states that a necessary and sufficient condition is the nonemptyness of a certain set, containing all possible sequences of moves such that a player wins independently of how the other player plays. But should this set be empty, the best a player could achieve would be a draw. So Zermelo defines another set containing all possible sequences of moves such that a player can postpone his loss for an infinite number of moves, which implies a draw. This set may also be empty, i.e., the player can avoid his loss for only finitely many moves if his opponent plays correctly. But this is equivalent to the opponent being able to force a win. This is the basis for all modern versions of Zermelo's theorem.
About the second question, Zermelo claimed that it will never take more moves than there are positions in the game. His proof is a proof by contradiction: Assume that a player can win in a number of moves larger than the number of positions. By the pigeonhole principle, at least one winning position must have appeared twice. So the player could have played at the first occurrence in the same way as he does at the second and thus could have won in fewer moves than there are positions.
In 1927, a Hungarian mathematician Dénes Kőnig revised Zermelo's paper and pointed out some gaps in the original work. First of all, Kőnig argues that Zermelo did not prove that a player, for example White, who is in a winning position is always able to force a win by making moves smaller than the number of positions in the game. Zermelo argued that White can change its behaviour at the first possibility of any related winning position and win without repetition. However, Kőnig maintains that this argument is not correct as it is not enough to reduce the number of moves in a single game below the number of possible positions. Thus, Zermelo claimed, but did not show, that a winning player can always win without repetition. The second objection by Kőnig is that the strategy 'do the same at the first occurrence of a position as at the second and thus win in fewer moves' cannot be made if it is Black's turn to move in this position. However, this argument is not correct because Zermelo considered two positions different whether Black or White makes a move.
== Zermelo's theorem and backward induction ==
It has been believed that Zermelo used backward induction as his method of proof. However, recent research on the Zermelo's theorem demonstrates that backward induction was not used to explain the strategy behind chess. Contrary to the popular belief, chess is not a finite game without at least one of the fifty move rule or threefold repetition rule. Strictly speaking, chess is an infinite game therefore backward induction does not provide the minmax theorem in this game.
Backward induction is a process of reasoning backward in time. It is used to analyse and solve extensive form games of perfect information. This method analyses the game starting at the end, and then works backwards to reach the beginning. In the process, backward induction determines the best strategy for the player that made the last move. Then the ultimate strategy is determined for the next-to last moving player of the game. The process is repeated again determining the best action for every point in the game has been found. Therefore, backward induction determines the Nash equilibrium of every subgame in the original game.
There is a number of reasons as to why backward induction is not present in the Zermelo's original paper:
Firstly, a recent study by Schwalbe and Walker (2001) demonstrated that Zermelo's paper contained basic idea of backward induction; however Zermelo did not make a formal statement on the theorem. Zermelo's original method was the idea of non-repetition. The first mention of backward induction was provided by László Kalmár in 1928. Kalmár generalised the work of Zermelo and Kőnig in his paper "On the Theory of Abstract Games". Kalmár was concerned with the question: "Given a winning position, how quickly can a win be forced?". His paper showed that winning without repetition is possible given that a player is a winning position. Kalmár's proof of non-repetition was proof by backward induction. In his paper, Kalmár introduced the concept of subgame and tactic. Kalmár's central argument was that a position can be a winning position only if a player can win in a finite number of moves. Also, a winning position for player A is always a losing position for player B.
== References ==
== External links ==
Original Paper(in German)
Ulrich Schwalbe, Paul Walker, Zermelo and the Early History of Game Theory, Games and Economic Behavior, Volume 34, 2001, 123-137, online | Wikipedia/Zermelo's_theorem_(game_theory) |
Dispute resolution or dispute settlement is the process of resolving disputes between parties. The term dispute resolution is conflict resolution through legal means.
Prominent venues for dispute settlement in international law include the International Court of Justice (formerly the Permanent Court of International Justice); the United Nations Human Rights Committee (which operates under the ICCPR) and European Court of Human Rights; the Panels and Appellate Body of the World Trade Organization; and the International Tribunal for the Law of the Sea. Half of all international agreements include a dispute settlement mechanism.
States are also known to form their own arbitration tribunals to settle disputes. Prominent private international courts, which adjudicate disputes between commercial private entities, include the International Court of Arbitration (of the International Chamber of Commerce) and the London Court of International Arbitration.
== Methods ==
Methods of dispute resolution include:
lawsuits (litigation) (legislative)
arbitration
collaborative law
mediation
conciliation
negotiation
facilitation
avoidance
One could theoretically include violence or even war as part of this spectrum, but dispute resolution practitioners do not usually do so; violence rarely ends disputes effectively, and indeed, often only escalates them. Also, violence rarely causes the parties involved in the dispute to no longer disagree on the issue that caused the violence. For example, a country successfully winning a war to annex part of another country's territory does not cause the former waring nations to no longer seriously disagree to whom the territory rightly belongs to and tensions may still remain high between the two nations.
Dispute resolution processes fall into two major types:
Adjudicative processes, such as litigation or arbitration, in which a judge, jury or arbitrator determines the outcome.
Consensual processes, such as collaborative law, mediation, conciliation, or negotiation, in which the parties attempt to reach agreement.
Not all disputes, even those in which skilled intervention occurs, end in resolution. Such intractable disputes form a special area in dispute resolution studies.
Dispute resolution is an important requirement in international trade, including negotiation, mediation, arbitration and litigation.
== Legal dispute resolution ==
The legal system provides resolutions for many different types of disputes. Some disputants will not reach agreement through a collaborative process. Some disputes need the coercive power of the state to enforce a resolution. Perhaps more importantly, many people want a professional advocate when they become involved in a dispute, particularly if the dispute involves perceived legal rights, legal wrongdoing, or threat of legal action against them.
The most common form of judicial dispute resolution is litigation. Litigation is initiated when one party files suit against another. In the United States, litigation is facilitated by the government within federal, state, and municipal courts. While litigation is often used to resolve disputes, it is strictly speaking a form of conflict adjudication and not a form of conflict resolution per se. This is because litigation only determines the legal rights and obligations of parties involved in a dispute and does not necessarily solve the disagreement between the parties involved in the dispute. For example, supreme court cases can rule on whether US states have the constitutional right to criminalize abortion but will not cause the parties involved in the case to no longer disagree on whether states do indeed have the constitutional authority to restrict access to abortion as one of the parties may disagree with the supreme courts reasoning and still disagree with the party that the supreme court sided with. Litigation proceedings are very formal and are governed by rules, such as rules of evidence and procedure, which are established by the legislature. Outcomes are decided by an impartial judge and/or jury, based on the factual questions of the case and the application law. The verdict of the court is binding, not advisory; however, both parties have the right to appeal the judgment to a higher court. Judicial dispute resolution is typically adversarial in nature, for example, involving antagonistic parties or opposing interests seeking an outcome most favorable to their position.
Due to the antagonistic nature of litigation, collaborators frequently opt for solving disputes privately.
Retired judges or private lawyers often become arbitrators or mediators; however, trained and qualified non-legal dispute resolution specialists form a growing body within the field of alternative dispute resolution (ADR). In the United States, many states now have mediation or other ADR programs annexed to the courts, to facilitate settlement of lawsuits.
== Extrajudicial dispute resolution ==
Some use the term dispute resolution to refer only to alternative dispute resolution (ADR), that is, extrajudicial processes such as arbitration, collaborative law, and mediation used to resolve conflict and potential conflict between and among individuals, business entities, governmental agencies, and (in the public international law context) states. ADR generally depends on agreement by the parties to use ADR processes, either before or after a dispute has arisen. ADR has experienced steadily increasing acceptance and utilization because of a perception of greater flexibility, costs below those of traditional litigation, and speedy resolution of disputes, among other perceived advantages. However, some have criticized these methods as taking away the right to seek redress of grievances in the courts, suggesting that extrajudicial dispute resolution may not offer the fairest way for parties not in an equal bargaining relationship, for example in a dispute between a consumer and a large corporation. In addition, in some circumstances, arbitration and other ADR processes may become as expensive as litigation or more so.
== See also ==
== References ==
== Further reading ==
Sherwyn, David, Tracey, Bruce & Zev Eigen, "In Defense of Mandatory Arbitration of Employment Disputes: Saving the Baby, Tossing out the Bath Water, and Constructing a New Sink in the Process", 2 U. Pa. J. Lab. & Emp. L. 73 (1999)
Ury, William, 2000. The Third Side: Why We Fight and How We Can Stop. New York: Penguin Putnam. ISBN 0-14-029634-4
Alés, Javier y Mata, Juan Diego " manual práctico para mediadores: el misterio de la mediacion" éxito Atelier. Barcelona 2016 | Wikipedia/Dispute_resolution |
In game theory, a focal point (or Schelling point) is a solution that people tend to choose by default in the absence of communication in order to avoid coordination failure. The concept was introduced by the American economist Thomas Schelling in his book The Strategy of Conflict (1960). Schelling states that "[p]eople can often concert their intentions or expectations with others if each knows that the other is trying to do the same" in a cooperative situation (p. 57), so their action would converge on a focal point which has some kind of prominence compared with the environment. However, the conspicuousness of the focal point depends on time, place and people themselves. It may not be a definite solution.
== Existence ==
The existence of the focal point is first demonstrated by Schelling with a series of questions. Here is one example: to determine the time and place to meet a stranger in New York City, but without being able to communicate in person beforehand. In this coordination game, any place and time in the city could be an equilibrium solution. Schelling asked a group of students this question and found that the most common answer was "noon at (the information booth at) Grand Central Terminal". There is nothing that makes Grand Central Terminal a location with a higher payoff because people could just as easily meet at another public location, such as a bar or a library, but its tradition as a meeting place raises its salience and therefore makes it a natural "focal point". Later, Schelling's informal experiments have been replicated under controlled conditions with monetary incentives by Judith Mehta.
The existence of focal points can help explain the use of social norms, including traditional gender roles, in order to ensure coordination, and why changing said norms can be difficult.
== Theories ==
Although the concept of a focal point has been widely accepted in game theory, it is still unclear how a focal point forms. The researchers have proposed theories from two aspects.
=== The level-n theory ===
Stahl and Wilson argue that a focal point is formed because players would try to predict how other players act. They model the level of "rational expectation" players by their ability to
form priors (models) about the behavior of other players;
choose the best responses given these priors.
A level-0 player will choose actions regardless of the actions of other players. A level-1 player believes that all other players are level-0 types. A level-n player estimates that all other players are level-0, 1, 2, ..., n − 1 types. Based on experimental data, most of the players only use one model to predict the behavior of all the other players. Although the hierarchy of types could be indefinite, the benefits of higher levels would decrease substantially while incurring a much greater cost. Because of the limit of players' expectation level and players' priors, it is possible to reach an equilibrium in games without communication.
==== The cognitive hierarchy theory ====
The cognitive hierarchy (CH) theory is a derivation of level-n theory. A level-n player from the CH model would assume that their strategy is the most sophisticated and that the levels 0, 1, 2, ..., n − 1 on which their opponents play follow a normalized Poisson distribution. This model works well in multi-player games where the players need to estimate a number in a given range, such as the Guess 2/3 of the average game. A player would be able to determine the value which they should play based on the assumed distribution of lower-level players described by the Poisson distribution. Another example of a game involving CH theory is the Keynesian beauty contest.
=== The team reasoning ===
Bacharach argued that people could find a focal point because they act as members of a team instead of individuals in a cooperative game. With the identity changed, the player follows the prescription of an imaginary group leader to maximize the group interest.
== Examples ==
=== Schelling's questions ===
Here is a subset of the questions raised by Schelling to prove the existence of a focal point.
Head-tail game: Name "heads" or "tails". If the two players name the same, they win an award, otherwise, they get nothing.
Letter order game: Give an order to letters A, B, and C. If the three players give the same order, they win an award, otherwise they get nothing.
Split money game: Two players share $100. They first write down their individual claims on a sheet of paper. If their claims add to $100 or less, both of them will get exactly what they claimed, but if the sum is higher than $100, they get nothing.
The results of the informal experiments are
For the two players, A and B, in head-tail game. 16 out of 22 A and 15 out of 22 B chose "heads".
For the three players, A, B, and C, in letter order game. 9 out of 12 A, 10 out of 12 B, and 14 out of 16 C wrote "ABC".
For the players to claim part of the $100. 36 out of 40 chose $50. 2 of the remainder chose $49 and $49.99.
These games suggest that focal points have some saliency. These characteristics make them preferable choices to people. Furthermore, people would assume each other has also noticed the saliency and make the same decision.
=== In coordination game ===
In a simple example, two people unable to communicate with each other are each shown a panel of four squares and asked to select one; if and only if they both select the same one, they will each receive a prize. Three of the squares are blue and one is red. Assuming they each know nothing about the other player, but that they each do want to win the prize, then they will, reasonably, both choose the red square.
The red square is not in a sense a better square; they could win by both choosing any square and in this sense, all squares are technically a Nash equilibrium. The red square is the "right" square to select only if a player can be sure that the other player has selected it, but by hypothesis neither can. However, it is the most salient and notable square, so—lacking any other one—most people will choose it, and this will in fact (often) work.
=== Collision game ===
Focal points can also have real-life applications. For example, imagine two bicycles headed towards each other and in danger of crashing. Avoiding collision becomes a coordination game where each player's winning choice depends on the other player's choice. Each player, in this case, has the choice to go straight, swerve to the left or swerve to the right. Both players want to avoid crashing, but neither knows what the other will do. In this case, the decision to swerve right can serve as a focal point which leads to the winning right–right outcome. It seems a natural focal point in places using right-hand traffic.
This idea of anti-coordination game is also apparent in the game of chicken, which involves two cars racing toward each other on a collision course and in which the driver who first decides to swerve is seen as a coward, while no driver swerving results in a fatal collision for both.
=== “Guess 2/3 of the average” game ===
The Guess 2/3 of the average game shows the level-n theory in practice. In this game, players are tasked with guessing an integer from 0 to 100 inclusive which they believe is closest to 2/3 of the average of all players’ guesses. A Nash equilibrium can be found by thinking through each level:
Level 0: The average can be in [0, 100].
Level 1: The average can be in [0, 67], which is 2/3 of the maximum average of level 0.
Level 2: The average can be in [0, 45], which is 2/3 of the maximum average of level 1.
Level N: Assuming all other players reason similarly, 2/3 of the maximum average will never be higher than
100
⋅
(
2
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3
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N
.
{\displaystyle 100\cdot (2/3)^{N}.}
As N grows, 2/3 of the average will trend towards zero. At this point, the only Nash equilibrium is for all players to guess 0.
Adding repetition to the game introduces a focal point at the Nash equilibrium solution of 0. This was shown by Camerer as, “[when] the game is played multiple times with the same group, the average moves close to 0”. Introducing the iterative aspect to the game forces all players onto higher levels of thinking, which allows them all to play guesses trending towards 0.
== See also ==
Game theory
Coordination failure (economics)
Coordination game
Simultaneous game
Surprisingly popular
Equilibrium selection
Rendezvous problem, the mathematical problem of maximising the probability of two people meeting
== References ==
== External links ==
Rare Entries Contests (an example) and Common Entries Contests, games of respectively avoiding and seeking out focal points
TED community experiment on focal / Schelling points
Schelling Points for Alien Contact https://youtu.be/3lwlNWMl86M | Wikipedia/Focal_point_(game_theory) |
A business can use a variety of pricing strategies when selling a product or service. To determine the most effective pricing strategy for a company, senior executives need to first identify the company's pricing position, pricing segment, pricing capability and their competitive pricing reaction strategy. Pricing strategies, tactics and roles vary from company to company, and also differ across countries, cultures, industries and over time, with the maturing of industries and markets and changes in wider economic conditions.
Pricing strategies determine the price companies set for their products. The price can be set to maximize profitability for each unit sold or from the market overall. It can also be used to defend an existing market from new entrants, to increase market share within a market or to enter a new market. Pricing strategies can bring both competitive advantages and disadvantages to its firm and often dictate the success or failure of a business; thus, it is crucial to choose the right strategy.
== Models of pricing ==
=== Absorption pricing ===
This pricing method aims to recover all the costs of producing a product. The price of a product includes the variable cost of each item plus a proportionate amount of the fixed costs:
Unit Variable Costs
+
Overhead
+
Managing Costs
Number of Units Produced
=
Absorption Price
{\displaystyle {\text{Unit Variable Costs}}+{\frac {{\text{Overhead}}+{\text{Managing Costs}}}{\text{Number of Units Produced}}}={\text{Absorption Price}}}
Fixed or variable costs, direct or indirect costs, employee salaries, utility costs, and other types of costs can also be calculated by applying the absorption pricing method.
=== Contribution margin-based pricing ===
Contribution margin-based pricing maximizes the profit derived from an individual product, based on the difference between the product's price and variable costs (the product's contribution margin per unit), and on one's assumptions regarding the relationship between the product's price and the number of units that can be sold at that price. The product's contribution to total firm profit (i.e. to operating income) is maximized when a price is chosen that maximizes the following:
(contribution margin per unit) × (number of units sold)
In cost-plus pricing, a company first determines its break-even price for the product. This is done by calculating all the costs involved in the production such as raw materials used in transportation etc., marketing and distribution of the product. Then a markup is set for each unit, based on the profit the company needs to make, its sales objectives and the price it believes customers will pay. For example, if a product's price is $10, and the contribution margin (also known as the profit margin) is 30 percent, then the price will be set at $10 * 1.30 = $13.
=== Cost plus pricing ===
Cost plus pricing is a cost-based method for setting the prices of goods and services. Under this approach, the direct material cost, direct labor cost, and overhead costs for a product are added up and added to a markup percentage (to create a profit margin) in order to derive the price of the product.
=== Creaming or skimming ===
Price skimming occurs when goods are priced higher so that fewer sales are needed to break even. Selling a product at a high price, and sacrificing high sales to gain a high profit is therefore "skimming" the market. Skimming is usually employed to reimburse the cost of investment of the original research into the product: commonly used in electronic markets when a new range, such as DVD players, are first sold at a high price. This strategy is often used to target "early adopters" of a product or service. Early adopters generally have a relatively lower price sensitivity—this can be attributed to: their need for the product outweighing their need to economize; a greater understanding of the product's value; or simply having a higher disposable income.
This strategy is employed only for a limited duration to recover most of the investment made to build the product. To gain further market share, a seller must use other pricing tactics such as economy or penetration. This method can have some setbacks as it could leave the product at a high price against the competition.
=== Decoy pricing ===
Method of pricing where the seller offers at least three products, and where two of them have a similar or equal price. The two products with similar prices should be the most expensive ones, and one of the two should be less attractive than the other. This strategy will make people compare the options with similar prices; as a result, sales of the more attractive high-priced item will increase.
=== Differential pricing ===
Differential pricing occurs when firms set various prices for the same product depending on their consumer's portfolio, geographic areas, demographic segments and the intensity of competition in the region.
=== Double ticketing ===
A form of deceptive pricing strategy that sells a product at the higher of two prices communicated to the consumer on, accompanying, or promoting the product.
=== Freemium ===
Freemium is a revenue model that works by offering a product or service free of charge (typically digital offerings such as software) while charging a premium for advanced features, functionality, or related products and services. The word "freemium" is a portmanteau combining the two aspects of the business model: "free" and "premium". It has become a highly popular model, with notable successes.
=== Good–better–best ===
A seller offers three prices for variations of the same good or service: a "good" no frills version, a "best" premium version, and a "better" version in the middle. Invoking the Goldilocks principle, customers may choose the "better" version because they are willing to pay more than the "good" price, but they are not willing to pay for the "best" version. A notable practitioner of the good–better–best pricing strategy is Apple Inc., which originally sold one model of iPhone in 2007, but by 2020, had adopted the practice of introducing good, better, and best models of iPhone and Apple Watch. Apple's competitors, such as Samsung Electronics, followed suit.
=== High-low pricing ===
A practice whereby many items are regularly priced higher than by competitors, but through promotions, advertisements, and or coupons, lower prices are offered on key items. The lower promotional prices designed to bring customers to the organization where the customer is offered the promotional product as well as the regular higher priced products.
=== Keystone pricing ===
A retail pricing strategy where retail price is set at double the wholesale price. For example, if a cost of a product for a retailer is £100, then the sale price would be £200. In a competitive industry, it is often not recommended to use keystone pricing as a pricing strategy due to its relatively high profit margin and the fact that other variables need to be taken into account.
=== Limit pricing ===
A limit price is the price set by a monopolist to discourage economic entry into a market. The limit price is the price that the entrant would face upon entering as long as the incumbent firm did not decrease output. The limit price is often lower than the average cost of production or just low enough to make entering not profitable.
The quantity produced by the incumbent firm to act as a deterrent to entry is usually larger than would be optimal for a monopolist, but might still produce higher economic profits than would be earned under perfect competition.
The problem with limit pricing as a strategy is that once the entrant has entered the market, the quantity used as a threat to deter entry is no longer the incumbent firm's best response. This means that for limit pricing to be an effective deterrent to entry, the threat must in some way be made credible. A way to achieve this is for the incumbent firm to constrain itself to produce a certain quantity whether entry occurs or not. An example of this would be if the firm signed a union contract to employ a certain (high) level of labor for a long period of time. In this strategy price of the product becomes the limit according to budget.
=== Loss leader ===
A loss leader or leader is a product sold at a low price (i.e. at cost or below cost) to stimulate other profitable sales. This would help the companies to expand its market share as a whole. Loss leader strategy is commonly used by retailers in order to lead the customers into buying products with higher marked-up prices to produce an increase in profits rather than purchasing the leader product which is sold at a lower cost. When a "featured brand" is priced to be sold at a lower cost, retailers tend not to sell large quantities of the loss leader products and also they tend to purchase less quantities from the supplier as well to prevent loss for the firm. Supermarkets and restaurants are an excellent example of retail firms that apply the strategy of loss leader.
=== Marginal-cost pricing ===
In business, the practice of setting the price of a product to equal the extra cost of producing an extra unit of output. By this policy, a producer charges, for each product unit sold, only the addition to total cost resulting from materials and direct labor. Businesses often set prices close to marginal cost during periods of poor sales. If, for example, an item has a marginal cost of $1.00 and a normal selling price is $2.00, the firm selling the item might wish to lower the price to $1.10 if demand has waned. The business would choose this approach because the incremental profit of 10 cents from the transaction is better than no sale at all.
=== Odd-Even pricing ===
Odd-Even pricing is often used by sellers to portray their products to be either cheaper or more expensive than their actual value. Sellers competing for price-sensitive consumers, will fix their product price to be odd. A good example of this can be noticed in most supermarkets where instead of pricing milk at £5, it would be written as £4.99. Contrarily, sellers competing for consumers with low price sensitivity, will fix their product price to be even. For example, often in upscale retail stores, handbags will be priced at £1250 instead of £1249.99.
=== Pay what you want ===
Pay what you want is a pricing system where buyers pay any desired amount for a given commodity, sometimes including zero. In some cases, a minimum (floor) price may be set, and/or a suggested price may be indicated as guidance for the buyer. The buyer can also select an amount higher than the standard price for the commodity.
Giving buyers the freedom to pay what they want may seem to not make much sense for a seller, but in some situations it can be very successful. While most uses of pay what you want have been at the margins of the economy, or for special promotions, there are emerging efforts to expand its utility to broader and more regular use.
=== Penetration pricing ===
Penetration pricing includes setting the price low with the goals of attracting customers and gaining market share. The price will be raised later once this market share is gained.
A firm that uses a penetration pricing strategy prices a product or a service at a smaller amount than its usual, long range market price in order to increase more rapid market recognition or to increase their existing market share. This strategy can sometimes discourage new competitors from entering a market position if they incorrectly observe the penetration price as a long range price.
Companies do their pricing in diverse ways. In small companies, prices are often set by the boss. In large companies, pricing is handled by division and the product line managers. In industries where pricing is a key influence, pricing departments are set to support others in determining suitable prices.
Penetration pricing strategy is usually used by firms or businesses who are just entering the market. In marketing it is a theoretical method that is used to lower the prices of the goods and services causing high demand for them in the future. This strategy of penetration pricing is vital and highly recommended to be applied over multiple situations that the firm may face. Such as, when the production rate of the firm is lower when compared to other firms in the market and also sometimes when firms face hardship into releasing their product in the market due to extremely large rate of competition. In these situations it is appropriate for a firm to use the penetration strategy to gain consumer attention.
This technique is very common in internet companies, which often don't turn a profit until they've acquired monopoly status, if then, instead putting all their money into expanding market share. It is very cheap to reuse a piece of software, once written, so there are substantial economies of scale that favour this approach, as does the social trap effect (it's hard to leave Facebook).
=== Performance-based pricing ===
A pricing strategy in which the seller is paid based on the effectiveness of its product or service. Examples of sellers who often use performance-based pricing are real estate agents, online advertising platforms, and personal injury attorneys. Performance-based pricing increases the risk of the seller but it creates opportunities for greater rewards. Sellers who use this pricing strategy have an advantage in attracting customers. Performance-based pricing has fewer chances to work if the desired outcome is not clearly defined and quantified between the two parties.
=== Predatory pricing ===
Predatory pricing, also known as aggressive pricing (also known as "undercutting"), intended to drive out competitors from a market. It is illegal in some countries.
Companies or firms that tend to get involved with the strategy of predatory pricing often have the goal to place restrictions or a barrier for other new businesses from entering the applicable market. This strategy may contradict anti–trust law, attempting to establish within the market a monopoly by the imposing company. Predatory pricing mainly occurs during price competitions in the market as it is easier to obfuscate the act. Using this strategy, in the short term consumers will benefit and be satisfied with lower cost products. In the long run, firms often will not benefit as this strategy will continue to be used by other businesses to undercut competitors' margins, causing an increase in competition within the field and facilitating major losses. This strategy is dangerous as it could be destructive to a firm in terms of losses and even lead to complete business failure.
=== Premium decoy pricing ===
Method of pricing where an organization artificially sets one product price high, in order to boost sales of a lower-priced product. Let's say there are two products, beef, and pork. The organization may increase the price of beef so that it becomes expensive in the eyes of the customers. Subsequently, pork becomes cheaper. Customers will then opt for cheaper pork. A limited-edition handbag can be considered as another example of the Premium Decoy Pricing that many bag manufacturers have provided a limited edition choice of bags for customers. The price is usually expensive that most customers would not able to purchase. However, it gives a luxury brand image and helps those manufacturers to build a more affordable handbag.
=== Premium pricing ===
Premium pricing is the practice of keeping the price of a product or service artificially high in order to encourage favorable perceptions among buyers, based solely on the price. The practice is intended to exploit the (not necessarily justifiable) tendency for buyers to assume that expensive items enjoy an exceptional reputation, are more reliable or desirable, or represent exceptional quality and distinction. Moreover, a premium price may portray the meaning of better quality in the eyes of the consumer.
Consumers are willing to pay more for trends, which is a key motive for premium pricing, and are not afraid of how much a product or service costs. The novelty of consumers wanting to have the latest trends is a challenge for marketers as they are having to entertain their consumers.
The aspiration of consumers and the feeling of treating themselves is the key factor of purchasing a good or service. Consumers are looking for constant change as they are constantly evolving and moving.
Examples of premium pricing:
Ethical consumption
Fair traders
Voluntarism
These are important drivers and examples of premium pricing, which help guide and distinguish of how a product or service is marketed and priced within today's market.
=== Price discrimination ===
Price discrimination is the practice of setting a different price for the same product in different segments to the market. For example, this can be for different classes, such as ages, or for different opening times.
Price discrimination may improve consumer surplus. When a firm price discriminates, it will sell up to the point where marginal cost meets the demand curve. Some conditions are required for price discrimination to exist:
Firms must face a downward-sloping demand curve, i.e. the demand for a product is inversely proportional to its price.
Accurately segment the market, i.e Two or more buying groups must be distinguished at a cost that does not exceed the revenue that distinguishes them.
Prevent resale
Have market power
There are three different types of price discrimination that revolve around the same strategy and same goal – maximize profit by segmenting the market, and extracting additional consumer surplus.
First-degree price discrimination
The business charges every consumer exactly how much they are willing to pay for the product. Assume the monopolist determines the price of the product based on the maximum amount of money a consumer is known to pay for any quantity of product that is exactly equal to the demand price for the product in order to obtain the total consumer surplus of each consumer.
Second-degree price discrimination
The business uses volume discounts which allow buyers to purchase a higher inventory at a reduced price. While this benefits the high-inventory buyer, it obviously hurts the low-inventory buyer who is forced to pay a higher price. This buyer may then be less competitive in the downstream market. For instance, telecommunications companies charge different prices for customers' monthly Internet access time. They charge a higher price for customers who have small usage whilst charging a lower price for customers who have large usage. In this way, the monopoly seller appropriates a portion of the buyer's consumer surplus for himself.
Third-degree price discrimination
This occurs when firms segment the market into high demand and low demand groups.That is, for the same commodity, a complete monopoly firm implements different prices relying on the different price elasticity of demand in different markets. i.e. the Power plants implement lower prices for more elastic industrial electricity and higher prices for less elastic household electricity.
Firms need to ensure they are aware of several factors of their business before proceeding with the strategy of price discrimination. Firms must have control over the changes they make regarding the price of their product by which they can gain profitability depending on the amount of sales made. The price can be increased or decreased at any point depending on the fluctuation of the rate of buyers and consumers. Price discrimination strategy is not feasible for all firms as there are many consequences that the firms may face due to the action. For example: if a firm sells a product to their customer for a cheaper price and that customer resells the product demanding a higher price from another buyer then the chances of the firm failing to make a higher profit is predicted because they could have sold their product at a higher rate than the re-seller and made further profit.
=== Price leadership ===
An observation made of oligopolistic business behavior in which one company, usually the dominant competitor among several, leads the way in determining prices, the others soon following. The context is a state of limited competition, in which a market is shared by a small number of producers or sellers.
=== Psychological pricing ===
Pricing designed to have a positive psychological impact. For example, there are often benefits to selling a product at $3.95 or $3.99, rather than $4.00. If the price of a product is $100 and the company prices it at $99, then it is using the psychological technique of just-below pricing. In most consumers' minds, $99 gives the impression of being considerably less than $100. A minor distinction in pricing can make a big difference in sales. The company that succeeds in finding appropriate psychological price points can improve sales and maximize revenue.
=== Sliding scale ===
The economic concept of sliding scale at its most basic: people pay as they are able to for services, events and items. Those with access to more resources pay more and thus provide the cushion for those with less access to pay less, creating a sustainable economic underpinning for said services, events and items.
=== Target pricing business ===
Pricing method whereby the selling price of a product is calculated to produce a particular rate of return on investment for a specific volume of production. The target pricing method is used most often by public utilities, like electric and gas companies, and companies whose capital investment is high, like automobile manufacturers.
Target pricing is not useful for companies whose capital investment is low because, according to this formula, the selling price will be understated. Also the target pricing method is not keyed to the demand for the product, and if the entire volume is not sold, a company might sustain an overall budgetary loss on the product.
=== Time-based pricing ===
A flexible pricing mechanism made possible by advances in information technology and employed mostly by Internet-based companies. By responding to market fluctuations or large amounts of data gathered from customers – ranging from where they live to what they buy to how much they have spent on past purchases – dynamic pricing allows online companies to adjust the prices of identical goods to correspond to a customer's willingness to pay. The airline industry is often cited as a dynamic pricing success story. In fact, it employs the technique so artfully that most of the passengers on any given airplane have paid different ticket prices for the same flight. As of 2018, several third-party tools have allowed merchants to take advantage of a time based dynamic pricing including Pricemole, SweetPricing, BeyondPricing, etc.
=== Time-sensitive pricing ===
Time-sensitive pricing is a cost-based method for setting prices for goods that have a short shelf life. Careful consideration has to be taken to the "Use By" and "Best Before" dates of the products, in relation to the "Mark Up" or "Return" of the products. That is to say the shorter period of time should have a lower Mark-up/Return margin, thus increasing the Turnover/sales of the product, and decreasing the Wastage/loss of products.
=== Value-based pricing ===
Pricing a product based on the value the product has for the customer and not on its costs of production or any other factor.
This pricing strategy is frequently used where the value to the customer is many times the cost of producing the item or service. For instance, the cost of producing a software CD is about the same independent of the software on it, but the prices vary with the perceived value the customers are expected to have. The perceived value will depend on the alternatives open to the customer. In business these alternatives are using a competitor's software, using a manual work around, or not doing an activity. In order to employ value-based pricing, one must know its customers' business, one's business costs, and one's perceived alternatives. It is also known as perceived-value pricing.
Value-based pricing have many effects on the business and consumer of the product. Value-based pricing is a fundamental business activity and is the process of developing product strategies and pricing them properly to establish the product within the market. This is a key concept for a relatively new product within the market, because without the correct price, there would be no sale. Having an overly high price for an average product would have negative effects on the business as the consumer would not buy the product. Having a low price on a luxury product would also have a negative impact on the business as in the long run the business would not be profitable. This can be seen as a positive for the consumer as they are not needing to pay extreme prices for the luxury product.
=== Variable pricing strategies ===
Variable pricing strategy sums up the total cost of the variable characteristics associated in the production of the product. Examples of variable characteristics are: interest rates, location, date, and region of production. The sum total of the following characteristics is then included within the original price of the product during marketing. Variable pricing enables product prices to have a balance "between sales volume and income per unit sold". Variable pricing strategy has the advantage of ensuring the sum total of the cost businesses would face in order to develop a new product. However, variable pricing strategy excludes the cost of fixed pricing. Fixed pricing includes the price of dedication received from manufactures in the production of developing the product and other involvement of factors.
=== Yield management strategies ===
Yield management is a strategy which aims to monitor consumer behaviour to gain and achieve maximum profit through selling goods and services that are perishable. The theory behind this strategy is to focus on the following aspects: buying behaviour patterns of consumers, external environmental factors and market price to successfully gain the most profit. This strategy of yield management is commonly used by the firms associated within the airlines industry. For example, a customer may purchase an airline ticket in the day time for $600 and another customer may purchase the same airline ticket on the same day in the evening for $800 – the reason being that during the day time the airline contained many seats that were spare which needed to be occupied and sold. Thus, prices were decreased in order to attract and manipulate the customers into buying an airline ticket with great deals or offers. However, during the evening time most seats were filled and the firm decided to increase the price of the airline ticket for the desperate customers who needed to purchase the spare seats that were available. This type of strategy is a vigilant way of connecting with the target consumers as well as flourishing the business.
== Pricing roles ==
Some organizations delegate pricing responsibilities to a range of departmental staff. In other cases, a Chief Pricing Officer plays a strategic role advising on market conditions on competitive opportunities which can be exploited via pricing decision-making.
== Nine laws of price sensitivity and consumer psychology ==
In their book, The Strategy and Tactics of Pricing (2002), Thomas Nagle and Reed Holden outline nine "laws" or factors that influence how a consumer perceives a given price and how price-sensitive they are likely to be with respect to different purchase decisions.
They are:
Reference price effect – buyer's price sensitivity for a given product increases the higher the product's price relative to perceived alternatives. Perceived alternatives can vary by buyer segment, by occasion, and other factors.
Difficult comparison effect – buyers are less sensitive to the price of a known or more reputable product when they have difficulty comparing it to potential alternatives.
Switching costs effect – the higher the product-specific investment a buyer must make to switch suppliers, the less price-sensitive that buyer is when choosing between alternatives.
Price-quality effect – buyers are less sensitive to price the more that higher prices signal higher quality. Products for which this effect is particularly relevant include: image products, exclusive products, and products with minimal cues for quality.
Expenditure effect – buyers are more price-sensitive when the expense accounts for a large percentage of buyers' available income or budget.
End-benefit effect – the effect refers to the relationship a given purchase has to a larger overall benefit, and is divided into two parts: Derived demand: The more sensitive buyers are to the price of the end benefit, the more sensitive they will be to the prices of those products that contribute to that benefit. Price proportion cost: The price proportion cost refers to the percent of the total cost of the end benefit accounted for by a given component that helps to produce the end benefit (e.g., think CPU and PCs). The smaller the given components share of the total cost of the end benefit, the less sensitive buyers will be to the components' price.
Shared-cost effect – the smaller the portion of the purchase price buyers must pay for themselves, the less price-sensitive they will be.
Fairness effect – buyers are more sensitive to the price of a product when the price is outside the range they perceive as "fair" or "reasonable" given the purchase context.
The framing effect – buyers are more price-sensitive when they perceive the price as a loss rather than a forgone gain, and they have greater price sensitivity when the price is paid separately rather than as part of a bundle.
== References == | Wikipedia/Pricing_strategy |
In management, a strategy map is a diagram that documents the strategic goals being pursued by an organization or management team. It is an element of the documentation associated with the Balanced Scorecard, and in particular is characteristic of the second generation of Balanced Scorecard designs that first appeared during the mid-1990s. The first diagrams of this type appeared in the early 1990s, and the idea of using this type of diagram to help document Balanced Scorecard was discussed in a paper by Robert S. Kaplan and David P. Norton in 1996.
The strategy map idea featured in several books and articles during the late 1990s by Robert S. Kaplan and David P. Norton. Their original book in 1996, "The Balanced Scorecard, Translating strategy into action", contained diagrams which are later called strategy maps, but at this time they did not refer to them as such. Kaplan & Norton's second book, The Strategy Focused Organization, explicitly refers to strategy maps and includes a chapter on how to build them. At this time, they said that "the relationship between the drivers and the desired outcomes constitute the hypotheses that define the strategy". Their Third book, Strategy Maps, goes into further detail about how to describe and visualise the strategy using strategy maps.
The Kaplan and Norton approach to strategy maps has:
An underlying framework of horizontal perspectives arranged in a cause and effect relationship, typically Financial, Customer, Process and Learning & Growth
Objectives within those perspectives. Each objective as text appearing within a shape (usually an oval or rectangle). Relatively few objectives (usually fewer than 20)
Vertical sets of linked objectives that span the perspectives. These are called strategic themes.
Clear cause-and-effect relationships between these objectives, across the perspectives. The strategic themes represent hypotheses about how the strategy will bring about change to the outcomes of the organisation.
Across a broader range of published sources, a looser approach is sometimes used. In these approaches, there are only a few common attributes. Some approaches use a more broad causal relationships between objectives shown with arrows that either join objectives together, or placed in a way not linked with specific objectives but to provide general euphemistic indications of where causality lies. For instance, Olve and Wetter, in their 1999 book Performance Drivers, also describe early performance driver models, but do not refer to them as strategy maps.
The purpose of the strategy map in Balanced Scorecard design, and its emergence as a design aid, is discussed in some detail in a research paper on the evolution of Balanced Scorecard designs during the 1990s by Lawrie & Cobbold.
== Origin of strategy maps ==
The Balanced Scorecard is a framework that is used to help in the design and implementation of strategic performance management tools within organizations.
One of the big challenges faced in the design of Balanced Scorecard-based performance management systems is deciding what activities and outcomes to monitor. By providing a simple visual representation of the strategic objectives to be focused on, along with additional visual cues in the form of the perspectives and causal arrows, the strategy map has been found useful in enabling discussion within a management team about what objectives to choose, and subsequently to support discussion of the actual performance achieved.
== Perspectives ==
Very early Balanced Scorecard articles by Robert S. Kaplan and David P. Norton proposed a simple design method for choosing the content of the Balanced Scorecard based on answers to four generic questions about the strategy to be pursued by the organization. These four questions, one about finances, one about marketing, one about processes, and one about organizational development evolved quickly into a standard set of "perspectives" ("Financial", "Customer", "Internal Business Processes", "Learning & Growth").
Design of a Balanced Scorecard became a process of selecting a small number of objectives in each perspective, and then choosing measures and targets to inform on progress against this objective. But very quickly it was realized that the perspective headings chosen only worked for specific organisations (small to medium-sized firms in North America - the target market of the Harvard Business Review), and during the mid to late 1990s papers began to be published arguing that other sets of headings would make more sense for specific organization types, and that some organisations would benefit from using more or less than four headings.
Despite these concerns, the 'standard' set of perspectives remains the most common, and traditionally is arrayed on the strategy map in the sequence (from bottom to top) "Learning & Growth", "Internal Business Processes", "Customer", "Financial" with causal arrows tending to flow "up" the page.
The 'standard' set of perspectives is arranged in the sequence (bottom to top) to allow a sequential linkage from the investment perspective of Learning and Growth to the outcome perspectives of Customer and Financial. A well-designed strategy map will allow the author to 'tell a story' from the bottom to the top along the lines of "If we invest in (training, education etc) it will allow us to improve our Internal Processes which will impact our Customers (in a positive way) and improve our Financial results. In other words, Learning and Growth drive Internal Process change which impacts Customer Satisfaction which in turn improves Profitability. In the early part of the century, it was recognized that the perspective Learning and Growth was lacking significant investment areas, that of infrastructure and IT. Today, the lower-most perspective has been renamed to Organisational Capacity.
== Links between the strategy map and strategy development ==
The strategy map is a device that promotes three stages of conversation during the strategy development, implementation and learning process
First to capture a strategy from a management team. To promote discussion amongst that team on the strategy, so they all leave the room telling the same story of their strategy.
Secondly to communicate the strategy, focus organization efforts, and choose appropriate measures to report on an organisation's progress in implementing a strategy.
Finally to provide a basis to review and potentially revise the strategy, (not simply the measures or targets) and support conversations and decision making, as the team learn from the strategy's implementation.
Over the years many have suggested that it can be used as, in part, a strategy development tool. Kaplan & Norton in their book The Strategy Focused Organisation argue that organisations could adopt 'industry standard' templates (basically a set of pre-determined strategic objectives).
== See also ==
Third-generation balanced scorecard
== References ==
== External links == | Wikipedia/Strategy_map |
In mathematics, the replicator equation is a type of dynamical system used in evolutionary game theory to model how the frequency of strategies in a population changes over time. It is a deterministic, monotone, non-linear, and non-innovative dynamic that captures the principle of natural selection in strategic interactions.
The replicator equation describes how strategies with higher-than-average fitness increase in frequency, while less successful strategies decline. Unlike other models of replication—such as the quasispecies model—the replicator equation allows the fitness of each type to depend dynamically on the distribution of population types, making the fitness function an endogenous component of the system. This allows it to model frequency-dependent selection, where the success of a strategy depends on its prevalence relative to others.
Another key difference from the quasispecies model is that the replicator equation does not include mechanisms for mutation or the introduction of new strategies, and is thus considered non-innovative. It assumes all strategies are present from the outset and models only the relative growth or decline of their proportions over time.
Replicator dynamics have been widely applied in fields such as biology (to study evolution and population dynamics), economics (to analyze bounded rationality and strategy evolution), and machine learning (particularly in multi-agent systems and reinforcement learning).
== Equation ==
The most general continuous form of the replicator equation is given by the differential equation:where
x
i
{\displaystyle x_{i}}
is the proportion of type
i
{\displaystyle i}
in the population,
is the vector of the distribution of types in the population,
f
i
(
x
)
{\displaystyle f_{i}(x)}
is the fitness of type
i
{\displaystyle i}
(which is dependent on the population), and
ϕ
(
x
)
{\displaystyle \phi (x)}
is the average population fitness (given by the weighted average of the fitness of the
n
{\displaystyle n}
types in the population). Since the elements of the population vector
x
{\displaystyle x}
sum to unity by definition, the equation is defined on the n-dimensional simplex.
The replicator equation assumes a uniform population distribution; that is, it does not incorporate population structure into the fitness. The fitness landscape does incorporate the population distribution of types, in contrast to other similar equations, such as the quasispecies equation.
In application, populations are generally finite, making the discrete version more realistic. The analysis is more difficult and computationally intensive in the discrete formulation, so the continuous form is often used, although there are significant properties that are lost due to this smoothing. Note that the continuous form can be obtained from the discrete form by a limiting process.
To simplify analysis, fitness is often assumed to depend linearly upon the population distribution, which allows the replicator equation to be written in the form:
x
i
˙
=
x
i
(
(
A
x
)
i
−
x
T
A
x
)
{\displaystyle {\dot {x_{i}}}=x_{i}\left(\left(Ax\right)_{i}-x^{T}Ax\right)}
where the payoff matrix
holds all the fitness information for the population: the expected payoff can be written as
(
A
x
)
i
{\displaystyle \left(Ax\right)_{i}}
and the mean fitness of the population as a whole can be written as
x
T
A
x
{\displaystyle x^{T}Ax}
. It can be shown that the change in the ratio of two proportions
x
i
/
x
j
{\displaystyle x_{i}/x_{j}}
with respect to time is:
d
d
t
(
x
i
x
j
)
=
x
i
x
j
[
f
i
(
x
)
−
f
j
(
x
)
]
{\displaystyle {d \over {dt}}\left({x_{i} \over {x_{j}}}\right)={x_{i} \over {x_{j}}}\left[f_{i}(x)-f_{j}(x)\right]}
In other words, the change in the ratio is driven entirely by the difference in fitness between types.
=== Derivation of deterministic and stochastic replicator dynamics ===
Suppose that the number of individuals of type
is
N
i
{\displaystyle N_{i}}
and that the total number of individuals is
N
{\displaystyle N}
. Define the proportion of each type to be
x
i
=
N
i
/
N
{\displaystyle x_{i}=N_{i}/N}
. Assume that the change in each type is governed by geometric Brownian motion:
d
N
i
=
f
i
N
i
d
t
+
σ
i
N
i
d
W
i
{\displaystyle dN_{i}=f_{i}N_{i}dt+\sigma _{i}N_{i}dW_{i}}
where
is the fitness associated with type
i
{\displaystyle i}
. The average fitness of the types
ϕ
=
x
T
f
{\displaystyle \phi =x^{T}f}
. The Wiener processes are assumed to be uncorrelated. For
x
i
(
N
1
,
.
.
.
,
N
m
)
{\displaystyle x_{i}(N_{1},...,N_{m})}
, Itô's lemma then gives us:
d
x
i
(
N
1
,
.
.
.
,
N
m
)
=
∂
x
i
∂
N
j
d
N
j
+
1
2
∂
2
x
i
∂
N
j
∂
N
k
d
N
j
d
N
k
=
∂
x
i
∂
N
j
d
N
j
+
1
2
∂
2
x
i
∂
N
j
2
(
d
N
j
)
2
{\displaystyle {\begin{aligned}dx_{i}(N_{1},...,N_{m})&={\partial x_{i} \over {\partial N_{j}}}dN_{j}+{1 \over {2}}{\partial ^{2}x_{i} \over {\partial N_{j}\partial N_{k}}}dN_{j}dN_{k}\\&={\partial x_{i} \over {\partial N_{j}}}dN_{j}+{1 \over {2}}{\partial ^{2}x_{i} \over {\partial N_{j}^{2}}}(dN_{j})^{2}\end{aligned}}}
The partial derivatives are then:
∂
x
i
∂
N
j
=
1
N
δ
i
j
−
x
i
N
∂
2
x
i
∂
N
j
2
=
−
2
N
2
δ
i
j
+
2
x
i
N
2
{\displaystyle {\begin{aligned}{\partial x_{i} \over {\partial N_{j}}}&={1 \over {N}}\delta _{ij}-{x_{i} \over {N}}\\{\partial ^{2}x_{i} \over {\partial N_{j}^{2}}}&=-{2 \over {N^{2}}}\delta _{ij}+{2x_{i} \over {N^{2}}}\end{aligned}}}
where
δ
i
j
{\displaystyle \delta _{ij}}
is the Kronecker delta function. These relationships imply that:
d
x
i
=
d
N
i
N
−
x
i
∑
j
d
N
j
N
−
(
d
N
i
)
2
N
2
+
x
i
∑
j
(
d
N
j
)
2
N
2
{\displaystyle dx_{i}={dN_{i} \over {N}}-x_{i}\sum _{j}{dN_{j} \over {N}}-{(dN_{i})^{2} \over {N^{2}}}+x_{i}\sum _{j}{(dN_{j})^{2} \over {N^{2}}}}
Each of the components in this equation may be calculated as:
d
N
i
N
=
f
i
x
i
d
t
+
σ
i
x
i
d
W
i
−
x
i
∑
j
d
N
j
N
=
−
x
i
(
ϕ
d
t
+
∑
j
σ
j
x
j
d
W
j
)
−
(
d
N
i
)
2
N
2
=
−
σ
i
2
x
i
2
d
t
x
i
∑
j
(
d
N
j
)
2
N
2
=
x
i
(
∑
j
σ
j
2
x
j
2
)
d
t
{\displaystyle {\begin{aligned}{dN_{i} \over {N}}&=f_{i}x_{i}dt+\sigma _{i}x_{i}dW_{i}\\-x_{i}\sum _{j}{dN_{j} \over {N}}&=-x_{i}\left(\phi dt+\sum _{j}\sigma _{j}x_{j}dW_{j}\right)\\-{(dN_{i})^{2} \over {N^{2}}}&=-\sigma _{i}^{2}x_{i}^{2}dt\\x_{i}\sum _{j}{(dN_{j})^{2} \over {N^{2}}}&=x_{i}\left(\sum _{j}\sigma _{j}^{2}x_{j}^{2}\right)dt\end{aligned}}}
Then the stochastic replicator dynamics equation for each type is given by:
Assuming that the
terms are identically zero, the deterministic replicator dynamics equation is recovered.
== Analysis ==
The analysis differs in the continuous and discrete cases: in the former, methods from differential equations are utilized, whereas in the latter the methods tend to be stochastic. Since the replicator equation is non-linear, an exact solution is difficult to obtain (even in simple versions of the continuous form) so the equation is usually analyzed in terms of stability. The replicator equation (in its continuous and discrete forms) satisfies the folk theorem of evolutionary game theory which characterizes the stability of equilibria of the equation. The solution of the equation is often given by the set of evolutionarily stable states of the population.
In general nondegenerate cases, there can be at most one interior evolutionary stable state (ESS), though there can be many equilibria on the boundary of the simplex. All the faces of the simplex are forward-invariant which corresponds to the lack of innovation in the replicator equation: once a strategy becomes extinct there is no way to revive it.
Phase portrait solutions for the continuous linear-fitness replicator equation have been classified in the two and three dimensional cases. Classification is more difficult in higher dimensions because the number of distinct portraits increases rapidly.
== Relationships to other equations ==
The continuous replicator equation on
n
{\displaystyle n}
types is equivalent to the Generalized Lotka–Volterra equation in
n
−
1
{\displaystyle n-1}
dimensions. The transformation is made by the change of variables:
x
i
=
y
i
1
+
∑
j
=
1
n
−
1
y
j
i
=
1
,
…
,
n
−
1
{\displaystyle x_{i}={\frac {y_{i}}{1+\sum _{j=1}^{n-1}{y_{j}}}}\quad i=1,\ldots ,n-1}
x
n
=
1
1
+
∑
j
=
1
n
−
1
y
j
,
{\displaystyle x_{n}={\frac {1}{1+\sum _{j=1}^{n-1}{y_{j}}}},}
where
y
i
{\displaystyle y_{i}}
is the Lotka–Volterra variable. The continuous replicator dynamic is also equivalent to the Price equation.
== Discrete replicator equation ==
When one considers an unstructured infinite population with non-overlapping generations, one should work with the discrete forms of the replicator equation. Mathematically, two simple phenomenological versions---
x
i
′
=
x
i
+
x
i
[
(
A
x
)
i
−
x
T
A
x
]
(
t
y
p
e
I
)
,
{\displaystyle x'_{i}=x_{i}+x_{i}\left[\left(Ax\right)_{i}-x^{T}Ax\right]\,({\rm {type~I),}}}
x
i
′
=
x
i
[
(
A
x
)
i
x
T
A
x
]
(
t
y
p
e
I
I
)
,
{\displaystyle x'_{i}=x_{i}\left[{\frac {\left(Ax\right)_{i}}{x^{T}Ax}}\right]\,({\rm {type~II),}}}
---are consistent with the Darwinian tenet of natural selection or any analogous evolutionary phenomena. Here, prime stands for the next time step. However, the discrete nature of the equations puts bounds on the payoff-matrix elements. Interestingly, for the simple case of two-player-two-strategy games, the type I replicator map is capable of showing period doubling bifurcation leading to chaos and it also gives a hint on how to generalize the concept of the evolutionary stable state to accommodate the periodic solutions of the map.
== Generalizations ==
A generalization of the replicator equation which incorporates mutation is given by the replicator-mutator equation, which takes the following form in the continuous version:
x
i
˙
=
∑
j
=
1
n
x
j
f
j
(
x
)
Q
j
i
−
ϕ
(
x
)
x
i
,
{\displaystyle {\dot {x_{i}}}=\sum _{j=1}^{n}{x_{j}f_{j}(x)Q_{ji}}-\phi (x)x_{i},}
where the matrix
Q
{\displaystyle Q}
gives the transition probabilities for the mutation of type
j
{\displaystyle j}
to type
i
{\displaystyle i}
,
f
i
{\displaystyle f_{i}}
is the fitness of the
i
t
h
{\displaystyle i^{th}}
and
ϕ
{\displaystyle \phi }
is the mean fitness of the population. This equation is a simultaneous generalization of the replicator equation and the quasispecies equation, and is used in the mathematical analysis of language.
The discrete version of the replicator-mutator equation may have two simple types in line with the two replicator maps written above:
x
i
′
=
x
i
+
∑
j
=
1
n
x
j
f
j
(
x
)
Q
j
i
−
ϕ
(
x
)
x
i
,
{\displaystyle x'_{i}=x_{i}+\sum _{j=1}^{n}{x_{j}f_{j}(x)Q_{ji}}-\phi (x)x_{i},}
and
x
i
′
=
∑
j
=
1
n
x
j
f
j
(
x
)
Q
j
i
ϕ
(
x
)
,
{\displaystyle x'_{i}={\frac {\sum _{j=1}^{n}{x_{j}f_{j}(x)Q_{ji}}}{\phi (x)}},}
respectively.
The replicator equation or the replicator-mutator equation can be extended to include the effect of delay that either corresponds to the delayed information about the population state or in realizing the effect of interaction among players. The replicator equation can also easily be generalized to asymmetric games. A recent generalization that incorporates population structure is used in evolutionary graph theory.
== References ==
== Further reading ==
Cressman, R. (2003). Evolutionary Dynamics and Extensive Form Games The MIT Press.
Taylor, P.D.; Jonker, L. (1978). "Evolutionary Stable Strategies and Game Dynamics". Mathematical Biosciences, 40: 145–156.
Sandholm, William H. (2010). Population Games and Evolutionary Dynamics. Economic Learning and Social Evolution, The MIT Press. | Wikipedia/Replicator_equation |
An economic model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical, framework designed to illustrate complex processes. Frequently, economic models posit structural parameters. A model may have various exogenous variables, and those variables may change to create various responses by economic variables. Methodological uses of models include investigation, theorizing, and fitting theories to the world.
== Overview ==
In general terms, economic models have two functions: first as a simplification of and abstraction from observed data, and second as a means of selection of data based on a paradigm of econometric study.
Simplification is particularly important for economics given the enormous complexity of economic processes. This complexity can be attributed to the diversity of factors that determine economic activity; these factors include: individual and cooperative decision processes, resource limitations, environmental and geographical constraints, institutional and legal requirements and purely random fluctuations. Economists therefore must make a reasoned choice of which variables and which relationships between these variables are relevant and which ways of analyzing and presenting this information are useful.
Selection is important because the nature of an economic model will often determine what facts will be looked at and how they will be compiled. For example, inflation is a general economic concept, but to measure inflation requires a model of behavior, so that an economist can differentiate between changes in relative prices and changes in price that are to be attributed to inflation.
In addition to their professional academic interest, uses of models include:
Forecasting economic activity in a way in which conclusions are logically related to assumptions;
Proposing economic policy to modify future economic activity;
Presenting reasoned arguments to politically justify economic policy at the national level, to explain and influence company strategy at the level of the firm, or to provide intelligent advice for household economic decisions at the level of households.
Planning and allocation, in the case of centrally planned economies, and on a smaller scale in logistics and management of businesses.
In finance, predictive models have been used since the 1980s for trading (investment and speculation). For example, emerging market bonds were often traded based on economic models predicting the growth of the developing nation issuing them. Since the 1990s many long-term risk management models have incorporated economic relationships between simulated variables in an attempt to detect high-exposure future scenarios (often through a Monte Carlo method).
A model establishes an argumentative framework for applying logic and mathematics that can be independently discussed and tested and that can be applied in various instances. Policies and arguments that rely on economic models have a clear basis for soundness, namely the validity of the supporting model.
Economic models in current use do not pretend to be theories of everything economic; any such pretensions would immediately be thwarted by computational infeasibility and the incompleteness or lack of theories for various types of economic behavior. Therefore, conclusions drawn from models will be approximate representations of economic facts. However, properly constructed models can remove extraneous information and isolate useful approximations of key relationships. In this way more can be understood about the relationships in question than by trying to understand the entire economic process.
The details of model construction vary with type of model and its application, but a generic process can be identified. Generally, any modelling process has two steps: generating a model, then checking the model for accuracy (sometimes called diagnostics). The diagnostic step is important because a model is only useful to the extent that it accurately mirrors the relationships that it purports to describe. Creating and diagnosing a model is frequently an iterative process in which the model is modified (and hopefully improved) with each iteration of diagnosis and respecification. Once a satisfactory model is found, it should be double checked by applying it to a different data set.
== Types of models ==
According to whether all the model variables are deterministic, economic models can be classified as stochastic or non-stochastic models; according to whether all the variables are quantitative, economic models are classified as discrete or continuous choice model; according to the model's intended purpose/function, it can be classified as
quantitative or qualitative; according to the model's ambit, it can be classified as a general equilibrium model, a partial equilibrium model, or even a non-equilibrium model; according to the economic agent's characteristics, models can be classified as rational agent models, representative agent models etc.
Stochastic models are formulated using stochastic processes. They model economically observable values over time. Most of econometrics is based on statistics to formulate and test hypotheses about these processes or estimate parameters for them. A widely used bargaining class of simple econometric models popularized by Tinbergen and later Wold are autoregressive models, in which the stochastic process satisfies some relation between current and past values. Examples of these are autoregressive moving average models and related ones such as autoregressive conditional heteroskedasticity (ARCH) and GARCH models for the modelling of heteroskedasticity.
Non-stochastic models may be purely qualitative (for example, relating to social choice theory) or quantitative (involving rationalization of financial variables, for example with hyperbolic coordinates, and/or specific forms of functional relationships between variables). In some cases economic predictions in a coincidence of a model merely assert the direction of movement of economic variables, and so the functional relationships are used only stoical in a qualitative sense: for example, if the price of an item increases, then the demand for that item will decrease. For such models, economists often use two-dimensional graphs instead of functions.
Qualitative models – although almost all economic models involve some form of mathematical or quantitative analysis, qualitative models are occasionally used. One example is qualitative scenario planning in which possible future events are played out. Another example is non-numerical decision tree analysis. Qualitative models often suffer from lack of precision.
At a more practical level, quantitative modelling is applied to many areas of economics and several methodologies have evolved more or less independently of each other. As a result, no overall model taxonomy is naturally available. We can nonetheless provide a few examples that illustrate some particularly relevant points of model construction.
An accounting model is one based on the premise that for every credit there is a debit. More symbolically, an accounting model expresses some principle of conservation in the form
algebraic sum of inflows = sinks − sources
This principle is certainly true for money and it is the basis for national income accounting. Accounting models are true by convention, that is any experimental failure to confirm them, would be attributed to fraud, arithmetic error or an extraneous injection (or destruction) of cash, which we would interpret as showing the experiment was conducted improperly.
Optimality and constrained optimization models – Other examples of quantitative models are based on principles such as profit or utility maximization. An example of such a model is given by the comparative statics of taxation on the profit-maximizing firm. The profit of a firm is given by
π
(
x
,
t
)
=
x
p
(
x
)
−
C
(
x
)
−
t
x
{\displaystyle \pi (x,t)=xp(x)-C(x)-tx\quad }
where
p
(
x
)
{\displaystyle p(x)}
is the price that a product commands in the market if it is supplied at the rate
x
{\displaystyle x}
,
x
p
(
x
)
{\displaystyle xp(x)}
is the revenue obtained from selling the product,
C
(
x
)
{\displaystyle C(x)}
is the cost of bringing the product to market at the rate
x
{\displaystyle x}
, and
t
{\displaystyle t}
is the tax that the firm must pay per unit of the product sold.
The profit maximization assumption states that a firm will produce at the output rate x if that rate maximizes the firm's profit. Using differential calculus we can obtain conditions on x under which this holds. The first order maximization condition for x is
∂
π
(
x
,
t
)
∂
x
=
∂
(
x
p
(
x
)
−
C
(
x
)
)
∂
x
−
t
=
0
{\displaystyle {\frac {\partial \pi (x,t)}{\partial x}}={\frac {\partial (xp(x)-C(x))}{\partial x}}-t=0}
Regarding x as an implicitly defined function of t by this equation (see implicit function theorem), one concludes that the derivative of x with respect to t has the same sign as
∂
2
(
x
p
(
x
)
−
C
(
x
)
)
∂
2
x
=
∂
2
π
(
x
,
t
)
∂
x
2
,
{\displaystyle {\frac {\partial ^{2}(xp(x)-C(x))}{\partial ^{2}x}}={\partial ^{2}\pi (x,t) \over \partial x^{2}},}
which is negative if the second order conditions for a local maximum are satisfied.
Thus the profit maximization model predicts something about the effect of taxation on output, namely that output decreases with increased taxation. If the predictions of the model fail, we conclude that the profit maximization hypothesis was false; this should lead to alternate theories of the firm, for example based on bounded rationality.
Borrowing a notion apparently first used in economics by Paul Samuelson, this model of taxation and the predicted dependency of output on the tax rate, illustrates an operationally meaningful theorem; that is one requiring some economically meaningful assumption that is falsifiable under certain conditions.
Aggregate models. Macroeconomics needs to deal with aggregate quantities such as output, the price level, the interest rate and so on. Now real output is actually a vector of goods and services, such as cars, passenger airplanes, computers, food items, secretarial services, home repair services etc. Similarly price is the vector of individual prices of goods and services. Models in which the vector nature of the quantities is maintained are used in practice, for example Leontief input–output models are of this kind. However, for the most part, these models are computationally much harder to deal with and harder to use as tools for qualitative analysis. For this reason, macroeconomic models usually lump together different variables into a single quantity such as output or price. Moreover, quantitative relationships between these aggregate variables are often parts of important macroeconomic theories. This process of aggregation and functional dependency between various aggregates usually is interpreted statistically and validated by econometrics. For instance, one ingredient of the Keynesian model is a functional relationship between consumption and national income: C = C(Y). This relationship plays an important role in Keynesian analysis.
== Problems with economic models ==
Most economic models rest on a number of assumptions that are not entirely realistic. For example, agents are often assumed to have perfect information, and markets are often assumed to clear without friction. Or, the model may omit issues that are important to the question being considered, such as externalities. Any analysis of the results of an economic model must therefore consider the extent to which these results may be compromised by inaccuracies in these assumptions, and a large literature has grown up discussing problems with economic models, or at least asserting that their results are unreliable.
== History ==
One of the major problems addressed by economic models has been understanding economic growth. An early attempt to provide a technique to approach this came from the French physiocratic school in the eighteenth century. Among these economists, François Quesnay was known particularly for his development and use of tables he called Tableaux économiques. These tables have in fact been interpreted in more modern terminology as a Leontiev model, see the Phillips reference below.
All through the 18th century (that is, well before the founding of modern political economy, conventionally marked by Adam Smith's 1776 Wealth of Nations), simple probabilistic models were used to understand the economics of insurance. This was a natural extrapolation of the theory of gambling, and played an important role both in the development of probability theory itself and in the development of actuarial science. Many of the giants of 18th century mathematics contributed to this field. Around 1730, De Moivre addressed some of these problems in the 3rd edition of The Doctrine of Chances. Even earlier (1709), Nicolas Bernoulli studies problems related to savings and interest in the Ars Conjectandi. In 1730, Daniel Bernoulli studied "moral probability" in his book Mensura Sortis, where he introduced what would today be called "logarithmic utility of money" and applied it to gambling and insurance problems, including a solution of the paradoxical Saint Petersburg problem. All of these developments were summarized by Laplace in his Analytical Theory of Probabilities (1812). Thus, by the time David Ricardo came along he had a well-established mathematical basis to draw from.
== Tests of macroeconomic predictions ==
In the late 1980s, the Brookings Institution compared 12 leading macroeconomic models available at the time. They compared the models' predictions for how the economy would respond to specific economic shocks (allowing the models to control for all the variability in the real world; this was a test of model vs. model, not a test against the actual outcome). Although the models simplified the world and started from a stable, known common parameters the various models gave significantly different answers. For instance, in calculating the impact of a monetary loosening on output some models estimated a 3% change in GDP after one year, and one gave almost no change, with the rest spread between.
Partly as a result of such experiments, modern central bankers no longer have as much confidence that it is possible to 'fine-tune' the economy as they had in the 1960s and early 1970s. Modern policy makers tend to use a less activist approach, explicitly because they lack confidence that their models will actually predict where the economy is going, or the effect of any shock upon it. The new, more humble, approach sees danger in dramatic policy changes based on model predictions, because of several practical and theoretical limitations in current macroeconomic models; in addition to the theoretical pitfalls, (listed above) some problems specific to aggregate modelling are:
Limitations in model construction caused by difficulties in understanding the underlying mechanisms of the real economy. (Hence the profusion of separate models.)
The law of unintended consequences, on elements of the real economy not yet included in the model.
The time lag in both receiving data and the reaction of economic variables to policy makers attempts to 'steer' them (mostly through monetary policy) in the direction that central bankers want them to move. Milton Friedman has vigorously argued that these lags are so long and unpredictably variable that effective management of the macroeconomy is impossible.
The difficulty in correctly specifying all of the parameters (through econometric measurements) even if the structural model and data were perfect.
The fact that all the model's relationships and coefficients are stochastic, so that the error term becomes very large quickly, and the available snapshot of the input parameters is already out of date.
Modern economic models incorporate the reaction of the public and market to the policy maker's actions (through game theory), and this feedback is included in modern models (following the rational expectations revolution and Robert Lucas, Jr.'s Lucas critique of non-microfounded models). If the response to the decision maker's actions (and their credibility) must be included in the model then it becomes much harder to influence some of the variables simulated.
=== Comparison with models in other sciences ===
Complex systems specialist and mathematician David Orrell wrote on this issue in his book Apollo's Arrow and explained that the weather, human health and economics use similar methods of prediction (mathematical models). Their systems—the atmosphere, the human body and the economy—also have similar levels of complexity. He found that forecasts fail because the models suffer from two problems: (i) they cannot capture the full detail of the underlying system, so rely on approximate equations; (ii) they are sensitive to small changes in the exact form of these equations. This is because complex systems like the economy or the climate consist of a delicate balance of opposing forces, so a slight imbalance in their representation has big effects. Thus, predictions of things like economic recessions are still highly inaccurate, despite the use of enormous models running on fast computers.
See Unreasonable ineffectiveness of mathematics § Economics and finance.
=== Effects of deterministic chaos on economic models ===
Economic and meteorological simulations may share a fundamental limit to their predictive powers: chaos. Although the modern mathematical work on chaotic systems began in the 1970s the danger of chaos had been identified and defined in Econometrica as early as 1958:
"Good theorising consists to a large extent in avoiding assumptions ... [with the property that] a small change in what is posited will seriously affect the conclusions."
(William Baumol, Econometrica, 26 see: Economics on the Edge of Chaos).
It is straightforward to design economic models susceptible to butterfly effects of initial-condition sensitivity.
However, the econometric research program to identify which variables are chaotic (if any) has largely concluded that aggregate macroeconomic variables probably do not behave chaotically. This would mean that refinements to the models could ultimately produce reliable long-term forecasts. However, the validity of this conclusion has generated two challenges:
In 2004 Philip Mirowski challenged this view and those who hold it, saying that chaos in economics is suffering from a biased "crusade" against it by neo-classical economics in order to preserve their mathematical models.
The variables in finance may well be subject to chaos. Also in 2004, the University of Canterbury study Economics on the Edge of Chaos concludes that after noise is removed from S&P 500 returns, evidence of deterministic chaos is found.
More recently, chaos (or the butterfly effect) has been identified as less significant than previously thought to explain prediction errors. Rather, the predictive power of economics and meteorology would mostly be limited by the models themselves and the nature of their underlying systems (see Comparison with models in other sciences above).
=== Critique of hubris in planning ===
A key strand of free market economic thinking is that the market's invisible hand guides an economy to prosperity more efficiently than central planning using an economic model. One reason, emphasized by Friedrich Hayek, is the claim that many of the true forces shaping the economy can never be captured in a single plan. This is an argument that cannot be made through a conventional (mathematical) economic model because it says that there are critical systemic-elements that will always be omitted from any top-down analysis of the economy.
== Examples of economic models ==
Cobb–Douglas model of production
Solow–Swan model of economic growth
Lucas islands model of money supply
Heckscher–Ohlin model of international trade
Black–Scholes model of option pricing
AD–AS model a macroeconomic model of aggregate demand– and supply
IS–LM model the relationship between interest rates and assets markets
Ramsey–Cass–Koopmans model of economic growth
Gordon–Loeb model for cyber security investments
== See also ==
Economic methodology
Computational economics
Agent-based computational economics
Endogeneity
Financial model
== Notes ==
== References ==
Baumol, William & Blinder, Alan (1982), Economics: Principles and Policy (2nd ed.), New York: Harcourt Brace Jovanovich, ISBN 0-15-518839-9.
Caldwell, Bruce (1994), Beyond Positivism: Economic Methodology in the Twentieth Century (Revised ed.), New York: Routledge, ISBN 0-415-10911-6.
Holcombe, R. (1989), Economic Models and Methodology, New York: Greenwood Press, ISBN 0-313-26679-4. Defines model by analogy with maps, an idea borrowed from Baumol and Blinder. Discusses deduction within models, and logical derivation of one model from another. Chapter 9 compares the neoclassical school and the Austrian School, in particular in relation to falsifiability.
Lange, Oskar (1945), "The Scope and Method of Economics", Review of Economic Studies, 13 (1), The Review of Economic Studies Ltd.: 19–32, doi:10.2307/2296113, JSTOR 2296113, S2CID 4140287. One of the earliest studies on methodology of economics, analysing the postulate of rationality.
de Marchi, N. B. & Blaug, M. (1991), Appraising Economic Theories: Studies in the Methodology of Research Programs, Brookfield, VT: Edward Elgar, ISBN 1-85278-515-2. A series of essays and papers analysing questions about how (and whether) models and theories in economics are empirically verified and the current status of positivism in economics.
Morishima, Michio (1976), The Economic Theory of Modern Society, New York: Cambridge University Press, ISBN 0-521-21088-7. A thorough discussion of many quantitative models used in modern economic theory. Also a careful discussion of aggregation.
Orrell, David (2007), Apollo's Arrow: The Science of Prediction and the Future of Everything, Toronto: Harper Collins Canada, ISBN 978-0-00-200740-5.
Phillips, Almarin (1955), "The Tableau Économique as a Simple Leontief Model", Quarterly Journal of Economics, 69 (1), The MIT Press: 137–44, doi:10.2307/1884854, JSTOR 1884854.
Samuelson, Paul A. (1948), "The Simple Mathematics of Income Determination", in Metzler, Lloyd A. (ed.), Income, Employment and Public Policy; essays in honor of Alvin Hansen, New York: W. W. Norton.
Samuelson, Paul A. (1983), Foundations of Economic Analysis (Enlarged ed.), Cambridge: Harvard University Press, ISBN 0-674-31301-1. This is a classic book carefully discussing comparative statics in microeconomics, though some dynamics is studied as well as some macroeconomic theory. This should not be confused with Samuelson's popular textbook.
Tinbergen, Jan (1939), Statistical Testing of Business Cycle Theories, Geneva: League of Nations.
Walsh, Vivian (1987), "Models and theory", The New Palgrave: A Dictionary of Economics, vol. 3, New York: Stockton Press, pp. 482–83, ISBN 0-935859-10-1.
Wold, H. (1938), A Study in the Analysis of Stationary Time Series, Stockholm: Almqvist and Wicksell.
Wold, H. & Jureen, L. (1953), Demand Analysis: A Study in Econometrics, New York: Wiley.
Gordon, Lawrence A.; Loeb, Martin P. (November 2002). "The Economics of Information Security Investment". ACM Transactions on Information and System Security. 5 (4): 438–457. doi:10.1145/581271.581274. S2CID 1500788.
== External links ==
R. Frigg and S. Hartmann, Models in Science. Entry in the Stanford Encyclopedia of Philosophy.
H. Varian How to build a model in your spare time The author makes several unexpected suggestions: Look for a model in the real world, not in journals. Look at the literature later, not sooner.
Elmer G. Wiens: Classical & Keynesian AD-AS Model – An on-line, interactive model of the Canadian Economy.
IFs Economic Sub-Model [1]: Online Global Model
Economic attractor | Wikipedia/Economic_model |
Strategic thinking is a mental or thinking process applied by individuals and within organizations in the context of achieving a goal or set of goals.
When applied in an organizational strategic management process, strategic thinking involves the generation and application of unique business insights and opportunities intended to create competitive advantage for a firm or organization. It can be done individually, as well as collaboratively among key people who can positively alter an organization's future. Group strategic thinking may create more value by enabling a proactive and creative dialogue, where individuals gain other people's perspectives on critical and complex issues. This is regarded as a benefit in highly competitive and fast-changing business landscapes.
== Overview ==
There is a generally accepted definition for strategic thinking, a common agreement as to its role or importance, and a standardised list of key competencies of strategic thinkers. There is also a consensus on whether strategic thinking is an uncommon ideal or a common and observable property of strategy. It includes finding and developing a strategic foresight capacity for an organization, by exploring all possible organizational futures, and challenging conventional thinking to foster decision making today. Research on strategic thought indicates that the critical strategic question is not the conventional "What?", but "Why?" or "How?". The work of Henry Mintzberg and other authors, further support the conclusion; and also draw a clear distinction between strategic thinking and strategic planning, another important strategic management thought process.
General Andre Beaufre wrote in 1963 that strategic thinking "is a mental process, at once abstract and rational, which must be capable of synthesizing both psychological and material data. The strategist must have a great capacity for both analysis and synthesis; analysis is necessary to assemble the data on which he makes his diagnosis, synthesis in order to produce from these data the diagnosis itself—and the diagnosis in fact amounts to a choice between alternative courses of action."
Most agree that traditional models of strategy making, which are primarily based on strategic planning, are not working. Strategy in today's competitive business landscape is moving away from the basic ‘strategic planning’ to more of ‘strategic thinking’ in order to remain competitive. However, both thought processes must work hand-in-hand in order to reap maximum benefit. It has been argued that the real heart of strategy is the 'strategist'; and for a better strategy execution requires a strategic thinker who can discover novel, imaginative strategies which can re-write the rules of the competitive game; and set in motion the chain of events that will shape and "define the future".
== Strategic thinking vs. strategic planning ==
In the view of F. Graetz, strategic thinking and planning are “distinct, but interrelated and complementary thought processes” that must sustain and support one another for effective strategic management. Graetz's model holds that the role of strategic thinking is "to seek innovation and imagine new and very different futures that may lead the company to redefine its core strategies and even its industry". Strategic planning's role is "to realise and to support strategies developed through the strategic thinking process and to integrate these back into the business".
Henry Mintzberg wrote in 1994 that strategic thinking is more about synthesis (i.e., "connecting the dots") than analysis (i.e., "finding the dots"). It is about "capturing what the manager learns from all sources (both the soft insights from his or her personal experiences and the experiences of others throughout the organization and the hard data from market research and the like) and then synthesizing that learning into a vision of the direction that the business should pursue." Mintzberg argued that strategic thinking cannot be systematized and is the critical part of strategy formation, as opposed to strategic planning exercises. In his view, strategic planning happens around the strategy formation or strategic thinking activity, by providing inputs for the strategist to consider and providing plans for controlling the implementation of the strategy after it is formed.
According to Jeanne Liedtka, strategic thinking differs from strategic planning along the following dimensions of strategic management:
== Strategic thinking and complexity ==
In a complex scenario, organizational actions are intensified by a global network of interactions, leading to diverse environmental, economic, and social challenges. This complexity is characterized by intricate networks and recursive cause-and-effect relationships, diverging from the linear logic of Cartesian thought and the punctual logic of dialectical thought. Within such systems, seemingly trivial actions can produce unexpected outcomes or be magnified by intricate relationship networks, resulting in entirely unpredictable consequences
To address this context, Terra and Passador advocate for strategic thinking capable of: (1) reconnecting phenomena across different levels and disciplines and treating them holistically; (2) addressing objects of study subjected to recursive causality; (3) understanding facts through their dynamics; (4) approaching problems through mappings and negative approaches; (5) integrating non-empirical elements; and (6) incorporating a new mathematical rationale to navigate the non-linearity of such systems and the continuous transition between certainty and uncertainty inherent in their dynamics.
In the realm of academic research, Stacey suggests that this reality demands studies in the field of strategic thinking to focus on explanation, hypotheses about whole systems, their dynamics, and the relationship between dynamic behavior and innovative success. In this type of study, methods such as scheme construction, phenomenological approaches based on deductions and metaphors and integrative frameworks have been employed to understand the dynamics of various organizational problems by assimilating concepts common to several fields of science. In the field of studies on strategic thinking, several authors have developed multidisciplinary approaches based on these premises, utilizing systems thinking and cybernetics, integrative approaches, new mathematics of chaos, and concepts such as order through noise, autopoiesis, and self-organization.
== Strategic thinking competencies ==
Liedtka observed five “major attributes of strategic thinking in practice” that resemble competencies:
Systems perspective, refers to being able to understand implications of strategic actions. "A strategic thinker has a mental model of the complete end-to-end system of value creation, his or her role within it, and an understanding of the competencies it contains."
Intent focused which means more determined and less distractible than rivals in the marketplace. Crediting Hamel and Prahalad with popularising the concept, Liedtka describes strategic intent as "the focus that allows individuals within an organization to marshal and leverage their energy, to focus attention, to resist distraction, and to concentrate for as long as it takes to achieve a goal."
Thinking in time means being able to hold past, present and future in mind at the same time to create better decision making and speed implementation. "Strategy is not driven by future intent alone. It is the gap between today’s reality and intent for the future that is critical." Scenario planning is a practical application for incorporating "thinking in time" into strategy making.
Hypothesis driven, ensuring that both creative and critical thinking are incorporated into strategy making. This competency explicitly incorporates the scientific method into strategic thinking.
Intelligent opportunism, which means being responsive to good opportunities. "The dilemma involved in using a well-articulated strategy to channel organisational efforts effectively and efficiently must always be balanced against the risks of losing sight of alternative strategies better suited to a changing environment."
== Strategic thinking opportunities ==
The main focus of strategic thinking is on long-term opportunities to achieve a purpose, goal, or set of goals, the broad view of opportunities includes taking a look at the entirety of a concept instead of merely focusing on individual details and seeing beyond the details to focus on the larger vision or organizational goals to produce long-term successes.
Big-picture thinking involves:
Envisioning the long term implications of decisions. Thinking about the long-term impact or implications means we take a wide-angle lense to discover all opportunities to achieve a purpose, goal, or a set of goals.
Identifying patterns. Looking for connections and patterns beyond the immediate situation.
Creating psychological distance. Creating distance from a decision, either in time or situationally.
Goal-directed behaviour and action towards an ideal future. Focusing on the overall vision and macro-level aspects of a project or decision.
Cross-functional collaboration. Working with multiple teams to put the big-picture thinking into practice.
There are six exercises to do strategic thinking, which include; reflection, ditch perfectionism, consider other perspectives, use a mind mapping tool, ask tough questions and brainstorm, stay focused on purpose.
Strategic thinking is one type of thinking, the ability to develop and implement long-term plans to achieve goals, analytical thinking is a foundation of strategic thinking, and many of the types of thinking that we could utilise include:
Analytical thinking.
Strategic thinking.
Creative thinking.
Intuitive thinking.
Systems thinking.
== See also ==
U.S. Army Strategist
== References ==
== External links ==
What is strategic thinking?, harvardbusiness.org
6 Habits of True Strategic Thinkers by Paul J. H. Schoemaker, Inc.com
For Great Leadership, Clear Your Head by Joshua Ehrlich, Harvard Business Review
How to Think Strategically by Michael Watkins, Harvard Business Review
Strategic Thinking: Success Secrets of Big Business Projects Dr David Stevens, McGraw Hill, 1997
Strategy Execution – Ensure your culture provides for common sense by i-nexus | Wikipedia/Strategic_thinking |
Porter's Five Forces Framework is a method of analysing the competitive environment of a business. It draws from industrial organization (IO) economics to derive five forces that determine the competitive intensity and, therefore, the attractiveness (or lack thereof) of an industry in terms of its profitability. An "unattractive" industry is one in which the effect of these five forces reduces overall profitability. The most unattractive industry would be one approaching "pure competition", in which available profits for all firms are driven to normal profit levels. The five-forces perspective is associated with its originator, Michael E. Porter of Harvard University. This framework was first published in Harvard Business Review in 1979.
Porter refers to these forces as the microenvironment, to contrast it with the more general term macroenvironment. They consist of those forces close to a company that affects its ability to serve its customers and make a profit. A change in any of the forces normally requires a business unit to re-assess the marketplace given the overall change in industry information. The overall industry attractiveness does not imply that every firm in the industry will return the same profitability. Firms are able to apply their core competencies, business model or network to achieve a profit above the industry average. A clear example of this is the airline industry. As an industry, profitability is low because the industry's underlying structure of high fixed costs and low variable costs afford enormous latitude in the price of airline travel. Airlines tend to compete on cost, and that drives down the profitability of individual carriers as well as the industry itself because it simplifies the decision by a customer to buy or not buy a ticket. This underscores the need for businesses to continuously evaluate their competitive landscape and adapt strategies in response to changes in industry dynamics, exemplified by the airline industry's struggle with profitability despite varying approaches to differentiation. A few carriers – Richard Branson's Virgin Atlantic is one – have tried, with limited success, to use sources of differentiation in order to increase profitability.
Porter's five forces include three forces from 'horizontal competition' – the threat of substitute products or services, the threat of established rivals, and the threat of new entrants – and two others from 'vertical' competition – the bargaining power of suppliers and the bargaining power of customers.
Porter developed his five forces framework in reaction to the then-popular SWOT analysis, which he found both lacking in rigor and ad hoc. Porter's five-forces framework is based on the structure–conduct–performance paradigm in industrial organizational economics. Other Porter's strategy tools include the value chain and generic competitive strategies.
== Five forces that shape competition ==
=== Threat of new entrants ===
New entrants put pressure on current within an industry through their desire to gain market share. This in turn puts pressure on prices, costs, and the rate of investment needed to sustain a business within the industry. The threat of new entrants is particularly intense if they are diversifying from another market as they can leverage existing expertise, cash flow, and brand identity which puts a strain on existing companies
profitability.
Barriers to entry restrict the threat of new entrants. If the barriers are high, the threat of new entrants is reduced, and conversely, if the barriers are low, the risk of new companies venturing into a given market is high. Barriers to entry are advantages that existing, established companies have over new entrants.
Michael E. Porter differentiates two factors that can have an effect on how much of a threat new entrants may pose:
Barriers to entry
The most attractive segment is one in which entry barriers are high and exit barriers are low. It is worth noting, however, that high barriers to entry almost always make exit more difficult.
Michael E. Porter lists seven major sources of entry barriers:
Supply-side economies of scale – spreading the fixed costs over a larger volume of units thus reducing the cost per unit. This can discourage a new entrant because they either have to start trading at a smaller volume of units and accept a price disadvantage over larger companies, or risk coming into the market on a large scale in an attempt to displace the existing market leader.
Demand-side benefits of scale – this occurs when a buyer's willingness to purchase a particular product or service increases with other people's willingness to purchase it. Also known as the network effect, people tend to value being in a 'network' with a larger number of people who use the same company.
Customer switching costs – These are well illustrated by structural market characteristics such as supply chain integration but also can be created by firms. Airline frequent flyer programs are an example.
Capital requirements – clearly the Internet has influenced this factor dramatically. Websites and apps can be launched cheaply and easily as opposed to the brick-and-mortar industries of the past.
Incumbency advantages independent of size (e.g., customer loyalty and brand equity).
Unequal access to distribution channels – if there are a limited number of distribution channels for a certain product/service, new entrants may struggle to find a retail or wholesale channel to sell through as existing competitors will have a claim on them.
Government policy such as sanctioned monopolies, legal franchise requirements, patents, and regulatory requirements.
Expected retaliation
For example, a specific characteristic of oligopoly markets is that prices generally settle at an equilibrium because any price rises or cuts are easily matched by the competition.
=== Threat of substitutes ===
A substitute product uses a different technology to try to solve the same economic need. Examples of substitutes are meat, poultry, and fish; landlines and cellular telephones; airlines, automobiles, trains, and ships; beer and wine; and so on. For example, tap water is a substitute for Coke, but Pepsi is a product that uses the same technology (albeit different ingredients) to compete head-to-head with Coke, so it is not a substitute. Increased marketing for drinking tap water might "shrink the pie" for both Coke and Pepsi, whereas increased Pepsi advertising would likely "grow the pie" (increase consumption of all soft drinks), while giving Pepsi a larger market share at Coke's expense.
Potential factors:
Buyer propensity to substitute. This aspect incorporated both tangible and intangible factors. Brand loyalty can be very important as in the Coke and Pepsi example above; however, contractual and legal barriers are also effective.
Relative price performance of substitute
Buyer's switching costs. This factor is well illustrated by the mobility industry. Uber and its many competitors took advantage of the incumbent taxi industry's dependence on legal barriers to entry and when those fell away, it was trivial for customers to switch. There were no costs as every transaction was atomic, with no incentive for customers not to try another product.
Perceived level of product differentiation which is classic Michael Porter in the sense that there are only two basic mechanisms for competition – lowest price or differentiation. Developing multiple products for niche markets is one way to mitigate this factor.
Number of substitute products available in the market
Ease of substitution
Availability of close substitutes
=== Bargaining power of customers ===
The bargaining power of customers is also described as the market of outputs: the ability of customers to put the firm under pressure, which also affects the customer's sensitivity to price changes. Firms can take measures to reduce buyer power, such as implementing a loyalty program. Buyers' power is high if buyers have many alternatives. It is low if they have few choices.
Potential factors:
Buyer concentration to firm concentration ratio
Degree of dependency upon existing channels of distribution
Bargaining leverage, particularly in industries with high fixed costs
Buyer switching costs
Buyer information availability
Availability of existing substitute products
Buyer price sensitivity
Differential advantage (uniqueness) of industry products
RFM (customer value) Analysis
=== Bargaining power of suppliers ===
The bargaining power of suppliers is also described as the market of inputs. Suppliers of raw materials, components, labour, and services (such as expertise) to the firm can be a source of power over the firm when there are few substitutes. If you are making biscuits and there is only one person who sells flour, you have no alternative but to buy it from them. Suppliers may refuse to work with the firm or charge excessively high prices for unique resources.
Potential factors are:
Supplier switching costs relative to firm switching costs
Degree of differentiation of inputs
Impact of inputs on cost and differentiation
Presence of substitute inputs
Strength of the distribution channel
Supplier concentration to the firm concentration ratio
Employee solidarity (e.g. labor unions)
Supplier competition: the ability to forward vertically integrate and cut out the buyer.
=== Competitive rivalry ===
Competitive rivalry is a measure of the extent of competition among existing firms. Price cuts, increased advertising expenditures, or investing in service/product enhancements and innovation are all examples of competitive moves that might limit profitability and lead to competitive moves. For most industries, the intensity of competitive rivalry is the biggest determinant of the competitiveness of the industry. Understanding industry rivals is vital to successfully marketing a product. Positioning depends on how the public perceives a product and distinguishes it from that of competitors. An organization must be aware of its competitors' marketing strategies and pricing and also be reactive to any changes made. Rivalry among competitors tends to be cutthroat and industry profitability is low while having the potential factors below:
Potential factors:
Competitive advantage through innovation
Competition between online and offline organizations
Level of advertising expense
Powerful competitive strategy which could potentially be realized by adhering to Porter's work on low cost versus differentiation.
Firm concentration ratio
== Factors, not forces ==
Other factors below should also be considered as they can contribute in evaluating a firm's strategic position. These factors can commonly be mistaken for being the underlying structure of the firm; however, the underlying structure consists of the five factors above.
=== Industry growth rate ===
Sometimes bad strategy decisions can be made when a narrow focus is kept on the growth rate of an industry. While rapid growth in an industry can seem attractive, it can also attract new entrants especially if entry barriers are low and suppliers are powerful. Furthermore, profitability is not guaranteed if powerful substitutes become available to the customers.
For example, Blockbuster dominated the rental market throughout 1990s. In 1998, Reed Hastings founded Netflix and entered the market. Netflix's CEO was famously laughed out of the room. While Blockbuster was thriving and expanding rapidly, its key pitfall was ignoring its competitors and focusing on its growth in the industry.
=== Technology and innovation ===
Technology in itself is a rapidly growing industry. Regardless of the advanced growth, it presents its limitations; such as customers not being able to physically touch/test products. Technology stand alone cannot always provide a desirable experience for a customer. "Boring" companies that are in high entry barrier industries with high switching costs and price-sensitive buyers can be more profitable than "tech savvy" companies.
For example, quite commonly websites with menus and online booking options attract customers to a restaurant. But the restaurant experience cannot be delivered online with the use of technology. Food delivery companies like Uber Eats can deliver food to customers but cannot replace the restaurant's atmospheric experience.
=== Government ===
Government cannot be a standalone force as it is a factor that can affect the firms structure of five forces above. It is neither good or bad for the industry's profitability.
For instance,
patents can raise barriers to entry
supplier power can be raised by union favoritism from government policies
failing companies reorganizing due to bankruptcy laws
=== Complementary products and services ===
Similar to the government above, complementary products/services cannot be a standalone factor because it's not necessarily bad or good for the industry's profitability. Complements occur when a customer benefits from multiple products combined. Individually those standalone products can be redundant. For example, a car would be unusable without petrol/gas and a driver. Or for example, a computer is best used with computer software. This factor is controversial (as discussed below in Criticisms) as many believe it to be 6th Force. However, complements influence the forces more than they form the underlying structure of the market.
For instance, complements can
influence barriers of entry by either lowering or raising it e.g. Apple providing set of tools to develop apps, lowers barriers to entry;
make substitution easier e.g. Spotify replacing CDs
A strategy consultant's job is to identify complements and apply them to the forces above.
== Usage ==
Strategy consultants occasionally use Porter's five forces framework when making a qualitative evaluation of a firm's strategic position. However, for most consultants, the framework is only a starting point and value chain analysis or another type of analysis may be used in conjunction with this model. Like all general frameworks, an analysis that uses it to the exclusion of specifics about a particular situation is considered naïve .
According to Porter, the five forces framework should be used at the line-of-business industry level; it is not designed to be used at the industry group or industry sector level. An industry is defined at a lower, more basic level: a market in which similar or closely related products and/or services are sold to buyers (see industry information). A firm that competes in a single industry should develop, at a minimum, one five forces analysis for its industry. Porter makes clear that for diversified companies, the primary issue in corporate strategy is the selection of industries (lines of business) in which the company will compete. The average Fortune Global 1,000 company competes in 52 industries.
== Criticisms ==
Porter's framework has been challenged by other academics and strategists. For instance, Kevin P. Coyne and Somu Subramaniam claim that three dubious assumptions underlie the five forces:
That buyers, competitors, and suppliers are unrelated and do not interact and collude.
That the source of value is a structural advantage (creating barriers to entry).
That uncertainty is low, allowing participants in a market to plan for and respond to changes in competitive behavior.
An important extension to Porter's work came from Adam Brandenburger and Barry Nalebuff of Yale School of Management in the mid-1990s. Using game theory, they added the concept of complementors (also called "the 6th force") to try to explain the reasoning behind strategic alliances. Complementors are known as the impact of related products and services already in the market. The idea that complementors are the sixth force has often been credited to Andrew Grove, former CEO of Intel Corporation. Martyn Richard Jones, while consulting at Groupe Bull, developed an augmented five forces model in Scotland in 1993. It is based on Porter's Framework and includes Government (national and regional) as well as pressure groups as the notional 6th force. This model was the result of work carried out as part of Groupe Bull's Knowledge Asset Management Organisation initiative.
Porter indirectly rebutted the assertions of other forces, by referring to innovation, government, and complementary products and services as "factors" that affect the five forces.
It is also perhaps not feasible to evaluate the attractiveness of an industry independently of the resources that a firm brings to that industry. It is thus argued (Wernerfelt 1984) that this theory be combined with the resource-based view (RBV) in order for the firm to develop a sounder framework.
Other criticisms include:
It places too much weight on the macro-environment and does not assess more specific areas of the business that also impact competitiveness and profitability
It does not provide any actions to help deal with high or low force threats (e.g., what should management do if there is a high threat of substitution?)
== See also ==
Competition
Economics of Strategy
Industry classification
Marketing Strategy
National Diamond
Strategic management
Porter's four corners model
Nonmarket forces
Value chain
Marketing management
Enshittification
== References ==
== Further reading ==
Coyne, K.P. and Sujit Balakrishnan (1996),Bringing discipline to strategy, The McKinsey Quarterly, No.4.
Porter, M.E. (March–April 1979) How Competitive Forces Shape Strategy, Harvard Business Review.
Porter, M.E. (1980) Competitive Strategy, Free Press, New York.
Porter, M.E. (January 2008) The Five Competitive Forces That Shape Strategy, Harvard Business Review.
Ireland, R. D., Hoskisson, R. and Hitt, M. (2008). Understanding business strategy: Concepts and cases. Cengage Learning.
Rainer R.K. and Turban E. (2009), Introduction to Information Systems (2nd edition), Wiley, pp 36–41.
Kotler P. (1997), Marketing Management, Prentice-Hall, Inc.
Mintzberg, H., Ahlstrand, B. and Lampel J. (1998) Strategy Safari, Simon & Schuster. | Wikipedia/Porter's_five_forces_analysis |
Algorithmic game theory (AGT) is an interdisciplinary field at the intersection of game theory and computer science, focused on understanding and designing algorithms for environments where multiple strategic agents interact. This research area combines computational thinking with economic principles to address challenges that emerge when algorithmic inputs come from self-interested participants.
In traditional algorithm design, inputs are assumed to be fixed and reliable. However, in many real-world applications—such as online auctions, internet routing, digital advertising, and resource allocation systems—inputs are provided by multiple independent agents who may strategically misreport information to manipulate outcomes in their favor. AGT provides frameworks to analyze and design systems that remain effective despite such strategic behavior.
The field can be approached from two complementary perspectives:
Analysis: Evaluating existing algorithms and systems through game-theoretic tools to understand their strategic properties. This includes calculating and proving properties of Nash equilibria (stable states where no participant can benefit by changing only their own strategy), measuring price of anarchy (efficiency loss due to selfish behavior), and analyzing best-response dynamics (how systems evolve when players sequentially optimize their strategies).
Design: Creating mechanisms and algorithms with both desirable computational properties and game-theoretic robustness. This sub-field, known as algorithmic mechanism design, develops systems that incentivize truthful behavior while maintaining computational efficiency.
Algorithm designers in this domain must satisfy traditional algorithmic requirements (such as polynomial-time running time and good approximation ratio) while simultaneously addressing incentive constraints that ensure participants act according to the system's intended design.
== History ==
=== Nisan-Ronen: a new framework for studying algorithms ===
In 1999, the seminal paper of Noam Nisan and Amir Ronen drew the attention of the Theoretical Computer Science community to designing algorithms for selfish (strategic) users. As they claim in the abstract:
We consider algorithmic problems in a distributed setting where the participants cannot be assumed to follow the algorithm but rather their own self-interest. As such participants, termed agents, are capable of manipulating the algorithm, the algorithm designer should ensure in advance that the agents’ interests are best served by behaving correctly.
Following notions from the field of mechanism design, we suggest a framework for studying such algorithms. In this model the algorithmic solution is adorned with payments to the participants and is termed a mechanism. The payments should be carefully chosen as to motivate all participants to act as the algorithm designer wishes. We apply the standard tools of mechanism design to algorithmic problems and in particular to the shortest path problem.
This paper coined the term algorithmic mechanism design and was recognized by the 2012 Gödel Prize committee as one of "three papers laying foundation of growth in Algorithmic Game Theory".
=== Price of Anarchy ===
The other two papers cited in the 2012 Gödel Prize for fundamental contributions to Algorithmic Game Theory introduced and developed the concept of "Price of Anarchy".
In their 1999 paper "Worst-case Equilibria", Koutsoupias and Papadimitriou proposed a new measure of the degradation of system efficiency due to the selfish behavior of its agents: the ratio of between system efficiency at an optimal configuration, and its efficiency at the worst Nash equilibrium. (The term "Price of Anarchy" only appeared a couple of years later.)
=== The Internet as a catalyst ===
The Internet created a new economy—both as a foundation for exchange and commerce, and in its own right. The computational nature of the Internet allowed for the use of computational tools in this new emerging economy. On the other hand, the Internet itself is the outcome of actions of many. This was new to the classic, ‘top-down’ approach to computation that held till then. Thus, game theory is a natural way to view the Internet and interactions within it, both human and mechanical.
Game theory studies equilibria (such as the Nash equilibrium). An equilibrium is generally defined as a state in which no player has an incentive to change their strategy. Equilibria are found in several fields related to the Internet, for instance financial interactions and communication load-balancing. Game theory provides tools to analyze equilibria, and a common approach is then to ‘find the game’—that is, to formalize specific Internet interactions as a game, and to derive the associated equilibria.
Rephrasing problems in terms of games allows the analysis of Internet-based interactions and the construction of mechanisms to meet specified demands. If equilibria can be shown to exist, a further question must be answered: can an equilibrium be found, and in reasonable time? This leads to the analysis of algorithms for finding equilibria. Of special importance is the complexity class PPAD, which includes many problems in algorithmic game theory.
== Areas of research ==
=== Algorithmic mechanism design ===
Mechanism design is the subarea of economics that deals with optimization under incentive constraints. Algorithmic mechanism design considers the optimization of economic systems under computational efficiency requirements. Typical objectives studied include revenue maximization and social welfare maximization.
=== Inefficiency of equilibria ===
The concepts of price of anarchy and price of stability were introduced to capture the loss in performance of a system due to the selfish behavior of its participants. The price of anarchy captures the worst-case performance of the system at equilibrium relative to the optimal performance possible. The price of stability, on the other hand, captures the relative performance of the best equilibrium of the system. These concepts are counterparts to the notion of approximation ratio in algorithm design.
=== Complexity of finding equilibria ===
The existence of an equilibrium in a game is typically established using non-constructive fixed point theorems. There are no efficient algorithms known for computing Nash equilibria. The problem is complete for the complexity class PPAD even in 2-player games. In contrast, correlated equilibria can be computed efficiently using linear programming, as well as learned via no-regret strategies.
=== Computational social choice ===
Computational social choice studies computational aspects of social choice, the aggregation of individual agents' preferences. Examples include algorithms and computational complexity of voting rules and coalition formation.
Other topics include:
Algorithms for computing Market equilibria
Fair division
Multi-agent systems
And the area counts with diverse practical applications:
Sponsored search auctions
Spectrum auctions
Cryptocurrencies
Prediction markets
Reputation systems
Sharing economy
Matching markets such as kidney exchange and school choice
Crowdsourcing and peer grading
Economics of the cloud
== Journals and newsletters ==
ACM Transactions on Economics and Computation (TEAC)
SIGEcom Exchanges
Algorithmic Game Theory papers are often also published in Game Theory journals such as GEB, Economics journals such as Econometrica, and Computer Science journals such as SICOMP.
== See also ==
Auction Theory
Computational social choice
Gamification
Load balancing (computing)
Mechanism design
Multi-agent system
Voting in game theory
== References ==
John von Neumann, Oskar Morgenstern (1944) Theory of Games and Economic Behavior. Princeton Univ. Press. 2007 edition: ISBN 978-0-691-13061-3
Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007), Algorithmic Game Theory (PDF), Cambridge, UK: Cambridge University Press, ISBN 978-0-521-87282-9.
== External links ==
gambit.sourceforge.net - a library of game theory software and tools for the construction and analysis of finite extensive and strategic games.
gamut.stanford.edu - a suite of game generators designated for testing game-theoretic algorithms. | Wikipedia/Algorithmic_game_theory |
Game Theory was an American power pop band, founded in 1982 by singer/songwriter Scott Miller, combining melodic jangle pop with dense experimental production and hyperliterate lyrics. MTV described their sound as "still visceral and vital" in 2013, with records "full of sweetly psychedelic-tinged, appealingly idiosyncratic gems" that continued "influencing a new generation of indie artists." Between 1982 and 1990, Game Theory released five studio albums and two EPs, which had long been out of print until 2014, when Omnivore Recordings began a series of remastered reissues of the entire Game Theory catalog. Miller's posthumously completed Game Theory album, Supercalifragile, was released in August 2017 in a limited first pressing.
Miller was the group's leader and sole constant member, presiding over frequently changing line-ups. During its early years in Davis, California, Game Theory was often associated with the Paisley Underground movement, but remained in northern California, moving to the Bay Area in 1985, while similarly aligned local bands moved to Los Angeles.
The group became known for its fusion of catchy musical hooks with musical complexity, as well as for Miller's lyrics that often featured self-described "young-adult-hurt-feeling-athons," along with literary references (e.g., Real Nighttime's allusions to James Joyce), and pop culture references ranging from Peanuts ("The Red Baron") to Star Trek quotes ("One More for St. Michael").
== Musical career ==
=== Transition from Alternate Learning (1982) ===
Prior to founding Game Theory, Scott Miller had been the lead singer and songwriter of Alternate Learning, which issued an EP in 1979 and an LP in 1981. Alternate Learning was based in Sacramento and Davis, California, and frequently performed at U.C. Davis. Two members of the band, Jozef Becker and Nancy Becker, would join Miller in Game Theory. Alternate Learning was disbanded in early 1982.
=== Meaning of "Game Theory" ===
Scott Miller chose to name his new band "Game Theory" as an allusion to the mathematical field of game theory, which he described as "the study of calculating the most appropriate action given an adversary, ... someone who was thinking against you, and you had to organize what his moves could be, and what your moves should be, to give yourself the minimum amount of failure." In a 1988 interview, Miller stated, "It's a theory of probability that's a mathematical discipline that more or less has been applied improperly to real-life situations. It's just that idea of a set of rules that gets misused that intrigued me about it ... kind of a telling comment on life in general—that you just have to have some sort of set of rules, but who knows what the set of rules should be." That theme, according to Miller, was what many Game Theory songs were about: "Always be wary of the superstructure of whatever situation you're in. It may just be that the whole game that you're into is something very bogus and you should get out."
=== Early Davis-based years (1982–1985) ===
By mid-1982, Scott Miller had assembled the first iteration of Game Theory, with himself as lead guitarist and vocalist. The group consisted of Miller, Nancy Becker (keyboards, vocals), Fred Juhos (bass, guitar, vocals), and Michael Irwin (drums).
The first Game Theory album was the Blaze of Glory LP, released on Rational Records in 1982. Due to a lack of funds to both press the album and print a jacket, a thousand copies of the LP were packaged in white plastic trash bags with Xeroxed cover art glued to each bag.
Nearly thirty years after the release of Blaze of Glory, Harvard professor Stephanie Burt described it as "true to the wordy awkwardness ... of the nerd stereotype, and yet true to the visceral power, the sexual charge, in guitar-based Anglo-American pop. The songs, and the people depicted in the songs, attempted to have fun, to act on instinct, but they knew they were too cerebral to make it so, except with like-minded small circles of puzzle-solvers."
With Dave Gill replacing Michael Irwin on drums, two 12-inch EPs followed. In 1983, the group released the six-song EP Pointed Accounts of People You Know, recorded at Samurai Sound Studio, which was co-owned by Gill. The group then recorded the five-song Distortion EP in December 1983 (released 1984), with The Three O'Clock's Michael Quercio producing. The first three releases, originally released on Rational, were anthologized by Alias Records in 1993 as the Distortion of Glory CD.
The early Game Theory was described as a "pseudo-psychedelic pop quartet" for which Miller sang and wrote "almost all of the material." On the first three releases, Miller shared co-writing credits on "The Young Drug" with Alternate Learning's Carolyn O'Rourke, and on "Life in July" with Nancy Becker. Miller also included three songs that were written by Fred Juhos, and later defended the decision to record Juhos's songs as a Beatles-like "relief from seriousness", though only one was included in the Distortion of Glory compilation. Juhos's contributions were criticized as failing to mesh with Miller's, and Miller later mused, "It's funny that his stuff wasn't popular. We all had the impression that no one was ever going to get into my stuff and that his one or two would be the ones to catapult us to fame."
Reviewers of Distortion of Glory wrote that the band had improved with each successive EP, both featuring "some stellar material." Notable songs included "The Red Baron", cited as "heartbreaking ... an anguished acoustic lost-love song leavened by keyboardist Nancy Becker's mocking 'fifty or more' backing vocal," as well as "Shark Pretty," which featured guest lead guitar by Bowie sideman Earl Slick (credited as Ernie Smith).
In 1984, the Dead Center LP was released in France, on the Lolita label. Dead Center was a compilation of selected tracks from Pointed Accounts of People You Know and Distortion, with three additional tracks including the group's cover of "The Letter" (a 1967 hit for the Box Tops with Alex Chilton's vocals).
Real Nighttime, recorded in July 1984, marked the entrance of Mitch Easter as producer for the band's remaining releases. Easter was also credited as a guest musician on Real Nighttime, along with Quercio and Jozef Becker.
The album was well-reviewed, appearing in the Village Voice's annual poll of 1984's best releases. One critic said the album walked "a fine line between pretension and genius." Miller contributed liner notes he penned in the style of James Joyce's Finnegans Wake, and the record sported "chiming guitars and great pop melodies" described as "breathtaking."
Reviewers wrote, and Miller later confirmed, that a recurring theme in the lyrics of Real Nighttime was life after college, which Miller paired with the intuition that "freedom had a strong aspect of being bad news." The song "24" placed the narrator at the cusp of a quarter-life crisis, as a self-conscious young adult whose mixed feelings established that he "doesn't know where he fits, or to how to live on his own, in a post-collegiate milieu." The theme continued with allusions to finding one's own direction and leaving the nest, as in "Curse of the Frontier Land" ("A year ago we called this a good time"), and "I Mean It This Time" ("Give me all the gin I need, for I may not be this strong when I call my parents and tell them they've been wrong.")
After commencing a national tour for Real Nighttime in October 1984, but before the album's 1985 release, the group went through a wholesale change in personnel, with only Miller remaining. According to Spin, the band had "lost one original member to motherhood and one to Jesus." As a result, a photograph of Miller was substituted for a photograph of the full group that had previously been taken for the album cover.
In 2013, after Scott Miller's death, the group's surviving members from this period (including both Irwin and Gill) briefly adopted the nickname "Game Theory 1.0," coined by Juhos during planning of the band's July 2013 reunion performance in a memorial tribute to Miller, to describe the pre-1985 version of the group's line-up.
=== The Big Shot Chronicles (1985–1986) ===
By early 1985, Miller had moved from Davis to the San Francisco Bay Area, where he assembled a new lineup featuring keyboardist Shelley LaFreniere, drummer Gil Ray and, on bass, Suzi Ziegler. The San Francisco version of Game Theory commenced a new national tour supporting Real Nighttime in 1985. The tour over, Ziegler left the band.
The Big Shot Chronicles was recorded in September 1985 at Mitch Easter's Drive-In Studio in Winston-Salem, during the middle of the band's tour. Twenty years later, Miller recalled the sessions as "the most effortless studio experience I've ever had," taking place "in a period of my life when being involved with the music business was surprisingly enjoyable."
Billboard pointed to The Big Shot Chronicles' "crisp, moody pop songs," taking note of Miller's high tenor vocals "sung in a self-described 'miserable whine'", and adding that Easter lent "an assured production touch" to this "collegiate fave."
According to Spin, the 1986 album sold more copies in its first few weeks of release, thanks to a distribution deal with Capitol Records, than all of Game Theory's previous records combined. Spin's review paired The Big Shot Chronicles with Real Nighttime by calling both albums "a rare commodity ... a pop record that can actually make you laugh and cry and squirm all at once." The Big Shot Chronicles was distinguished as "harsh, dense, and metallic-sounding," and "damned ambitious as pop fare goes nowadays, with difficult time signatures, criss-cross rhythms, off-beat chordings, and surreal, vertiginous lyrics."
Among college audiences, a contemporaneous review pointed to the band's originality in a genre "so codified that a little change in tradition is apocalyptic," citing the band's experimental notes as quirky and bizarre, yet "such loving care is taken with the obvious influences that you appreciate the music for simply reaffirming everything that's right about pop. It's one of the most important reasons for liking Game Theory, because any band with good taste is worth saving from obscurity."
Decades later, in the 2007 book Shake Some Action: The Ultimate Power Pop Guide, The Big Shot Chronicles was ranked No. 16 out of the "Top 200 power pop albums of all time." The reviewer noted, "Nowhere are Miller's eccentricities more consistently tuneful and genius-like than on The Big Shot Chronicles," citing the song "Regenisraen" as "absolutely gorgeous, hymn-like," among other "top-shelfers." The release was, however, "surprisingly passed over by the buying public."
=== Lolita Nation and Two Steps from the Middle Ages (1986–1988) ===
For the band's October–November 1986 national tour supporting the release of The Big Shot Chronicles, Game Theory took on two new members, resulting in the line-up of Scott Miller (lead vocal, guitars), Shelley LaFreniere (keyboards), Gil Ray (drums), Guillaume Gassuan (bass), and Donnette Thayer (backing vocal, guitars). Thayer, who was then Miller's girlfriend, had been a guest musician on Game Theory's first album, Blaze of Glory. This iteration of the band recorded two albums, released in 1987 and 1988.
In a review of the double set Lolita Nation, Spin cited it as "some of the gutsiest, most distinctive rock 'n' roll heard in 1987," with "sumptuous melodic hooks ... played with startling intensity and precision," while simultaneously noting that the band "elected to shinny way out on an aesthetic limb" with "a thoroughly perplexing conglomeration of brief instrumental shards and stabs". Miller told the San Francisco Chronicle that, with Lolita Nation, he "wanted to throw away some of the givens. It's meant to have a lot of unexpected things happening on it without being abrasive or industrial," labeling the music "experimental pop." The CD version of Lolita Nation, long out of print, has since become a collector's item.
The group's 1988 release, Two Steps from the Middle Ages, took a less experimental approach, but despite numerous positive reviews and airplay on college radio, the album failed to reach a mainstream audience. Spin wrote:
Good — even great — pop songs are Scott Miller's specialty ... creating essential California rock 'n' roll for the 80s – tense, bristling energy, ingenious hooks and haunting melodies that ought to spell commercial potential. But the albums have remained stuck in the cultist-critic-college DJ loop.
One problem is that Game Theory's obvious debt to Alex Chilton ... and their association with Mitch Easter ... got them lumped in with a whole genre of pop-for-pop's-sake smarty-pants, too coyly clever for their own good. But Game Theory has always rocked harder and thought bigger than the other "quirky popsters."
Practical factors also got in the way of greater success. Soon after the release of Two Steps, their record label, Enigma Records, went out of business. In addition, there were conflicts within the group. After the 1988 tour, Donnette Thayer left the group to form Hex with Steve Kilbey of The Church. LaFreniere and Gassuan left the group at that time as well, and Ray sustained a disabling back injury that rendered him temporarily unable to play drums.
=== Touring and final recordings (1989–1990) ===
In 1989, Miller convened another new version of Game Theory, which toured in 1989 and 1990. The line-up consisted of Miller (lead vocal, guitars), Michael Quercio (bass, drums, backing vocals), Jozef Becker (drums, bass), and Gil Ray, who was shifted by Miller from drums to playing guitar and keyboards. Jozef Becker had been a member of Miller's previous band Alternate Learning, and had played as a guest musician on earlier Game Theory releases. Quercio, best known for his previous work as frontman of The Three O'Clock, also had a long affiliation with Game Theory, having produced the 1984 Distortion EP, and having appeared as a guest musician on Real Nighttime and Lolita Nation.
Prior to the group's 1989 "mini-tour" of the Northwestern United States, Ray was a victim of random street violence in San Francisco, resulting in a serious eye injury. Ray ultimately left the group in 1990, and the group briefly continued as a trio.
Game Theory's penultimate recording sessions took place in April 1989, when Nancy Becker, the group's original keyboard player and backup vocalist in the early 1980s, returned to record new versions of three songs for the compilation Tinker to Evers to Chance. The re-recorded songs included one Alternate Learning song, and two from the band's first LP, Blaze of Glory.
In late 1989, the line-up of Miller, Quercio, Ray, and Jozef Becker recorded a demo in San Francisco, co-produced by Miller and Dan Vallor, with four songs that included "Inverness" and "Idiot Son" (both later to be performed by the Loud Family) and, with Quercio taking on lead vocals, "My Free Ride.": 90 The London-based tabloid Bucketfull of Brains wrote, "One listen to this latest demo ... and you can't help but wonder if pop music can get any better than this."
In a 1990 interview promoting the release of Tinker to Evers to Chance, Miller laughed that Game Theory stood at "a rocky pitfall-ridden crossroad," and Quercio noted, "When a major label hears someone like Scott or me sing, they say, 'That doesn't really sound like anybody,' and don't know what market to plug it into ... Sometimes originality is your worst enemy."
=== Transition to the Loud Family (1991) ===
By 1991, Quercio had left Game Theory, opting to return to Los Angeles to form the band Permanent Green Light. With Jozef Becker remaining as drummer, Miller recruited three new members to join Game Theory in 1991. This new line-up had rehearsed several times as Game Theory before Miller decided that the differences in sound and energy warranted a new name for the group, which began performing in the Bay Area in 1991 as the Loud Family. Game Theory's Gil Ray later returned to drumming as a member of the Loud Family, beginning with their 1998 album Days for Days.
== Game Theory after Scott Miller ==
=== Reunion of Game Theory (2013) ===
Scott Miller had been making preparations to reunite Game Theory before he died unexpectedly on April 15, 2013.
The surviving original members of Game Theory reunited on July 20, 2013, to perform a memorial concert in Miller's hometown of Sacramento. Game Theory's 2013 line-up included Nancy Becker (keyboards, backing vocals), Fred Juhos (bass, piano), Michael Irwin (drums), Dave Gill (drums), and lead vocalist Alison Faith Levy of the Loud Family. Guest performers included Steve Harris of Urban Sherpas (lead guitar), and Bradley Skaught of The Bye Bye Blackbirds (vocals). An acoustic opening set was performed by Game Theory members Gil Ray (guitar, vocals) and Suzi Ziegler (vocals), with Alison Faith Levy (vocals).
=== Supercalifragile (2017) ===
Miller's record label, 125 Records, revealed after Miller's death in April 2013 that "Scott had been planning to start recording a new Game Theory album, Supercalifragile, this summer, and was looking forward to getting back into the studio and reuniting with some of his former collaborators." Supercalifragile was to be the band's first album of new material since Two Steps from the Middle Ages in 1988.
In September 2015, Miller's wife Kristine Chambers announced that she and Ken Stringfellow had teamed to produce a finished recording from the source material for Supercalifragile that Miller had left behind in various stages of completion, "including fully-formed songs and many other ideas, sketches, lyrics, even musical gestures and snippets of found sound." A preliminary decision to release the album under Scott Miller's name, using the title I Love You All, was later reconsidered in favor of Miller's original plans for a Game Theory project.
On May 5, 2016, it was announced that the project, now under Miller's planned title Supercalifragile as the sixth and final Game Theory album, would be released in early 2017. A Kickstarter campaign, created to fund the pressing and other expenses involved with completing the album, was fully funded within two weeks.
Recording sessions that included Anton Barbeau, Jozef Becker, Stéphane Schück, and Stringfellow took place in the summer of 2015 at Abbey Road Studios in London. Sessions with Game Theory members Nan Becker, Dave Gill, Gil Ray, and Suzi Ziegler, in late May and early June 2016, were held at Sharkbite Studio in Oakland. Additional members of Game Theory who appeared included Fred Juhos, Donnette Thayer, and Shelley LaFreniere, along with The Loud Family's Alison Faith Levy.
Other friends and former collaborators involved as performers and co-songwriters included Aimee Mann, Jon Auer of the Posies, Doug Gillard, Ted Leo, Will Sheff, and Matt LeMay. The contributors also included Peter Buck of R.E.M., John Moremen, and Jonathan Segel. Mitch Easter, Game Theory's former producer, played guitar, drums, and synth on the song "Laurel Canyon," and mixed two tracks.
Drummer Gil Ray died on January 24, 2017, at the age of 60.
Supercalifragile was released in August 2017, first to Kickstarter backers and then publicly through Bandcamp on August 24.
== Reissues of Game Theory albums ==
=== Rarity and unavailability ===
In 1993, Alias Records (which had recently signed the Loud Family) re-released the Game Theory albums Real Nighttime and The Big Shot Chronicles on CD, with additional bonus tracks. Alias also released the CD compilation Distortion of Glory, combining Game Theory's Blaze of Glory LP and material from the Pointed Accounts and Distortion EPs.
For over 25 years, from the time of their initial release on Enigma until after Miller's death, the albums Lolita Nation (1987) and Two Steps from the Middle Ages (1988), and the compilation Tinker to Evers to Chance (1990), were not re-issued on CD and became rare collectors' items. Despite approaches by more than one label and Miller's public offer of cooperation, Game Theory's catalog remained out of print until 2014, due to what Miller understood to be rights issues that prevented physical access to the original master recordings.
Over the decades, the increasing difficulty of finding copies of Game Theory albums contributed to the band's inability to transcend what Miller described as "national obscurity, as opposed to regional obscurity." In 2013, MTV wrote of "Miller's indelible output" and "Game Theory's transcendent tunes" as a "legacy ... ready and waiting for discovery."
=== Reissues on Omnivore Recordings (2014–) ===
In July 2014, Omnivore Recordings announced their commitment to reissue Game Theory's recordings, remastered from the original tapes. Noting that Miller's work with Game Theory had been out of print and "missing for decades," Omnivore stated that they were "pleased to right that audio wrong" with a series of expanded reissues of the group's catalog. The reissue series is produced by Pat Thomas, Dan Vallor (Game Theory's tour manager and sound engineer during the 1980s), and Grammy-winning producer Cheryl Pawelski.
The first in the series, an expanded version of Game Theory's 1982 debut album Blaze of Glory, was released in September 2014, on CD and vinyl. In addition to the 12 original tracks, the reissue was supplemented with 15 bonus tracks (four from Alternate Learning, and 11 previously unissued recordings). The first pressing of the reissued vinyl LP was on translucent pink vinyl, with black to follow. The reissue also included a booklet with essays and remembrances from band members and colleagues, including Steve Wynn of The Dream Syndicate. The booklet also included previously unreleased images by photographer Robert Toren, some of which appeared in Omnivore's promotional video for the release launch.
Omnivore's November 2014 expanded reissue of Dead Center, on CD only, included material from the Game Theory EPs Pointed Accounts of People You Know (1983) and Distortion (1984), reissued on vinyl only.
The reissue of Real Nighttime (1985), the first of Game Theory's albums to be produced by Mitch Easter, was released in 2015 on CD and red vinyl, with 13 bonus tracks and liner notes that included new essays by Byron Coley and The New Pornographers' A.C. Newman, as well as an interview with Easter.
Departing from chronological order, Omnivore's February 2016 reissue of Lolita Nation was a double CD set, with the second disc featuring 21 bonus tracks. A concurrent double LP release, with its first run in a limited edition on dark green translucent vinyl, included a download card providing the full 48-track CD program.
Omnivore followed with reissues of The Big Shot Chronicles in September 2016 and Two Steps from the Middle Ages in June 2017.
In 2020, Omnivore concluded the series of reissues by releasing Across the Barrier of Sound: PostScript, a compilation album consisting of material recorded in 1989 and 1990, featuring previously unreleased songs from Game Theory's final lineup.
== Timeline ==
== Discography ==
Studio albums
Blaze of Glory (1982)
Real Nighttime (1985)
The Big Shot Chronicles (1986)
Lolita Nation (1987)
Two Steps from the Middle Ages (1988)
Supercalifragile (2017)
== References ==
== External links ==
Official website
Game Theory at AllMusic
Game Theory discography at Discogs | Wikipedia/Game_Theory_(band) |
The foundations of negotiation theory are decision analysis, behavioral decision-making, game theory, and negotiation analysis.
Another classification of theories distinguishes between Structural Analysis, Strategic Analysis, Process Analysis, Integrative Analysis, and behavioral analysis of negotiations.
Negotiation is a strategic discussion that resolves an issue in a way that both parties find acceptable. Individuals should make separate, interactive decisions; and negotiation analysis considers how groups of reasonably bright individuals should and could make joint, collaborative decisions. These theories are interleaved and should be approached from the synthetic perspective.
== Common assumptions of most theories ==
Negotiation is a specialized and formal version of conflict resolution, most frequently employed when important issues must be agreed upon. Negotiation is necessary when one party requires the other party's agreement to achieve its aim. The aim of negotiating is to build a shared environment leading to long-term trust, and it often involves a third, neutral party to extract the issues from the emotions and keep the individuals concerned focused. It is a powerful method for resolving conflict and requires skill and experience. Henry Kissinger defined negotiation as "a process of combining conflicting positions into a common position under a decision rule of unanimity, a phenomenon in which the outcome is determined by the process." Druckman adds that negotiations pass through stages that consist of agenda-setting, a search for guiding principles, defining the issues, bargaining for favorable concession exchanges, and a search for implementing details. Transitions between stages are referred to as turning points.
Most theories of negotiations share the notion of negotiations as a process, but they differ in their description of the process.
Structural, strategic, and procedural analysis builds on rational actors, who are able to prioritize clear goals, are able to make trade-offs between conflicting values, are consistent in their behavioral patterns, and are able to take uncertainty into account.
Negotiations differ from mere coercion, in that negotiating parties have the theoretical possibility to withdraw from negotiations. It is easier to study bilateral negotiations, as opposed to multilateral negotiations.
== Structural analysis ==
Structural Analysis is based on a distribution of empowering elements among two negotiating parties. Structural theory moves away from traditional Realist notions of power in that it does not only consider power to be a possession, manifested for example in economic or military resources, but also thinks of power as a relation.
Based on the distribution of elements, in structural analysis we find either power-symmetry between equally strong parties or power-asymmetry between a stronger and a weaker party. All elements from which the respective parties can draw power constitute structure. They may be of material nature, i.e., hard power (such as weapons), or of social nature, i.e., soft power (such as norms, contracts, or precedents).
These instrumental elements of power, are either defined as parties’ relative position (resources position) or as their relative ability to make their options prevail.
According to structural analysis, negotiations can be described with matrices, such as the Prisoner's dilemma, a concept taken from game theory. Another common example is the game of Chicken.
Structural analysis is easy to criticize, because it predicts that the strongest will always win. This, however, does not always hold true.
== Strategic analysis ==
Strategic analysis starts with the assumption that both parties have a veto. Thus, in essence, negotiating parties can cooperate (C) or defect (D). Structural analysis then evaluates Á outcomes of negotiations (C, C; C, D; D, D; D, C), by assigning values to each of the possible outcomes.
Often, cooperation of both sides yields the best outcome. The problem is that the parties can never be sure that the other is going to cooperate, mainly because of two reasons: first, decisions are made at the same time or, second, concessions of one side might not be returned. Therefore, the parties have contradicting incentives to cooperate or defect. If one party cooperates or makes a concession and the other does not, the defecting party might relatively gain more.
Trust may be built only in repetitive games through the emergence of reliable patterns of behavior, such as tit-for-tat.
== Process analysis ==
Process analysis is the theory closest to haggling.
Process Analysis focuses on the study of the dynamics of processes. E.g., both Zeuthen and Cross tried to find a formula in order to predict the behavior of the other party in finding a rate of concession, in order to predict the likely outcome. Process analysis is the main resource in this chapter of negotiation.
The process of negotiation, therefore, is considered to unfold between fixed points: starting point of discord, endpoint of convergence. The so-called security point, which is the result of optional withdrawal, is also taken into account.
An important feature of negotiation processes is the idea of turning points (TPs). A considerable amount of research has been devoted to analyses of TPs in single and comparative case studies, as well as experiments. Considered as departures in the process, Druckman has proposed a three-part framework for analysis in which precipitating events precede (and cause) departures which have immediate and delayed consequences. Precipitating events can be external as when a mediator becomes involved, substantive as when a new idea is proposed, or procedural as when the formal plenary structure becomes divided into committees. Departures can be abrupt or relatively slow and consequences can escalate, moving away from agreement, or they might move in the direction of agreement. Using this framework in a comparative study of 34 cases, Druckman discovered that external events were needed to move talks on security or arms control toward agreement. However, new ideas or changed procedures were more important for progress in trade or political negotiations. Different patterns were also found for interest-based, cognitive-based, and values-based conflicts and between domestic and international negotiations.
Turning points are also analyzed in relation to negotiation crises or disruptions in the flow of the talks. Earlier research showed that TPs are more likely to occur in the context of crises, often in the form of changes that put the talks back on track and transition to a new stage (Druckman, 1986, 2001). A key to resolving crises is reframing the issues being discussed. The choice to reframe was shown to occur more frequently among negotiators when their trust is low and transaction costs are high. The research to date on TPs has generated ideas likely to stimulate further studies. Some of these ideas include a search for the underlying mechanisms that can explain the emergence of TPs. Foremost among these are flexibility and adaptability in response to crises or violations of expected behavior. The key challenge is to discover the conditions that foster progress toward a solution to the dilemma of balancing the desire to agree with the desire to come out favorably. For a review of the research on turning points, see Druckman and Olekalns.
== Integrative analysis ==
Integrative analysis divides the process into successive stages, rather than talking about fixed points. It extends analysis to pre-negotiations stages, in which parties make first contacts. The outcome is explained as the performance of the actors at different stages. Stages may include pre-negotiations, finding a formula of distribution, crest behavior, settlement
== Bad faith negotiation ==
Bad faith is a concept in negotiation theory whereby parties pretend to reason to reach settlement, but have no intention to do so, for example, one political party may pretend to negotiate, with no intention to compromise, for political effect.
=== Inherent bad faith model in international relations and political psychology ===
Bad faith in political science and political psychology refers to negotiating strategies in which there is no real intention to reach compromise, or a model of information processing. The "inherent bad faith model" of information processing is a theory in political psychology that was first put forth by Ole Holsti to explain the relationship between John Foster Dulles’ beliefs and his model of information processing. It is the most widely studied model of one's opponent. A state is presumed to be implacably hostile, and contra-indicators of this are ignored. They are dismissed as propaganda ploys or signs of weakness. Examples are John Foster Dulles’ position regarding the Soviet Union, or Israel's initial position on the Palestine Liberation Organization.
== See also ==
Argumentation theory
Dispute resolution
List of books about negotiation
Morphological analysis
Negotiation
Vicente Blanco Gaspar
== References == | Wikipedia/Negotiation_theory |
Algorithmic mechanism design (AMD) lies at the intersection of economic game theory, optimization, and computer science. The prototypical problem in mechanism design is to design a system for multiple self-interested participants, such that the participants' self-interested actions at equilibrium lead to good system performance. Typical objectives studied include revenue maximization and social welfare maximization. Algorithmic mechanism design differs from classical economic mechanism design in several respects. It typically employs the analytic tools of theoretical computer science, such as worst case analysis and approximation ratios, in contrast to classical mechanism design in economics which often makes distributional assumptions about the agents. It also considers computational constraints to be of central importance: mechanisms that cannot be efficiently implemented in polynomial time are not considered to be viable solutions to a mechanism design problem. This often, for example, rules out the classic economic mechanism, the Vickrey–Clarke–Groves auction.
== History ==
Noam Nisan and Amir Ronen first coined "Algorithmic mechanism design" in a research paper published in 1999.
== See also ==
Algorithmic game theory
Computational social choice
Metagame
Incentive compatible
Vickrey–Clarke–Groves mechanism
== References and notes ==
== Further reading ==
Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.
Dütting, Paul; Geiger, Andreas (May 9, 2007), Algorithmic Mechanism Design (PDF), Seminar Report, University of Karlsruhe, Fakultät für Informatik, archived from the original (PDF) on June 13, 2015, retrieved June 11, 2015. | Wikipedia/Algorithmic_mechanism_design |
In philosophy of science, idealization is the process by which scientific models assume facts about the phenomenon being modeled that are strictly false but make models easier to understand or solve. That is, it is determined whether the phenomenon approximates an "ideal case," then the model is applied to make a prediction based on that ideal case.
If an approximation is accurate, the model will have high predictive power; for example, it is not usually necessary to account for air resistance when determining the acceleration of a falling bowling ball, and doing so would be more complicated. In this case, air resistance is idealized to be zero. Although this is not strictly true, it is a good approximation because its effect is negligible compared to that of gravity.
Idealizations may allow predictions to be made when none otherwise could be. For example, the approximation of air resistance as zero was the only option before the formulation of Stokes' law allowed the calculation of drag forces. Many debates surrounding the usefulness of a particular model are about the appropriateness of different idealizations.
== Early use ==
Galileo used the concept of idealization in order to formulate the law of free fall. Galileo, in his study of bodies in motion, set up experiments that assumed frictionless surfaces and spheres of perfect roundness. The crudity of ordinary objects has the potential to obscure their mathematical essence, and idealization is used to combat this tendency.
The most well-known example of idealization in Galileo's experiments is in his analysis of motion. Galileo predicted that if a perfectly round and smooth ball were rolled along a perfectly smooth horizontal plane, there would be nothing to stop the ball (in fact, it would slide instead of roll, because rolling requires friction). This hypothesis is predicated on the assumption that there is no air resistance.
== Other examples ==
=== Mathematics ===
Geometry involves the process of idealization because it studies ideal entities, forms and figures. Perfect circles, spheres, straight lines and angles are abstractions that help us think about and investigate the world.
=== Science ===
An example of the use of idealization in physics is in Boyle's Gas Law:
Given any x and any y, if all the molecules in y are perfectly elastic and spherical, possess equal masses and volumes, have negligible size, and exert no forces on one another except during collisions, then if x is a gas and y is a given mass of x which is trapped in a vessel of variable size and the temperature of y is kept constant, then any decrease of the volume of y increases the pressure of y proportionally, and vice versa.
In physics, people will often solve for Newtonian systems without friction. While we know that friction is present in actual systems, solving the model without friction can provide insights to the behavior of actual systems where the force of friction is negligible.
=== Social science ===
It has been argued by the "Poznań School" (in Poland) that Karl Marx used idealization in the social sciences (see the works written by Leszek Nowak). Similarly, in economic models individuals are assumed to make maximally rational choices. This assumption, although known to be violated by actual humans, can often lead to insights about the behavior of human populations.
In psychology, idealization refers to a defence mechanism in which a person perceives another to be better (or have more desirable attributes) than would actually be supported by the evidence. This sometimes occurs in child custody conflicts. The child of a single parent frequently may imagine ("idealize") the (ideal) absent parent to have those characteristics of a perfect parent. However, the child may find imagination is favorable to reality. Upon meeting that parent, the child may be happy for a while, but disappointed later when learning that the parent does not actually nurture, support and protect as the former caretaker parent had.
A notable proponent of idealization in both the natural sciences and the social sciences was the economist Milton Friedman. In his view, the standard by which we should assess an empirical theory is the accuracy of the predictions that that theory makes. This amounts to an instrumentalist conception of science, including social science. He also argues against the criticism that we should reject an empirical theory if we find that the assumptions of that theory are not realistic, in the sense of being imperfect descriptions of reality. This criticism is wrongheaded, Friedman claims, because the assumptions of any empirical theory are necessarily unrealistic, since such a theory must abstract from the particular details of each instance of the phenomenon that the theory seeks to explain. This leads him to the conclusion that “[t]ruly important and significant hypotheses will be found to have ‘assumptions’ that are wildly inaccurate descriptive representations of reality, and, in general, the more significant the theory, the more unrealistic the assumptions (in this sense).” Consistently with this, he makes the case for seeing the assumptions of neoclassical positive economics as not importantly different from the idealizations that are employed in natural science, drawing a comparison between treating a falling body as if it were falling in a vacuum and viewing firms as if they were rational actors seeking to maximize expected returns.
Against this instrumentalist conception, which judges empirical theories on the basis of their predictive success, the social theorist Jon Elster has argued that an explanation in the social sciences is more convincing when it ‘opens the black box’ — that is to say, when the explanation specifies a chain of events leading from the independent variable to the dependent variable. The more detailed this chain, argues Elster, the less likely it is that the explanation specifying that chain is neglecting a hidden variable that could account for both the independent variable and the dependent variable. Relatedly, he also contends that social-scientific explanations should be formulated in terms of causal mechanisms, which he defines as “frequently occurring and easily recognizable causal patterns that are triggered under generally unknown conditions or with indeterminate consequences.” All this informs Elster's disagreement with rational-choice theory in general and Friedman in particular. On Elster's analysis, Friedman is right to argue that criticizing the assumptions of an empirical theory as unrealistic is misguided, but he is mistaken to defend on this basis the value of rational-choice theory in social science (especially economics). Elster presents two reasons for why this is the case: first, because rational-choice theory does not illuminate “a mechanism that brings about non-intentionally the same outcome that a superrational agent could have calculated intentionally”, a mechanism “that would simulate rationality”; and second, because rational-choice explanations do not provide precise, pinpoint predictions, comparable to those of quantum mechanics. When a theory can predict outcomes that precisely, then, Elster contends, we have reason to believe that theory is true. Accordingly, Elster wonders whether the as-if assumptions of rational-choice theory help explain any social or political phenomenon.
=== Science education ===
In science education, idealized science can be thought of as engaging students in the practices of science and doing so authentically, which means allowing for the messiness of scientific work without needing to be immersed in the complexity of professional science and its esoteric content. This helps the student develop the mindset of a scientist as well as their habits and dispositions. Idealized science is especially important for learning science because of the deeply cognitively and materially distributed nature of modern science, where most science is done by larger groups of scientists. One example is a 2016 gravitational waves paper listing over a thousand authors and more than a hundred science institutions. By simplifying the content, students can engage in all aspects of scientific work and not just add one small piece of the whole project. Idealized Science also helps to dispel the notion that science simply follows a single set scientific method. Instead, idealized science provides a framework for the iterative nature of scientific work, the reliance on critique, and the social aspects that help continually guide the work.
== Limits on use ==
While idealization is used extensively by certain scientific disciplines, it has been rejected by others. For instance, Edmund Husserl recognized the importance of idealization but opposed its application to the study of the mind, holding that mental phenomena do not lend themselves to idealization.
Although idealization is considered one of the essential elements of modern science, it is nonetheless the source of continued controversy in the literature of the philosophy of science. For example, Nancy Cartwright suggested that Galilean idealization presupposes tendencies or capacities in nature and that this allows for generalization beyond what is the ideal case.
There is continued philosophical concern over how Galileo's idealization method assists in the description of the behavior of individuals or objects in the real world. Since the laws created through idealization (such as the ideal gas law) describe only the behavior of ideal bodies, these laws can only be used to predict the behavior of real bodies when a considerable number of factors have been physically eliminated (e.g. through shielding conditions) or ignored. Laws that account for these factors are usually more complicated and in some cases have not yet been developed.
== See also ==
Spherical cow
== References ==
== Further reading ==
William F, Barr, A Pragmatic Analysis of Idealization in Physics, Philosophy of Science, Vol. 41, No. 1, pg 48, Mar. 1974.
Krzysztof Brzechczyn, (ed.), Idealization XIII: Modeling in History, Amsterdam-New York: Rodopi, 2009.
Nancy Cartwright, How the Laws of Physics Lie, Clarendon Press:Oxford 1983
Francesco Coniglione, Between Abstraction and Idealization: Scientific Practice and Philosophical Awareness, in F. Coniglione, R. Poli and R. Rollinger (Eds.), Idealization XI: Historical Studies on Abstraction, Atlanta-Amsterdam:Rodopi 2004, pp. 59–110.
Craig Dilworth, The Metaphysics of Science: An Account of Modern Science in Terms of Principles, Laws and Theories, Springer:Dordrecht 2007 (2a ed.)
Andrzej Klawiter, Why Did Husserl Not Become the Galileo of the Science of Consciousness?, in F. Coniglione, R. Poli and R. Rollinger, (Eds.), Idealization XI: Historical Studies on Abstraction, Poznań Studies in the Philosophy of the Sciences and the Humanities, Vol. 82, Rodopi:Atlanta-Amsterdam 2004, pp. 253–271.
Mansoor Niaz, The Role of Idealization in Science and Its Implications for Science Education, Journal of Science Education and Technology, Vol. 8, No. 2, 1999, pp. 145–150.
Leszek Nowak, The Structure of Idealization. Towards a Systematic Interpretation of the Marxian Idea of Science, Dordrecht:Reidel 1980
Leszek Nowak and Izabella Nowakowa, Idealization X: The Richness of Idealization, Amsterdam / Atlanta: Rodopi 2000. | Wikipedia/Idealization_(science_philosophy) |
In game theory, the outcome of a game is the ultimate result of a strategic interaction with one or more people, dependant on the choices made by all participants in a certain exchange. It represents the final payoff resulting from a set of actions that individuals can take within the context of the game. Outcomes are pivotal in determining the payoffs and expected utility for parties involved. Game theorists commonly study how the outcome of a game is determined and what factors affect it.
A strategy is a set of actions that a player can take in response to the actions of others. Each player’s strategy is based on their expectation of what the other players are likely to do, often explained in terms of probability. Outcomes are dependent on the combination of strategies chosen by involved players and can be represented in a number of ways; one common way is a payoff matrix showing the individual payoffs for each players with a combination of strategies, as seen in the payoff matrix example below. Outcomes can be expressed in terms of monetary value or utility to a specific person. Additionally, a game tree can be used to deduce the actions leading to an outcome by displaying possible sequences of actions and the outcomes associated.
A commonly used theorem in relation to outcomes is the Nash equilibrium. This theorem is a combination of strategies in which no player can improve their payoff or outcome by changing their strategy, given the strategies of the other players. In other words, a Nash equilibrium is a set of strategies in which each player is doing the best possible, assuming what the others are doing to receive the most optimal outcome for themselves. Not all games have a unique nash equilibrium and if they do, it may not be the most desirable outcome. Additionally, the desired outcomes is greatly affected by individuals chosen strategies, and their beliefs on what they believe other players will do under the assumption that players will make the most rational decision for themselves. A common example of the nash equilibrium and undesirable outcomes is the Prisoner’s Dilemma game.
== Choosing among outcomes ==
Many different concepts exist to express how players might interact. An optimal interaction may be one in which no player's payoff can be made greater, without making any other player's payoff lesser. Such a payoff is described as Pareto efficient, and the set of such payoffs is called the Pareto frontier.
Many economists study the ways in which payoffs are in some sort of economic equilibrium. One example of such an equilibrium is the Nash equilibrium, where each player plays a strategy such that their payoff is maximized given the strategy of the other players.
Players are persons who make logical economic decisions. It is assumed that human people make all of their economic decisions based only on the idea that they are irrational. A player's rewards (utilities, profits, income, or subjective advantages) are assumed to be maximised. The purpose of game-theoretic analysis, when applied to a rational approach, is to provide recommendations on how to make choices against other rational players. First, it reduces the possible outcomes; logical action is more predictable than irrational. Second, it provides a criterion for assessing an economic system's efficiency.
In a Prisoner's Dilemma game between two players, player one and player two can choose the utilities that are the best response to maximise their outcomes. "A best response to a coplayer’s strategy is a strategy that yields the highest payoff against that particular strategy". A matrix is used to present the payoff of both players in the game. For example, the best response of player one is the highest payoff for player one’s move, and vice versa. For player one, they will pick the payoffs from the column strategies. For player two, they will choose their moves based on the two row strategies. Assuming both players do not know the opponents strategies. It is a dominant strategy for the first player to choose a payoff of 5 rather than a payoff of 3 because strategy D is a better response than strategy C.
== Applications ==
Outcome optimisation in game theory has many real world applications that can help predict actions and economic behaviours by other players. Examples of this include stock trades and investments, cost of goods in business, corporate behaviour and even social sciences.
Equilibria are not always Pareto efficient, and a number of game theorists design ways to enforce Pareto efficient play, or play that satisfies some other sort of social optimality. The theory of this is called implementation theory.
== References == | Wikipedia/Outcome_(game_theory) |
Strategic planning is the activity undertaken by an organization through which it seeks to define its future direction and makes decisions such as resource allocation aimed at achieving its intended goals. "Strategy" has many definitions, but it generally involves setting major goals, determining actions to achieve these goals, setting a timeline, and mobilizing resources to execute the actions. A strategy describes how the ends (goals) will be achieved by the means (resources) in a given span of time. Often, Strategic planning is long term and organizational action steps are established from two to five years in the future. Strategy can be planned ("intended") or can be observed as a pattern of activity ("emergent") as the organization adapts to its environment or competes in the market.
The senior leadership of an organization is generally tasked with determining strategy. It is executed by strategic planners or strategists, who involve many parties and research sources in their analysis of the organization and its relationship to the environment in which it competes.
Strategy includes processes of formulation and implementation; strategic planning helps coordinate both. However, strategic planning is analytical in nature (i.e., it involves "finding the dots"); strategy formation itself involves synthesis (i.e., "connecting the dots") via strategic thinking. As such, strategic planning occurs around the strategy formation activity.
== History ==
Strategic planning became prominent in corporations during the 1960s and remains an important aspect of strategic management.
McKinsey & Company developed a capability maturity model in the 1970s to describe the sophistication of planning processes, with strategic management ranked the highest. The four stages include:
Financial planning, which is primarily about annual budgets and a functional focus, with limited regard for the environment;
Forecast-based planning, which includes multi-year financial plans and more robust capital allocation across business units;
Externally oriented planning, where a thorough situation analysis and competitive assessment is performed;
Strategic management, where widespread strategic thinking occurs and a well-defined strategic framework is used.
Categories 3 and 4 are strategic planning, while the first two categories are non-strategic or essentially financial planning. Each stage builds on the previous stages; that is, a stage 4 organization completes activities in all four categories.
In 1993, President Bill Clinton signed into law the Government Performance and Results Act, which required US federal agencies to develop strategic plans for how they would deliver high quality products and services to the American people.
In the business sector, McKinsey research undertaken and published in 2006 found that, although many companies had a formal strategic-planning process, the process was not being used for their "most important decisions".
For Michael C. Sekora, Project Socrates founder in the Reagan White House, during the Cold War the economically challenged Soviet Union was able to keep on western military capabilities by using technology-based planning while the U.S. was slowed by finance-based planning, until the Reagan administration launched the Socrates Project, which should be revived to keep up with China as an emerging superpower.
== Process ==
=== Overview ===
Strategic planning is a process and thus has inputs, activities, outputs and outcomes. This process, like all processes, has constraints. It may be formal or informal and is typically iterative, with feedback loops throughout the process. Some elements of the process may be continuous and others may be executed as discrete projects with a definitive start and end during a period. Strategic planning provides inputs for strategic thinking: these are best seen as distinct but complementary activities. Strategic thinking guides the actual strategy formation. Typical strategic planning efforts include the evaluation of the organization's mission and strategic issues to strengthen current practices and determine the need for new programming. The end result is the organization's strategy, including a diagnosis of the environment and competitive situation, a guiding policy on what the organization intends to accomplish, and key initiatives or action plans for achieving the guiding policy.
Michael Porter wrote in 1980 that formulation of competitive strategy includes consideration of four key elements:
Company strengths and weaknesses;
Personal values of the key implementers (i.e., management and the board);
Industry opportunities and threats;
Broader societal expectations.
The first two elements relate to factors internal to the company (i.e., the internal environment), while the latter two relate to factors external to the company (i.e., the external environment). These elements are considered throughout the strategic planning process.
=== Inputs ===
Data is gathered from various sources, such as interviews with key executives, review of publicly available documents on the competition or market, primary research (e.g., visiting or observing competitor places of business or comparing prices), industry studies, reports of the organization's performance, etc. This may be part of a competitive intelligence program. Inputs are gathered to help establish a baseline, support an understanding of the competitive environment and its opportunities and risks. Other inputs include an understanding of the values of key stakeholders, such as the board, shareholders, and senior management. These values may be captured in an organization's vision and mission statements.
=== Activities ===
Strategic planning activities include meetings and other communication among the organization's leaders and personnel to develop a common understanding regarding the competitive environment and what the organization's response to that environment should be. A variety of strategic planning tools may be completed as part of strategic planning activities.
The organization's leaders may have a series of questions they want to be answered in formulating the strategy and gathering inputs.
=== Outputs ===
The output of strategic planning includes documentation and communication describing the organization's strategy and how it should be implemented, sometimes referred to as the strategic plan. The strategy may include a diagnosis of the competitive situation, a guiding policy for achieving the organization's goals, and specific action plans to be implemented. A strategic plan may cover multiple years and be updated periodically.
The organization may use a variety of methods of measuring and monitoring progress towards the strategic objectives and measures established, such as a balanced scorecard or strategy map. Organizations may also plan their financial statements (i.e., balance sheets, income statements, and cash flows) for several years when developing their strategic plan, as part of the goal-setting activity. The term operational budget is often used to describe the expected financial performance of an organization for the upcoming year. Capital budgets very often form the backbone of a strategic plan, especially as it increasingly relates to Information and Communications Technology (ICT).
=== Outcomes ===
While the planning process produces outputs, strategy implementation or execution of the strategic plan produces outcomes. These outcomes will invariably differ from the strategic goals. How close they are to the strategic goals and vision will determine the success or failure of the strategic plan. Unintended outcomes might also be an issue. They need to be attended to and understood for strategy development and execution to be a true learning process.
== Tools and approaches ==
=== Analytical tools ===
A variety of analytical tools and techniques are used in strategic planning. These were developed by companies and management consulting firms to help provide a framework for strategic planning. Such tools include:
PEST analysis, which covers the remote external environment elements such as political, economic, social and technological (PESTLE adds legal/regulatory and ecological/environmental);
Scenario planning, which was originally used in the military and recently used by large corporations to analyze future scenarios.
Porter five forces analysis, which addresses industry attractiveness and rivalry through the bargaining power of buyers and suppliers and the threat of substitute products and new market entrants;
SWOT analysis, which addresses internal strengths and weaknesses relative to the external opportunities and threats;
Growth-share matrix, which involves portfolio decisions about which businesses to retain or divest; and
Balanced scorecards and strategy maps, which creates a systematic framework for measuring and controlling strategy.
Responsive evaluation, which uses a constructivist evaluation approach to identify the outcomes of objectives, which then supports future strategic planning exercises.
VRIO Framework, which determines the competitive advantage of a product or a service through evaluating four elements such as value, rarity, imitability and organization.
=== Specific contexts ===
Strategic planning can be used in project management with a focus on the development of a standard repeatable methodology adding to the likelihood of achieving project objectives. This requires a lot of thinking process and interaction among stakeholders. Strategic planning in Project Management provides an organization the framework and consistency of action. In addition, it ensures communication of overall goals and understanding roles of teams or individual to achieve them. The commitment of top management must be evident throughout the process to reduce resistance to change, ensure acceptance, and avoid common pitfalls. Strategic planning does not guarantee success but will help improve likelihood of success of an organization.
strategic planning is also desirable within educational institutions. We are already in a transitional period in which old practices are no longer permanent but require revision to meet the needs of academia, which is frustrating in the educational sector. To meet the changing needs of this new society, educational institutions must reorganize. Finding ways to maintain achievements while improving effectiveness can be difficult for educational institutions. Keeping up with society's rapid changes. Some strategic planners are hesitant to address societal outcomes, so they often ignore them and assume they will happen on their own. Instead of defining the vision for how we want our children to live, they direct their attention to courses, content, and resources with the mistaken belief that societally useful outcomes will follow. When this occurs, the true strategic plan is never developed or implemented.
== Strategic planning vs. financial planning ==
Simply extending financial statement projections into the future without consideration of the competitive environment is a form of financial planning or budgeting, not strategic planning. In business, the term "financial plan" is often used to describe the expected financial performance of an organization for future periods. The term "budget" is used for a financial plan for the upcoming year. A "forecast" is typically a combination of actual performance year-to-date plus expected performance for the remainder of the year, so is generally compared against plan or budget and prior performance. The financial plans accompanying a strategic plan may include three–five years of projected performance.
== Criticism ==
=== Strategic planning vs. strategic thinking ===
Strategic planning has been criticized for attempting to systematize strategic thinking and strategy formation, which Henry Mintzberg argues are inherently creative activities involving synthesis or "connecting the dots" which cannot be systematized. Mintzberg argues that strategic planning can help coordinate planning efforts and measure progress on strategic goals, but that it occurs "around" the strategy formation process rather than within it. It functions remote from the "front lines" or contact with the competitive environment (i.e., in business, facing the customer where the effect of competition is most clearly evident) may not be effective at supporting strategy efforts.
=== Evidence on strategic planning's impact ===
While much criticism surrounds strategic planning, evidence suggests that it does work. In a 2019 meta-analysis including data from almost 9,000 public and private organizations, strategic planning is found to have a positive impact on organizational performance. Strategic planning is particularly potent in enhancing an organization's capacity to achieve its goals (i.e., effectiveness). However, the study argues that just having a plan is not enough. For strategic planning to work, it needs to include some formality (i.e., including an analysis of the internal and external environment and the stipulation of strategies, goals and plans based on these analyses), comprehensiveness (i.e., producing many strategic options before selecting the course to follow) and careful stakeholder management (i.e., thinking carefully about whom to involve during the different steps of the strategic planning process, how, when and why).
=== Strategic plans as tools to communicate and control ===
Henry Mintzberg in the article "The Fall and Rise of Strategic Planning" (1994), argued that the lesson that should be accepted is that managers will never be able to take charge of strategic planning through a formalized process. Therefore, he underscored the role of plans as tools to communicate and control. It ensures that there is coordination wherein everyone in the organization is moving in the same direction. The plans are the prime media communicating the management's strategic intentions, thereby promoting a common direction instead of individual discretion. It is also the tool to secure the support of the organization's external sphere, such as financiers, suppliers or government agencies, who are helping achieve the organization's plans and goals.
== The strategic plan genre of communication ==
Cornut et al (2012) studied the particular features of the strategic plan genre of communication by examining a corpus of strategic plans from public and non-profit organizations. They defined strategic plans as the "key material manifestation" of organizations' strategies and argued that, even though strategic plans are specific to an organization, there is a generic quality that draws on shared institutional understanding on the substance, form and communicative purposes of the strategic plan. Hence, they posit that strategic plan is a genre of organizational communication (Bhatia, 2004; Yates and Orlikowski, 1992 as cited in Cornut et al., 2012). In this sense, genre is defined as the "conventionalized discursive actions in which participating individuals or institutions have shared perceptions of communicative purposes as well as those of constraints operating their construction, interpretation and conditions of use" (Bhatia, 2004: 87; see also Frow, 2005; Swales, 1990 as cited in Cornut et al., 2012).
The authors compared the corpus of strategic plans with nine other corpora. This included annual reports from the public sector and nongovernment organizations, research articles, project plans, executive speeches, State of the Union addresses, horoscopes, religious sermons, business magazine articles and annual reports for-profit corporations included in the Standard & Poor's 500 largest companies (S&P 500).
The authors used textual analysis, including content analysis and corpus linguistics. Content analysis was used to identify themes and concepts, such as values and cognition; while corpus linguistics was used to identify naturally occurring texts and patterns (Biber et al., 1998 as cited in Cornut et al., 2012).
=== Taking a stance ===
The strategic plans showed significantly less self-reference than all other corpora, with the exemption of project plans and S&P 500 annual reports. The results indicated that strategic plans have more moderate verbs of deontic value. This was interpreted as an indication that "commands and commitments are not overtly hedged, but neither are they particularly strong".
Guidance on the sections of a strategic plan abound but there are few studies about the nature of language used for these documents. Cornut et al.'s (2012) study showed that writers of strategic plans have a shared understanding of what is the appropriate language. Thus, the authors argued, a true strategist is one who is able to instantiate the genre strategic plan through appropriate application of language.
=== Strategic planning as communicative process ===
Spee et. al. (2011) explored the strategic planning as communicative process based on Ricoeur's concepts of decontextualization and recontextualization, they conceptualize strategic planning activities as being constituted through the iterative and recursive relationship of talk and text, this elaborate the construction of a strategic plan as a communicative process. This study looks at the way that texts within the planning process, such as PowerPoint presentations, planning documents and targets that are part of a strategic plan, are constructed in preparation, through a series of communicative interface. Throughout the process, strategy documents were essential in detaining the developing strategy as they were constantly revised up until an ultimate plan was accepted.
A book edited by Mandeville-Gamble (2015) sees the roles of managers as important in terms of communicating the strategic vision of the organization. Many of the authors in the book by Mandeville-Gamble agree that a strategic plan is merely an unrealized vision unless it is widely shared and sparks the willingness to change within individuals in the organization. Similarly, Goodman in 2017 emphasized that the advent of the internet and social media has become one of the most important vehicle to which corporate strategic plan can be distributed to an organizations internal and external stakeholders. This distribution of knowledge allows for staff of organization to access and share the institutional thinking this able to reformulate it in their own words.
=== Strategic planning through control mechanisms ===
Strategic planning through control mechanisms (mostly by the way of a communication program) is set in the hopes of coming to desired outcomes that reflect company or organizational goals. As further supplement to this idea, controls can also be realized in both measurable and intangible controls, specifically output controls, behavioural controls, and clan controls. By way of simple definition, output controls work toward to tangible and quantifiable results; behavioural controls are geared toward behaviours of people in an organization; and clan controls are dependent and are executed while keeping in mind norms, traditions, and organizational culture. All these three are implemented in order to keep systems and strategies running and focused toward desired results (n.d.).
=== Strategic planning, learning organizations, and communication ===
Strategic planning is both the impetus for and result of critical thinking, optimization, and motivation for the growth and development of organizations. The core disciplines, which are inherent in systems thinking, personal and organizational mastery, mental models, building a shared vision, and team learning. In a time of machine learning and data analytics, these core disciplines remain to be relevant in so far as having human resource and human interest become the driving force behind organizations.
Moreover, it cannot be denied that communication plays a role in the realization of learning organizations and strategic planning. In a study by Barker and Camarata (1998), the authors noted that there are theories that could explain the invaluable role of communication, and these are from Rational Choice Theory to Social Exchange Theory where costs, rewards, and outcomes are valued in maintaining communication and thus relationships to serve the ends of an organization and its members. Thus, while many organizations and companies try their best to become learning organizations and exercise strategic planning, without communication, relationships fail and the core disciplines are never truly met (Barker & Camarata, 1998).
== See also ==
== References ==
== Further reading ==
Michael Allison and Jude Kaye (2005). Strategic Planning for Nonprofit Organizations. Second Edition. John Wiley and Sons.
John Argenti (1968). Corporate Planning – A Practical Guide. Allen & Unwin.
John Argenti (1974). Systematic Corporate Planning. Wiley.
Bradford and Duncan (2000). Simplified Strategic Planning. Chandler House.
Patrick J. Burkhart and Suzanne Reuss (1993). Successful Strategic Planning: A Guide for Nonprofit Agencies and Organizations. Newbury Park: Sage Publications.
L. Fahey and V. K. Narayman (1986). Macroenvironmental Analysis for Strategic Management. West Publishing.
Stephen G. Haines (2004). ABCs of strategic management: an executive briefing and plan-to-plan day on strategic management in the 21st century.
T. Kono (1994) "Changing a Company's Strategy and Culture", Long Range Planning, 27, 5 (October 1994), pp. 85–97
Philip Kotler (1986), "Megamarketing" In: Harvard Business Review. (March–April 1986)
Theodore Levitt (1960) "Marketing myopia", In: Harvard Business Review, (July–August 1960)
M. Lorenzen (2006). "Strategic Planning for Academic Library Instructional Programming." In: Illinois Libraries 86, no. 2 (Summer 2006): 22–29.
R. F. Lusch and V. N. Lusch (1987). Principles of Marketing. Kent Publishing,
Max Mckeown (2012), The Strategy Book, FT Prentice Hall.
John Naisbitt (1982). Megatrends: Ten New Directions Transforming our Lives. Macdonald.
Erica Olsen (2012). Strategic Planning Kit for Dummies, 2nd Edition. John Wiley & Sons, Inc.
Brian Tracy (2000). The 100 Absolutely Unbreakable Laws of Business Success. Berrett, Koehler Publishers. | Wikipedia/Strategic_planning |
The pirate game is a simple mathematical game. It is a multi-player version of the ultimatum game.
== The game ==
There are five rational pirates (in strict decreasing order of seniority A, B, C, D and E) who found 100 gold coins. They must decide how to distribute them.
The pirate world's rules of distribution say that the most senior pirate first proposes a plan of distribution. The pirates, including the proposer, then vote on whether to accept this distribution. If the majority accepts the plan, the coins are disbursed and the game ends. In case of a tie vote, the proposer has the casting vote. If the majority rejects the plan, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again. The process repeats until a plan is accepted or if there is one pirate left.
Pirates base their decisions on four factors:
Each pirate wants to survive.
Given survival, each pirate wants to maximize the number of gold coins he receives.
Each pirate would prefer to throw another overboard, if all other results would otherwise be equal.
The pirates do not trust each other, and will neither make nor honor any promises between pirates apart from a proposed distribution plan that gives a whole number of gold coins to each pirate.
== The result ==
To increase the chance of their plan being accepted, one might expect that Pirate A will have to offer the other pirates most of the gold. However, this is far from the theoretical result. When each of the pirates votes, they will not just be thinking about the current proposal, but also other outcomes down the line. In addition, the order of seniority is known in advance so each of them can accurately predict how the others might vote in any scenario. This becomes apparent if we work backwards.
The final possible scenario would have all the pirates except D and E thrown overboard. Since D is senior to E, they have the casting vote; so, D would propose to keep 100 for themself and 0 for E.
If there are three left (C, D and E), C knows that D will offer E 0 in the next round; therefore, C has to offer E one coin in this round to win E's vote. Therefore, when only three are left the allocation is C:99, D:0, E:1.
If B, C, D and E remain, B can offer 1 to D; because B has the casting vote, only D's vote is required. Thus, B proposes B:99, C:0, D:1, E:0.
(In the previous round, one might consider proposing B:99, C:0, D:0, E:1, as E knows it won't be possible to get more coins, if any, if E throws B overboard. But, as each pirate is eager to throw the others overboard, E would prefer to kill B, to get the same amount of gold from C.)
With this knowledge, A can count on C and E's support for the following allocation, which is the final solution:
A: 98 coins
B: 0 coins
C: 1 coin
D: 0 coins
E: 1 coin
(Note: A:98, B:0, C:0, D:1, E:1 or other variants are not good enough, as D would rather throw A overboard to get the same amount of gold from B.)
== Extension ==
The solution follows the same general pattern for other numbers of pirates and/or coins. However, the game changes in character when it is extended beyond there being twice as many pirates as there are coins. Ian Stewart wrote about Steve Omohundro's extension to an arbitrary number of pirates in the May 1999 edition of Scientific American and described the rather intricate pattern that emerges in the solution.
Supposing there are just 100 gold pieces, then:
Pirate #201 as captain can stay alive only by offering all the gold one each to the lowest odd-numbered pirates, keeping none.
Pirate #202 as captain can stay alive only by taking no gold and offering one gold each to 100 pirates who would not receive a gold coin from #201. Therefore, there are 101 possible recipients of these one gold coin bribes being the 100 even-numbered pirates up to 200 and number #201. Since there are no constraints as to which 100 of these 101 they will choose, any choice is equally good and they can be thought of as choosing at random. This is how chance begins to enter the considerations for higher-numbered pirates.
Pirate #203 as captain will not have enough gold available to bribe a majority, and so will die.
Pirate #204 as captain has #203's vote secured without bribes: #203 will only survive if #204 also survives. So #204 can remain safe by reaching 102 votes by bribing 100 pirates with one gold coin each. This seems most likely to work by bribing odd-numbered pirates optionally including #202, who will get nothing from #203. However, it may also be possible to bribe others instead as they only have a 100/101 chance of being offered a gold coin by pirate #202.
With 205 pirates, all pirates bar #205 prefer to kill #205 unless given gold, so #205 is doomed as captain.
Similarly with 206 or 207 pirates, only votes of #205 to #206/7 are secured without gold which is insufficient votes, so #206 and #207 are also doomed.
For 208 pirates, the votes of self-preservation from #205, #206, and #207 without any gold are enough to allow #208 to reach 104 votes and survive.
In general, if G is the number of gold pieces and N (> 2G) is the number of pirates, then
All pirates whose number is less than or equal to 2G + M will survive, where M is the highest power of 2 that does not exceed N – 2G.
Any pirates whose number exceeds 2G + M will die.
Any pirate whose number is greater than 2G + M/2 will receive no gold.
There is no unique solution as to who gets one gold coin and who does not if the number of pirates is 2G+2 or greater. A simple solution dishes out one gold to the odd or even pirates up to 2G depending whether M is an even or odd power of 2.
Another way to see this is to realize that every pirate M will have the vote of all the pirates from M/2 + 1 to M out of self preservation since their survival is secured only with the survival of the pirate M. Because the highest ranking pirate can break the tie, the captain only needs the votes of half of the pirates over 2G, which only happens each time (2G + a Power of 2) is reached. For instance, with 100 gold pieces and 500 pirates, pirates #500 through #457 die, and then #456 survives (as 456 = 200 + 28) as they have the 128 guaranteed self-preservation votes of pirates #329 through #456, plus 100 votes from the pirates they bribe, making up the 228 votes that they need. The numbers of pirates past #200 who can guarantee their survival as captain with 100 gold pieces are #201, #202, #204, #208, #216, #232, #264, #328, #456, #712, etc.: they are separated by longer and longer strings of pirates who are doomed no matter what division they propose.
== See also ==
Creative problem solving
Lateral thinking
== Notes ==
== References ==
Robert E. Goodin, ed. (1998). "Chapter 3: Second best theories". The Theory of Institutional Design. Cambridge University Press. pp. 90–102. ISBN 978-0-521-63643-8. | Wikipedia/Pirate_game |
The labor theory of property, also called the labor theory of appropriation, labor theory of ownership, labor theory of entitlement, and principle of first appropriation, is a theory of natural law that holds that property originally comes about by the exertion of labor upon natural resources. The theory has been used to justify the homestead principle, which holds that one may gain whole permanent ownership of an unowned natural resource by performing an act of original appropriation.
In his Second Treatise on Government, the philosopher John Locke asked by what right an individual can claim to own one part of the world, when, according to the Bible, God gave the world to all humanity in common. He answered that, although persons belong to God, they own the fruits of their labor. When a person works, that labor enters into the object upon which they are working. Thus, the object becomes the property of that person; however, Locke held that one may only appropriate property in this fashion if the Lockean proviso held true, that is, "... there is enough, and as good, left in common for others".
== Locke's formulation ==
Though the earth, and all inferiour creatures, be common to all men, yet every man has a property in his own person: this nobody has any right to but himself. The labour of his body, and the work of his hands, we may say, are properly his. Whatsoever then he removes out of the state that nature hath provided, and left it in, he hath mixed his labour with, and joined to it something that is his own, and thereby makes it his property. It being by him removed from the common state nature hath placed it in, it hath by this labour something annexed to it, that excludes the common right of other men. For this labour being the unquestionable property of the labourer, no man but he can have a right to what that is once joined to, at least where there is enough, and as good, left in common for others.
== Exclusive ownership and creation ==
Locke argued in support of individual property rights as natural rights. Following the argument, the fruits of one's labor are one's own because one worked for it. Thus, any form of income tax would be hostile to natural law. Furthermore, the laborer must also hold a natural property right in the resource itself because exclusive ownership was immediately necessary for production.
Jean-Jacques Rousseau later criticized this second step in Discourse on Inequality, where he argues that the natural right argument does not extend to resources that one did not create. Both philosophers hold that the relation between labor and ownership pertains only to property that was significantly unused before such labor took place.
== Enclosure vs mixing labor ==
Land in its original state would be considered unowned by anyone, but if an individual applied his labor to the land by farming it, for example, it becomes his property. Merely placing a fence around land rather than using the land enclosed would not bring property into being according to most natural law theorists.
Economist Murray Rothbard put it this way:
If Columbus lands on a new continent, is it legitimate for him to proclaim all the new continent his own, or even that sector 'as far as his eye can see'? Clearly, this would not be the case in the free society that we are postulating. Columbus or Crusoe would have to use the land, to 'cultivate' it in some way, before he could be asserted to own it.... If there is more land than can be used by a limited labor supply, then the unused land must simply remain unowned until a first user arrives on the scene. Any attempt to claim a new resource that someone does not use would have to be considered invasive of the property right of whoever the first user will turn out to be.
== Acquisition vs mixing labor ==
The labor theory of property does not only apply to land itself, but to any application of labor to nature. For example, natural rights thinker Lysander Spooner, says that an apple taken from an unowned tree would become the property of the person who plucked it, as he has labored to acquire it. He says the "only way, in which ["the wealth of nature"] can be made useful to mankind, is by their taking possession of it individually, and thus making it private property."
However, some, such as Benjamin Tucker have not seen this as creating property in all things. Tucker argued that "in the case of land, or of any other material the supply of which is so limited that all cannot hold it in unlimited quantities, these should only be considered owned while the individual is in the act of using or occupying these things." This is a rejection of Absentee ownership for land.
== Lockean proviso ==
Locke held that individuals have a right to homestead private property from nature by working on it, but that they can do so only "...at least where there is enough, and as good, left in common for others". The proviso maintains that appropriation of unowned resources is a diminution of the rights of others to it and would be acceptable only so long as it does not make anyone worse off than they would have been before. The phrase "Lockean Proviso" was coined by political philosopher Robert Nozick and is based on the ideas elaborated by John Locke in his Second Treatise of Government.
== Criticism ==
Aside from critiques of natural rights as a whole, Locke's labor theory of property has been singled out for critique by modern academics who doubt the idea that mixing something owned with something unowned could imbue the object with ownership:
[W]hy isn't mixing what I own with what I don't own a way of losing what I own rather than a way of gaining what I don't? If I own a can of tomato juice and spill it in the sea so that its molecules (made radioactive, so I can check this) mingle evenly throughout the sea, do I thereby come to own the sea, or have I foolishly dissipated my tomato juice?
Jeremy Waldron believes that Locke has made a category mistake, as only objects can be mixed with other objects and laboring is not an object, but an activity.
Stephan Kinsella also agrees with this line of thinking, criticizing the Objectivists for trying to defend intellectual property in terms of mixing labor, and arguing that we should think in terms of the First possession theory of property rather than the Labor theory of property:
By focusing on first occupancy, rather than on labor, as the key to homesteading, there is no need to place creation as the fount of property rights, as Objectivists and others do. Instead, property rights must be recognized in first-comers (or their contractual transferees) in order to avoid the omnipresent problem of conflict over scarce resources. Creation itself is neither necessary nor sufficient to gain rights in unowned resources. Further, there is no need to maintain the strange view that one “owns” one s labor in order to own things one first occupies. Labor is a type of action, and action is not ownable; rather, it is the way that some tangible things (e.g., bodies) act in the world.
Judith Jarvis Thomson points out that the act of laboring makes Locke's argument either an appeal to desert, in which case the reward is arbitrary-"Why not instead a medal and a handshake from the president?"-or little different than first possession theories that existed before Locke.
Ellen Meiksins Wood provides a number of critiques of Locke's labor theory of property from a Marxist perspective. Wood notes that Locke is not actually concerned with the act of labor or improving the use value of property, but rather is focused on the creation of exchange value as the basis of property.
For one thing, it turns out that there is no direct correspondence between labour and property, because one man can appropriate the labour of another. He can acquire a right of property in something by 'mixing' with it not his own labour but the labour of someone else whom he employs. It appears that the issue for Locke has less to do with the activity of labour as such than with its profitable use. In calculating the value of the acre in America, for instance, he talks not about the Indian's expenditure of effort, labour, but about the Indian's failure to realize a profit. The issue, in other words, is not the labour of a human being but the productivity of property, its exchange value and its application to commercial profit.
In addition to the theoretical deficiencies of Locke's theory of property, Wood also argues that Locke also provides a justification for the dispossession of indigenous land. The idea that making land productive serves as the basis of property rights establishes the corollary that the failure to improve land could mean forfeiting property rights. Under Locke's theory, "[e]ven if land is occupied by indigenous peoples, and even if they make use of the land themselves, their land is still open to legitimate colonial expropriation." Locke's notion that property "derives from the creation of value, from 'improvement' that enhances exchange value, implies not only that mere occupancy is not enough to establish property rights, or even that hunting-gathering cannot establish the right of property while agriculture can, but also that insufficiently productive and profitable agriculture, by the standards of English agrarian capitalism, effectively constitutes waste."
Economist John Quiggin argues that this fits into a larger fundamental criticism of Locke's labor theory of property which values a particular type of labor and land use (i.e., agriculture) over all others. It thus does not recognize usage of land, for example, by hunter-gatherer societies as granting rights to ownership. In essence, the Lockean proviso depends on "the existence of a frontier, beyond which lies boundless usable land. This in turn requires the erasure (mentally and usually in brutal reality) of the people already living beyond the frontier and drawing their sustenance from the land in question." Locke's theories of property rights are often interpreted in the context of his support for chattel slavery of "prisoners captured in war" as a philosophical justification for the enslavement of Black Africans and expulsion or killing of Native Americans by early American colonists to gain their land.
== See also ==
Entitlement theory
First possession theory of property
Georgism
Homestead principle
Propertarianism
== External links ==
John Locke's Theory of Property, and the Dispossession of Indigenous Peoples in the Settler-Colony Indigenous Peoples in the Settler-Colony - American Indian Law Journal
== References == | Wikipedia/Labor_theory_of_property |
Proponents of democratic peace theory argue that both electoral and republican forms of democracy are hesitant to engage in armed conflict with other identified democracies. Different advocates of this theory suggest that several factors are responsible for motivating peace between democratic states. Individual theorists maintain "monadic" forms of this theory (democracies are in general more peaceful in their international relations); "dyadic" forms of this theory (democracies do not go to war with other democracies); and "systemic" forms of this theory (more democratic states in the international system makes the international system more peaceful).
In terms of norms and identities, it is hypothesized that democratic publics are more dovish in their interactions with other democracies, and that democratically elected leaders are more likely to resort to peaceful resolution in disputes (both in domestic politics and international politics). In terms of structural or institutional constraints, it is hypothesized that institutional checks and balances, accountability of leaders to the public, and larger winning coalitions make it harder for democratic leaders to go to war unless there are clearly favorable ratio of benefits to costs.
These structural constraints, along with the transparent nature of democratic politics, make it harder for democratic leaders to mobilize for war and initiate surprise attacks, which reduces fear and inadvertent escalation to war. The transparent nature of democratic political systems, as well as deliberative debates (involving opposition parties, the media, experts, and bureaucrats), make it easier for democratic states to credibly signal their intentions. The concept of audience costs entails that threats issued by democratic leaders are taken more seriously because democratic leaders will be electorally punished by their publics from backing down from threats, which reduces the risk of misperception and miscalculation by states.
The connection between peace and democracy has long been recognized, but theorists disagree about the direction of causality. The democratic peace theory posits that democracy causes peace, while the territorial peace theory makes the opposite claim that peace causes democracy. Other theories argue that omitted variables explain the correlation better than democratic peace theory. Alternative explanations for the correlation of peace among democracies include arguments revolving around institutions, commerce, interdependence, alliances, US world dominance and political stability.
== History ==
Though the democratic peace theory was not rigorously or scientifically studied until the 1960s, the basic principles of the concept had been argued as early as the 18th century in the works of philosopher Immanuel Kant and political theorist Thomas Paine. Kant foreshadowed the theory in his essay Perpetual Peace: A Philosophical Sketch written in 1795, although he thought that a world with only constitutional republics was only one of several necessary conditions for a perpetual peace. In earlier but less cited works, Thomas Paine made similar or stronger claims about the peaceful nature of republics. Paine wrote in "Common Sense" in 1776: "The Republics of Europe are all (and we may say always) in peace." Paine argued that kings would go to war out of pride in situations where republics would not. French historian and social scientist Alexis de Tocqueville also argued, in Democracy in America (1835–1840), that democratic nations were less likely to wage war. Herbert Spencer also argued for a relationship between democracy and peace.
Dean Babst, a criminologist, was the first to do statistical research on this topic. His academic paper supporting the theory was published in 1964 in Wisconsin Sociologist; he published a slightly more popularized version, in 1972, in the trade journal Industrial Research. Both versions initially received little attention.
Melvin Small and J. David Singer responded; they found an absence of wars between democratic states with two "marginal exceptions", but denied that this pattern had statistical significance. This paper was published in the Jerusalem Journal of International Relations which finally brought more widespread attention to the theory, and started the academic debate. A 1983 paper by political scientist Michael W. Doyle contributed further to popularizing the theory.
Maoz and Abdolali extended the research to lesser conflicts than wars. Bremer, Maoz and Russett found the correlation between democracy and peacefulness remained significant after controlling for many possible confounding variables. This moved the theory into the mainstream of social science. Supporters of realism in international relations and others responded by raising many new objections. Other researchers attempted more systematic explanations of how democracy might cause peace, and of how democracy might also affect other aspects of foreign relations such as alliances and collaboration.
There have been numerous further studies in the field since these pioneering works. Most studies have found some form of democratic peace exists, although neither methodological disputes nor doubtful cases are entirely resolved.
== Definitions ==
Research on the democratic peace theory has to define "democracy" and "peace" (or, more often, "war").
=== Defining democracy ===
Democracies have been defined differently by different theorists and researchers; this accounts for some of the variations in their findings. Some examples:
Small and Singer define democracy as a nation that (1) holds periodic elections in which the opposition parties are as free to run as government parties, (2) allows at least 10% of the adult population to vote, and (3) has a parliament that either controls or enjoys parity with the executive branch of the government.
Doyle requires (1) that "liberal regimes" have market or private property economics, (2) they have policies that are internally sovereign, (3) they have citizens with juridical rights, and (4) they have representative governments. Either 30% of the adult males were able to vote or it was possible for every man to acquire voting rights as by attaining enough property. He allows greater power to hereditary monarchs than other researchers; for example, he counts the rule of Louis-Philippe of France as a liberal regime.
Ray requires that at least 50% of the adult population is allowed to vote and that there has been at least one peaceful, constitutional transfer of executive power from one independent political party to another by means of an election. This definition excludes long periods often viewed as democratic. For example, the United States until 1800, India from independence until 1979, and Japan until 1993 were all under a dominant-party system, and thus would not be counted under this definition.
Rummel states that "By democracy is meant liberal democracy, where those who hold power are elected in competitive elections with a secret ballot and wide franchise (loosely understood as including at least 2/3 of adult males); where there is freedom of speech, religion, and organization; and a constitutional framework of law to which the government is subordinate and that guarantees equal rights."
==== Non-binary classifications ====
The above definitions are binary, classifying nations into either democracies or non-democracies. Many researchers have instead used more finely grained scales. One example is the Polity data series which scores each state on two scales, one for democracy and one for autocracy, for each year since 1800; as well as several others. The use of the Polity Data has varied. Some researchers have done correlations between the democracy scale and belligerence; others have treated it as a binary classification by (as its maker does) calling all states with a high democracy score and a low autocracy score democracies; yet others have used the difference of the two scores, sometimes again making this into a binary classification.
==== Young democracies ====
Several researchers have observed that many of the possible exceptions to the democratic peace have occurred when at least one of the involved democracies was very young. Many of them have therefore added a qualifier, typically stating that the peacefulness apply to democracies older than three years. Rummel argues that this is enough time for "democratic procedures to be accepted, and democratic culture to settle in." Additionally, this may allow for other states to actually come to the recognition of the state as a democracy.
Mansfield and Snyder, while agreeing that there have been no wars between mature liberal democracies, state that countries in transition to democracy are especially likely to be involved in wars. They find that democratizing countries are even more warlike than stable democracies, stable autocracies or even countries in transition towards autocracy. So, they suggest caution in eliminating these wars from the analysis, because this might hide a negative aspect of the process of democratization. A reanalysis of the earlier study's statistical results emphasizes that the above relationship between democratization and war can only be said to hold for those democratizing countries where the executive lacks sufficient power, independence, and institutional strength. A review cites several other studies finding that the increase in the risk of war in democratizing countries happens only if many or most of the surrounding nations are undemocratic. If wars between young democracies are included in the analysis, several studies and reviews still find enough evidence supporting the stronger claim that all democracies, whether young or established, go into war with one another less frequently; while some do not.
=== Defining war ===
Quantitative research on international wars usually defines war as a military conflict with more than 1000 killed in battle in one year. This is the definition used in the Correlates of War Project which has also supplied the data for many studies on war. It turns out that most of the military conflicts in question fall clearly above or below this threshold.
Some researchers have used different definitions. For example, Weart defines war as more than 200 battle deaths. Russett, when looking at Ancient Greece, only requires some real battle engagement, involving on both sides forces under state authorization.
Militarized Interstate Disputes (MIDs), in the Correlates of War Project classification, are lesser conflicts than wars. Such a conflict may be no more than military display of force with no battle deaths. MIDs and wars together are "militarized interstate conflicts" or MICs. MIDs include the conflicts that precede a war; so the difference between MIDs and MICs may be less than it appears.
Statistical analysis and concerns about degrees of freedom are the primary reasons for using MID's instead of actual wars. Wars are relatively rare. An average ratio of 30 MIDs to one war provides a richer statistical environment for analysis.
=== Monadic vs. dyadic peace ===
Most research is regarding the dyadic peace, that democracies do not fight one another. Very few researchers have supported the monadic peace, that democracies are more peaceful in general. There are some recent papers that find a slight monadic effect. Müller and Wolff, in listing them, agree "that democracies on average might be slightly, but not strongly, less warlike than other states," but general "monadic explanations are neither necessary nor convincing." They note that democracies have varied greatly in their belligerence against non-democracies.
== Possible exceptions ==
Some scholars support the democratic peace on probabilistic grounds: since many wars have been fought since democracies first arose, we might expect a proportionate number of wars to have occurred between democracies, if democracies fought each other as freely as other pairs of states; but proponents of democratic peace theory claim that the number is much less than might be expected. However, opponents of the theory argue this is mistaken and claim there are numerous examples of wars between democracies.
Historically, troublesome cases for the Democratic peace theory include the Sicilian Expedition, the War of 1812, the U.S. Civil War, the Fashoda Crisis, conflicts between Ecuador and Peru, the Cod Wars, the Spanish–American War, and the Kargil War. Doyle cites the Paquisha War and the Lebanese air force's intervention in the Six-Day War. The total number of cases suggested in the literature is at least 50. The data set Bremer was using showed one exception, the French-Thai War of 1940; Gleditsch sees the state of war between Finland and United Kingdom during World War II, as a special case, which should probably be treated separately: an incidental state of war between democracies during large and complex war with hundreds of belligerents and the constant shifting of geopolitical and diplomatic boundaries. However, the British did conduct a few military actions of minor scope against the Finns, more to demonstrate their alliance with the Soviets than to actually engage in war with Finland. Page Fortna discusses the 1974 Turkish invasion of Cyprus and the Kargil War as exceptions, finding the latter to be the most significant.
== Conflict initiation ==
According to a 2017 review study, "there is enough evidence to conclude that democracy does cause peace at least between democracies, that the observed correlation between democracy and peace is not spurious".
Most studies have looked only at who is involved in the conflicts and ignored the question of who initiated the conflict. In many conflicts both sides argue that the other side was the initiator. Several researchers have argued that studying conflict initiation is of limited value, because existing data about conflict initiation may be especially unreliable. Even so, several studies have examined this. Reitner and Stam argue that autocracies initiate conflicts against democracies more frequently than democracies do against autocracies. Quackenbush and Rudy, while confirming Reiter and Stam's results, find that democracies initiate wars against non-democracies more frequently than non-democracies do to each other. Several following studies have studied how different types of autocracies with different institutions vary regarding conflict initiation. Personalistic and military dictatorships may be particularly prone to conflict initiation, as compared to other types of autocracy such as one-party states, but also more likely to be targeted in a war having other initiators.
One 2017 study found that democracies are no more likely to settle border disputes peacefully than non-democracies.
== Internal violence and genocide ==
Most of this article discusses research on relations between states. However, there is also evidence that democracies have less internal systematic violence. For instance, one study finds that the most democratic and the most authoritarian states have few civil wars, and intermediate regimes the most. The probability for a civil war is also increased by political change, regardless whether toward greater democracy or greater autocracy. Intermediate regimes continue to be the most prone to civil war, regardless of the time since the political change. In the long run, since intermediate regimes are less stable than autocracies, which in turn are less stable than democracies, durable democracy is the most probable end-point of the process of democratization. Abadie's study finds that the most democratic nations have the least terrorism. Harff finds that genocide and politicide are rare in democracies. Rummel finds that the more democratic a regime, the less its democide. He finds that democide has killed six times as many people as battles.
Davenport and Armstrong II list several other studies and states: "Repeatedly, democratic political systems have been found to decrease political bans, censorship, torture, disappearances and mass killing, doing so in a linear fashion across diverse measurements, methodologies, time periods, countries, and contexts." It concludes: "Across measures and methodological techniques, it is found that below a certain level, democracy has no impact on human rights violations, but above this level democracy influences repression in a negative and roughly linear manner." They also state that thirty years worth of statistical research has revealed that only two variables decrease human rights violations: political democracy and economic development.
Abulof and Goldman add a caveat, focusing on the contemporary Middle East and North Africa (MENA). Statistically, a MENA democracy makes a country more prone to both the onset and incidence of civil war, and the more democratic a MENA state is, the more likely it is to experience violent intrastate strife. Moreover, anocracies do not seem to be predisposed to civil war, either worldwide or in MENA. Looking for causality beyond correlation, they suggest that democracy's pacifying effect is partly mediated through societal subscription to self-determination and popular sovereignty. This may turn “democratizing nationalism” to a long-term prerequisite, not just an immediate hindrance, to peace and democracy.
== Explanations ==
These theories have traditionally been categorized into two groups: explanations that focus on democratic norms and explanations that focus on democratic political structures. They usually are meant to be explanations for little violence between democracies, not for a low level of internal violence in democracies.
Several of these mechanisms may also apply to countries of similar systems. The book Never at War finds evidence that the oligarchic republics common in ancient Greece and medieval and early modern Europe hardly ever made war on one another. One example is the Polish–Lithuanian Commonwealth, in which the Sejm resisted and vetoed most royal proposals for war, like those of Władysław IV Vasa.
A study by V-Dem Institute found both interbranch constraint on the executive and civil society activism as the mechanism for democratic peace but found accountability provided directly by elections not as crucial.
=== Democratic norms ===
One example from the first group is that liberal democratic culture may make the leaders accustomed to negotiation and compromise. Policy makers who have built their careers within a political culture of non-violent accommodations with domestic rivals, unlike autocrats who typically hold power through the threat of coercion, will be inclined toward non-violent methods abroad. Another that a belief in human rights may make people in democracies reluctant to go to war, especially against other democracies. The decline in colonialism, also by democracies, may be related to a change in perception of non-European peoples and their rights.
Bruce Russett also argues that the democratic culture affects the way leaders resolve conflicts. In addition, he holds that a social norm emerged toward the end of the nineteenth century; that democracies should not fight each other, which strengthened when the democratic culture and the degree of democracy increased, for example by widening the franchise. Increasing democratic stability allowed partners in foreign affairs to perceive a nation as reliably democratic. The alliances between democracies during the two World Wars and the Cold War also strengthened the norms. He sees less effective traces of this norm in Greek antiquity.
Hans Köchler relates the question of transnational democracy to empowering the individual citizen by involving him, through procedures of direct democracy, in a country's international affairs, and he calls for the restructuring of the United Nations Organization according to democratic norms. He refers in particular to the Swiss practice of participatory democracy.
Mousseau argues that it is market-oriented development that creates the norms and values that explain both democracy and the peace. In less developed countries individuals often depend on social networks that impose conformity to in-group norms and beliefs, and loyalty to group leaders. When jobs are plentiful on the market, in contrast, as in market-oriented developed countries, individuals depend on a strong state that enforces contracts equally. Cognitive routines emerge of abiding by state law rather than group leaders, and, as in contracts, tolerating differences among individuals. Voters in marketplace democracies thus accept only impartial ‘liberal’ governments, and constrain leaders to pursue their interests in securing equal access to global markets and in resisting those who distort such access with force. Marketplace democracies thus share common foreign policy interests in the supremacy—and predictability—of international law over brute power politics, and equal and open global trade over closed trade and imperial preferences. When disputes do originate between marketplace democracies, they are less likely than others to escalate to violence because both states, even the stronger one, perceive greater long-term interests in the supremacy of law over power politics.
Braumoeller argues that liberal norms of conflict resolution vary because liberalism takes many forms. By examining survey results from the newly independent states of the former Soviet Union, the author demonstrates that liberalism in that region bears a stronger resemblance to 19th-century liberal nationalism than to the sort of universalist, Wilsonian liberalism described by democratic peace theorists, and that, as a result, liberals in the region are more, not less, aggressive than non-liberals.
A 2013 study by Jessica Weeks and Michael Tomz found through survey experiments that the public was less supportive of war in cases involving fellow democracies.
=== Democratic political structures ===
The case for institutional constraints goes back to Immanuel Kant, who wrote:
[I]f the consent of the citizens is required in order to decide that war should be declared (and in this constitution it cannot but be the case), nothing is more natural than that they would be very cautious in commencing such a poor game, decreeing for themselves all the calamities of war. Among the latter would be: having to fight, having to pay the costs of war from their own resources, having painfully to repair the devastation war leaves behind, and, to fill up the measure of evils, load themselves with a heavy national debt that would embitter peace itself and that can never be liquidated on account of constant wars in the future.
Democracy thus gives influence to those most likely to be killed or wounded in wars, and their relatives and friends (and to those who pay the bulk of the war taxes.) This monadic theory must, however, explain why democracies do attack non-democratic states. One explanation is that these democracies were threatened or otherwise were provoked by the non-democratic states. Doyle argued that the absence of a monadic peace is only to be expected: the same ideologies that cause liberal states to be at peace with each other inspire idealistic wars with the illiberal, whether to defend oppressed foreign minorities or avenge countrymen settled abroad. Doyle also notes liberal states do conduct covert operations against each other; the covert nature of the operation, however, prevents the publicity otherwise characteristic of a free state from applying to the question.
Charles Lipson argues that four factors common in democracies give them a "contracting advantage" that leads to a dyadic democratic peace: (1) Greater transparency, (2) Greater continuity, (3) Electoral incentives for leaders to keep promises, and (4) Constitutional governance.
Studies show that democratic states are more likely than autocratic states to win the wars that they start. One explanation is that democracies, for internal political and economic reasons, have greater resources. This might mean that democratic leaders are unlikely to select other democratic states as targets because they perceive them to be particularly formidable opponents. One study finds that interstate wars have important impacts on the fate of political regimes, and that the probability that a political leader will fall from power in the wake of a lost war is particularly high in democratic states.
As described by Gelpi and Griesdorf, several studies have argued that liberal leaders face institutionalized constraints that impede their capacity to mobilize the state's resources for war without the consent of a broad spectrum of interests. Survey results that compare the attitudes of citizens and elites in the Soviet successor states are consistent with this argument. Moreover, these constraints are readily apparent to other states and cannot be manipulated by leaders. Thus, democracies send credible signals to other states of an aversion to using force. These signals allow democratic states to avoid conflicts with one another, but they may attract aggression from non-democratic states. Democracies may be pressured to respond to such aggression—perhaps even preemptively—through the use of force. Also as described by Gelpi and Griesdorf, studies have argued that when democratic leaders do choose to escalate international crises, their threats are taken as highly credible, since there must be a relatively large public opinion for these actions. In disputes between liberal states, the credibility of their bargaining signals allows them to negotiate a peaceful settlement before mobilization. A 2017 study by Jeff Carter found evidence that democratic states are slower to mobilize for war.
An explanation based on game theory similar to the last two above is that the participation of the public and the open debate send clear and reliable information regarding the intentions of democracies to other states. In contrast, it is difficult to know the intentions of non-democratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy.
The risk factors for certain types of state have, however, changed since Kant's time. In the quote above, Kant points to the lack of popular support for war – first that the populace will directly or indirectly suffer in the event of war – as a reason why republics will not tend to go to war. The number of American troops killed or maimed versus the number of Iraqi soldiers and civilians maimed and killed in the American-Iraqi conflict is indicative. This may explain the relatively great willingness of democratic states to attack weak opponents: the Iraq war was, initially at least, highly popular in the United States. The case of the Vietnam War might, nonetheless, indicate a tipping point where publics may no longer accept continuing attrition of their soldiers (even while remaining relatively indifferent to the much higher loss of life on the part of the populations attacked).
Coleman uses economic cost-benefit analysis to reach conclusions similar to Kant's. Coleman examines the polar cases of autocracy and liberal democracy. In both cases, the costs of war are assumed to be borne by the people. In autocracy, the autocrat receives the entire benefits of war, while in a liberal democracy the benefits are dispersed among the people. Since the net benefit to an autocrat exceeds the net benefit to a citizen of a liberal democracy, the autocrat is more likely to go to war. The disparity of benefits and costs can be so high that an autocrat can launch a welfare-destroying war when his net benefit exceeds the total cost of war. Contrarily, the net benefit of the same war to an individual in a liberal democracy can be negative so that he would not choose to go to war. This disincentive to war is increased between liberal democracies through their establishment of linkages, political and economic, that further raise the costs of war between them. Therefore, liberal democracies are less likely to go war, especially against each other. Coleman further distinguishes between offensive and defensive wars and finds that liberal democracies are less likely to fight defensive wars that may have already begun due to excessive discounting of future costs.
Brad LeVeck and Neil Narang argue that democratic states are less likely to produce decision-making errors in crises due to a larger and more diverse set of actors who are involved in the foreign policy decision-making process.
Using selectorate theory, Bruce Bueno de Mesquita, James D. Morrow, Randolph M. Siverson and Alastair Smith argue that the democratic peace stems in part from the fact that democratic leaders sustain their power through large winning coalitions, which means that democratic leaders devote more resources to war, have an advantage in war, and choose wars that they are highly likely to win. These leads democratic states to avoid one another, but war with weak non-democratic states.
=== Audience costs ===
A prominent rational choice argument for the democratic peace is that democracies carry greater audience costs than authoritarian states, which makes them better at signaling their intentions in interstate disputes. Arguments regarding the credibility of democratic states in disputes has been subject to debate among international relations scholars. Two studies from 2001, using the MID and ICB datasets, provided empirical support for the notion that democracies were more likely to issue effective threats. However, a 2012 study by Alexander B. Downes and Todd S. Sechser found that existing datasets were not suitable to draw any conclusions as to whether democratic states issued more effective threats. They constructed their own dataset specifically for interstate military threats and outcomes, which found no relationship between regime type and effective threats. A 2017 study which recoded flaws in the MID dataset ultimately conclude, "that there are no regime-based differences in dispute reciprocation, and prior findings may be based largely on poorly coded data." Other scholars have disputed the democratic credibility argument, questioning its causal logic and empirical validity. Research by Jessica Weeks argued that some authoritarian regime types have similar audience costs as in democratic states.
A 2021 study found that Americans perceived democracies to be more likely to back down in crises, which contradicts the expectations of the audience costs literature.
=== Democracy differences ===
One general criticism motivating research of different explanations is that actually the theory cannot claim that "democracy causes peace", because the evidence for democracies being, in general, more peaceful is very slight or nonexistent; it only can support the claim that "joint democracy causes peace". According to Rosato, this casts doubts on whether democracy is actually the cause because, if so, a monadic effect would be expected.
Perhaps the simplest explanation to such perceived anomaly (but not the one the Realist Rosato prefers, see the section on Realist explanations below) is that democracies are not peaceful to each other because they are democratic, but rather because they are similar in democratic scores. This line of thought started with several independent observations of an "Autocratic Peace" effect, a reduced probability of war (obviously no author claims its absence) between states which are both non-democratic, or both highly so. This has led to the hypothesis that democratic peace emerges as a particular case when analyzing a subset of states which are, in fact, similar. Or, that similarity in general does not solely affect the probability of war, but only coherence of strong political regimes such as full democracies and stark autocracies.
Autocratic peace and the explanation based on democratic similarity. is a relatively recent development, and opinions about its value are varied. Henderson builds a model considering political similarity, geographic distance and economic interdependence as its main variables, and concludes that democratic peace is a statistical artifact which disappears when the above variables are taken into account. Werner finds a conflict reducing effect from political similarity in general, but with democratic dyads being particularly peaceful, and noting some differences in behavior between democratic and autocratic dyads with respect to alliances and power evaluation. Beck, King, and Zeng use neural networks to show two distinct low probability zones, corresponding to high democracy and high autocracy. Petersen uses a different statistical model and finds that autocratic peace is not statistically significant, and that the effect attributed to similarity is mostly driven by the pacifying effect of joint democracy. Ray similarly disputes the weight of the argument on logical grounds, claiming that statistical analysis on "political similarity" uses a main variable which is an extension of "joint democracy" by linguistic redefinition, and so it is expected that the war reducing effects are carried on in the new analysis. Bennett builds a direct statistical model based on a triadic classification of states into "democratic", "autocratic" and "mixed". He finds that autocratic dyads have a 35% reduced chance of going into any type of armed conflict with respect to a reference mixed dyad. Democratic dyads have a 55% reduced chance. This effect gets stronger when looking at more severe conflicts; for wars (more than 1000 battle deaths), he estimates democratic dyads to have an 82% lower risk than autocratic dyads. He concludes that autocratic peace exists, but democratic peace is clearly stronger. However, he finds no relevant pacifying effect of political similarity, except at the extremes of the scale.
To summarize a rather complex picture, there are no less than four possible stances on the value of this criticism:
Political similarity, plus some complementary variables, explains everything. Democratic peace is a statistical artifact. Henderson subscribes to this view.
Political similarity has a pacifying effect, but democracy makes it stronger. Werner would probably subscribe to this view.
Political similarity in general has little or no effect, except at the extremes of the democracy-autocracy scale: a democratic peace and an autocratic peace exist separately, with the first one being stronger, and may have different explanations. Bennett holds this view, and Kinsella mentions this as a possibility
Political similarity has little or no effect and there is no evidence for autocratic peace. Petersen and Ray are among defendants of this view.
==== Interactive model of democratic peace ====
The interactive model of democratic peace is a combination of democratic similarity with the traditional model of democratic peace theory demonstrated on V-Dem Democracy Indices.
== Criticism ==
There are several logically distinguishable classes of criticism. They usually apply to no wars or few MIDs between democracies, not to little systematic violence in established democracies. In addition, there have been a number of wars between democracies. The 1987–1989 JVP insurrection in Sri Lanka is an example in which politicide was committed by a democratic regime, resulting in the deaths of at least 13,000 and 30,000 suspected JVP members or alleged supporters.
=== Statistical significance ===
One study has argued that there have been as many wars between democracies as one would expect between any other couple of states. Its authors conclude that the argument for democratic peace "rests in an ambiguity", since empirical evidence not confirm neither deny democratic pacifism, and strongly relies upon what degree of democracy makes a government democratic; according to them "because perfect democracy is infeasible, one can always sidestep counter-evidence by raising the bar of democracy".
Others state that, although there may be some evidence for democratic peace, the data sample or the time span may be too small to assess any definitive conclusions. For example, Gowa finds evidence for democratic peace to be insignificant before 1939, because of the too small number of democracies, and offers an alternate realist explanation for the following period. Gowa's use of statistics has been criticized, with several other studies and reviews finding different or opposing results. However, this can be seen as the longest-lasting criticism to the theory; as noted earlier, also some supporters agree that the statistical sample for assessing its validity is limited or scarce, at least if only full-scale wars are considered.
According to one study, which uses a rather restrictive definition of democracy and war, there were no wars between jointly democratic couples of states in the period from 1816 to 1992. Assuming a purely random distribution of wars between states, regardless of their democratic character, the predicted number of conflicts between democracies would be around ten. So, Ray argues that the evidence is statistically significant, but that it is still conceivable that, in the future, even a small number of inter-democratic wars would cancel out such evidence.
=== Peace comes before democracy ===
The territorial peace theory argues that peace leads to democracy more than democracy leads to peace. This argument is supported by historical studies showing that peace almost always comes before democracy and that states do not develop democracy until all border disputes have been settled. These studies indicate that there is strong evidence that peace causes democracy but little evidence that democracy causes peace.
The hypothesis that peace causes democracy is supported by psychological and cultural theories. Christian Welzel's human empowerment theory posits that existential security leads to emancipative cultural values and support for a democratic political organization. This also follows from the so-called regality theory based on evolutionary psychology.
The territorial peace theory explains why countries in conflict with their neighbor countries are unlikely to develop democracy. The democratic peace theory is more relevant for peace between non-neighbor countries and for relations between countries that are already at peace with each other.
=== Third factors causing both democracy and peace ===
Several other theories argue that omitted variables explain both peace and democracy.
Variables that may explain both democracy and peace include institutions, commerce, interdependence, alliances, US world dominance and political stability.
These theories are further explained under Other explanations.
=== Wars against non-democracies ===
Critics of Democratic Peace theory note that liberal states often engage in conflicts with non-liberal states they deem "rogue," "failed," or "evil." Several studies fail to confirm that democracies are less likely to wage war than autocracies if wars against non-democracies are included.
Edward Gibbon stressed that the principal conquests of the Romans were achieved under the republic; and the emperors, for the most part, were satisfied with preserving those dominions which had been acquired by the policy of the Senate, the active emulation of the consuls, and the martial enthusiasm of the people.
=== Signalling ===
The notion that democracies can signal intentions more credibly has been disputed.
=== Criticism of definitions, methodology and data ===
Some authors criticize the definition of democracy by arguing that states continually reinterpret other states' regime types as a consequence of their own objective interests and motives, such as economic and security concerns. For example, one study reports that Germany was considered a democratic state by Western opinion leaders at the end of the 19th century; yet in the years preceding World War I, when its relations with the United States, France and Britain started deteriorating, Germany was gradually reinterpreted as an autocratic state, in absence of any actual regime change. Shimmin moves a similar criticism regarding the western perception of Milosevic's Serbia between 1989 and 1999. Rummel replies to this criticism by stating that, in general, studies on democratic peace do not focus on other countries' perceptions of democracy; and in the specific case of Serbia, by arguing that the limited credit accorded by western democracies to Milosevic in the early 1990s did not amount to a recognition of democracy, but only to the perception that possible alternative leaders could be even worse.
Some democratic peace researchers have been criticized for post hoc reclassifying some specific conflicts as non-wars or political systems as non-democracies without checking and correcting the whole data set used similarly. Supporters and opponents of the democratic peace agree that this is bad use of statistics, even if a plausible case can be made for the correction. A military affairs columnist of the newspaper Asia Times has summarized the above criticism in a journalist's fashion describing the theory as subject to the no true Scotsman problem: exceptions are explained away as not being between "real" democracies or "real" wars.
Some democratic peace researchers require that the executive result from a substantively contested election. This may be a restrictive definition: For example, the National Archives of the United States notes that "For all intents and purposes, George Washington was unopposed for election as President, both in 1789 and 1792". (Under the original provisions for the Electoral College, there was no distinction between votes for president and Vice-president: each elector was required to vote for two distinct candidates, with the runner-up to be vice-president. Every elector cast one of his votes for Washington, John Adams received a majority of the other votes; there were several other candidates: so the election for vice president was contested.)
Spiro made several other criticisms of the statistical methods used. Russett and a series of papers described by Ray responded to this, for example with different methodology.
Sometimes the datasets used have also been criticized. For example, some authors have criticized the Correlates of War data for not including civilian deaths in the battle deaths count, especially in civil wars. Cohen and Weeks argue that most fishing disputes, which include no deaths and generally very limited threats of violence, should be excluded even from the list of military disputes. Gleditsch made several criticisms to the Correlates of War data set, and produced a revised set of data. Maoz and Russett made several criticisms to the Polity I and II data sets, which have mostly been addressed in later versions.
The most comprehensive critique points out that "democracy" is rarely defined, never refers to substantive democracy, is unclear about causation, has been refuted in more than 100 studies, fails to account for some 200 deviant cases, and has been promoted ideologically to justify one country seeking to expand democracy abroad. Most studies treat the complex concept of "democracy" as a bivariate variable rather than attempting to dimensionalize the concept. Studies also fail to take into account the fact that there are dozens of types of democracy, so the results are meaningless unless articulated to a particular type of democracy or claimed to be true for all types, such as consociational or economic democracy, with disparate datasets.
=== Microfoundations ===
Recent work into the democratic norms explanations shows that the microfoundations on which this explanation rest do not find empirical support. Within most earlier studies, the presence of liberal norms in democratic societies and their subsequent influence on the willingness to wage war was merely assumed, never measured. Moreover, it was never investigated whether or not these norms are absent within other regime-types. Two recent studies measured the presence of liberal norms and investigated the assumed effect of these norms on the willingness to wage war. The results of both studies show that liberal democratic norms are not only present within liberal democracies, but also within other regime-types. Moreover, these norms are not of influence on the willingness to attack another state during an interstate conflict at the brink of war.
Sebastian Rosato argues that democratic peace theory makes several false assumptions. Firstly, it assumes that democratic populaces will react negatively to the costs of war upon them. However, in modern wars casualties tend to be fairly low and soldiers are largely volunteers, meaning they accept the risks of fighting, so their families and friends, whom the cost of their death falls on heaviest, are less likely to criticise the government than the families and friends of conscripted soldiers. Secondly, democratic peace theory ignores the role of nationalism; democratic populaces are just as likely to be influenced by nationalist sentiment as anyone else and if a democratic populace believes that a war is necessary for their nation, the populace will support it. Lastly, democratic leaders are as likely to guide public opinion as they are to follow it. Democratic leaders are often aware of the power of nationalist sentiment and thus seek to encourage it when it comes to war, arguing that war is necessary to defend or spread the nation's way of life. Democratic leaders may even have an advatange over authoritarians in this regard, as they can be seen as more legitimately representative. Rosato argues that this does not just apply to wars of defence but also aggression; democratic populaces can be roused by nationalist feelings to support aggressive wars if they are seen as in the national interest.
Rosato also argues that authoritarian leaders have a reduced incentive to go to war because civilian control over the military is less guaranteed in autocracies; there is always the risk the military could subvert civilian leadership and a war which results in defeat could swiftly result in a coup. Even military dictators run the risk of internal dissent within the armed forces. Autocratic leaders in general also risk unleashing political and social turmoil that could destroy them if they go to war. Conversely, bellicose democratic leaders can rely on the acknowledgement of the legitimacy of the democratic process, as pacifist actors in democracies will need to respect the legitimacy of a democratically elected government. If pro-war groups can capture the organs of the state in a democracy legitimately, then anti-war groups will have little means of opposing them outside of extra-constitutional means, which would likely backfire and cause the anti-war groups to lose legitimacy.
A 2017 study found that public opinion in China showed the same reluctance in going to war as publics in democratic states, which suggests that publics in democratic states are not generally more opposed to war than publics in authoritarian states.
=== Limited consequences ===
The peacefulness may have various limitations and qualifiers and may not actually mean very much in the real world.
Democratic peace researchers do in general not count as wars conflicts which do not kill a thousand on the battlefield; thus they exclude for example the bloodless Cod Wars. However, research has also found a peacefulness between democracies when looking at lesser conflicts.
Liberal democracies have less of these wars than other states after 1945. This might be related to changes in the perception of non-European peoples, as embodied in the Universal Declaration of Human Rights.
Related to this is the human rights violations committed against native people, sometimes by liberal democracies. One response is that many of the worst crimes were committed by non-democracies, like in the European colonies before the nineteenth century, in King Leopold II of Belgium's privately owned Congo Free State, and in Joseph Stalin's Soviet Union. The United Kingdom abolished slavery in British territory in 1833, immediately after the Reform Act 1832 had significantly enlarged the franchise. (Of course, the abolition of the slave trade had been enacted in 1807; and many DPT supporters would deny that the UK was a liberal democracy in 1833 when examining interstate wars.)
Hermann and Kegley Jr. argue that interventions between democracies are more likely to happen than projected by an expected model. They further argue that democracies are more likely to intervene in other liberal states than against countries that are non-democracies. Finally, they argue that these interventions between democracies have been increasing over time and that the world can expect more of these interventions in the future. The methodology used has been criticized and more recent studies have found opposing results.
Rummel argues that the continuing increase in democracy worldwide will soon lead to an end to wars and democide, possibly around or even before the middle of this century. The fall of Communism and the increase in the number of democratic states were accompanied by a sudden and dramatic decline in total warfare, interstate wars, ethnic wars, revolutionary wars, and the number of refugees and displaced persons. One report claims that the two main causes of this decline in warfare are the end of the Cold War itself and decolonization; but also claims that the three Kantian factors have contributed materially.
=== Historical periods ===
Economic historians Joel Mokyr and Hans-Joachim Voth argue that democratic states may have been more vulnerable to conquest because the rulers in those states were too heavily constrained. Absolutist rulers in other states could, however, operate more freely.
=== Covert operations and proxy wars ===
Critics of the democratic peace theory have pointed to covert operations and military interventions between democracies, and argued that these interventions indicate that democracies do not necessarily trust and respect each other. Alexander B. Downes and Lary Lauren Lilley argue that covert operations conducted by democratic states has different implications depending on which version of democratic peace theory one adheres to. They argue that covert operations are inconsistent with variants of democratic peace theory that emphasize norms and checks-and-balances, but that covert operations may be more consistent with versions of democratic peace theory that rely on selectorate theory's notion of large versus small winning coalitions.
A 2015 study by Michael Poznansky reconciles findings that democracies engage in covert interventions against one another by arguing that democracies do so when they expect another state's democratic character to break down or decay.
A 2022 study found that democracies rarely wage proxy wars against fellow democracies: "strong democratic institutions prevent elected leaders from engaging in proxy war against sister regimes, and embargo violations tend to occur when democratic institutions are weak."
=== Information manipulation ===
Chaim Kaufmann argues that the lead-up to the Iraq War demonstrates that constraints on war in democracies may hinge on whether democratic governments can control and manipulate information, and suppress intelligence findings that run counter to administration rhetoric, as well as whether there is a strong opposition party and powerful media.
=== Coup by provoking a war ===
Many democracies become non-democratic by war, as being aggressed or as aggressor (quickly after a coup), sometimes the coup leader worked to provoke that war.
Carl Schmitt wrote on how to overrule a Constitution: "Sovereign is he who decides on the exception." Schmitt, again on the need for internal (and foreign) enemies because they are useful to persuade the people not to trust anyone more than the Leader: "As long as the state is a political entity this requirement for internal peace compels it in critical situations to decide also upon the domestic enemy. Every state provides, therefore, some kind of formula for the declaration of an internal enemy." Whatever opposition will be pictured and intended as the actual foreign enemy's puppet.
== Other explanations ==
=== Economic factors ===
The capitalist peace, or capitalist peace theory, posits that according to given criteria for economic development (capitalism), developed economies have not engaged in war with each other, and rarely enter into low-level disputes. These theories have been proposed as an explanation for democratic peace by accounting for both democracy and the peace among democratic nations. The exact nature of the causality depends upon both the proposed variable and the measure of the indicator for the concept used.
A majority of researchers on the determinants of democracy agree that economic development is a primary factor which allows the formation of a stable and healthy democracy. Thus, some researchers have argued that economic development also plays a factor in the establishment of peace.
Mousseau argues that a culture of contracting in advanced market-oriented economies may cause both democracy and peace. These studies indicate that democracy, alone, is an unlikely cause of the democratic peace. A low level of market-oriented economic development may hinder development of liberal institutions and values. Hegre and Souva confirmed these expectations. Mousseau finds that democracy is a significant factor only when both democracies have levels of economic development well above the global median. In fact, the poorest 21% of the democracies studied, and the poorest 4–5% of current democracies, are significantly more likely than other kinds of countries to fight each other. Mousseau, Hegre, and Oneal confirm that if at least one of the democracies involved has a very low level of economic development, democracy is ineffective in preventing war; however, they find that when also controlling for trade, 91% of all the democratic pairs had high enough development for the pacifying effect of democracy to be important during the 1885–1992 period and all in 1992. The difference in results of these two studies may be due to sampling: Mousseau's 2005 study observed only neighboring states where poor countries actually can fight each other. In fact, fully 89% of militarized conflicts between less developed countries from 1920 and 2000 were among directly contiguous neighbors. He argues that it is not likely that the results can be explained by trade: Because developed states have large economies, they do not have high levels of trade interdependence. In fact, the correlation of developed democracy with trade interdependence is a scant 0.06 (Pearson's r – considered substantively no correlation by statisticians.)
Both World Wars were fought between countries which can be considered economically developed. Mousseau argues that both Germany and Japan – like the USSR during the Cold War and Saudi Arabia today – had state-managed economies and thus lacked his market norms. Hegre finds that democracy is correlated with civil peace only for developed countries, and for countries with high levels of literacy. Conversely, the risk of civil war decreases with development only for democratic countries.
Gartzke argues that economic freedom (a quite different concept from Mousseau's market norms) or financial dependence explains the developed democratic peace, and these countries may be weak on these dimensions too. Rummel criticizes Gartzke's methodology and argues that his results are invalid.
Allan Dafoe, John R. Oneal, and Bruce Russett have challenged Gartzke and Mousseau's research.
Several studies find that democracy, more trade causing greater economic interdependence, and membership in more intergovernmental organizations reduce the risk of war. This is often called the Kantian peace theory since it is similar to Kant's earlier theory about a perpetual peace; it is often also called "liberal peace" theory, especially when one focuses on the effects of trade and democracy. (The theory that free trade can cause peace is quite old and referred to as Cobdenism.) Many researchers agree that these variables positively affect each other but each has a separate pacifying effect. For example, in countries exchanging a substantial amount of trade, economic interest groups may exist that oppose a reciprocal disruptive war, but in democracy such groups may have more power, and the political leaders be more likely to accept their requests. Weede argues that the pacifying effect of free trade and economic interdependence may be more important than that of democracy, because the former affects peace both directly and indirectly, by producing economic development and ultimately, democracy. Weede also lists some other authors supporting this view. However, some recent studies find no effect from trade but only from democracy.
None of the authors listed argues that free trade alone causes peace. Even so, the issue of whether free trade or democracy is more important in maintaining peace may have potentially significant practical consequences, for example on evaluating the effectiveness of applying economic sanctions and restrictions to autocratic countries.
It was Michael Doyle who reintroduced Kant's three articles into democratic peace theory. He argued that a pacific union of liberal states has been growing for the past two centuries. He denies that a pair of states will be peaceful simply because they are both liberal democracies; if that were enough, liberal states would not be aggressive towards weak non-liberal states (as the history of American relations with Mexico shows they are). Rather, liberal democracy is a necessary condition for international organization and hospitality (which are Kant's other two articles)—and all three are sufficient to produce peace. Other Kantians have not repeated Doyle's argument that all three in the triad must be present, instead stating that all three reduce the risk of war.
Immanuel Wallerstein has argued that it is the global capitalist system that creates shared interests among the dominant parties, thus inhibiting potentially harmful belligerence.
Toni Negri and Michael Hardt take a similar stance, arguing that the intertwined network of interests in the global capitalism leads to the decline of individual nation states, and the rise of a global Empire which has no outside, and no external enemies. As a result, they write, "The era of imperialist, interimperialist, and anti-imperialist wars is over. (...) we have entered the era of minor and internal conflicts. Every imperial war is a civil war, a police action".
=== Other explanations ===
Many studies supporting the theory have controlled for many possible alternative causes of the peace. Examples of factors controlled for are geographic distance, geographic contiguity, power status, alliance ties, militarization, economic wealth and economic growth, power ratio, and political stability. These studies have often found very different results depending on methodology and included variables, which has caused criticism. DPT does not state democracy is the only thing affecting the risk of military conflict. Many of the mentioned studies have found that other factors are also important.
Several studies have also controlled for the possibility of reverse causality from peace to democracy. For example, one study supports the theory of simultaneous causation, finding that dyads involved in wars are likely to experience a decrease in joint democracy, which in turn increases the probability of further war. So they argue that disputes between democratizing or democratic states should be resolved externally at a very early stage, in order to stabilize the system. Another study finds that peace does not spread democracy, but spreading democracy is likely to spread peace. A different kind of reverse causation lies in the suggestion that impending war could destroy or decrease democracy, because the preparation for war might include political restrictions, which may be the cause for the findings of democratic peace. However, this hypothesis has been statistically tested in a study whose authors find, depending on the definition of the pre-war period, no such effect or a very slight one. So, they find this explanation unlikely. This explanation would predict a monadic effect, although weaker than the dyadic one.
Weart argues that the peacefulness appears and disappears rapidly when democracy appears and disappears. This in his view makes it unlikely that variables that change more slowly are the explanation. Weart, however, has been criticized for not offering any quantitative analysis supporting his claims.
Wars tend very strongly to be between neighboring states. Gleditsch showed that the average distance between democracies is about 8000 miles, the same as the average distance between all states. He believes that the effect of distance in preventing war, modified by the democratic peace, explains the incidence of war as fully as it can be explained.
A 2020 study in International Organization found that it was not democracy per se that reduces the prospects for conflict, but whether women's suffrage was ensured. The study argued, "women's more pacific preferences generate a dyadic democratic peace (i.e., between democracies), as well as a monadic peace."
According to Azar Gat's War in Human Civilization, there are several related and independent factors that contribute to democratic societies being more peaceful than other forms of governments:
Wealth and comfort: Increased prosperity in democratic societies has been associated with peace because civilians are less willing to endure hardship of war and military service due to a more luxurious life at home than in pre-modern times. Increased wealth has worked to decrease war through comfort.
Metropolitan service society: The majority of army recruits come from the countryside or factory workers. Many believe that these types of people are suited for war. But as technology progressed the army turned more towards advanced services in information that rely more on computerized data which urbanized people are recruited more for this service.
Sexual revolution: The availability of sex due to the pill and women joining the labor market could be another factor that has led to less enthusiasm for men to go to war. Young men are more reluctant leave behind the pleasures of life for the rigors and chastity of the army.
Fewer young males: There is greater life expectancy which leads to fewer young males. Young males are the most aggressive and the ones that join the army the most. With fewer younger males in developed societies could help explain more pacificity.
Fewer children per family (lower fertility rate): During pre modern times it was always hard for families to lose a child but in modern times it has become more difficult due to more families having only one or two children. It has become even harder for parents to risk the loss of a child in war. However, Gat recognizes that this argument is a difficult one because during pre modern times the life expectancy was not high for children and bigger families were necessary.
Women's franchise: Women are less overtly aggressive than men. Therefore, women are less inclined to serious violence and do not support it as much as men do. In liberal democracies women have been able to influence the government by getting elected. Electing more women could have an effect on whether liberal democracies take a more aggressive approach on certain issues.
Nuclear weapons: Nuclear weapons could be the reason for not having a great power war. Many believe that a nuclear war would result in mutually assured destruction (MAD) which means that both countries involved in a nuclear war have the ability to strike the other until both sides are wiped out. This results in countries not wanting to strike the other for fear of being wiped out.
=== Realist explanations ===
Supporters of realism in international relations in general argue that not democracy or its absence, but considerations and evaluations of power, cause peace or war. Specifically, many realist critics claim that the effect ascribed to democratic, or liberal, peace, is in fact due to alliance ties between democratic states which in turn are caused, one way or another, by realist factors.
For example, Farber and Gowa find evidence for peace between democracies to be statistically significant only in the period from 1945 on, and consider such peace an artifact of the Cold War, when the threat from the communist states forced democracies to ally with one another. Mearsheimer offers a similar analysis of the Anglo-American peace before 1945, caused by the German threat. Spiro finds several instances of wars between democracies, arguing that evidence in favor of the theory might be not so vast as other authors report, and claims that the remaining evidence consists of peace between allied states with shared objectives. He acknowledges that democratic states might have a somewhat greater tendency to ally with one another, and regards this as the only real effect of democratic peace. Rosato argues that most of the significant evidence for democratic peace has been observed after World War II; and that it has happened within a broad alliance, which can be identified with NATO and its satellite nations, imposed and maintained by American dominance as part of Pax Americana. One of the main points in Rosato's argument is that, although never engaged in open war with another liberal democracy during the Cold War, the United States intervened openly or covertly in the political affairs of democratic states several times, for example in the Chilean coup of 1973, the Operation Ajax (1953 coup in Iran) and Operation PBSuccess (1954 coup in Guatemala); in Rosato's view, these interventions show the United States' determination to maintain an "imperial peace".
The most direct counter arguments to such criticisms have been studies finding peace between democracies to be significant even when controlling for "common interests" as reflected in alliance ties. Regarding specific issues, Ray objects that explanations based on the Cold War should predict that the Communist bloc would be at peace within itself also, but exceptions include the Soviet Invasion of Afghanistan, the Cambodian-Vietnamese War, and the Sino-Vietnamese War. Ray also argues that the external threat did not prevent conflicts in the Western bloc when at least one of the involved states was a non-democracy, such as the Turkish Invasion of Cyprus (against Greek Junta supported Cypriot Greeks), the Falklands War, and the Football War. Also, one study notes that the explanation "goes increasingly stale as the post-Cold War world accumulates an increasing number of peaceful dyad-years between democracies". Rosato's argument about American dominance has also been criticized for not giving supporting statistical evidence.
Some realist authors also criticize in detail the explanations first by supporters of democratic peace, pointing to supposed inconsistencies or weaknesses.
Rosato criticizes most explanations to how democracy might cause peace. Arguments based on normative constraints, he argues, are not consistent with the fact that democracies do go to war no less than other states, thus violating norms preventing war; for the same reason he refutes arguments based on the importance of public opinion. Regarding explanations based on greater accountability of leaders, he finds that historically autocratic leaders have been removed or punished more often than democratic leaders when they get involved in costly wars. Finally, he also criticizes the arguments that democracies treat each other with trust and respect even during crises; and that democracy might be slow to mobilize its composite and diverse groups and opinions, hindering the start of a war, drawing support from other authors. Another realist, Layne, analyzes the crises and brinkmanship that took place between non-allied democratic great powers, during the relatively brief period when such existed. He finds no evidence either of institutional or cultural constraints against war; indeed, there was popular sentiment in favor of war on both sides. Instead, in all cases, one side concluded that it could not afford to risk that war at that time, and made the necessary concessions.
Rosato's objections have been criticized for claimed logical and methodological errors, and for being contradicted by existing statistical research. Russett replies to Layne by re-examining some of the crises studied in his article, and reaching different conclusions; Russett argues that perceptions of democracy prevented escalation, or played a major role in doing so. Also, a recent study finds that, while in general the outcome of international disputes is highly influenced by the contenders' relative military strength, this is not true if both contenders are democratic states; in this case the authors find the outcome of the crisis to be independent of the military capabilities of contenders, which is contrary to realist expectations. Finally, both the realist criticisms here described ignore new possible explanations, like the game-theoretic one discussed below.
=== Nuclear deterrent ===
A different kind of realist criticism stresses the role of nuclear weapons in maintaining peace. In realist terms, this means that, in the case of disputes between nuclear powers, respective evaluation of power might be irrelevant because of Mutual assured destruction preventing both sides from foreseeing what could be reasonably called a "victory". The 1999 Kargil War between India and Pakistan has been cited as a counterexample to this argument, though this was a small, regional conflict and the threat of WMDs being used contributed to its de-escalation.
Some supporters of the democratic peace do not deny that realist factors are also important. Research supporting the theory has also shown that factors such as alliance ties and major power status influence interstate conflict behavior.
== Statistical difficulties due to newness of democracy ==
One problem with the research on wars is that, as the Realist John Mearsheimer put it, "democracies have been few in number over the past two centuries, and thus there have been few opportunities where democracies were in a position to fight one another". Democracies have been very rare until recently. Even looser definitions of democracy, such as Doyle's, find only a dozen democracies before the late nineteenth century, and many of them short-lived or with limited franchise. Freedom House finds no independent state with universal suffrage in 1900.
Wayman, a supporter of the theory, states that "If we rely solely on whether there has been an inter-democratic war, it is going to take many more decades of peace to build our confidence in the stability of the democratic peace".
=== Studying lesser conflicts ===
Many researchers have reacted to this limitation by studying lesser conflicts instead, since they have been far more common. There have been many more MIDs than wars; the Correlates of War Project counts several thousand during the last two centuries. A review lists many studies that have reported that democratic pairs of states are less likely to be involved in MIDs than other pairs of states.
Another study finds that after both states have become democratic, there is a decreasing probability for MIDs within a year and this decreases almost to zero within five years.
When examining the inter-liberal MIDs in more detail, one study finds that they are less likely to involve third parties, and that the target of the hostility is less likely to reciprocate, if the target reciprocates the response is usually proportional to the provocation, and the disputes are less likely to cause any loss of life. The most common action was "Seizure of Material or Personnel".
Studies find that the probability that disputes between states will be resolved peacefully is positively affected by the degree of democracy exhibited by the lesser democratic state involved in that dispute. Disputes between democratic states are significantly shorter than disputes involving at least one undemocratic state. Democratic states are more likely to be amenable to third party mediation when they are involved in disputes with each other.
In international crises that include the threat or use of military force, one study finds that if the parties are democracies, then relative military strength has no effect on who wins. This is different from when non-democracies are involved. These results are the same also if the conflicting parties are formal allies. Similarly, a study of the behavior of states that joined ongoing militarized disputes reports that power is important only to autocracies: democracies do not seem to base their alignment on the power of the sides in the dispute.
== Academic relevance and derived studies ==
Democratic peace theory is a well established research field with more than a hundred authors having published articles about it. Several peer-reviewed studies mention in their introduction that most researchers accept the theory as an empirical fact. According to a 2021 study by Kosuke Imai and James Lo, "overturning the negative association between democracy and conflict would require a confounder that is forty-seven times more prevalent in democratic dyads than in other dyads. To put this number in context, the relationship between democracy and peace is at least five times as robust as that between smoking and lung cancer. To explain away the democratic peace, therefore, scholars would have to find far more powerful confounders than those already identified in the literature."
Imre Lakatos suggested that what he called a "progressive research program" is better than a "degenerative" one when it can explain the same phenomena as the "degenerative" one, but is also characterized by growth of its research field and the discovery of important novel facts. In contrast, the supporters of the "degenerative" program do not make important new empirical discoveries, but instead mostly apply adjustments to their theory in order to defend it from competitors. Some researchers argue that democratic peace theory is now the "progressive" program in international relations. According to these authors, the theory can explain the empirical phenomena previously explained by the earlier dominant research program, realism in international relations; in addition, the initial statement that democracies do not, or rarely, wage war on one another, has been followed by a rapidly growing literature on novel empirical regularities.
Other examples are several studies finding that democracies are more likely to ally with one another than with other states, forming alliances which are likely to last longer than alliances involving non-democracies; several studies showing that democracies conduct diplomacy differently and in a more conciliatory way compared to non-democracies; one study finding that democracies with proportional representation are in general more peaceful regardless of the nature of the other party involved in a relationship; and another study reporting that proportional representation system and decentralized territorial autonomy is positively associated with lasting peace in postconflict societies.
== Influence ==
The democratic peace theory has been extremely divisive among political scientists. It is rooted in the idealist and classical liberalist traditions and is opposed to the dominant theory of realism.
In the United States, presidents from both major parties have expressed support for the theory. In his 1994 State of the Union address, then-President Bill Clinton, a member of the Democratic Party, said: "Ultimately, the best strategy to ensure our security and to build a durable peace is to support the advance of democracy elsewhere. Democracies don't attack each other". In a 2004 press conference, then-President George W. Bush, a member of the Republican Party, said: "And the reason why I'm so strong on democracy is democracies don't go to war with each other. And the reason why is the people of most societies don't like war, and they understand what war means.... I've got great faith in democracies to promote peace. And that's why I'm such a strong believer that the way forward in the Middle East, the broader Middle East, is to promote democracy."
In a 1999 speech, Chris Patten, the then-European Commissioner for External Relations, said: "Inevitable because the EU was formed partly to protect liberal values, so it is hardly surprising that we should think it appropriate to speak out. But it is also sensible for strategic reasons. Free societies tend not to fight one another or to be bad neighbours". The A Secure Europe in a Better World, European Security Strategy states: "The best protection for our security is a world of well-governed democratic states." Tony Blair has also claimed the theory is correct.
=== As justification for initiating war ===
Some fear that the democratic peace theory may be used to justify wars against non-democracies in order to bring lasting peace, in a democratic crusade. Woodrow Wilson in 1917 asked Congress to declare war against Imperial Germany, citing Germany's sinking of American ships due to unrestricted submarine warfare and the Zimmermann telegram, but also stating that "A steadfast concert for peace can never be maintained except by a partnership of democratic nations" and "The world must be made safe for democracy." R. J. Rummel was a notable proponent of war for the purpose of spreading democracy, based on this theory.
Some point out that the democratic peace theory has been used to justify the 2003 Iraq War, others argue that this justification was used only after the war had already started. Furthermore, Weede has argued that the justification is extremely weak, because forcibly democratizing a country completely surrounded by non-democracies, most of which are full autocracies, as Iraq was, is at least as likely to increase the risk of war as it is to decrease it (some studies show that dyads formed by one democracy and one autocracy are the most warlike, and several find that the risk of war is greatly increased in democratizing countries surrounded by non-democracies). According to Weede, if the United States and its allies wanted to adopt a rationale strategy of forced democratization based on democratic peace, which he still does not recommend, it would be best to start intervening in countries which border with at least one or two stable democracies, and expand gradually. Also, research shows that attempts to create democracies by using external force has often failed. Gleditsch, Christiansen, and Hegre argue that forced democratization by interventionism may initially have partial success, but often create an unstable democratizing country, which can have dangerous consequences in the long run. Those attempts which had a permanent and stable success, like democratization in Austria, West Germany and Japan after World War II, mostly involved countries which had an advanced economic and social structure already, and implied a drastic change of the whole political culture. Supporting internal democratic movements and using diplomacy may be far more successful and less costly. Thus, the theory and related research, if they were correctly understood, may actually be an argument against a democratic crusade.
Michael Haas has written perhaps the most trenchant critique of a hidden normative agenda. Among the points raised: Due to sampling manipulation, the research creates the impression that democracies can justifiably fight non-democracies, snuff out budding democracies, or even impose democracy. And due to sloppy definitions, there is no concern that democracies continue undemocratic practices yet remain in the sample as if pristine democracies.
This criticism is confirmed by David Keen who finds that almost all historical attempts to impose democracy by violent means have failed.
== Related theories ==
=== Republican liberalism ===
Republican liberalism is a variation of Democratic Peace Theory which claims that liberal and republican democracies will rarely go to war with each other. It argues that these governments are more peaceful than non-democracies and will avoid conflict when possible. According to Micheal Doyle, there are three main reasons for this: Democracies tend to have similar domestic political cultures, they share common morals, and their economic systems are interdependent. Liberal democracies (republics) that trade with each other, are economically dependent on one another and therefore, will always attempt to maintain diplomatic relations as to not disrupt their economies.
Liberalism, as an overarching theory, holds that diplomacy and cooperation is the most effective way to avoid war and maintain peace. This is contrasting to the theory of realism, which states that conflict will always be recurrent in the international system, whether due to human nature or the anarchic international system.
The concept of Republican liberalism is thought to have initially originated from Immanuel Kant's book "Perpetual Peace: A Philosophical Sketch" (1795). The term "Perpetual Peace" refers to the permanent establishment of peace, and was made notorious by the book. Democratic peace, commercial peace and institutional peace were all advanced in the book as well. It takes a rather utopian view, that humanities' desire for peace will out compete humanities' desire for war.
=== Kantian liberalism ===
Kant and the liberal school of thought view international co-operation as a more rational option for states than resorting to war. However, the neo-liberal approach concedes to the realist school of thought, that when states cooperate it is simply because it is in their best interest. Kant insisted that a world with only peace was possible, and he offered three definitive articles that would create the pathway for it. Each went on to become a dominant strain of post–World War II liberal international relations theory.
I: "The Civil Constitution of Every State should be Republican"
Kant believed that every state should have Republican style form of government. As in, a state where "supreme power is held by the people and their elected representatives." Kant saw this in Ancient Rome, where they began to move away from Athenian democracy (direct democracy) and towards a representative democracy. Kant believed giving the citizens the right to vote and decide for themselves would lead to shorter wars and less wars. He also thought it was important to "check the power of monarchs", to establish a system of checks and balances where no one person holds absolute power. Peace is always dependent on the internal character of governments. Republics, with a legislative body that will be able to hold the executive leader in check and maintain the peace.
II: "The Law of Nations shall be founded on a Federation of Free States"
Kant argues nations, like individuals can be tempted to harm each other at any given moment. So, rule of law should be established internationally. Without international laws and courts of judgement, then force would be the only way to settle disputes. States ought to instead develop international organisations and rules that facilitate cooperation. In any case, some kind of federation is necessary in order to maintain peace between nations. Contemporary examples include the United Nations and the European Union, which try to maintain peace and encourage cooperation among nations.
III: "The Law of World Citizenship shall be Limited to Conditions of Universal Hospitality"
Kant is referring to "the right of the stranger to not to be treated as an enemy when he arrives in the land of another.". So long "stranger" is peaceful, he should not be treated with hostility. However, this is not the right to be a "permanent visitor", simply as a temporary stay. This is applicable in the contemporary world when a country is receiving a world leader. The host country usually holds a state welcoming ceremony which strengthens diplomatic relations.
=== European peace ===
There is significant debate over whether the lack of any major European general wars since 1945 is due to cooperation and integration of liberal-democratic European states themselves (as in the European Union or Franco-German cooperation), an enforced peace due to the intervention of the Soviet Union and the United States until 1989 and the United States alone thereafter, or a combination of both. The debate over this theory was thrust in the public eye when the 2012 Nobel Peace Prize was awarded to the European Union for its role in creating peace in Europe. Notable major wars in Europe after 1945 are Yugoslav Wars and Russian invasion of Ukraine, which follows the prediction of the interactive model of democratic peace.
== See also ==
Global Peace Index
Peace and conflict studies
== Notes and references ==
=== Notes ===
=== References ===
== Bibliography ==
== Further reading ==
== External links == | Wikipedia/Democratic_peace_theory |
In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from the beginning and is required to output an answer which solves the problem at hand.
In operations research, the area in which online algorithms are developed is called online optimization.
As an example, consider the sorting algorithms selection sort and insertion sort: selection sort repeatedly selects the minimum element from the unsorted remainder and places it at the front, which requires access to the entire input; it is thus an offline algorithm. On the other hand, insertion sort considers one input element per iteration and produces a partial solution without considering future elements. Thus insertion sort is an online algorithm.
Note that the final result of an insertion sort is optimum, i.e., a correctly sorted list. For many problems, online algorithms cannot match the performance of offline algorithms. If the ratio between the performance of an online algorithm and an optimal offline algorithm is bounded, the online algorithm is called competitive.
Not every offline algorithm has an efficient online counterpart.
In grammar theory they are associated with Straight-line grammars.
== Definition ==
Because it does not know the whole input, an online algorithm is forced to make decisions that may later turn out not to be optimal, and the study of online algorithms has focused on the quality of decision-making that is possible in this setting. Competitive analysis formalizes this idea by comparing the relative performance of an online and offline algorithm for the same problem instance. Specifically, the competitive ratio of an algorithm, is defined as the worst-case ratio of its cost divided by the optimal cost, over all possible inputs. The competitive ratio of an online problem is the best competitive ratio achieved by an online algorithm. Intuitively, the competitive ratio of an algorithm gives a measure on the quality of solutions produced by this algorithm, while the competitive ratio of a problem shows the importance of knowing the future for this problem.
=== Other interpretations ===
For other points of view on online inputs to algorithms, see
streaming algorithm: focusing on the amount of memory needed to accurately represent past inputs;
dynamic algorithm: focusing on the time complexity of maintaining solutions to problems with online inputs.
=== Examples ===
Some online algorithms:
Insertion sort
Perceptron
Reservoir sampling
Greedy algorithm
Adversary model
Metrical task systems
Odds algorithm
Page replacement algorithm
Algorithms for calculating variance
Ukkonen's algorithm
== Online problems ==
A problem exemplifying the concepts of online algorithms is the Canadian traveller problem. The goal of this problem is to minimize the cost of reaching a target in a weighted graph where some of the edges are unreliable and may have been removed from the graph. However, that an edge has been removed (failed) is only revealed to the traveller when she/he reaches one of the edge's endpoints. The worst case for this problem is simply that all of the unreliable edges fail and the problem reduces to the usual shortest path problem. An alternative analysis of the problem can be made with the help of competitive analysis. For this method of analysis, the offline algorithm knows in advance which edges will fail and the goal is to minimize the ratio between the online and offline algorithms' performance. This problem is PSPACE-complete.
There are many formal problems that offer more than one online algorithm as solution:
k-server problem
Job shop scheduling problem
List update problem
Bandit problem
Secretary problem
Search games
Ski rental problem
Linear search problem
Portfolio selection problem
== See also ==
Dynamic algorithm
Prophet inequality
Real-time computing
Streaming algorithm
Sequential algorithm
Online machine learning/Offline learning
== References ==
Borodin, A.; El-Yaniv, R. (1998). Online Computation and Competitive Analysis. Cambridge University Press. ISBN 0-521-56392-5.
== External links ==
Bibliography of papers on online algorithms | Wikipedia/Online_algorithm |
Alternative dispute resolution (ADR), or external dispute resolution (EDR), typically denotes a wide range of dispute resolution processes and techniques that parties can use to settle disputes with the help of a third party. They are used for disagreeing parties who cannot come to an agreement short of litigation. However, ADR is also increasingly being adopted as a tool to help settle disputes within the court system.
Despite historic resistance to ADR by many popular parties and their advocates, ADR has gained widespread acceptance among both the general public and the legal profession in recent years. In 2008, some courts required some parties to resort to ADR of some type like mediation, before permitting the parties' cases to be tried (the European Mediation Directive (2008) expressly contemplates so-called "compulsory" mediation. This means that attendance is compulsory, not that settlement must be reached through mediation). Additionally, parties to merger and acquisition transactions are increasingly turning to ADR to resolve post-acquisition disputes. In England and Wales, ADR is now more commonly referred to as ‘NCDR’ (Non Court Dispute Resolution), in an effort to promote this as the normal (rather than alternative) way to resolve disputes. A 2023 judgment of the Court of Appeal called Churchill v Merthyr confirmed that in the right case the Court can order (i) the parties to engage in NCDR and / or (ii) stay the proceedings to allow for NCDR to take place. This overturns the previous orthodoxy (the 2004 Court of Appeal decision of Halsey v. Milton Keynes General NHS
Trust) which was that unwilling parties could not be obliged to participate in NCDR.
The rising popularity of ADR can be explained by the increasing caseload of traditional courts, the perception that ADR imposes fewer costs than litigation, a preference for confidentiality, and the desire of some parties to have greater control over the selection of the individual or individuals who will decide their dispute. Some of the senior judiciary in certain jurisdictions (of which England and Wales is one) are strongly in favour of this use of mediation and other NCDR processes to settle disputes. Since the 1990s many American courts have also increasingly advocated for the use of ADR to settle disputes. However, it is not clear as to whether litigants can properly identify and then use the ADR programmes available to them, thereby potentially limiting their effectiveness.
== History ==
The term "alternative dispute resolution" arose from Frank Sander's paper, "Varieties of Dispute Processing".
Traditional arbitration involved heads of trade guilds or other dominant authorities settling disputes. The modern innovation was to have commercial vendors of arbitrators, often ones with little or no social or political dominance over the parties. The advantage was that such persons were much more readily available. The disadvantage is that it does not involve the community of the parties. When wool contract arbitration was conducted by senior guild officials, the arbitrator combined a seasoned expert on the subject matter with a socially dominant individual whose patronage, goodwill and opinion were important.
ADR can increasingly be conducted online, which is known as online dispute resolution (ODR, which is mostly a buzzword and an attempt to create a distinctive product). ODR services can be provided by government entities, and as such may form part of the litigation process. Moreover, they can be provided on a global scale, where no effective domestic remedies are available to disputing parties, as in the case of the UDRP and domain name disputes. In this respect, ODR might not satisfy the "alternative" element of ADR. In England and Wales, the Online Procedure Rule Committee was set up under the Judicial Review and Courts Act 2022 to make rules governing the practice and procedure for specific types of online court and tribunal proceedings across the Civil, Family and Tribunal jurisdictions. OPRC is an advisory non-departmental public body, sponsored by the Ministry of Justice. The committee is chaired by the Master of the Rolls, Head of Civil Justice.
== Definition ==
ADR has historically been divided between methods of resolving disputes outside of official judicial mechanisms and informal methods attached to official judicial mechanisms. Regardless of whether they are part of an overarching proceeding, the mechanisms are generally similar. There are four general classes of ADR: negotiation, mediation, collaborative law, and arbitration. In some contexts, such as in the settlement of investment disputes, arbitration is not considered as a form of ADR, since it is the principal means of settling these disputes. Some academics include conciliation as a fifth category, but others include this within the definition of mediation.
Conflict resolution is one major goal of all the ADR processes. If a process leads to resolution, it is a dispute resolution process. "Alternative" dispute resolution is usually considered to be alternative to litigation. For example, corporate dispute resolution can involve a customer service department handling disputes about its own products; addressing concerns between consumers and independent, third-party sellers; and participating in a reputation-based enforcement mechanism. It also can be used as a colloquialism for allowing a dispute to drop or as an alternative to violence.
In recent years, there has been more discussion about taking a systems approach in order to offer different kinds of options to people who are in conflict and to foster "appropriate" dispute resolution. That is, some cases and some complaints, in fact, ought to go to a formal grievance, to a court, to the police, to a compliance officer, or to a government IG. Other conflicts could be settled by the parties if they had enough support and coaching, and yet other cases need mediation or arbitration. Thus "alternative" dispute resolution usually means a method that is not the courts. "Appropriate" dispute resolution considers all the possible responsible options for conflict resolution that are relevant to a given issue.
=== Negotiation ===
In negotiation, participation is voluntary and there is no third party who facilitates the resolution process or imposes a resolution. (NB – a third party like a chaplain or organizational ombudsperson or social worker or a skilled friend may be coaching one or both of the parties behind the scenes, a process called "Helping People Help Themselves" – see Helping People Help Themselves, in Negotiation Journal July 1990, pp. 239–248, which includes a section on helping someone draft a letter to someone who is perceived to have wronged them.)
=== Mediation ===
In mediation, there is a third party, a mediator, who facilitates the resolution process (and may even suggest a resolution, typically known as a "mediator's proposal"), but does not impose a resolution on the parties. In some countries (for example, the United Kingdom), ADR is synonymous with what is generally referred to as mediation in other countries. Structured transformative mediation as used by the U.S. Postal Service is a formal process.
Traditional people's mediation has always involved the parties remaining in contact for most or all of the mediation sessions. The innovation of separating the parties after (or sometimes before) a joint session and conducting the rest of the process without the parties in the same area was a major innovation and one that dramatically improved mediation's success rate.
==== Lawyer-supported mediation ====
Lawyer-supported mediation is a "non-adversarial method of alternative dispute resolution to resolves disputes, such as to settle family issues at a time of divorce or separation, including child support, custody issues and division of property".
==== Party-directed mediation ====
Party-directed mediation (PDM) is an approach to mediation that seeks to empower each party in a dispute, enabling each party to have a more direct influence upon the resolution of a conflict, by offering both means and processes for enhancing the negotiation skills of contenders. The intended prospect of party-directed mediation is to improve upon the ability and willingness of disputants to deal with subsequent differences.
=== Collaborative law ===
In collaborative law or collaborative divorce, each party has an attorney who facilitates the resolution process within specifically contracted terms. The parties reach an agreement with the support of the attorneys (who are trained in the process) and mutually agreed experts. No one imposes a resolution on the parties.
=== Arbitration ===
In arbitration, participation is typically voluntary, and there is a third party who, as a private judge, imposes a resolution. Arbitrations often occur because parties to contracts agree that any future dispute concerning the agreement will be resolved by arbitration. This is known as a 'Scott Avery Clause'. In recent years, the enforceability of arbitration clauses, particularly in the context of consumer agreements (e.g., credit card agreements), has drawn scrutiny from courts. Although parties may appeal arbitration outcomes to courts, such appeals face an exacting standard of review.
=== Conciliation ===
Conciliation is an alternative dispute resolution (ADR) process whereby the parties to a dispute use a conciliator, who meets with the parties both separately and together in an attempt to resolve their differences. They do this by lowering tensions, improving communications, interpreting issues, encouraging parties to explore potential solutions and assisting parties in finding a mutually acceptable outcome.
=== Other methods ===
Beyond the basic types of alternative dispute resolutions, there are other different forms of ADR.
==== Case evaluation ====
Case evaluation is a non-binding process in which parties present the facts and the issues to a neutral case evaluator who advises the parties on the strengths and weaknesses of their respective positions, and assesses how the dispute is likely to be decided by a jury or other adjudicator.
==== Early neutral evaluation ====
Early neutral evaluation is a process that takes place soon after a case has been filed in court. The case is referred to an expert who is asked to provide a balanced and neutral evaluation of the dispute. The evaluation of the expert can assist the parties in assessing their case and may influence them towards a settlement.
==== One Couple One Lawyer ====
One Couple One Lawyer, or Single Lawyer, is a family law process developed in England and Wales where a separating couple shares one lawyer who advises them both, impartially and together, as to how a judge would view their case, and the likely outcome were they to litigate, thus enabling them to reach a fair settlement on separation or divorce. This differs from early neutral evaluation as it is designed so that parties never require separate representation, are assisted throughout by one legal team and the process has no adversarial features at all, either at the financial disclosure or advice stages.
In April 2024, a new definition of NCDR was set out in the Family Procedure (Amendments No 2) Rules 2023/1324 as “methods of resolving a dispute other than through the court process, including but not limited to mediation, arbitration, evaluation by a neutral third party (such as a private Financial Dispute Resolution process) and collaborative law.” In the accompanying Pre-application Protocol (Annex to PD9A), the One Couple One Lawyer process was also referenced “The court may also consider the parties having obtained legal advice via the “single lawyer” or a “one couple, one lawyer” scheme as good evidence of a constructive attempt to obtain advice and avoid unnecessary proceedings […]”
==== Family group conference ====
A family group conference is a meeting between members of a family and members of their extended related group. At this meeting (or often a series of meetings) the family becomes involved in learning skills for interaction and in making a plan to stop the abuse or other ill-treatment between its members.
==== Neutral fact-finding ====
Neutral fact-finding is a process where a neutral third party, selected either by the disputing parties or by the court, investigates an issue and reports or testifies in court. The neutral fact-finding process is particularly useful for resolving complex scientific and factual disputes.
==== Expert determination ====
Expert determination is a procedure where a dispute or a difference between the parties is submitted, by mutual agreement of the parties, to one or more experts who make a determination on the matter referred to them. The determination is binding, unless the parties agreed otherwise, and is a confidential procedure.
==== Ombudsmen ====
Ombudsmen are a third party selected by an institution—for example, a university, hospital, corporation or government agency—to deal with complaints by employees, clients or constituents. An organizational ombudsman works within the institution to look into complaints independently and impartially. Calling an organizational ombudsman is always voluntary; according to the International Ombudsman Association Standards of Practice, no one can be compelled to use an ombudsman office. Organizational ombudsman offices refer people to all conflict management options in the organization: formal and informal, rights-based and interest-based. But, in addition, in part because they have no decision-making authority, ombudsman offices can, themselves, offer a wide spectrum of informal options.
== Advantages and disadvantages ==
Suitable for multi-party disputes
Lower costs: in many cases dispute resolution is available to consumers free of charge
Likelihood and speed of settlements
Flexibility of process
Parties' control of process
Parties' choice of forum
Practical solutions
Wider range of issues can be considered
Shared future interests may be protected
Confidentiality
Risk management
Generally no need for lawyers
Can be a less confrontational alternative to the court system
However, ADR is less suitable than litigation when there is:
A need for precedent
A need for court orders
A need for interim orders
A need for evidential rules
A need for enforcement
Power imbalance between parties
Quasi-criminal allegations
Complexity in the case
The need for live evidence or analysis of complex evidence
The need for expert evidence
== Country-specific examples ==
=== Canada ===
In the 1980s and 1990s Canada saw the beginning of a "cultural shift" in their experience with ADR practices. During this time, the need was recognized for an alternative to the more adversarial approach to dispute settlement that is typical in traditional court proceedings. This growth continued over the coming decades, with ADR now being widely recognized as a legitimate and effective approach to dispute resolution. In 2014, the Supreme Court of Canada stated in Hryniak v Mauldin that "meaningful access to justice is now the greatest challenge to the rule of law in Canada today... [The] balance between procedure and access struck by our justice system must reflect modern reality and recognize that new models of adjudication can be fair and just." However, in the decades leading up to this declaration there had already been a number of experiments in ADR practices across the provinces.
One of the first and most notable ADR initiatives in Canada began on 4 January 1999, with the creation of the Ontario Mandatory Mediation Program. This program included the implementation of Rule 24.1, which established mandatory mediation for non-family civil case-managed actions. Beginning in a selection of courts across Ontario and Ottawa in 1999, the program would be expanded in 2002 to cover Windsor, Ontario's third-largest judicial area. Until this point, opposition to mandatory mediation in place of traditional litigation had been grounded in the idea that mediation practices are effective when disputing parties voluntarily embrace the process. However, reports analyzing the effectiveness of Ontario's experiment concluded that overall mandatory mediation as a form of ADR was able to reduce both the cost and time delay of finding a dispute resolution, compared to a control group. In addition to this, 2/3's of the parties surveyed from this study outlined the benefits to mandatory mediation, these included:(i) providing one or more parties with new information they considered relevant;
(ii) identifying matters important to one or more of the parties;
(iii) setting priorities among issues;
(iv) facilitating discussion of new settlement offers;
(v) achieving better awareness of the potential monetary savings from settling earlier in the litigation process;
(vi) at least one of the parties gaining a better understanding of his or her own ADR in Administrative Litigation 157 case; and
(vii) at least one of the parties gaining a better understanding of his or her opponent's case.In other provinces, the need for ADR to at least be examined as an alternative to traditional court proceedings has also been expressed. For instance, in 2015 Quebec implemented the New Code, which mandated that parties must at least consider mediation before moving to settle a dispute in court. The New Code also codified the role of the mediator in the courtroom, outlining that mediators must remain impartial and cannot give evidence on either party's behalf should the dispute progress to a judicial proceeding. In 2009, a report showed that Manitoba's experience with their Judicially Assisted Dispute Resolution program, an ADR initiative where the court appoints a judge to act as a mediator between two disputing parties who both voluntarily wish to pursue JADR.
One of the main arguments for ADR practices in Canada cites the over-clogged judicial system. This is one of the main arguments for ADR across many regions; however, Alberta, in particular, suffers from this issue. With a rising population, in 2018 Alberta had the highest ratio for the population to Superior Court Justices, 63,000:1. The national average on the other hand is nearly half that, with one Justice being counted for every 35,000 Canadians.
To become qualified as a mediator in Canada, it is possible to gain mediation training through certain private organizations or post-secondary institutions. The ADR Institute of Canada (ADRIC) is the preeminent ADR training organization in Canada.
=== India ===
Alternative dispute resolution in India is not new and it was in existence even under the previous Arbitration Act of 1940. The Arbitration and Conciliation Act, 1996 has been enacted to accommodate the harmonization mandates of UNCITRAL Model. To streamline the Indian legal system, the traditional civil law known as Code of Civil Procedure, (CPC) 1908 has also been amended, and Section 89 has been introduced. Section 89(1) of CPC provides an option for the settlement of disputes outside the court. It provides that where it appears to the court that there exist elements that may be acceptable to the parties, the court may formulate the terms of a possible settlement and refer the same for arbitration, conciliation, mediation or judicial settlement. India has also enacted The Mediation Act, 2023 to provide for the law with respect to mediation in India.
Due to the extremely slow judicial process, there has been a large emphasis on alternate dispute resolution mechanisms in India. While the Arbitration and Conciliation Act of 1996 is a fairly standard Western approach towards ADR, the Lok Adalat system constituted under the National Legal Services Authority Act, 1987 is a uniquely Indian approach.
A study on commercial dispute resolution in south India has been done by a think tank organization based in Kochi, Centre for Public Policy Research. The study reveals that the Court-annexed Mediation Centre in Bangalore has a success rate of 64%, while its counterpart in Kerala has an average success rate of 27.7%. Furthermore, amongst the three southern states (Karnataka, Tamil Nadu, and Kerala), Tamil Nadu is said to have the highest adoption of dispute resolution, Kerala the least.
==== Arbitration and Conciliation Act, 1996 ====
An Act to consolidate and amend the law relating to domestic arbitration, international commercial arbitration and enforcement of foreign arbitral awards as also to define the law relating to conciliation and for matters connected therewith or incidental thereto.
===== Arbitration =====
The process of arbitration can start only if there exists a valid Arbitration Agreement between the parties prior to the emergence of the dispute. As per Section 7, such an agreement must be in writing. The contract regarding which the dispute exists, must either contain an arbitration clause or must refer to a separate document signed by the parties containing the arbitration agreement. The existence of an arbitration agreement can also be inferred by written correspondence such as letters, telex, or telegrams which provide a record of an agreement. An exchange of statement of claim and defence in which the existence of an arbitration agreement is alleged by one party and not denied by other is also considered as a valid written arbitration agreement.
Any party to the dispute can start the process of appointing an arbitrator and if the other party does not cooperate, the party can approach the office of Chief Justice for the appointment of an arbitrator. There are only two grounds upon which a party can challenge the appointment of an arbitrator – reasonable doubt in the impartiality of the arbitrator and the lack of proper qualification of the arbitrator as required by the arbitration agreement. A sole arbitrator or a panel of arbitrators so appointed constitute the Arbitration Tribunal.
Except for some interim measures, there is very little scope for judicial intervention in the arbitration process. The arbitration tribunal has jurisdiction over its own jurisdiction. Thus, if a party wants to challenge the jurisdiction of the arbitration tribunal, it can do so only before the tribunal itself. If the tribunal rejects the request, there is little the party can do except to approach a court after the tribunal makes an award. Section 34 provides certain grounds upon which a party can appeal to the principal civil court of original jurisdiction for setting aside the award.
The period for filing an appeal for setting aside an award is over, or if such an appeal is rejected, the award is binding on the parties and is considered as a decree of the court.
===== Conciliation =====
Conciliation is a less formal form of arbitration. This process does not require the existence of any prior agreement. Any party can request the other party to appoint a conciliator. One conciliator is preferred but two or three are also allowed. In the case of multiple conciliators, all must act jointly. If a party rejects an offer to conciliate, there can be no conciliation.
Parties may submit statements to the conciliator describing the general nature of the dispute and the points at issue. Each party sends a copy of the statement to the other. The conciliator may request further details, may ask to meet the parties, or communicate with the parties orally or in writing. Parties may even submit suggestions for the settlement of the dispute to the conciliator.
When it appears to the conciliator that elements of settlement exist, he may draw up the terms of the settlement and send it to the parties for their acceptance. If both the parties sign the settlement document, it shall be final and binding on both.
This process is similar to the US practice of mediation. However, in India, mediation is different from conciliation and is a completely informal type of ADR mechanism.
==== Lok Adalat ====
Etymologically, Lok Adalat means "people's court". India has had a long history of resolving disputes through the mediation of village elders. The current system of Lok Adalats is an improvement on that and is based on Gandhian principles. This is a non-adversarial system, whereby mock courts (called Lok Adalats) are held by the State Authority, District Authority, Supreme Court Legal Services Committee, High Court Legal Services Committee, or Taluk Legal Services Committee, periodically for exercising such jurisdiction as they think fit. These are usually presided by a retired judge, social activists, or members of the legal profession. It does not have jurisdiction on matters related to non-compoundable offences.
While in regular suits, the plaintiff is required to pay the prescribed court fee, in Lok Adalat, there is no court fee and no rigid procedural requirement (i.e. no need to follow the process given by [Indian] Civil Procedure Code or Indian Evidence Act), which makes the process very fast. Parties can directly interact with the judge, which is not possible in regular courts.
Cases that are pending in regular courts can be transferred to a Lok Adalat if both the parties agree. A case can also be transferred to a Lok Adalat if one party applies to the court and the court sees some chance of settlement after giving an opportunity of being heard to the other party.
The focus in Lok Adalats is on compromise. When no compromise is reached, the matter goes back to the court. However, if a compromise is reached, an award is made and is binding on the parties. It is enforced as a decree of a civil court. An important aspect is that the award is final and cannot be appealed, not even under Article 226 of the Constitution of India [which empowers the litigants to file Writ Petition before High Courts] because it is a judgement by consent.
All proceedings of a Lok Adalat are deemed to be judicial proceedings and every Lok Adalat is deemed to be a Civil Court.
=== Pakistan ===
The relevant laws (or parlour provisions) dealing with the ADR are summarized as under:
S.89-A of the Civil Procedure Code, 1908 (Indian but amended in 2002) read with Order X Rule 1-A (deals with alternative dispute resolution methods).
The Small Claims and Minor Offences Courts Ordinance, 2002.
Sections 102–106 of the Local Government Ordinance, 2001.
Sections 10 and 12 of the Family Courts Act, 1964.
Chapter XXII of the Code of Criminal Procedure, 1898 (summary trial provisions).
The Arbitration Act, 1940 (Indian).
Articles 153–154 of the Constitution of Pakistan, 1973 (Council of Common Interest)
Article 156 of the Constitution of Pakistan, 1973 (National Economic Council)
Article 160 of the Constitution of Pakistan, 1973 (National Finance Commission)
Article 184 of the Constitution of Pakistan, 1973 (Original Jurisdiction when federal or provincial governments are at dispute with one another)
Arbitration (International Investment Disputes) Act, 2011
Recognition and Enforcement (Arbitration Agreements and Foreign Arbitral Awards) Act, 2011
Alternative Dispute Resolution Act. 2017
=== Somalia ===
Somalia has a cultural and historic mediation and justice system known as Xeer, which is an informal justice system. It is a kind of justice system in which the arbiter listens to both sides of a dispute and then concludes a solution that both sides will accept.
=== Sub-Saharan Africa ===
Before modern state law was introduced under colonialism, African customary legal systems mainly relied on mediation and conciliation. In many countries, these traditional mechanisms have been integrated into the official legal system. In Benin, specialised tribunaux de conciliation hear cases on a broad range of civil law matters. Results are then transmitted to the court of the first instance where either a successful conciliation is confirmed or jurisdiction is assumed by the higher court. Similar tribunals also operate, in varying modes, in other francophone African countries.
=== United Kingdom ===
In the United Kingdom, ADR is encouraged as a means of resolving taxpayers' disputes with His Majesty's Revenue and Customs.
ADR providers exist in the regulated finance, telecoms and energy sectors. Outside these regulated areas, there are schemes in many sectors which provide schemes for voluntary membership. Two sets of regulations, in March and June 2015, were laid in Parliament to implement the European Directive on alternative dispute resolution in the UK.
Alternative Dispute Resolution is now widely used in the UK across many sectors. In the communications, energy, finance and legal sectors, it is compulsory for traders to signpost to approved ADR schemes when they are unable to resolve disputes with consumers. In the aviation sector there is a quasi-compulsory ADR landscape, where airlines have an obligation to signpost to either an approved ADR scheme or PACT - which is operated by the Civil Aviation Authority.
The UK adopted the Alternative Dispute Resolution for Consumer Disputes (Competent Authorities and Information) Regulations 2015 on 1 October 2015, which set out rules in relation to ADR and put measures into place to widen the use and application of ADR in disputes with consumers after any available internal procedures have been exhausted.
==== England and Wales ====
Judges in England and Wales often encourage use of ADR in appropriate legal cases, and such encouragement is endorsed in the Civil Procedure Rules (CPR 1.4). Halsey v Milton Keynes General NHS Trust provided guidance on cases where one party is willing to take part in ADR and the other refuses to do so on grounds which might be considered unreasonable. In a case which followed shortly after Halsey between Burchell, a builder, and Mr and Mrs Bullard, his customer, the Bullards and their solicitors had "blithely battle[d] on" with litigation where the Appeal Court found that ADR would have been a speedier and less costly means of resolving the parties' dispute. In a 2013 appeal case which has been described as "com[ing] a long way" since Halsey, the Court of Appeal strengthened the argument for using mediation and asserted that "mediation works". In PGF II SA v OMFS Company 1 Ltd., PGF II issued several invitations to OFMS to take part in mediation to resolve a dispute on dilapidations between them, which received no response. The trial court and appeal court agreed that "no response" amounted to an "unreasonable refusal to participate" in ADR. The issues were resolved by a settlement immediately prior to the trial date and a cost sanction imposed on OFMS. The Appeal Court upheld the guidance in the ADR Handbook, which stated that "silence in the face of an invitation to participate in ADR is, as a general rule, of itself unreasonable", and thus endorsed the value of the ADR Handbook itself.
In England and Wales, ADR is now more commonly referred to as ‘NCDR’ (Non Court Dispute Resolution), in an effort to promote this as the normal (rather than alternative) way to resolve disputes. In a 2023 judgment, Churchill v Merthyr Tydfil County Borough Council, the Court of Appeal confirmed that in the right case courts can order (i) the parties to engage in NCDR and / or (ii) stay the proceedings to allow for NCDR to take place. This overturns the previous orthodoxy (the 2004 Court of Appeal decision in Halsey) which was that unwilling parties could not be obliged to participate in NCDR.
The Senior Judiciary in England and Wales are strongly in favour of greater use of NCDR. The Online Procedure Rule Committee was set up under the Judicial Review and Courts Act 2022 to make rules governing the practice and procedure for specific types of online court and tribunal proceedings across the Civil, Family and Tribunal jurisdictions. OPRC is an advisory non-departmental public body, sponsored by the Ministry of Justice. The committee is chaired by the Master of the Rolls, Head of Civil Justice. The aim is to deliver more integrated, efficient and digital approach to justice. Its work will support the use of innovative methods of resolving disputes and help define the operation of pre-action dispute resolution
In the Family Division, there has been a prevailing judicial view that the court should be the last resort for families. High Court judgments have followed the Court of Appeal decision in Churchill v Merthyr, confirming that the courts can stay proceedings to require parties to attend NCDR (Re X and NA v LA).
In April 2024, a new definition of NCDR was set out in the Family Procedure (Amendments No 2) Rules 2023/1324 as “methods of resolving a dispute other than through the court process, including but not limited to mediation, arbitration, evaluation by a neutral third party (such as a private Financial Dispute Resolution process) and collaborative law.”
In the accompanying Pre-application Protocol (Annex to PD9A), the One Couple One Lawyer process was also referenced “The court may also consider the parties having obtained legal advice via the “single lawyer” or a “one couple, one lawyer” scheme as good evidence of a constructive attempt to obtain advice and avoid unnecessary proceedings […]” One Couple One Lawyer, or Single Lawyer, is a family law process developed in England and Wales where a separating couple shares one lawyer who advises them both, impartially and together, as to how a judge would view their case, and the likely outcome were they to litigate, thus enabling them to reach a fair settlement on separation or divorce. This differs from early neutral evaluation as it is designed so that parties never require separate representation, are assisted throughout by one legal team and the process has no adversarial features at all, either at the financial disclosure or advice stages.
The new Family Procedure rules also gave the courts two new powers:
1. to require parties to set out their views on using NCDR in Form FM5; and
2. to consider whether a failure, without good reason, to engage in NCDR should impact on who pays the costs of the litigation.
=== United States ===
==== U.S. Navy ====
SECNAVINST 5800.13A established the DON ADR Program Office with the following missions:
Coordinate ADR policy and initiatives;
Assist activities in securing or creating cost-effective ADR techniques or local programs;
Promote the use of ADR, and provide training in negotiation and ADR methods;
Serve as legal counsel for in-house neutrals used on ADR matters; and,
For matters that do not use in-house neutrals, the program assists DON attorneys and other representatives concerning issues in controversy that are amenable to using ADR.
The ADR Office also serves as the point of contact for questions regarding the use of ADR. The Assistant General Counsel (ADR) serves as the "Dispute Resolution Specialist" for the DON, as required by the Administrative Dispute Resolution Act of 1996. Members of the office represent the DON's interests on a variety of DoD and interagency working groups that promote the use of ADR within the Federal Government.
One example of ADR in the government after ADR act of 1996 is the Alternative Dispute Resolution Program which is used by the USDA to respond to conflict that may result in destructive outcomes by offering employees different options to combat discrepancies. They also offer complaint processes that are used for situations that may need to be ended by an outside interest. These are based on the court system meaning they are "Rights based".
ADR has also been input in all fifty states with a wide range of administrative provisions that offer different ways of dissolving conflict. While many states have adopted some version of the Uniform Arbitration Act, the Revised Uniform Arbitration Act, or the Uniform Mediation Act, there are also many laws and regulations that create or mandate various forms of dispute resolution unique or particular to the specific state in which it was enacted. There are multiple rules and laws associated with ADR so much that a database filled with these laws has been created. The primary goal of this compilation is to provide the researcher with free and easy access to each state's statutes addressing ADR on the map found here: ADR Laws Per State.
==== Structured negotiation ====
Structured negotiation is a type of collaborative and solution-driven alternative dispute resolution that differs from traditional ADR options in that it does not rely on a third-party mediator and is not initiated by a legal complaint. The process is often implemented in cases in which a party or parties seek injunctive relief. Structured negotiation has been used to arrange agreements that typically arise from would-be Americans with Disabilities Act (ADA) legal complaints. The technique can be contrasted with certain types of lawsuits often referred to as "drive-by lawsuits" where a long strings of lawsuits about the ADA are filed publicly by a single lawyer and settled quickly and confidentially, a practice which can undermine the struggle to adopt more inclusive accessibility practices.
Structured negotiation was first used in 1999 to settle the first legal agreement in the United States in which Citibank agreed to install Talking ATMs, and was quickly followed by similar agreements with several other financial institutions, including Bank of America and Wells Fargo. The Bank of America agreement in structured negotiation in 2000 was the first settlement in the United States to reference the Web Content Accessibility Guidelines (WCAG). Subsequently, structured negotiation has been used to settle various digital disability access and disability rights agreements with a variety of American businesses, universities, and local governments. Structured negotiation has also been used in other civil rights resolutions to alter business practices, including a policy by the Lyft ride-sharing service regarding the acceptance of LGBTQ passengers.
== See also ==
Construction law
Family therapy
National Academy of Arbitrators
Ombudsman
Society of Construction Arbitrators
Teen courts
Turnaround ADR
Federal Arbitration Act
== References ==
== Further reading ==
Gary Born. "International Commercial Arbitration" (2009 Kluwer).
Lynch, J. "ADR and Beyond: A Systems Approach to Conflict Management", Negotiation Journal, Volume 17, Number 3, July 2001, Volume, p. 213.
Mackie, Karl J. (ed.). "A Handbook of Dispute Resolution: ADR in action" (1991 Routledge).
William Ury, Roger Fisher, Bruce Patton. "Getting to Yes" (1981 Penguin Group).
== External links ==
Party-Directed Mediation: Facilitating Dialogue Between Individuals by Gregorio Billikopf, free complete book PDF download, at the University of California (3rd Edition, posted 24 March 2014)
Party-Directed Mediation: Facilitating Dialogue Between Individuals by Gregorio Billikopf, free complete book download, from Internet Archive (3rd Edition, multiple file formats including PDF, EPUB, and others)
https://nationalaglawcenter.org/research-by-topic/alternative-dispute-resolution-2/ | Wikipedia/Alternative_dispute_resolution |
Demographic economics or population economics is the application of economic analysis to demography, the study of human populations, including size, growth, density, distribution, and vital statistics.
== Aspects ==
Aspects of the subject include:
marriage and fertility
the family
divorce
morbidity and life expectancy/mortality
dependency ratios
migration
population growth
population size
public policy
the demographic transition from "population explosion" to (dynamic) stability or decline.
Other subfields include measuring value of life and the economics of the elderly and the handicapped and of gender, race, minorities, and non-labor discrimination. In coverage and subfields, it complements labor economics and implicates a variety of other economics subjects.
== Subareas ==
The Journal of Economic Literature classification codes are a way of categorizing subjects in economics. There, demographic economics is paired with labour economics as one of 19 primary classifications at JEL: J. It has eight subareas:
General
Demographic Trends and Forecasts
Marriage; Marital Dissolution; Family Structure
Fertility; Family Planning; Child Care; Children; Youth
Economics of the Elderly; Economics of the Handicapped
Economics of Minorities and Races; Non-labor Discrimination
Economics of Gender; Non-labor Discrimination
Value of life; Foregone Income
Public Policy
== See also ==
Cost of raising a child
Family economics
Generational accounting
Growth economics
Retirement age, international comparison
Related:
Income and fertility
Demographic dividend
Demographic transition
Demographic gift
Demographic window
Demographic trap
Preston curve
Development economics
== Notes ==
== References ==
John Eatwell, Murray Milgate, and Peter Newman, ed. ([1987] 1989. Social Economics: The New Palgrave, pp. v-vi. Arrow-page searchable links to entries for:
"Ageing Populations," pp. 1-3, by Robert L. Clark
"Declining Population," pp. 10-15, by Robin Barlow
"Demographic Transition," pp. 16-23, by Ansley J. Coale
"Extended Family," pp. 58-63, by Oliva Harris
"Family," pp. 65-76, by Gary S. Becker
"Fertility," pp.77-89, by Richard A. Easterlin
"Gender," pp. 95-108, by Francine D. Blau
"Race and Economics," pp. 215-218, by H. Stanback
"Value of Life," pp.289-76, by Thomas C. Schelling
Nathan Keyfitz, 1987. "demography," The New Palgrave: A Dictionary of Economics, v. 1, pp. 796–802.
T. Paul Schultz, 1981. Economics of Population. Addison-Wesley. Book review.
John B. Shoven, ed., 2011. Demography and the Economy, University of Chicago Press. Scroll-down description and preview.
Julian L. Simon, 1977. The Economics of Population Growth. Princeton,
_____, [1981] 1996. The Ultimate Resource 2, rev. and expanded. Princeton. Description and preview links.
Dennis A. Ahlburg, 1998. "Julian Simon and the Population Growth Debate," Population and Development Review, 24(2), pp. 317-327.
M. Perlman, 1982. [Untitled review of Simon, 1977 & 1981], Population Studies, 36(3), pp. 490-494.
Julian L. Simon, ed., 1997. The Economics Of Population: Key Modern Writings. Description.
_____, ed., 1998. The Economics of Population: Classic Writings. Description and scroll to chapter-preview links.
Joseph J. Spengler 1951. "The Population Obstacle to Economic Betterment," American Economic Review, 41(2), pp. 343-354.
_____, 1966. "The Economist and the Population Question," American Economic Review, 56(1/2), pp. 1–24.
== Journals ==
Demography – Scope and links to issue contents & abstracts.
Journal of Population Economics – Aims and scope and 20th Anniversary statement, 2006.
Population and Development Review – Aims and abstract & supplement links.
Population Bulletin – Each issue on a current population topic.
Population Studies —Aims and scope.
Review of Economics of the Household | Wikipedia/Demographic_economics |
In fair division problems, spite is a phenomenon that occurs when a player's value of an allocation decreases when one or more other players' valuation increases. Thus, other things being equal, a player exhibiting spite will prefer an allocation in which other players receive less than more (if more of the good is desirable).
In this language, spite is difficult to analyze because one has to assess two sets of preferences. For example, in the divide and choose method, a spiteful player would have to make a trade-off between depriving his opponent of cake, and getting more himself.
Within the field of sociobiology, spite is used to describe those social behaviors that have a negative impact on both the actor and recipient(s). Spite can be favored by kin selection when: (a) it leads to an indirect benefit to some third party that is sufficiently related to the actor (Wilsonian spite); or (b) when it is directed primarily at negatively related individuals (Hamiltonian spite). Negative relatedness occurs when two individuals are less related than average.
== In game theory ==
The iterated prisoner's dilemma provides an example where players may "punish" each other for failing to cooperate in previous rounds, even if doing so would cause negative consequences for both players. For example, the simple "tit for tat" strategy has been shown to be effective in round-robin tournaments of iterated prisoner's dilemma.
== In industrial relations ==
There is always difficulty in fairly dividing the proceeds of a business between the business owners and the employees.
When a trade union decides to call a strike, both employer and the union members lose money (and may damage the national economy). The unionists hope that the employer will give in to their demands before such losses have destroyed the business.
In the reverse direction, an employer may terminate the employment of certain productive workers who are agitating for higher wages or organising a trade union. Losing productive workers is a setback to both the business and the employees but this can serve as an example to others and thus maximise employer power.
== In behavioral ecology ==
Polyembryonic wasps, including C. floridanum, exhibit spite through instances of precocious larval development. Spite provides an explanation for how natural selection can favor harmful behaviors that are costly to both the actor and the recipient; spite is typically considered a form of altruism that benefits a secondary recipient. Two criteria demonstrate that spite is truly occurring: (i) the behavior is truly costly to the actor and does not provide a long-term direct benefit; and (ii) harming behaviors are directed toward relatively unrelated individuals.
== See also ==
Appeal to spite
Hamilton's rule
Spite (sentiment)
== References ==
Foster, K.R., Wenseleers, T. & Ratnieks, F.L.W. (2001) Spite: Hamilton's unproven theory. Annales Zoologici Fennici, p. 38,229-238. [1]
Gardner, A. & West, S.A. (2006) Spite. Current Biology, p. 16, R662-R664.[2] | Wikipedia/Spite_(game_theory) |
Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory.
Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of variations by Edward J. McShane. Optimal control can be seen as a control strategy in control theory.
== General method ==
Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost function. The optimal control can be derived using Pontryagin's maximum principle (a necessary condition also known as Pontryagin's minimum principle or simply Pontryagin's principle), or by solving the Hamilton–Jacobi–Bellman equation (a sufficient condition).
We begin with a simple example. Consider a car traveling in a straight line on a hilly road. The question is, how should the driver press the accelerator pedal in order to minimize the total traveling time? In this example, the term control law refers specifically to the way in which the driver presses the accelerator and shifts the gears. The system consists of both the car and the road, and the optimality criterion is the minimization of the total traveling time. Control problems usually include ancillary constraints. For example, the amount of available fuel might be limited, the accelerator pedal cannot be pushed through the floor of the car, speed limits, etc.
A proper cost function will be a mathematical expression giving the traveling time as a function of the speed, geometrical considerations, and initial conditions of the system. Constraints are often interchangeable with the cost function.
Another related optimal control problem may be to find the way to drive the car so as to minimize its fuel consumption, given that it must complete a given course in a time not exceeding some amount. Yet another related control problem may be to minimize the total monetary cost of completing the trip, given assumed monetary prices for time and fuel.
A more abstract framework goes as follows. Minimize the continuous-time cost functional
J
[
x
(
⋅
)
,
u
(
⋅
)
,
t
0
,
t
f
]
:=
E
[
x
(
t
0
)
,
t
0
,
x
(
t
f
)
,
t
f
]
+
∫
t
0
t
f
F
[
x
(
t
)
,
u
(
t
)
,
t
]
d
t
{\displaystyle J[{\textbf {x}}(\cdot ),{\textbf {u}}(\cdot ),t_{0},t_{f}]:=E\,[{\textbf {x}}(t_{0}),t_{0},{\textbf {x}}(t_{f}),t_{f}]+\int _{t_{0}}^{t_{f}}F\,[{\textbf {x}}(t),{\textbf {u}}(t),t]\,\mathrm {d} t}
subject to the first-order dynamic constraints (the state equation)
x
˙
(
t
)
=
f
[
x
(
t
)
,
u
(
t
)
,
t
]
,
{\displaystyle {\dot {\textbf {x}}}(t)={\textbf {f}}\,[\,{\textbf {x}}(t),{\textbf {u}}(t),t],}
the algebraic path constraints
h
[
x
(
t
)
,
u
(
t
)
,
t
]
≤
0
,
{\displaystyle {\textbf {h}}\,[{\textbf {x}}(t),{\textbf {u}}(t),t]\leq {\textbf {0}},}
and the endpoint conditions
e
[
x
(
t
0
)
,
t
0
,
x
(
t
f
)
,
t
f
]
=
0
{\displaystyle {\textbf {e}}[{\textbf {x}}(t_{0}),t_{0},{\textbf {x}}(t_{f}),t_{f}]=0}
where
x
(
t
)
{\displaystyle {\textbf {x}}(t)}
is the state,
u
(
t
)
{\displaystyle {\textbf {u}}(t)}
is the control,
t
{\displaystyle t}
is the independent variable (generally speaking, time),
t
0
{\displaystyle t_{0}}
is the initial time, and
t
f
{\displaystyle t_{f}}
is the terminal time. The terms
E
{\displaystyle E}
and
F
{\displaystyle F}
are called the endpoint cost and the running cost respectively. In the calculus of variations,
E
{\displaystyle E}
and
F
{\displaystyle F}
are referred to as the Mayer term and the Lagrangian, respectively. Furthermore, it is noted that the path constraints are in general inequality constraints and thus may not be active (i.e., equal to zero) at the optimal solution. It is also noted that the optimal control problem as stated above may have multiple solutions (i.e., the solution may not be unique). Thus, it is most often the case that any solution
[
x
∗
(
t
)
,
u
∗
(
t
)
,
t
0
∗
,
t
f
∗
]
{\displaystyle [{\textbf {x}}^{*}(t),{\textbf {u}}^{*}(t),t_{0}^{*},t_{f}^{*}]}
to the optimal control problem is locally minimizing.
== Linear quadratic control ==
A special case of the general nonlinear optimal control problem given in the previous section is the linear quadratic (LQ) optimal control problem. The LQ problem is stated as follows. Minimize the quadratic continuous-time cost functional
J
=
1
2
x
T
(
t
f
)
S
f
x
(
t
f
)
+
1
2
∫
t
0
t
f
[
x
T
(
t
)
Q
(
t
)
x
(
t
)
+
u
T
(
t
)
R
(
t
)
u
(
t
)
]
d
t
{\displaystyle J={\tfrac {1}{2}}\mathbf {x} ^{\mathsf {T}}(t_{f})\mathbf {S} _{f}\mathbf {x} (t_{f})+{\tfrac {1}{2}}\int _{t_{0}}^{t_{f}}[\,\mathbf {x} ^{\mathsf {T}}(t)\mathbf {Q} (t)\mathbf {x} (t)+\mathbf {u} ^{\mathsf {T}}(t)\mathbf {R} (t)\mathbf {u} (t)]\,\mathrm {d} t}
Subject to the linear first-order dynamic constraints
x
˙
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
,
{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),}
and the initial condition
x
(
t
0
)
=
x
0
{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}
A particular form of the LQ problem that arises in many control system problems is that of the linear quadratic regulator (LQR) where all of the matrices (i.e.,
A
{\displaystyle \mathbf {A} }
,
B
{\displaystyle \mathbf {B} }
,
Q
{\displaystyle \mathbf {Q} }
, and
R
{\displaystyle \mathbf {R} }
) are constant, the initial time is arbitrarily set to zero, and the terminal time is taken in the limit
t
f
→
∞
{\displaystyle t_{f}\rightarrow \infty }
(this last assumption is what is known as infinite horizon). The LQR problem is stated as follows. Minimize the infinite horizon quadratic continuous-time cost functional
J
=
1
2
∫
0
∞
[
x
T
(
t
)
Q
x
(
t
)
+
u
T
(
t
)
R
u
(
t
)
]
d
t
{\displaystyle J={\tfrac {1}{2}}\int _{0}^{\infty }[\mathbf {x} ^{\mathsf {T}}(t)\mathbf {Q} \mathbf {x} (t)+\mathbf {u} ^{\mathsf {T}}(t)\mathbf {R} \mathbf {u} (t)]\,\mathrm {d} t}
Subject to the linear time-invariant first-order dynamic constraints
x
˙
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
,
{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t),}
and the initial condition
x
(
t
0
)
=
x
0
{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}
In the finite-horizon case the matrices are restricted in that
Q
{\displaystyle \mathbf {Q} }
and
R
{\displaystyle \mathbf {R} }
are positive semi-definite and positive definite, respectively. In the infinite-horizon case, however, the matrices
Q
{\displaystyle \mathbf {Q} }
and
R
{\displaystyle \mathbf {R} }
are not only positive-semidefinite and positive-definite, respectively, but are also constant. These additional restrictions on
Q
{\displaystyle \mathbf {Q} }
and
R
{\displaystyle \mathbf {R} }
in the infinite-horizon case are enforced to ensure that the cost functional remains positive. Furthermore, in order to ensure that the cost function is bounded, the additional restriction is imposed that the pair
(
A
,
B
)
{\displaystyle (\mathbf {A} ,\mathbf {B} )}
is controllable. Note that the LQ or LQR cost functional can be thought of physically as attempting to minimize the control energy (measured as a quadratic form).
The infinite horizon problem (i.e., LQR) may seem overly restrictive and essentially useless because it assumes that the operator is driving the system to zero-state and hence driving the output of the system to zero. This is indeed correct. However the problem of driving the output to a desired nonzero level can be solved after the zero output one is. In fact, it can be proved that this secondary LQR problem can be solved in a very straightforward manner. It has been shown in classical optimal control theory that the LQ (or LQR) optimal control has the feedback form
u
(
t
)
=
−
K
(
t
)
x
(
t
)
{\displaystyle \mathbf {u} (t)=-\mathbf {K} (t)\mathbf {x} (t)}
where
K
(
t
)
{\displaystyle \mathbf {K} (t)}
is a properly dimensioned matrix, given as
K
(
t
)
=
R
−
1
B
T
S
(
t
)
,
{\displaystyle \mathbf {K} (t)=\mathbf {R} ^{-1}\mathbf {B} ^{\mathsf {T}}\mathbf {S} (t),}
and
S
(
t
)
{\displaystyle \mathbf {S} (t)}
is the solution of the differential Riccati equation. The differential Riccati equation is given as
S
˙
(
t
)
=
−
S
(
t
)
A
−
A
T
S
(
t
)
+
S
(
t
)
B
R
−
1
B
T
S
(
t
)
−
Q
{\displaystyle {\dot {\mathbf {S} }}(t)=-\mathbf {S} (t)\mathbf {A} -\mathbf {A} ^{\mathsf {T}}\mathbf {S} (t)+\mathbf {S} (t)\mathbf {B} \mathbf {R} ^{-1}\mathbf {B} ^{\mathsf {T}}\mathbf {S} (t)-\mathbf {Q} }
For the finite horizon LQ problem, the Riccati equation is integrated backward in time using the terminal boundary condition
S
(
t
f
)
=
S
f
{\displaystyle \mathbf {S} (t_{f})=\mathbf {S} _{f}}
For the infinite horizon LQR problem, the differential Riccati equation is replaced with the algebraic Riccati equation (ARE) given as
0
=
−
S
A
−
A
T
S
+
S
B
R
−
1
B
T
S
−
Q
{\displaystyle \mathbf {0} =-\mathbf {S} \mathbf {A} -\mathbf {A} ^{\mathsf {T}}\mathbf {S} +\mathbf {S} \mathbf {B} \mathbf {R} ^{-1}\mathbf {B} ^{\mathsf {T}}\mathbf {S} -\mathbf {Q} }
Understanding that the ARE arises from infinite horizon problem, the matrices
A
{\displaystyle \mathbf {A} }
,
B
{\displaystyle \mathbf {B} }
,
Q
{\displaystyle \mathbf {Q} }
, and
R
{\displaystyle \mathbf {R} }
are all constant. It is noted that there are in general multiple solutions to the algebraic Riccati equation and the positive definite (or positive semi-definite) solution is the one that is used to compute the feedback gain. The LQ (LQR) problem was elegantly solved by Rudolf E. Kálmán.
== Numerical methods for optimal control ==
Optimal control problems are generally nonlinear and therefore, generally do not have analytic solutions (e.g., like the linear-quadratic optimal control problem). As a result, it is necessary to employ numerical methods to solve optimal control problems. In the early years of optimal control (c. 1950s to 1980s) the favored approach for solving optimal control problems was that of indirect methods. In an indirect method, the calculus of variations is employed to obtain the first-order optimality conditions. These conditions result in a two-point (or, in the case of a complex problem, a multi-point) boundary-value problem. This boundary-value problem actually has a special structure because it arises from taking the derivative of a Hamiltonian. Thus, the resulting dynamical system is a Hamiltonian system of the form
x
˙
=
∂
H
∂
λ
λ
˙
=
−
∂
H
∂
x
{\displaystyle {\begin{aligned}{\dot {\textbf {x}}}&={\frac {\partial H}{\partial {\boldsymbol {\lambda }}}}\\[1.2ex]{\dot {\boldsymbol {\lambda }}}&=-{\frac {\partial H}{\partial {\textbf {x}}}}\end{aligned}}}
where
H
=
F
+
λ
T
f
−
μ
T
h
{\displaystyle H=F+{\boldsymbol {\lambda }}^{\mathsf {T}}{\textbf {f}}-{\boldsymbol {\mu }}^{\mathsf {T}}{\textbf {h}}}
is the augmented Hamiltonian and in an indirect method, the boundary-value problem is solved (using the appropriate boundary or transversality conditions). The beauty of using an indirect method is that the state and adjoint (i.e.,
λ
{\displaystyle {\boldsymbol {\lambda }}}
) are solved for and the resulting solution is readily verified to be an extremal trajectory. The disadvantage of indirect methods is that the boundary-value problem is often extremely difficult to solve (particularly for problems that span large time intervals or problems with interior point constraints). A well-known software program that implements indirect methods is BNDSCO.
The approach that has risen to prominence in numerical optimal control since the 1980s is that of so-called direct methods. In a direct method, the state or the control, or both, are approximated using an appropriate function approximation (e.g., polynomial approximation or piecewise constant parameterization). Simultaneously, the cost functional is approximated as a cost function. Then, the coefficients of the function approximations are treated as optimization variables and the problem is "transcribed" to a nonlinear optimization problem of the form:
Minimize
F
(
z
)
{\displaystyle F(\mathbf {z} )}
subject to the algebraic constraints
g
(
z
)
=
0
h
(
z
)
≤
0
{\displaystyle {\begin{aligned}\mathbf {g} (\mathbf {z} )&=\mathbf {0} \\\mathbf {h} (\mathbf {z} )&\leq \mathbf {0} \end{aligned}}}
Depending upon the type of direct method employed, the size of the nonlinear optimization problem can be quite small (e.g., as in a direct shooting or quasilinearization method), moderate (e.g. pseudospectral optimal control) or may be quite large (e.g., a direct collocation method). In the latter case (i.e., a collocation method), the nonlinear optimization problem may be literally thousands to tens of thousands of variables and constraints. Given the size of many NLPs arising from a direct method, it may appear somewhat counter-intuitive that solving the nonlinear optimization problem is easier than solving the boundary-value problem. It is, however, the fact that the NLP is easier to solve than the boundary-value problem. The reason for the relative ease of computation, particularly of a direct collocation method, is that the NLP is sparse and many well-known software programs exist (e.g., SNOPT) to solve large sparse NLPs. As a result, the range of problems that can be solved via direct methods (particularly direct collocation methods which are very popular these days) is significantly larger than the range of problems that can be solved via indirect methods. In fact, direct methods have become so popular these days that many people have written elaborate software programs that employ these methods. In particular, many such programs include DIRCOL, SOCS, OTIS, GESOP/ASTOS, DITAN. and PyGMO/PyKEP. In recent years, due to the advent of the MATLAB programming language, optimal control software in MATLAB has become more common. Examples of academically developed MATLAB software tools implementing direct methods include RIOTS, DIDO, DIRECT, FALCON.m, and GPOPS, while an example of an industry developed MATLAB tool is PROPT. These software tools have increased significantly the opportunity for people to explore complex optimal control problems both for academic research and industrial problems. Finally, it is noted that general-purpose MATLAB optimization environments such as TOMLAB have made coding complex optimal control problems significantly easier than was previously possible in languages such as C and FORTRAN.
== Discrete-time optimal control ==
The examples thus far have shown continuous time systems and control solutions. In fact, as optimal control solutions are now often implemented digitally, contemporary control theory is now primarily concerned with discrete time systems and solutions. The Theory of Consistent Approximations provides conditions under which solutions to a series of increasingly accurate discretized optimal control problem converge to the solution of the original, continuous-time problem. Not all discretization methods have this property, even seemingly obvious ones. For instance, using a variable step-size routine to integrate the problem's dynamic equations may generate a gradient which does not converge to zero (or point in the right direction) as the solution is approached. The direct method RIOTS is based on the Theory of Consistent Approximation.
== Examples ==
A common solution strategy in many optimal control problems is to solve for the costate (sometimes called the shadow price)
λ
(
t
)
{\displaystyle \lambda (t)}
. The costate summarizes in one number the marginal value of expanding or contracting the state variable next turn. The marginal value is not only the gains accruing to it next turn but associated with the duration of the program. It is nice when
λ
(
t
)
{\displaystyle \lambda (t)}
can be solved analytically, but usually, the most one can do is describe it sufficiently well that the intuition can grasp the character of the solution and an equation solver can solve numerically for the values.
Having obtained
λ
(
t
)
{\displaystyle \lambda (t)}
, the turn-t optimal value for the control can usually be solved as a differential equation conditional on knowledge of
λ
(
t
)
{\displaystyle \lambda (t)}
. Again it is infrequent, especially in continuous-time problems, that one obtains the value of the control or the state explicitly. Usually, the strategy is to solve for thresholds and regions that characterize the optimal control and use a numerical solver to isolate the actual choice values in time.
=== Finite time ===
Consider the problem of a mine owner who must decide at what rate to extract ore from their mine. They own rights to the ore from date
0
{\displaystyle 0}
to date
T
{\displaystyle T}
. At date
0
{\displaystyle 0}
there is
x
0
{\displaystyle x_{0}}
ore in the ground, and the time-dependent amount of ore
x
(
t
)
{\displaystyle x(t)}
left in the ground declines at the rate of
u
(
t
)
{\displaystyle u(t)}
that the mine owner extracts it. The mine owner extracts ore at cost
u
(
t
)
2
/
x
(
t
)
{\displaystyle u(t)^{2}/x(t)}
(the cost of extraction increasing with the square of the extraction speed and the inverse of the amount of ore left) and sells ore at a constant price
p
{\displaystyle p}
. Any ore left in the ground at time
T
{\displaystyle T}
cannot be sold and has no value (there is no "scrap value"). The owner chooses the rate of extraction varying with time
u
(
t
)
{\displaystyle u(t)}
to maximize profits over the period of ownership with no time discounting.
== See also ==
== References ==
== Further reading ==
Bertsekas, D. P. (1995). Dynamic Programming and Optimal Control. Belmont: Athena. ISBN 1-886529-11-6.
Bryson, A. E.; Ho, Y.-C. (1975). Applied Optimal Control: Optimization, Estimation and Control (Revised ed.). New York: John Wiley and Sons. ISBN 0-470-11481-9.
Fleming, W. H.; Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. New York: Springer. ISBN 0-387-90155-8.
Kamien, M. I.; Schwartz, N. L. (1991). Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management (Second ed.). New York: Elsevier. ISBN 0-444-01609-0.
Kirk, D. E. (1970). Optimal Control Theory: An Introduction. Englewood Cliffs: Prentice-Hall. ISBN 0-13-638098-0.
== External links ==
Victor M. Becerra, ed. (2008). "Optimal control". Scholarpedia. Retrieved 31 December 2022.
Computational Optimal Control
Dr. Benoît CHACHUAT: Automatic Control Laboratory – Nonlinear Programming, Calculus of Variations and Optimal Control.
DIDO - MATLAB tool for optimal control
GEKKO - Python package for optimal control
GESOP – Graphical Environment for Simulation and OPtimization
GPOPS-II – General-Purpose MATLAB Optimal Control Software
CasADi – Free and open source symbolic framework for optimal control
PROPT – MATLAB Optimal Control Software
OpenOCL – Open Optimal Control Library Archived 20 April 2019 at the Wayback Machine
acados – open-source software framework for nonlinear optimal control
Rockit (Rapid Optimal Control kit) – a software framework to quickly prototype optimal control problems
Elmer G. Wiens: Optimal Control – Applications of Optimal Control Theory Using the Pontryagin Maximum Principle with interactive models.
On Optimal Control by Yu-Chi Ho
Pseudospectral optimal control: Part 1
Pseudospectral optimal control: Part 2
Lecture Recordings and Script by Prof. Moritz Diehl, University of Freiburg on Numerical Optimal Control | Wikipedia/Optimal_control |
In economics and game theory, the decisions of two or more players are called strategic complements if they mutually reinforce one another, and they are called strategic substitutes if they mutually offset one another. These terms were originally coined by Bulow, Geanakoplos, and Klemperer (1985).
To see what is meant by 'reinforce' or 'offset', consider a situation in which the players all have similar choices to make, as in the paper of Bulow et al., where the players are all imperfectly competitive firms that must each decide how much to produce. Then the production decisions are strategic complements if an increase in the production of one firm increases the marginal revenues of the others, because that gives the others an incentive to produce more too. This tends to be the case if there are sufficiently strong aggregate increasing returns to scale and/or the demand curves for the firms' products have a sufficiently low own-price elasticity. On the other hand, the production decisions are strategic substitutes if an increase in one firm's output decreases the marginal revenues of the others, giving them an incentive to produce less.
According to Russell Cooper and Andrew John, strategic complementarity is the basic property underlying examples of multiple equilibria in coordination games.
== Calculus formulation ==
Mathematically, consider a symmetric game with two players that each have payoff function
Π
(
x
i
,
x
j
)
{\displaystyle \,\Pi (x_{i},x_{j})}
, where
x
i
{\displaystyle \,x_{i}}
represents the player's own decision, and
x
j
{\displaystyle \,x_{j}}
represents the decision of the other player. Assume
Π
{\displaystyle \,\Pi }
is increasing and concave in the player's own strategy
x
i
{\displaystyle \,x_{i}}
. Under these assumptions, the two decisions are strategic complements if an increase in each player's own decision
x
i
{\displaystyle \,x_{i}}
raises the marginal payoff
∂
Π
j
∂
x
j
{\displaystyle {\frac {\partial \Pi _{j}}{\partial x_{j}}}}
of the other player. In other words, the decisions are strategic complements if the second derivative
∂
2
Π
j
∂
x
j
∂
x
i
{\displaystyle {\frac {\partial ^{2}\Pi _{j}}{\partial x_{j}\partial x_{i}}}}
is positive for
i
≠
j
{\displaystyle i\neq j}
. Equivalently, this means that the function
Π
{\displaystyle \,\Pi }
is supermodular.
On the other hand, the decisions are strategic substitutes if
∂
2
Π
j
∂
x
j
∂
x
i
{\displaystyle {\frac {\partial ^{2}\Pi _{j}}{\partial x_{j}\partial x_{i}}}}
is negative, that is, if
Π
{\displaystyle \,\Pi }
is submodular.
== Example ==
In their original paper, Bulow et al. use a simple model of competition between two firms to illustrate their ideas.
The revenue for firm x with production rates
(
x
1
,
x
2
)
{\displaystyle (x_{1},x_{2})}
is given by
U
x
(
x
1
,
x
2
;
y
2
)
=
p
1
x
1
+
(
1
−
x
2
−
y
2
)
x
2
−
(
x
1
+
x
2
)
2
/
2
−
F
{\displaystyle U_{x}(x_{1},x_{2};y_{2})=p_{1}x_{1}+(1-x_{2}-y_{2})x_{2}-(x_{1}+x_{2})^{2}/2-F}
while the revenue for firm y with production rate
y
2
{\displaystyle y_{2}}
in market 2 is given by
U
y
(
y
2
;
x
1
,
x
2
)
=
(
1
−
x
2
−
y
2
)
y
2
−
y
2
2
/
2
−
F
{\displaystyle U_{y}(y_{2};x_{1},x_{2})=(1-x_{2}-y_{2})y_{2}-y_{2}^{2}/2-F}
At any interior equilibrium,
(
x
1
∗
,
x
2
∗
,
y
2
∗
)
{\displaystyle (x_{1}^{*},x_{2}^{*},y_{2}^{*})}
, we must have
∂
U
x
∂
x
1
=
0
,
∂
U
x
∂
x
2
=
0
,
∂
U
y
∂
y
2
=
0.
{\displaystyle {\dfrac {\partial U_{x}}{\partial x_{1}}}=0,{\dfrac {\partial U_{x}}{\partial x_{2}}}=0,{\dfrac {\partial U_{y}}{\partial y_{2}}}=0.}
Using vector calculus, geometric algebra, or differential geometry, Bulow et al. showed that the sensitivity
of the Cournot equilibrium to changes in
p
1
{\displaystyle p_{1}}
can be calculated in terms of second partial derivatives
of the payoff functions:
[
d
x
1
∗
d
p
1
d
x
2
∗
d
p
1
d
y
2
∗
d
p
1
]
=
[
∂
2
U
x
∂
x
1
∂
x
1
∂
2
U
x
∂
x
1
∂
x
2
∂
2
U
x
∂
x
1
∂
y
2
∂
2
U
x
∂
x
1
∂
x
2
∂
2
U
x
∂
x
2
∂
x
2
∂
2
U
x
∂
y
2
∂
x
2
∂
2
U
y
∂
x
1
∂
y
2
∂
2
U
y
∂
x
2
∂
y
2
∂
2
U
y
∂
y
2
∂
y
2
]
−
1
[
−
∂
2
U
x
∂
p
1
∂
x
1
−
∂
2
U
x
∂
p
1
∂
x
2
−
∂
2
U
y
∂
p
1
∂
y
2
]
{\displaystyle {\begin{bmatrix}{\dfrac {dx_{1}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dx_{2}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dy_{2}^{*}}{dp_{1}}}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial ^{2}U_{x}}{\partial x_{1}\partial x_{1}}}&{\dfrac {\partial ^{2}U_{x}}{\partial x_{1}\partial x_{2}}}&{\dfrac {\partial ^{2}U_{x}}{\partial x_{1}\partial y_{2}}}\\[2.2ex]{\dfrac {\partial ^{2}U_{x}}{\partial x_{1}\partial x_{2}}}&{\dfrac {\partial ^{2}U_{x}}{\partial x_{2}\partial x_{2}}}&{\dfrac {\partial ^{2}U_{x}}{\partial y_{2}\partial x_{2}}}\\[2.2ex]{\dfrac {\partial ^{2}U_{y}}{\partial x_{1}\partial y_{2}}}&{\dfrac {\partial ^{2}U_{y}}{\partial x_{2}\partial y_{2}}}&{\dfrac {\partial ^{2}U_{y}}{\partial y_{2}\partial y_{2}}}\end{bmatrix}}^{-1}{\begin{bmatrix}-{\dfrac {\partial ^{2}U_{x}}{\partial p_{1}\partial x_{1}}}\\[2.2ex]-{\dfrac {\partial ^{2}U_{x}}{\partial p_{1}\partial x_{2}}}\\[2.2ex]-{\dfrac {\partial ^{2}U_{y}}{\partial p_{1}\partial y_{2}}}\end{bmatrix}}}
When
1
/
4
≤
p
1
≤
2
/
3
{\displaystyle 1/4\leq p_{1}\leq 2/3}
,
[
d
x
1
∗
d
p
1
d
x
2
∗
d
p
1
d
y
2
∗
d
p
1
]
=
[
−
1
−
1
0
−
1
−
3
−
1
0
−
1
−
3
]
−
1
[
−
1
0
0
]
=
1
5
[
8
−
3
1
]
{\displaystyle {\begin{bmatrix}{\dfrac {dx_{1}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dx_{2}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dy_{2}^{*}}{dp_{1}}}\end{bmatrix}}={\begin{bmatrix}-1&-1&0\\-1&-3&-1\\0&-1&-3\end{bmatrix}}^{-1}{\begin{bmatrix}-1\\0\\0\end{bmatrix}}={\frac {1}{5}}{\begin{bmatrix}8\\-3\\1\end{bmatrix}}}
This, as price is increased in market 1, Firm x sells more in market 1 and less in market 2, while firm y sells more in market 2. If the Cournot equilibrium of this model is calculated explicitly, we find
x
1
∗
=
max
{
0
,
8
p
1
−
2
5
}
,
x
2
∗
=
max
{
0
,
2
−
3
p
1
5
}
,
y
2
∗
=
p
1
+
1
5
.
{\displaystyle x_{1}^{*}=\max \left\{0,{\frac {8p_{1}-2}{5}}\right\},x_{2}^{*}=\max \left\{0,{\frac {2-3p_{1}}{5}}\right\},y_{2}^{*}={\frac {p_{1}+1}{5}}.}
== Supermodular games ==
A game with strategic complements is also called a supermodular game. This was first formalized by Topkis, and studied by Vives. There are efficient algorithms for finding pure-strategy Nash equilibria in such games.
== See also ==
Supermodular
Coordination game
Coordination failure (economics)
Uniqueness or multiplicity of equilibrium
Multiplier (economics)
== References == | Wikipedia/Strategic_complements |
In game theory, and particularly mechanism design, participation constraints or individual rationality constraints are said to be satisfied if a mechanism leaves all participants at least as well-off as they would have been if they hadn't participated.
In terms of information structure, there are 3 types of Participation Constraints:
1. Ex-post, this is the strongest form. Assuming every player knows all others' types and its own type, it makes the decision to participate the game.
2. Interim, this is in the middle. Assuming every player only knows its own type, it decides to participate the game given that its expected utility is greater than its outside option.
3. Ex-ante, this the weakest form. Assuming every player have no knowledge for neither others and itself, the player decide to participate based on the prior distribution of the players type (and then calculate its expected utility).
Unfortunately, it can frequently be shown that participation constraints are incompatible with other desirable properties of mechanisms for many purposes. One of the classic result is Gibbard-Satterthwaite Theorem.
One kind of participation constraint is the participation criterion for voting systems. It requires that by voting, a voter should not decrease the odds of their preferred candidates winning.
== See also ==
Individual rationality
Compensating variation
== References == | Wikipedia/Participation_constraint_(mechanism_design) |
Deterrence theory refers to the scholarship and practice of how threats of using force by one party can convince another party to refrain from initiating some other course of action. The topic gained increased prominence as a military strategy during the Cold War with regard to the use of nuclear weapons and is related to but distinct from the concept of mutual assured destruction, according to which a full-scale nuclear attack on a power with second-strike capability would devastate both parties. The central problem of deterrence revolves around how to credibly threaten military action or nuclear punishment on the adversary despite its costs to the deterrer. Deterrence in an international relations context is the application of deterrence theory to avoid conflict.
Deterrence is widely defined as any use of threats (implicit or explicit) or limited force intended to dissuade an actor from taking an action (i.e. maintain the status quo). Deterrence is unlike compellence, which is the attempt to get an actor (such as a state) to take an action (i.e. alter the status quo). Both are forms of coercion. Compellence has been characterized as harder to successfully implement than deterrence. Deterrence also tends to be distinguished from defense or the use of full force in wartime.
Deterrence is most likely to be successful when a prospective attacker believes that the probability of success is low and the costs of attack are high. Central problems of deterrence include the credible communication of threats and assurance. Deterrence does not necessarily require military superiority.
"General deterrence" is considered successful when an actor who might otherwise take an action refrains from doing so due to the consequences that the deterrer is perceived likely to take. "Immediate deterrence" is considered successful when an actor seriously contemplating immediate military force or action refrains from doing so. Scholars distinguish between "extended deterrence" (the protection of allies) and "direct deterrence" (protection of oneself). Rational deterrence theory holds that an attacker will be deterred if they believe that:(Probability of deterrer carrying out deterrent threat × Costs if threat carried out) > (Probability of the attacker accomplishing the action × Benefits of the action)This model is frequently simplified in game-theoretic terms as:Costs × P(Costs) > Benefits × P(Benefits)
== History ==
=== World War II ===
During World War II, some historians have argued that deterrence prevented the Western Allies and Axis from extensive chemical warfare, as had been used in World War I. Nonetheless, Nazi Germany used chemical weapons during the Siege of Sevastopol, Siege of Odessa, and Battle of the Kerch Peninsula, while Imperial Japan frequently used chemical weapons against Chinese troops. Conversely, during the Nuremberg trials, Hermann Göring stated that initiating an exchange of chemical weapons during the Operation Overlord would have immobilized the Wehrmacht, which widely relied on horse-drawn transport, and a suitable gas mask for horses had not been designed.
=== Cold War ===
==== Concept ====
While the concept of deterrence precedes the Cold War, it was during the Cold War that the concept evolved into a clearly articulated objective in strategic planning and diplomacy, with considerable analysis by scholars.
Most of the innovative work on deterrence theory occurred from the late 1940s to mid-1960s. Historically, scholarship on deterrence has tended to focus on nuclear deterrence. Since the end of the Cold War, there has been an extension of deterrence scholarship to areas that are not specifically about nuclear weapons.
NATO was founded in 1949 with deterring aggression as one of its goals.
A distinction is sometimes made between nuclear deterrence and "conventional deterrence."
The two most prominent deterrent strategies are "denial" (denying the attacker the benefits of attack) and "punishment" (inflicting costs on the attacker).
Lesson of Munich, where appeasement failed, contributes to deterrence theory. In the words of scholars Frederik Logevall and Kenneth Osgood, "Munich and appeasement have become among the dirtiest words in American politics, synonymous with naivete and weakness, and signifying a craven willingness to barter away the nation's vital interests for empty promises." They claimed that the success of US foreign policy often depends upon a president withstanding "the inevitable charges of appeasement that accompany any decision to negotiate with hostile powers.
==== Examples ====
By November 1945 general Curtis LeMay, who led American air raids on Japan during World War II, was thinking about how the next war would be fought. He said in a speech that month to the Ohio Society of New York that since "No air attack, once it is launched, can be completely stopped", his country needed an air force that could immediately retaliate: "If we are prepared it may never come. It is not immediately conceivable that any nation will dare to attack us if we are prepared".
In pursuit of nuclear deterrence, the superpowers of the USSR and US engaged in a nuclear arms race. Warheads themselves evolved from fission weapons to thermonuclear weapons, and were extensively miniaturized for both strategic and tactical use. Nuclear weapons delivery was equally important, such as the perceived bomber gap and missile gap. Deterrence was a primary factor in the ultimate proliferation of nuclear weapons to ten nations in total. Generally this was the form of the threat perceived from a nearby recently nuclear-armed neighbor. In the case of Israel and South Africa deterrence was against the threat of conventional attack.
Additionally, chemical weapons were a component of deterrence for both sides, and large stockpiles were maintained until their destruction began following the 1993 Chemical Weapons Convention. Offensive biological weapons programs were pursued by both countries in the first two decades of the Cold War, but the United States program was ended by president Richard Nixon in 1969.
== Concept ==
The concept of deterrence can be defined as the use of threats in limited force by one party to convince another party to refrain from initiating some course of action. In Arms and Influence (1966), Schelling offers a broader definition of deterrence, as he defines it as "to prevent from action by fear of consequences." Glenn Snyder also offers a broad definition of deterrence, as he argues that deterrence involves both the threat of sanction and the promise of reward.
A threat serves as a deterrent to the extent that it convinces its target not to carry out the intended action because of the costs and losses that target would incur. In international security, a policy of deterrence generally refers to threats of military retaliation directed by the leaders of one state to the leaders of another in an attempt to prevent the other state from resorting to the use of military force in pursuit of its foreign policy goals.
As outlined by Huth, a policy of deterrence can fit into two broad categories: preventing an armed attack against a state's own territory (known as direct deterrence) or preventing an armed attack against another state (known as extended deterrence). Situations of direct deterrence often occur if there is a territorial dispute between neighboring states in which major powers like the United States do not directly intervene. On the other hand, situations of extended deterrence often occur when a great power becomes involved. The latter case has generated most interest in academic literature. Building on the two broad categories, Huth goes on to outline that deterrence policies may be implemented in response to a pressing short-term threat (known as immediate deterrence) or as strategy to prevent a military conflict or short-term threat from arising (known as general deterrence).
A successful deterrence policy must be considered in military terms but also political terms: International relations, foreign policy and diplomacy. In military terms, deterrence success refers to preventing state leaders from issuing military threats and actions that escalate peacetime diplomatic and military co-operation into a crisis or militarized confrontation that threatens armed conflict and possibly war. The prevention of crises of wars, however, is not the only aim of deterrence. In addition, defending states must be able to resist the political and the military demands of a potential attacking nation. If armed conflict is avoided at the price of diplomatic concessions to the maximum demands of the potential attacking nation under the threat of war, it cannot be claimed that deterrence has succeeded.
Furthermore, as Jentleson et al. argue, two key sets of factors for successful deterrence are important: a defending state strategy that balances credible coercion and deft diplomacy consistent with the three criteria of proportionality, reciprocity, and coercive credibility and minimizes international and domestic constraints and the extent of an attacking state's vulnerability as shaped by its domestic political and economic conditions. In broad terms, a state wishing to implement a strategy of deterrence is most likely to succeed if the costs of noncompliance that it can impose on and the benefits of compliance it can offer to another state are greater than the benefits of noncompliance and the costs of compliance.
Deterrence theory holds that nuclear weapons are intended to deter other states from attacking with their nuclear weapons, through the promise of retaliation and possibly mutually assured destruction. Nuclear deterrence can also be applied to an attack by conventional forces. For example, the doctrine of massive retaliation threatened to launch US nuclear weapons in response to Soviet attacks.
A successful nuclear deterrent requires a country to preserve its ability to retaliate by responding before its own weapons are destroyed or ensuring a second-strike capability. A nuclear deterrent is sometimes composed of a nuclear triad, as in the case of the nuclear weapons owned by the United States, Russia, China and India. Other countries, such as the United Kingdom and France, have only sea-based and air-based nuclear weapons.
=== Proportionality ===
Jentleson et al. provides further detail in relation to those factors. Proportionality refers to the relationship between the defending state's scope and nature of the objectives being pursued and the instruments available for use to pursue them. The more the defending state demands of another state, the higher that state's costs of compliance and the greater need for the defending state's strategy to increase the costs of noncompliance and the benefits of compliance. That is a challenge, as deterrence is by definition a strategy of limited means. George (1991) goes on to explain that deterrence sometimes goes beyond threats to the actual use of military force, but if force is actually used, it must be limited and fall short of full-scale use to succeed.
The main source of disproportionality is an objective that goes beyond policy change to regime change, which has been seen in Libya, Iraq, and North Korea. There, defending states have sought to change the leadership of a state and to policy changes relating primarily to their nuclear weapons programs.
=== Reciprocity ===
Secondly, Jentleson et al. outlines that reciprocity involves an explicit understanding of linkage between the defending state's carrots and the attacking state's concessions. The balance lies in not offering too little, too late or for too much in return and not offering too much, too soon, or for too little return.
=== Coercive credibility ===
Finally, coercive credibility requires that in addition to calculations about costs and benefits of co-operation, the defending state convincingly conveys to the attacking state that failure to co-operate has consequences. Threats, uses of force, and other coercive instruments such as economic sanctions must be sufficiently credible to raise the attacking state's perceived costs of noncompliance. A defending state having a superior military capability or economic strength in itself is not enough to ensure credibility. Indeed, all three elements of a balanced deterrence strategy are more likely to be achieved if other major international actors like the UN or NATO are supportive, and opposition within the defending state's domestic politics is limited.
The other important considerations outlined by Jentleson et al. that must be taken into consideration is the domestic political and economic conditions in the attacking state affecting its vulnerability to deterrence policies and the attacking state's ability to compensate unfavourable power balances. The first factor is whether internal political support and regime security are better served by defiance, or there are domestic political gains to be made from improving relations with the defending state. The second factor is an economic calculation of the costs that military force, sanctions, and other coercive instruments can impose and the benefits that trade and other economic incentives may carry. That is partly a function of the strength and flexibility of the attacking state's domestic economy and its capacity to absorb or counter the costs being imposed. The third factor is the role of elites and other key domestic political figures within the attacking state. To the extent that such actors' interests are threatened with the defending state's demands, they act to prevent or block the defending state's demands.
== Rational deterrence theory ==
One approach to theorizing about deterrence has entailed the use of rational choice and game-theoretic models of decision making (see game theory). Rational deterrence theory entails:
Rationality: actors are rational
Unitary actor assumption: actors are understood as unitary
Dyads: interactions tend to be between dyads (or triads) of states
Strategic interactions: actors consider the choices of other actors
Cost-benefit calculations: outcomes reflect actors' cost-benefit calculations
Deterrence theorists have consistently argued that deterrence success is more likely if a defending state's deterrent threat is credible to an attacking state. Huth outlines that a threat is considered credible if the defending state possesses both the military capabilities to inflict substantial costs on an attacking state in an armed conflict, and the attacking state believes that the defending state is resolved to use its available military forces. Huth goes on to explain the four key factors for consideration under rational deterrence theory: the military balance, signaling and bargaining power, reputations for resolve, interests at stake.
The American economist Thomas Schelling brought his background in game theory to the subject of studying international deterrence. Schelling's (1966) classic work on deterrence presents the concept that military strategy can no longer be defined as the science of military victory. Instead, it is argued that military strategy was now equally, if not more, the art of coercion, intimidation and deterrence. Schelling says the capacity to harm another state is now used as a motivating factor for other states to avoid it and influence another state's behavior. To be coercive or deter another state, violence must be anticipated and avoidable by accommodation. It can therefore be summarized that the use of the power to hurt as bargaining power is the foundation of deterrence theory and is most successful when it is held in reserve.
In an article celebrating Schelling's Nobel Memorial Prize for Economics, Michael Kinsley, Washington Post op‑ed columnist and one of Schelling's former students, anecdotally summarizes Schelling's reorientation of game theory thus: "[Y]ou're standing at the edge of a cliff, chained by the ankle to someone else. You'll be released, and one of you will get a large prize, as soon as the other gives in. How do you persuade the other guy to give in, when the only method at your disposal—threatening to push him off the cliff—would doom you both? Answer: You start dancing, closer and closer to the edge. That way, you don't have to convince him that you would do something totally irrational: plunge him and yourself off the cliff. You just have to convince him that you are prepared to take a higher risk than he is of accidentally falling off the cliff. If you can do that, you win."
=== Military balance ===
Deterrence is often directed against state leaders who have specific territorial goals that they seek to attain either by seizing disputed territory in a limited military attack or by occupying disputed territory after the decisive defeat of the adversary's armed forces. In either case, the strategic orientation of potential attacking states generally is for the short term and is driven by concerns about military cost and effectiveness. For successful deterrence, defending states need the military capacity to respond quickly and strongly to a range of contingencies. Deterrence often fails if either a defending state or an attacking state underestimates or overestimates the other's ability to undertake a particular course of action.
=== Signaling and bargaining power ===
The central problem for a state that seeks to communicate a credible deterrent threat by diplomatic or military actions is that all defending states have an incentive to act as if they are determined to resist an attack in the hope that the attacking state will back away from military conflict with a seemingly resolved adversary. If all defending states have such incentives, potential attacking states may discount statements made by defending states along with any movement of military forces as merely bluffs. In that regard, rational deterrence theorists have argued that costly signals are required to communicate the credibility of a defending state's resolve. Those are actions and statements that clearly increase the risk of a military conflict and also increase the costs of backing down from a deterrent threat. States that bluff are unwilling to cross a certain threshold of threat and military action for fear of committing themselves to an armed conflict.
=== Reputations for resolve ===
There are three different arguments that have been developed in relation to the role of reputations in influencing deterrence outcomes. The first argument focuses on a defending state's past behavior in international disputes and crises, which creates strong beliefs in a potential attacking state about the defending state's expected behaviour in future conflicts. The credibilities of a defending state's policies are arguably linked over time, and reputations for resolve have a powerful causal impact on an attacking state's decision whether to challenge either general or immediate deterrence. The second approach argues that reputations have a limited impact on deterrence outcomes because the credibility of deterrence is heavily determined by the specific configuration of military capabilities, interests at stake, and political constraints faced by a defending state in a given situation of attempted deterrence. The argument of that school of thought is that potential attacking states are not likely to draw strong inferences about a defending states resolve from prior conflicts because potential attacking states do not believe that a defending state's past behaviour is a reliable predictor of future behavior. The third approach is a middle ground between the first two approaches and argues that potential attacking states are likely to draw reputational inferences about resolve from the past behaviour of defending states only under certain conditions. The insight is the expectation that decisionmakers use only certain types of information when drawing inferences about reputations, and an attacking state updates and revises its beliefs when a defending state's unanticipated behavior cannot be explained by case-specific variables.
An example shows that the problem extends to the perception of the third parties as well as main adversaries and underlies the way in which attempts at deterrence can fail and even backfire if the assumptions about the others' perceptions are incorrect.
=== Interests at stake ===
Although costly signaling and bargaining power are more well established arguments in rational deterrence theory, the interests of defending states are not as well known. Attacking states may look beyond the short-term bargaining tactics of a defending state and seek to determine what interests are at stake for the defending state that would justify the risks of a military conflict. The argument is that defending states that have greater interests at stake in a dispute are more resolved to use force and more willing to endure military losses to secure those interests. Even less well-established arguments are the specific interests that are more salient to state leaders such as military interests and economic interests.
Furthermore, Huth argues that both supporters and critics of rational deterrence theory agree that an unfavorable assessment of the domestic and international status quo by state leaders can undermine or severely test the success of deterrence. In a rational choice approach, if the expected utility of not using force is reduced by a declining status quo position, deterrence failure is more likely since the alternative option of using force becomes relatively more attractive.
=== Tripwires ===
Tripwires entail that small forces are deployed abroad with the assumption that an attack on them will trigger a greater deployment of forces. Dan Reiter and Paul Poast have argued that tripwires do not deter aggression. Dan Altman has argued that tripwires do work to deter aggression, citing the Western deployment of forces to Berlin in 1948–1949 to deter Soviet aggression as a successful example.
A 2022 study by Brian Blankenship and Erik Lin-Greenberg found that high-resolve, low-capability signals (such as tripwires) were not viewed as more reassuring to allies than low-resolve, high-capability alternatives (such as forces stationed offshore). Their study cast doubt on the reassuring value of tripwires.
== Nuclear deterrence theory ==
In 1966, Schelling is prescriptive in outlining the impact of the development of nuclear weapons in the analysis of military power and deterrence. In his analysis, before the widespread use of assured second strike capability, or immediate reprisal, in the form of SSBN submarines, Schelling argues that nuclear weapons give nations the potential to destroy their enemies but also the rest of humanity without drawing immediate reprisal because of the lack of a conceivable defense system and the speed with which nuclear weapons can be deployed. A nation's credible threat of such severe damage empowers their deterrence policies and fuels political coercion and military deadlock, which can produce proxy warfare.
According to Kenneth Waltz, there are three requirements for successful nuclear deterrence:
Part of a state's nuclear arsenal must appear to be able to survive an attack by the adversary and be used for a retaliatory second strike
The state must not respond to false alarms of a strike by the adversary
The state must maintain command and control
The stability–instability paradox is a key concept in rational deterrence theory. It states that when two countries each have nuclear weapons, the probability of a direct war between them greatly decreases, but the probability of minor or indirect conflicts between them increases. This occurs because rational actors want to avoid nuclear wars, and thus they neither start major conflicts nor allow minor conflicts to escalate into major conflicts—thus making it safe to engage in minor conflicts. For instance, during the Cold War the United States and the Soviet Union never engaged each other in warfare, but fought proxy wars in Korea, Vietnam, Angola, the Middle East, Nicaragua and Afghanistan and spent substantial amounts of money and manpower on gaining relative influence over the third world.
Bernard Brodie wrote in 1959 that a credible nuclear deterrent must be always ready. An extended nuclear deterrence guarantee is also called a nuclear umbrella.
Scholars have debated whether having a superior nuclear arsenal provides a deterrent against other nuclear-armed states with smaller arsenals. Matthew Kroenig has argued that states with nuclear superiority are more likely to win nuclear crises, whereas Todd Sechser, Matthew Fuhrmann and David C. Logan have challenged this assertion. A 2023 study found that a state with nuclear weapons is less likely to be targeted by non-nuclear states, but that a state with nuclear weapons is not less likely to target other nuclear states in low-level conflict. A 2022 study by Kyungwon Suh suggests that nuclear superiority may not reduce the likelihood that nuclear opponents will initiate nuclear crises.
Proponents of nuclear deterrence theory argue that newly nuclear-armed states may pose a short- or medium-term risk, but that "nuclear learning" occurs over time as states learn to live with new nuclear-armed states. Mark S. Bell and Nicholas L. Miller have however argued that there is a weak theoretical and empirical basis for notions of "nuclear learning."
=== Stages of US policy of deterrence ===
The US policy of deterrence during the Cold War underwent significant variations.
==== Containment ====
The early stages of the Cold War were generally characterized by the containment of communism, an aggressive stance on behalf of the US especially on developing nations under its sphere of influence. The period was characterized by numerous proxy wars throughout most of the globe, particularly Africa, Asia, Central America, and South America. One notable conflict was the Korean War. George F. Kennan, who is taken to be the founder of this policy in his Long Telegram, asserted that he never advocated military intervention, merely economic support, and that his ideas were misinterpreted as espoused by the general public.
==== Détente ====
With the US drawdown from Vietnam, the normalization of US relations with China, and the Sino-Soviet Split, the policy of containment was abandoned and a new policy of détente was established, with peaceful co-existence was sought between the United States and the Soviet Union. Although all of those factors contributed to this shift, the most important factor was probably the rough parity achieved in stockpiling nuclear weapons with the clear capability of mutual assured destruction (MAD). Therefore, the period of détente was characterized by a general reduction in the tension between the Soviet Union and the United States and a thawing of the Cold War, which lasted from the late 1960s until the start of the 1980s. The doctrine of mutual nuclear deterrence then characterized relations between the United States and the Soviet Union and relations with Russia until the onset of the New Cold War in the early 2010s. Since then, relations have been less clear.
==== Reagan era ====
A third shift occurred with US President Ronald Reagan's arms build-up during the 1980s. Reagan attempted to justify the policy by concerns of growing Soviet influence in Latin America and the post-1979 revolutionary government of Iran. Similar to the old policy of containment, the US funded several proxy wars, including support for Saddam Hussein of Iraq during the Iran–Iraq War, support for the mujahideen in Afghanistan, who were fighting for independence from the Soviet Union, and several anticommunist movements in Latin America such as the overthrow of the Sandinista government in Nicaragua. The funding of the Contras in Nicaragua led to the Iran-Contra Affair, while overt support led to a ruling from the International Court of Justice against the United States in Nicaragua v. United States.
The final expression of the full impact of deterrence during the cold war can be seen in the agreement between Reagan and Mikhail Gorbachev in 1985. They "agreed that a nuclear war cannot be won and must never be fought. Recognizing that any conflict between the USSR and the U.S. could have catastrophic consequences, they emphasized the importance of preventing any war between them, whether nuclear or conventional. They will not seek to achieve military superiority.".
While the army was dealing with the breakup of the Soviet Union and the spread of nuclear technology to other nations beyond the United States and Russia, the concept of deterrence took on a broader multinational dimension. The US policy on deterrence after the Cold War was outlined in 1995 in the document called "Essentials of Post–Cold War Deterrence". It explains that while relations with Russia continue to follow the traditional characteristics of MAD, but the US policy of deterrence towards nations with minor nuclear capabilities should ensure by threats of immense retaliation (or even pre-emptive action) not to threaten the United States, its interests, or allies. The document explains that such threats must also be used to ensure that nations without nuclear technology refrain from developing nuclear weapons and that a universal ban precludes any nation from maintaining chemical or biological weapons. The current tensions with Iran and North Korea over their nuclear programs are caused partly by the continuation of the policy of deterrence.
=== Post-Cold War period ===
By the beginning of the 2022 Russian invasion of Ukraine, many western hawks expressed the view that deterrence worked in that war but only in one way – in favor of Russia. Former US security advisor, John Bolton, said: Deterrence is working in the Ukraine crisis, just not for the right side. The United States and its allies failed to deter Russia from invading. The purpose of deterrence strategy is to prevent the conflict entirely, and there Washington failed badly. On the other hand, Russian deterrence is enjoying spectacular success. Russia has convinced the West that even a whisper of NATO military action in Ukraine would bring disastrous consequences. Putin threatens, blusters, uses the word “nuclear,” and the West wilts.
When Elon Musk prevented Ukraine from carrying drone attacks on the Russian Black Sea fleet by denying to enable needed Starlink communications in Crimea, Anne Applebaum argued Musk had been deterred by Russia after the country's ambassador warned him an attack on Crimea would be met with a nuclear response. Later Ukrainian attacks on the same fleet using a different communications system also caused deterrence, this time to the Russian Navy.
Timo S. Koster who served at NATO as Director of Defence Policy & Capabilities similarly argued: A massacre is taking place in Europe and the strongest military alliance in the world is staying out of it. We are deterred and Russia is not. Philip Breedlove, a retired four-star U.S. Air Force general and a former SACEUR, said that Western fears about nuclear weapons and World War III have left it "fully deterred" and Putin "completely undeterred." The West have "ceded the initiative to the enemy." No attempt was made by NATO to deter Moscow with the threat of military force, wondered another expert. To the contrary, it was Russia’s deterrence that proved to be successful.
== Cyber deterrence ==
Since the early 2000s, there has been an increased focus on cyber deterrence. Cyber deterrence has two meanings:
The use of cyber actions to deter other states
The deterrence of an adversary's cyber operations
Scholars have debated how cyber capabilities alter traditional understandings of deterrence, given that it may be harder to attribute responsibility for cyber attacks, the barriers to entry may be lower, the risks and costs may be lower for actors who conduct cyber attacks, it may be harder to signal and interpret intentions, the advantage of offense over defense, and weak actors and non-state actors can develop considerable cyber capabilities. Scholars have also debated the feasibility of launching highly damaging cyber attacks and engaging in destructive cyber warfare, with most scholars expressing skepticism that cyber capabilities have enhanced the ability of states to launch highly destructive attacks. The most prominent cyber attack to date is the Stuxnet attack on Iran's nuclear program. By 2019, the only publicly acknowledged case of a cyber attack causing a power outage was the 2015 Ukraine power grid hack.
There are various ways to engage in cyber deterrence:
Denial: preventing adversaries from achieving military objectives by defending against them
Punishment: the imposition of costs on the adversary
Norms: the establishment and maintenance of norms that establish appropriate standards of behavior
Escalation: raising the probability that costs will be imposed on the adversary
Entanglement and interdependence: interdependence between actors can have a deterrent effect
There is a risk of unintended escalation in cyberspace due to difficulties in discerning the intent of attackers, and complexities in state-hacker relationships. According to political scientists Joseph Brown and Tanisha Fazal, states frequently neither confirm nor deny responsibility for cyber operations so that they can avoid the escalatory risks (that come with public credit) while also signaling that they have cyber capabilities and resolve (which can be achieved if intelligence agencies and governments believe they were responsible).
According to Lennart Maschmeyer, cyber weapons have limited coercive effectiveness due to a trilemma "whereby speed, intensity, and control are negatively correlated. These constraints pose a trilemma for actors because a gain in one variable tends to produce losses across the other two variables."
== Intrawar deterrence ==
Intrawar deterrence is deterrence within a war context. It means that war has broken out but actors still seek to deter certain forms of behavior. In the words of Caitlin Talmadge, "intra-war deterrence failures... can be thought of as causing wars to get worse in some way." Examples of intrawar deterrence include deterring adversaries from resorting to nuclear, chemical and biological weapons attacks or attacking civilian populations indiscriminately. Broadly, it involves any prevention of escalation.
== Latent nuclear deterrence ==
Matthew Fuhrmann refers to the ability of some states to rapidly develop or gain nuclear weapons as "latent nuclear deterrence". These states do not necessarily aim to go all the way in building nuclear weapons, but they may develop the civilian nuclear technology that would rapidly enable them to build a nuclear weapon. They can use this nuclear latency status for coercive purposes, as they can deter adversaries who do not wish to see the state develop nuclear weapons or potentially use those nuclear weapons.
== Criticism ==
=== Deterrence failures ===
Deterrence theory has been criticized by numerous scholars for various reasons, the most basic being skepticism that decision makers are rational. A prominent strain of criticism argues that rational deterrence theory is contradicted by frequent deterrence failures, which may be attributed to misperceptions. Here it's argued that misestimations of perceived costs and benefits by analysts contribute to deterrence failures, as exemplified in case of Russian invasion of Ukraine. Frozen conflicts can be seen as rewarding aggression.
=== Misprediction of behavior ===
Scholars have also argued that leaders do not behave in ways that are consistent with the predictions of nuclear deterrence theory. Scholars have also argued that rational deterrence theory does not grapple sufficiently with emotions and psychological biases that make accidents, loss of self-control, and loss of control over others likely. Frank C. Zagare has argued that deterrence theory is logically inconsistent and empirically inaccurate. In place of classical deterrence, rational choice scholars have argued for perfect deterrence, which assumes that states may vary in their internal characteristics and especially in the credibility of their threats of retaliation.
=== Suicide attacks ===
Advocates for nuclear disarmament, such as Global Zero, have criticized nuclear deterrence theory. Sam Nunn, William Perry, Henry Kissinger, and George Shultz have all called upon governments to embrace the vision of a world free of nuclear weapons, and created the Nuclear Security Project to advance that agenda. In 2010, the four were featured in a documentary film entitled Nuclear Tipping Point where proposed steps to achieve nuclear disarmament. Kissinger has argued, "The classical notion of deterrence was that there was some consequences before which aggressors and evildoers would recoil. In a world of suicide bombers, that calculation doesn't operate in any comparable way." Shultz said, "If you think of the people who are doing suicide attacks, and people like that get a nuclear weapon, they are almost by definition not deterrable."
=== Stronger deterrent ===
Paul Nitze argued in 1994 that nuclear weapons were obsolete in the "new world disorder" after the dissolution of the Soviet Union, and he advocated reliance on precision guided munitions to secure a permanent military advantage over future adversaries.
=== Minimum deterrence ===
As opposed to the extreme mutually assured destruction form of deterrence, the concept of minimum deterrence in which a state possesses no more nuclear weapons than is necessary to deter an adversary from attacking is presently the most common form of deterrence practiced by nuclear weapon states, such as China, India, Pakistan, Britain, and France. Pursuing minimal deterrence during arms negotiations between the United States and Russia allows each state to make nuclear stockpile reductions without the state becoming vulnerable, but it has been noted that there comes a point that further reductions may be undesirable, once minimal deterrence is reached, as further reductions beyond that point increase a state's vulnerability and provide an incentive for an adversary to expand its nuclear arsenal secretly.
France has developed and maintained its own nuclear deterrent under the belief that the United States will refuse to risk its own cities by assisting Western Europe in a nuclear war.
=== Ethical objections ===
In the post cold war era, philosophical objections to the reliance upon deterrence theories in general have also been raised on purely ethical grounds. Scholars such as Robert L. Holmes have noted that the implementation of such theories is inconsistent with a fundamental deontological presumption which prohibits the killing of innocent life. Consequently, such theories are prima facie immoral in nature. In addition, he observes that deterrence theories serve to perpetuate a state of mutual assured destruction between nations over time. Holmes further argues that it is therefore both irrational and immoral to utilize a methodology for perpetuating international peace which relies exclusively upon the continuous development of new iterations of the very weapons which it is designed to prohibit.
== See also ==
== Notes ==
== References ==
== Further reading ==
Schultz, George P. and Goodby, James E. The War that Must Never be Fought, Hoover Press, ISBN 978-0-8179-1845-3, 2015.
Freedman, Lawrence. 2004. Deterrence. New York: Polity Press.
Jervis, Robert, Richard N. Lebow and Janice G. Stein. 1985. Psychology and Deterrence. Baltimore: Johns Hopkins University Press. 270 pp.
Morgan, Patrick. 2003. Deterrence Now. Cambridge University Press.
T.V. Paul, Patrick M. Morgan, James J. Wirtz, Complex Deterrence: Strategy In the Global Age (University of Chicago Press, 2009) ISBN 978-0-226-65002-9.
Waltz, Kenneth N. "Nuclear Myths and Political Realities". The American Political Science Review. Vol. 84, No. 3 (Sep, 1990), pp. 731–746.
== External links == | Wikipedia/Deterrence_theory |
The strategic grid model is a contingency approach that can be used to determine the strategic relevance of IT to an organization. The model was proposed by F. Warren McFarlan and James L. McKenney in 1983, and takes the impact of the information technology on the strategy in future planning as the horizontal axis, and the current impact of the information technology on corporate strategy as the vertical axis, which is divided into four types: support, turnaround, factory, and strategic.
== Overview ==
Strategic grid model has four quadrants built around two straightforward questions:
How important the management feels the current IT systems are to the company.
How important the company thinks future developments in IT will be, ie the impact of future IT developments on its way of doing business.
Depending on the responses to these questions, a company can be placed in the four quadrants as follows:
== Analysis ==
In order to assess the strategic impact of IT, McFarlan proposed the analysis of five basic questions about IT applications, related to the competitive forces:
If IT applications can build barriers to the entry of new competitors in the industry
If IT applications can build switching costs for suppliers
If IT applications can change the basis of competition
If IT applications can change the balance of power in supplier relationships
If IT applications can create new products
Nevertheless, these questions should take into account current and future planned circumstances. Thus, IT may present a smaller or greater importance, according to the kind of company and industry operations.
== References == | Wikipedia/Strategic_Grid_Model |
Rational choice modeling refers to the use of decision theory (the theory of rational choice) as a set of guidelines to help understand economic and social behavior. The theory tries to approximate, predict, or mathematically model human behavior by analyzing the behavior of a rational actor facing the same costs and benefits.
Rational choice models are most closely associated with economics, where mathematical analysis of behavior is standard. However, they are widely used throughout the social sciences, and are commonly applied to cognitive science, criminology, political science, and sociology.
== Overview ==
The basic premise of rational choice theory is that the decisions made by individual actors will collectively produce aggregate social behaviour. The theory also assumes that individuals have preferences out of available choice alternatives. These preferences are assumed to be complete and transitive. Completeness refers to the individual being able to say which of the options they prefer (i.e. individual prefers A over B, B over A or are indifferent to both). Alternatively, transitivity is where the individual weakly prefers option A over B and weakly prefers option B over C, leading to the conclusion that the individual weakly prefers A over C. The rational agent will then perform their own cost–benefit analysis using a variety of criterion to perform their self-determined best choice of action.
One version of rationality is instrumental rationality, which involves achieving a goal using the most cost effective method without reflecting on the worthiness of that goal. Duncan Snidal emphasises that the goals are not restricted to self-regarding, selfish, or material interests. They also include other-regarding, altruistic, as well as normative or ideational goals.
Rational choice theory does not claim to describe the choice process, but rather it helps predict the outcome and pattern of choice. It is consequently assumed that the individual is a self-interested or “homo economicus”. Here, the individual comes to a decision that optimizes their preferences by balancing costs and benefits.
Rational choice theory has proposed that there are two outcomes of two choices regarding human action. Firstly, the feasible region will be chosen within all the possible and related action. Second, after the preferred option has been chosen, the feasible region that has been selected was picked based on restriction of financial, legal, social, physical or emotional restrictions that the agent is facing. After that, a choice will be made based on the preference order.
The concept of rationality used in rational choice theory is different from the colloquial and most philosophical use of the word. In this sense, "rational" behaviour can refer to "sensible", "predictable", or "in a thoughtful, clear-headed manner." Rational choice theory uses a much more narrow definition of rationality. At its most basic level, behavior is rational if it is reflective and consistent (across time and different choice situations). More specifically, behavior is only considered irrational if it is logically incoherent, i.e. self-contradictory.
Early neoclassical economists writing about rational choice, including William Stanley Jevons, assumed that agents make consumption choices so as to maximize their happiness, or utility. Contemporary theory bases rational choice on a set of choice axioms that need to be satisfied, and typically does not specify where the goal (preferences, desires) comes from. It mandates just a consistent ranking of the alternatives.: 501 Individuals choose the best action according to their personal preferences and the constraints facing them.
== Actions, assumptions, and individual preferences ==
Rational choice theory can be viewed in different contexts. At an individual level, the theory suggests that the agent will decide on the action (or outcome) they most prefer. If the actions (or outcomes) are evaluated in terms of costs and benefits, the choice with the maximum net benefit will be chosen by the rational individual. Rational behaviour is not solely driven by monetary gain, but can also be driven by emotional motives.
The theory can be applied to general settings outside of those identified by costs and benefits. In general, rational decision making entails choosing among all available alternatives the alternative that the individual most prefers. The "alternatives" can be a set of actions ("what to do?") or a set of objects ("what to choose/buy"). In the case of actions, what the individual really cares about are the outcomes that results from each possible action. Actions, in this case, are only an instrument for obtaining a particular outcome.
=== Formal statement ===
The available alternatives are often expressed as a set of objects, for example a set of j exhaustive and exclusive actions:
A
=
{
a
1
,
…
,
a
i
,
…
,
a
j
}
{\displaystyle A=\{a_{1},\ldots ,a_{i},\ldots ,a_{j}\}}
For example, if a person can choose to vote for either Roger or Sara or to abstain, their set of possible alternatives is:
A
=
{
Vote for Roger, Vote for Sara, Abstain
}
{\displaystyle A=\{{\text{Vote for Roger, Vote for Sara, Abstain}}\}}
The theory makes two technical assumptions about individuals' preferences over alternatives:
Completeness – for any two alternatives ai and aj in the set, either ai is preferred to aj, or aj is preferred to ai, or the individual is indifferent between ai and aj. In other words, all pairs of alternatives can be compared with each other.
Transitivity – if alternative a1 is preferred to a2, and alternative a2 is preferred to a3, then a1 is preferred to a3.
Together these two assumptions imply that given a set of exhaustive and exclusive actions to choose from, an individual can rank the elements of this set in terms of his preferences in an internally consistent way (the ranking constitutes a total ordering, minus some assumptions), and the set has at least one maximal element.
The preference between two alternatives can be:
Strict preference occurs when an individual prefers a1 to a2 and does not view them as equally preferred.
Weak preference implies that individual either strictly prefers a1 over a2 or is indifferent between them.
Indifference occurs when an individual neither prefers a1 to a2, nor a2 to a1. Since (by completeness) the individual does not refuse a comparison, they must therefore be indifferent in this case.
Research since the 1980s sought to develop models that weaken these assumptions and argue some cases of this behaviour can be considered rational. However, the Dutch book theorems show that this comes at a major cost of internal coherence, such that weakening any of the Von Neumann–Morgenstern axioms makes. The most severe consequences are associated with violating independence of irrelevant alternatives, and transitive preferences, or fully abandoning completeness rather than weakening it to "asymptotic" completeness.
== Utility maximization ==
Often preferences are described by their utility function or payoff function. This is an ordinal number that an individual assigns over the available actions, such as:
u
(
a
i
)
>
u
(
a
j
)
.
{\displaystyle u\left(a_{i}\right)>u\left(a_{j}\right).}
The individual's preferences are then expressed as the relation between these ordinal assignments. For example, if an individual prefers the candidate Sara over Roger over abstaining, their preferences would have the relation:
u
(
Sara
)
>
u
(
Roger
)
>
u
(
abstain
)
.
{\displaystyle u\left({\text{Sara}}\right)>u\left({\text{Roger}}\right)>u\left({\text{abstain}}\right).}
A preference relation that as above satisfies completeness, transitivity, and, in addition, continuity, can be equivalently represented by a utility function.
== Benefits ==
The rational choice approach allows preferences to be represented as real-valued utility functions. Economic decision making then becomes a problem of maximizing this utility function, subject to constraints (e.g. a budget). This has many advantages. It provides a compact theory that makes empirical predictions with a relatively sparse model – just a description of the agent's objectives and constraints. Furthermore, optimization theory is a well-developed field of mathematics. These two factors make rational choice models tractable compared to other approaches to choice. Most importantly, this approach is strikingly general. It has been used to analyze not only personal and household choices about
traditional economic matters like consumption and savings, but also choices about education, marriage, child-bearing, migration, crime and so on, as well as business decisions about output, investment, hiring, entry, exit, etc. with varying degrees of success.
In the field of political science rational choice theory has been used to help predict human decision making and model for the future; therefore it is useful in creating effective public policy, and enables the government to develop solutions quickly and efficiently.
Despite the empirical shortcomings of rational choice theory, the flexibility and tractability of rational choice models (and the lack of equally powerful alternatives) lead to them still being widely used.
== Applications ==
Rational choice theory has become increasingly employed in social sciences other than economics, such as sociology, evolutionary theory and political science in recent decades. It has had far-reaching impacts on the study of political science, especially in fields like the study of interest groups, elections, behaviour in legislatures, coalitions, and bureaucracy. In these fields, the use of rational choice theory to explain broad social phenomena is the subject of controversy.
=== Rational choice theory in political science ===
Rational choice theory provides a framework to explain why groups of rational individuals can come to collectively irrational decisions. For example, while at the individual level a group of people may have common interests, applying a rational choice framework to their individually rational preferences can explain group-level outcomes that fail to accomplish any one individual's preferred objectives. Rational choice theory provides a framework to describe outcomes like this as the product of rational agents performing their own cost–benefit analysis to maximize their self-interests, a process that doesn't always align with the group's preferences.
==== Rational choice in voting behavior ====
Voter behaviour shifts significantly thanks to rational theory, which is ingrained in human nature, the most significant of which occurs when there are times of economic trouble. An example in economic policy, economist Anthony Downs concluded that a high income voter ‘votes for whatever party he believes would provide him with the highest utility income from government action’, using rational choice theory to explain people's income as their justification for their preferred tax rate.
Downs' work provides a framework for analyzing tax-rate preference in a rational choice framework. He argues that an individual votes if it is in their rational interest to do so. Downs models this utility function as B + D > C, where B is the benefit of the voter winning, D is the satisfaction derived from voting and C is the cost of voting. It is from this that we can determine that parties have moved their policy outlook to be more centric in order to maximise the number of voters they have for support. It is from this very simple framework that more complex adjustments can be made to describe the success of politicians as an outcome of their ability or failure to satisfy the utility function of individual voters.
==== Rational choice theory in international relations ====
Rational choice theory has become one of the major tools used to study international relations. Proponents of its use in this field typically assume that states and the policies crafted at the national outcome are the outcome of self-interested, politically shrewd actors including, but not limited to, politicians, lobbyists, businesspeople, activists, regular voters and any other individual in the national audience. The use of rational choice theory as a framework to predict political behavior has led to a rich literature that describes the trajectory of policy to varying degrees of success. For example, some scholars have examined how states can make credible threats to deter other states from a (nuclear) attack. Others have explored under what conditions states wage war against each other. Yet others have investigated under what circumstances the threat and imposition of international economic sanctions tend to succeed and when they are likely to fail.
=== Rational choice theory in social interactions ===
Rational choice theory and social exchange theory involves looking at all social relations in the form of costs and rewards, both tangible and non tangible.
According to Abell, Rational Choice Theory is "understanding individual actors... as acting, or more likely interacting, in a manner such that they can be deemed to be doing the best they can for themselves, given their objectives, resources, circumstances, as they seem them". Rational Choice Theory has been used to comprehend the complex social phenomena, of which derives from the actions and motivations of an individual. Individuals are often highly motivated by their wants and needs.
By making calculative decisions, it is considered as rational action. Individuals are often making calculative decisions in social situations by weighing out the pros and cons of an action taken towards a person. The decision to act on a rational decision is also dependent on the unforeseen benefits of the friendship. Homan mentions that actions of humans are motivated by punishment or rewards. This reinforcement through punishments or rewards determines the course of action taken by a person in a social situation as well. Individuals are motivated by mutual reinforcement and are also fundamentally motivated by the approval of others. Attaining the approval of others has been a generalized character, along with money, as a means of exchange in both Social and Economic exchanges. In Economic exchanges, it involves the exchange of goods or services. In Social exchange, it is the exchange of approval and certain other valued behaviors.
Rational Choice Theory in this instance, heavily emphasizes the individual's interest as a starting point for making social decisions. Despite differing view points about Rational choice theory, it all comes down to the individual as a basic unit of theory. Even though sharing, cooperation and cultural norms emerge, it all stems from an individual's initial concern about the self.
G.S Becker offers an example of how Rational choice can be applied to personal decisions, specifically regarding the rationale that goes behind decisions on whether to marry or divorce another individual. Due to the self-serving drive on which the theory of rational choice is derived, Becker concludes that people marry if the expected utility from such marriage exceeds the utility one would gain from remaining single, and in the same way couples would separate should the utility of being together be less than expected and provide less (economic) benefit than being separated would. Since the theory behind rational choice is that individuals will take the course of action that best serves their personal interests, when considering relationships it is still assumed that they will display such mentality due to deep-rooted, self-interested aspects of human nature.
Social Exchange and Rational Choice Theory both comes down to an individual's efforts to meet their own personal needs and interests through the choices they make. Even though some may be done sincerely for the welfare of others at that point of time, both theories point to the benefits received in return. These returns may be received immediately or in the future, be it tangible or not.
Coleman discussed a number of theories to elaborate on the premises and promises of rational choice theory. One of the concepts that He introduced was Trust. It is where "individuals place trust, in both judgement and performance of others, based on rational considerations of what is best, given the alternatives they confront". In a social situation, there has to be a level of trust among the individuals. He noted that this level of trust is a consideration that an individual takes into concern before deciding on a rational action towards another individual. It affects the social situation as one navigates the risks and benefits of an action. By assessing the possible outcomes or alternatives to an action for another individual, the person is making a calculated decision. In another situation such as making a bet, you are calculating the possible lost and how much can be won. If the chances of winning exceeds the cost of losing, the rational decision would be to place the bet. Therefore, the decision to place trust in another individual involves the same rational calculations that are involved in the decision of making a bet.
Even though rational theory is used in Economics and Social settings, there are some similarities and differences. The concept of reward and reinforcement is parallel to each other while the concept of cost is also parallel to the concept of punishment. However, there is a difference of underlying assumptions in both contexts. In a social setting, the focus is often on the current or past reinforcements, with no guarantee of immediate tangible or intangible returns from another individual in the future. In Economics, decisions are made with heavier emphasis on future rewards.
Despite having both perspectives differ in focus, they primarily reflect on how individuals make different rational decisions when given an immediate or long-term circumstances to consider in their rational decision making.
== Criticism ==
Both the assumptions and the behavioral predictions of rational choice theory have sparked criticism from various camps.
=== The limits of rationality ===
As mentioned above, some economists have developed models of bounded rationality, such as Herbert Simon, which hope to be more psychologically plausible without completely abandoning the idea that reason underlies decision-making processes. Simon argues factors such as imperfect information, uncertainty and time constraints all affect and limit our rationality, and therefore our decision-making skills. Furthermore, his concepts of 'satisficing' and 'optimizing' suggest sometimes because of these factors, we settle for a decision which is good enough, rather than the best decision. Other economists have developed more theories of human decision-making that allow for the roles of uncertainty, institutions, and determination of individual tastes by their socioeconomic environment (cf. Fernandez-Huerga, 2008).
=== Philosophical critiques ===
Martin Hollis and Edward J. Nell's 1975 book offers both a philosophical critique of neo-classical economics and an innovation in the field of economic methodology. Further, they outlined an alternative vision to neo-classicism based on a rationalist theory of knowledge. Within neo-classicism, the authors addressed consumer behaviour (in the form of indifference curves and simple versions of revealed preference theory) and marginalist producer behaviour in both product and factor markets. Both are based on rational optimizing behaviour. They consider imperfect as well as perfect markets since neo-classical thinking embraces many market varieties and disposes of a whole system for their classification. However, the authors believe that the issues arising from basic maximizing models have extensive implications for econometric methodology (Hollis and Nell, 1975, p. 2). In particular it is this class of models – rational behavior as maximizing behaviour – which provide support for specification and identification. And this, they argue, is where the flaw is to be found. Hollis and Nell (1975) argued that positivism (broadly conceived) has provided neo-classicism with important support, which they then show to be unfounded. They base their critique of neo-classicism not only on their critique of positivism but also on the alternative they propose, rationalism. Indeed, they argue that rationality is central to neo-classical economics – as rational choice – and that this conception of rationality is misused. Demands are made of it that it cannot fulfill. Ultimately, individuals do not always act rationally or conduct themselves in a utility maximising manner.
Duncan K. Foley (2003, p. 1) has also provided an important criticism of the concept of rationality and its role in economics. He argued that“Rationality” has played a central role in shaping and establishing the hegemony of contemporary mainstream economics. As the specific claims of robust neoclassicism fade into the history of economic thought, an orientation toward situating explanations of economic phenomena in relation to rationality has increasingly become the touchstone by which mainstream economists identify themselves and recognize each other. This is not so much a question of adherence to any particular conception of rationality, but of taking rationality of individual behavior as the unquestioned starting point of economic analysis.
Foley (2003, p. 9) went on to argue thatThe concept of rationality, to use Hegelian language, represents the relations of modern capitalist society one-sidedly. The burden of rational-actor theory is the assertion that ‘naturally’ constituted individuals facing existential conflicts over scarce resources would rationally impose on themselves the institutional structures of modern capitalist society, or something approximating them. But this way of looking at matters systematically neglects the ways in which modern capitalist society and its social relations in fact constitute the ‘rational’, calculating individual. The well-known limitations of rational-actor theory, its static quality, its logical antinomies, its vulnerability to arguments of infinite regress, its failure to develop a progressive concrete research program, can all be traced to this starting-point.
More recently Edward J. Nell and Karim Errouaki (2011, Ch. 1) argued that:The DNA of neoclassical economics is defective. Neither the induction problem nor the problems of methodological individualism can be solved within the framework of neoclassical assumptions. The neoclassical approach is to call on rational economic man to solve both. Economic relationships that reflect rational choice should be ‘projectible’. But that attributes a deductive power to ‘rational’ that it cannot have consistently with positivist (or even pragmatist) assumptions (which require deductions to be simply analytic). To make rational calculations projectible, the agents may be assumed to have idealized abilities, especially foresight; but then the induction problem is out of reach because the agents of the world do not resemble those of the model. The agents of the model can be abstract, but they cannot be endowed with powers actual agents could not have. This also undermines methodological individualism; if behaviour cannot be reliably predicted on the basis of the ‘rational choices of agents’, a social order cannot reliably follow from the choices of agents.
=== Psychological critiques ===
The validity of Rational Choice Theory has been generally refuted by the results of research in behavioral psychology. The revision or alternative theory that arises from these discrepancies is called Prospect Theory.
The 'doubly-divergent' critique of Rational Choice Theory implicit in Prospect Theory has sometimes been presented as a revision or alternative. Daniel Kahneman's work has been notably elaborated by research undertaken and supervised by Jonathan Haidt and other scholars.
=== Empirical critiques ===
In their 1994 work, Pathologies of Rational Choice Theory, Donald P. Green and Ian Shapiro argue that the empirical outputs of rational choice theory have been limited. They contend that much of the applicable literature, at least in political science, was done with weak statistical methods and that when corrected many of the empirical outcomes no longer hold. When taken in this perspective, rational choice theory has provided very little to the overall understanding of political interaction – and is an amount certainly disproportionately weak relative to its appearance in the literature. Yet, they concede that cutting-edge research, by scholars well-versed in the general scholarship of their fields (such as work on the U.S. Congress by Keith Krehbiel, Gary Cox, and Mat McCubbins) has generated valuable scientific progress.
=== Methodological critiques ===
Schram and Caterino (2006) contains a fundamental methodological criticism of rational choice theory for promoting the view that the natural science model is the only appropriate methodology in social science and that political science should follow this model, with its emphasis on quantification and mathematization. Schram and Caterino argue instead for methodological pluralism. The same argument is made by William E. Connolly, who in his work Neuropolitics shows that advances in neuroscience further illuminate some of the problematic practices of rational choice theory.
=== Sociological critiques ===
Pierre Bourdieu fiercely opposed rational choice theory as grounded in a misunderstanding of how social agents operate. Bourdieu argued that social agents do not continuously calculate according to explicit rational and economic criteria. According to Bourdieu, social agents operate according to an implicit practical logic – a practical sense – and bodily dispositions. Social agents act according to their "feel for the game" (the "feel" being, roughly, habitus, and the "game" being the field).
Other social scientists, inspired in part by Bourdieu's thinking have expressed concern about the inappropriate use of economic metaphors in other contexts, suggesting that this may have political implications. The argument they make is that by treating everything as a kind of "economy" they make a particular vision of the way an economy works seem more natural. Thus, they suggest, rational choice is as much ideological as it is scientific.
==== Criticism based on motivational assumptions ====
Rational choice theorists discuss individual values and structural elements as equally important determinants of outcomes. However, for methodological reasons in the empirical application, more emphasis is usually placed on social structural determinants. Therefore, in line with structural functionalism and social network analysis perspectives, rational choice explanations are considered mainstream in sociology .
==== Criticism based on the assumption of realism ====
Some of the scepticism among sociologists regarding rational choice stems from a misunderstanding of the lack of realist assumptions. Social research has shown that social agents usually act solely based on habit or impulse, the power of emotion. Social Agents predict the expected consequences of options in stock markets and economic crises and choose the best option through collective "emotional drives," implying social forces rather than "rational" choices.
However, sociology commonly misunderstands rational choice in its critique of rational choice theory. Rational choice theory does not explain what rational people would do in a given situation, which falls under decision theory. Theoretical choice focuses on social outcomes rather than individual outcomes. Social outcomes are identified as stable equilibria in which individuals have no incentive to deviate from their course of action. This orientation of others' behaviour toward social outcomes may be unintended or undesirable. Therefore, the conclusions generated in such cases are relegated to the "study of irrational behaviour".
=== Criticism based on the biopolitical paradigm ===
The basic assumptions of rational choice theory do not take into account external factors (social, cultural, economic) that interfere with autonomous decision-making. Representatives of the biopolitical paradigm such as Michel Foucault drew attention to the micro-power structures that shape the soul, body and mind and thus top-down impose certain decisions on individuals. Humans – according to the assumptions of the biopolitical paradigm – therefore conform to dominant social and cultural systems rather than to their own subjectively defined goals, which they would seek to achieve through rational and optimal decisions.
=== Critiques on the basis of evolutionary psychology ===
An evolutionary psychology perspective suggests that many of the seeming contradictions and biases regarding rational choice can be explained as being rational in the context of maximizing biological fitness in the ancestral environment but not necessarily in the current one. Thus, when living at subsistence level where a reduction of resources may have meant death it may have been rational to place a greater value on losses than on gains. Proponents argue it may also explain differences between groups.
=== Critiques on the basis of emotion research ===
Proponents of emotional choice theory criticize the rational choice paradigm by drawing on new findings from emotion research in psychology and neuroscience. They point out that rational choice theory is generally based on the assumption that decision-making is a conscious and reflective process based on thoughts and beliefs. It presumes that people decide on the basis of calculation and deliberation. However, cumulative research in neuroscience suggests that only a small part of the brain's activities operate at the level of conscious reflection. The vast majority of its activities consist of unconscious appraisals and emotions. The significance of emotions in decision-making has generally been ignored by rational choice theory, according to these critics. Moreover, emotional choice theorists contend that the rational choice paradigm has difficulty incorporating emotions into its models, because it cannot account for the social nature of emotions. Even though emotions are felt by individuals, psychologists and sociologists have shown that emotions cannot be isolated from the social environment in which they arise. Emotions are inextricably intertwined with people's social norms and identities, which are typically outside the scope of standard rational choice models. Emotional choice theory seeks to capture not only the social but also the physiological and dynamic character of emotions. It represents a unitary action model to organize, explain, and predict the ways in which emotions shape decision-making.
=== The difference between public and private spheres ===
Herbert Gintis has also provided an important criticism to rational choice theory. He argued that rationality differs between the public and private spheres. The public sphere being what you do in collective action and the private sphere being what you do in your private life. Gintis argues that this is because “models of rational choice in the private sphere treat agents’ choices as instrumental”. “Behaviour in the public sphere, by contrast, is largely non-instrumental because it is non-consequential". Individuals make no difference to the outcome, “much as single molecules make no difference to the properties of the gas" (Herbert, G). This is a weakness of rational choice theory as it shows that in situations such as voting in an election, the rational decision for the individual would be to not vote as their vote makes no difference to the outcome of the election. However, if everyone were to act in this way the democratic society would collapse as no one would vote. Therefore, we can see that rational choice theory does not describe how everything in the economic and political world works, and that there are other factors of human behaviour at play.
== See also ==
== Notes ==
== References ==
Abella, Alex (2008). Soldiers of Reason: The RAND Corporation and the Rise of the American Empire. New York: Harcourt.
Allingham, Michael (2002). Choice Theory: A Very Short Introduction, Oxford, ISBN 978-0192803030.
Anand, P. (1993)."Foundations of Rational Choice Under Risk", Oxford: Oxford University Press.
Amadae, S.M.(2003). Rationalizing Capitalist Democracy: The Cold War Origins of Rational Choice Liberalism, Chicago: University of Chicago Press.
Arrow, Kenneth J. ([1987] 1989). "Economic Theory and the Hypothesis of Rationality," in The New Palgrave: Utility and Probability, pp. 25–39.
Bicchieri, Cristina (1993). Rationality and Coordination. Cambridge University Press
Bicchieri, Cristina (2003). “Rationality and Game Theory”, in The Handbook of Rationality, The Oxford Reference Library of Philosophy, Oxford University Press.
Cristian Maquieira, Jan 2019, Japan's Withdrawal from the International Whaling Commission: A Disaster that Could Have Been Avoided, Available at: [2], November 2019
Downs, Anthony (1957). "An Economic Theory of Democracy." Harper.
Anthony Downs, 1957, An Economic Theory of Political Action in a Democracy, Journal of Political Economy, Vol. 65, No. 2, pp. 135–150
Coleman, James S. (1990). Foundations of Social Theory
Dixon, Huw (2001), Surfing Economics, Pearson. Especially chapters 7 and 8
Elster, Jon (1979). Ulysses and the Sirens, Cambridge University Press.
Elster, Jon (1989). Nuts and Bolts for the Social Sciences, Cambridge University Press.
Elster, Jon (2007). Explaining Social Behavior – more Nuts and Bolts for the Social Sciences, Cambridge University Press.
Fernandez-Huerga (2008.) The Economic Behavior of Human Beings: The Institutionalist//Post-Keynesian Model Journal of Economic Issues. vol. 42 no. 3, September.
Schram, Sanford F. and Brian Caterino, eds. (2006). Making Political Science Matter: Debating Knowledge, Research, and Method. New York and London: New York University Press.
Walsh, Vivian (1996). Rationality, Allocation, and Reproduction, Oxford. Description and scroll to chapter-preview links.
Martin Hollis and Edward J. Nell (1975) Rational Economic Man. Cambridge: Cambridge University Press.
Foley, D. K. (1989) Ideology and Methodology. An unpublished lecture to Berkeley graduate students in 1989 discussing personal and collective survival strategies for non-mainstream economists.
Foley, D.K. (1998). Introduction (chapter 1) in Peter S. Albin, Barriers and Bounds to Rationality: Essays on Economic Complexity and Dynamics in Interactive Systems. Princeton: Princeton University Press.
Foley, D. K. (2003) Rationality and Ideology in Economics. lecture in the World Political Economy course at the Graduate Faculty of New School UM, New School.
Boland, L. (1982) The Foundations of Economic Method. London: George Allen & Unwin
Edward J. Nell and Errouaki, K. (2011) Rational Econometric Man. Cheltenham: E. Elgar.
Pierre Bourdieu (2005) The Social Structures of the Economy, Polity 2005
Calhoun, C. et al. (1992) "Pierre Bourdieu: Critical Perspectives." University of Chicago Press.
Gary Browning, Abigail Halcli, Frank Webster, 2000, Understanding Contemporary Society: Theories of the Present, London, Sage Publications
Grenfell, M (2011) "Bourdieu, Language and Linguistics" London, Continuum.
Grenfell, M. (ed) (2008) "Pierre Bourdieu: Key concepts" London, Acumen Press
Herbert Gintis. Centre for the study of Governance and Society CSGS(Rational Choice and Political Behaviour: A lecture by Herbert Gintis. YouTube video. 23:57. Nov 21, 2018)
== Further reading ==
Gilboa, Itzhak (2010). Rational Choice. Cambridge, MA: MIT Press.
Green, Donald P., and Justin Fox (2007). "Rational Choice Theory," in The SAGE Handbook of Social Science Methodology, edited by William Outhwaite and Stephen P. Turner. London: Sage, pp. 269–282.
Kydd, Andrew H. (2008). "Methodological Individualism and Rational Choice," The Oxford Handbook of International Relations, edited by Christian Reus-Smit and Duncan Snidal. Oxford: Oxford University Press, pp. 425–443.
Mas-Colell, A., M. D. Whinston, and J. R. Green (1995). Microeconomic Theory. Oxford: Oxford University Press.
Nedergaard, Peter (July 2006). "The 2003 reform of the Common Agricultural Policy: against all odds or rational explanations?" (PDF). Journal of European Integration. 28 (3): 203–223. doi:10.1080/07036330600785749. S2CID 154437960.
== External links ==
Rational Choice Theory at the Stanford Encyclopedia of Philosophy
Rational Choice Theory – Article by John Scott
The New Nostradamus – on the use by Bruce Bueno de Mesquita of rational choice theory in political forecasting
To See The Future, Use The Logic Of Self-Interest – NPR audio clip | Wikipedia/Rational_agent_model |
In game theory, a strategy A dominates another strategy B if A will always produce a better result than B, regardless of how any other player plays. Some very simple games (called straightforward games) can be solved using dominance.
== Terminology ==
A player can compare two strategies, A and B, to determine which one is better. The result of the comparison is one of:
B strictly dominates (>) A: choosing B always gives a better outcome than choosing A, no matter what the other players do.
B weakly dominates (≥) A: choosing B always gives at least as good an outcome as choosing A, no matter what the other players do, and there is at least one set of opponents' actions for which B gives a better outcome than A. (Notice that if B strictly dominates A, then B weakly dominates A. Therefore, we can say "B dominates A" to mean "B weakly dominates A".)
B is weakly dominated by A: there is at least one set of opponents' actions for which B gives a worse outcome than A, while all other sets of opponents' actions give B the same payoff as A. (Strategy A weakly dominates B).
B is strictly dominated by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. (Strategy A strictly dominates B).
Neither A nor B dominates the other: B and A are not equivalent, and B neither dominates, nor is dominated by, A. Choosing A is better in some cases, while choosing B is better in other cases, depending on exactly how the opponent chooses to play. For example, B is "throw rock" while A is "throw scissors" in Rock, Paper, Scissors.
This notion can be generalized beyond the comparison of two strategies.
Strategy B is strictly dominant if strategy B strictly dominates every other possible strategy.
Strategy B is weakly dominant if strategy B weakly dominates every other possible strategy.
Strategy B is strictly dominated if some other strategy exists that strictly dominates B.
Strategy B is weakly dominated if some other strategy exists that weakly dominates B.
Strategy: A complete contingent plan for a player in the game. A complete contingent plan is a full specification of a player's behavior, describing each action a player would take at every possible decision point. Because information sets represent points in a game where a player must make a decision, a player's strategy describes what that player will do at each information set.
Rationality: The assumption that each player acts in a way that is designed to bring about what he or she most prefers given probabilities of various outcomes; von Neumann and Morgenstern showed that if these preferences satisfy certain conditions, this is mathematically equivalent to maximizing a payoff. A straightforward example of maximizing payoff is that of monetary gain, but for the purpose of a game theory analysis, this payoff can take any desired outcome—cash reward, minimization of exertion or discomfort, or promoting justice can all be modeled as amassing an overall “utility” for the player. The assumption of rationality states that players will always act in the way that best satisfies their ordering from best to worst of various possible outcomes.
Common Knowledge: The assumption that each player has knowledge of the game, knows the rules and payoffs associated with each course of action, and realizes that every other player has this same level of understanding. This is the premise that allows a player to make a value judgment on the actions of another player, backed by the assumption of rationality, into consideration when selecting an action.
== Dominance and Nash equilibria ==
If a strictly dominant strategy exists for one player in a game, that player will play that strategy in each of the game's Nash equilibria. If both players have a strictly dominant strategy, the game has only one unique Nash equilibrium, referred to as a "dominant strategy equilibrium". However, that Nash equilibrium is not necessarily "efficient", meaning that there may be non-equilibrium outcomes of the game that would be better for both players. The classic game used to illustrate this is the Prisoner's Dilemma.
Strictly dominated strategies cannot be a part of a Nash equilibrium, and as such, it is irrational for any player to play them. On the other hand, weakly dominated strategies may be part of Nash equilibria. For instance, consider the payoff matrix pictured at the right.
Strategy C weakly dominates strategy D. Consider playing C: If one's opponent plays C, one gets 1; if one's opponent plays D, one gets 0. Compare this to D, where one gets 0 regardless. Since in one case, one does better by playing C instead of D and never does worse, C weakly dominates D. Despite this,
(
D
,
D
)
{\displaystyle (D,D)}
is a Nash equilibrium. Suppose both players choose D. Neither player will do any better by unilaterally deviating—if a player switches to playing C, they will still get 0. This satisfies the requirements of a Nash equilibrium. Suppose both players choose C. Neither player will do better by unilaterally deviating—if a player switches to playing D, they will get 0. This also satisfies the requirements of a Nash equilibrium.
== Iterated elimination of strictly dominated strategies ==
The iterated elimination (or deletion, or removal) of dominated strategies (also denominated as IESDS, or IDSDS, or IRSDS) is one common technique for solving games that involves iteratively removing dominated strategies. In the first step, all dominated strategies are removed from the strategy space of each of the players, since no rational player would ever play these strategies. This results in a new, smaller game. Some strategies—that were not dominated before—may be dominated in the smaller game. The first step is repeated, creating a new even smaller game, and so on.
This process is valid since it is assumed that rationality among players is common knowledge, that is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on ad infinitum (see Aumann, 1976).
== See also ==
Max-dominated strategy
Risk dominance
Winning strategy
== References ==
Fudenberg, Drew; Tirole, Jean (1993). Game Theory. MIT Press.
Gibbons, Robert (1992). Game Theory for Applied Economists. Princeton University Press. ISBN 0-691-00395-5.
Gintis, Herbert (2000). Game Theory Evolving. Princeton University Press. ISBN 0-691-00943-0.
Leyton-Brown, Kevin; Shoham, Yoav (2008). Essentials of Game Theory: A Concise, Multidisciplinary Introduction. San Rafael, CA: Morgan & Claypool Publishers. ISBN 978-1-59829-593-1.. An 88-page mathematical introduction; see Section 3.3. Free online at many universities.
Rapoport, A. (1966). Two-Person Game Theory: The Essential Ideas. University of Michigan Press.
Jim Ratliff's Game Theory Course: Strategic Dominance
Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7. A comprehensive reference from a computational perspective; see Sections 3.4.3, 4.5. Downloadable free online.
"Strict Dominance in Mixed Strategies – Game Theory 101". gametheory101.com. Retrieved 2021-12-17.
Watson Joel. Strategy : An Introduction to Game Theory. Third ed. W.W. Norton & Company 2013.
This article incorporates material from Dominant strategy on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. | Wikipedia/Dominant_strategy |
Behavioral strategy is an interdisciplinary field within strategic management that integrates insights from psychology, behavioral economics, and cognitive science to better understand how individuals and groups make strategic decisions. It challenges the assumptions of traditional economic models that presume perfect rationality, instead emphasizing how real-world decision-making is shaped by cognitive biases, emotions, social dynamics, and bounded rationality.
Emerging in response to the limitations of purely rational models of strategy, behavioral strategy seeks to incorporate psychologically realistic assumptions into both the theory and practice of strategic management. It applies behavioral perspectives to core strategic topics such as CEO and top management team behavior, market entry decisions, competitive dynamics, and organizational change. It is typically characterized by the following features:
It is microfoundational, drawing on individual-level psychological processes to explain firm-level outcomes;
It draws broadly from psychological subfields—including cognitive, social, and organizational psychology—as well as from behavioral economics and sociology;
It emphasizes empirically grounded assumptions, relying on evidence from laboratory and field experiments rather than abstract models or mathematical tractability.
Methodologically, behavioral strategy embraces pluralism, employing qualitative research, experiments, surveys, agent-based modeling, and traditional formal and statistical methods. Common research topics include cognitive bias in strategic decisions, the use of heuristics in uncertainty, bounded rationality in competitive interactions, and the influence of organizational culture on strategic behavior.
== Historical foundations ==
=== Early contributions ===
Herbert Simon's research on cognitive decision making and the concept of bounded rationality contributed to early developments in behavioral strategy. Simon identified four categorical observations on variations in ability to solve complex problems and make decisions:
Problem representation is crucial to problem-solving. Effective problem solvers accurately represent problems, highlighting their nature and utilizing the most pertinent information for solutions.
Pattern recognition facilitates problem-solving. When analyzing problems and solutions, patterns emerge that translate to 'if/then' solutions. For example, Porter's five Forces model exemplifies 'if/then' patterns that connect strategic problems with effective solutions. For example, "if your supplier has high bargaining power, then seek alternative sources".
Patterns enhance memory encoding and recall. A connection to a memory allows faster recall from long term memory, and patterns increase this connection making memories easier to recall.
Practice builds expertise. Repeatedly engaging in decision-making and problem-solving correlates with increased skill. Using representation, recognizing patterns, and recalling these patterns enhances one's abilities.
These observations provided foundational support for the development of behavioral strategy research.
=== Evolution and development ===
The application of psychological insights to research on firm behavior and performance has a rich history. This includes research on the behavioral theory of the firm (Cyert & March, 1963; Gavetti, Levinthal, and Ocasio, 2007), aspirations (Greve, 1998), attention (Ocasio, 1997), emotions (Nickerson & Zenger, 2008), goals (Lindenberg & Foss, 2011), cognitive schema, maps, sensemaking, and cognitive rivalry (Porac and Thomas, 1990; Reger and Huff, 1993; Lant and Baum, 1995; Weick, 1995), routines (Cyert & March, 1963), decision theory (Kahneman and Lovallo, 1993), escalation (Staw and Cummings, 1981), motivation (Foss & Weber, 2016), hubris (Bollaert and Petit, 2010), top management teams (Hambrick and Mason, 1984), dominant logic (Prahalad & Bettis, 1986), competitive interaction (Chen, Smith & Grimm, 1992), and organizational learning (Levinthal and March, 1993).
However, the first explicit use of the term "behavioral strategy" in a journal appears to be in Lovallo and Sibony (2010), which links the concept to behavioral economics literature and the underlying heuristics and biases research. While this was published in a practitioner journal, Powell, Lovallo and Fox (2011) later edited a special issue on "Psychological Foundations of Strategic Management" in the Strategic Management Journal. Retrospectively, this may be seen as the key event in launching behavioral strategy as a coherent, institutionalized research effort rather than a multitude of relatively unconnected research streams.
In their editorial essay, Powell et al. outlined three reasons why there was a need for a concerted research effort in behavioral strategy:
Strategy research had been slow to incorporate relevant results from psychology
The field lacked adequate psychological grounding (e.g., firm heterogeneity was assumed rather than explained through reasoning and decision-making processes)
Recent developments (such as advances in cognitive neuroscience linking strategic decision-making and brain activity) had created opportunities for closer integration of cognitive sciences and strategy research
The following year, Rindova, Reger, and Dalpiaz (2012) referred to a "'sociocognitive' perspective" in strategy which, "while varied in its theoretical framings, focuses on the roles of managers' and observers' attention; the bounded rationality of their cognitions, intuitions, and emotions; and the use of biases and heuristics to socially construct 'perceptual answers' to traditional strategic management questions about how firms obtain and sustain competitive advantage."
=== Defining the field ===
The growing interest in behavioral strategy has prompted several attempts to define the field (Powell et al., 2011; Rindova et al., 2012; Hambrick and Crossland, 2019) and surveys of closely related theoretical approaches, such as the behavioral theory of the firm (Gavetti, Levinthal, Greve, & Ocasio, 2012) and problemistic search (Posen et al., 2018). Hambrick and Crossland (2019) proposed conceptualizing behavioral strategy using an imagery of differently sized "tents":
Small tent: "A direct transposition of the logic of behavioral economics (and behavioral finance) to the field of strategic management"
Medium tent: "A commitment to understanding the psychology of strategists"
Large tent: "All forms and styles of research that consider any psychological, social, or political ingredients in strategic management"
Today, behavioral strategy has evolved into a significant subfield within strategic management. It applies insights from social psychology and cognitive science to enhance strategic decision-making by deepening understanding of social dynamics and human cognition. The field places particular emphasis on top managers' cognitive processes and the patterns of collaboration and communication within organizations, with its foundation in behavioral decision theory. Strategic cognition, a key component of behavioral strategy, focuses on understanding cognitive structures within organizations and their decision-making processes. Both effective analytical reasoning and intuition play significant roles in strategy formulation, influencing organizational and managerial cognition. The field has gained substantial attention in academic circles, with dedicated issues and volumes in prestigious conferences and publications. Despite this growth, behavioral strategy remains somewhat fragmented. To address this challenge, scholars have proposed integrating theoretical and empirical approaches to provide a more comprehensive understanding of how behavior impacts strategic outcomes.
== Application in Covid-19 ==
Behavioral strategy affected decisions made during the COVID-19 disruption. Behavioral strategy provides psychologically based interpretations that can illuminate how individuals and organizations respond to such disruptions. It suggests that strategists may not be good at using formal models, rules, or forecasts because they are not statisticians. There is supporting evidence of this observed during the disruption caused by Covid-19. Some decision-makers treated extreme model projections as deterministic predictions rather than recognizing them as improbable worst-case scenarios. An example of this was the societal lockdown. It was impossible to forecast the economic and social consequences of the lockdown, and its effectiveness, and yet decision-makers decided to implement this worst-case scenario. Another example of worst-case scenario being implemented is when the CDC gave guidance on wearing masks outdoors as this was an example of extreme caution. Decision-makers appeared to overlook the consequences of or misunderstand the lack of error margins around initial forecasts. Also of relevance, decision-makers may rely too much on models, forecasts, and data that are available. When decision-making problems are ill-structured and require quick action, relying solely on formal models and forecasts can be problematic. It becomes necessary to incorporate intuition and soft data into the decision-making process in these cases.
== Limitations of behavioral strategy ==
Strategy making is a deeply social process and strategy research doesn't sufficiently account for this. Different experts' social standards vary, and this will influence what information is collected. COVID-19 highlighted how behavioral strategy frameworks don’t allow dealing with uncertainty beyond standard treatments of risky decision-making. Behavioral strategy is useful in extreme circumstances, however, there is more research to be done on the weaknesses present for disruptions like this.
== See also ==
Adaptive market hypothesis – Economic theory
Behavioral operations management
Economic sociology – Branch of sociology
Market sentiment – General attitude of investors to market price development
Socioeconomics – Branch of sociologyPages displaying short descriptions of redirect targets
== References ==
Bollaert, Helen & Petit, Valérie. 2010. Beyond the dark side of executive psychology: Current research and new directions. European Management Journal. 28(5): 362–376.
Bridoux, Flore & Stoelhorst, J.W. 2016. Stakeholder Relationships and Social Welfare: A Behavioral Theory of Contributions to Joint Value Creation. Academy of Management Review. 41(2): 229–251.
Chen, M-J., Smith, Ken G., & Grimm, Curtis M. 1992. Action Characteristics as Predictors of Competitive Responses. Management Science. 38(3): 307–458.
Cyert, Richard M. & March, James G. 1963. A Behavioral Theory of the Firm. University of Illionois at Urbana-Champaign's Academy for Entrepreneurial Leadership Historical Research Reference in Entrepreneurship.
Felin, T., Foss, N.J., & Ployhardt, R. 2015. Microfoundations for Management Research." Academy of Management Annals 9: 575–632.
Foss, N.J. & Weber, L. 2016. Putting Opportunism in the Back Seat: Bounded Rationality, Costly Conflict and Hierarchical Forms. Academy of Management Review, 41: 41–79.
Garbuio, M., Porac, J., Lovallo, D. & Dong, A. 2015. A Design Cognition Perspective on Strategic Option Generation. Advances in Strategic Management. 32(1):437-465.
Gavetti, Levinthal, & Ocasio. 2007. Neo-Carnegie: The Carnegie School's Past, Present, and Reconstructing for the Future. Organization Science. 18(3): 523–536.
Gavetti, G., Levinthal, D., Greve, H. & Ocasio, W. 2012. The Behavioral Theory of the Firm: Assessment and Prospects. The Academy of Management Annals 6(1):1-40.
Greve, Henrich R. 1998. Performance, Aspirations, and Risky Organizational Change. Administrative Science Quarterly. 43(1): 58–86.
Hambrick, Donald C. & Crossland, Craig. 2018. A strategy for behavioral strategy: Appraisal of small, midsize, and large tent conceptions of this embryonic community. In M. Augier, C. Fang & V. Rindova, eds., Behavioral Strategy in Perspective (Advances in Strategic Management) 39: 22–39. Emerald Publishing.
Hambrick, Donald C. & Mason, Phyllis A. 1984. Upper Echelons: The Organization as a Reflection of Its Top Managers. The Academy of Management Review. 9 (2): 193–206.
Kahneman, Daniel & Lovallo, Dan. 1993. Timid Choices and Bold Forecasts: A Cognitive Perspective on Risk Taking. Management Science. 39 (1): 17–31.
Kruglanski, A. W., & Kopetz, C. 2009. What is so special (and nonspecial) about goals? A view from the cognitive perspective. In G. B. Moskowitz & H. Grant, eds., The psychology of goals (p. 27–55). Guilford Press.
Lant, T.K. & Baum J.A.C. 1995. Cognitive sources of socially constructed competitive groups: Examples from the Manhattan hotel industry. In: W. R. Scott & S. Christensen, eds., The Institutional Construcdtion of Organizations. 15–38. Sage Publications.
Levinthal, Daniel A. & March, James G. 1993. The Myopia of Learning. Strategic Management Journal. 14 (S2): 95–112.
Lindenberg, S. & Foss, N.J. 2011. Managing Motivation for Joint Production: The Role of Goal Framing and Governance Mechanisms. Academy of Management Review 36: 500–525.
Lovallo, Dan & Sibony, Olivier. 2010. The Case For Behavioral Strategy. McKinsey Quarterly: 30–40.
Nickerson, Jack A. & Zenger, Todd R. 2008. Envy, Comparison Costs, and the Economic Theory of the Firm. Strategic Management Journal. 29(13): 1429–1449.
Ocasio, William. 1997. Towards an Attention-Based View of The Firm. Strategic Management Journal. 18(S1): 187–206.
Porac, Joseph F. & Thomas, Howard. 1990. Taxonomic Mental Models in Competitor Definition. The Academy of Management Review. 15(2): 224–240.
Powell, Thomas C., Lovallo, Dan & Fox Craig R. 2011. Behavioral Strategy. Strategic Management Journal. 32(13): 1369–1386.
Prahalad, C. K. & Bettis, Richard A. 1986. The dominant logic: A new linkage between diversity and performance. Strategic Management Journal 7(6): 485–501.
Reger, Rhonda K. & Huff, Anne Sigismund. 1993. Strategic groups: A cognitive perspective. Strategic Management Journal. 14(2): 103–123.
Rindova, Violina P., Reger, Rhonda K., & Dalpiaz, Elena. 2012. The mind of the strategist and the eye of the beholder: The Socio-cognitive perspective in strategy research. In G.B. Dagnino, eds., Handbook of Research on Comptetive Strategy. Edward Elgar Publishing.
Ryan, Richard M. & Deci, Edward L. 2017. Self-Determination Theory: Basic Psychological Needs in Motivation, Development, and Wellness. Guildford Publications.
Seminowicz, D.A., Mikulis, D. J.; Davis, K. D. 2004. Cognitive modulation of pain-related brain responses depends on behavioral strategy. Pain 112(1): 48–58.
Staw, Barry M & Cummings, Larry L. 1981. Research in Organizational Behavior. JAI Press.
Weick, Karl E.1995. Sensemaking in Organizations. Sage Publications. University of Michigan.
== External links ==
Behavioral Economics The Behavioral Economics Guide
Behavioral Finance Overview of Behavioral Finance | Wikipedia/Behavioral_strategy |
In combinatorial game theory, the strategy-stealing argument is a general argument that shows, for many two-player games, that the second player cannot have a guaranteed winning strategy. The strategy-stealing argument applies to any symmetric game (one in which either player has the same set of available moves with the same results, so that the first player can "use" the second player's strategy) in which an extra move can never be a disadvantage. A key property of a strategy-stealing argument is that it proves that the first player can win (or possibly draw) the game without actually constructing such a strategy. So, although it might prove the existence of a winning strategy, the proof gives no information about what that strategy is.
The argument works by obtaining a contradiction. A winning strategy is assumed to exist for the second player, who is using it. But then, roughly speaking, after making an arbitrary first move – which by the conditions above is not a disadvantage – the first player may then also play according to this winning strategy. The result is that both players are guaranteed to win – which is absurd, thus contradicting the assumption that such a strategy exists.
Strategy-stealing was invented by John Nash in the 1940s to show that the game of hex is always a first-player win, as ties are not possible in this game. However, Nash did not publish this method, and József Beck credits its first publication to Alfred W. Hales and Robert I. Jewett, in the 1963 paper on tic-tac-toe in which they also proved the Hales–Jewett theorem. Other examples of games to which the argument applies include the m,n,k-games such as gomoku. In the game of Chomp strategy stealing shows that the first player has a winning strategy in any rectangular board (other than 1x1). In the game of Sylver coinage, strategy stealing has been used to show that the first player can win in certain positions called "enders". In all of these examples the proof reveals nothing about the actual strategy.
== Example ==
A strategy-stealing argument can be used on the example of the game of tic-tac-toe, for a board and winning rows of any size. Suppose that the second player (P2) is using a strategy S which guarantees a win. The first player (P1) places an X in an arbitrary position. P2 responds by placing an O according to S. But if P1 ignores the first random X, P1 is now in the same situation as P2 on P2's first move: a single enemy piece on the board. P1 may therefore make a move according to S – that is, unless S calls for another X to be placed where the ignored X is already placed. But in this case, P1 may simply place an X in some other random position on the board, the net effect of which will be that one X is in the position demanded by S, while another is in a random position, and becomes the new ignored piece, leaving the situation as before. Continuing in this way, S is, by hypothesis, guaranteed to produce a winning position (with an additional ignored X of no consequence). But then P2 has lost – contradicting the supposition that P2 had a guaranteed winning strategy. Such a winning strategy for P2, therefore, does not exist, and tic-tac-toe is either a forced win for P1 or a tie. (Further analysis shows it is in fact a tie.)
The same proof holds for any strong positional game.
== Chess ==
There is a class of chess positions called Zugzwang in which the player obligated to move would prefer to "pass" if this were allowed. Because of this, the strategy-stealing argument cannot be applied to chess. It is not currently known whether White or Black can force a win with optimal play, or if both players can force a draw. However, virtually all students of chess consider White's first move to be an advantage and White wins more often than black in high-level games.
== Go ==
In Go passing is allowed. When the starting position is symmetrical (empty board, neither player has any points), this means that the first player could steal the second player's winning strategy simply by giving up the first move. Since the 1930s, however, the second player is typically awarded some compensation points, which makes the starting position asymmetrical, and the strategy-stealing argument will no longer work.
An elementary strategy in the game is "mirror go", where the second player performs moves which are diagonally opposite those of this opponent. This approach may be defeated using ladder tactics, ko fights, or successfully competing for control of the board's central point.
== Constructivity ==
The strategy-stealing argument shows that the second player cannot win, by means of deriving a contradiction from any hypothetical winning strategy for the second player. The argument is commonly employed in games where there can be no draw, by means of the law of the excluded middle. However, it does not provide an explicit strategy for the first player, and because of this it has been called non-constructive. This raises the question of how to actually compute a winning strategy.
For games with a finite number of reachable positions, such as chomp, a winning strategy can be found by exhaustive search. However, this might be impractical if the number of positions is large.
In 2019, Greg Bodwin and Ofer Grossman proved that the problem of finding a winning strategy is PSPACE-hard in two kinds of games in which strategy-stealing arguments were used: the minimum poset game and the symmetric Maker-Maker game.
== References == | Wikipedia/Strategy_stealing_argument |
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed two-person zero-sum games, in which a participant's gains or losses are exactly balanced by the losses and gains of the other participant. In the 1950s, it was extended to the study of non zero-sum games, and was eventually applied to a wide range of behavioral relations. It is now an umbrella term for the science of rational decision making in humans, animals, and computers.
Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by Theory of Games and Economic Behavior (1944), co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.
Game theory was developed extensively in the 1950s, and was explicitly applied to evolution in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory in 1999, and fifteen game theorists have won the Nobel Prize in economics as of 2020, including most recently Paul Milgrom and Robert B. Wilson.
== History ==
=== Earliest results ===
In 1713, a letter attributed to Charles Waldegrave, an active Jacobite and uncle to British diplomat James Waldegrave, analyzed a game called "le her". Waldegrave provided a minimax mixed strategy solution to a two-person version of the card game, and the problem is now known as the Waldegrave problem.
In 1838, Antoine Augustin Cournot provided a model of competition in oligopolies. Though he did not refer to it as such, he presented a solution that is the Nash equilibrium of the game in his Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth). In 1883, Joseph Bertrand critiqued Cournot's model as unrealistic, providing an alternative model of price competition which would later be formalized by Francis Ysidro Edgeworth.
In 1913, Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels (On an Application of Set Theory to the Theory of the Game of Chess), which proved that the optimal chess strategy is strictly determined.
=== Foundation ===
The work of John von Neumann established game theory as its own independent field in the early-to-mid 20th century, with von Neumann publishing his paper On the Theory of Games of Strategy in 1928. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. Von Neumann's work in game theory culminated in his 1944 book Theory of Games and Economic Behavior, co-authored with Oskar Morgenstern. The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of money) as an independent discipline. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.
In his 1938 book Applications aux Jeux de Hasard and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix is symmetric and provided a solution to a non-trivial infinite game (known in English as Blotto game). Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games, a conjecture that was proved false by von Neumann.
In 1950, John Nash developed a criterion for mutual consistency of players' strategies known as the Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. Nash proved that every finite n-player, non-zero-sum (not just two-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium in mixed strategies.
Game theory experienced a flurry of activity in the 1950s, during which the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed. The 1950s also saw the first applications of game theory to philosophy and political science. The first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by mathematicians Merrill M. Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications to global nuclear strategy.
==== Prize-winning achievements ====
In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium. Later he would introduce trembling hand perfection as well. In 1994 Nash, Selten and Harsanyi became Economics Nobel Laureates for their contributions to economic game theory.
In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection and common knowledge were introduced and analyzed.
In 1994, John Nash was awarded the Nobel Memorial Prize in the Economic Sciences for his contribution to game theory. Nash's most famous contribution to game theory is the concept of the Nash equilibrium, which is a solution concept for non-cooperative games, published in 1951. A Nash equilibrium is a set of strategies, one for each player, such that no player can improve their payoff by unilaterally changing their strategy.
In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten, and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing equilibrium coarsening and correlated equilibria, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.
In 2007, Leonid Hurwicz, Eric Maskin, and Roger Myerson were awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory". Myerson's contributions include the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict. Hurwicz introduced and formalized the concept of incentive compatibility.
In 2012, Alvin E. Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design". In 2014, the Nobel went to game theorist Jean Tirole.
== Different types of games ==
=== Cooperative / non-cooperative ===
A game is cooperative if the players are able to form binding commitments externally enforced (e.g. through contract law). A game is non-cooperative if players cannot form alliances or if all agreements need to be self-enforcing (e.g. through credible threats).
Cooperative games are often analyzed through the framework of cooperative game theory, which focuses on predicting which coalitions will form, the joint actions that groups take, and the resulting collective payoffs. It is different from non-cooperative game theory which focuses on predicting individual players' actions and payoffs by analyzing Nash equilibria.
Cooperative game theory provides a high-level approach as it describes only the structure and payoffs of coalitions, whereas non-cooperative game theory also looks at how strategic interaction will affect the distribution of payoffs. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory (the converse does not hold) provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation.
=== Symmetric / asymmetric ===
A symmetric game is a game where each player earns the same payoff when making the same choice. In other words, the identity of the player does not change the resulting game facing the other player. Many of the commonly studied 2×2 games are symmetric. The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games.
The most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured in this section's graphic is asymmetric despite having identical strategy sets for both players.
=== Zero-sum / non-zero-sum ===
Zero-sum games (more generally, constant-sum games) are games in which choices by players can neither increase nor decrease the available resources. In zero-sum games, the total benefit goes to all players in a game, for every combination of strategies, and always adds to zero (more informally, a player benefits only at the equal expense of others). Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero-sum games include matching pennies and most classical board games including Go and chess.
Many games studied by game theorists (including the famed prisoner's dilemma) are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.
Furthermore, constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade. It is possible to transform any constant-sum game into a (possibly asymmetric) zero-sum game by adding a dummy player (often called "the board") whose losses compensate the players' net winnings.
=== Simultaneous / sequential ===
Simultaneous games are games where both players move simultaneously, or instead the later players are unaware of the earlier players' actions (making them effectively simultaneous). Sequential games (a type of dynamic games) are games where players do not make decisions simultaneously, and player's earlier actions affect the outcome and decisions of other players. This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while they do not know which of the other available actions the first player actually performed.
The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection.
In short, the differences between sequential and simultaneous games are as follows:
=== Perfect information and imperfect information ===
An important subset of sequential games consists of games of perfect information. A game with perfect information means that all players, at every move in the game, know the previous history of the game and the moves previously made by all other players. An imperfect information game is played when the players do not know all moves already made by the opponent such as a simultaneous move game. Examples of perfect-information games include tic-tac-toe, checkers, chess, and Go.
Many card games are games of imperfect information, such as poker and bridge. Perfect information is often confused with complete information, which is a similar concept pertaining to the common knowledge of each player's sequence, strategies, and payoffs throughout gameplay. Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken, whereas perfect information is knowledge of all aspects of the game and players. Games of incomplete information can be reduced, however, to games of imperfect information by introducing "moves by nature".
=== Bayesian game ===
One of the assumptions of the Nash equilibrium is that every player has correct beliefs about the actions of the other players. However, there are many situations in game theory where participants do not fully understand the characteristics of their opponents. Negotiators may be unaware of their opponent's valuation of the object of negotiation, companies may be unaware of their opponent's cost functions, combatants may be unaware of their opponent's strengths, and jurors may be unaware of their colleague's interpretation of the evidence at trial. In some cases, participants may know the character of their opponent well, but may not know how well their opponent knows his or her own character.
Bayesian game means a strategic game with incomplete information. For a strategic game, decision makers are players, and every player has a group of actions. A core part of the imperfect information specification is the set of states. Every state completely describes a collection of characteristics relevant to the player such as their preferences and details about them. There must be a state for every set of features that some player believes may exist.
For example, where Player 1 is unsure whether Player 2 would rather date her or get away from her, while Player 2 understands Player 1's preferences as before. To be specific, supposing that Player 1 believes that Player 2 wants to date her under a probability of 1/2 and get away from her under a probability of 1/2 (this evaluation comes from Player 1's experience probably: she faces players who want to date her half of the time in such a case and players who want to avoid her half of the time). Due to the probability involved, the analysis of this situation requires to understand the player's preference for the draw, even though people are only interested in pure strategic equilibrium.
=== Combinatorial games ===
Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and Go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve some particular problems and answer some general questions.
Games of perfect information have been studied in combinatorial game theory, which has developed novel representations, e.g. surreal numbers, as well as combinatorial and algebraic (and sometimes non-constructive) proof methods to solve games of certain types, including "loopy" games that may result in infinitely long sequences of moves. These methods address games with higher combinatorial complexity than those usually considered in traditional (or "economic") game theory. A typical game that has been solved this way is Hex. A related field of study, drawing from computational complexity theory, is game complexity, which is concerned with estimating the computational difficulty of finding optimal strategies.
Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures (like chess, go, or backgammon) for which no provable optimal strategies have been found. The practical solutions involve computational heuristics, like alpha–beta pruning or use of artificial neural networks trained by reinforcement learning, which make games more tractable in computing practice.
=== Discrete and continuous games ===
Much of game theory is concerned with finite, discrete games that have a finite number of players, moves, events, outcomes, etc. Many concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set. For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.
=== Differential games ===
Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations. The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory. In particular, there are two types of strategies: the open-loop strategies are found using the Pontryagin maximum principle while the closed-loop strategies are found using Bellman's Dynamic Programming method.
A particular case of differential games are the games with a random time horizon. In such games, the terminal time is a random variable with a given probability distribution function. Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.
=== Evolutionary game theory ===
Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted. In general, the evolution of strategies over time according to such rules is modeled as a Markov chain with a state variable such as the current strategy profile or how the game has been played in the recent past. Such rules may feature imitation, optimization, or survival of the fittest.
In biology, such models can represent evolution, in which offspring adopt their parents' strategies and parents who play more successful strategies (i.e. corresponding to higher payoffs) have a greater number of offspring. In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.
=== Stochastic outcomes (and relation to other fields) ===
Individual decision problems with stochastic outcomes are sometimes considered "one-player games". They may be modeled using similar tools within the related disciplines of decision theory, operations research, and areas of artificial intelligence, particularly AI planning (with uncertainty) and multi-agent system. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using Markov decision processes (MDP).
Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" ("moves by nature"). This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dice where required by the game.
For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the minimax solution is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution. The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely (but costly) events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen. (See Black swan theory for more discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses in investment banking.)
General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability (of moves by other players) have also been studied. The "gold standard" is considered to be partially observable stochastic game (POSG), but few realistic problems are computationally feasible in POSG representation.
=== Metagames ===
These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed. The theory of metagames is related to mechanism design theory.
The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard, whereby a situation is framed as a strategic game in which stakeholders try to realize their objectives by means of the options available to them. Subsequent developments have led to the formulation of confrontation analysis.
=== Mean field game theory ===
Mean field game theory is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal, in the engineering literature by Peter E. Caines, and by mathematicians Pierre-Louis Lions and Jean-Michel Lasry.
== Representation of games ==
The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: the players of the game, the information and actions available to each player at each decision point, and the payoffs for each outcome. (Eric Rasmusen refers to these four "essential elements" by the acronym "PAPI".) A game theorist typically uses these elements, along with a solution concept of their choosing, to deduce a set of equilibrium strategies for each player such that, when these strategies are employed, no player can profit by unilaterally deviating from their strategy. These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability.
Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.
=== Extensive form ===
The extensive form can be used to formalize games with a time sequencing of moves. Extensive form games can be visualized using game trees (as pictured here). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a decision tree. To solve any extensive form game, backward induction must be used. It involves working backward up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached.
The game pictured consists of two players. The way this particular game is structured (i.e., with sequential decision making and perfect information), Player 1 "moves" first by choosing either F or U (fair or unfair). Next in the sequence, Player 2, who has now observed Player 1's move, can choose to play either A or R (accept or reject). Once Player 2 has made their choice, the game is considered finished and each player gets their respective payoff, represented in the image as two numbers, where the first number represents Player 1's payoff, and the second number represents Player 2's payoff. Suppose that Player 1 chooses U and then Player 2 chooses A: Player 1 then gets a payoff of "eight" (which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players) and Player 2 gets a payoff of "two".
The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set (i.e. the players do not know at which point they are), or a closed line is drawn around them. (See example in the imperfect information section.)
=== Normal form ===
The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.
When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.
Every extensive-form game has an equivalent normal-form game, however, the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.
=== Characteristic function form ===
In cooperative game theory the characteristic function lists the payoff of each coalition. The origin of this formulation is in John von Neumann and Oskar Morgenstern's book.
Formally, a characteristic function is a function
v
:
2
N
→
R
{\displaystyle v:2^{N}\to \mathbb {R} }
from the set of all possible coalitions of players to a set of payments, and also satisfies
v
(
∅
)
=
0
{\displaystyle v(\emptyset )=0}
. The function describes how much collective payoff a set of players can gain by forming a coalition.
=== Alternative game representations ===
Alternative game representation forms are used for some subclasses of games or adjusted to the needs of interdisciplinary research. In addition to classical game representations, some of the alternative representations also encode time related aspects.
== General and applied uses ==
As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game-theoretic analysis was by Antoine Augustin Cournot in 1838 with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.
Although pre-twentieth-century naturalists such as Charles Darwin made game-theoretic kinds of statements, the use of game-theoretic analysis in biology began with Ronald Fisher's studies of animal behavior during the 1930s. This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his 1982 book Evolution and the Theory of Games.
In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior. In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behavior. Game-theoretic approaches have also been suggested in the philosophy of language and philosophy of science. Game-theoretic arguments of this type can be found as far back as Plato. An alternative version of game theory, called chemical game theory, represents the player's choices as metaphorical chemical reactant molecules called "knowlecules". Chemical game theory then calculates the outcomes as equilibrium solutions to a system of chemical reactions.
=== Description and modeling ===
The primary use of game theory is to describe and model how human populations behave. Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations. Game theorists usually assume players act rationally, but in practice, human rationality and/or behavior often deviates from the model of rationality as used in game theory. Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists. However, empirical work has shown that in some classic games, such as the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.
Some game theorists, following the work of John Maynard Smith and George R. Price, have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).
=== Prescriptive or normative analysis ===
Some scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players – provided they are in (the same) Nash equilibrium – playing a strategy that is part of a Nash equilibrium seems appropriate. This normative use of game theory has also come under criticism.
=== Economics ===
Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents. Applications include a wide array of economic phenomena and approaches, such as auctions, bargaining, mergers and acquisitions pricing, fair division, duopolies, oligopolies, social network formation, agent-based computational economics, general equilibrium, mechanism design, and voting systems; and across such broad areas as experimental economics, behavioral economics, information economics, industrial organization, and political economy.
This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria". A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.
The payoffs of the game are generally taken to represent the utility of individual players.
A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Economists and business professors suggest two primary uses (noted above): descriptive and prescriptive.
==== Managerial economics ====
Game theory also has an extensive use in a specific branch or stream of economics – Managerial Economics. One important usage of it in the field of managerial economics is in analyzing strategic interactions between firms. For example, firms may be competing in a market with limited resources, and game theory can help managers understand how their decisions impact their competitors and the overall market outcomes. Game theory can also be used to analyze cooperation between firms, such as in forming strategic alliances or joint ventures. Another use of game theory in managerial economics is in analyzing pricing strategies. For example, firms may use game theory to determine the optimal pricing strategy based on how they expect their competitors to respond to their pricing decisions. Overall, game theory serves as a useful tool for analyzing strategic interactions and decision making in the context of managerial economics.
=== Business ===
The Chartered Institute of Procurement & Supply (CIPS) promotes knowledge and use of game theory within the context of business procurement. CIPS and TWS Partners have conducted a series of surveys designed to explore the understanding, awareness and application of game theory among procurement professionals. Some of the main findings in their third annual survey (2019) include:
application of game theory to procurement activity has increased – at the time it was at 19% across all survey respondents
65% of participants predict that use of game theory applications will grow
70% of respondents say that they have "only a basic or a below basic understanding" of game theory
20% of participants had undertaken on-the-job training in game theory
50% of respondents said that new or improved software solutions were desirable
90% of respondents said that they do not have the software they need for their work.
=== Project management ===
Sensible decision-making is critical for the success of projects. In project management, game theory is used to model the decision-making process of players, such as investors, project managers, contractors, sub-contractors, governments and customers. Quite often, these players have competing interests, and sometimes their interests are directly detrimental to other players, making project management scenarios well-suited to be modeled by game theory.
Piraveenan (2019) in his review provides several examples where game theory is used to model project management scenarios. For instance, an investor typically has several investment options, and each option will likely result in a different project, and thus one of the investment options has to be chosen before the project charter can be produced. Similarly, any large project involving subcontractors, for instance, a construction project, has a complex interplay between the main contractor (the project manager) and subcontractors, or among the subcontractors themselves, which typically has several decision points. For example, if there is an ambiguity in the contract between the contractor and subcontractor, each must decide how hard to push their case without jeopardizing the whole project, and thus their own stake in it. Similarly, when projects from competing organizations are launched, the marketing personnel have to decide what is the best timing and strategy to market the project, or its resultant product or service, so that it can gain maximum traction in the face of competition. In each of these scenarios, the required decisions depend on the decisions of other players who, in some way, have competing interests to the interests of the decision-maker, and thus can ideally be modeled using game theory.
Piraveenan summarizes that two-player games are predominantly used to model project management scenarios, and based on the identity of these players, five distinct types of games are used in project management.
Government-sector–private-sector games (games that model public–private partnerships)
Contractor–contractor games
Contractor–subcontractor games
Subcontractor–subcontractor games
Games involving other players
In terms of types of games, both cooperative as well as non-cooperative, normal-form as well as extensive-form, and zero-sum as well as non-zero-sum are used to model various project management scenarios.
=== Political science ===
The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.
Early examples of game theory applied to political science are provided by Anthony Downs. In his 1957 book An Economic Theory of Democracy, he applies the Hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence. Game theory was applied in 1962 to the Cuban Missile Crisis during the presidency of John F. Kennedy.
It has also been proposed that game theory explains the stability of any form of political government. Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects. Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king (or other established government) as the person whose orders will be followed. Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime. Thus, in a process that can be modeled by variants of the prisoner's dilemma, during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively.
A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy.
However, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting. War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting. Moreover, war may arise because of commitment problems: if two countries wish to settle a dispute via peaceful means, but each wishes to go back on the terms of that settlement, they may have no choice but to resort to warfare. Finally, war may result from issue indivisibilities.
Game theory could also help predict a nation's responses when there is a new rule or law to be applied to that nation. One example is Peter John Wood's (2013) research looking into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce greenhouse gas emissions. However, he concluded that this idea could not work because it would create a prisoner's dilemma for the nations.
=== Defence science and technology ===
Game theory has been used extensively to model decision-making scenarios relevant to defence applications. Most studies that has applied game theory in defence settings are concerned with Command and Control Warfare, and can be further classified into studies dealing with (i) Resource Allocation Warfare (ii) Information Warfare (iii) Weapons Control Warfare, and (iv) Adversary Monitoring Warfare. Many of the problems studied are concerned with sensing and tracking, for example a surface ship trying to track a hostile submarine and the submarine trying to evade being tracked, and the interdependent decision making that takes place with regards to bearing, speed, and the sensor technology activated by both vessels.
The tool, for example, automates the transformation of public vulnerability data into models, allowing defenders to synthesize optimal defence strategies through Stackelberg equilibrium analysis. This approach enhances cyber resilience by enabling defenders to anticipate and counteract attackers’ best responses, making game theory increasingly relevant in adversarial cybersecurity environments.
Ho et al. provide a broad summary of game theory applications in defence, highlighting its advantages and limitations across both physical and cyber domains.
=== Biology ===
Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces. The best-known equilibrium in biology is known as the evolutionarily stable strategy (ESS), first introduced in (Maynard Smith & Price 1973). Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.
In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. (Fisher 1930) suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.
Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication. The analysis of signaling games and other communication games has provided insight into the evolution of communication among animals. For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization. Ants have also been shown to exhibit feed-forward behavior akin to fashion (see Paul Ormerod's Butterfly Economics).
Biologists have used the game of chicken to analyze fighting behavior and territoriality.
According to Maynard Smith, in the preface to Evolution and the Theory of Games, "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed". Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature.
One such phenomenon is known as biological altruism. This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself. This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness. Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival. All of these actions increase the overall fitness of a group, but occur at a cost to the individual.
Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives. Hamilton's rule explains the evolutionary rationale behind this selection with the equation c < b × r, where the cost c to the altruist must be less than the benefit b to the recipient multiplied by the coefficient of relatedness r. The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles. This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on through survival of its offspring, can forgo the option of having offspring itself because the same number of alleles are passed on. For example, helping a sibling (in diploid animals) has a coefficient of 1⁄2, because (on average) an individual shares half of the alleles in its sibling's offspring. Ensuring that enough of a sibling's offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring. The coefficient values depend heavily on the scope of the playing field; for example if the choice of whom to favor includes all genetic living things, not just all relatives, we assume the discrepancy between all humans only accounts for approximately 1% of the diversity in the playing field, a coefficient that was 1⁄2 in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g. epigenetics, religion, science, etc.) persisted through time the playing field becomes larger still, and the discrepancies smaller.
=== Computer science and logic ===
Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations. Also, game theory provides a theoretical basis to the field of multi-agent systems.
Separately, game theory has played a role in online algorithms; in particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games. Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms, especially online algorithms.
The emergence of the Internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets. Algorithmic game theory and within it algorithmic mechanism design combine computational algorithm design and analysis of complex systems with economic theory.
Game theory has multiple applications in the field of artificial intelligence and machine learning. It is often used in developing autonomous systems that can make complex decisions in uncertain environment. Some other areas of application of game theory in AI/ML context are as follows - multi-agent system formation, reinforcement learning, mechanism design etc. By using game theory to model the behavior of other agents and anticipate their actions, AI/ML systems can make better decisions and operate more effectively.
=== Philosophy ===
Game theory has been put to several uses in philosophy. Responding to two papers by W.V.O. Quine (1960, 1967), Lewis (1969) used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games. In addition, he first suggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued by several philosophers since Lewis. Following Lewis (1969) game-theoretic account of conventions, Edna Ullmann-Margalit (1977) and Bicchieri (2006) have developed theories of social norms that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game.
Game theory has also challenged philosophers to think in terms of interactive epistemology: what it means for a collective to have common beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from the interactions of agents. Philosophers who have worked in this area include Bicchieri (1989, 1993), Skyrms (1990), and Stalnaker (1999).
The synthesis of game theory with ethics was championed by R. B. Braithwaite. The hope was that rigorous mathematical analysis of game theory might help formalize the more imprecise philosophical discussions. However, this expectation was only materialized to a limited extent.
In ethics, some (most notably David Gauthier, Gregory Kavka, and Jean Hampton) authors have attempted to pursue Thomas Hobbes' project of deriving morality from self-interest. Since games like the prisoner's dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project. This general strategy is a component of the general social contract view in political philosophy (for examples, see Gauthier (1986) and Kavka (1986)).
Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. These authors look at several games including the prisoner's dilemma, stag hunt, and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality (see, e.g., Skyrms (1996, 2004) and Sober and Wilson (1998)).
=== Epidemiology ===
Since the decision to take a vaccine for a particular disease is often made by individuals, who may consider a range of factors and parameters in making this decision (such as the incidence and prevalence of the disease, perceived and real risks associated with contracting the disease, mortality rate, perceived and real risks associated with vaccination, and financial cost of vaccination), game theory has been used to model and predict vaccination uptake in a society.
== Well known examples of games ==
=== Prisoner's dilemma ===
William Poundstone described the game in his 1993 book Prisoner's Dilemma:
Two members of a criminal gang, A and B, are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communication with their partner. The principal charge would lead to a sentence of ten years in prison; however, the police do not have the evidence for a conviction. They plan to sentence both to two years in prison on a lesser charge but offer each prisoner a Faustian bargain: If one of them confesses to the crime of the principal charge, betraying the other, they will be pardoned and free to leave while the other must serve the entirety of the sentence instead of just two years for the lesser charge.
The dominant strategy (and therefore the best response to any possible opponent strategy), is to betray the other, which aligns with the sure-thing principle. However, both prisoners staying silent would yield a greater reward for both of them than mutual betrayal.
=== Battle of the sexes ===
The "battle of the sexes" is a term used to describe the perceived conflict between men and women in various areas of life, such as relationships, careers, and social roles. This conflict is often portrayed in popular culture, such as movies and television shows, as a humorous or dramatic competition between the genders. This conflict can be depicted in a game theory framework. This is an example of non-cooperative games.
An example of the "battle of the sexes" can be seen in the portrayal of relationships in popular media, where men and women are often depicted as being fundamentally different and in conflict with each other. For instance, in some romantic comedies, the male and female protagonists are shown as having opposing views on love and relationships, and they have to overcome these differences in order to be together.
In this game, there are two pure strategy Nash equilibria: one where both the players choose the same strategy and the other where the players choose different options. If the game is played in mixed strategies, where each player chooses their strategy randomly, then there is an infinite number of Nash equilibria. However, in the context of the "battle of the sexes" game, the assumption is usually made that the game is played in pure strategies.
=== Ultimatum game ===
The ultimatum game is a game that has become a popular instrument of economic experiments. An early description is by Nobel laureate John Harsanyi in 1961.
One player, the proposer, is endowed with a sum of money. The proposer is tasked with splitting it with another player, the responder (who knows what the total sum is). Once the proposer communicates his decision, the responder may accept it or reject it. If the responder accepts, the money is split per the proposal; if the responder rejects, both players receive nothing. Both players know in advance the consequences of the responder accepting or rejecting the offer. The game demonstrates how social acceptance, fairness, and generosity influence the players decisions.
Ultimatum game has a variant, that is the dictator game. They are mostly identical, except in dictator game the responder has no power to reject the proposer's offer.
=== Trust game ===
The Trust Game is an experiment designed to measure trust in economic decisions. It is also called "the investment game" and is designed to investigate trust and demonstrate its importance rather than "rationality" of self-interest. The game was designed by Berg Joyce, John Dickhaut and Kevin McCabe in 1995.
In the game, one player (the investor) is given a sum of money and must decide how much of it to give to another player (the trustee). The amount given is then tripled by the experimenter. The trustee then decides how much of the tripled amount to return to the investor. If the recipient is completely self interested, then he/she should return nothing. However that is not true as the experiment conduct. The outcome suggest that people are willing to place a trust, by risking some amount of money, in the belief that there would be reciprocity.
=== Cournot Competition ===
The Cournot competition model involves players choosing quantity of a homogenous product to produce independently and simultaneously, where marginal cost can be different for each firm and the firm's payoff is profit. The production costs are public information and the firm aims to find their profit-maximizing quantity based on what they believe the other firm will produce and behave like monopolies. In this game firms want to produce at the monopoly quantity but there is a high incentive to deviate and produce more, which decreases the market-clearing price. For example, firms may be tempted to deviate from the monopoly quantity if there is a low monopoly quantity and high price, with the aim of increasing production to maximize profit. However this option does not provide the highest payoff, as a firm's ability to maximize profits depends on its market share and the elasticity of the market demand. The Cournot equilibrium is reached when each firm operates on their reaction function with no incentive to deviate, as they have the best response based on the other firms output. Within the game, firms reach the Nash equilibrium when the Cournot equilibrium is achieved.
=== Bertrand Competition ===
The Bertrand competition assumes homogenous products and a constant marginal cost and players choose the prices. The equilibrium of price competition is where the price is equal to marginal costs, assuming complete information about the competitors' costs. Therefore, the firms have an incentive to deviate from the equilibrium because a homogenous product with a lower price will gain all of the market share, known as a cost advantage.
== In popular culture ==
Based on the 1998 book by Sylvia Nasar, the life story of game theorist and mathematician John Nash was turned into the 2001 biopic A Beautiful Mind, starring Russell Crowe as Nash.
The 1959 military science fiction novel Starship Troopers by Robert A. Heinlein mentioned "games theory" and "theory of games". In the 1997 film of the same name, the character Carl Jenkins referred to his military intelligence assignment as being assigned to "games and theory".
The 1964 film Dr. Strangelove satirizes game theoretic ideas about deterrence theory. For example, nuclear deterrence depends on the threat to retaliate catastrophically if a nuclear attack is detected. A game theorist might argue that such threats can fail to be credible, in the sense that they can lead to subgame imperfect equilibria. The movie takes this idea one step further, with the Soviet Union irrevocably committing to a catastrophic nuclear response without making the threat public.
The 1980s power pop band Game Theory was founded by singer/songwriter Scott Miller, who described the band's name as alluding to "the study of calculating the most appropriate action given an adversary ... to give yourself the minimum amount of failure".
Liar Game, a 2005 Japanese manga and 2007 television series, presents the main characters in each episode with a game or problem that is typically drawn from game theory, as demonstrated by the strategies applied by the characters.
The 1974 novel Spy Story by Len Deighton explores elements of game theory in regard to cold war army exercises.
The 2008 novel The Dark Forest by Liu Cixin explores the relationship between extraterrestrial life, humanity, and game theory.
Joker, the prime antagonist in the 2008 film The Dark Knight presents game theory concepts—notably the prisoner's dilemma in a scene where he asks passengers in two different ferries to bomb the other one to save their own.
In the 2018 film Crazy Rich Asians, the female lead Rachel Chu is a professor of economics and game theory at New York University. At the beginning of the film she is seen in her NYU classroom playing a game of poker with her teaching assistant and wins the game by bluffing; then in the climax of the film, she plays a game of mahjong with her boyfriend's disapproving mother Eleanor, losing the game to Eleanor on purpose but winning her approval as a result.
In the 2017 film Molly's Game, Brad, an inexperienced poker player, makes an irrational betting decision without realizing and causes his opponent Harlan to deviate from his Nash Equilibrium strategy, resulting in a significant loss when Harlan loses the hand.
== See also ==
Applied ethics – Practical application of moral considerations
Bandwidth-sharing game – Type of resource allocation game
Chainstore paradox – Game theory paradox
Collective intentionality – Intentionality that occurs when two or more individuals undertake a task together
Core (game theory) – term in game theoryPages displaying wikidata descriptions as a fallback
Glossary of game theory
Intra-household bargaining – negotiations between members of a household to reach decisionsPages displaying wikidata descriptions as a fallback
Kingmaker scenario – Endgame situation in game theory
Law and economics – Application of economic theory to analysis of legal systems
Mutual assured destruction – Doctrine of military strategy
Outline of artificial intelligence – Overview of and topical guide to artificial intelligence
Parrondo's paradox – Paradox in game theory
Precautionary principle – Risk management strategy
Quantum refereed game
Risk management – Identification, evaluation and control of risks
Self-confirming equilibrium
Tragedy of the commons – Self-interests causing depletion of a shared resource
Traveler's dilemma – non-zero-sum game thought experimentPages displaying wikidata descriptions as a fallback
Wilson doctrine (economics) – Argument in economic theory
Compositional game theory
Lists
List of cognitive biases
List of emerging technologies
List of games in game theory
== Notes ==
== References ==
== Sources ==
Ben-David, S.; Borodin, A.; Karp, R.; Tardos, G.; Wigderson, A. (January 1994). "On the power of randomization in on-line algorithms". Algorithmica. 11 (1): 2–14. doi:10.1007/BF01294260. S2CID 26771869.
Downs, Anthony (1957), An Economic theory of Democracy, New York: Harper
Fisher, Sir Ronald Aylmer (1930). The Genetical Theory of Natural Selection. Clarendon Press.
Gauthier, David (1986), Morals by agreement, Oxford University Press, ISBN 978-0-19-824992-4
Grim, Patrick; Kokalis, Trina; Alai-Tafti, Ali; Kilb, Nicholas; St Denis, Paul (2004), "Making meaning happen", Journal of Experimental & Theoretical Artificial Intelligence, 16 (4): 209–243, doi:10.1080/09528130412331294715, S2CID 5737352
Harper, David; Maynard Smith, John (2003), Animal signals, Oxford University Press, ISBN 978-0-19-852685-8
Howard, Nigel (1971), Paradoxes of Rationality: Games, Metagames, and Political Behavior, Cambridge, MA: The MIT Press, ISBN 978-0-262-58237-7
Kavka, Gregory S. (1986). Hobbesian Moral and Political Theory. Princeton University Press. ISBN 978-0-691-02765-4.
Lewis, David (1969), Convention: A Philosophical Study, ISBN 978-0-631-23257-5 (2002 edition)
Maynard Smith, John; Price, George R. (1973), "The logic of animal conflict", Nature, 246 (5427): 15–18, Bibcode:1973Natur.246...15S, doi:10.1038/246015a0, S2CID 4224989
Osborne, Martin J.; Rubinstein, Ariel (1994), A course in game theory, MIT Press, ISBN 978-0-262-65040-3. A modern introduction at the graduate level.
Poundstone, William (1993). Prisoner's Dilemma (1st Anchor Books ed.). New York: Anchor. ISBN 0-385-41580-X.
Quine, W.v.O (1967), "Truth by Convention", Philosophica Essays for A.N. Whitehead, Russel and Russel Publishers, ISBN 978-0-8462-0970-6
Quine, W.v.O (1960), "Carnap and Logical Truth", Synthese, 12 (4): 350–374, doi:10.1007/BF00485423, S2CID 46979744
Skyrms, Brian (1996), Evolution of the social contract, Cambridge University Press, ISBN 978-0-521-55583-8
Skyrms, Brian (2004), The stag hunt and the evolution of social structure, Cambridge University Press, ISBN 978-0-521-53392-8
Sober, Elliott; Wilson, David Sloan (1998), Unto others: the evolution and psychology of unselfish behavior, Harvard University Press, ISBN 978-0-674-93047-6
Webb, James N. (2007), Game theory: decisions, interaction and evolution, Undergraduate mathematics, Springer, ISBN 978-1-84628-423-6 Consistent treatment of game types usually claimed by different applied fields, e.g. Markov decision processes.
== Further reading ==
=== Textbooks and general literature ===
Aumann, Robert J (1987), "game theory", The New Palgrave: A Dictionary of Economics, vol. 2, pp. 460–82.
Camerer, Colin (2003), "Introduction", Behavioral Game Theory: Experiments in Strategic Interaction, Russell Sage Foundation, pp. 1–25, ISBN 978-0-691-09039-9, archived from the original on 14 May 2011, retrieved 9 February 2011, Description.
Dutta, Prajit K. (1999), Strategies and games: theory and practice, MIT Press, ISBN 978-0-262-04169-0. Suitable for undergraduate and business students.
Fernandez, L F.; Bierman, H S. (1998), Game theory with economic applications, Addison-Wesley, ISBN 978-0-201-84758-1. Suitable for upper-level undergraduates.
Gaffal, Margit; Padilla Gálvez, Jesús (2014). Dynamics of Rational Negotiation: Game Theory, Language Games and Forms of Life. Springer.
Gibbons, Robert D. (1992), Game theory for applied economists, Princeton University Press, ISBN 978-0-691-00395-5. Suitable for advanced undergraduates.
Published in Europe as Gibbons, Robert (2001), A Primer in Game Theory, London: Harvester Wheatsheaf, ISBN 978-0-7450-1159-2.
Gintis, Herbert (2000), Game theory evolving: a problem-centered introduction to modeling strategic behavior, Princeton University Press, ISBN 978-0-691-00943-8
Green, Jerry R.; Mas-Colell, Andreu; Whinston, Michael D. (1995), Microeconomic theory, Oxford University Press, ISBN 978-0-19-507340-9. Presents game theory in formal way suitable for graduate level.
Joseph E. Harrington (2008) Games, strategies, and decision making, Worth, ISBN 0-7167-6630-2. Textbook suitable for undergraduates in applied fields; numerous examples, fewer formalisms in concept presentation.
Isaacs, Rufus (1999), Differential Games: A Mathematical Theory With Applications to Warfare and Pursuit, Control and Optimization, New York: Dover Publications, ISBN 978-0-486-40682-4
Michael Maschler; Eilon Solan; Shmuel Zamir (2013), Game Theory, Cambridge University Press, ISBN 978-1-108-49345-1. Undergraduate textbook.
Miller, James H. (2003), Game theory at work: how to use game theory to outthink and outmaneuver your competition, New York: McGraw-Hill, ISBN 978-0-07-140020-6. Suitable for a general audience.
Shoham, Yoav; Leyton-Brown, Kevin (2009), Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations, New York: Cambridge University Press, ISBN 978-0-521-89943-7, retrieved 8 March 2016
Watson, Joel (2013), Strategy: An Introduction to Game Theory (3rd edition), New York: W.W. Norton and Co., ISBN 978-0-393-91838-0. A leading textbook at the advanced undergraduate level.
McCain, Roger A. (2010). Game Theory: A Nontechnical Introduction to the Analysis of Strategy. World Scientific. ISBN 978-981-4289-65-8.
=== Historically important texts ===
Aumann, R. J.; Shapley, L. S. (1974), Values of Non-Atomic Games, Princeton University Press
Cournot, A. Augustin (1838), "Recherches sur les principles mathematiques de la théorie des richesses", Libraire des Sciences Politiques et Sociales
Edgeworth, Francis Y. (1881), Mathematical Psychics, London: Kegan Paul
Farquharson, Robin (1969), Theory of Voting, Blackwell (Yale U.P. in the U.S.), ISBN 978-0-631-12460-3
Luce, R. Duncan; Raiffa, Howard (1957), Games and decisions: introduction and critical survey, New York: Wiley
reprinted edition: R. Duncan Luce; Howard Raiffa (1989), Games and decisions: introduction and critical survey, New York: Dover Publications, ISBN 978-0-486-65943-5
Maynard Smith, John (1982), Evolution and the theory of games, Cambridge University Press, ISBN 978-0-521-28884-2
Nash, John (1950), "Equilibrium points in n-person games", Proceedings of the National Academy of Sciences of the United States of America, 36 (1): 48–49, Bibcode:1950PNAS...36...48N, doi:10.1073/pnas.36.1.48, PMC 1063129, PMID 16588946
Shapley, L.S. (1953), A Value for n-person Games, In: Contributions to the Theory of Games volume II, H. W. Kuhn and A. W. Tucker (eds.)
Shapley, L. S. (October 1953). "Stochastic Games". Proceedings of the National Academy of Sciences. 39 (10): 1095–1100. Bibcode:1953PNAS...39.1095S. doi:10.1073/pnas.39.10.1095. PMC 1063912. PMID 16589380.
von Neumann, John (1928), "Zur Theorie der Gesellschaftsspiele", Mathematische Annalen, 100 (1): 295–320, doi:10.1007/bf01448847, S2CID 122961988 English translation: "On the Theory of Games of Strategy," in A. W. Tucker and R. D. Luce, ed. (1959), Contributions to the Theory of Games, v. 4, p. 42. Princeton University Press.
von Neumann, John; Morgenstern, Oskar (1944), "Theory of games and economic behavior", Nature, 157 (3981), Princeton University Press: 172, Bibcode:1946Natur.157..172R, doi:10.1038/157172a0, S2CID 29754824
Zermelo, Ernst (1913), "Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels", Proceedings of the Fifth International Congress of Mathematicians, 2: 501–4
=== Other material ===
Allan Gibbard, "Manipulation of voting schemes: a general result", Econometrica, Vol. 41, No. 4 (1973), pp. 587–601.
McDonald, John (1950–1996), Strategy in Poker, Business & War, W. W. Norton, ISBN 978-0-393-31457-1 {{citation}}: ISBN / Date incompatibility (help). A layman's introduction.
Papayoanou, Paul (2010), Game Theory for Business: A Primer in Strategic Gaming, Probabilistic, ISBN 978-0-9647938-7-3.
Satterthwaite, Mark Allen (April 1975). "Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions" (PDF). Journal of Economic Theory. 10 (2): 187–217. doi:10.1016/0022-0531(75)90050-2.
Siegfried, Tom (2006), A Beautiful Math, Joseph Henry Press, ISBN 978-0-309-10192-9
Skyrms, Brian (1990), The Dynamics of Rational Deliberation, Harvard University Press, ISBN 978-0-674-21885-7
Thrall, Robert M.; Lucas, William F. (1963), "
n
{\displaystyle n}
-person games in partition function form", Naval Research Logistics Quarterly, 10 (4): 281–298, doi:10.1002/nav.3800100126
Dolev, Shlomi; Panagopoulou, Panagiota N.; Rabie, Mikaël; Schiller, Elad M.; Spirakis, Paul G. (2011). "Rationality authority for provable rational behavior". Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing. pp. 289–290. doi:10.1145/1993806.1993858. ISBN 978-1-4503-0719-2.
Chastain, Erick; Livnat, Adi; Papadimitriou, Christos; Vazirani, Umesh (June 2014), "Algorithms, games, and evolution", Proceedings of the National Academy of Sciences of the United States of America, 111 (29): 10620–10623, Bibcode:2014PNAS..11110620C, doi:10.1073/pnas.1406556111, PMC 4115542, PMID 24979793
== External links ==
James Miller (2015): Introductory Game Theory Videos.
"Games, theory of", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Paul Walker: History of Game Theory Page.
David Levine: Game Theory. Papers, Lecture Notes and much more stuff.
Alvin Roth:"Game Theory and Experimental Economics page". Archived from the original on 15 August 2000. Retrieved 13 September 2003. — Comprehensive list of links to game theory information on the Web
Adam Kalai: Game Theory and Computer Science — Lecture notes on Game Theory and Computer Science
Mike Shor: GameTheory.net — Lecture notes, interactive illustrations and other information.
Jim Ratliff's Graduate Course in Game Theory (lecture notes).
Don Ross: Review Of Game Theory in the Stanford Encyclopedia of Philosophy.
Bruno Verbeek and Christopher Morris: Game Theory and Ethics
Elmer G. Wiens: Game Theory — Introduction, worked examples, play online two-person zero-sum games.
Marek M. Kaminski: Game Theory and Politics Archived 20 October 2006 at the Wayback Machine — Syllabuses and lecture notes for game theory and political science.
Websites on game theory and social interactions
Kesten Green's Conflict Forecasting at the Wayback Machine (archived 11 April 2011) — See Papers for evidence on the accuracy of forecasts from game theory and other methods Archived 15 September 2019 at the Wayback Machine.
McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2007) Gambit: Software Tools for Game Theory.
Benjamin Polak: Open Course on Game Theory at Yale Archived 3 August 2010 at the Wayback Machine videos of the course
Benjamin Moritz, Bernhard Könsgen, Danny Bures, Ronni Wiersch, (2007) Spieltheorie-Software.de: An application for Game Theory implemented in JAVA.
Antonin Kucera: Stochastic Two-Player Games.
Yu-Chi Ho: What is Mathematical Game Theory; What is Mathematical Game Theory (#2); What is Mathematical Game Theory (#3); What is Mathematical Game Theory (#4)-Many person game theory; What is Mathematical Game Theory ?( #5) – Finale, summing up, and my own view | Wikipedia/Strategic_interaction |
Strategic studies is an interdisciplinary academic field centered on the study of peace and conflict strategies, often devoting special attention to the relationship between military history, international politics, geostrategy, international diplomacy, international economics, and military power. In the scope of the studies are also subjects such as the role of intelligence, diplomacy, and international cooperation for security and defense. The subject is normally taught at the post-graduate academic or professional, usually strategic-political and strategic-military levels.
Strategic studies is closely associated with grand strategy, which a state's strategy of how means (military and nonmilitary) can be used to advance and achieve national interests in the long-term.
The academic foundations of the subject began with analysis of texts such as Sun Tzu’s Art of War and Carl von Clausewitz’s On War. In recent times, the major conflicts of the nineteenth century and the two World Wars have spurred strategic thinkers such as Mahan, Corbett, Giulio Douhet, Liddell Hart and, later, André Beaufre. The Cold War with its danger of degenerating into a nuclear war produced an expansion of the discipline, with authors like Bernard Brodie, Michael Howard, Raymond Aron, Lucien Poirier, Lawrence Freedman, Colin Gray, and many others.
== See also ==
Grand strategy
Combat effectiveness
U.S. Army Strategist
== References == | Wikipedia/Strategic_studies |
In cooperative game theory, the core is the set of feasible allocations or imputations where no coalition of agents can benefit by breaking away from the grand coalition. One can think of the core corresponding to situations where it is possible to sustain cooperation among all agents. A coalition is said to improve upon or block a feasible allocation if the members of that coalition can generate more value among themselves than they are allocated in the original allocation. As such, that coalition is not incentivized to stay with the grand coalition.
An allocation is said to be in the core of a game if there is no coalition that can improve upon it. The core is then the set of all feasible allocations.
== Origin ==
The idea of the core already appeared in the writings of Edgeworth (1881), at the time referred to as the contract curve. Even though von Neumann and Morgenstern considered it an interesting concept, they only worked with zero-sum games where the core is always empty. The modern definition of the core is due to Gillies.
== Definition ==
Consider a transferable utility cooperative game
(
N
,
v
)
{\displaystyle (N,v)}
where
N
{\displaystyle N}
denotes the set of players and
v
{\displaystyle v}
is the characteristic function. An imputation
x
∈
R
N
{\displaystyle x\in \mathbb {R} ^{N}}
is dominated by another imputation
y
{\displaystyle y}
if there exists a coalition
C
{\displaystyle C}
, such that each player in
C
{\displaystyle C}
weakly-prefers
y
{\displaystyle y}
(
x
i
≤
y
i
{\displaystyle x_{i}\leq y_{i}}
for all
i
∈
C
{\displaystyle i\in C}
) and there exists
i
∈
C
{\displaystyle i\in C}
that strictly-prefers
y
{\displaystyle y}
(
x
i
<
y
i
{\displaystyle x_{i}<y_{i}}
), and
C
{\displaystyle C}
can enforce
y
{\displaystyle y}
by threatening to leave the grand coalition to form
C
{\displaystyle C}
(
∑
i
∈
C
y
i
≤
v
(
C
)
{\displaystyle \sum _{i\in C}y_{i}\leq v(C)}
). The core is the set of imputations that are not dominated by any other imputation.
=== Weak core ===
An imputation
x
∈
R
N
{\displaystyle x\in \mathbb {R} ^{N}}
is strongly-dominated by another imputation
y
{\displaystyle y}
if there exists a coalition
C
{\displaystyle C}
, such that each player in
C
{\displaystyle C}
strictly-prefers
y
{\displaystyle y}
(
x
i
<
y
i
{\displaystyle x_{i}<y_{i}}
for all
i
∈
C
{\displaystyle i\in C}
). The weak core is the set of imputations that are not strongly-dominated.
== Properties ==
Another definition, equivalent to the one above, states that the core is a set of payoff allocations
x
∈
R
N
{\displaystyle x\in \mathbb {R} ^{N}}
satisfying
Efficiency:
∑
i
∈
N
x
i
=
v
(
N
)
{\displaystyle \sum _{i\in N}x_{i}=v(N)}
,
Coalitional rationality:
∑
i
∈
C
x
i
≥
v
(
C
)
{\displaystyle \sum _{i\in C}x_{i}\geq v(C)}
for all subsets (coalitions)
C
⊆
N
{\displaystyle C\subseteq N}
.
The core is always well-defined, but can be empty.
The core is a set which satisfies a system of weak linear inequalities. Hence the core is closed and convex.
The Bondareva–Shapley theorem: the core of a game is nonempty if and only if the game is "balanced".
Every Walrasian equilibrium has the core property, but not vice versa. The Edgeworth conjecture states that, given additional assumptions, the limit of the core as the number of consumers goes to infinity is a set of Walrasian equilibria.
Let there be n players, where n is odd. A game that proposes to divide one unit of a good among a coalition having at least (n+1)/2 members has an empty core. That is, no stable coalition exists.
== Example ==
=== Example 1: Miners ===
Consider a group of n miners, who have discovered large bars of gold. If two miners can carry one piece of gold, then the payoff of a coalition S is
v
(
S
)
=
{
|
S
|
/
2
,
if
|
S
|
is even
;
(
|
S
|
−
1
)
/
2
,
if
|
S
|
is odd
.
{\displaystyle v(S)={\begin{cases}|S|/2,&{\text{if }}|S|{\text{ is even}};\\(|S|-1)/2,&{\text{if }}|S|{\text{ is odd}}.\end{cases}}}
If there are more than two miners and there is an even number of miners, then the core consists of the single payoff where each miner gets 1/2. If there is an odd number of miners, then the core is empty.
=== Example 2: Gloves ===
Mr A and Mr B are knitting gloves. The gloves are one-size-fits-all, and two gloves make a pair that they sell for €5. They have each made three gloves. How to share the proceeds from the sale? The problem can be described by a characteristic function form game with the following characteristic function: Each man has three gloves, that is one pair with a market value of €5. Together, they have 6 gloves or 3 pair, having a market value of €15. Since the singleton coalitions (consisting of a single man) are the only non-trivial coalitions of the game all possible distributions of this sum belong to the core, provided both men get at least €5, the amount they can achieve on their own. For instance (7.5, 7.5) belongs to the core, but so does (5, 10) or (9, 6).
=== Example 3: Shoes ===
For the moment ignore shoe sizes: a pair consists of a left and a right shoe, which can then be sold for €10. Consider a game with 2001 players: 1000 of them have 1 left shoe, 1001 have 1 right shoe. The core of this game is somewhat surprising: it consists of a single imputation that gives 10 to those having a (scarce) left shoe, and 0 to those owning an (oversupplied) right shoe. No coalition can block this outcome, because no left shoe owner will accept less than 10, and any imputation that pays a positive amount to any right shoe owner must pay less than 10000 in total to the other players, who can get 10000 on their own. So, there is just one imputation in the core.
The message remains the same, even if we increase the numbers as long as left shoes are scarcer. The core has been criticized for being so extremely sensitive to oversupply of one type of player.
== The core in general equilibrium theory ==
The Walrasian equilibria of an exchange economy in a general equilibrium model, will lie in the core of the cooperation game between the agents. Graphically, and in a two-agent economy (see Edgeworth Box), the core is the set of points on the contract curve (the set of Pareto optimal allocations) lying between each of the agents' indifference curves defined at the initial endowments.
== The core in voting theory ==
When alternatives are allocations (list of consumption bundles), it is natural to assume that any nonempty subsets of individuals can block a given allocation.
When alternatives are public (such as the amount of a certain public good), however, it is more appropriate to assume that only the coalitions that are large enough can block a given alternative. The collection of such large ("winning") coalitions is called a simple game.
The core of a simple game with respect to a profile of preferences is based on the idea that only winning coalitions can reject an alternative
x
{\displaystyle x}
in favor of another alternative
y
{\displaystyle y}
. A necessary and sufficient condition for the core to be nonempty for all profile of preferences, is provided in terms of the Nakamura number for the simple game.
== See also ==
Cooperative bargaining
Welfare economics
Pareto efficiency
Knaster–Kuratowski–Mazurkiewicz–Shapley theorem - instrumental in proving the non-emptiness of the core.
== References ==
=== Works cited ===
Edgeworth, Francis Ysidro (1881). Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. London: C. K. Paul.
== Further reading ==
Ichiishi, Tatsuro (1983). "Cooperative Behavior and Stability". Game Theory for Economic Analysis. New York: Academic Press. pp. 77–117. ISBN 0-12-370180-5.
Osborne, Martin J.; Rubinstein, Ariel (1994). A Course in Game Theory. The MIT Press.
Peleg, B (1992). "Axiomatizations of the Core". In Aumann, Robert J.; Hart, Sergiu (eds.). Handbook of Game Theory with Economic Applications. Vol. I. Amsterdam: Elsevier. pp. 397–412. ISBN 978-0-444-88098-7.
Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7.
Telser, Lester G. (1994). "The Usefulness of Core Theory in Economics". Journal of Economic Perspectives. 8 (2): 151–164. doi:10.1257/jep.8.2.151. | Wikipedia/Core_(game_theory) |
Conflict resolution is conceptualized as the methods and processes involved in facilitating the peaceful ending of conflict and retribution. Committed group members attempt to resolve group conflicts by actively communicating information about their conflicting motives or ideologies to the rest of group (e.g., intentions; reasons for holding certain beliefs) and by engaging in collective negotiation. Dimensions of resolution typically parallel the dimensions of conflict in the way the conflict is processed. Cognitive resolution is the way disputants understand and view the conflict, with beliefs, perspectives, understandings and attitudes. Emotional resolution is in the way disputants feel about a conflict, the emotional energy. Behavioral resolution is reflective of how the disputants act, their behavior. Ultimately a wide range of methods and procedures for addressing conflict exist, including negotiation, mediation, mediation-arbitration, diplomacy, and creative peacebuilding.
== Characteristics ==
Wallensteen defines conflict resolution (for peace and conflict studies) as:
[S]ocial situation where the armed conflicting parties in a (voluntarily) agreement resolve to live peacefully with – and/or dissolve – their basic incompatibilities and henceforth cease to use arms against one another.
The "conflicting parties" concerned in this definition are formally or informally organized groups engaged in intrastate or interstate conflict.
'Basic incompatibility' refers to a severe disagreement between at least two sides where their demands cannot be met by the same resources at the same time.
=== Territoriality ===
According to conflict database Uppsala Conflict Data Program's definition, war may occur between parties who contest an incompatibility. The nature of an incompatibility can be territorial or governmental, but a warring party must be a "government of a state or any opposition organization or alliance of organizations that uses armed force to promote its position in the incompatibility in an intrastate or an interstate armed conflict". Wars can conclude with a peace agreement, which is a "formal agreement... which addresses the disputed incompatibility, either by settling all or part of it, or by clearly outlining a process for how [...] to regulate the incompatibility."
A ceasefire is another form of agreement made by warring parties; unlike a peace agreement, it only "regulates the conflict behaviour of warring parties", and does not resolve the issue that brought the parties to war in the first place.
Peacekeeping measures may be deployed to avoid violence in solving such incompatibilities. Beginning in the last century, political theorists have been developing the theory of a global peace system that relies upon broad social and political measures to avoid war in the interest of achieving world peace. The Blue Peace approach developed by Strategic Foresight Group facilitates cooperation between countries over shared water resources, thus reducing the risk of war and enabling sustainable development.
The escalating costs of conflict have increased use of third parties who may serve as a conflict specialists to resolve conflicts. In fact, relief and development organizations have added peace-building specialists to their teams. Many major international non-governmental organizations have seen a growing need to hire practitioners trained in conflict analysis and resolution. Furthermore, this expansion has resulted in the need for conflict resolution practitioners to work in a variety of settings such as in businesses, court systems, government agencies, nonprofit organizations, and educational institutions throughout the world. Democracy has a positive influence on conflict resolution.
== Models ==
=== Modes ===
Ruble and Thomas transposed the managerial grid model in terms of conflict resolution. They adapted the classification scheme to dimensions identified in conflict research that represent a range of behaviors beyond the dichotomy between cooperation and competition. The X-axis evaluates cooperativity, the extent by which mutual goals are achieved. The Y-axis evaluates assertiveness, how parties insist on carrying their own objectives.
Thomas and Kilmann extended that grid with a rating system for five modes of behavior. When parties are assertive but their objectives lack compatibility, they become competitive; when parties are assertive toward compatible objectives, they can be collaborating; when no party prioritizes objectives that are mutually exclusive, they can display avoidance; parties can be accommodating when assertiveness is low but cooperativity is high; when there is no real bias toward assertiveness and cooperativity, compromising can obtain.
However, not every style leads to an acceptable result in every situation. For example, a collaboration does not work if the goals of the two conflict parties are immutable and mutually exclusive. The different styles have different advantages and disadvantages. Depending on the situation, different conflict styles can be considered desirable to achieve the best results.
=== Dual concern ===
The dual concern model of conflict resolution is a conceptual perspective that assumes individuals' preferred method of dealing with conflict is based on two underlying themes or dimensions: concern for self (assertiveness) and concern for others (empathy). According to the model, group members balance their concern for satisfying personal needs and interests with their concern for satisfying the needs and interests of others in different ways. The intersection of these two dimensions ultimately leads individuals towards exhibiting different styles of conflict resolution. The dual model identifies five group conflict resolution styles or strategies that individuals may use depending on their dispositions toward pro-self or pro-social goals.
Avoidance
Characterized by joking, changing or avoiding the topic, or even denying that a problem exists, the conflict avoidance style is used when an individual has withdrawn in dealing with the other party, when one is uncomfortable with conflict, or due to cultural contexts. During conflict, these avoiders adopt a "wait and see" attitude, often allowing conflict to phase out on its own without any personal involvement. By neglecting to address high-conflict situations, avoiders risk allowing problems to fester or spin out of control.
Accommodating
In contrast, yielding, "accommodating", smoothing or suppression conflict styles are characterized by a high level of concern for others and a low level of concern for oneself. This passive pro-social approach emerges when individuals derive personal satisfaction from meeting the needs of others and have a general concern for maintaining stable, positive social relationships. When faced with conflict, individuals with an accommodating conflict style tend to harmonize into others' demands out of respect for the social relationship. With this sense of yielding to the conflict, individuals fall back to others' input instead of finding solutions with their own intellectual resolution.
Competitive
The competitive, "fighting" or forcing conflict style maximizes individual assertiveness (i.e., concern for self) and minimizes empathy (i.e., concern for others). Groups consisting of competitive members generally enjoy seeking domination over others, and typically see conflict as a "win or lose" predicament. Fighters tend to force others to accept their personal views by employing competitive power tactics (arguments, insults, accusations or even violence) that foster intimidation.
Conciliation
The conciliation, "compromising", bargaining or negotiation conflict style is typical of individuals who possess an intermediate level of concern for both personal and others' outcomes. Compromisers value fairness and, in doing so, anticipate mutual give-and-take interactions. By accepting some demands put forth by others, compromisers believe this agreeableness will encourage others to meet them halfway, thus promoting conflict resolution. This conflict style can be considered an extension of both "yielding" and "cooperative" strategies.
Cooperation
Characterized by an active concern for both pro-social and pro-self behavior, the cooperation, integration, confrontation or problem-solving conflict style is typically used when an individual has elevated interests in their own outcomes as well as in the outcomes of others. During conflict, cooperators collaborate with others in an effort to find an amicable solution that satisfies all parties involved in the conflict. Individuals using this type of conflict style tend to be both highly assertive and highly empathetic. By seeing conflict as a creative opportunity, collaborators willingly invest time and resources into finding a "win-win" solution. According to the literature on conflict resolution, a cooperative conflict resolution style is recommended above all others. This resolution may be achieved by lowering the aggressor's guard while raising the ego.
=== Regret analysis ===
The conflict resolution curve derived from an analytical model that offers a peaceful solution by motivating conflicting entities. Forced resolution of conflict might invoke another conflict in the future.
Conflict resolution curve (CRC) separates conflict styles into two separate domains: domain of competing entities and domain of accommodating entities. There is a sort of agreement between targets and aggressors on this curve. Their judgements of badness compared to goodness of each other are analogous on CRC. So, arrival of conflicting entities to some negotiable points on CRC is important before peace building. CRC does not exist (i.e., singular) in reality if the aggression of the aggressor is certain. Under such circumstances it might lead to apocalypse with mutual destruction.
The curve explains why nonviolent struggles ultimately toppled repressive regimes and sometimes forced leaders to change the nature of governance. Also, this methodology has been applied to capture conflict styles on the Korean Peninsula and dynamics of negotiation processes.
=== Four-sides ===
In the third step, the actual conflict of interest is identified and mutual understanding for the interest of the other party is developed. This requires understanding and respecting the underlying values and motivations. According to the four-sides model by Friedemann Schulz von Thun, there are two levels of information in every statement: the content level and the emotional or relationship level. Both levels contain interests, the differences of which to the other conflict party should be balanced as much as possible. Then a win-win solution for the conflict can be developed together.
=== Circle of Conflict ===
Christopher W. Moore's "Circle of conflict" model, first published in 1986, emphasizes five sources of conflict:
data: information, interpretation, incompleteness;
relationship: personal dynamics, miscommunication, misbehaviors;
value: incompatible beliefs, principles, or priorities;
structure: organization failures, power imbalances, resource constraints;
interests: needs, desires, incentives, procedures.
Conflicts may have multiple sources. Identifying the source of the conflict ought to facilitate its resolution.
=== Nonviolent communication (NVC) ===
== Theories ==
=== Relational dialectics ===
The main concepts of relational dialectics are:
Contradictions – The concept is that the contrary has the characteristics of its opposite. People can seek to be in a relationship but still need their space.
Totality – The totality comes when the opposites unite. Thus, the relationship is balanced with contradictions and only then it reaches totality
Process – Comprehended through various social processes. These processes simultaneously continue within a relationship in a recurring manner.
Praxis – The relationship progresses with experience and both people interact and communicate effectively to meet their needs. Praxis is a concept of practicability in making decisions in a relationship despite opposing wants and needs
=== Strategy of conflict ===
Thomas Schelling applied game theory to situations where the outcome is not zero-sum.
Conflict is a contest. Rational behavior, in this contest, is a matter of judgment and perception.
Strategy makes predictions using "rational behavior – behavior motivated by a serious calculation of advantages, a calculation that in turn is based on an explicit and internally consistent value system".
Cooperation is always temporary, interests will change.
=== Ripeness ===
=== Mechanisms ===
One theory discussed within the field of peace and conflict studies is conflict resolution mechanisms: independent procedures in which the conflicting parties can have confidence. They can be formal or informal arrangements with the intention of resolving the conflict. In Understanding Conflict Resolution Wallensteen draws from the works of Lewis A. Coser, Johan Galtung and Thomas Schelling, and presents seven distinct theoretical mechanisms for conflict resolutions:
A shift in priorities for one of the conflicting parties. While it is rare that a party completely changes its basic positions, it can display a shift in to what it gives highest priority. In such an instance new possibilities for conflict resolutions may arise.
The contested resource is divided. In essence, this means both conflicting parties display some extent of shift in priorities which then opens up for some form of "meeting the other side halfway" agreement.
Horse-trading between the conflicting parties. This means that one side gets all of its demands met on one issue, while the other side gets all of its demands met on another issue.
The parties decide to share control, and rule together over the contested resource. It could be permanent, or a temporary arrangement for a transition period that, when over, has led to a transcendence of the conflict.
The parties agree to leave control to someone else. In this mechanism the primary parties agree, or accept, that a third party takes control over the contested resource.
The parties resort to conflict resolution mechanisms, notably arbitration or other legal procedures. This means finding a procedure for resolving the conflict through some of the previously mentioned five ways, but with the added quality that it is done through a process outside of the parties' immediate control.
Some issues can be left for later. The argument for this is that political conditions and popular attitudes can change, and some issues can gain from being delayed, as their significance may pale with time.
Nicholson notes that a conflict is resolved when the inconsistency between wishes and actions of parties is resolved. Negotiation is an important part of conflict resolution, and any design of a process which tries to incorporate positive conflict from the start needs to be cautious not to let it degenerate into the negative types of conflict. Actual conflict resolutions range from discussions between the parties involved, such as in mediations or collective bargaining, to violent confrontations such as in interstate wars or civil wars. "Between" these are the variants of lawful or courtly clarification, which by no means have to take the form of "mud fights", but can be handled as "professional delegation" of the problem to lawyers, in order to relieve oneself from the time-consuming and strenuous clarification procedure. Many conflicts can be resolved without escalation by the parties involved. If the conflict parties do not come to a solution themselves, accompanying measures can be taken by third parties.
The goal of conflict resolution is an effective and lasting solution to the conflict. This is achieved through the satisfaction of all parties involved, which ideally results in constructively working together on the problem (collaboration, cooperation). In addition, a regulation of the conflict can occur through a decision by an authority, e.g., by an arbitrator, a court, a parent, or a supervisor. Unprocessed conflicts generate frustration and aggression, which can result in cost, damage, and scapegoats.
== Praxis ==
=== De-escalation ===
The first step in a dispute is usually de-escalation (e.g., cessation of hostilities, reduction of open aggression). A reciprocal tit for tat strategy ("an eye for an eye") can build trust between groups in the case of mutually collaborative or mutually competitive conflict styles. To facilitate a change of positions in a conflict party, face-saving bridges should be built, e.g., by discussing what has already changed since the beginning of conflict resolution or by introducing common fair behavioral norms.
Escalating behavior should not be reacted to immediately, to give the person or persons time to regain emotional self-control, making them more accessible to arguments and avoiding mutual escalation. Anger can be reduced by an apology, humor, a recess, common behavioral norms, greater distance (switch to online discussion), or by background information that the escalation of the other side was not intended. Afterwards, the problematic behavior can be addressed in a calm manner, followed by an acknowledgment of those substantive points of the escalating person that are correct. Alternatively, a feedback sandwich can be used.
In the case of avoiding behavior, more questions should be asked and more attention should be paid to the participation of these persons in the conflict resolution and to their immaterial interests (such as recognition and autonomy). In the conversation, a reminder can be given for motivation that the processing of the conflict serves the satisfaction of the interests of both sides.
=== Regulated communication ===
The second step is the initiation of communication between the conflicting parties, often through mediation. Accompanying conditions are described by the Harvard Concept. Alternatively, the moderation cycle according to Josef W. Seifert can be followed. Furthermore, I-messages can be alternated with active listening according to Thomas Gordon or nonviolent communication according to Marshall B. Rosenberg can be used to depersonalize a discussion.
=== Glasl's management strategies ===
Glasl, on the other hand, assigns six strategies for conflict management to the nine escalation stages of Friedrich Glasl's model of conflict escalation.
Level 1-3 (hardening, polarization & debate, actions instead of words): Moderation
Level 3-5 (actions instead of words, concern about image & coalitions, loss of face): Process support
Level 4-6 (concern about image & coalitions, loss of face, threatening strategies): socio-therapeutic process support
Level 5-7 (loss of face, threatening strategies, limited destructive strikes): conciliation/mediation
Level 6-8 (threatening strategies, limited destructive strikes, fragmentation): arbitration/judicial proceedings
Level 7-9 (limited destructive strikes, fragmentation, together into the abyss): power intervention
=== Interest-based relational approach (IBR) ===
Developed by Roger Fisher and William Ury in their 1981 seminal book Getting to Yes: Negotiating Agreement Without Giving In, the IBR approach originated from work at the Harvard Negotiation Project. Its has four core tactics:
separate the people from the problem;
focus on interests, not positions;
find options for mutual gain;
insist on using objective criteria.
The Harvard Negotiation Project was one of the founding entities of the Program on Negotiation (PON) at Harvard Law School in 1983.
=== Forcing ===
When one of the conflict's parts firmly pursues his or her own concerns despite the resistance of the other(s). This may involve pushing one viewpoint at the expense of another or maintaining firm resistance to the counterpart's actions; it is also commonly known as "competing".
Forcing may be appropriate when all other, less forceful methods, do not work or are ineffective; when someone needs to stand up for his/her own rights (or the represented group/organization's rights), resist aggression and pressure. It may be also considered a suitable option when a quick resolution is required and using force is justified (e.g. in a life-threatening situation, to stop an aggression), and as a very last resort to resolve a long-lasting conflict.
However, forcing may also negatively affect the relationship with the opponent in the long run; may intensified the conflict if the opponent decides to react in the same way (even if it was not the original intention); it does not allow to take advantage in a productive way of the other side's position and, last but not least, taking this approach may require a lot of energy and be exhausting to some individuals.
=== Win-win / collaborating ===
Collaboration involves an attempt to work with the other part involved in the conflict to find a win-win solution to the problem in hand, or at least to find a solution that most satisfies the concerns of both parties. The win-win approach sees conflict resolution as an opportunity to come to a mutually beneficial result; and it includes identifying the underlying concerns of the opponents and finding an alternative which meets each party's concerns. From that point of view, it is the most desirable outcome when trying to solve a problem for all partners.
Collaborating may be the best solution when consensus and commitment of other parties is important; when the conflict occurs in a collaborative, trustworthy environment and when it is required to address the interests of multiple stakeholders. But more specially, it is the most desirable outcome when a long-term relationship is important so that people can continue to collaborate in a productive way; collaborating is in few words, sharing responsibilities and mutual commitment. For parties involved, the outcome of the conflict resolution is less stressful; however, the process of finding and establishing a win-win solution may be longer and should be very involving.
It may require more effort and more time than some other methods; for the same reason, collaborating may not be practical when timing is crucial and a quick solution or fast response is required.
=== Compromising ===
Different from the win-win solution, in this outcome the conflict parties find a mutually acceptable solution which partially satisfies both parties. This can occur as both parties converse with one another and seek to understand the other's point of view. Compromising may be an optimal solution when the goals are moderately important and not worth the use of more assertive or more involving approaches. It may be useful when reaching temporary settlement on complex issues and as a first step when the involved parties do not know each other well or have not yet developed a high level of mutual trust. Compromising may be a faster way to solve things when time is a factor. The level of tensions can be lower as well, but the result of the conflict may be also less satisfactory.
If this method is not well managed, and the factor time becomes the most important one, the situation may result in both parties being not satisfied with the outcome (i.e. a lose-lose situation). Moreover, it does not contribute to building trust in the long run and it may require a closer monitoring of the kind of partially satisfactory compromises acquired.
=== Withdrawing ===
This technique consists on not addressing the conflict, postpone it or simply withdrawing; for that reason, it is also known as Avoiding. This outcome is suitable when the issue is trivial and not worth the effort or when more important issues are pressing, and one or both the parties do not have time to deal with it. Withdrawing may be also a strategic response when it is not the right time or place to confront the issue, when more time is needed to think and collect information before acting or when not responding may bring still some winnings for at least some of the involves parties. Moreover, withdrawing may be also employed when someone know that the other party is totally engaged with hostility and does not want (can not) to invest further unreasonable efforts.
Withdrawing may give the possibility to see things from a different perspective while gaining time and collecting further information, and specially is a low stress approach particularly when the conflict is a short time one. However, not acting may be interpreted as an agreement and therefore it may lead to weakening or losing a previously gained position with one or more parties involved. Furthermore, when using withdrawing as a strategy more time, skills and experiences together with other actions may need to be implemented.
=== Smoothing ===
Smoothing is accommodating the concerns of others first of all, rather than one's own concerns. This kind of strategy may be applied when the issue of the conflict is much more important for the counterparts whereas for the other is not particularly relevant. It may be also applied when someone accepts that he/she is wrong and furthermore there are no other possible options than continuing an unworthy competing-pushing situation. Just as withdrawing, smoothing may be an option to find at least a temporal solution or obtain more time and information, however, it is not an option when priority interests are at stake.
There is a high risk of being abused when choosing the smoothing option. Therefore, it is important to keep the right balance and to not give up one own interests and necessities. Otherwise, confidence in one's ability, mainly with an aggressive opponent, may be seriously damaged, together with credibility by the other parties involved. Needed to say, in these cases a transition to a Win-Win solution in the future becomes particularly more difficult when someone.
== Between organizations ==
Relationships between organizations, such as strategic alliances, buyer-supplier partnerships, organizational networks, or joint ventures are prone to conflict. Conflict resolution in inter-organizational relationships has attracted the attention of business and management scholars. They have related the forms of conflict (e.g., integrity-based vs. competence-based conflict) to the mode of conflict resolution and the negotiation and repair approaches used by organizations. They have also observed the role of important moderating factors such as the type of contractual arrangement, the level of trust between organizations, or the type of power asymmetry.
=== Conflict management ===
Conflict management refers to the long-term management of intractable conflicts. It is the label for the variety of ways by which people handle grievances—standing up for what they consider to be right and against what they consider to be wrong. Those ways include such diverse phenomena as gossip, ridicule, lynching, terrorism, warfare, feuding, genocide, law, mediation, and avoidance. Which forms of conflict management will be used in any given situation can be somewhat predicted and explained by the social structure—or social geometry—of the case.
Conflict management is often considered to be distinct from conflict resolution.
In order for actual conflict to occur, there should be an expression of exclusive patterns which explain why and how the conflict was expressed the way it was. Conflict is often connected to a previous issue. Resolution refers to resolving a dispute to the approval of one or both parties, whereas management is concerned with an ongoing process that may never have a resolution. Neither is considered the same as conflict transformation, which seeks to reframe the positions of the conflict parties.
=== Counseling ===
When personal conflict leads to frustration and loss of efficiency, counseling may prove helpful. Although few organizations can afford to have professional counselors on staff, given some training, managers may be able to perform this function. Nondirective counseling, or "listening with understanding", is little more than being a good listener—something often considered to be important in a manager.
Sometimes simply being able to express one's feelings to a concerned and understanding listener is enough to relieve frustration and make it possible for an individual to advance to a problem-solving frame of mind. The nondirective approach is one effective way for managers to deal with frustrated subordinates and coworkers.
There are other, more direct and more diagnostic, methods that could be used in appropriate circumstances. However, the great strength of the nondirective approach lies in its simplicity, its effectiveness, and that it deliberately avoids the manager-counselor's diagnosing and interpreting emotional problems, which would call for special psychological training. Listening to staff with sympathy and understanding is unlikely to escalate the problem, and is a widely used approach for helping people cope with problems that interfere with their effectiveness in the workplace.
== Cultural issues ==
Conflict resolution as both a professional practice and academic field is highly sensitive to cultural practices. In Western cultural contexts, such as Canada and the United States, successful conflict resolution usually involves fostering communication among disputants, problem solving, and drafting agreements that meet underlying needs. In these situations, conflict resolvers often talk about finding a mutually satisfying ("win-win") solution for everyone involved.
In many non-Western cultural contexts, such as Afghanistan, Vietnam, and China, it is also important to find "win-win" solutions; however, the routes taken to find them may be very different. In these contexts, direct communication between disputants that explicitly addresses the issues at stake in the conflict can be perceived as very rude, making the conflict worse and delaying resolution. It can make sense to involve religious, tribal, or community leaders; communicate difficult truths through a third party; or make suggestions through stories. Intercultural conflicts are often the most difficult to resolve because the expectations of the disputants can be very different, and there is much occasion for misunderstanding.
== In animals ==
Conflict resolution has also been studied in non-humans, including dogs, cats, monkeys, snakes, elephants, and primates. Aggression is more common among relatives and within a group than between groups. Instead of creating distance between the individuals, primates tend to be more intimate in the period after an aggressive incident. These intimacies consist of grooming and various forms of body contact. Stress responses, including increased heart rates, usually decrease after these reconciliatory signals. Different types of primates, as well as many other species who live in groups, display different types of conciliatory behavior. Resolving conflicts that threaten the interaction between individuals in a group is necessary for survival, giving it a strong evolutionary value. A further focus of this is among species that have stable social units, individual relationships, and the potential for intragroup aggression that may disrupt beneficial relationships. The role of these reunions in negotiating relationships is examined along with the susceptibility of these relationships to partner value asymmetries and biological market effects. These findings contradict previous existing theories about the general function of aggression, i.e. creating space between individuals (first proposed by Konrad Lorenz), which seems to be more the case in conflicts between groups than it is within groups.
In addition to research in primates, biologists are beginning to explore reconciliation in other animals. Until recently, the literature dealing with reconciliation in non-primates has consisted of anecdotal observations and very little quantitative data. Although peaceful post-conflict behavior had been documented going back to the 1960s, it was not until 1993 that Rowell made the first explicit mention of reconciliation in feral sheep. Reconciliation has since been documented in spotted hyenas, lions, bottlenose dolphins, dwarf mongoose, domestic goats, domestic dogs, and, recently, in red-necked wallabies.
== See also ==
=== Organizations ===
Center for the Study of Genocide, Conflict Resolution, and Human Rights
Conscience: Taxes for Peace not War is a London organisation that promotes peacebuilding as an alternative to military security
Crisis Management Initiative (CMI)
Heidelberg Institute for International Conflict Research
Peninsula Conflict Resolution Center
Jimmy and Rosalynn Carter School for Peace and Conflict Resolution
Search for Common Ground is one of the world's largest non-government organisations dedicated to conflict resolution
Seeds of Peace develops and empowers young leaders from regions of conflict to work towards peace through coexistence
United Network of Young Peacebuilders (UNOY) is a global non-governmental organization and youth network dedicated to the role of youth in peacebuilding and conflict resolution
University for Peace is a United Nations mandated organization and graduate school dedicated to conflict resolution and peace studies
Uppsala Conflict Data Program is an academic data collection project that provides descriptions of political violence and conflict resolution
== Footnotes ==
== References ==
== Works cited ==
Bannon, I. & Paul Collier (Eds.). (2003). Natural resources and violent conflict: Options and actions. WThe World Bank.
Ury, F. & Rodger Fisher. (1981). Getting to yes: Negotiating agreement without giving in. Penguin.
Wilmot, W. & Jouyce Hocker. (2007). Interpersonal conflict. McGraw-Hill.
Bercovitch, Jacob and Jackson, Richard. 2009. Conflict Resolution in the Twenty-first Century: Principles, Methods, and Approaches. Archived 8 June 2010 at the Wayback Machine University of Michigan Press, Ann Arbor.
de Waal, Frans B. M. and Angeline van Roosmalen. 1979. Reconciliation and consolation among chimpanzees. Behavioral Ecology and Sociobiology 5: 55–66.
de Waal, Frans B. M. 1989. Peacemaking Among Primates. Harvard University Press.
Judge, Peter G.; de Waal, Frans B.M. (1993). "Conflict avoidance among rhesus monkeys: coping with short-term crowding". Animal Behaviour. 46 (2): 221–232. doi:10.1006/anbe.1993.1184. S2CID 53175846.
Veenema, Hans; et al. (1994). "Methodological improvements for the study of reconciliation". Behavioural Processes. 31 (1): 29–38. doi:10.1016/0376-6357(94)90035-3. PMID 24897415. S2CID 25126127.
de Waal, Frans B. M. and Filippo Aureli. 1996. Consolation, reconciliation, and a possible cognitive difference between macaques and chimpanzees. Reaching into thought: The minds of the great apes (Eds. Anne E. Russon, Kim A. Bard, Sue Taylor Parker), Cambridge University Press, New York, NY: 80–110.
Aureli, Filippo (1997). "Post-conflict anxiety in non-human primates: the mediating role of emotion in conflict resolution". Aggressive Behavior. 23 (5): 315–328. doi:10.1002/(sici)1098-2337(1997)23:5<315::aid-ab2>3.0.co;2-h.
Castles, Duncan L.; Whiten, Andrew (1998). "Post-conflict behaviour of wild olive baboons, I. Reconciliation, redirection, and consolation". Ethology. 104 (2): 126–147. Bibcode:1998Ethol.104..126C. doi:10.1111/j.1439-0310.1998.tb00057.x.
Aureli, Filippo and Frans B. M. de Waal, eds. 2000. Natural Conflict Resolution. University of California Press.
de Waal, Frans B. M. 2000. Primates––A natural heritage of conflict resolution. Science 289: 586–590.
Hicks, Donna. 2011. Dignity: The Essential Role It Plays in Resolving Conflict. Yale University Press
Silk, Joan B. (2002). "The form and function of reconciliation in primates". Annual Review of Anthropology. 31: 21–44. doi:10.1146/annurev.anthro.31.032902.101743.
Weaver, Ann; de Waal, Frans B. M. (2003). "The mother-offspring relationship as a template in social development: reconciliation in captive brown capuchins (Cebus apella)". Journal of Comparative Psychology. 117 (1): 101–110. doi:10.1037/0735-7036.117.1.101. PMID 12735370. S2CID 9632420.
Palagi, Elisabetta; et al. (2004). "Reconciliation and consolation in captive bonobos (Pan paniscus)". American Journal of Primatology. 62 (1): 15–30. doi:10.1002/ajp.20000. PMID 14752810. S2CID 22452710.
Palagi, Elisabetta; et al. (2005). "Aggression and reconciliation in two captive groups of Lemur catta". International Journal of Primatology. 26 (2): 279–294. doi:10.1007/s10764-005-2925-x. S2CID 22639928.
Bar-Siman-Tov, Yaacov (Ed.) (2004). From Conflict Resolution to Reconciliation. Oxford University Press
== Further readings ==
Coleman, Peter T. (2011). The Five Percent: Finding Solutions to Seemingly Impossible Conflicts. PublicAffairs. ISBN 978-1-58648-921-2.
Staniland, Paul (2021). Ordering Violence: Explaining Armed Group-state Relations from Conflict to Cooperation. Cornell University Press. ISBN 978-1-5017-6110-2. | Wikipedia/Conflict_resolution |
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a position evolves through alternating moves, each governed by well-defined rules, with the aim of achieving a specific winning condition. Unlike economic game theory, combinatorial game theory generally avoids the study of games of chance or games involving imperfect information, preferring instead games in which the current state and the full set of available moves are always known to both players. However, as mathematical techniques develop, the scope of analyzable games expands, and the boundaries of the field continue to evolve. Authors typically define the term "game" at the outset of academic papers, with definitions tailored to the specific game under analysis rather than reflecting the field’s full scope.
Combinatorial games include well-known examples such as chess, checkers, and Go, which are considered complex and non-trivial, as well as simpler, "solved" games like tic-tac-toe. Some combinatorial games, such as infinite chess, may feature an unbounded playing area. In the context of combinatorial game theory, the structure of such games is typically modeled using a game tree. The field also encompasses single-player puzzles like Sudoku, and zero-player automata such as Conway's Game of Life—although these are sometimes more accurately categorized as mathematical puzzles or automata, given that the strictest definitions of "game" imply the involvement of multiple participants.
A key concept in combinatorial game theory is that of the solved game. For instance, tic-tac-toe is solved in that optimal play by both participants always results in a draw. Determining such outcomes for more complex games is significantly more difficult. Notably, in 2007, checkers was announced to be weakly solved, with perfect play by both sides leading to a draw; however, this result required a computer-assisted proof. Many real-world games remain too complex for complete analysis, though combinatorial methods have shown some success in the study of Go endgames. Analyzing a position using combinatorial game theory involves identifying the optimal sequence of moves for both players until the game's conclusion, but this process becomes prohibitively difficult for anything beyond simple games.
It is useful to distinguish between combinatorial "mathgames"—games of primary interest to mathematicians and scientists for theoretical exploration—and "playgames," which are more widely played for entertainment and competition. Some games, such as Nim, straddle both categories. Nim played a foundational role in the development of combinatorial game theory and was among the earliest games to be programmed on a computer. Tic-tac-toe continues to be used in teaching fundamental concepts of game AI design to computer science students.
== Difference with traditional game theory ==
Combinatorial game theory contrasts with "traditional" or "economic" game theory, which, although it can address sequential play, often incorporates elements of probability and incomplete information. While economic game theory employs utility theory and equilibrium concepts, combinatorial game theory is primarily concerned with two-player perfect-information games and has pioneered novel techniques for analyzing game trees, such as through the use of surreal numbers, which represent a subset of all two-player perfect-information games. The types of games studied in this field are of particular interest in areas such as artificial intelligence, especially for tasks in automated planning and scheduling. However, there is a distinction in emphasis: while economic game theory tends to focus on practical algorithms—such as the alpha–beta pruning strategy commonly taught in AI courses—combinatorial game theory places greater weight on theoretical results, including the analysis of game complexity and the existence of optimal strategies through methods like the strategy-stealing argument.
== History ==
Combinatorial game theory arose in relation to the theory of impartial games, in which any play available to one player must be available to the other as well. One such game is Nim, which can be solved completely. Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses. In the 1930s, the Sprague–Grundy theorem showed that all impartial games are equivalent to heaps in Nim, thus showing that major unifications are possible in games considered at a combinatorial level, in which detailed strategies matter, not just pay-offs.
In the 1960s, Elwyn R. Berlekamp, John H. Conway and Richard K. Guy jointly introduced the theory of a partisan game, in which the requirement that a play available to one player be available to both is relaxed. Their results were published in their book Winning Ways for your Mathematical Plays in 1982. However, the first work published on the subject was Conway's 1976 book On Numbers and Games, also known as ONAG, which introduced the concept of surreal numbers and the generalization to games. On Numbers and Games was also a fruit of the collaboration between Berlekamp, Conway, and Guy.
Combinatorial games are generally, by convention, put into a form where one player wins when the other has no moves remaining. It is easy to convert any finite game with only two possible results into an equivalent one where this convention applies. One of the most important concepts in the theory of combinatorial games is that of the sum of two games, which is a game where each player may choose to move either in one game or the other at any point in the game, and a player wins when his opponent has no move in either game. This way of combining games leads to a rich and powerful mathematical structure.
Conway stated in On Numbers and Games that the inspiration for the theory of partisan games was based on his observation of the play in Go endgames, which can often be decomposed into sums of simpler endgames isolated from each other in different parts of the board.
== Examples ==
The introductory text Winning Ways introduced a large number of games, but the following were used as motivating examples for the introductory theory:
Blue–Red Hackenbush - At the finite level, this partisan combinatorial game allows constructions of games whose values are dyadic rational numbers. At the infinite level, it allows one to construct all real values, as well as many infinite ones that fall within the class of surreal numbers.
Blue–Red–Green Hackenbush - Allows for additional game values that are not numbers in the traditional sense, for example, star.
Toads and Frogs - Allows various game values. Unlike most other games, a position is easily represented by a short string of characters.
Domineering - Various interesting games, such as hot games, appear as positions in Domineering, because there is sometimes an incentive to move, and sometimes not. This allows discussion of a game's temperature.
Nim - An impartial game. This allows for the construction of the nimbers. (It can also be seen as a green-only special case of Blue-Red-Green Hackenbush.)
The classic game Go was influential on the early combinatorial game theory, and Berlekamp and Wolfe subsequently developed an endgame and temperature theory for it (see references). Armed with this they were able to construct plausible Go endgame positions from which they could give expert Go players a choice of sides and then defeat them either way.
Another game studied in the context of combinatorial game theory is chess. In 1953 Alan Turing wrote of the game, "If one can explain quite unambiguously in English, with the aid of mathematical symbols if required, how a calculation is to be done, then it is always possible to programme any digital computer to do that calculation, provided the storage capacity is adequate." In a 1950 paper, Claude Shannon estimated the lower bound of the game-tree complexity of chess to be 10120, and today this is referred to as the Shannon number. Chess remains unsolved, although extensive study, including work involving the use of supercomputers has created chess endgame tablebases, which shows the result of perfect play for all end-games with seven pieces or less. Infinite chess has an even greater combinatorial complexity than chess (unless only limited end-games, or composed positions with a small number of pieces are being studied).
== Overview ==
A game, in its simplest terms, is a list of possible "moves" that two players, called left and right, can make. The game position resulting from any move can be considered to be another game. This idea of viewing games in terms of their possible moves to other games leads to a recursive mathematical definition of games that is standard in combinatorial game theory. In this definition, each game has the notation {L|R}. L is the set of game positions that the left player can move to, and R is the set of game positions that the right player can move to; each position in L and R is defined as a game using the same notation.
Using Domineering as an example, label each of the sixteen boxes of the four-by-four board by A1 for the upper leftmost square, C2 for the third box from the left on the second row from the top, and so on. We use e.g. (D3, D4) to stand for the game position in which a vertical domino has been placed in the bottom right corner. Then, the initial position can be described in combinatorial game theory notation as
{
(
A
1
,
A
2
)
,
(
B
1
,
B
2
)
,
…
|
(
A
1
,
B
1
)
,
(
A
2
,
B
2
)
,
…
}
.
{\displaystyle \{(\mathrm {A} 1,\mathrm {A} 2),(\mathrm {B} 1,\mathrm {B} 2),\dots |(\mathrm {A} 1,\mathrm {B} 1),(\mathrm {A} 2,\mathrm {B} 2),\dots \}.}
In standard Cross-Cram play, the players alternate turns, but this alternation is handled implicitly by the definitions of combinatorial game theory rather than being encoded within the game states.
{
(
A
1
,
A
2
)
|
(
A
1
,
B
1
)
}
=
{
{
|
}
|
{
|
}
}
.
{\displaystyle \{(\mathrm {A} 1,\mathrm {A} 2)|(\mathrm {A} 1,\mathrm {B} 1)\}=\{\{|\}|\{|\}\}.}
The above game describes a scenario in which there is only one move left for either player, and if either player makes that move, that player wins. (An irrelevant open square at C3 has been omitted from the diagram.) The {|} in each player's move list (corresponding to the single leftover square after the move) is called the zero game, and can actually be abbreviated 0. In the zero game, neither player has any valid moves; thus, the player whose turn it is when the zero game comes up automatically loses.
The type of game in the diagram above also has a simple name; it is called the star game, which can also be abbreviated ∗. In the star game, the only valid move leads to the zero game, which means that whoever's turn comes up during the star game automatically wins.
An additional type of game, not found in Domineering, is a loopy game, in which a valid move of either left or right is a game that can then lead back to the first game. Checkers, for example, becomes loopy when one of the pieces promotes, as then it can cycle endlessly between two or more squares. A game that does not possess such moves is called loopfree.
There are also transfinite games, which have infinitely many positions—that is, left and right have lists of moves that are infinite rather than finite.
== Game abbreviations ==
=== Numbers ===
Numbers represent the number of free moves, or the move advantage of a particular player. By convention positive numbers represent an advantage for Left, while negative numbers represent an advantage for Right. They are defined recursively with 0 being the base case.
0 = {|}
1 = {0|}, 2 = {1|}, 3 = {2|}
−1 = {|0}, −2 = {|−1}, −3 = {|−2}
The zero game is a loss for the first player.
The sum of number games behaves like the integers, for example 3 + −2 = 1.
Any game number is in the class of the surreal numbers.
=== Star ===
Star, written as ∗ or {0|0}, is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.
∗ + ∗ = 0, because the first player must turn one copy of ∗ to a 0, and then the other player will have to turn the other copy of ∗ to a 0 as well; at this point, the first player would lose, since 0 + 0 admits no moves.
The game ∗ is neither positive nor negative; it and all other games in which the first player wins (regardless of which side the player is on) are said to be fuzzy or confused with 0; symbolically, we write ∗ || 0.
The game ∗n is notation for {0, ∗, …, ∗(n−1)| 0, ∗, …, ∗(n−1)}, which is also representative of normal-play Nim with a single heap of n objects. (Note that ∗0 = 0 and ∗1 = ∗.)
=== Up and down ===
Up, written as ↑, is a position in combinatorial game theory. In standard notation, ↑ = {0|∗}. Its negative is called down.
−↑ = ↓ (down)
Up is strictly positive (↑ > 0), and down is strictly negative (↓ < 0), but both are infinitesimal. Up and down are defined in Winning Ways for your Mathematical Plays.
=== "Hot" games ===
Consider the game {1|−1}. Both moves in this game are an advantage for the player who makes them; so the game is said to be "hot;" it is greater than any number less than −1, less than any number greater than 1, and fuzzy with any number in between. It is written as ±1. Note that a subclass of hot games, referred to as ±n for some numerical game n is a switch game. Switch games can be added to numbers, or multiplied by positive ones, in the expected fashion; for example, 4 ± 1 = {5|3}.
== Nimbers ==
An impartial game is one where, at every position of the game, the same moves are available to both players. For instance, Nim is impartial, as any set of objects that can be removed by one player can be removed by the other. However, domineering is not impartial, because one player places horizontal dominoes and the other places vertical ones. Likewise Checkers is not impartial, since the players own different colored pieces. For any ordinal number, one can define an impartial game generalizing Nim in which, on each move, either player may replace the number with any smaller ordinal number; the games defined in this way are known as nimbers. The Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a nimber.
The "smallest" nimbers – the simplest and least under the usual ordering of the ordinals – are 0 and ∗.
== See also ==
Alpha–beta pruning, an optimised algorithm for searching the game tree
Backward induction, reasoning backwards from a final situation
Cooling and heating (combinatorial game theory), various transformations of games making them more amenable to the theory
Connection game, a type of game where players attempt to establish connections
Endgame tablebase, a database saying how to play endgames
Expectiminimax tree, an adaptation of a minimax game tree to games with an element of chance
Extensive-form game, a game tree enriched with payoffs and information available to players
Game classification, an article discussing ways of classifying games
Game complexity, an article describing ways of measuring the complexity of games
Grundy's game, a mathematical game in which heaps of objects are split
Multi-agent system, a type of computer system for tackling complex problems
Positional game, a type of game where players claim previously-unclaimed positions
Solving chess
Sylver coinage, a mathematical game of choosing positive integers that are not the sum of non-negative multiples of previously chosen integers
Wythoff's game, a mathematical game of taking objects from one or two piles
Topological game, a type of mathematical game played in a topological space
Zugzwang, being obliged to play when this is disadvantageous
== Notes ==
== References ==
Albert, Michael H.; Nowakowski, Richard J.; Wolfe, David (2007). Lessons in Play: An Introduction to Combinatorial Game Theory. A K Peters Ltd. ISBN 978-1-56881-277-9.
Beck, József (2008). Combinatorial games: tic-tac-toe theory. Cambridge University Press. ISBN 978-0-521-46100-9.
Berlekamp, E.; Conway, J. H.; Guy, R. (1982). Winning Ways for your Mathematical Plays: Games in general. Academic Press. ISBN 0-12-091101-9. 2nd ed., A K Peters Ltd (2001–2004), ISBN 1-56881-130-6, ISBN 1-56881-142-X
Berlekamp, E.; Conway, J. H.; Guy, R. (1982). Winning Ways for your Mathematical Plays: Games in particular. Academic Press. ISBN 0-12-091102-7. 2nd ed., A K Peters Ltd (2001–2004), ISBN 1-56881-143-8, ISBN 1-56881-144-6.
Berlekamp, Elwyn; Wolfe, David (1997). Mathematical Go: Chilling Gets the Last Point. A K Peters Ltd. ISBN 1-56881-032-6.
Bewersdorff, Jörg (2021). Luck, Logic and White Lies: The Mathematics of Games (2nd ed.). A K Peters/CRC Press. doi:10.1201/9781003092872. ISBN 978-1-003-09287-2. See especially sections 21–26.
Conway, John Horton (1976). On Numbers and Games. Academic Press. ISBN 0-12-186350-6. 2nd ed., A K Peters Ltd (2001), ISBN 1-56881-127-6.
Robert A. Hearn; Erik D. Demaine (2009). Games, Puzzles, and Computation. A K Peters, Ltd. ISBN 978-1-56881-322-6.
== External links ==
List of combinatorial game theory links at the homepage of David Eppstein
An Introduction to Conway's games and numbers by Dierk Schleicher and Michael Stoll
Combinational Game Theory terms summary by Bill Spight
Combinatorial Game Theory Workshop, Banff International Research Station, June 2005 | Wikipedia/Combinatorial_Game_Theory |
Compositional game theory is a branch of game theory and computer science, which aims to present large complex games as a composition of simple small games.
== Motivation ==
A major theme in computer science is the ability to construct simple building-blocks (e.g. functions or procedures in a programming language), and compose them into larger structures (e.g. more complex functions or programs). This principle is also called modularity.
In contrast, in classic game theory, even complex games are treated as single, monolithic objects. This makes the analysis of games hard to scale.
Compositional game theory (CGT) aims to apply the modularity principle to game theory. The main motivation is to make it easier to analyze large games using software tools.
== Higher-order game ==
A higher-order simultaneous game is a generalization of a Simultaneous game in which players are defined by selection functions rather than by utility functions. Formally, a higher-order simultaneous game for n players contains the following elements:
A set R of outcomes.
For each player i, a set Xi of choices (possible actions).
We define Σ as the Cartesian product of all Xi, and call it the set of strategy profiles.
An outcome function, from Σ to R. This function determines, for each combination of actions of the players, what the outcome will be.
For each player i, there is a selection function denoted di. The selection function takes as input a context, which is a function from Xi to R; and returns a set of best-responses, which is a subset of Xi.
The term "higher-order" comes from the latter element. The best-response correspondence of each player is a higher-order function, as is input is itself a function. Every strategy-profile s1 in Σ, defines for each player i a function from Xi to R: the function maps each possible action xi in Xi to the outcome that would result if all players except i play as in s1, whereas player i switches his action to xi. In other words, s1 defines the context in which player i operates.
Given two strategy-tuples s1 and s2 in Σ, we say that s2 is a best-response to s1 if, for each player i, s2,i is contained in the output of di on the context generated by s1. The best-response relation is a binary relation contained in Σ x Σ, denoted by B.
In a standard game, instead of the selection function, there is a utility function ui for each player i. A utility function takes as input an outcome from R, and returns a real number. Such a game can be represented as a higher-order game as follows. For each player i, the selection function returns the set of actions from Xi that maximize the utility of agent i, given the context.
== Open games ==
The main object of study in CGT is the open game. An open game has the following elements:
A set X of observations;
A set Y of outcomes;
A set Σ of strategy profiles.
A play function P, which is a function from Σ x X to Y;
A coplay function C, which is a function from Σ x X x R to S;
A best-response function B, which is a function from X x (Y -> R) to a relation in Σ x Σ.
It is an abstraction of a higher-order game.
Open games can be decomposed in two ways:
In sequence - yielding a sequential game;
In parallel - yielding a simultaneous game.
== See also ==
Bayesian open games.
== External links ==
Open game engine - Haskell code for constructing and analyzing open games.
== References == | Wikipedia/Compositional_game_theory |
The business model canvas is a strategic management template that is used for developing new business models and documenting existing ones. It offers a visual chart with elements describing a firm's or product's value proposition, infrastructure, customers, and finances, assisting businesses to align their activities by illustrating potential trade-offs.
The nine "building blocks" of the business model design template that came to be called the business model canvas were initially proposed in 2005 by Alexander Osterwalder, based on his PhD work supervised by Yves Pigneur on business model ontology. Since the release of Osterwalder's work around 2008, the authors have developed related tools such as the Value Proposition Canvas and the Culture Map, and new canvases for specific niches have also appeared.
== Description ==
Formal descriptions of a business become the building blocks for its activities. Many different business conceptualizations exist; Osterwalder's 2004 thesis and co-authored 2010 book propose a single reference model based on the similarities of a wide range of business model conceptualizations. With his business model design template, an enterprise can easily describe its business model.
Osterwalder's canvas has nine boxes: customer segments, value propositions, channels, customer relationships, revenue streams, key resources, key activities, key partnerships, and cost structure.: 16–17 Descriptions below are based largely on the 2010 book Business Model Generation.: 20–41
Infrastructure
Key activities: The most important activities in executing a company's value proposition. An example for Bic, the pen manufacturer, would be creating an efficient supply chain to drive down costs.
Key resources: The resources that are necessary to create value for the customer. They are considered assets to a company that are needed to sustain and support the business. These resources could be human, financial, physical and intellectual.
Partner network: In order to optimize operations and reduce risks of a business model, organizations usually cultivate buyer-supplier relationships so they can focus on their core activity. Complementary business alliances also can be considered through joint ventures or strategic alliances between competitors or non-competitors.
Offering
Value propositions: The collection of products and services a business offers to meet the needs of its customers. According to Osterwalder, a company's value proposition is what distinguishes it from its competitors. The value proposition provides value through various elements such as newness, performance, customization, "getting the job done", design, brand/status, price, cost reduction, risk reduction, accessibility, and convenience/usability.
The value propositions may be:
Quantitative – price and efficiency
Qualitative – overall customer experience and outcome
Customers
Customer segments: To build an effective business model, a company must identify which customers it tries to serve. Various sets of customers can be segmented based on their different needs and attributes to ensure appropriate implementation of corporate strategy to meet the characteristics of selected groups of clients. The different types of customer segments include:
Mass market: There is no specific segmentation for a company that follows the mass market element as the organization displays a wide view of potential clients: e.g. car.
Niche market: Customer segmentation based on specialized needs and characteristics of its clients: e.g. Rolex.
Segmented: A company applies additional segmentation within existing customer segment. In the segmented situation, the business may further distinguish its clients based on gender, age, and/or income.
Diversify: A business serves multiple customer segments with different needs and characteristics.
Multi-sided platform/market: For a smooth day-to-day business operation, some companies will serve mutually dependent customer segments. A credit card company will provide services to credit card holders while simultaneously assisting merchants who accept those credit cards.
Channels: A company can deliver its value proposition to its targeted customers through different channels. Effective channels will distribute a company's value proposition in ways that are fast, efficient and cost-effective. An organization can reach its clients through its own channels (store front), partner channels (major distributors), or a combination of both.
Customer relationships: To ensure the survival and success of any businesses, companies must identify the type of relationship they want to create with their customer segments. That element should address three critical steps of a customer's relationship: How the business will get new customers, how the business will keep customers purchasing or using its services and how the business will grow its revenue from its current customers. Various forms of customer relationships include:
Personal assistance: Assistance in a form of employee-customer interaction. Such assistance is performed during sales and/or after sales.
Dedicated personal assistance: The most intimate and hands-on personal assistance in which a sales representative is assigned to handle all the needs and questions of a special set of clients.
Self service: The type of relationship that translates from the indirect interaction between the company and the clients. Here, an organization provides the tools needed for the customers to serve themselves easily and effectively.
Automated services: A system similar to self-service but more personalized as it has the ability to identify individual customers and their preferences. An example of this would be Amazon.com making book suggestions based on the characteristics of previous book purchases.
Communities: Creating a community allows for direct interactions among different clients and the company. The community platform produces a scenario where knowledge can be shared and problems are solved between different clients.
Co-creation: A personal relationship is created through the customer's direct input to the final outcome of the company's products/services.
Finances
Cost structure: This describes the most important monetary consequences while operating under different business models.
Classes of business structures:
Cost-driven – This business model focuses on minimizing all costs and having no frills: e.g. low-cost airlines.
Value-driven – Less concerned with cost, this business model focuses on creating value for products and services: e.g. Louis Vuitton, Rolex.
Characteristics of cost structures:
Fixed costs – Costs are unchanged across different applications: e.g. salary, rent.
Variable costs – Costs vary depending on the amount of production of goods or services: e.g. music festivals.
Economies of scale – Costs go down as the amount of goods are ordered or produced.
Economies of scope – Costs go down due to incorporating other businesses which have a direct relation to the original product.
Revenue streams: The way a company makes income from each customer segment. Several ways to generate a revenue stream:
Asset sale – (the most common type) Selling ownership rights to a physical good: e.g. retail corporations.
Usage fee – Money generated from the use of a particular service: e.g. UPS.
Subscription fees – Revenue generated by selling access to a continuous service: e.g. Netflix.
Lending/leasing/renting – Giving exclusive right to an asset for a particular period of time: e.g. leasing a car.
Licensing – Revenue generated from charging for the use of a protected intellectual property.
Brokerage fees – Revenue generated from an intermediate service between 2 parties: e.g. broker selling a house for commission.
Advertising – Revenue generated from charging fees for product advertising.
== Application ==
The business model canvas can be printed out on a large surface so that groups of people can jointly start sketching and discussing business model elements with post-it notes or board markers. It is a hands-on tool that aims to foster understanding, discussion, creativity, and analysis. It is distributed under a Creative Commons license from Strategyzer AG and can be used without any restrictions for modeling businesses. It is also available in web-based software format.
== Alternative forms ==
The business model canvas has been used and adapted to suit specific business scenarios and applications, such as Ash Maurya's Lean Canvas for startup companies.
== Criticism ==
The business model canvas has been characterized as static because it does not capture changes in strategy or the evolution of the model nor much detail about the interaction between the components and how this makes the model work. Some limits of the template are its focus on organizations and its consequent conceptual isolation from its environment, whether this is related to the industry structure or to stakeholders such as society and natural environment.
== See also ==
Business process modeling
Business plan
Business reference model
Minimum viable product § Business Model Canvas
Nine windows – systems-engineering matrix diagram with nine boxes
Product/market fit
Unique selling proposition
== Citations ==
== References ==
Project Management Institute (2021). A guide to the project management body of knowledge (PMBOK guide). Project Management Institute (7th ed.). Newtown Square, PA. ISBN 978-1-62825-664-2.{{cite book}}: CS1 maint: location missing publisher (link)
== External links ==
Media related to Business Model Canvas at Wikimedia Commons
Alexander Osterwalder: Tools for Business Model Generation, a 53-minute video discussing the business model canvas in detail. Stanford Entrepreneurship Corner, 26 January 2012 | Wikipedia/Business_Model_Canvas |
The term conceptual model refers to any model that is formed after a conceptualization or generalization process. Conceptual models are often abstractions of things in the real world, whether physical or social. Semantic studies are relevant to various stages of concept formation. Semantics is fundamentally a study of concepts, the meaning that thinking beings give to various elements of their experience.
== Overview ==
=== Concept models and conceptual models ===
The value of a conceptual model is usually directly proportional to how well it corresponds to a past, present, future, actual or potential state of affairs. A concept model (a model of a concept) is quite different because in order to be a good model it need not have this real world correspondence. In artificial intelligence, conceptual models and conceptual graphs are used for building expert systems and knowledge-based systems; here the analysts are concerned to represent expert opinion on what is true not their own ideas on what is true.
=== Type and scope of conceptual models ===
Conceptual models range in type from the more concrete, such as the mental image of a familiar physical object, to the formal generality and abstractness of mathematical models which do not appear to the mind as an image. Conceptual models also range in terms of the scope of the subject matter that they are taken to represent. A model may, for instance, represent a single thing (e.g. the Statue of Liberty), whole classes of things (e.g. the electron), and even very vast domains of subject matter such as the physical universe. The variety and scope of conceptual models is due to the variety of purposes for using them.
Conceptual modeling is the activity of formally describing some aspects of the physical and social world around us for the purposes of understanding and communication.
=== Fundamental objectives ===
A conceptual model's primary objective is to convey the fundamental principles and basic functionality of the system which it represents. Also, a conceptual model must be developed in such a way as to provide an easily understood system interpretation for the model's users. A conceptual model, when implemented properly, should satisfy four fundamental objectives.
Enhance an individual's understanding of the representative system
Facilitate efficient conveyance of system details between stakeholders
Provide a point of reference for system designers to extract system specifications
Document the system for future reference and provide a means for collaboration
The conceptual model plays an important role in the overall system development life cycle. Figure 1 below, depicts the role of the conceptual model in a typical system development scheme. It is clear that if the conceptual model is not fully developed, the execution of fundamental system properties may not be implemented properly, giving way to future problems or system shortfalls. These failures do occur in the industry and have been linked to; lack of user input, incomplete or unclear requirements, and changing requirements. Those weak links in the system design and development process can be traced to improper execution of the fundamental objectives of conceptual modeling. The importance of conceptual modeling is evident when such systemic failures are mitigated by thorough system development and adherence to proven development objectives/techniques.
== Modelling techniques ==
Numerous techniques can be applied across multiple disciplines to increase the user's understanding of the system to be modeled. A few techniques are briefly described in the following text, however, many more exist or are being developed. Some commonly used conceptual modeling techniques and methods include: workflow modeling, workforce modeling, rapid application development, object-role modeling, and the Unified Modeling Language (UML).
=== Data flow modeling ===
Data flow modeling (DFM) is a basic conceptual modeling technique that graphically represents elements of a system. DFM is a fairly simple technique; however, like many conceptual modeling techniques, it is possible to construct higher and lower level representative diagrams. The data flow diagram usually does not convey complex system details such as parallel development considerations or timing information, but rather works to bring the major system functions into context. Data flow modeling is a central technique used in systems development that utilizes the structured systems analysis and design method (SSADM).
=== Entity relationship modeling ===
Entity–relationship modeling (ERM) is a conceptual modeling technique used primarily for software system representation. Entity-relationship diagrams, which are a product of executing the ERM technique, are normally used to represent database models and information systems. The main components of the diagram are the entities and relationships. The entities can represent independent functions, objects, or events. The relationships are responsible for relating the entities to one another. To form a system process, the relationships are combined with the entities and any attributes needed to further describe the process. Multiple diagramming conventions exist for this technique; IDEF1X, Bachman, and EXPRESS, to name a few. These conventions are just different ways of viewing and organizing the data to represent different system aspects.
=== Event-driven process chain ===
The event-driven process chain (EPC) is a conceptual modeling technique which is mainly used to systematically improve business process flows. Like most conceptual modeling techniques, the event driven process chain consists of entities/elements and functions that allow relationships to be developed and processed. More specifically, the EPC is made up of events which define what state a process is in or the rules by which it operates. In order to progress through events, a function/ active event must be executed. Depending on the process flow, the function has the ability to transform event states or link to other event driven process chains. Other elements exist within an EPC, all of which work together to define how and by what rules the system operates. The EPC technique can be applied to business practices such as resource planning, process improvement, and logistics.
=== Joint application development ===
The dynamic systems development method uses a specific process called JEFFF to conceptually model a systems life cycle. JEFFF is intended to focus more on the higher level development planning that precedes a project's initialization. The JAD process calls for a series of workshops in which the participants work to identify, define, and generally map a successful project from conception to completion. This method has been found to not work well for large scale applications, however smaller applications usually report some net gain in efficiency.
=== Place/transition net ===
Also known as Petri nets, this conceptual modeling technique allows a system to be constructed with elements that can be described by direct mathematical means. The petri net, because of its nondeterministic execution properties and well defined mathematical theory, is a useful technique for modeling concurrent system behavior, i.e. simultaneous process executions.
=== State transition modeling ===
State transition modeling makes use of state transition diagrams to describe system behavior. These state transition diagrams use distinct states to define system behavior and changes. Most current modeling tools contain some kind of ability to represent state transition modeling. The use of state transition models can be most easily recognized as logic state diagrams and directed graphs for finite-state machines.
=== Technique evaluation and selection ===
Because the conceptual modeling method can sometimes be purposefully vague to account for a broad area of use, the actual application of concept modeling can become difficult. To alleviate this issue, and shed some light on what to consider when selecting an appropriate conceptual modeling technique, the framework proposed by Gemino and Wand will be discussed in the following text. However, before evaluating the effectiveness of a conceptual modeling technique for a particular application, an important concept must be understood; Comparing conceptual models by way of specifically focusing on their graphical or top level representations is shortsighted. Gemino and Wand make a good point when arguing that the emphasis should be placed on a conceptual modeling language when choosing an appropriate technique. In general, a conceptual model is developed using some form of conceptual modeling technique. That technique will utilize a conceptual modeling language that determines the rules for how the model is arrived at. Understanding the capabilities of the specific language used is inherent to properly evaluating a conceptual modeling technique, as the language reflects the techniques descriptive ability. Also, the conceptual modeling language will directly influence the depth at which the system is capable of being represented, whether it be complex or simple.
==== Considering affecting factors ====
Building on some of their earlier work, Gemino and Wand acknowledge some main points to consider when studying the affecting factors: the content that the conceptual model must represent, the method in which the model will be presented, the characteristics of the model's users, and the conceptual model languages specific task. The conceptual model's content should be considered in order to select a technique that would allow relevant information to be presented. The presentation method for selection purposes would focus on the technique's ability to represent the model at the intended level of depth and detail. The characteristics of the model's users or participants is an important aspect to consider. A participant's background and experience should coincide with the conceptual model's complexity, else misrepresentation of the system or misunderstanding of key system concepts could lead to problems in that system's realization. The conceptual model language task will further allow an appropriate technique to be chosen. The difference between creating a system conceptual model to convey system functionality and creating a system conceptual model to interpret that functionality could involve two completely different types of conceptual modeling languages.
==== Considering affected variables ====
Gemino and Wand go on to expand the affected variable content of their proposed framework by considering the focus of observation and the criterion for comparison. The focus of observation considers whether the conceptual modeling technique will create a "new product", or whether the technique will only bring about a more intimate understanding of the system being modeled. The criterion for comparison would weigh the ability of the conceptual modeling technique to be efficient or effective. A conceptual modeling technique that allows for development of a system model which takes all system variables into account at a high level may make the process of understanding the system functionality more efficient, but the technique lacks the necessary information to explain the internal processes, rendering the model less effective.
When deciding which conceptual technique to use, the recommendations of Gemino and Wand can be applied in order to properly evaluate the scope of the conceptual model in question. Understanding the conceptual models scope will lead to a more informed selection of a technique that properly addresses that particular model. In summary, when deciding between modeling techniques, answering the following questions would allow one to address some important conceptual modeling considerations.
What content will the conceptual model represent?
How will the conceptual model be presented?
Who will be using or participating in the conceptual model?
How will the conceptual model describe the system?
What is the conceptual models focus of observation?
Will the conceptual model be efficient or effective in describing the system?
Another function of the simulation conceptual model is to provide a rational and factual basis for assessment of simulation application appropriateness.
== Models in philosophy and science ==
=== Mental model ===
In cognitive psychology and philosophy of mind, a mental model is a representation of something in the mind, but a mental model may also refer to a nonphysical external model of the mind itself.
=== Metaphysical models ===
A metaphysical model is a type of conceptual model which is distinguished from other conceptual models by its proposed scope; a metaphysical model intends to represent reality in the broadest possible way. This is to say that it explains the answers to fundamental questions such as whether matter and mind are one or two substances; or whether or not humans have free will.
=== Epistemological models ===
An epistemological model is a type of conceptual model whose proposed scope is the known and the knowable, and the believed and the believable.
=== Logical models ===
In logic, a model is a type of interpretation under which a particular statement is true. Logical models can be broadly divided into ones which only attempt to represent concepts, such as mathematical models; and ones which attempt to represent physical objects, and factual relationships, among which are scientific models.
Model theory is the study of (classes of) mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A system that gives meaning to the sentences of a formal language is called a model for the language. If a model for a language moreover satisfies a particular sentence or theory (set of sentences), it is called a model of the sentence or theory. Model theory has close ties to algebra and universal algebra.
=== Mathematical models ===
Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures.
=== Scientific models ===
A scientific model is a simplified abstract view of a complex reality. A scientific model represents empirical objects, phenomena, and physical processes in a logical way. Attempts to formalize the principles of the empirical sciences use an interpretation to model reality, in the same way logicians axiomatize the principles of logic. The aim of these attempts is to construct a formal system that will not produce theoretical consequences that are contrary to what is found in reality. Predictions or other statements drawn from such a formal system mirror or map the real world only insofar as these scientific models are true.
== Statistical models ==
A statistical model is a probability distribution function proposed as generating data. In a parametric model, the probability distribution function has variable parameters, such as the mean and variance in a normal distribution, or the coefficients for the various exponents of the independent variable in linear regression. A nonparametric model has a distribution function without parameters, such as in bootstrapping, and is only loosely confined by assumptions. Model selection is a statistical method for selecting a distribution function within a class of them; e.g., in linear regression where the dependent variable is a polynomial of the independent variable with parametric coefficients, model selection is selecting the highest exponent, and may be done with nonparametric means, such as with cross validation.
In statistics there can be models of mental events as well as models of physical events. For example, a statistical model of customer behavior is a model that is conceptual (because behavior is physical), but a statistical model of customer satisfaction is a model of a concept (because satisfaction is a mental not a physical event).
== Social and political models ==
=== Economic models ===
In economics, a model is a theoretical construct that represents economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified framework designed to illustrate complex processes, often but not always using mathematical techniques. Frequently, economic models use structural parameters. Structural parameters are underlying parameters in a model or class of models. A model may have various parameters and those parameters may change to create various properties.
== Models in systems architecture ==
A system model is the conceptual model that describes and represents the structure, behavior, and more views of a system. A system model can represent multiple views of a system by using two different approaches. The first one is the non-architectural approach and the second one is the architectural approach. The non-architectural approach respectively picks a model for each view. The architectural approach, also known as system architecture, instead of picking many heterogeneous and unrelated models, will use only one integrated architectural model.
=== Business process modelling ===
In business process modelling the enterprise process model is often referred to as the business process model. Process models are core concepts in the discipline of process engineering. Process models are:
Processes of the same nature that are classified together into a model.
A description of a process at the type level.
Since the process model is at the type level, a process is an instantiation of it.
The same process model is used repeatedly for the development of many applications and thus, has many instantiations.
One possible use of a process model is to prescribe how things must/should/could be done in contrast to the process itself which is really what happens. A process model is roughly an anticipation of what the process will look like. What the process shall be will be determined during actual system development.
== Models in information system design ==
=== Conceptual models of human activity systems ===
Conceptual models of human activity systems are used in soft systems methodology (SSM), which is a method of systems analysis concerned with the structuring of problems in management. These models are models of concepts; the authors specifically state that they are not intended to represent a state of affairs in the physical world. They are also used in information requirements analysis (IRA) which is a variant of SSM developed for information system design and software engineering.
=== Logico-linguistic models ===
Logico-linguistic modeling is another variant of SSM that uses conceptual models. However, this method combines models of concepts with models of putative real world objects and events. It is a graphical representation of modal logic in which modal operators are used to distinguish statement about concepts from statements about real world objects and events.
=== Data models ===
==== Entity–relationship model ====
In software engineering, an entity–relationship model (ERM) is an abstract and conceptual representation of data. Entity–relationship modeling is a database modeling method, used to produce a type of conceptual schema or semantic data model of a system, often a relational database, and its requirements in a top-down fashion. Diagrams created by this process are called entity-relationship diagrams, ER diagrams, or ERDs.
Entity–relationship models have had wide application in the building of information systems intended to support activities involving objects and events in the real world. In these cases they are models that are conceptual. However, this modeling method can be used to build computer games or a family tree of the Greek Gods, in these cases it would be used to model concepts.
==== Domain model ====
A domain model is a type of conceptual model used to depict the structural elements and their conceptual constraints within a domain of interest (sometimes called the problem domain). A domain model includes the various entities, their attributes and relationships, plus the constraints governing the conceptual integrity of the structural model elements comprising that problem domain. A domain model may also include a number of conceptual views, where each view is pertinent to a particular subject area of the domain or to a particular subset of the domain model which is of interest to a stakeholder of the domain model.
Like entity–relationship models, domain models can be used to model concepts or to model real world objects and events.
== See also ==
== References ==
== Further reading ==
J. Parsons, L. Cole (2005), "What do the pictures mean? Guidelines for experimental evaluation of representation fidelity in diagrammatical conceptual modeling techniques", Data & Knowledge Engineering 55: 327–342; doi:10.1016/j.datak.2004.12.008
A. Gemino, Y. Wand (2005), "Complexity and clarity in conceptual modeling: Comparison of mandatory and optional properties", Data & Knowledge Engineering 55: 301–326; doi:10.1016/j.datak.2004.12.009
D. Batra (2005), "Conceptual Data Modeling Patterns", Journal of Database Management 16: 84–106
Papadimitriou, Fivos. (2010). "Conceptual Modelling of Landscape Complexity". Landscape Research, 35(5):563-570. doi:10.1080/01426397.2010.504913
== External links ==
Models article in the Internet Encyclopedia of Philosophy | Wikipedia/Conceptual_model |
Theory of Games and Economic Behavior, published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinary research field of game theory. In the introduction of its 60th anniversary commemorative edition from the Princeton University Press, the book is described as "the classic work upon which modern-day game theory is based."
== Overview ==
The book is based partly on earlier research by von Neumann, published in 1928 under the German title "Zur Theorie der Gesellschaftsspiele" ("On the Theory of Board Games").
The derivation of expected utility from its axioms appeared in an appendix to the Second Edition (1947). Von Neumann and Morgenstern used objective probabilities, supposing that all the agents had the same probability distribution, as a convenience. However, Neumann and Morgenstern mentioned that a theory of subjective probability could be provided, and this task was completed by Jimmie Savage in 1954 and Johann Pfanzagl in 1967. Savage extended von Neumann and Morgenstern's axioms of rational preferences to endogenize probability and make it subjective. He then used Bayes' theorem to update these subject probabilities in light of new information, thus linking rational choice and inference.
== Reception ==
Herbert A. Simon praised the book.
== See also ==
Pfanzagl, J (1967). "Subjective Probability Derived from the Morgenstern-von Neumann Utility Theory". In Martin Shubik (ed.). Essays in Mathematical Economics In Honor of Oskar Morgenstern. Princeton University Press. pp. 237–251.
Pfanzagl, J. in cooperation with V. Baumann and H. Huber (1968). "Events, Utility and Subjective Probability". Theory of Measurement. Wiley. pp. 195–220.
Morgenstern, Oskar (1976). "Some Reflections on Utility". In Andrew Schotter (ed.). Selected Economic Writings of Oskar Morgenstern. New York University Press. pp. 65–70.
Morgenstern Oskar (1976). "The Collaboration Between Oskar Morgenstern and John von Neumann on the Theory of Games". Journal of Economic Literature. 14 (3): 805–816. JSTOR 2722628.
Commemorative edition of the book Theory of Games and Economic Behavior
Copeland A. H. (1945). "Review of 'The Theory of Games and Economic Behavior". Bulletin of the American Mathematical Society. 51: 498–504. doi:10.1090/s0002-9904-1945-08391-8.
Hurwicz Leonid (1945). "The Theory of Economic Behavior". American Economic Review. 35 (5): 909–925. JSTOR 1812602.
Kaysen Carl (1946). "A Revolution in Economic Theory?". Review of Economic Studies. 14 (1): 1–15. doi:10.2307/2295753. JSTOR 2295753.
Marschak Jacob (1946). "Neumann's and Morgenstern's New Approach to Static Economics" (PDF). Journal of Political Economy. 54 (2): 97–115. doi:10.1086/256327. S2CID 154536775.
Stone Richard (1948). "The Theory of Games". Economic Journal. 58 (230): 185–201. doi:10.2307/2225934. JSTOR 2225934.
== References ==
== External links ==
Theory of Games and Economic Behavior, full text at archive.org (public domain) | Wikipedia/Theory_of_games_and_economic_behavior |
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an
n
{\displaystyle n}
-dimensional simplex contains a cell whose vertices all have different colors.
The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms, and are applied in fair division (cake cutting) algorithms.
According to the Soviet Mathematical Encyclopaedia (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had also become known as the Sperner lemma – this point is discussed in the English translation (ed. M. Hazewinkel). It is now commonly known as the Knaster–Kuratowski–Mazurkiewicz lemma.
== Statement ==
=== One-dimensional case ===
In one dimension, Sperner's Lemma can be regarded as a discrete version of the intermediate value theorem. In this case, it essentially says that if a discrete function takes only the values 0 and 1, begins at the value 0 and ends at the value 1, then it must switch values an odd number of times.
=== Two-dimensional case ===
The two-dimensional case is the one referred to most frequently. It is stated as follows:
Subdivide a triangle ABC arbitrarily into a triangulation consisting of smaller triangles meeting edge to edge. Then a Sperner coloring of the triangulation is defined as an assignment of three colors to the vertices of the triangulation such that
Each of the three vertices A, B, and C of the initial triangle has a distinct color
The vertices that lie along any edge of triangle ABC have only two colors, the two colors at the endpoints of the edge. For example, each vertex on AC must have the same color as A or C.
Then every Sperner coloring of every triangulation has at least one "rainbow triangle", a smaller triangle in the triangulation that has its vertices colored with all three different colors. More precisely, there must be an odd number of rainbow triangles.
=== Multidimensional case ===
In the general case the lemma refers to a n-dimensional simplex:
A
=
A
1
A
2
…
A
n
+
1
.
{\displaystyle {\mathcal {A}}=A_{1}A_{2}\ldots A_{n+1}.}
Consider any triangulation T, a disjoint division of
A
{\displaystyle {\mathcal {A}}}
into smaller n-dimensional simplices, again meeting face-to-face. Denote the coloring function as:
f
:
S
→
{
1
,
2
,
3
,
…
,
n
,
n
+
1
}
,
{\displaystyle f:S\to \{1,2,3,\dots ,n,n+1\},}
where S is the set of vertices of T. A coloring function defines a Sperner coloring when:
The vertices of the large simplex are colored with different colors, that is, without loss of generality, f(Ai) = i for 1 ≤ i ≤ n + 1.
Vertices of T located on any k-dimensional subface of the large simplex
A
i
1
A
i
2
…
A
i
k
+
1
{\displaystyle A_{i_{1}}A_{i_{2}}\ldots A_{i_{k+1}}}
are colored only with the colors
i
1
,
i
2
,
…
,
i
k
+
1
.
{\displaystyle i_{1},i_{2},\ldots ,i_{k+1}.}
Then every Sperner coloring of every triangulation of the n-dimensional simplex has an odd number of instances of a rainbow simplex, meaning a simplex whose vertices are colored with all n + 1 colors. In particular, there must be at least one rainbow simplex.
== Proofs ==
=== Proof by induction ===
We shall first address the two-dimensional case. Consider a graph G built from the triangulation T as follows:
The vertices of G are the members of T plus the area outside the triangle. Two vertices are connected with an edge if their corresponding areas share a common border with one endpoint colored 1 and the other colored 2.
Note that on the interval AB there is an odd number of borders colored 1-2 (simply because A is colored 1, B is colored 2; and as we move along AB, there must be an odd number of color changes in order to get different colors at the beginning and at the end). On the intervals BC and CA, there are no borders colored 1-2 at all. Therefore, the vertex of G corresponding to the outer area has an odd degree. But it is known (the handshaking lemma) that in a finite graph there is an even number of vertices with odd degree. Therefore, the remaining graph, excluding the outer area, has an odd number of vertices with odd degree corresponding to members of T.
It can be easily seen that the only possible degree of a triangle from T is 0, 1, or 2, and that the degree 1 corresponds to a triangle colored with the three colors 1, 2, and 3.
Thus we have obtained a slightly stronger conclusion, which says that in a triangulation T there is an odd number (and at least one) of full-colored triangles.
A multidimensional case can be proved by induction on the dimension of a simplex. We apply the same reasoning, as in the two-dimensional case, to conclude that in a n-dimensional triangulation there is an odd number of full-colored simplices.
=== Commentary ===
Here is an elaboration of the proof given previously, for a reader new to graph theory.
This diagram numbers the colors of the vertices of the example given previously. The small triangles whose vertices all have different numbers are shaded in the graph. Each small triangle becomes a node in the new graph derived from the triangulation. The small letters identify the areas, eight inside the figure, and area i designates the space outside of it.
As described previously, those nodes that share an edge whose endpoints are numbered 1 and 2 are joined in the derived graph. For example, node d shares an edge with the outer area i, and its vertices all have different numbers, so it is also shaded. Node b is not shaded because two vertices have the same number, but it is joined to the outer area.
One could add a new full-numbered triangle, say by inserting a node numbered 3 into the edge between 1 and 1 of node a, and joining that node to the other vertex of a. Doing so would have to create a pair of new nodes, like the situation with nodes f and g.
=== Proof without induction ===
Andrew McLennan and Rabee Tourky presented a different proof, using the volume of a simplex. It proceeds in one step, with no induction.
== Computing a Sperner simplex ==
Suppose there is a d-dimensional simplex of side-length N, and it is triangulated into sub-simplices of side-length 1. There is a function that, given any vertex of the triangulation, returns its color. The coloring is guaranteed to satisfy Sperner's boundary condition. How many times do we have to call the function in order to find a rainbow simplex? Obviously, we can go over all the triangulation vertices, whose number is O(Nd), which is polynomial in N when the dimension is fixed. But, can it be done in time O(poly(log N)), which is polynomial in the binary representation of N?
This problem was first studied by Christos Papadimitriou. He introduced a complexity class called PPAD, which contains this as well as related problems (such as finding a Brouwer fixed point). He proved that finding a Sperner simplex is PPAD-complete even for d=3. Some 15 years later, Chen and Deng proved PPAD-completeness even for d=2. It is believed that PPAD-hard problems cannot be solved in time O(poly(log N)).
== Generalizations ==
=== Subsets of labels ===
Suppose that each vertex of the triangulation may be labeled with multiple colors, so that the coloring function is F : S → 2[n+1].
For every sub-simplex, the set of labelings on its vertices is a set-family over the set of colors [n + 1]. This set-family can be seen as a hypergraph.
If, for every vertex v on a face of the simplex, the colors in f(v) are a subset of the set of colors on the face endpoints, then there exists a sub-simplex with a balanced labeling – a labeling in which the corresponding hypergraph admits a perfect fractional matching. To illustrate, here are some balanced labeling examples for n = 2:
({1}, {2}, {3}) - balanced by the weights (1, 1, 1).
({1,2}, {2,3}, {3,1}) - balanced by the weights (1/2, 1/2, 1/2).
({1,2}, {2,3}, {1}) - balanced by the weights (0, 1, 1).
This was proved by Shapley in 1973. It is a combinatorial analogue of the KKMS lemma.
=== Polytopal variants ===
Suppose that we have a d-dimensional polytope P with n vertices. P is triangulated, and each vertex of the triangulation is labeled with a label from {1, …, n}. Every main vertex i is labeled i. A sub-simplex is called fully-labeled if it is d-dimensional, and each of its d + 1 vertices has a different label. If every vertex in a face F of P is labeled with one of the labels on the endpoints of F, then there are at least n – d fully-labeled simplices. Some special cases are:
d = n – 1. In this case, P is a simplex. The polytopal Sperner lemma guarantees that there is at least 1 fully-labeled simplex. That is, it reduces to Sperner's lemma.
d = 2. Suppose a two-dimensional polygon with n vertices is triangulated and labeled using the labels 1, …, n such that, on each face between vertex i and vertex i + 1 (mod n), only the labels i and i + 1 are used. Then, there are at least n – 2 sub-triangles in which three different labels are used.
The general statement was conjectured by Atanassov in 1996, who proved it for the case d = 2. The proof of the general case was first given by de Loera, Peterson, and Su in 2002. They provide two proofs: the first is non-constructive and uses the notion of pebble sets; the second is constructive and is based on arguments of following paths in graphs.
Meunier extended the theorem from polytopes to polytopal bodies, which need not be convex or simply-connected. In particular, if P is a polytope, then the set of its faces is a polytopal body. In every Sperner labeling of a polytopal body with vertices v1, …, vn, there are at least:
n
−
d
−
1
+
⌈
min
i
=
1
n
deg
B
(
P
)
(
v
i
)
d
⌉
{\displaystyle n-d-1+\left\lceil {\frac {\min _{i=1}^{n}\deg _{B(P)}(v_{i})}{d}}\right\rceil }
fully-labeled simplices such that any pair of these simplices receives two different labelings. The degree degB(P)(vi) is the number of edges of B(P) to which vi belongs. Since the degree is at least d, the lower bound is at least n – d. But it can be larger. For example, for the cyclic polytope in 4 dimensions with n vertices, the lower bound is:
n
−
4
−
1
+
⌈
n
−
1
4
⌉
≈
5
4
n
.
{\displaystyle n-4-1+\left\lceil {\frac {n-1}{4}}\right\rceil \approx {\frac {5}{4}}n.}
Musin further extended the theorem to d-dimensional piecewise-linear manifolds, with or without a boundary.
Asada, Frick, Pisharody, Polevy, Stoner, Tsang and Wellner further extended the theorem to pseudomanifolds with boundary, and improved the lower bound on the number of facets with pairwise-distinct labels.
=== Cubic variants ===
Suppose that, instead of a simplex triangulated into sub-simplices, we have an n-dimensional cube partitioned into smaller n-dimensional cubes.
Harold W. Kuhn proved the following lemma. Suppose the cube [0,M]n, for some integer M, is partitioned into Mn unit cubes. Suppose each vertex of the partition is labeled with a label from {1, …, n + 1}, such that for every vertex v: (1) if vi = 0 then the label on v is at most i; (2) if vi=M then the label on v is not i. Then there exists a unit cube with all the labels {1, …, n + 1} (some of them more than once). The special case n = 2 is: suppose a square is partitioned into sub-squares, and each vertex is labeled with a label from {1,2,3}. The left edge is labeled with 1 (= at most 1); the bottom edge is labeled with 1 or 2 (= at most 2); the top edge is labeled with 1 or 3 (= not 2); and the right edge is labeled with 2 or 3 (= not 1). Then there is a square labeled with 1,2,3.
Another variant, related to the Poincaré–Miranda theorem, is as follows. Suppose the cube [0,M]n is partitioned into Mn unit cubes. Suppose each vertex is labeled with a binary vector of length n, such that for every vertex v: (1) if vi = 0 then the coordinate i of label on v is 0; (2) if vi = M then coordinate i of the label on v is 1; (3) if two vertices are neighbors, then their labels differ by at most one coordinate. Then there exists a unit cube in which all 2n labels are different. In two dimensions, another way to formulate this theorem is: in any labeling that satisfies conditions (1) and (2), there is at least one cell in which the sum of labels is 0 [a 1-dimensional cell with (1,1) and (-1,-1) labels, or a 2-dimensional cells with all four different labels].
Wolsey strengthened these two results by proving that the number of completely-labeled cubes is odd.
Musin extended these results to general quadrangulations.
=== Rainbow variants ===
Suppose that, instead of a single labeling, we have n different Sperner labelings. We consider pairs (simplex, permutation) such that, the label of each vertex of the simplex is chosen from a different labeling (so for each simplex, there are n! different pairs). Then there are at least n! fully labeled pairs. This was proved by Ravindra Bapat for any triangulation. A simpler proof, which only works for specific triangulations, was presented later by Su.
Another way to state this lemma is as follows. Suppose there are n people, each of whom produces a different Sperner labeling of the same triangulation. Then, there exists a simplex, and a matching of the people to its vertices, such that each vertex is labeled by its owner differently (one person labels its vertex by 1, another person labels its vertex by 2, etc.). Moreover, there are at least n! such matchings. This can be used to find an envy-free cake-cutting with connected pieces.
Asada, Frick, Pisharody, Polevy, Stoner, Tsang and Wellner extended this theorem to pseudomanifolds with boundary.
More generally, suppose we have m different Sperner labelings, where m may be different than n. Then:: Thm 2.1
For any positive integers k1, …, km whose sum is m + n – 1, there exists a baby-simplex on which, for every i ∈ {1, …, m}, labeling number i uses at least ki (out of n) distinct labels. Moreover, each label is used by at least one (out of m) labeling.
For any positive integers I1, …, Imwhose sum is m + n – 1, there exists a baby-simplex on which, for every j ∈ {1, …, n},, the label j is used by at least lj (out of m) different labelings.
Both versions reduce to Sperner's lemma when m = 1, or when all m labelings are identical.
See for similar generalizations.
=== Oriented variants ===
Brown and Cairns strengthened Sperner's lemma by considering the orientation of simplices. Each sub-simplex has an orientation that can be either +1 or -1 (if it is fully-labeled), or 0 (if it is not fully-labeled). They proved that the sum of all orientations of simplices is +1. In particular, this implies that there is an odd number of fully-labeled simplices.
As an example for n = 3, suppose a triangle is triangulated and labeled with {1,2,3}. Consider the cyclic sequence of labels on the boundary of the triangle. Define the degree of the labeling as the number of switches from 1 to 2, minus the number of switches from 2 to 1. See examples in the table at the right. Note that the degree is the same if we count switches from 2 to 3 minus 3 to 2, or from 3 to 1 minus 1 to 3.
Musin proved that the number of fully labeled triangles is at least the degree of the labeling. In particular, if the degree is nonzero, then there exists at least one fully labeled triangle.
If a labeling satisfies the Sperner condition, then its degree is exactly 1: there are 1-2 and 2-1 switches only in the side between vertices 1 and 2, and the number of 1-2 switches must be one more than the number of 2-1 switches (when walking from vertex 1 to vertex 2). Therefore, the original Sperner lemma follows from Musin's theorem.
=== Trees and cycles ===
There is a similar lemma about finite and infinite trees and cycles.
== Related results ==
Mirzakhani and Vondrak study a weaker variant of a Sperner labeling, in which the only requirement is that label i is not used on the face opposite to vertex i. They call it Sperner-admissible labeling. They show that there are Sperner-admissible labelings in which every cell contains at most 4 labels. They also prove an optimal lower bound on the number of cells that must have at least two different labels in each Sperner-admissible labeling. They also prove that, for any Sperner-admissible partition of the regular simplex, the total area of the boundary between the parts is minimized by the Voronoi partition.
== Applications ==
Sperner colorings have been used for effective computation of fixed points. A Sperner coloring can be constructed such that fully labeled simplices correspond to fixed points of a given function. By making a triangulation smaller and smaller, one can show that the limit of the fully labeled simplices is exactly the fixed point. Hence, the technique provides a way to approximate fixed points.
A related application is the numerical detection of periodic orbits and symbolic dynamics. Sperner's lemma can also be used in root-finding algorithms and fair division algorithms; see Simmons–Su protocols.
Sperner's lemma is one of the key ingredients of the proof of Monsky's theorem, that a square cannot be cut into an odd number of equal-area triangles.
Sperner's lemma can be used to find a competitive equilibrium in an exchange economy, although there are more efficient ways to find it.: 67
Fifty years after first publishing it, Sperner presented a survey on the development, influence and applications of his combinatorial lemma.
== Equivalent results ==
There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in
the top row can be deduced from the one below it in the same column.
== See also ==
Topological combinatorics
== References ==
== External links ==
Proof of Sperner's Lemma at cut-the-knot
Sperner's lemma and the Triangle Game, at the n-rich site.
Sperner's lemma in 2D, a web-based game at itch.io. | Wikipedia/Sperner's_lemma |
A weak evolutionarily stable strategy (WESS) is a more broad form of evolutionarily stable strategy (ESS). Like ESS, a WESS is able to defend against an invading "mutant" strategy. This means the WESS cannot be entirely eliminated from the population.
The definition of WESS is similar to ESS. Any strategy s is a weakly evolutionarily stable strategy (WESS) if for any strategy s*≠s:
(i) u(s, s) > u(s*, s) or
(ii) u(s, s) = u(s*, s) and u(s, s*) ≥ u(s*, s*).
One example of WESS, in a prisoner's dilemma, is Tit-for-tat (a strategy that cooperates in the first interaction and then reciprocates the other player's action from the previous turn in all other iterations).
== References == | Wikipedia/Weak_evolutionarily_stable_strategy |
The black swan theory or theory of black swan events is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalized after the fact with the benefit of hindsight. The term arose from Latin expression which was based on the presumption that black swans did not exist. The expression was used in the original manner until around 1697 when Dutch mariners saw black swans living in Australia. After this, the term was reinterpreted to mean an unforeseen and consequential event.
The reinterpreted theory was articulated by Nassim Nicholas Taleb, starting in 2001, to explain:
The disproportionate role of high-profile, hard-to-predict, and rare events that are beyond the realm of normal expectations in history, science, finance, and technology.
The non-computability of the probability of consequential rare events using scientific methods (owing to the very nature of small probabilities).
The psychological biases that blind people, both individually and collectively, to uncertainty and to the substantial role of rare events in historical affairs.
Taleb's "black swan theory" (which differs from the earlier philosophical versions of the problem) refers only to statistically unexpected events of large magnitude and consequence and their dominant role in history. Such events, considered extreme outliers, collectively play vastly larger roles than regular occurrences.: xxi More technically, in the scientific monograph "Silent Risk", Taleb mathematically defines the black swan problem as "stemming from the use of degenerate metaprobability".
== Background ==
The phrase "black swan" derives from a Latin expression; its oldest known occurrence is from the 2nd-century Roman poet Juvenal's characterization in his Satire VI of something being "rara avis in terris nigroque simillima cygno" ("a bird as rare upon the earth as a black swan").: 165 When the phrase was coined, the black swan was presumed by Romans not to exist.
Juvenal's phrase was a common expression in 16th century London as a statement of impossibility. The London expression derives from the Old World presumption that all swans must be white because all historical records of swans reported that they had white feathers. In that context, a black swan was impossible or at least nonexistent.
However, in 1697, Dutch explorers led by Willem de Vlamingh became the first Europeans to see black swans, in Western Australia. The term subsequently metamorphosed to connote the idea that a perceived impossibility might later be disproved. Taleb notes that in the 19th century, John Stuart Mill used the black swan logical fallacy as a new term to identify falsification.
Black swan events were discussed by Taleb in his 2001 book Fooled By Randomness, which concerned financial events. His 2007 book The Black Swan extended the metaphor to events outside financial markets. Taleb regards almost all major scientific discoveries, historical events, and artistic accomplishments as "black swans"—undirected and unpredicted. He gives the rise of the Internet, the personal computer, World War I, the dissolution of the Soviet Union, and the September 11, 2001 attacks as examples of black swan events.: prologue
Taleb asserts:What we call here a Black Swan (and capitalize it) is an event with the following three attributes.
First, it is an outlier, as it lies outside the realm of regular expectations, because nothing in the past can convincingly point to its possibility. Second, it carries an extreme 'impact'. Third, in spite of its outlier status, human nature makes us concoct explanations for its occurrence after the fact, making it explainable and predictable.
I stop and summarize the triplet: rarity, extreme 'impact', and retrospective (though not prospective) predictability. A small number of Black Swans explains almost everything in our world, from the success of ideas and religions, to the dynamics of historical events, to elements of our own personal lives.
== Identifying ==
Based on the author's criteria:
The event is a surprise (to the observer).
The event has a major effect.
After the first recorded instance of the event, it is rationalized by hindsight, as if it could have been expected; that is, the relevant data were available but unaccounted for in risk mitigation programs. The same is true for the personal perception by individuals.
According to Taleb, the COVID-19 pandemic was not a black swan, as it was expected with great certainty that a global pandemic would eventually take place. Instead, it is considered a white swan—such an event has a major effect, but is compatible with statistical properties.
== Coping with black swans ==
The practical aim of Taleb's book is not to attempt to predict events which are unpredictable, but to build robustness against negative events while still exploiting positive events. Taleb contends that banks and trading firms are very vulnerable to hazardous black swan events and are exposed to unpredictable losses. On the subject of business, and quantitative finance in particular, Taleb critiques the widespread use of the normal distribution model employed in financial engineering, calling it a Great Intellectual Fraud. Taleb elaborates the robustness concept as a central topic of his later book, Antifragile: Things That Gain From Disorder.
In the second edition of The Black Swan, Taleb provides "Ten Principles for a Black-Swan-Robust Society".: 374–78
Taleb states that a black swan event depends on the observer. For example, what may be a Black Swan surprise for a turkey is not a Black Swan surprise to its butcher; hence the objective should be to "avoid being the turkey" by identifying areas of vulnerability to "turn the Black Swans white".
== Epistemological approach ==
Taleb claims that his black swan is different from the earlier philosophical versions of the problem, specifically in epistemology (as associated with David Hume, John Stuart Mill, Karl Popper, and others), as it concerns a phenomenon with specific statistical properties which he calls, "the fourth quadrant".
Taleb's problem is about epistemic limitations in some parts of the areas covered in decision making. These limitations are twofold: philosophical (mathematical) and empirical (human-known) epistemic biases. The philosophical problem is about the decrease in knowledge when it comes to rare events because these are not visible in past samples and therefore require a strong a priori (extrapolating) theory; accordingly, predictions of events depend more and more on theories when their probability is small. In the "fourth quadrant", knowledge is uncertain and consequences are large, requiring more robustness.
According to Taleb, thinkers who came before him who dealt with the notion of the improbable (such as Hume, Mill, and Popper) focused on the problem of induction in logic, specifically, that of drawing general conclusions from specific observations. The central and unique attribute of Taleb's black swan event is that it is high-impact. His claim is that almost all consequential events in history come from the unexpected – yet humans later convince themselves that these events are explainable in hindsight.
One problem, labeled the ludic fallacy by Taleb, is the belief that the unstructured randomness found in life resembles the structured randomness found in games. This stems from the assumption that the unexpected may be predicted by extrapolating from variations in statistics based on past observations, especially when these statistics are presumed to represent samples from a normal distribution. These concerns often are highly relevant in financial markets, where major players sometimes assume normal distributions when using value at risk models, although market returns typically have fat tail distributions.
Taleb said:I don't particularly care about the usual. If you want to get an idea of a friend's temperament, ethics, and personal elegance, you need to look at him under the tests of severe circumstances, not under the regular rosy glow of daily life. Can you assess the danger a criminal poses by examining only what he does on an ordinary day? Can we understand health without considering wild diseases and epidemics? Indeed the normal is often irrelevant. Almost everything in social life is produced by rare but consequential shocks and jumps; all the while almost everything studied about social life focuses on the 'normal,' particularly with 'bell curve' methods of inference that tell you close to nothing. Why? Because the bell curve ignores large deviations, cannot handle them, yet makes us confident that we have tamed uncertainty. Its nickname in this book is GIF, Great Intellectual Fraud.More generally, decision theory, which is based on a fixed universe or a model of possible outcomes, ignores and minimizes the effect of events that are "outside the model". For instance, a simple model of daily stock market returns may include extreme moves such as Black Monday (1987), but might not model the breakdown of markets following the September 11, 2001 attacks. Consequently, the New York Stock Exchange and Nasdaq exchange remained closed till September 17, 2001, the most protracted shutdown since the Great Depression. A fixed model considers the "known unknowns", but ignores the "unknown unknowns", made famous by a statement of Donald Rumsfeld. The term "unknown unknowns" appeared in a 1982 New Yorker article on the aerospace industry, which cites the example of metal fatigue, the cause of crashes in Comet airliners in the 1950s.
Deterministic chaotic dynamics reproducing the Black Swan Event have been researched in economics. That is in agreement with Taleb's comment regarding some distributions which are not usable with precision, but which are more descriptive, such as the fractal, power law, or scalable distributions and that awareness of these might help to temper expectations. Beyond this, Taleb emphasizes that many events simply are without precedent, undercutting the basis of this type of reasoning altogether.
Taleb also argues for the use of counterfactual reasoning when considering risk.: p. xvii
== See also ==
== References ==
== Bibliography ==
Taleb, Nassim Nicholas (2010) [2007], The Black Swan: The Impact of the Highly Improbable (2nd ed.), London: Penguin, ISBN 978-0-14103459-1, retrieved 26 February 2017.
Taleb, Nassim Nicholas (September 2008), "The Fourth Quadrant: A Map of the Limits of Statistics", Third Culture, The Edge Foundation, retrieved 23 May 2012.
The U.S. response to NEOs- avoiding a black swan event
== External links ==
McGee, Suzanne (5 December 2012), Black Swan Stocks Could Make Your Portfolio a Turkey, Fiscal Times, CNBC, retrieved 20 January 2016. | Wikipedia/Black_swan_theory |
In operations research, drama theory is one of several problem structuring methods. It is based on game theory and adapts the use of games to complex organisational situations, accounting for emotional responses that can provoke irrational reactions and lead the players to redefine the game. In a drama, emotions trigger rationalizations that create changes in the game, and so change follows change until either all conflicts are resolved or action becomes necessary. The game as redefined is then played.
Drama theory was devised by professor Nigel Howard in the early 1990s and, since then, has been turned to defense, political, health, industrial relations and commercial applications. Drama theory is an extension of Howard's metagame analysis work developed at the University of Pennsylvania in the late 1960s, and presented formally in his book Paradoxes of Rationality, published by MIT Press. Metagame analysis was originally used to advise on the Strategic Arms Limitation Talks (SALT).
== Basics of drama theory ==
A drama unfolds through episodes in which characters interact. The episode is a period of preplay communication between characters who, after communicating, act as players in a game that is constructed through the dialogue between them. The action that follows the episode is the playing out of this game; it sets up the next episode. Most drama-theoretic terminology is derived from a theatrical model applied to real life interactions; thus, an episode goes through phases of scene-setting, build-up, climax and decision. This is followed by denouement, which is the action that sets up the next episode. The term drama theory and the use of theatrical terminology is justified by the fact that the theory applies to stage plays and fictional plots as well as to politics, war, business, personal and community relations, psychology, history and other kinds of human interaction. It was applied to help with the structuring of The Prisoner's Dilemma, a West End play by David Edgar about the problems of peace-keeping.
In the build-up phase of an episode, the characters exchange ideas and opinions in some form or another and try to advocate their preferred position – the game outcome that they are hoping to see realised. The position each character takes may be influenced by others' positions. Each character also presents a fallback or stated intention. This is the action (i.e., individual strategy) a character says it will implement if current positions and stated intentions do not change. Taken together, the stated intentions form what is called a threatened future if they contradict some character's position; if they do not – i.e., if they implement every position – they form what is called an agreement.
When it is common knowledge among the characters that positions and stated intentions are seen by their presenters as 'final', the build-up ends and the parties reach a moment of truth. Here they usually face dilemmas arising from the fact that their threats or promises are incredible or inadequate. Different dilemmas are possible depending on whether or not there is an agreement. If there is an agreement (i.e., stated intentions implement every position), the possible dilemmas resemble those found in the prisoner's dilemma game; they arise from characters distrusting each other's declared intention to implement the agreement. If there is no agreement, more dilemmas are possible, resembling those in the game of chicken; they arise from the fact that a character's threat or its determination to stick to its position and reject other positions may be incredible to another character.
Drama theory asserts that a character faced with a dilemma feels specific positive or negative emotions that it tries to rationalize by persuading itself and others that the game should be redefined in a way that eliminates the dilemma; for example, a character with an incredible threat makes it credible by becoming angry and finding reasons why it should prefer to carry out the threat; likewise, a character with an incredible promise feels positive emotion toward the other as it looks for reasons why it should prefer to carry its promise. Emotional tension leads to the climax, where characters re-define the moment of truth by finding rationalizations for changing positions, stated intentions, preferences, options or the set of characters. There is some experimental evidence to confirm this assertion of drama theory.
Six dilemmas (formerly called paradoxes) are defined, and it is proved that if none of them exist then the characters have an agreement that they fully trust each other to carry out. This is the fundamental theorem of drama theory. Until a resolution meeting these conditions is arrived at, the characters are under emotional pressure to rationalize re-definitions of the game that they will play. Re-definitions inspired by new dilemmas then follow each other until eventually, with or without a resolution, characters become players in the game they have defined for themselves. In game-theoretic terms, this is a game with a focal point – i.e., it is a game in which each player has stated its intention to implement a certain strategy. This strategy is its threat (part of the threatened future) if an agreement has not been reached, and its promise (part of the agreement), if an agreement has been reached. At this point, players (since they are playing a game) decide whether to believe each other, and so to predict what others will do in order to decide what to do themselves.
== Dilemmas defined in drama theory ==
The dilemmas that character A may face with respect to another character B at a moment of truth are as follows.
A's cooperation dilemma: B does not believe A would carry out its actual or putative promise to implement B's position
A's trust dilemma: A does not believe B would carry out its actual or putative promise to implement A's position
A's persuasion (also known as "deterrence") dilemma: B certainly prefers the threatened future to A's position
A's rejection (also known as "inducement") dilemma: A may prefer B's position to the threatened future
A's threat dilemma: B does not believe A would carry out its threat not to implement B's position
A's positioning dilemma: A prefers B's position to its own, but rejects it (usually because A considers B's position to be unrealistic)
== Relationship to game theory ==
Drama-theorists build and analyze models (called card tables or options boards) that are isomorphic to game models, but unlike game theorists and most other model-builders, do not do so with the aim of finding a 'solution'. Instead, the aim is to find the dilemmas facing characters and so help to predict how they will re-define the model itself – i.e., the game that will be played. Such prediction requires not only analysis of the model and its dilemmas, but also exploration of the reality outside the model; without this it is impossible to decide which ways of changing the model in order to eliminate dilemmas might be rationalized by the characters.
The relation between drama theory and game theory is complementary in nature. Game theory does not explain how the game that is played is arrived at – i.e., how players select a small number of players and strategies from the virtually infinite set they could select, and how they arrive at common knowledge about each other's selections and preferences for the resulting combinations of strategies. Drama theory tries to explain this, and also to explain how the focal point is arrived at for the game with a focal point that is finally played. However, drama theory does not explain how players will act when they finally have to play a particular game with a focal point, even though it has to make assumptions about this. This is what game theory tries to explain and predict.
== See also ==
Confrontation analysis
== References ==
N. Howard, 'Confrontation Analysis', CCRP Publications, 1999. Available from the CCRP website.
P. Bennett, J. Bryant and N. Howard, 'Drama Theory and Confrontation Analysis' – can be found (along with other recent PSM methods) in: J. V. Rosenhead and J. Mingers (eds) Rational Analysis for a Problematic World Revisited: problem structuring methods for complexity, uncertainty and conflict, Wiley, 2001.
== Further reading ==
J. Bryant, The Six Dilemmas of Collaboration: inter-organisational relationships as drama, Wiley, 2003.
N. Howard, Paradoxes of Rationality', MIT Press, 1971. | Wikipedia/Drama_theory |
Behavioral game theory seeks to examine how people's strategic decision-making behavior is shaped by social preferences, social utility and other psychological factors. Behavioral game theory analyzes interactive strategic decisions and behavior using the methods of game theory, experimental economics, and experimental psychology. Experiments include testing deviations from typical simplifications of economic theory such as the independence axiom and neglect of altruism, fairness, and framing effects. As a research program, the subject is a development of the last three decades.
Traditional game theory is a critical principle of economic theory, and assumes that people's strategic decisions are shaped by rationality, selfishness and utility maximisation. It focuses on the mathematical structure of equilibria, and tends to use basic rational choice theory and utility maximization as the primary principles within economic models. At the same time rational choice theory is an ideal model that assumes that individuals will actively choose the option with the greatest benefit. The fact is that consumers have different preferences and rational choice theory is not accurate in its assumptions about consumer behavior. In contrast to traditional game theory, behavioral game theory examines how actual human behavior tends to deviate from standard predictions and models. In order to more accurately understand these deviations and determine the factors and conditions involved in strategic decision making, behavioral game theory aims to create new models that incorporate psychological principles. Studies of behavioral game theory demonstrate that choices are not always rational and do not always represent the utility maximizing choice.
Behavioral game theory largely utilizes empirical and theoretical research to understand human behavior. It also uses laboratory and field experiments, as well as modeling – both theoretical and computational. Recently, methods from machine learning have been applied in work at the intersection of economics, psychology, and computer science to improve both prediction and understanding of behavior in games.
== History ==
Behavioral game theory began with the work of Allais in 1953 and Ellsberg in 1961. They discovered the Allais paradox and the Ellsberg paradox, respectively. Both paradoxes show that choices made by participants in a game do not reflect the benefit they expect to receive from making those choices. In 1956, the work of Vernon Smith showed that economic markets could be examined experimentally rather than only theoretically, and reinforced the importance of rationality and self-interest within economic models. According to rational choice theory, consumers' behavior depends on three reasons. The first reason is that the degree of emotional pleasure consumers derive from their purchases depends on their preferences. The second reason is that consumers do not have enough choices. The third reason is that consumers derive greater pleasure from a limited number of choices. Later in the 1970s, economists Tversky and Kahneman, as well as several other co-workers, conducted experiments that discovered variations of traditional decision-making models such as regret theory, prospect theory, and hyperbolic discounting. These discoveries showed that decision makers consider many factors when making choices. For example, a person may seek to minimize the amount of regret they will feel after making a decision and weigh their options based on the amount of regret they anticipate from each. Due to the fact that these theories were not previously examined by traditional economic theory, factors such as regret along with many others fueled further research on the subject of social preferences and decision making.
Beginning in the 1980s experimenters started examining the conditions that cause divergence from rational choice. Ultimatum and bargaining games examined the effect of emotions on predictions of opponent behavior. One of the most well known examples of an ultimatum game is the television show Deal or No Deal in which participants must make decisions to sell or continue playing based on monetary ultimatums given to them by "the banker." These games also explored the effect of trust on decision-making outcomes and utility maximizing behavior. Common resource games were used to experimentally test how cooperation and social desirability affect subject's choices. A real-life example of a common resource game might be a party guest's decision to take from a food platter. The guest's decisions be affected by how hungry they are, how much of the shared resource (the food) is left and if the guest believes others would judge them for taking more. Experimenters believed that any behavior that did not maximize utility as the result of participant's flawed reasoning. By the turn of the century economists and psychologists expanded this research. Models based on the rational choice theory were adapted to reflect decision maker preferences and attempt to rationalize choices that did not maximize utility.
== Comparison to traditional game theory ==
There are various distinctions between traditional game theory and behavioral game theory. Traditional game theory uses theoretical and mathematical models to determine the most beneficial choice of all players in a game. Game theory uses rational choice theory to predict people's decisions in conditions of uncertainty. It understands strategic behavior to be influenced by utility-maximising preferences, as well as player's assumed knowledge of their opponents and material constraints. It also allows for players to predict their opponents' strategies. Also consumers' decisions are affected by psychological issues, and inattentional blindness is important in influencing the outcome of decisions. This is due to the fact that when consumers' attention is focused on one thing, they ignore other choices. Inattentional blindness believes that human attention and cognition are limited, which explains why consumers will make choices based on their personal preferences. Traditional game theory is a primarily normative theory as it seeks to pinpoint the decision that rational players should choose, but does not attempt to explain why that decision was made. Rationality is a primary assumption of game theory, so there are not explanations for different forms of rational decisions or irrational decisions.
In contrast to traditional game theory, behavioral game theory uses empirical models to explain how social preferences, such as ideals of fairness, efficiency or equity, influence human decisions and strategic reasoning. Behavioral game theory attempts to explain factors that influence real-world decisions. These factors are not explored in the area of traditional game theory, but can be postulated and observed using empirical data. Findings from behavioral game theory will tend to have higher external validity and can be better applied to real world decision-making behavior. Behavioral game theory is a primarily positive theory rather than a normative theory. A positive theory seeks to describe phenomena rather than prescribe a correct action. Positive theories must be testable and can be proven true or false. A normative theory is subjective and based on opinions. Because of this, normative theories cannot be proven true or false. Behavioral game theory attempts to explain decision making using experimental data. The theory allows for rational and irrational decisions because both are examined using real-life experiments in the form of simple games. Simple games are often used in behavioral game theory research as a way of analyzing unexplored phenomena, such as social preferences and social utility, that are not explored in traditional game theory.
== Examining social utility and preferences through games ==
Simple games are regularly utilized in behavioral game theory experiments in order to examine player's social utility. The simplicity of these games means that players do not face intellectual challenges, and player's choices are not impacted or altered by the game itself. This makes the games extremely useful in understanding social preferences. Games often include monetary rewards to easily calculate how players will act if their choices are driven by monetary incentives and payoffs. Player's actions are often shaped by the social utility function, whereby their choices are shaped by the benefits that both they and their opponent would receive. Traditional game theory would expect rational players to attempt to maximise their monetary rewards. If these calculations were wrong, however, and if players choose not to maximise their utility, then players would be exhibiting a social preference for a particular action. Behavioral game theory explains how players often deviate from traditional norms, and quite regularly consider factors such as social welfare when making their strategic decisions. For example, players are known to sacrifice high monetary rewards in order to maintain fairness within the game.
Different games demonstrate different social preferences. For example, the ultimatum game is known to demonstrate negative reciprocity. The premise of the ultimatum game is that Player 1 is given a certain amount of money, and is then forced to offer a certain amount to Player 2. Player 2 can then choose to either accept or reject Player 1's offer. If Player 2 accepts the offer, then both players are able to enjoy the amount offered. If Player 2 rejects the offer, then neither player is able to receive the money. Results from ultimatum game experiments demonstrate that players value being treated fairly and do not react well when one player is attempting to receive better payoffs than the other. Studies show that people are more likely sacrifice all monetary rewards if they are offered less than 20 percent of the original amount. This represents negative reciprocity preferences, as players would rather sacrifice their payoff in order to punish their opponent for their unkind behavior. However, being scared of having their offers rejected, people often give Player 2 around 40-50 percent of the original amount.
Another example of a social preference is positive reciprocity, which is displayed in the gift exchange game. The gift exchange game involves Player 1 either keeping set amount of money, or offering an even larger amount to Player 2. Player 2 is then able to decide how they wish to divide the money between the two of them. In this game, Player 1 trusts that Player 2 will return a certain amount of money to them. Findings from this game often show that if Player 2 is offered a generous amount of money from Player 1, then they are more likely to return the favour and give Player 1 back an equally generous amount. This demonstrates how players appreciate being treated kindly, and are more likely to treat their opponent kindly in return. The concept of positive reciprocity can be seen in real-life examples, such as the workplace. If an employer offers a large wage to their employees, then the employees often pay back the favour by working harder.
Altruism is another social preference seen in the dictator game. This game is similar to the ultimatum and gift exchange games. In this game, however, Player 1 is given an amount of money, and can then offer however much they would like to Player 2. Unlike the ultimatum game, Player 2 cannot reject the amount they have been offered. As a result, people are more likely to reduce the amount of money they offer to Player 2. Despite this, results show that people still offer Player 2 a sum of around 20-30 percent of the original amount. The dictator game shows how people are willing to share their rewards with people, despite not being forced to.
The prisoner's dilemma game is effective in examining the social preference of cooperation. The logic behind the prisoner's dilemma is that every players rational choice is to defect, rather than cooperate. As it is in each player's best interest to defect, both players would rationally choose to defect. This results in a worse payoff for both players. The ultimatum game requires two players to agree on the allocation of money, yet what is reflected by the game is that humans are more concerned with whether the allocation is fair than whether the benefits are maximized. This behavior also illustrates that behavioral game theory is more well thought out than traditional game theory. However, in an attempt to reach a fair equilibrium for both players, results from the prisoner's dilemma game show that people cooperate much more than traditionally thought. When one player decides to cooperate, then the other players are more likely to cooperate too. This goes against the traditional beliefs that people only make decisions that maximise their utility.
== Examples of games used in behavioral game theory research ==
Signaling game
Dictator Game
Ultimatum Game
Keynesian beauty contest
Normal form game
Cooperative game
Gift-exchange game
Prisoner's Dilemma
Zero-sum Games
== Factors that affect rationality in games ==
=== Learning ===
Learning models are a way of explaining and predicting strategic decisions in behavioral game theory. More specifically, they aim to explain how player's choices may change when given the chance to learn about their opponents or the game. There are three different types of learning models. The first is reinforcement learning. Reinforcement learning suggests that if a player received a high reward from choosing a certain behavior or strategy, then that player would be more inclined to use the same strategy again. If a particular strategy has not been used before however, then the strategy would not appear to be more or less appealing to the player. Another learning model is belief learning. Belief learning assumes that players often remember their opponents previous strategies in games, and will henceforth change their own strategies based on their opponents past behavior. Lastly, experience weighted attraction learning uses a mixture of belief learning and reinforcement learning in its model. This model accounts for the strategies and payoffs that have been played and unplayed. The experience weighted attraction learning framework posits that people learn from past experiences as well as by questioning what they could've done differently. Furthermore, it also believes that players evaluate their past rewards half as much as their actual rewards.
=== Beliefs ===
Beliefs about other people in a decision-making game are expected to influence ones ability to make rational choices. However, beliefs of others can also cause experimental results to deviate from equilibrium, utility-maximizing decisions. In an experiment by Costa-Gomez (2008) participants were questioned about their first order beliefs about their opponent's actions prior to completing a series of normal-form games with other participants. Participants complied with Nash Equilibrium only 35% of the time. Further, participants only stated beliefs that their opponents would comply with traditional game theory equilibrium 15% of the time. This means participants believed their opponents would be less rational than they really were. The results of this study show that participants do not choose the utility-maximizing action and they expect their opponents to do the same. Also, the results show that participants did not choose the utility-maximizing action that corresponded to their beliefs about their opponent's action. While participants may have believed their opponent was more likely to make a certain decision, they still made decisions as if their opponent was choosing randomly. Another study that examined participants from the TV show Deal or No Deal found divergence from rational choice. Participants were more likely to base their decisions on previous outcomes when progressing through the game. Risk aversion decreased when participants' expectations were not met within the game. For example, a subject that experienced a string of positive outcomes was less likely to accept the deal and end the game. The same was true for a subject that experienced primarily negative outcomes early in the game.
=== Social cooperation ===
Social behavior and cooperation with other participants are two factors that are not modeled in traditional game theory, but are often seen in an experimental setting. The evolution of social norms has been neglected in decision-making models, but these norms influence the ways in which real people interact with one another and make choices. One tendency is for a person to be a strong reciprocator. This type of person enters a game with the predisposition to cooperate with other players. They will increase their cooperation levels in response to cooperation from other players and decrease their cooperation levels, even at their own expense, to punish players who do not cooperate. This is not payoff-maximizing behavior, as a strong reciprocator is willing to reduce their payoff in order to encourage cooperation from others. Rational choice theory has limitations in interactive decision making, and it is also difficult to accurately predict human behavior in social cooperation. Behavioral games not only require players to make rational choices, but also require players to be able to predict the decisions of other players in their interactions. In game experiments, rational choice conflicts with individual decision making, and individual behavior may be able to achieve greater gains than rational choice. Rational choice theory has limitations and uncertainties for social interaction decisions, so the predicted results are not the same as the experimental results.
Dufwenberg and Kirchsteiger (2004) developed a model based on reciprocity called the sequential reciprocity equilibrium. This model adapts traditional game theory logic to the idea that players reciprocate actions in order to cooperate. The model had been used to more accurately predict experimental outcomes of classic games such as the prisoner's dilemma and the centipede game. Rabin (1993) also created a fairness equilibrium that measures altruism's effect on choices. He found that when a player is altruistic to another player the second player is more likely to reciprocate that altruism. This is due to the idea of fairness. Fairness equilibriums take the form of mutual maximum, where both players choose an outcome that benefits both of them the most, or mutual minimum, where both players choose an outcome that hurts both of them the most. These equilibriums are also Nash equilibriums, but they incorporate the willingness of participants to cooperate and play fair.
=== Incentives, consequences, and deception ===
The role of incentives and consequences in decision-making is interesting to behavioral game theorists because it affects rational behavior. Post (2008) analyzed Deal or no Deal contestant behavior in order to reach conclusions about decision-making when stakes are high. Studying the contestant's choices formed the conclusion that, in a sequential game with high stakes, decisions were based on previous outcomes rather than rationality. Players who face a succession of good outcomes, in this case they eliminate the low-value cases from play, or players who face a succession of poor outcomes become less risk averse. This means that players who are having exceptionally good or exceptionally bad outcomes are more likely to gamble and continue playing than average players. The lucky or unlucky players were willing to reject offers of over one hundred percent of the expected value of their case in order to continue playing. This shows a shift from risk avoiding behavior to risk seeking behavior. This study highlights behavioral biases that are not accounted for by traditional game theory. Riskier behavior in unlucky contestants can be attributed to the break-even effect, which states that gamblers will continue to make risky decisions in order to win back money. On the other hand, riskier behavior in lucky contestants can be explained by the house-money effect, which states that winning gamblers are more likely to make risky decisions because they perceive that they are not gambling with their own money. This analysis shows that incentives influence rational choice, especially when players make a series of decisions.
Incentives and consequences also play a large role in deception in games. Gneezy (2005) studied deception using a cheap talk sender-receiver game. In this type of game player one receives information about the payouts of option A and option B. Then, player one gives a recommendation to player two about which option to take. Player one can choose to deceive player two, and player two can choose to reject player one's advice. Gneezy found that participants were more sensitive to their gain from lying than to their opponent's loss. He also found that participants were not wholly selfish and cared about how much their opponents lost from their deception, but this effect diminished as their own payout increased. These findings show that decision makers examine both incentives to lie and consequences of lying in order to decide whether or not to lie. In general people are averse to lying, but given the right incentives they tend to ignore consequences. Wang (2009) also used a cheap talk game to study deception in participants with an incentive to deceive. Using eye tracking he found that participants who received information about payoffs focused on their own payoff twice as often as their opponents. This suggests minimal strategic thinking. Further, participants' pupils dilated when they sent a deceiving, and they dilated more when telling a bigger lie. Through these physical cues Wang concluded that deception is cognitively difficult. These findings show that factors such as incentives, consequences, and deception can create irrational decisions and affect the way games unfold.
A consequence of the game theory is its lack of use of empirical data to predict outcomes. "game theory will be no substitute for an empirically grounded behavioral theory when we want to predict what people will actually do in a competitive situation" Predicting rational behavior is possible with game theory but it can be improved if the theory is open to adjustment. The predicted result of the game can be improved and long-lasting if the discipline expands its knowledge of behavioral theory. How people act, think, and feel affect their decisions to play a role in this theory.,. Ken Binmore makes an excellent point that when empirically sound data is present, game theory should not hold the final decision outcome. That this is good for trying to understand if the rational decision being made is due to game theory or is just a consistent behavioral decision being made. The field of economics should try to take every step in improving empirical information in that there is little reliance on just a theory. Businesses value game theory, and the economic discipline must improve the strength of game theory by trying to establish an empirical database. Society will be able to advance its knowledge of behavioral game theory just by expanding the economic discipline of data. Alvin E Roth states, "if we do not take steps in the direction of adding a solid empirical base to game theory, but instead continue to rely on game theory primarily for conceptual insights, then it is likely that long before a hundred-year game theory will have experienced sharply diminishing return"
=== Group decisions ===
Behavioral game theory considers the effects of groups on rationality. In the real world many decisions are made by teams, yet traditional game theory uses an individual as a decision maker. Milton Friedman argues that usually people ignore individual behavior and focus more on group behavior, so group behavior is often perceived as more rational. This created a need to model group decision-making behavior. Bornstein and Yaniv (1998) examined the difference in rationality between groups and individuals in an ultimatum game. In this game player one (or group one) decides what percentage of a payout to give to player two (or group two) and then player two decides whether to accept or reject this offer. Participants in the group condition were put in groups of three and allowed to deliberate on their decisions. Perfect rationality in this game would be player one offering player two none of the payout, but that is almost never the case in observed offers. Bornstein and Yaniv found that groups were less generous, willing to give up a smaller portion of the payoff, in the player one condition and more accepting of low offers in the player two condition than individuals. These results suggest that groups are more rational than individuals.
Kocher and Sutter (2005) used a beauty contest game to study and compare individual and group behavior. A beauty contest game is one where all participants choose a number between zero and one hundred. The winner is the participant who chooses a number closest to two thirds of the average number. In the first round the rational choice would be thirty-three, as it is two thirds of the average number, fifty. Given an infinite number of rounds all participants should choose zero according to game theory. Kocher and Sutter found that groups did not perform more rationally than individuals in the first round of the game. However, groups performed more rationally than individuals in subsequent rounds. This shows that groups are able to learn the game and adapt their strategy faster than individuals.
== See also ==
Behavioral economics
Experimental economics
Game theory
== References == | Wikipedia/Behavioral_game_theory |
An Economic Theory of Democracy is a treatise of economics written by Anthony Downs, published in 1957. The book set forth a model with precise conditions under which economic theory could be applied to non-market political decision-making. It also suggested areas of empirical research that could be tested to confirm the validity of his conclusions in the model. Much of this offshoot research eventually became integrated into public choice theory. Downs' theory abstains from making normative statements about public policy choices and instead focuses on what is rational, given the relevant incentives, for government to do.
== Contents ==
In chapter eight of the book Downs explains how the concept of ideology is central to his theory. Depending on the ideological distribution of voters in a given political community, electoral outcomes can be stable and peaceful or wildly varied and even result in violent revolution. The likely number of political parties can also be identified if one also considers the electoral structure. If the ideological positions of voters are displayed in the form of a graph and if that graph shows a single peak, then a median voter can be identified and in a representative democracy, the choice of candidates and the choice of policies will gravitate toward the positions of the median voter. Conversely, if the graph of ideological distribution is double-peaked, indicating that most voters are either extremely liberal or extremely conservative, the tendency toward political consensus or political equilibrium is difficult to attain because legislators representing each mode are penalized by voters for attempting to achieve consensus with the other side by supporting policies representative of a middle position. Here is a list of the key propositions Downs attempts to prove in chapter eight:
A two-party democracy cannot provide stable and effective government unless there is a large measure of ideological consensus among its citizens.
Parties in a two-party system deliberately change their platforms so that they resemble one another; whereas parties in a multi-party system try to remain as ideologically distinct from each other as possible.
If the distribution of ideologies in a society's citizenry remains constant, its political system will move toward a position of equilibrium in which the number of parties and their ideological positions are stable over time.
New parties can be most successfully launched immediately after some significant change in the distribution of ideological views among eligible voters.
In a two-party system, it is rational for each party to encourage voters to be irrational by making its platform vague and ambiguous.
The conditions under which his theory prevails are outlined in chapter two. Many of these conditions have been challenged by later scholarship. In anticipation of such criticism, Downs quotes Milton Friedman in chapter two that: “Theoretical models should be tested primarily by the accuracy of their predictions rather than by the reality of their assumptions”.
In a 2004 study, Bernard Grofman argued that "A careful reading of Downs offers a much more sophisticated and nuanced portrait of the factors affecting party differentiation than the simplistic notion that, in plurality elections, we ought to expect party convergence to the views of the median voter." According to Grofman, recent research in the Downsian tradition expected nonconvergence of parties in a two-party democracy.
== See also ==
Mancur Olson
== References ==
== External links ==
Downs: An economic theory of democracy. Article at WikiSummary. | Wikipedia/An_Economic_Theory_of_Democracy |
Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. The use of the term "mean field" is inspired by mean-field theory in physics, which considers the behavior of systems of large numbers of particles where individual particles have negligible impacts upon the system. In other words, each agent acts according to his minimization or maximization problem taking into account other agents’ decisions and because their population is large we can assume the number of agents goes to infinity and a representative agent exists.
In traditional game theory, the subject of study is usually a game with two players and discrete time space, and extends the results to more complex situations by induction. However, for games in continuous time with continuous states (differential games or stochastic differential games) this strategy cannot be used because of the complexity that the dynamic interactions generate. On the other hand with MFGs we can handle large numbers of players through the mean representative agent and at the same time describe complex state dynamics.
This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal, in the engineering literature by Minyi Huang, Roland Malhame, and Peter E. Caines and independently and around the same time by mathematicians Jean-Michel Lasry and Pierre-Louis Lions.
In continuous time a mean-field game is typically composed of a Hamilton–Jacobi–Bellman equation that describes the optimal control problem of an individual and a Fokker–Planck equation that describes the dynamics of the aggregate distribution of agents. Under fairly general assumptions it can be proved that a class of mean-field games is the limit as
N
→
∞
{\displaystyle N\to \infty }
of an N-player Nash equilibrium.
A related concept to that of mean-field games is "mean-field-type control". In this case, a social planner controls the distribution of states and chooses a control strategy. The solution to a mean-field-type control problem can typically be expressed as a dual adjoint Hamilton–Jacobi–Bellman equation coupled with Kolmogorov equation. Mean-field-type game theory is the multi-agent generalization of the single-agent mean-field-type control.
== General Form of a Mean-field Game ==
The following system of equations can be used to model a typical Mean-field game:
{
−
∂
t
u
−
ν
Δ
u
+
H
(
x
,
m
,
D
u
)
=
0
(
1
)
∂
t
m
−
ν
Δ
m
−
div
(
D
p
H
(
x
,
m
,
D
u
)
m
)
=
0
(
2
)
m
(
0
)
=
m
0
(
3
)
u
(
x
,
T
)
=
G
(
x
,
m
(
T
)
)
(
4
)
{\displaystyle {\begin{cases}-\partial _{t}u-\nu \Delta u+H(x,m,Du)=0&(1)\\\partial _{t}m-\nu \Delta m-\operatorname {div} (D_{p}H(x,m,Du)m)=0&(2)\\m(0)=m_{0}&(3)\\u(x,T)=G(x,m(T))&(4)\end{cases}}}
The basic dynamics of this set of Equations can be explained by an average agent's optimal control problem. In a mean-field game, an average agent can control their movement
α
{\displaystyle \alpha }
to influence the population's overall location by:
d
X
t
=
α
t
d
t
+
2
ν
d
B
t
{\displaystyle dX_{t}=\alpha _{t}dt+{\sqrt {2\nu }}dB_{t}}
where
ν
{\displaystyle \nu }
is a parameter and
B
t
{\displaystyle B_{t}}
is a standard Brownian motion. By controlling their movement, the agent aims to minimize their overall expected cost
C
{\displaystyle C}
throughout the time period
[
0
,
T
]
{\displaystyle [0,T]}
:
C
=
E
[
∫
0
T
L
(
X
s
,
α
s
,
m
(
s
)
)
d
s
+
G
(
X
T
,
m
(
T
)
)
]
{\displaystyle C=\mathbb {E} \left[\int _{0}^{T}L(X_{s},\alpha _{s},m(s))ds+G(X_{T},m(T))\right]}
where
L
(
X
s
,
α
s
,
m
(
s
)
)
{\displaystyle L(X_{s},\alpha _{s},m(s))}
is the running cost at time
s
{\displaystyle s}
and
G
(
X
T
,
m
(
T
)
)
{\displaystyle G(X_{T},m(T))}
is the terminal cost at time
T
{\displaystyle T}
. By this definition, at time
t
{\displaystyle t}
and position
x
{\displaystyle x}
, the value function
u
(
t
,
x
)
{\displaystyle u(t,x)}
can be determined as:
u
(
t
,
x
)
=
inf
α
E
[
∫
t
T
L
(
X
s
,
α
s
,
m
(
s
)
)
d
s
+
G
(
X
T
,
m
(
T
)
)
]
{\displaystyle u(t,x)=\inf _{\alpha }\mathbb {E} \left[\int _{t}^{T}L(X_{s},\alpha _{s},m(s))ds+G(X_{T},m(T))\right]}
Given the definition of the value function
u
(
t
,
x
)
{\displaystyle u(t,x)}
, it can be tracked by the Hamilton-Jacobi equation (1). The optimal action of the average players
α
∗
(
x
,
t
)
{\displaystyle \alpha ^{*}(x,t)}
can be determined as
α
∗
(
x
,
t
)
=
D
p
H
(
x
,
m
,
D
u
)
{\displaystyle \alpha ^{*}(x,t)=D_{p}H(x,m,Du)}
. As all agents are relatively small and cannot single-handedly change the dynamics of the population, they will individually adapt the optimal control and the population would move in that way. This is similar to a Nash Equilibrium, in which all agents act in response to a specific set of others' strategies. The optimal control solution then leads to the Kolmogorov-Fokker-Planck equation (2).
== Finite State Games ==
A prominent category of mean field is games with a finite number of states and a finite number of actions per player. For those games, the analog of the Hamilton-Jacobi-Bellman equation is the Bellman equation, and the discrete version of the Fokker-Planck equation is the Kolmogorov equation. Specifically, for discrete-time models, the players' strategy is the Kolmogorov equation's probability matrix. In continuous time models, players have the ability to control the transition rate matrix.
A discrete mean field game can be defined by a tuple
G
=
(
E
,
A
,
{
Q
a
}
,
m
0
,
{
c
a
}
,
β
)
{\displaystyle {\mathcal {G}}=({\mathcal {E}},{\mathcal {A}},\{Q_{a}\},{\bf {m}}_{0},\{c_{a}\},\beta )}
, where
E
{\displaystyle {\mathcal {E}}}
is the state space,
A
{\displaystyle {\mathcal {A}}}
the action set,
Q
a
{\displaystyle Q_{a}}
the transition rate matrices,
m
0
{\displaystyle {\bf {m}}_{0}}
the initial state,
{
c
a
}
{\displaystyle \{c_{a}\}}
the cost functions and
β
{\displaystyle \beta }
∈
R
{\displaystyle \in \mathbb {R} }
a discount factor. Furthermore, a mixed strategy is a measurable function
π
:
E
×
R
+
→
P
(
A
)
{\displaystyle \pi :\mathbb {E} \times \mathbb {R} ^{+}{\xrightarrow[{}]{}}{\mathcal {P(A)}}}
, that associates to each state
i
∈
E
{\displaystyle i\in {\mathcal {E}}}
and each time
t
≥
0
{\displaystyle t\geq 0}
a probability measure
π
i
(
t
)
∈
P
(
A
)
{\displaystyle \pi _{i}(t)\in {\mathcal {P(A)}}}
on the set of possible actions. Thus
π
i
,
a
(
t
)
{\displaystyle \pi _{i,a}(t)}
is the probability that, at time
t
{\displaystyle t}
a player in state
i
{\displaystyle i}
takes action
a
{\displaystyle a}
, under strategy
π
{\displaystyle \pi }
. Additionally, rate matrices
{
Q
a
(
m
π
(
t
)
)
}
a
∈
A
{\displaystyle \{Q_{a}({\bf {m}}^{\pi }(t))\}_{a\in {\mathcal {A}}}}
define the evolution over the time of population distribution, where
m
π
(
t
)
∈
P
(
E
)
{\displaystyle {\bf {m}}^{\pi }(t)\in {\mathcal {P({\mathcal {E}})}}}
is the population distribution at time
t
{\displaystyle t}
.
== Linear-quadratic Gaussian game problem ==
From Caines (2009), a relatively simple model of large-scale games is the linear-quadratic Gaussian model. The individual agent's dynamics are modeled as a stochastic differential equation
d
X
i
=
(
a
i
X
i
+
b
i
u
i
)
d
t
+
σ
i
d
W
i
,
i
=
1
,
…
,
N
,
{\displaystyle dX_{i}=(a_{i}X_{i}+b_{i}u_{i})\,dt+\sigma _{i}\,dW_{i},\quad i=1,\dots ,N,}
where
X
i
{\displaystyle X_{i}}
is the state of the
i
{\displaystyle i}
-th agent,
u
i
{\displaystyle u_{i}}
is the control of the
i
{\displaystyle i}
-th agent, and
W
i
{\displaystyle W_{i}}
are independent Wiener processes for all
i
=
1
,
…
,
N
{\displaystyle i=1,\dots ,N}
. The individual agent's cost is
J
i
(
u
i
,
ν
)
=
E
{
∫
0
∞
e
−
ρ
t
[
(
X
i
−
ν
)
2
+
r
u
i
2
]
d
t
}
,
ν
=
Φ
(
1
N
∑
k
≠
i
N
X
k
+
η
)
.
{\displaystyle J_{i}(u_{i},\nu )=\mathbb {E} \left\{\int _{0}^{\infty }e^{-\rho t}\left[(X_{i}-\nu )^{2}+ru_{i}^{2}\right]\,dt\right\},\quad \nu =\Phi \left({\frac {1}{N}}\sum _{k\neq i}^{N}X_{k}+\eta \right).}
The coupling between agents occurs in the cost function.
== General and Applied Use ==
The paradigm of Mean Field Games has become a major connection between distributed decision-making and stochastic modeling. Starting out in the stochastic control literature, it is gaining rapid adoption across a range of applications, including:
a. Financial market
Carmona reviews applications in financial engineering and economics that can be cast and tackled within the framework of the MFG paradigm. Carmona argues that models in macroeconomics, contract theory, finance, …, greatly benefit from the switch to continuous time from the more traditional discrete-time models. He considers only continuous time models in his review chapter, including systemic risk, price impact, optimal execution, models for bank runs, high-frequency trading, and cryptocurrencies.
b. Crowd motions
MFG assumes that individuals are smart players which try to optimize their strategy and path with respect to certain costs (equilibrium with rational expectations approach). MFG models are useful to describe the anticipation phenomenon: the forward part describes the crowd evolution while the backward gives the process of how the anticipations are built. Additionally, compared to multi-agent microscopic model computations, MFG only requires lower computational costs for the macroscopic simulations. Some researchers have turned to MFG in order to model the interaction between populations and study the decision-making process of intelligent agents, including aversion and congestion behavior between two groups of pedestrians, departure time choice of morning commuters, and decision-making processes for autonomous vehicle.
c. Control and mitigation of Epidemics
Since the epidemic has affected society and individuals significantly, MFG and mean-field controls (MFCs) provide a perspective to study and understand the underlying population dynamics, especially in the context of the Covid-19 pandemic response. MFG has been used to extend the SIR-type dynamics with spatial effects or allowing for individuals to choose their behaviors and control their contributions to the spread of the disease. MFC is applied to design the optimal strategy to control the virus spreading within a spatial domain, control individuals’ decisions to limit their social interactions, and support the government’s nonpharmaceutical interventions.
== See also ==
== References ==
== External links ==
Mean Field Stochastic Control (Slides), 2009 IEEE Control Systems Society Bode Prize Lecture by Peter E. Caines
Caines, Peter E. (2013). "Mean Field Games". Encyclopedia of Systems and Control. pp. 1–6. doi:10.1007/978-1-4471-5102-9_30-1. ISBN 978-1-4471-5102-9. S2CID 33954904.
Notes on Mean Field Games, from Pierre-Louis Lions' lectures at Collège de France
(in French) Video lectures by Pierre-Louis Lions
Mean field games and applications by Olivier Guéant, Jean-Michel Lasry, and Pierre-Louis Lions | Wikipedia/Mean_field_game_theory |
An evolutionarily stable strategy (ESS) is a strategy (or set of strategies) that is impermeable when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy (or set of strategies) which may be novel or initially rare. Introduced by John Maynard Smith and George R. Price in 1972/3, it is an important concept in behavioural ecology, evolutionary psychology, mathematical game theory and economics, with applications in other fields such as anthropology, philosophy and political science.
In game-theoretical terms, an ESS is an equilibrium refinement of the Nash equilibrium, being a Nash equilibrium that is also "evolutionarily stable." Thus, once fixed in a population, natural selection alone is sufficient to prevent alternative (mutant) strategies from replacing it (although this does not preclude the possibility that a better strategy, or set of strategies, will emerge in response to selective pressures resulting from environmental change).
== History ==
Evolutionarily stable strategies were defined and introduced by John Maynard Smith and George R. Price in a 1973 Nature paper. Such was the time taken in peer-reviewing the paper for Nature that this was preceded by a 1972 essay by Maynard Smith in a book of essays titled On Evolution. The 1972 essay is sometimes cited instead of the 1973 paper, but university libraries are much more likely to have copies of Nature. Papers in Nature are usually short; in 1974, Maynard Smith published a longer paper in the Journal of Theoretical Biology. Maynard Smith explains further in his 1982 book Evolution and the Theory of Games. Sometimes these are cited instead. In fact, the ESS has become so central to game theory that often no citation is given, as the reader is assumed to be familiar with it.
Maynard Smith mathematically formalised a verbal argument made by Price, which he read while peer-reviewing Price's paper. When Maynard Smith realized that the somewhat disorganised Price was not ready to revise his article for publication, he offered to add Price as co-author.
The concept was derived from R. H. MacArthur and W. D. Hamilton's work on sex ratios, derived from Fisher's principle, especially Hamilton's (1967) concept of an unbeatable strategy. Maynard Smith was jointly awarded the 1999 Crafoord Prize for his development of the concept of evolutionarily stable strategies and the application of game theory to the evolution of behaviour.
Uses of ESS:
The ESS was a major element used to analyze evolution in Richard Dawkins' bestselling 1976 book The Selfish Gene.
The ESS was first used in the social sciences by Robert Axelrod in his 1984 book The Evolution of Cooperation. Since then, it has been widely used in the social sciences, including anthropology, economics, philosophy, and political science.
In the social sciences, the primary interest is not in an ESS as the end of biological evolution, but as an end point in cultural evolution or individual learning.
In evolutionary psychology, ESS is used primarily as a model for human biological evolution.
== Motivation ==
The Nash equilibrium is the traditional solution concept in game theory. It depends on the cognitive abilities of the players. It is assumed that players are aware of the structure of the game and consciously try to predict the moves of their opponents and to maximize their own payoffs. In addition, it is presumed that all the players know this (see common knowledge). These assumptions are then used to explain why players choose Nash equilibrium strategies.
Evolutionarily stable strategies are motivated entirely differently. Here, it is presumed that the players' strategies are biologically encoded and heritable. Individuals have no control over their strategy and need not be aware of the game. They reproduce and are subject to the forces of natural selection, with the payoffs of the game representing reproductive success (biological fitness). It is imagined that alternative strategies of the game occasionally occur, via a process like mutation. To be an ESS, a strategy must be resistant to these alternatives.
Given the radically different motivating assumptions, it may come as a surprise that ESSes and Nash equilibria often coincide. In fact, every ESS corresponds to a Nash equilibrium, but some Nash equilibria are not ESSes.
== Nash equilibrium ==
An ESS is a refined or modified form of a Nash equilibrium. (See the next section for examples which contrast the two.) In a Nash equilibrium, if all players adopt their respective parts, no player can benefit by switching to any alternative strategy. In a two player game, it is a strategy pair. Let E(S,T) represent the payoff for playing strategy S against strategy T. The strategy pair (S, S) is a Nash equilibrium in a two player game if and only if for both players, for any strategy T:
E(S,S) ≥ E(T,S)
In this definition, a strategy T≠S can be a neutral alternative to S (scoring equally well, but not better).
A Nash equilibrium is presumed to be stable even if T scores equally, on the assumption that there is no long-term incentive for players to adopt T instead of S. This fact represents the point of departure of the ESS.
Maynard Smith and Price specify two conditions for a strategy S to be an ESS. For all T≠S, either
E(S,S) > E(T,S), or
E(S,S) = E(T,S) and E(S,T) > E(T,T)
The first condition is sometimes called a strict Nash equilibrium. The second is sometimes called "Maynard Smith's second condition". The second condition means that although strategy T is neutral with respect to the payoff against strategy S, the population of players who continue to play strategy S has an advantage when playing against T.
There is also an alternative, stronger definition of ESS, due to Thomas. This places a different emphasis on the role of the Nash equilibrium concept in the ESS concept. Following the terminology given in the first definition above, this definition requires that for all T≠S
E(S,S) ≥ E(T,S), and
E(S,T) > E(T,T)
In this formulation, the first condition specifies that the strategy is a Nash equilibrium, and the second specifies that Maynard Smith's second condition is met. Note that the two definitions are not precisely equivalent: for example, each pure strategy in the coordination game below is an ESS by the first definition but not the second.
In words, this definition looks like this: The payoff of the first player when both players play strategy S is higher than (or equal to) the payoff of the first player when he changes to another strategy T and the second player keeps his strategy S and the payoff of the first player when only his opponent changes his strategy to T is higher than his payoff in case that both of players change their strategies to T.
This formulation more clearly highlights the role of the Nash equilibrium condition in the ESS. It also allows for a natural definition of related concepts such as a weak ESS or an evolutionarily stable set.
=== Examples of differences between Nash equilibria and ESSes ===
In most simple games, the ESSes and Nash equilibria coincide perfectly. For instance, in the prisoner's dilemma there is only one Nash equilibrium, and its strategy (Defect) is also an ESS.
Some games may have Nash equilibria that are not ESSes. For example, in harm thy neighbor (whose payoff matrix is shown here) both (A, A) and (B, B) are Nash equilibria, since players cannot do better by switching away from either. However, only B is an ESS (and a strong Nash). A is not an ESS, so B can neutrally invade a population of A strategists and predominate, because B scores higher against B than A does against B. This dynamic is captured by Maynard Smith's second condition, since E(A, A) = E(B, A), but it is not the case that E(A,B) > E(B,B).
Nash equilibria with equally scoring alternatives can be ESSes. For example, in the game Harm everyone, C is an ESS because it satisfies Maynard Smith's second condition. D strategists may temporarily invade a population of C strategists by scoring equally well against C, but they pay a price when they begin to play against each other; C scores better against D than does D. So here although E(C, C) = E(D, C), it is also the case that E(C,D) > E(D,D). As a result, C is an ESS.
Even if a game has pure strategy Nash equilibria, it might be that none of those pure strategies are ESS. Consider the Game of chicken. There are two pure strategy Nash equilibria in this game (Swerve, Stay) and (Stay, Swerve). However, in the absence of an uncorrelated asymmetry, neither Swerve nor Stay are ESSes. There is a third Nash equilibrium, a mixed strategy which is an ESS for this game (see Hawk-dove game and Best response for explanation).
This last example points to an important difference between Nash equilibria and ESS. Nash equilibria are defined on strategy sets (a specification of a strategy for each player), while ESS are defined in terms of strategies themselves. The equilibria defined by ESS must always be symmetric, and thus have fewer equilibrium points.
== Vs. evolutionarily stable state ==
In population biology, the two concepts of an evolutionarily stable strategy (ESS) and an evolutionarily stable state are closely linked but describe different situations.
In an evolutionarily stable strategy, if all the members of a population adopt it, no mutant strategy can invade. Once virtually all members of the population use this strategy, there is no 'rational' alternative. ESS is part of classical game theory.
In an evolutionarily stable state, a population's genetic composition is restored by selection after a disturbance, if the disturbance is not too large. An evolutionarily stable state is a dynamic property of a population that returns to using a strategy, or mix of strategies, if it is perturbed from that initial state. It is part of population genetics, dynamical system, or evolutionary game theory. This is now called convergent stability.
B. Thomas (1984) applies the term ESS to an individual strategy which may be mixed, and evolutionarily stable population state to a population mixture of pure strategies which may be formally equivalent to the mixed ESS.
Whether a population is evolutionarily stable does not relate to its genetic diversity: it can be genetically monomorphic or polymorphic.
== Stochastic ESS ==
In the classic definition of an ESS, no mutant strategy can invade. In finite populations, any mutant could in principle invade, albeit at low probability, implying that no ESS can exist. In an infinite population, an ESS can instead be defined as a strategy which, should it become invaded by a new mutant strategy with probability p, would be able to counterinvade from a single starting individual with probability >p, as illustrated by the evolution of bet-hedging.
== Prisoner's dilemma ==
A common model of altruism and social cooperation is the Prisoner's dilemma. Here a group of players would collectively be better off if they could play Cooperate, but since Defect fares better each individual player has an incentive to play Defect. One solution to this problem is to introduce the possibility of retaliation by having individuals play the game repeatedly against the same player. In the so-called iterated Prisoner's dilemma, the same two individuals play the prisoner's dilemma over and over. While the Prisoner's dilemma has only two strategies (Cooperate and Defect), the iterated Prisoner's dilemma has a huge number of possible strategies. Since an individual can have different contingency plan for each history and the game may be repeated an indefinite number of times, there may in fact be an infinite number of such contingency plans.
Three simple contingency plans which have received substantial attention are Always Defect, Always Cooperate, and Tit for Tat. The first two strategies do the same thing regardless of the other player's actions, while the latter responds on the next round by doing what was done to it on the previous round—it responds to Cooperate with Cooperate and Defect with Defect.
If the entire population plays Tit-for-Tat and a mutant arises who plays Always Defect, Tit-for-Tat will outperform Always Defect. If the population of the mutant becomes too large — the percentage of the mutant will be kept small. Tit for Tat is therefore an ESS, with respect to only these two strategies. On the other hand, an island of Always Defect players will be stable against the invasion of a few Tit-for-Tat players, but not against a large number of them. If we introduce Always Cooperate, a population of Tit-for-Tat is no longer an ESS. Since a population of Tit-for-Tat players always cooperates, the strategy Always Cooperate behaves identically in this population. As a result, a mutant who plays Always Cooperate will not be eliminated. However, even though a population of Always Cooperate and Tit-for-Tat can coexist, if there is a small percentage of the population that is Always Defect, the selective pressure is against Always Cooperate, and in favour of Tit-for-Tat. This is due to the lower payoffs of cooperating than those of defecting in case the opponent defects.
This demonstrates the difficulties in applying the formal definition of an ESS to games with large strategy spaces, and has motivated some to consider alternatives.
== Human behavior ==
The fields of sociobiology and evolutionary psychology attempt to explain animal and human behavior and social structures, largely in terms of evolutionarily stable strategies. Sociopathy (chronic antisocial or criminal behavior) may be a result of a combination of two such strategies.
Evolutionarily stable strategies were originally considered for biological evolution, but they can apply to other contexts. In fact, there are stable states for a large class of adaptive dynamics. As a result, they can be used to explain human behaviours that lack any genetic influences.
== See also ==
Antipredator adaptation
Behavioral ecology
Evolutionary psychology
Fitness landscape
Hawk–dove game
Koinophilia
Sociobiology
War of attrition (game)
Farsightedness (game theory)
== References ==
== Further reading ==
Weibull, Jörgen (1997). Evolutionary game theory. MIT Press. ISBN 978-0-262-73121-8. Classic reference textbook.
Hines, W. G. S. (1987). "Evolutionary stable strategies: a review of basic theory". Theoretical Population Biology. 31 (2): 195–272. Bibcode:1987TPBio..31..195H. doi:10.1016/0040-5809(87)90029-3. PMID 3296292.
Leyton-Brown, Kevin; Shoham, Yoav (2008). Essentials of Game Theory: A Concise, Multidisciplinary Introduction. San Rafael, CA: Morgan & Claypool Publishers. ISBN 978-1-59829-593-1.. An 88-page mathematical introduction; see Section 3.8. Free online Archived 2000-08-15 at the Wayback Machine at many universities.
Parker, G. A. (1984) Evolutionary stable strategies. In Behavioural Ecology: an Evolutionary Approach (2nd ed) Krebs, J. R. & Davies N.B., eds. pp 30–61. Blackwell, Oxford.
Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7. Archived from the original on 2011-05-01. Retrieved 2008-12-17.. A comprehensive reference from a computational perspective; see Section 7.7. Downloadable free online.
Maynard Smith, John. (1982) Evolution and the Theory of Games. ISBN 0-521-28884-3. Classic reference.
== External links ==
Evolutionarily Stable Strategies at Animal Behavior: An Online Textbook by Michael D. Breed.
Game Theory and Evolutionarily Stable Strategies, Kenneth N. Prestwich's site at College of the Holy Cross.
Evolutionarily stable strategies knol Archived: https://web.archive.org/web/20091005015811/http://knol.google.com/k/klaus-rohde/evolutionarily-stable-strategies-and/xk923bc3gp4/50# | Wikipedia/Evolutionarily_stable_strategy |
In combinatorial game theory, star, written as ∗ or ∗1, is the value given to the game where both players have only the option of moving to the zero game. Star may also be denoted as the surreal form {0|0}. This game is an unconditional first-player win.
Star, as defined by John Conway in Winning Ways for your Mathematical Plays, is a value, but not a number in the traditional sense. Star is not zero, but neither positive nor negative, and is therefore said to be fuzzy and confused with (a fourth alternative that means neither "less than", "equal to", nor "greater than") 0. It is less than all positive rational numbers, and greater than all negative rationals.
Games other than {0 | 0} may have value ∗. For example, the game
∗
2
+
∗
3
{\displaystyle *2+*3}
, where the values are nimbers, has value ∗ despite each player having more options than simply moving to 0.
== Why ∗ ≠ 0 ==
A combinatorial game has a positive and negative player; which player moves first is left ambiguous. The combinatorial game 0, or { | }, leaves no options and is a second-player win. Likewise, a combinatorial game is won (assuming optimal play) by the second player if and only if its value is 0. Therefore, a game of value ∗, which is a first-player win, is neither positive nor negative. However, ∗ is not the only possible value for a first-player win game (see nimbers).
Star does have the property that the sum ∗ + ∗, has value 0, because the first-player's only move is to the game ∗, which the second-player will win.
== Example of a value-∗ game ==
Nim, with one pile and one piece, has value ∗. The first player will remove the piece, and the second player will lose. A single-pile Nim game with one pile of n pieces (also a first-player win) is defined to have value ∗n. The numbers ∗z for integers z form an infinite field of characteristic 2, when addition is defined in the context of combinatorial games and multiplication is given a more complex definition.
== See also ==
Nimbers
Surreal numbers
== References ==
Conway, J. H., On Numbers and Games, Academic Press Inc. (London) Ltd., 1976 | Wikipedia/Star_(game_theory) |
In economics, industrial organization is a field that builds on the theory of the firm by examining the structure of (and, therefore, the boundaries between) firms and markets. Industrial organization adds real-world complications to the perfectly competitive model, complications such as transaction costs, limited information, and barriers to entry of new firms that may be associated with imperfect competition. It analyzes determinants of firm and market organization and behavior on a continuum between competition and monopoly, including from government actions.
There are different approaches to the subject. One approach is descriptive in providing an overview of industrial organization, such as measures of competition and the size-concentration of firms in an industry. A second approach uses microeconomic models to explain internal firm organization and market strategy, which includes internal research and development along with issues of internal reorganization and renewal. A third aspect is oriented to public policy related to economic regulation, antitrust law, and, more generally, the economic governance of law in defining property rights, enforcing contracts, and providing organizational infrastructure.
The extensive use of game theory in industrial economics has led to the export of this tool to other branches of microeconomics, such as behavioral economics and corporate finance. Industrial organization has also had significant practical impacts on antitrust law and competition policy.
The development of industrial organization as a separate field owes much to Edward Chamberlin, Joan Robinson, Edward S. Mason, J. M. Clark, Joe S. Bain and Paolo Sylos Labini, among others.
== Subareas ==
The Journal of Economic Literature (JEL) classification codes are one way of representing the range of economics subjects and subareas. There, Industrial Organization, one of 20 primary categories, has 9 secondary categories, each with multiple tertiary categories. The secondary categories are listed below with corresponding available article-preview links of The New Palgrave Dictionary of Economics Online and footnotes to their respective JEL-tertiary categories and associated New-Palgrave links.
JEL: L1 – Market Structure, Firm Strategy, and Market Performance
JEL: L2 – Firm Objectives, Organization, and Behavior
JEL: L3 – Non-profit organizations and Public enterprise
JEL: L4 – Antitrust Issues and Policies
JEL: L5 – Regulation and Industrial policy
JEL: L6 – Industry Studies: Manufacturing
JEL: L7 – Industry Studies: Primary Products and Construction
JEL: L8 – Industry Studies: Services
JEL: L9 – Industry Studies: Transportation and Utilities
== Market structures ==
The common market structures studied in this field are: perfect competition, monopolistic competition, duopoly, oligopoly, oligopsony, monopoly and monopsony.
== Areas of study ==
Industrial organization investigates the outcomes of these market structures in environments with
Price discrimination
Product differentiation
Durable goods
Experience goods
Collusion
Signalling, such as warranties and advertising.
Mergers and acquisitions
Entry and Exit
== History of the field ==
A 2009 book Pioneers of Industrial Organization traces the development of the field from Adam Smith to recent times and includes dozens of short biographies of major figures in Europe and North America who contributed to the growth and development of the discipline.
Other reviews by publication year and earliest available cited works those in 1970/1937, 1972/1933, 1974, 1987/1937-1956 (3 cites), 1968–9 (7 cites), 2009/c. 1900, and 2010/1951.
== See also ==
== Notes ==
== References ==
Tirole, Jean (1988). The Theory of Industrial Organization, MIT press.
Belleflamme, Paul & Martin Peitz, 2010. Industrial Organization: Markets and Strategies. Cambridge University Press. Summary and Resources
Cabral, Luís M. B., 2000. Introduction to Industrial Organization. MIT Press. Links to Description and chapter-preview links.
Shepherd, William, 1985. The Economics of Industrial Organization, Prentice-Hall. ISBN 0-13-231481-9
Shy, Oz, 1995. Industrial Organization: Theory and Applications. Description and chapter-preview links. MIT Press.
Vives, Xavier, 2001. Oligopoly Pricing: Old Ideas and New Tools. MIT Press. Description and scroll to chapter-preview links.
Jeffrey Church & Roger Ware, 2005. "Industrial Organization: A Strategic Approach", (aka IOSA Archived 2016-12-08 at the Wayback Machine)”, Free Textbook
Nicolas Boccard, 2010. "Industrial Organization, a Contract Based approach (aka IOCB)”, Open Source Textbook
== Journals ==
The RAND Journal of Economics
International Journal of the Economics of Business and issue preview links
International Journal of Industrial Organization and issue-preview links
Journal of Industrial Economics, Aims and Scope, and issue-preview links.
Journal of Law, Economics, and Organization and issue-preview links.
Review of Industrial Organization
== External links ==
Quotations related to Industrial organization at Wikiquote | Wikipedia/Industrial_organization |
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a position evolves through alternating moves, each governed by well-defined rules, with the aim of achieving a specific winning condition. Unlike economic game theory, combinatorial game theory generally avoids the study of games of chance or games involving imperfect information, preferring instead games in which the current state and the full set of available moves are always known to both players. However, as mathematical techniques develop, the scope of analyzable games expands, and the boundaries of the field continue to evolve. Authors typically define the term "game" at the outset of academic papers, with definitions tailored to the specific game under analysis rather than reflecting the field’s full scope.
Combinatorial games include well-known examples such as chess, checkers, and Go, which are considered complex and non-trivial, as well as simpler, "solved" games like tic-tac-toe. Some combinatorial games, such as infinite chess, may feature an unbounded playing area. In the context of combinatorial game theory, the structure of such games is typically modeled using a game tree. The field also encompasses single-player puzzles like Sudoku, and zero-player automata such as Conway's Game of Life—although these are sometimes more accurately categorized as mathematical puzzles or automata, given that the strictest definitions of "game" imply the involvement of multiple participants.
A key concept in combinatorial game theory is that of the solved game. For instance, tic-tac-toe is solved in that optimal play by both participants always results in a draw. Determining such outcomes for more complex games is significantly more difficult. Notably, in 2007, checkers was announced to be weakly solved, with perfect play by both sides leading to a draw; however, this result required a computer-assisted proof. Many real-world games remain too complex for complete analysis, though combinatorial methods have shown some success in the study of Go endgames. Analyzing a position using combinatorial game theory involves identifying the optimal sequence of moves for both players until the game's conclusion, but this process becomes prohibitively difficult for anything beyond simple games.
It is useful to distinguish between combinatorial "mathgames"—games of primary interest to mathematicians and scientists for theoretical exploration—and "playgames," which are more widely played for entertainment and competition. Some games, such as Nim, straddle both categories. Nim played a foundational role in the development of combinatorial game theory and was among the earliest games to be programmed on a computer. Tic-tac-toe continues to be used in teaching fundamental concepts of game AI design to computer science students.
== Difference with traditional game theory ==
Combinatorial game theory contrasts with "traditional" or "economic" game theory, which, although it can address sequential play, often incorporates elements of probability and incomplete information. While economic game theory employs utility theory and equilibrium concepts, combinatorial game theory is primarily concerned with two-player perfect-information games and has pioneered novel techniques for analyzing game trees, such as through the use of surreal numbers, which represent a subset of all two-player perfect-information games. The types of games studied in this field are of particular interest in areas such as artificial intelligence, especially for tasks in automated planning and scheduling. However, there is a distinction in emphasis: while economic game theory tends to focus on practical algorithms—such as the alpha–beta pruning strategy commonly taught in AI courses—combinatorial game theory places greater weight on theoretical results, including the analysis of game complexity and the existence of optimal strategies through methods like the strategy-stealing argument.
== History ==
Combinatorial game theory arose in relation to the theory of impartial games, in which any play available to one player must be available to the other as well. One such game is Nim, which can be solved completely. Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses. In the 1930s, the Sprague–Grundy theorem showed that all impartial games are equivalent to heaps in Nim, thus showing that major unifications are possible in games considered at a combinatorial level, in which detailed strategies matter, not just pay-offs.
In the 1960s, Elwyn R. Berlekamp, John H. Conway and Richard K. Guy jointly introduced the theory of a partisan game, in which the requirement that a play available to one player be available to both is relaxed. Their results were published in their book Winning Ways for your Mathematical Plays in 1982. However, the first work published on the subject was Conway's 1976 book On Numbers and Games, also known as ONAG, which introduced the concept of surreal numbers and the generalization to games. On Numbers and Games was also a fruit of the collaboration between Berlekamp, Conway, and Guy.
Combinatorial games are generally, by convention, put into a form where one player wins when the other has no moves remaining. It is easy to convert any finite game with only two possible results into an equivalent one where this convention applies. One of the most important concepts in the theory of combinatorial games is that of the sum of two games, which is a game where each player may choose to move either in one game or the other at any point in the game, and a player wins when his opponent has no move in either game. This way of combining games leads to a rich and powerful mathematical structure.
Conway stated in On Numbers and Games that the inspiration for the theory of partisan games was based on his observation of the play in Go endgames, which can often be decomposed into sums of simpler endgames isolated from each other in different parts of the board.
== Examples ==
The introductory text Winning Ways introduced a large number of games, but the following were used as motivating examples for the introductory theory:
Blue–Red Hackenbush - At the finite level, this partisan combinatorial game allows constructions of games whose values are dyadic rational numbers. At the infinite level, it allows one to construct all real values, as well as many infinite ones that fall within the class of surreal numbers.
Blue–Red–Green Hackenbush - Allows for additional game values that are not numbers in the traditional sense, for example, star.
Toads and Frogs - Allows various game values. Unlike most other games, a position is easily represented by a short string of characters.
Domineering - Various interesting games, such as hot games, appear as positions in Domineering, because there is sometimes an incentive to move, and sometimes not. This allows discussion of a game's temperature.
Nim - An impartial game. This allows for the construction of the nimbers. (It can also be seen as a green-only special case of Blue-Red-Green Hackenbush.)
The classic game Go was influential on the early combinatorial game theory, and Berlekamp and Wolfe subsequently developed an endgame and temperature theory for it (see references). Armed with this they were able to construct plausible Go endgame positions from which they could give expert Go players a choice of sides and then defeat them either way.
Another game studied in the context of combinatorial game theory is chess. In 1953 Alan Turing wrote of the game, "If one can explain quite unambiguously in English, with the aid of mathematical symbols if required, how a calculation is to be done, then it is always possible to programme any digital computer to do that calculation, provided the storage capacity is adequate." In a 1950 paper, Claude Shannon estimated the lower bound of the game-tree complexity of chess to be 10120, and today this is referred to as the Shannon number. Chess remains unsolved, although extensive study, including work involving the use of supercomputers has created chess endgame tablebases, which shows the result of perfect play for all end-games with seven pieces or less. Infinite chess has an even greater combinatorial complexity than chess (unless only limited end-games, or composed positions with a small number of pieces are being studied).
== Overview ==
A game, in its simplest terms, is a list of possible "moves" that two players, called left and right, can make. The game position resulting from any move can be considered to be another game. This idea of viewing games in terms of their possible moves to other games leads to a recursive mathematical definition of games that is standard in combinatorial game theory. In this definition, each game has the notation {L|R}. L is the set of game positions that the left player can move to, and R is the set of game positions that the right player can move to; each position in L and R is defined as a game using the same notation.
Using Domineering as an example, label each of the sixteen boxes of the four-by-four board by A1 for the upper leftmost square, C2 for the third box from the left on the second row from the top, and so on. We use e.g. (D3, D4) to stand for the game position in which a vertical domino has been placed in the bottom right corner. Then, the initial position can be described in combinatorial game theory notation as
{
(
A
1
,
A
2
)
,
(
B
1
,
B
2
)
,
…
|
(
A
1
,
B
1
)
,
(
A
2
,
B
2
)
,
…
}
.
{\displaystyle \{(\mathrm {A} 1,\mathrm {A} 2),(\mathrm {B} 1,\mathrm {B} 2),\dots |(\mathrm {A} 1,\mathrm {B} 1),(\mathrm {A} 2,\mathrm {B} 2),\dots \}.}
In standard Cross-Cram play, the players alternate turns, but this alternation is handled implicitly by the definitions of combinatorial game theory rather than being encoded within the game states.
{
(
A
1
,
A
2
)
|
(
A
1
,
B
1
)
}
=
{
{
|
}
|
{
|
}
}
.
{\displaystyle \{(\mathrm {A} 1,\mathrm {A} 2)|(\mathrm {A} 1,\mathrm {B} 1)\}=\{\{|\}|\{|\}\}.}
The above game describes a scenario in which there is only one move left for either player, and if either player makes that move, that player wins. (An irrelevant open square at C3 has been omitted from the diagram.) The {|} in each player's move list (corresponding to the single leftover square after the move) is called the zero game, and can actually be abbreviated 0. In the zero game, neither player has any valid moves; thus, the player whose turn it is when the zero game comes up automatically loses.
The type of game in the diagram above also has a simple name; it is called the star game, which can also be abbreviated ∗. In the star game, the only valid move leads to the zero game, which means that whoever's turn comes up during the star game automatically wins.
An additional type of game, not found in Domineering, is a loopy game, in which a valid move of either left or right is a game that can then lead back to the first game. Checkers, for example, becomes loopy when one of the pieces promotes, as then it can cycle endlessly between two or more squares. A game that does not possess such moves is called loopfree.
There are also transfinite games, which have infinitely many positions—that is, left and right have lists of moves that are infinite rather than finite.
== Game abbreviations ==
=== Numbers ===
Numbers represent the number of free moves, or the move advantage of a particular player. By convention positive numbers represent an advantage for Left, while negative numbers represent an advantage for Right. They are defined recursively with 0 being the base case.
0 = {|}
1 = {0|}, 2 = {1|}, 3 = {2|}
−1 = {|0}, −2 = {|−1}, −3 = {|−2}
The zero game is a loss for the first player.
The sum of number games behaves like the integers, for example 3 + −2 = 1.
Any game number is in the class of the surreal numbers.
=== Star ===
Star, written as ∗ or {0|0}, is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.
∗ + ∗ = 0, because the first player must turn one copy of ∗ to a 0, and then the other player will have to turn the other copy of ∗ to a 0 as well; at this point, the first player would lose, since 0 + 0 admits no moves.
The game ∗ is neither positive nor negative; it and all other games in which the first player wins (regardless of which side the player is on) are said to be fuzzy or confused with 0; symbolically, we write ∗ || 0.
The game ∗n is notation for {0, ∗, …, ∗(n−1)| 0, ∗, …, ∗(n−1)}, which is also representative of normal-play Nim with a single heap of n objects. (Note that ∗0 = 0 and ∗1 = ∗.)
=== Up and down ===
Up, written as ↑, is a position in combinatorial game theory. In standard notation, ↑ = {0|∗}. Its negative is called down.
−↑ = ↓ (down)
Up is strictly positive (↑ > 0), and down is strictly negative (↓ < 0), but both are infinitesimal. Up and down are defined in Winning Ways for your Mathematical Plays.
=== "Hot" games ===
Consider the game {1|−1}. Both moves in this game are an advantage for the player who makes them; so the game is said to be "hot;" it is greater than any number less than −1, less than any number greater than 1, and fuzzy with any number in between. It is written as ±1. Note that a subclass of hot games, referred to as ±n for some numerical game n is a switch game. Switch games can be added to numbers, or multiplied by positive ones, in the expected fashion; for example, 4 ± 1 = {5|3}.
== Nimbers ==
An impartial game is one where, at every position of the game, the same moves are available to both players. For instance, Nim is impartial, as any set of objects that can be removed by one player can be removed by the other. However, domineering is not impartial, because one player places horizontal dominoes and the other places vertical ones. Likewise Checkers is not impartial, since the players own different colored pieces. For any ordinal number, one can define an impartial game generalizing Nim in which, on each move, either player may replace the number with any smaller ordinal number; the games defined in this way are known as nimbers. The Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a nimber.
The "smallest" nimbers – the simplest and least under the usual ordering of the ordinals – are 0 and ∗.
== See also ==
Alpha–beta pruning, an optimised algorithm for searching the game tree
Backward induction, reasoning backwards from a final situation
Cooling and heating (combinatorial game theory), various transformations of games making them more amenable to the theory
Connection game, a type of game where players attempt to establish connections
Endgame tablebase, a database saying how to play endgames
Expectiminimax tree, an adaptation of a minimax game tree to games with an element of chance
Extensive-form game, a game tree enriched with payoffs and information available to players
Game classification, an article discussing ways of classifying games
Game complexity, an article describing ways of measuring the complexity of games
Grundy's game, a mathematical game in which heaps of objects are split
Multi-agent system, a type of computer system for tackling complex problems
Positional game, a type of game where players claim previously-unclaimed positions
Solving chess
Sylver coinage, a mathematical game of choosing positive integers that are not the sum of non-negative multiples of previously chosen integers
Wythoff's game, a mathematical game of taking objects from one or two piles
Topological game, a type of mathematical game played in a topological space
Zugzwang, being obliged to play when this is disadvantageous
== Notes ==
== References ==
Albert, Michael H.; Nowakowski, Richard J.; Wolfe, David (2007). Lessons in Play: An Introduction to Combinatorial Game Theory. A K Peters Ltd. ISBN 978-1-56881-277-9.
Beck, József (2008). Combinatorial games: tic-tac-toe theory. Cambridge University Press. ISBN 978-0-521-46100-9.
Berlekamp, E.; Conway, J. H.; Guy, R. (1982). Winning Ways for your Mathematical Plays: Games in general. Academic Press. ISBN 0-12-091101-9. 2nd ed., A K Peters Ltd (2001–2004), ISBN 1-56881-130-6, ISBN 1-56881-142-X
Berlekamp, E.; Conway, J. H.; Guy, R. (1982). Winning Ways for your Mathematical Plays: Games in particular. Academic Press. ISBN 0-12-091102-7. 2nd ed., A K Peters Ltd (2001–2004), ISBN 1-56881-143-8, ISBN 1-56881-144-6.
Berlekamp, Elwyn; Wolfe, David (1997). Mathematical Go: Chilling Gets the Last Point. A K Peters Ltd. ISBN 1-56881-032-6.
Bewersdorff, Jörg (2021). Luck, Logic and White Lies: The Mathematics of Games (2nd ed.). A K Peters/CRC Press. doi:10.1201/9781003092872. ISBN 978-1-003-09287-2. See especially sections 21–26.
Conway, John Horton (1976). On Numbers and Games. Academic Press. ISBN 0-12-186350-6. 2nd ed., A K Peters Ltd (2001), ISBN 1-56881-127-6.
Robert A. Hearn; Erik D. Demaine (2009). Games, Puzzles, and Computation. A K Peters, Ltd. ISBN 978-1-56881-322-6.
== External links ==
List of combinatorial game theory links at the homepage of David Eppstein
An Introduction to Conway's games and numbers by Dierk Schleicher and Michael Stoll
Combinational Game Theory terms summary by Bill Spight
Combinatorial Game Theory Workshop, Banff International Research Station, June 2005 | Wikipedia/Combinatorial_game_theory |
In combinatorial game theory, the strategy-stealing argument is a general argument that shows, for many two-player games, that the second player cannot have a guaranteed winning strategy. The strategy-stealing argument applies to any symmetric game (one in which either player has the same set of available moves with the same results, so that the first player can "use" the second player's strategy) in which an extra move can never be a disadvantage. A key property of a strategy-stealing argument is that it proves that the first player can win (or possibly draw) the game without actually constructing such a strategy. So, although it might prove the existence of a winning strategy, the proof gives no information about what that strategy is.
The argument works by obtaining a contradiction. A winning strategy is assumed to exist for the second player, who is using it. But then, roughly speaking, after making an arbitrary first move – which by the conditions above is not a disadvantage – the first player may then also play according to this winning strategy. The result is that both players are guaranteed to win – which is absurd, thus contradicting the assumption that such a strategy exists.
Strategy-stealing was invented by John Nash in the 1940s to show that the game of hex is always a first-player win, as ties are not possible in this game. However, Nash did not publish this method, and József Beck credits its first publication to Alfred W. Hales and Robert I. Jewett, in the 1963 paper on tic-tac-toe in which they also proved the Hales–Jewett theorem. Other examples of games to which the argument applies include the m,n,k-games such as gomoku. In the game of Chomp strategy stealing shows that the first player has a winning strategy in any rectangular board (other than 1x1). In the game of Sylver coinage, strategy stealing has been used to show that the first player can win in certain positions called "enders". In all of these examples the proof reveals nothing about the actual strategy.
== Example ==
A strategy-stealing argument can be used on the example of the game of tic-tac-toe, for a board and winning rows of any size. Suppose that the second player (P2) is using a strategy S which guarantees a win. The first player (P1) places an X in an arbitrary position. P2 responds by placing an O according to S. But if P1 ignores the first random X, P1 is now in the same situation as P2 on P2's first move: a single enemy piece on the board. P1 may therefore make a move according to S – that is, unless S calls for another X to be placed where the ignored X is already placed. But in this case, P1 may simply place an X in some other random position on the board, the net effect of which will be that one X is in the position demanded by S, while another is in a random position, and becomes the new ignored piece, leaving the situation as before. Continuing in this way, S is, by hypothesis, guaranteed to produce a winning position (with an additional ignored X of no consequence). But then P2 has lost – contradicting the supposition that P2 had a guaranteed winning strategy. Such a winning strategy for P2, therefore, does not exist, and tic-tac-toe is either a forced win for P1 or a tie. (Further analysis shows it is in fact a tie.)
The same proof holds for any strong positional game.
== Chess ==
There is a class of chess positions called Zugzwang in which the player obligated to move would prefer to "pass" if this were allowed. Because of this, the strategy-stealing argument cannot be applied to chess. It is not currently known whether White or Black can force a win with optimal play, or if both players can force a draw. However, virtually all students of chess consider White's first move to be an advantage and White wins more often than black in high-level games.
== Go ==
In Go passing is allowed. When the starting position is symmetrical (empty board, neither player has any points), this means that the first player could steal the second player's winning strategy simply by giving up the first move. Since the 1930s, however, the second player is typically awarded some compensation points, which makes the starting position asymmetrical, and the strategy-stealing argument will no longer work.
An elementary strategy in the game is "mirror go", where the second player performs moves which are diagonally opposite those of this opponent. This approach may be defeated using ladder tactics, ko fights, or successfully competing for control of the board's central point.
== Constructivity ==
The strategy-stealing argument shows that the second player cannot win, by means of deriving a contradiction from any hypothetical winning strategy for the second player. The argument is commonly employed in games where there can be no draw, by means of the law of the excluded middle. However, it does not provide an explicit strategy for the first player, and because of this it has been called non-constructive. This raises the question of how to actually compute a winning strategy.
For games with a finite number of reachable positions, such as chomp, a winning strategy can be found by exhaustive search. However, this might be impractical if the number of positions is large.
In 2019, Greg Bodwin and Ofer Grossman proved that the problem of finding a winning strategy is PSPACE-hard in two kinds of games in which strategy-stealing arguments were used: the minimum poset game and the symmetric Maker-Maker game.
== References == | Wikipedia/Strategy-stealing_argument |
In game theory, normal form is a description of a game. Unlike extensive form, normal-form representations are not graphical per se, but rather represent the game by way of a matrix. While this approach can be of greater use in identifying strictly dominated strategies and Nash equilibria, some information is lost as compared to extensive-form representations. The normal-form representation of a game includes all perceptible and conceivable strategies, and their corresponding payoffs, for each player.
In static games of complete, perfect information, a normal-form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is the set of all strategies available to that player, whereas a strategy is a complete plan of action for every stage of the game, regardless of whether that stage actually arises in play. A payoff function for a player is a mapping from the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a cardinal or ordinal utility—often cardinal in the normal-form representation) of a player, i.e. the payoff function of a player takes as its input a strategy profile (that is a specification of strategies for every player) and yields a representation of payoff as its output.
== An example ==
The matrix provided is a normal-form representation of a game in which players move simultaneously (or at least do not observe the other player's move before making their own) and receive the payoffs as specified for the combinations of actions played. For example, if player 1 plays top and player 2 plays left, player 1 receives 4 and player 2 receives 3. In each cell, the first number represents the payoff to the row player (in this case player 1), and the second number represents the payoff to the column player (in this case player 2).
=== Other representations ===
Often, symmetric games (where the payoffs do not depend on which player chooses each action) are represented with only one payoff. This is the payoff for the row player. For example, the payoff matrices on the right and left below represent the same game.
The topological space of games with related payoff matrices can also be mapped, with adjacent games having the most similar matrices. This shows how incremental incentive changes can change the game.
== Uses of normal form ==
=== Dominated strategies ===
The payoff matrix facilitates elimination of dominated strategies, and it is usually used to illustrate this concept. For example, in the prisoner's dilemma, we can see that each prisoner can either "cooperate" or "defect". If exactly one prisoner defects, he gets off easily and the other prisoner is locked up for a long time. However, if they both defect, they will both be locked up for a shorter time. One can determine that Cooperate is strictly dominated by Defect. One must compare the first numbers in each column, in this case 0 > −1 and −2 > −5. This shows that no matter what the column player chooses, the row player does better by choosing Defect. Similarly, one compares the second payoff in each row; again 0 > −1 and −2 > −5. This shows that no matter what row does, column does better by choosing Defect. This demonstrates the unique Nash equilibrium of this game is (Defect, Defect).
=== Sequential games in normal form ===
These matrices only represent games in which moves are simultaneous (or, more generally, information is imperfect). The above matrix does not represent the game in which player 1 moves first, observed by player 2, and then player 2 moves, because it does not specify each of player 2's strategies in this case. In order to represent this sequential game we must specify all of player 2's actions, even in contingencies that can never arise in the course of the game. In this game, player 2 has actions, as before, Left and Right. Unlike before he has four strategies, contingent on player 1's actions. The strategies are:
Left if player 1 plays Top and Left otherwise
Left if player 1 plays Top and Right otherwise
Right if player 1 plays Top and Left otherwise
Right if player 1 plays Top and Right otherwise
On the right is the normal-form representation of this game.
== General formulation ==
In order for a game to be in normal form, we are provided with the following data:
There is a finite set I of players, each player is denoted by i. Each player i has a finite k number of pure strategies
S
i
=
{
1
,
2
,
…
,
k
}
.
{\displaystyle S_{i}=\{1,2,\ldots ,k\}.}
A pure strategy profile is an association of strategies to players, that is an I-tuple
s
→
=
(
s
1
,
s
2
,
…
,
s
I
)
{\displaystyle {\vec {s}}=(s_{1},s_{2},\ldots ,s_{I})}
such that
s
1
∈
S
1
,
s
2
∈
S
2
,
…
,
s
I
∈
S
I
{\displaystyle s_{1}\in S_{1},s_{2}\in S_{2},\ldots ,s_{I}\in S_{I}}
A payoff function is a function
u
i
:
S
1
×
S
2
×
…
×
S
I
→
R
.
{\displaystyle u_{i}:S_{1}\times S_{2}\times \ldots \times S_{I}\rightarrow \mathbb {R} .}
whose intended interpretation is the award given to a single player at the outcome of the game. Accordingly, to completely specify a game, the payoff function has to be specified for each player in the player set I= {1, 2, ..., I}.
Definition: A game in normal form is a structure
T
=
⟨
I
,
S
,
u
⟩
{\displaystyle \mathrm {T} =\langle I,\mathbf {S} ,\mathbf {u} \rangle }
where:
I
=
{
1
,
2
,
…
,
I
}
{\displaystyle I=\{1,2,\ldots ,I\}}
is a set of players,
S
=
{
S
1
,
S
2
,
…
,
S
I
}
{\displaystyle \mathbf {S} =\{S_{1},S_{2},\ldots ,S_{I}\}}
is an I-tuple of pure strategy sets, one for each player, and
u
=
{
u
1
,
u
2
,
…
,
u
I
}
{\displaystyle \mathbf {u} =\{u_{1},u_{2},\ldots ,u_{I}\}}
is an I-tuple of payoff functions.
== References ==
Fudenberg, D.; Tirole, J. (1991). Game Theory. MIT Press. ISBN 0-262-06141-4.
Leyton-Brown, Kevin; Shoham, Yoav (2008). Essentials of Game Theory: A Concise, Multidisciplinary Introduction. San Rafael, CA: Morgan & Claypool Publishers. ISBN 978-1-59829-593-1.. An 88-page mathematical introduction; free online at many universities.
Luce, R. D.; Raiffa, H. (1989). Games and Decisions. Dover Publications. ISBN 0-486-65943-7.
Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7.. A comprehensive reference from a computational perspective; see Chapter 3. Downloadable free online.
Weibull, J. (1996). Evolutionary Game Theory. MIT Press. ISBN 0-262-23181-6.
J. von Neumann and O. Morgenstern, Theory of games and Economic Behavior, John Wiley Science Editions, 1964. Which was originally published in 1944 by Princeton University Press. | Wikipedia/Strategic_form |
In game theory, a move, action, or play is any one of the options which a player can choose in a setting where the optimal outcome depends not only on their own actions but on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship.
The term strategy is typically used to mean a complete algorithm for playing a game, telling a player what to do for every possible situation. A player's strategy determines the action the player will take at any stage of the game. However, the idea of a strategy is often confused or conflated with that of a move or action, because of the correspondence between moves and pure strategies in most games: for any move X, "always play move X" is an example of a valid strategy, and as a result every move can also be considered to be a strategy. Other authors treat strategies as being a different type of thing from actions, and therefore distinct.
It is helpful to think about a "strategy" as a list of directions, and a "move" as a single turn on the list of directions itself. This strategy is based on the payoff or outcome of each action. The goal of each agent is to consider their payoff based on a competitors action. For example, competitor A can assume competitor B enters the market. From there, Competitor A compares the payoffs they receive by entering and not entering. The next step is to assume Competitor B does not enter and then consider which payoff is better based on if Competitor A chooses to enter or not enter. This technique can identify dominant strategies where a player can identify an action that they can take no matter what the competitor does to try to maximize the payoff.
A strategy profile (sometimes called a strategy combination) is a set of strategies for all players which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.
== Strategy set ==
A player's strategy set defines what strategies are available for them to play.
A player has a finite strategy set if they have a number of discrete strategies available to them. For instance, a game of rock paper scissors comprises a single move by each player—and each player's move is made without knowledge of the other's, not as a response—so each player has the finite strategy set {rock paper scissors}.
A strategy set is infinite otherwise. For instance the cake cutting game has a bounded continuum of strategies in the strategy set {Cut anywhere between zero percent and 100 percent of the cake}.
In a dynamic game, games that are played over a series of time, the strategy set consists of the possible rules a player could give to a robot or agent on how to play the game. For instance, in the ultimatum game, the strategy set for the second player would consist of every possible rule for which offers to accept and which to reject.
In a Bayesian game, or games in which players have incomplete information about one another, the strategy set is similar to that in a dynamic game. It consists of rules for what action to take for any possible private information.
=== Choosing a strategy set ===
In applied game theory, the definition of the strategy sets is an important part of the art of making a game simultaneously solvable and meaningful. The game theorist can use knowledge of the overall problem, that is the friction between two or more players, to limit the strategy spaces, and ease the solution.
For instance, strictly speaking in the Ultimatum game a player can have strategies such as: Reject offers of ($1, $3, $5, ..., $19), accept offers of ($0, $2, $4, ..., $20). Including all such strategies makes for a very large strategy space and a somewhat difficult problem. A game theorist might instead believe they can limit the strategy set to: {Reject any offer ≤ x, accept any offer > x; for x in ($0, $1, $2, ..., $20)}.
== Pure and mixed strategies ==
A pure strategy provides a complete and deterministic plan for how a player will act in every possible situation in a game. It specifies exactly what action the player will take at each decision point, given any information they may have. A player's strategy set consists of all the pure strategies available to them.
A mixed strategy is a probability distribution over the set of pure strategies. Rather than committing to a single course of action, the player randomizes among pure strategies according to specified probabilities. Mixed strategies are particularly useful in games where no pure strategy constitutes a best response, allowing players to avoid being predictable. Since the outcomes depend on probabilities, we refer to the resulting payoffs as expected payoffs.
A pure strategy can be viewed as a special case of a mixed strategy—one in which a single pure strategy is chosen with probability 1, and all others with probability 0.
A totally mixed strategy is a mixed strategy in which every pure strategy in the player's strategy set is assigned a strictly positive probability—that is, no pure strategy is excluded or played with zero probability. This means the player randomizes across all of their options, never fully ruling any one out. Totally mixed strategies are important in some advanced game theory concepts like trembling hand perfect equilibrium, where the idea is to model players as occasionally making small mistakes. In that context, assigning positive probability to every strategy—even suboptimal ones—helps capture how players might still end up choosing them due to small "trembles" in decision-making.
== Mixed strategy ==
=== Illustration ===
In a soccer penalty kick, the kicker must choose whether to kick to the right or left side of the goal, and simultaneously the goalie must decide which way to block it. Also, the kicker has a direction they are best at shooting, which is left if they are right-footed. The matrix for the soccer game illustrates this situation, a simplified form of the game studied by Chiappori, Levitt, and Groseclose (2002). It assumes that if the goalie guesses correctly, the kick is blocked, which is set to the base payoff of 0 for both players. If the goalie guesses wrong, the kick is more likely to go in if it is to the left (payoffs of +2 for the kicker and -2 for the goalie) than if it is to the right (the lower payoff of +1 to kicker and -1 to goalie).
This game has no pure-strategy equilibrium, because one player or the other would deviate from any profile of strategies—for example, (Left, Left) is not an equilibrium because the Kicker would deviate to Right and increase his payoff from 0 to 1.
The kicker's mixed-strategy equilibrium is found from the fact that they will deviate from randomizing unless their payoffs from Left Kick and Right Kick are exactly equal. If the goalie leans left with probability g, the kicker's expected payoff from Kick Left is g(0) + (1-g)(2), and from Kick Right is g(1) + (1-g)(0). Equating these yields g= 2/3. Similarly, the goalie is willing to randomize only if the kicker chooses mixed strategy probability k such that Lean Left's payoff of k(0) + (1-k)(-1) equals Lean Right's payoff of k(-2) + (1-k)(0), so k = 1/3. Thus, the mixed-strategy equilibrium is (Prob(Kick Left) = 1/3, Prob(Lean Left) = 2/3).
In equilibrium, the kicker kicks to their best side only 1/3 of the time. That is because the goalie is guarding that side more. Also, in equilibrium, the kicker is indifferent which way they kick, but for it to be an equilibrium they must choose exactly 1/3 probability.
Chiappori, Levitt, and Groseclose try to measure how important it is for the kicker to kick to their favored side, add center kicks, etc., and look at how professional players actually behave. They find that they do randomize, and that kickers kick to their favored side 45% of the time and goalies lean to that side 57% of the time. Their article is well-known as an example of how people in real life use mixed strategies.
=== Significance ===
In his famous paper, John Forbes Nash proved that there is an equilibrium for every finite game. One can divide Nash equilibria into two types. Pure strategy Nash equilibria are Nash equilibria where all players are playing pure strategies. Mixed strategy Nash equilibria are equilibria where at least one player is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies, see Matching pennies. However, many games do have pure strategy Nash equilibria (e.g. the Coordination game, the Prisoner's dilemma, the Stag hunt). Further, games can have both pure strategy and mixed strategy equilibria. An easy example is the pure coordination game, where in addition to the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both players play either strategy with probability 1/2.
=== Interpretations of mixed strategies ===
During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic", since they are weak Nash equilibria, and a player is indifferent about whether to follow their equilibrium strategy probability or deviate to some other probability.
Game theorist Ariel Rubinstein describes alternative ways of understanding the concept. The first, due to Harsanyi (1973), is called purification, and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors.
A second interpretation imagines the game players standing for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents.
Later, Aumann and Brandenburger (1995), re-interpreted Nash equilibrium as an equilibrium in beliefs, rather than actions. For instance, in rock paper scissors an equilibrium in beliefs would have each player believing the other was equally likely to play each strategy. This interpretation weakens the descriptive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to actually play a pure strategy of Rock in each play of the game, even though over time the probabilities are those of the mixed strategy.
== Behavior strategy ==
While a mixed strategy assigns a probability distribution over pure strategies, a behavior strategy (or behavioral strategy) assigns at each information set a probability distribution over the set of possible actions. While the two concepts are very closely related in the context of normal form games, they have very different implications for extensive form games. Roughly, a mixed strategy randomly chooses a deterministic path through the game tree, while a behavior strategy can be seen as a stochastic path.
The relationship between mixed and behavior strategies is the subject of Kuhn's theorem, a behavioral outlook on traditional game-theoretic hypotheses. The result establishes that in any finite extensive-form game with perfect recall, for any player and any mixed strategy, there exists a behavior strategy that, against all profiles of strategies (of other players), induces the same distribution over terminal nodes as the mixed strategy does. The converse is also true.
A famous example of why perfect recall is required for the equivalence is given by Piccione and Rubinstein (1997) with their Absent-Minded Driver game.
=== Outcome equivalence ===
Outcome equivalence combines the mixed and behavioral strategy of Player i in relation to the pure strategy of Player i’s opponent. Outcome equivalence is defined as the situation in which, for any mixed and behavioral strategy that Player i takes, in response to any pure strategy that Player I’s opponent plays, the outcome distribution of the mixed and behavioral strategy must be equal. This equivalence can be described by the following formula: (Q^(U(i), S(-i)))(z) = (Q^(β(i), S(-i)))(z), where U(i) describes Player i's mixed strategy, β(i) describes Player i's behavioral strategy, and S(-i) is the opponent's strategy.
=== Strategy with perfect recall ===
Perfect recall is defined as the ability of every player in game to remember and recall all past actions within the game. Perfect recall is required for equivalence as, in finite games with imperfect recall, there will be existing mixed strategies of Player I in which there is no equivalent behavior strategy. This is fully described in the Absent-Minded Driver game formulated by Piccione and Rubinstein. In short, this game is based on the decision-making of a driver with imperfect recall, who needs to take the second exit off the highway to reach home but does not remember which intersection they are at when they reach it. Figure [2] describes this game.
Without perfect information (i.e. imperfect information), players make a choice at each decision node without knowledge of the decisions that have preceded it. Therefore, a player’s mixed strategy can produce outcomes that their behavioral strategy cannot, and vice versa. This is demonstrated in the Absent-minded Driver game. With perfect recall and information, the driver has a single pure strategy, which is [continue, exit], as the driver is aware of what intersection (or decision node) they are at when they arrive to it. On the other hand, looking at the planning-optimal stage only, the maximum payoff is achieved by continuing at both intersections, maximized at p=2/3 (reference). This simple one player game demonstrates the importance of perfect recall for outcome equivalence, and its impact on normal and extended form games.
== See also ==
Nash equilibrium
Haven (graph theory)
Evolutionarily stable strategy
== References == | Wikipedia/Strategy_(game_theory) |
In combinatorial game theory, the paranoid algorithm is a game tree search algorithm designed to analyze multi-player games using a two-player adversarial framework. The algorithm assumes all opponents form a coalition to minimize the focal player’s payoff, transforming an n-player non-zero-sum game into a zero-sum game between the focal player and the coalition.
The paranoid algorithm significantly improves upon the maxn algorithm by enabling the use of alpha-beta pruning and other minimax-based optimization techniques that are less effective in standard multi-player game analysis. By treating opponents as a unified adversary whose payoff is the opposite of the focal player’s payoff, the algorithm can apply branch and bound techniques and achieve substantial performance improvements over traditional multi-player algorithms.
While the paranoid assumption may not accurately reflect the true strategic interactions in all multi-player scenarios—where players typically optimize their own payoffs—the algorithm has proven effective in practice for artificial intelligence applications in board games and other combinatorial multi-player games. The algorithm is particularly valuable in computer game AI where computational efficiency is crucial and the simplified opponent model provides adequate performance for real-time applications.
== See also ==
Maxn algorithm
Minimax algorithm
== References == | Wikipedia/Paranoid_algorithm |
The volunteer's dilemma is a game that models a situation in which each player can either make a small sacrifice that benefits everybody, or instead wait in hope of benefiting from someone else's sacrifice.
One example is a scenario in which the electricity supply has failed for an entire neighborhood. All inhabitants know that the electricity company will fix the problem as long as at least one person calls to notify them, at some cost. If no one volunteers, the worst possible outcome is obtained for all participants. If any one person elects to volunteer, the rest benefit by not doing so.
A public good is only produced if at least one person volunteers to pay an arbitrary cost. In this game, bystanders decide independently on whether to sacrifice themselves for the benefit of the group. Because the volunteer receives no benefit, there is a greater incentive for freeriding than to sacrifice oneself for the group. If no one volunteers, everyone loses. The social phenomena of the bystander effect and diffusion of responsibility heavily relate to the volunteer's dilemma.
== Payoff matrix ==
The payoff matrix for the game is shown below:
When the volunteer's dilemma takes place between only two players, the game gets the character of the game "chicken". As seen by the payoff matrix, there is no dominant strategy in the volunteer's dilemma. In a mixed-strategy Nash equilibrium, an increase in N players will decrease the likelihood that at least one person volunteers, which is consistent with the bystander effect.
== Examples in real life ==
=== The murder of Kitty Genovese ===
The story of Kitty Genovese is often cited as an example of the volunteer's dilemma. Genovese was stabbed to death outside her apartment building in Queens, New York, in 1964. According to a highly influential New York Times account, dozens of people witnessed the assault but did not get involved because they thought others would contact the police anyway and did not want to incur the personal cost of getting involved. Subsequent investigations have shown the original account to have been unfounded, and although it inspired sound scientific research, its use as a simplistic parable in psychology textbooks has been criticized.
=== The meerkat ===
The meerkat exhibits the volunteer's dilemma in nature. One or more meerkats act as sentries while the others forage for food. If a predator approaches, the sentry meerkat lets out a warning call so the others can burrow to safety. However, the altruism of this meerkat puts it at risk of being discovered by the predator.
== Quantum volunteer's dilemma ==
One significant volunteer's dilemma variant was introduced by Weesie and Franzen in 1998 and involves cost-sharing among volunteers. In this variant of the volunteer's dilemma, if there is no volunteer, all players receive a payoff of 0. If there is at least one volunteer, the reward of b units is distributed to all players. In contrast, the total cost of c units incurred by volunteering is divided equally among all the volunteers. It is shown that for classical mixed strategies setting, there is a unique symmetric Nash equilibrium and it is obtained by setting the probability of volunteering for each player to be the unique root in the open interval (0,1) of the degree-n polynomial
g
n
{\displaystyle g_{n}}
given by
g
n
(
α
)
=
(
1
−
α
)
n
−
1
(
2
n
α
+
1
−
α
)
−
1
{\displaystyle g_{n}(\alpha )=(1-\alpha )^{n-1}(2n\alpha +1-\alpha )-1}
In 2024, a quantum variant of the classical volunteer’s dilemma was introduced with b=2 and c=1. This generalizes the classical setting by allowing players to utilize quantum strategies. This is achieved by employing the Eisert–Wilkens–Lewenstein quantization framework. In this setting, the players receive an entangled n-qubit state with each player controlling one qubit. The decision of each player can be viewed as determining two angles. Symmetric Nash equilibria that attain a payoff value of
2
−
1
/
n
{\displaystyle 2-1/n}
for each player is shown, and each player volunteers at this Nash equilibrium. Furthermore, these Nash equilibria are Pareto optimal. It is shown that the payoff function of Nash equilibria in the quantum setting is higher than the payoff of Nash equilibria in the classical setting.
== See also ==
Bystander effect
Civil courage
Death of Cristina and Violetta Djeordsevic (Italy)
Death of Wang Yue (China)
Mamihlapinatapai
Prisoner's dilemma
Social loafing
Tragedy of the Commons
== References == | Wikipedia/Volunteer's_dilemma |
In multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal. Computing this decomposition is an open problem.
Canonical polyadic decomposition (CPD) is a variant of the tensor rank decomposition, in which the tensor is approximated as a sum of K rank-1 tensors for a user-specified K. The CP decomposition has found some applications in linguistics and chemometrics. It was introduced by Frank Lauren Hitchcock in 1927 and later rediscovered several times, notably in psychometrics.
The CP decomposition is referred to as CANDECOMP, PARAFAC, or CANDECOMP/PARAFAC (CP). Note that the PARAFAC2 rank decomposition is a variation of the CP decomposition.
Another popular generalization of the matrix SVD known as the higher-order singular value decomposition computes orthonormal mode matrices and has found applications in econometrics, signal processing, computer vision, computer graphics, and psychometrics.
== Notation ==
A scalar variable is denoted by lower case italic letters,
a
{\displaystyle a}
and an upper bound scalar is denoted by an upper case italic letter,
A
{\displaystyle A}
.
Indices are denoted by a combination of lowercase and upper case italic letters,
1
≤
i
≤
I
{\displaystyle 1\leq i\leq I}
. Multiple indices that one might encounter when referring to the multiple modes of a tensor are conveniently denoted by
1
≤
i
m
≤
I
m
{\displaystyle 1\leq i_{m}\leq I_{m}}
where
1
≤
m
≤
M
{\displaystyle 1\leq m\leq M}
.
A vector is denoted by a lower case bold Times Roman,
a
{\displaystyle \mathbf {a} }
and a matrix is denoted by bold upper case letters
A
{\displaystyle \mathbf {A} }
.
A higher order tensor is denoted by calligraphic letters,
A
{\displaystyle {\mathcal {A}}}
. An element of an
M
{\displaystyle M}
-order tensor
A
∈
C
I
1
×
I
2
×
…
I
m
×
…
I
M
{\displaystyle {\mathcal {A}}\in \mathbb {C} ^{I_{1}\times I_{2}\times \dots I_{m}\times \dots I_{M}}}
is denoted by
a
i
1
,
i
2
,
…
,
i
m
,
…
i
M
{\displaystyle a_{i_{1},i_{2},\dots ,i_{m},\dots i_{M}}}
or
A
i
1
,
i
2
,
…
,
i
m
,
…
i
M
{\displaystyle {\mathcal {A}}_{i_{1},i_{2},\dots ,i_{m},\dots i_{M}}}
.
== Definition ==
A data tensor
A
∈
F
I
0
×
I
1
×
…
×
I
C
{\displaystyle {\mathcal {A}}\in {\mathbb {F} }^{I_{0}\times I_{1}\times \ldots \times I_{C}}}
is a collection of multivariate observations organized into a M-way array where M=C+1. Every tensor may be represented with a suitably large
R
{\displaystyle R}
as a linear combination of
R
{\displaystyle R}
rank-1 tensors:
A
=
∑
r
=
1
R
λ
r
a
0
,
r
⊗
a
1
,
r
⊗
a
2
,
r
⋯
⊗
a
c
,
r
⊗
⋯
⊗
a
C
,
r
,
{\displaystyle {\mathcal {A}}=\sum _{r=1}^{R}\lambda _{r}\mathbf {a} _{0,r}\otimes \mathbf {a} _{1,r}\otimes \mathbf {a} _{2,r}\dots \otimes \mathbf {a} _{c,r}\otimes \cdots \otimes \mathbf {a} _{C,r},}
where
λ
r
∈
F
{\displaystyle \lambda _{r}\in {\mathbb {F} }}
and
a
m
,
r
∈
F
I
m
{\displaystyle \mathbf {a} _{m,r}\in {\mathbb {F} }^{I_{m}}}
where
1
≤
m
≤
M
{\displaystyle 1\leq m\leq M}
. When the number of terms
R
{\displaystyle R}
is minimal in the above expression, then
R
{\displaystyle R}
is called the rank of the tensor, and the decomposition is often referred to as a (tensor) rank decomposition, minimal CP decomposition, or Canonical Polyadic Decomposition (CPD). If the number of terms is not minimal, then the above decomposition is often referred to as CANDECOMP/PARAFAC, Polyadic decomposition'.
== Tensor rank ==
Contrary to the case of matrices, computing the rank of a tensor is NP-hard. The only notable well-understood case consists of tensors in
F
I
m
⊗
F
I
n
⊗
F
2
{\displaystyle F^{I_{m}}\otimes F^{I_{n}}\otimes F^{2}}
, whose rank can be obtained from the Kronecker–Weierstrass normal form of the linear matrix pencil that the tensor represents. A simple polynomial-time algorithm exists for certifying that a tensor is of rank 1, namely the higher-order singular value decomposition.
The rank of the tensor of zeros is zero by convention. The rank of a tensor
a
1
⊗
⋯
⊗
a
M
{\displaystyle \mathbf {a} _{1}\otimes \cdots \otimes \mathbf {a} _{M}}
is one, provided that
a
m
∈
F
I
m
∖
{
0
}
{\displaystyle \mathbf {a} _{m}\in F^{I_{m}}\setminus \{0\}}
.
=== Field dependence ===
The rank of a tensor depends on the field over which the tensor is decomposed. It is known that some real tensors may admit a complex decomposition whose rank is strictly less than the rank of a real decomposition of the same tensor. As an example, consider the following real tensor
A
=
x
1
⊗
x
2
⊗
x
3
+
x
1
⊗
y
2
⊗
y
3
−
y
1
⊗
x
2
⊗
y
3
+
y
1
⊗
y
2
⊗
x
3
,
{\displaystyle {\mathcal {A}}=\mathbf {x} _{1}\otimes \mathbf {x} _{2}\otimes \mathbf {x} _{3}+\mathbf {x} _{1}\otimes \mathbf {y} _{2}\otimes \mathbf {y} _{3}-\mathbf {y} _{1}\otimes \mathbf {x} _{2}\otimes \mathbf {y} _{3}+\mathbf {y} _{1}\otimes \mathbf {y} _{2}\otimes \mathbf {x} _{3},}
where
x
i
,
y
j
∈
R
2
{\displaystyle \mathbf {x} _{i},\mathbf {y} _{j}\in \mathbb {R} ^{2}}
. The rank of this tensor over the reals is known to be 3, while its complex rank is only 2 because it is the sum of a complex rank-1 tensor with its complex conjugate, namely
A
=
1
2
(
z
¯
1
⊗
z
2
⊗
z
¯
3
+
z
1
⊗
z
¯
2
⊗
z
3
)
,
{\displaystyle {\mathcal {A}}={\frac {1}{2}}({\bar {\mathbf {z} }}_{1}\otimes \mathbf {z} _{2}\otimes {\bar {\mathbf {z} }}_{3}+\mathbf {z} _{1}\otimes {\bar {\mathbf {z} }}_{2}\otimes \mathbf {z} _{3}),}
where
z
k
=
x
k
+
i
y
k
{\displaystyle \mathbf {z} _{k}=\mathbf {x} _{k}+i\mathbf {y} _{k}}
.
In contrast, the rank of real matrices will never decrease under a field extension to
C
{\displaystyle \mathbb {C} }
: real matrix rank and complex matrix rank coincide for real matrices.
=== Generic rank ===
The generic rank
r
(
I
1
,
…
,
I
M
)
{\displaystyle r(I_{1},\ldots ,I_{M})}
is defined as the least rank
r
{\displaystyle r}
such that the closure in the Zariski topology of the set of tensors of rank at most
r
{\displaystyle r}
is the entire space
F
I
1
⊗
⋯
⊗
F
I
M
{\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}
. In the case of complex tensors, tensors of rank at most
r
(
I
1
,
…
,
I
M
)
{\displaystyle r(I_{1},\ldots ,I_{M})}
form a dense set
S
{\displaystyle S}
: every tensor in the aforementioned space is either of rank less than the generic rank, or it is the limit in the Euclidean topology of a sequence of tensors from
S
{\displaystyle S}
. In the case of real tensors, the set of tensors of rank at most
r
(
I
1
,
…
,
I
M
)
{\displaystyle r(I_{1},\ldots ,I_{M})}
only forms an open set of positive measure in the Euclidean topology. There may exist Euclidean-open sets of tensors of rank strictly higher than the generic rank. All ranks appearing on open sets in the Euclidean topology are called typical ranks. The smallest typical rank is called the generic rank; this definition applies to both complex and real tensors. The generic rank of tensor spaces was initially studied in 1983 by Volker Strassen.
As an illustration of the above concepts, it is known that both 2 and 3 are typical ranks of
R
2
⊗
R
2
⊗
R
2
{\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}}
while the generic rank of
C
2
⊗
C
2
⊗
C
2
{\displaystyle \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}}
is 2. Practically, this means that a randomly sampled real tensor (from a continuous probability measure on the space of tensors) of size
2
×
2
×
2
{\displaystyle 2\times 2\times 2}
will be a rank-1 tensor with probability zero, a rank-2 tensor with positive probability, and rank-3 with positive probability. On the other hand, a randomly sampled complex tensor of the same size will be a rank-1 tensor with probability zero, a rank-2 tensor with probability one, and a rank-3 tensor with probability zero. It is even known that the generic rank-3 real tensor in
R
2
⊗
R
2
⊗
R
2
{\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}}
will be of complex rank equal to 2.
The generic rank of tensor spaces depends on the distinction between balanced and unbalanced tensor spaces. A tensor space
F
I
1
⊗
⋯
⊗
F
I
M
{\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}
, where
I
1
≥
I
2
≥
⋯
≥
I
M
{\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}}
,
is called unbalanced whenever
I
1
>
1
+
∏
m
=
2
M
I
m
−
∑
m
=
2
M
(
I
m
−
1
)
,
{\displaystyle I_{1}>1+\prod _{m=2}^{M}I_{m}-\sum _{m=2}^{M}(I_{m}-1),}
and it is called balanced otherwise.
==== Unbalanced tensor spaces ====
When the first factor is very large with respect to the other factors in the tensor product, then the tensor space essentially behaves as a matrix space. The generic rank of tensors living in an unbalanced tensor spaces is known to equal
r
(
I
1
,
…
,
I
M
)
=
min
{
I
1
,
∏
m
=
2
M
I
m
}
{\displaystyle r(I_{1},\ldots ,I_{M})=\min \left\{I_{1},\prod _{m=2}^{M}I_{m}\right\}}
almost everywhere. More precisely, the rank of every tensor in an unbalanced tensor space
F
I
1
×
⋯
×
I
M
∖
Z
{\displaystyle F^{I_{1}\times \cdots \times I_{M}}\setminus Z}
, where
Z
{\displaystyle Z}
is some indeterminate closed set in the Zariski topology, equals the above value.
==== Balanced tensor spaces ====
The expected generic rank of tensors living in a balanced tensor space is equal to
r
E
(
I
1
,
…
,
I
M
)
=
⌈
Π
Σ
+
1
⌉
{\displaystyle r_{E}(I_{1},\ldots ,I_{M})=\left\lceil {\frac {\Pi }{\Sigma +1}}\right\rceil }
almost everywhere for complex tensors and on a Euclidean-open set for real tensors, where
Π
=
∏
m
=
1
M
I
m
and
Σ
=
∑
m
=
1
M
(
I
m
−
1
)
.
{\displaystyle \Pi =\prod _{m=1}^{M}I_{m}\quad {\text{and}}\quad \Sigma =\sum _{m=1}^{M}(I_{m}-1).}
More precisely, the rank of every tensor in
C
I
1
×
⋯
×
I
M
∖
Z
{\displaystyle \mathbb {C} ^{I_{1}\times \cdots \times I_{M}}\setminus Z}
, where
Z
{\displaystyle Z}
is some indeterminate closed set in the Zariski topology, is expected to equal the above value. For real tensors,
r
E
(
I
1
,
…
,
I
M
)
{\displaystyle r_{E}(I_{1},\ldots ,I_{M})}
is the least rank that is expected to occur on a set of positive Euclidean measure. The value
r
E
(
I
1
,
…
,
I
M
)
{\displaystyle r_{E}(I_{1},\ldots ,I_{M})}
is often referred to as the expected generic rank of the tensor space
F
I
1
×
⋯
×
I
M
{\displaystyle F^{I_{1}\times \cdots \times I_{M}}}
because it is only conjecturally correct. It is known that the true generic rank always satisfies
r
(
I
1
,
…
,
I
M
)
≥
r
E
(
I
1
,
…
,
I
M
)
.
{\displaystyle r(I_{1},\ldots ,I_{M})\geq r_{E}(I_{1},\ldots ,I_{M}).}
The Abo–Ottaviani–Peterson conjecture states that equality is expected, i.e.,
r
(
I
1
,
…
,
I
M
)
=
r
E
(
I
1
,
…
,
I
M
)
{\displaystyle r(I_{1},\ldots ,I_{M})=r_{E}(I_{1},\ldots ,I_{M})}
, with the following exceptional cases:
F
(
2
m
+
1
)
×
(
2
m
+
1
)
×
3
with
m
=
1
,
2
,
…
{\displaystyle F^{(2m+1)\times (2m+1)\times 3}{\text{ with }}m=1,2,\ldots }
F
(
m
+
1
)
×
(
m
+
1
)
×
2
×
2
with
m
=
2
,
3
,
…
{\displaystyle F^{(m+1)\times (m+1)\times 2\times 2}{\text{ with }}m=2,3,\ldots }
In each of these exceptional cases, the generic rank is known to be
r
(
I
1
,
…
,
I
m
,
…
,
I
M
)
=
r
E
(
I
1
,
…
,
I
M
)
+
1
{\displaystyle r(I_{1},\ldots ,I_{m},\ldots ,I_{M})=r_{E}(I_{1},\ldots ,I_{M})+1}
. Note that while the set of tensors of rank 3 in
F
2
×
2
×
2
×
2
{\displaystyle F^{2\times 2\times 2\times 2}}
is defective (13 and not the expected 14), the generic rank in that space is still the expected one, 4. Similarly, the set of tensors of rank 5 in
F
4
×
4
×
3
{\displaystyle F^{4\times 4\times 3}}
is defective (44 and not the expected 45), but the generic rank in that space is still the expected 6.
The AOP conjecture has been proved completely in a number of special cases. Lickteig showed already in 1985 that
r
(
n
,
n
,
n
)
=
r
E
(
n
,
n
,
n
)
{\displaystyle r(n,n,n)=r_{E}(n,n,n)}
, provided that
n
≠
3
{\displaystyle n\neq 3}
. In 2011, a major breakthrough was established by Catalisano, Geramita, and Gimigliano who proved that the expected dimension of the set of rank
s
{\displaystyle s}
tensors of format
2
×
2
×
⋯
×
2
{\displaystyle 2\times 2\times \cdots \times 2}
is the expected one except for rank 3 tensors in the 4 factor case, yet the expected rank in that case is still 4. As a consequence,
r
(
2
,
2
,
…
,
2
)
=
r
E
(
2
,
2
,
…
,
2
)
{\displaystyle r(2,2,\ldots ,2)=r_{E}(2,2,\ldots ,2)}
for all binary tensors.
=== Maximum rank ===
The maximum rank that can be admitted by any of the tensors in a tensor space is unknown in general; even a conjecture about this maximum rank is missing. Presently, the best general upper bound states that the maximum rank
r
max
(
I
1
,
…
,
I
M
)
{\displaystyle r_{\mbox{max}}(I_{1},\ldots ,I_{M})}
of
F
I
1
⊗
⋯
⊗
F
I
M
{\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}
, where
I
1
≥
I
2
≥
⋯
≥
I
M
{\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}}
, satisfies
r
max
(
I
1
,
…
,
I
M
)
≤
min
{
∏
m
=
2
M
I
m
,
2
⋅
r
(
I
1
,
…
,
I
M
)
}
,
{\displaystyle r_{\mbox{max}}(I_{1},\ldots ,I_{M})\leq \min \left\{\prod _{m=2}^{M}I_{m},2\cdot r(I_{1},\ldots ,I_{M})\right\},}
where
r
(
I
1
,
…
,
I
M
)
{\displaystyle r(I_{1},\ldots ,I_{M})}
is the (least) generic rank of
F
I
1
⊗
⋯
⊗
F
I
M
{\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}
.
It is well-known that the foregoing inequality may be strict. For instance, the generic rank of tensors in
R
2
×
2
×
2
{\displaystyle \mathbb {R} ^{2\times 2\times 2}}
is two, so that the above bound yields
r
max
(
2
,
2
,
2
)
≤
4
{\displaystyle r_{\mbox{max}}(2,2,2)\leq 4}
, while it is known that the maximum rank equals 3.
=== Border rank ===
A rank-
s
{\displaystyle s}
tensor
A
{\displaystyle {\mathcal {A}}}
is called a border tensor if there exists a sequence of tensors of rank at most
r
<
s
{\displaystyle r<s}
whose limit is
A
{\displaystyle {\mathcal {A}}}
. If
r
{\displaystyle r}
is the least value for which such a convergent sequence exists, then it is called the border rank of
A
{\displaystyle {\mathcal {A}}}
. For order-2 tensors, i.e., matrices, rank and border rank always coincide, however, for tensors of order
≥
3
{\displaystyle \geq 3}
they may differ. Border tensors were first studied in the context of fast approximate matrix multiplication algorithms by Bini, Lotti, and Romani in 1980.
A classic example of a border tensor is the rank-3 tensor
A
=
u
⊗
u
⊗
v
+
u
⊗
v
⊗
u
+
v
⊗
u
⊗
u
,
with
‖
u
‖
=
‖
v
‖
=
1
and
⟨
u
,
v
⟩
≠
1.
{\displaystyle {\mathcal {A}}=\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {u} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {u} ,\quad {\text{with }}\|\mathbf {u} \|=\|\mathbf {v} \|=1{\text{ and }}\langle \mathbf {u} ,\mathbf {v} \rangle \neq 1.}
It can be approximated arbitrarily well by the following sequence of rank-2 tensors
A
m
=
m
(
u
+
1
m
v
)
⊗
(
u
+
1
m
v
)
⊗
(
u
+
1
m
v
)
−
m
u
⊗
u
⊗
u
=
u
⊗
u
⊗
v
+
u
⊗
v
⊗
u
+
v
⊗
u
⊗
u
+
1
m
(
u
⊗
v
⊗
v
+
v
⊗
u
⊗
v
+
v
⊗
v
⊗
u
)
+
1
m
2
v
⊗
v
⊗
v
{\displaystyle {\begin{aligned}{\mathcal {A}}_{m}&=m(\mathbf {u} +{\frac {1}{m}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{m}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{m}}\mathbf {v} )-m\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {u} \\&=\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {u} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {u} +{\frac {1}{m}}(\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {v} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {v} \otimes \mathbf {v} \otimes \mathbf {u} )+{\frac {1}{m^{2}}}\mathbf {v} \otimes \mathbf {v} \otimes \mathbf {v} \end{aligned}}}
as
m
→
∞
{\displaystyle m\to \infty }
. Therefore, its border rank is 2, which is strictly less than its rank. When the two vectors are orthogonal, this example is also known as a W state.
== Properties ==
=== Identifiability ===
It follows from the definition of a pure tensor that
A
=
a
1
⊗
a
2
⊗
⋯
⊗
a
M
=
b
1
⊗
b
2
⊗
⋯
⊗
b
M
{\displaystyle {\mathcal {A}}=\mathbf {a} _{1}\otimes \mathbf {a} _{2}\otimes \cdots \otimes \mathbf {a} _{M}=\mathbf {b} _{1}\otimes \mathbf {b} _{2}\otimes \cdots \otimes \mathbf {b} _{M}}
if and only if there exist
λ
k
{\displaystyle \lambda _{k}}
such that
λ
1
λ
2
⋯
λ
M
=
1
{\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{M}=1}
and
a
m
=
λ
m
b
m
{\displaystyle \mathbf {a} _{m}=\lambda _{m}\mathbf {b} _{m}}
for all m. For this reason, the parameters
{
a
m
}
m
=
1
M
{\displaystyle \{\mathbf {a} _{m}\}_{m=1}^{M}}
of a rank-1 tensor
A
{\displaystyle {\mathcal {A}}}
are called identifiable or essentially unique. A rank-
r
{\displaystyle r}
tensor
A
∈
F
I
1
⊗
F
I
2
⊗
⋯
⊗
F
I
M
{\displaystyle {\mathcal {A}}\in F^{I_{1}}\otimes F^{I_{2}}\otimes \cdots \otimes F^{I_{M}}}
is called identifiable if every of its tensor rank decompositions is the sum of the same set of
r
{\displaystyle r}
distinct tensors
{
A
1
,
A
2
,
…
,
A
r
}
{\displaystyle \{{\mathcal {A}}_{1},{\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{r}\}}
where the
A
i
{\displaystyle {\mathcal {A}}_{i}}
's are of rank 1. An identifiable rank-
r
{\displaystyle r}
thus has only one essentially unique decomposition
A
=
∑
i
=
1
r
A
i
,
{\displaystyle {\mathcal {A}}=\sum _{i=1}^{r}{\mathcal {A}}_{i},}
and all
r
!
{\displaystyle r!}
tensor rank decompositions of
A
{\displaystyle {\mathcal {A}}}
can be obtained by permuting the order of the summands. Observe that in a tensor rank decomposition all the
A
i
{\displaystyle {\mathcal {A}}_{i}}
's are distinct, for otherwise the rank of
A
{\displaystyle {\mathcal {A}}}
would be at most
r
−
1
{\displaystyle r-1}
.
==== Generic identifiability ====
Order-2 tensors in
F
I
1
⊗
F
I
2
≃
F
I
1
×
I
2
{\displaystyle F^{I_{1}}\otimes F^{I_{2}}\simeq F^{I_{1}\times I_{2}}}
, i.e., matrices, are not identifiable for
r
>
1
{\displaystyle r>1}
. This follows essentially from the observation
A
=
∑
i
=
1
r
a
i
⊗
b
i
=
∑
i
=
1
r
a
i
b
i
T
=
A
B
T
=
(
A
X
−
1
)
(
B
X
T
)
T
=
∑
i
=
1
r
c
i
d
i
T
=
∑
i
=
1
r
c
i
⊗
d
i
,
{\displaystyle {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}\otimes \mathbf {b} _{i}=\sum _{i=1}^{r}\mathbf {a} _{i}\mathbf {b} _{i}^{T}=AB^{T}=(AX^{-1})(BX^{T})^{T}=\sum _{i=1}^{r}\mathbf {c} _{i}\mathbf {d} _{i}^{T}=\sum _{i=1}^{r}\mathbf {c} _{i}\otimes \mathbf {d} _{i},}
where
X
∈
G
L
r
(
F
)
{\displaystyle X\in \mathrm {GL} _{r}(F)}
is an invertible
r
×
r
{\displaystyle r\times r}
matrix,
A
=
[
a
i
]
i
=
1
r
{\displaystyle A=[\mathbf {a} _{i}]_{i=1}^{r}}
,
B
=
[
b
i
]
i
=
1
r
{\displaystyle B=[\mathbf {b} _{i}]_{i=1}^{r}}
,
A
X
−
1
=
[
c
i
]
i
=
1
r
{\displaystyle AX^{-1}=[\mathbf {c} _{i}]_{i=1}^{r}}
and
B
X
T
=
[
d
i
]
i
=
1
r
{\displaystyle BX^{T}=[\mathbf {d} _{i}]_{i=1}^{r}}
. It can be shown that for every
X
∈
G
L
n
(
F
)
∖
Z
{\displaystyle X\in \mathrm {GL} _{n}(F)\setminus Z}
, where
Z
{\displaystyle Z}
is a closed set in the Zariski topology, the decomposition on the right-hand side is a sum of a different set of rank-1 tensors than the decomposition on the left-hand side, entailing that order-2 tensors of rank
r
>
1
{\displaystyle r>1}
are generically not identifiable.
The situation changes completely for higher-order tensors in
F
I
1
⊗
F
I
2
⊗
⋯
⊗
F
I
M
{\displaystyle F^{I_{1}}\otimes F^{I_{2}}\otimes \cdots \otimes F^{I_{M}}}
with
M
>
2
{\displaystyle M>2}
and all
I
m
≥
2
{\displaystyle I_{m}\geq 2}
. For simplicity in notation, assume without loss of generality that the factors are ordered such that
I
1
≥
I
2
≥
⋯
≥
I
M
≥
2
{\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}\geq 2}
. Let
S
r
⊂
F
I
1
⊗
⋯
F
I
m
⊗
⋯
⊗
F
I
M
{\displaystyle S_{r}\subset F^{I_{1}}\otimes \cdots F^{I_{m}}\otimes \cdots \otimes F^{I_{M}}}
denote the set of tensors of rank bounded by
r
{\displaystyle r}
. Then, the following statement was proved to be correct using a computer-assisted proof for all spaces of dimension
Π
<
15000
{\displaystyle \Pi <15000}
, and it is conjectured to be valid in general:
There exists a closed set
Z
r
{\displaystyle Z_{r}}
in the Zariski topology such that every tensor
A
∈
S
r
∖
Z
r
{\displaystyle {\mathcal {A}}\in S_{r}\setminus Z_{r}}
is identifiable (
S
r
{\displaystyle S_{r}}
is called generically identifiable in this case), unless either one of the following exceptional cases holds:
The rank is too large:
r
>
r
E
(
I
1
,
I
2
,
…
,
I
M
)
{\displaystyle r>r_{E}(I_{1},I_{2},\ldots ,I_{M})}
;
The space is identifiability-unbalanced, i.e.,
I
1
>
∏
m
=
2
M
i
m
−
∑
m
=
2
M
(
I
m
−
1
)
{\textstyle I_{1}>\prod _{m=2}^{M}i_{m}-\sum _{m=2}^{M}(I_{m}-1)}
, and the rank is too large:
r
≥
∏
m
=
2
M
I
m
−
∑
m
=
2
M
(
I
m
−
1
)
{\textstyle r\geq \prod _{m=2}^{M}I_{m}-\sum _{m=2}^{M}(I_{m}-1)}
;
The space is the defective case
F
4
⊗
F
4
⊗
F
3
{\displaystyle F^{4}\otimes F^{4}\otimes F^{3}}
and the rank is
r
=
5
{\displaystyle r=5}
;
The space is the defective case
F
n
⊗
F
n
⊗
F
2
⊗
F
2
{\displaystyle F^{n}\otimes F^{n}\otimes F^{2}\otimes F^{2}}
, where
n
≥
2
{\displaystyle n\geq 2}
, and the rank is
r
=
2
n
−
1
{\displaystyle r=2n-1}
;
The space is
F
4
⊗
F
4
⊗
F
4
{\displaystyle F^{4}\otimes F^{4}\otimes F^{4}}
and the rank is
r
=
6
{\displaystyle r=6}
;
The space is
F
6
⊗
F
6
⊗
F
3
{\displaystyle F^{6}\otimes F^{6}\otimes F^{3}}
and the rank is
r
=
8
{\displaystyle r=8}
; or
The space is
F
2
⊗
F
2
⊗
F
2
⊗
F
2
⊗
F
2
{\displaystyle F^{2}\otimes F^{2}\otimes F^{2}\otimes F^{2}\otimes F^{2}}
and the rank is
r
=
5
{\displaystyle r=5}
.
The space is perfect, i.e.,
r
E
(
I
1
,
I
2
,
…
,
I
M
)
=
Π
Σ
+
1
{\textstyle r_{E}(I_{1},I_{2},\ldots ,I_{M})={\frac {\Pi }{\Sigma +1}}}
is an integer, and the rank is
r
=
r
E
(
I
1
,
I
2
,
…
,
I
M
)
{\textstyle r=r_{E}(I_{1},I_{2},\ldots ,I_{M})}
.
In these exceptional cases, the generic (and also minimum) number of complex decompositions is
proved to be
∞
{\displaystyle \infty }
in the first 4 cases;
proved to be two in case 5;
expected to be six in case 6;
proved to be two in case 7; and
expected to be at least two in case 8 with exception of the two identifiable cases
F
5
⊗
F
4
⊗
F
3
{\displaystyle F^{5}\otimes F^{4}\otimes F^{3}}
and
F
3
⊗
F
2
⊗
F
2
⊗
F
2
{\displaystyle F^{3}\otimes F^{2}\otimes F^{2}\otimes F^{2}}
.
In summary, the generic tensor of order
M
>
2
{\displaystyle M>2}
and rank
r
<
Π
Σ
+
1
{\textstyle r<{\frac {\Pi }{\Sigma +1}}}
that is not identifiability-unbalanced is expected to be identifiable (modulo the exceptional cases in small spaces).
=== Ill-posedness of the standard approximation problem ===
The rank approximation problem asks for the rank-
r
{\displaystyle r}
decomposition closest (in the usual Euclidean topology) to some rank-
s
{\displaystyle s}
tensor
A
{\displaystyle {\mathcal {A}}}
, where
r
<
s
{\displaystyle r<s}
. That is, one seeks to solve
min
a
i
m
∈
F
I
m
‖
A
−
∑
i
=
1
r
a
i
1
⊗
a
i
2
⊗
⋯
⊗
a
i
M
‖
F
,
{\displaystyle \min _{\mathbf {a} _{i}^{m}\in F^{I_{m}}}\|{\mathcal {A}}-\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{M}\|_{F},}
where
‖
⋅
‖
F
{\displaystyle \|\cdot \|_{F}}
is the Frobenius norm.
It was shown in a 2008 paper by de Silva and Lim that the above standard approximation problem may be ill-posed. A solution to aforementioned problem may sometimes not exist because the set over which one optimizes is not closed. As such, a minimizer may not exist, even though an infimum would exist. In particular, it is known that certain so-called border tensors may be approximated arbitrarily well by a sequence of tensor of rank at most
r
{\displaystyle r}
, even though the limit of the sequence converges to a tensor of rank strictly higher than
r
{\displaystyle r}
. The rank-3 tensor
A
=
u
⊗
u
⊗
v
+
u
⊗
v
⊗
u
+
v
⊗
u
⊗
u
,
with
‖
u
‖
=
‖
v
‖
=
1
and
⟨
u
,
v
⟩
≠
1
{\displaystyle {\mathcal {A}}=\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {u} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {u} ,\quad {\text{with }}\|\mathbf {u} \|=\|\mathbf {v} \|=1{\text{ and }}\langle \mathbf {u} ,\mathbf {v} \rangle \neq 1}
can be approximated arbitrarily well by the following sequence of rank-2 tensors
A
n
=
n
(
u
+
1
n
v
)
⊗
(
u
+
1
n
v
)
⊗
(
u
+
1
n
v
)
−
n
u
⊗
u
⊗
u
{\displaystyle {\mathcal {A}}_{n}=n(\mathbf {u} +{\frac {1}{n}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{n}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{n}}\mathbf {v} )-n\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {u} }
as
n
→
∞
{\displaystyle n\to \infty }
. This example neatly illustrates the general principle that a sequence of rank-
r
{\displaystyle r}
tensors that converges to a tensor of strictly higher rank needs to admit at least two individual rank-1 terms whose norms become unbounded. Stated formally, whenever a sequence
A
n
=
∑
i
=
1
r
a
i
,
n
1
⊗
a
i
,
n
2
⊗
⋯
⊗
a
i
,
n
M
{\displaystyle {\mathcal {A}}_{n}=\sum _{i=1}^{r}\mathbf {a} _{i,n}^{1}\otimes \mathbf {a} _{i,n}^{2}\otimes \cdots \otimes \mathbf {a} _{i,n}^{M}}
has the property that
A
n
→
A
{\displaystyle {\mathcal {A}}_{n}\to {\mathcal {A}}}
(in the Euclidean topology) as
n
→
∞
{\displaystyle n\to \infty }
, then there should exist at least
1
≤
i
≠
j
≤
r
{\displaystyle 1\leq i\neq j\leq r}
such that
‖
a
i
,
n
1
⊗
a
i
,
n
2
⊗
⋯
⊗
a
i
,
n
M
‖
F
→
∞
and
‖
a
j
,
n
1
⊗
a
j
,
n
2
⊗
⋯
⊗
a
j
,
n
M
‖
F
→
∞
{\displaystyle \|\mathbf {a} _{i,n}^{1}\otimes \mathbf {a} _{i,n}^{2}\otimes \cdots \otimes \mathbf {a} _{i,n}^{M}\|_{F}\to \infty {\text{ and }}\|\mathbf {a} _{j,n}^{1}\otimes \mathbf {a} _{j,n}^{2}\otimes \cdots \otimes \mathbf {a} _{j,n}^{M}\|_{F}\to \infty }
as
n
→
∞
{\displaystyle n\to \infty }
. This phenomenon is often encountered when attempting to approximate a tensor using numerical optimization algorithms. It is sometimes called the problem of diverging components. It was, in addition, shown that a random low-rank tensor over the reals may not admit a rank-2 approximation with positive probability, leading to the understanding that the ill-posedness problem is an important consideration when employing the tensor rank decomposition.
A common partial solution to the ill-posedness problem consists of imposing an additional inequality constraint that bounds the norm of the individual rank-1 terms by some constant. Other constraints that result in a closed set, and, thus, well-posed optimization problem, include imposing positivity or a bounded inner product strictly less than unity between the rank-1 terms appearing in the sought decomposition.
== Calculating the CPD ==
Alternating algorithms:
alternating least squares (ALS)
alternating slice-wise diagonalisation (ASD)
Direct algorithms:
pencil-based algorithms
moment-based algorithms
General optimization algorithms:
simultaneous diagonalization (SD)
simultaneous generalized Schur decomposition (SGSD)
Levenberg–Marquardt (LM)
nonlinear conjugate gradient (NCG)
limited memory BFGS (L-BFGS)
General polynomial system solving algorithms:
homotopy continuation
== Applications ==
In machine learning, the CP-decomposition is the central ingredient in learning probabilistic latent variables models via the technique of moment-matching. For example, consider the multi-view model which is a probabilistic latent variable model. In this model, the generation of samples are posited as follows: there exists a hidden random variable that is not observed directly, given which, there are several conditionally independent random variables known as the different "views" of the hidden variable. For example, assume there are three views
x
1
,
x
2
,
x
3
{\displaystyle x_{1},x_{2},x_{3}}
of a
k
{\displaystyle k}
-state categorical hidden variable
h
{\displaystyle h}
. Then the empirical third moment of this latent variable model
E
[
x
1
⊗
x
2
⊗
x
3
]
{\displaystyle E[x_{1}\otimes x_{2}\otimes x_{3}]}
is a rank 3 tensor and can be decomposed as:
E
[
x
1
⊗
x
2
⊗
x
3
]
=
∑
i
=
1
k
P
r
(
h
=
i
)
E
[
x
1
|
h
=
i
]
⊗
E
[
x
2
|
h
=
i
]
⊗
E
[
x
3
|
h
=
i
]
{\displaystyle E[x_{1}\otimes x_{2}\otimes x_{3}]=\sum _{i=1}^{k}Pr(h=i)E[x_{1}|h=i]\otimes E[x_{2}|h=i]\otimes E[x_{3}|h=i]}
.
In applications such as topic modeling, this can be interpreted as the co-occurrence of words in a document. Then the coefficients in the decomposition of this empirical moment tensor can be interpreted as the probability of choosing a specific topic and each column of the factor matrix
E
[
x
|
h
=
i
]
{\displaystyle E[x|h=i]}
corresponds to probabilities of words in the vocabulary in the corresponding topic.
== See also ==
Latent class analysis
Multilinear subspace learning
Singular value decomposition
Tucker decomposition
Higher-order singular value decomposition
Tensor decomposition
== References ==
== Further reading ==
Kolda, Tamara G.; Bader, Brett W. (2009). "Tensor Decompositions and Applications". SIAM Rev. 51 (3): 455–500. Bibcode:2009SIAMR..51..455K. CiteSeerX 10.1.1.153.2059. doi:10.1137/07070111X. S2CID 16074195.
Landsberg, Joseph M. (2012). Tensors: Geometry and Applications. AMS.
== External links ==
PARAFAC Tutorial
Parallel Factor Analysis (PARAFAC)
FactoMineR (free exploratory multivariate data analysis software linked to R) | Wikipedia/Tensor_rank_decomposition |
Controllability is an important property of a control system and plays a crucial role in many regulation problems, such as the stabilization of unstable systems using feedback, tracking problems, obtaining optimal control strategies, or, simply prescribing an input that has a desired effect on the state.
Controllability and observability are dual notions. Controllability pertains to regulating the state by a choice of a suitable input, while observability pertains to being able to know the state by observing the output (assuming that the input is also being observed).
Broadly speaking, the concept of controllability relates to the ability to steer a system around in its configuration space using only certain admissible manipulations. The exact definition varies depending on the framework or the type of models dealt with.
The following are examples of variants of notions of controllability that have been introduced in the systems and control literature:
State controllability: the ability to steer the system between states
Strong controllability: the ability to steer between states over any specified time window
Collective controllability: the ability to simultaneously steer a collection of dynamical systems
Trajectory controllability: the ability to steer along a predefined trajectory rather than just to a desired final state
Output controllability: the ability to steer to specified values of the output
Controllability in the behavioural framework: a compatibility condition between past and future input and output trajectories
== State controllability ==
The state of a deterministic system, which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any given time. In particular, no information on the past of a system is needed to help in predicting the future, if the states at the present time are known and all current and future values of the control variables (those whose values can be chosen) are known.
Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any final state in a finite time interval.: 737
That is, we can informally define controllability as follows:
If for any initial state
x
0
{\displaystyle \mathbf {x_{0}} }
and any final state
x
f
{\displaystyle \mathbf {x_{f}} }
there exists an input sequence to transfer the system state from
x
0
{\displaystyle \mathbf {x_{0}} }
to
x
f
{\displaystyle \mathbf {x_{f}} }
in a finite time interval, then the system modeled by the state-space representation is controllable. For the simplest example of a continuous, LTI system, the row dimension of the state space expression
x
˙
=
A
x
(
t
)
+
B
u
(
t
)
{\displaystyle {\dot {\mathbf {x} }}=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}
determines the interval; each row contributes a vector in the state space of the system. If there are not enough such vectors to span the state space of
x
{\displaystyle \mathbf {x} }
, then the system cannot achieve controllability. It may be necessary to modify
A
{\displaystyle \mathbf {A} }
and
B
{\displaystyle \mathbf {B} }
to better approximate the underlying differential relationships it estimates to achieve controllability.
Controllability does not mean that a reached state can be maintained, merely that any state can be reached.
Controllability does not mean that arbitrary paths can be made through state space, only that there exists a path within the prescribed finite time interval.
== Continuous linear systems ==
Consider the continuous linear system
x
˙
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)}
y
(
t
)
=
C
(
t
)
x
(
t
)
+
D
(
t
)
u
(
t
)
.
{\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t)+D(t)\mathbf {u} (t).}
There exists a control
u
{\displaystyle u}
from state
x
0
{\displaystyle x_{0}}
at time
t
0
{\displaystyle t_{0}}
to state
x
1
{\displaystyle x_{1}}
at time
t
1
>
t
0
{\displaystyle t_{1}>t_{0}}
if and only if
x
1
−
ϕ
(
t
0
,
t
1
)
x
0
{\displaystyle x_{1}-\phi (t_{0},t_{1})x_{0}}
is in the column space of
W
(
t
0
,
t
1
)
=
∫
t
0
t
1
ϕ
(
t
0
,
t
)
B
(
t
)
B
(
t
)
T
ϕ
(
t
0
,
t
)
T
d
t
{\displaystyle W(t_{0},t_{1})=\int _{t_{0}}^{t_{1}}\phi (t_{0},t)B(t)B(t)^{T}\phi (t_{0},t)^{T}dt}
where
ϕ
{\displaystyle \phi }
is the state-transition matrix, and
W
(
t
0
,
t
1
)
{\displaystyle W(t_{0},t_{1})}
is the Controllability Gramian.
In fact, if
η
0
{\displaystyle \eta _{0}}
is a solution to
W
(
t
0
,
t
1
)
η
=
x
1
−
ϕ
(
t
0
,
t
1
)
x
0
{\displaystyle W(t_{0},t_{1})\eta =x_{1}-\phi (t_{0},t_{1})x_{0}}
then a control given by
u
(
t
)
=
−
B
(
t
)
T
ϕ
(
t
0
,
t
)
T
η
0
{\displaystyle u(t)=-B(t)^{T}\phi (t_{0},t)^{T}\eta _{0}}
would make the desired transfer.
Note that the matrix
W
{\displaystyle W}
defined as above has the following properties:
W
(
t
0
,
t
1
)
{\displaystyle W(t_{0},t_{1})}
is symmetric
W
(
t
0
,
t
1
)
{\displaystyle W(t_{0},t_{1})}
is positive semidefinite for
t
1
≥
t
0
{\displaystyle t_{1}\geq t_{0}}
W
(
t
0
,
t
1
)
{\displaystyle W(t_{0},t_{1})}
satisfies the linear matrix differential equation
d
d
t
W
(
t
,
t
1
)
=
A
(
t
)
W
(
t
,
t
1
)
+
W
(
t
,
t
1
)
A
(
t
)
T
−
B
(
t
)
B
(
t
)
T
,
W
(
t
1
,
t
1
)
=
0
{\displaystyle {\frac {d}{dt}}W(t,t_{1})=A(t)W(t,t_{1})+W(t,t_{1})A(t)^{T}-B(t)B(t)^{T},\;W(t_{1},t_{1})=0}
W
(
t
0
,
t
1
)
{\displaystyle W(t_{0},t_{1})}
satisfies the equation
W
(
t
0
,
t
1
)
=
W
(
t
0
,
t
)
+
ϕ
(
t
0
,
t
)
W
(
t
,
t
1
)
ϕ
(
t
0
,
t
)
T
{\displaystyle W(t_{0},t_{1})=W(t_{0},t)+\phi (t_{0},t)W(t,t_{1})\phi (t_{0},t)^{T}}
== Rank condition for controllability ==
The Controllability Gramian involves integration of the state-transition matrix of a system. A simpler condition for controllability is a rank condition analogous to the Kalman rank condition for time-invariant systems.
Consider a continuous-time linear system
Σ
{\displaystyle \Sigma }
smoothly varying in an interval
[
t
0
,
t
]
{\displaystyle [t_{0},t]}
of
R
{\displaystyle \mathbb {R} }
:
x
˙
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)}
y
(
t
)
=
C
(
t
)
x
(
t
)
+
D
(
t
)
u
(
t
)
.
{\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t)+D(t)\mathbf {u} (t).}
The state-transition matrix
ϕ
{\displaystyle \phi }
is also smooth. Introduce the n x m matrix-valued function
M
0
(
t
)
=
ϕ
(
t
0
,
t
)
B
(
t
)
{\displaystyle M_{0}(t)=\phi (t_{0},t)B(t)}
and define
M
k
(
t
)
{\displaystyle M_{k}(t)}
=
d
k
M
0
d
t
k
(
t
)
,
k
⩾
1
{\displaystyle {\frac {\mathrm {d^{k}} M_{0}}{\mathrm {d} t^{k}}}(t),k\geqslant 1}
.
Consider the matrix of matrix-valued functions obtained by listing all the columns of the
M
i
{\displaystyle M_{i}}
,
i
=
0
,
1
,
…
,
k
{\displaystyle i=0,1,\ldots ,k}
:
M
(
k
)
(
t
)
:=
[
M
0
(
t
)
,
…
,
M
k
(
t
)
]
{\displaystyle M^{(k)}(t):=\left[M_{0}(t),\ldots ,M_{k}(t)\right]}
.
If there exists a
t
¯
∈
[
t
0
,
t
]
{\displaystyle {\bar {t}}\in [t_{0},t]}
and a nonnegative integer k such that
rank
M
(
k
)
(
t
¯
)
=
n
{\displaystyle \operatorname {rank} M^{(k)}({\bar {t}})=n}
, then
Σ
{\displaystyle \Sigma }
is controllable.
If
Σ
{\displaystyle \Sigma }
is also analytically varying in an interval
[
t
0
,
t
]
{\displaystyle [t_{0},t]}
, then
Σ
{\displaystyle \Sigma }
is controllable on every nontrivial subinterval of
[
t
0
,
t
]
{\displaystyle [t_{0},t]}
if and only if there exists a
t
¯
∈
[
t
0
,
t
]
{\displaystyle {\bar {t}}\in [t_{0},t]}
and a nonnegative integer k such that
rank
M
(
k
)
(
t
i
)
=
n
{\displaystyle \operatorname {rank} M^{(k)}(t_{i})=n}
.
The above methods can still be complex to check, since it involves the computation of the state-transition matrix
ϕ
{\displaystyle \phi }
. Another equivalent condition is defined as follow. Let
B
0
(
t
)
=
B
(
t
)
{\displaystyle B_{0}(t)=B(t)}
, and for each
i
≥
0
{\displaystyle i\geq 0}
, define
B
i
+
1
(
t
)
{\displaystyle B_{i+1}(t)}
=
A
(
t
)
B
i
(
t
)
−
d
d
t
B
i
(
t
)
.
{\displaystyle A(t)B_{i}(t)-{\frac {\mathrm {d} }{\mathrm {d} t}}B_{i}(t).}
In this case, each
B
i
{\displaystyle B_{i}}
is obtained directly from the data
(
A
(
t
)
,
B
(
t
)
)
.
{\displaystyle (A(t),B(t)).}
The system is controllable if there exists a
t
¯
∈
[
t
0
,
t
]
{\displaystyle {\bar {t}}\in [t_{0},t]}
and a nonnegative integer
k
{\displaystyle k}
such that
rank
(
[
B
0
(
t
¯
)
,
B
1
(
t
¯
)
,
…
,
B
k
(
t
¯
)
]
)
=
n
{\displaystyle {\textrm {rank}}(\left[B_{0}({\bar {t}}),B_{1}({\bar {t}}),\ldots ,B_{k}({\bar {t}})\right])=n}
.
=== Example ===
Consider a system varying analytically in
(
−
∞
,
∞
)
{\displaystyle (-\infty ,\infty )}
and matrices
A
(
t
)
=
[
t
1
0
0
t
3
0
0
0
t
2
]
{\displaystyle A(t)={\begin{bmatrix}t&1&0\\0&t^{3}&0\\0&0&t^{2}\end{bmatrix}}}
,
B
(
t
)
=
[
0
1
1
]
.
{\displaystyle B(t)={\begin{bmatrix}0\\1\\1\end{bmatrix}}.}
Then
[
B
0
(
0
)
,
B
1
(
0
)
,
B
2
(
0
)
,
B
3
(
0
)
]
=
[
0
1
0
−
1
1
0
0
0
1
0
0
2
]
{\displaystyle [B_{0}(0),B_{1}(0),B_{2}(0),B_{3}(0)]={\begin{bmatrix}0&1&0&-1\\1&0&0&0\\1&0&0&2\end{bmatrix}}}
and since this matrix has rank 3, the system is controllable on every nontrivial interval of
R
{\displaystyle \mathbb {R} }
.
=== Continuous linear time-invariant (LTI) systems ===
Consider the continuous linear time-invariant system
x
˙
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
{\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+B\mathbf {u} (t)}
y
(
t
)
=
C
x
(
t
)
+
D
u
(
t
)
{\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)+D\mathbf {u} (t)}
where
x
{\displaystyle \mathbf {x} }
is the
n
×
1
{\displaystyle n\times 1}
"state vector",
y
{\displaystyle \mathbf {y} }
is the
m
×
1
{\displaystyle m\times 1}
"output vector",
u
{\displaystyle \mathbf {u} }
is the
r
×
1
{\displaystyle r\times 1}
"input (or control) vector",
A
{\displaystyle A}
is the
n
×
n
{\displaystyle n\times n}
"state matrix",
B
{\displaystyle B}
is the
n
×
r
{\displaystyle n\times r}
"input matrix",
C
{\displaystyle C}
is the
m
×
n
{\displaystyle m\times n}
"output matrix",
D
{\displaystyle D}
is the
m
×
r
{\displaystyle m\times r}
"feedthrough (or feedforward) matrix".
The
n
×
n
r
{\displaystyle n\times nr}
controllability matrix is given by
R
=
[
B
A
B
A
2
B
.
.
.
A
n
−
1
B
]
{\displaystyle R={\begin{bmatrix}B&AB&A^{2}B&...&A^{n-1}B\end{bmatrix}}}
The system is controllable if the controllability matrix has full row rank (i.e.
rank
(
R
)
=
n
{\displaystyle \operatorname {rank} (R)=n}
).
== Discrete linear time-invariant (LTI) systems ==
For a discrete-time linear state-space system (i.e. time variable
k
∈
Z
{\displaystyle k\in \mathbb {Z} }
) the state equation is
x
(
k
+
1
)
=
A
x
(
k
)
+
B
u
(
k
)
{\displaystyle {\textbf {x}}(k+1)=A{\textbf {x}}(k)+B{\textbf {u}}(k)}
where
A
{\displaystyle A}
is an
n
×
n
{\displaystyle n\times n}
matrix and
B
{\displaystyle B}
is a
n
×
r
{\displaystyle n\times r}
matrix (i.e.
u
{\displaystyle \mathbf {u} }
is
r
{\displaystyle r}
inputs collected in a
r
×
1
{\displaystyle r\times 1}
vector). The test for controllability is that the
n
×
n
r
{\displaystyle n\times nr}
matrix
C
=
[
B
A
B
A
2
B
⋯
A
n
−
1
B
]
{\displaystyle {\mathcal {C}}={\begin{bmatrix}B&AB&A^{2}B&\cdots &A^{n-1}B\end{bmatrix}}}
has full row rank (i.e.,
rank
(
C
)
=
n
{\displaystyle \operatorname {rank} ({\mathcal {C}})=n}
). That is, if the system is controllable,
C
{\displaystyle {\mathcal {C}}}
will have
n
{\displaystyle n}
columns that are linearly independent; if
n
{\displaystyle n}
columns of
C
{\displaystyle {\mathcal {C}}}
are linearly independent, each of the
n
{\displaystyle n}
states is reachable by giving the system proper inputs through the variable
u
(
k
)
{\displaystyle u(k)}
.
=== Derivation ===
Given the state
x
(
0
)
{\displaystyle {\textbf {x}}(0)}
at an initial time, arbitrarily denoted as k=0, the state equation gives
x
(
1
)
=
A
x
(
0
)
+
B
u
(
0
)
,
{\displaystyle {\textbf {x}}(1)=A{\textbf {x}}(0)+B{\textbf {u}}(0),}
then
x
(
2
)
=
A
x
(
1
)
+
B
u
(
1
)
=
A
2
x
(
0
)
+
A
B
u
(
0
)
+
B
u
(
1
)
,
{\displaystyle {\textbf {x}}(2)=A{\textbf {x}}(1)+B{\textbf {u}}(1)=A^{2}{\textbf {x}}(0)+AB{\textbf {u}}(0)+B{\textbf {u}}(1),}
and so on with repeated back-substitutions of the state variable, eventually yielding
x
(
n
)
=
B
u
(
n
−
1
)
+
A
B
u
(
n
−
2
)
+
⋯
+
A
n
−
1
B
u
(
0
)
+
A
n
x
(
0
)
{\displaystyle {\textbf {x}}(n)=B{\textbf {u}}(n-1)+AB{\textbf {u}}(n-2)+\cdots +A^{n-1}B{\textbf {u}}(0)+A^{n}{\textbf {x}}(0)}
or equivalently
Imposing any desired value of the state vector
x
(
n
)
{\displaystyle {\textbf {x}}(n)}
on the left side, this can always be solved for the stacked vector of control vectors if and only if the matrix of matrices at the beginning of the right side has full row rank.
=== Example ===
For example, consider the case when
n
=
2
{\displaystyle n=2}
and
r
=
1
{\displaystyle r=1}
(i.e. only one control input). Thus,
B
{\displaystyle B}
and
A
B
{\displaystyle AB}
are
2
×
1
{\displaystyle 2\times 1}
vectors. If
[
B
A
B
]
{\displaystyle {\begin{bmatrix}B&AB\end{bmatrix}}}
has rank 2 (full rank), and so
B
{\displaystyle B}
and
A
B
{\displaystyle AB}
are linearly independent and span the entire plane. If the rank is 1, then
B
{\displaystyle B}
and
A
B
{\displaystyle AB}
are collinear and do not span the plane.
Assume that the initial state is zero.
At time
k
=
0
{\displaystyle k=0}
:
x
(
1
)
=
A
x
(
0
)
+
B
u
(
0
)
=
B
u
(
0
)
{\displaystyle x(1)=A{\textbf {x}}(0)+B{\textbf {u}}(0)=B{\textbf {u}}(0)}
At time
k
=
1
{\displaystyle k=1}
:
x
(
2
)
=
A
x
(
1
)
+
B
u
(
1
)
=
A
B
u
(
0
)
+
B
u
(
1
)
{\displaystyle x(2)=A{\textbf {x}}(1)+B{\textbf {u}}(1)=AB{\textbf {u}}(0)+B{\textbf {u}}(1)}
At time
k
=
0
{\displaystyle k=0}
all of the reachable states are on the line formed by the vector
B
{\displaystyle B}
.
At time
k
=
1
{\displaystyle k=1}
all of the reachable states are linear combinations of
A
B
{\displaystyle AB}
and
B
{\displaystyle B}
.
If the system is controllable then these two vectors can span the entire plane and can be done so for time
k
=
2
{\displaystyle k=2}
.
The assumption made that the initial state is zero is merely for convenience.
Clearly if all states can be reached from the origin then any state can be reached from another state (merely a shift in coordinates).
This example holds for all positive
n
{\displaystyle n}
, but the case of
n
=
2
{\displaystyle n=2}
is easier to visualize.
=== Analogy for example of n = 2 ===
Consider an analogy to the previous example system.
You are sitting in your car on an infinite, flat plane and facing north.
The goal is to reach any point in the plane by driving a distance in a straight line, come to a full stop, turn, and driving another distance, again, in a straight line.
If your car has no steering then you can only drive straight, which means you can only drive on a line (in this case the north-south line since you started facing north).
The lack of steering case would be analogous to when the rank of
C
{\displaystyle C}
is 1 (the two distances you drove are on the same line).
Now, if your car did have steering then you could easily drive to any point in the plane and this would be the analogous case to when the rank of
C
{\displaystyle C}
is 2.
If you change this example to
n
=
3
{\displaystyle n=3}
then the analogy would be flying in space to reach any position in 3D space (ignoring the orientation of the aircraft).
You are allowed to:
fly in a straight line
turn left or right by any amount (Yaw)
direct the plane upwards or downwards by any amount (Pitch)
Although the 3-dimensional case is harder to visualize, the concept of controllability is still analogous.
== Nonlinear systems ==
Nonlinear systems in the control-affine form
x
˙
=
f
(
x
)
+
∑
i
=
1
m
g
i
(
x
)
u
i
{\displaystyle {\dot {\mathbf {x} }}=\mathbf {f(x)} +\sum _{i=1}^{m}\mathbf {g} _{i}(\mathbf {x} )u_{i}}
are locally accessible about
x
0
{\displaystyle x_{0}}
if the accessibility distribution
R
{\displaystyle R}
spans
n
{\displaystyle n}
space, when
n
{\displaystyle n}
equals the dimension of
x
{\displaystyle x}
and R is given by:
R
=
[
g
1
⋯
g
m
[
a
d
g
i
k
g
j
]
⋯
[
a
d
f
k
g
i
]
]
.
{\displaystyle R={\begin{bmatrix}\mathbf {g} _{1}&\cdots &\mathbf {g} _{m}&[\mathrm {ad} _{\mathbf {g} _{i}}^{k}\mathbf {\mathbf {g} _{j}} ]&\cdots &[\mathrm {ad} _{\mathbf {f} }^{k}\mathbf {\mathbf {g} _{i}} ]\end{bmatrix}}.}
Here,
[
a
d
f
k
g
]
{\displaystyle [\mathrm {ad} _{\mathbf {f} }^{k}\mathbf {\mathbf {g} } ]}
is the repeated Lie bracket operation defined by
[
a
d
f
k
g
]
=
[
f
⋯
j
⋯
[
f
,
g
]
]
.
{\displaystyle [\mathrm {ad} _{\mathbf {f} }^{k}\mathbf {\mathbf {g} } ]={\begin{bmatrix}\mathbf {f} &\cdots &j&\cdots &\mathbf {[\mathbf {f} ,\mathbf {g} ]} \end{bmatrix}}.}
The controllability matrix for linear systems in the previous section can in fact be derived from this equation.
== Controllability via state feedback ==
When control authority on a linear dynamical system is exerted through a choice of a time-varying feedback gain matrix
K
(
t
)
{\displaystyle K(t)}
, the system
x
˙
=
(
A
−
B
K
(
t
)
)
x
{\displaystyle {\dot {\mathbf {x} }}=(A-BK(t))\mathbf {x} }
is nonlinear, in that products of control parameters and states are present. The accessibility distribution
R
{\displaystyle R}
is, as before,
R
=
[
B
A
B
⋯
A
n
−
1
B
]
.
{\displaystyle R={\begin{bmatrix}B&AB&\cdots &A^{n-1}B\end{bmatrix}}.}
It is clear that for the system to be controllable, it is necessary that
R
{\displaystyle R}
has full column rank. It turns out that this condition is also sufficient. However, the (optimal) control strategy explained earlier needs to be slightly modified so that the trajectory when applying an optimal input to steer the system between the specified states, does not pass through the origin, else the regulating input cannot be written in feedback form
u
=
−
K
(
t
)
x
{\displaystyle u=-K(t)\mathbf {x} }
.
== Collective controllability -- Control of the state transition via feedback ==
Collective controllability represents the ability to steer
n
{\displaystyle n}
linear dynamical systems that obey identical dynamics
x
˙
(
i
)
(
t
)
=
A
x
(
i
)
(
t
)
+
B
u
(
i
)
(
t
)
{\displaystyle {\dot {\mathbf {x} }}^{(i)}(t)=A\mathbf {x} ^{(i)}(t)+B\mathbf {u} ^{(i)}(t)}
where
n
{\displaystyle n}
equals the dimension of
x
{\displaystyle \mathbf {x} }
,
between specified starting and ending configurations by way of a common state feedback gain matrix
K
(
t
)
{\displaystyle K(t)}
, and thereby, each instantiating a control input
u
(
i
)
(
t
)
=
K
(
t
)
x
(
i
)
(
t
)
{\displaystyle \mathbf {u} ^{(i)}(t)=K(t){\mathbf {x} }^{(i)}(t)}
for
i
∈
{
1
,
…
,
n
}
{\displaystyle i\in \{1,\ldots ,n\}}
, respectively.
The accessibility distribution
R
{\displaystyle R}
having full column rank is trivially a necessary condition. It is also sufficient, and in fact, the collective is strongly controllable, in that it can be steered from an initial
configuration
Φ
(
0
)
=
[
x
(
1
)
(
0
)
…
x
(
n
)
(
0
)
]
{\displaystyle \Phi (0)={\begin{bmatrix}\mathbf {x} ^{(1)}(0)\ldots \mathbf {x} ^{(n)}(0)\end{bmatrix}}}
to any specified terminal configuration
Φ
(
T
)
=
[
x
(
1
)
(
T
)
…
x
(
n
)
(
T
)
]
,
{\displaystyle \Phi (T)={\begin{bmatrix}\mathbf {x} ^{(1)}(T)\ldots \mathbf {x} ^{(n)}(T)\end{bmatrix}},}
provided
det
(
Φ
(
0
)
Φ
(
T
)
)
>
0
{\displaystyle \det(\Phi (0)\Phi (T))>0}
, over any specified time interval
[
0
,
T
]
{\displaystyle [0,T]}
through a choice of a common time-varying feedback gain matrix
K
(
t
)
{\displaystyle K(t)}
provided
R
{\displaystyle R}
has full column rank
== Null Controllability ==
If a discrete control system is null-controllable, it means that there exists a controllable
u
(
k
)
{\displaystyle u(k)}
so that
x
(
k
0
)
=
0
{\displaystyle x(k_{0})=0}
for some initial state
x
(
0
)
=
x
0
{\displaystyle x(0)=x_{0}}
. In other words, it is equivalent to the condition that there exists a matrix
F
{\displaystyle F}
such that
A
+
B
F
{\displaystyle A+BF}
is nilpotent.
This can be easily shown by controllable-uncontrollable decomposition.
== Output controllability ==
Output controllability is the related notion for the output of the system (denoted y in the previous equations); the output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. It is not necessary that there is any relationship between state controllability and output controllability. In particular:
A controllable system is not necessarily output controllable. For example, if matrix D = 0 and matrix C does not have full row rank, then some positions of the output are masked by the limiting structure of the output matrix, and therefore unachievable. Moreover, even though the system can be moved to any state in finite time, there may be some outputs that are inaccessible by all states. A trivial numerical example uses D=0 and a C matrix with at least one row of zeros; thus, the system is not able to produce a non-zero output along that dimension.
An output controllable system is not necessarily state controllable. For example, if the dimension of the state space is greater than the dimension of the output, then there will be a set of possible state configurations for each individual output. That is, the system can have significant zero dynamics, which are trajectories of the system that are not observable from the output. Consequently, being able to drive an output to a particular position in finite time says nothing about the state configuration of the system.
For a linear continuous-time system, like the example above, described by matrices
A
{\displaystyle A}
,
B
{\displaystyle B}
,
C
{\displaystyle C}
, and
D
{\displaystyle D}
, the
m
×
(
n
+
1
)
r
{\displaystyle m\times (n+1)r}
output controllability matrix
[
C
B
C
A
B
C
A
2
B
⋯
C
A
n
−
1
B
D
]
{\displaystyle {\begin{bmatrix}CB&CAB&CA^{2}B&\cdots &CA^{n-1}B&D\end{bmatrix}}}
has full row rank (i.e. rank
m
{\displaystyle m}
) if and only if the system is output controllable.: 742
== Controllability under input constraints ==
In systems with limited control authority, it is often no longer possible to move any initial state to any final state inside the controllable subspace. This phenomenon is caused by constraints on the input that could be inherent to the system (e.g. due to saturating actuator) or imposed on the system for other reasons (e.g. due to safety-related concerns). The controllability of systems with input and state constraints is studied in the context of reachability and viability theory.
== Controllability in the behavioral framework ==
In the so-called behavioral system theoretic approach due to Willems (see people in systems and control), models considered do not directly define an input–output structure. In this framework systems are described by admissible trajectories of a collection of variables, some of which might be interpreted as inputs or outputs.
A system is then defined to be controllable in this setting, if any past part of a behavior (trajectory of the external variables) can be concatenated with any future trajectory of the behavior in such a way that the concatenation is contained in the behavior, i.e. is part of the admissible system behavior.: 151
== Stabilizability ==
A slightly weaker notion than controllability is that of stabilizability. A system is said to be stabilizable when all uncontrollable state variables can be made to have stable dynamics. Thus, even though some of the state variables cannot be controlled (as determined by the controllability test above) all the state variables will still remain bounded during the system's behavior.
== Reachable set ==
Let T ∈ Т and x ∈ X (where X is the set of all possible states and Т is an interval of time). The reachable set from x in time T is defined as:
R
T
(
x
)
=
{
z
∈
X
:
x
→
T
z
}
{\displaystyle R^{T}{(x)}=\left\{z\in X:x{\overset {T}{\rightarrow }}z\right\}}
, where xT→z denotes that there exists a state transition from x to z in time T.
For autonomous systems the reachable set is given by :
I
m
(
R
)
=
I
m
(
B
)
+
I
m
(
A
B
)
+
.
.
.
.
+
I
m
(
A
n
−
1
B
)
{\displaystyle \mathrm {Im} (R)=\mathrm {Im} (B)+\mathrm {Im} (AB)+....+\mathrm {Im} (A^{n-1}B)}
,
where R is the controllability matrix.
In terms of the reachable set, the system is controllable if and only if
I
m
(
R
)
=
R
n
{\displaystyle \mathrm {Im} (R)=\mathbb {R} ^{n}}
.
Proof
We have the following equalities:
R
=
[
B
A
B
.
.
.
.
A
n
−
1
B
]
{\displaystyle R=[B\ AB....A^{n-1}B]}
I
m
(
R
)
=
I
m
(
[
B
A
B
.
.
.
.
A
n
−
1
B
]
)
{\displaystyle \mathrm {Im} (R)=\mathrm {Im} ([B\ AB....A^{n-1}B])}
d
i
m
(
I
m
(
R
)
)
=
r
a
n
k
(
R
)
{\displaystyle \mathrm {dim(Im} (R))=\mathrm {rank} (R)}
Considering that the system is controllable, the columns of R should be linearly independent. So:
d
i
m
(
I
m
(
R
)
)
=
n
{\displaystyle \mathrm {dim(Im} (R))=n}
r
a
n
k
(
R
)
=
n
{\displaystyle \mathrm {rank} (R)=n}
I
m
(
R
)
=
R
n
◼
{\displaystyle \mathrm {Im} (R)=\mathbb {R} ^{n}\quad \blacksquare }
A related set to the reachable set is the controllable set, defined by:
C
T
(
x
)
=
{
z
∈
X
:
z
→
T
x
}
{\displaystyle C^{T}{(x)}=\left\{z\in X:z{\overset {T}{\rightarrow }}x\right\}}
.
The relation between reachability and controllability is presented by Sontag:
(a) An n-dimensional discrete linear system is controllable if and only if:
R
(
0
)
=
R
k
(
0
)
=
X
{\displaystyle R(0)=R^{k}{(0)=X}}
(Where X is the set of all possible values or states of x and k is the time step).
(b) A continuous-time linear system is controllable if and only if:
R
(
0
)
=
R
e
(
0
)
=
X
{\displaystyle R(0)=R^{e}{(0)=X}}
for all e>0.
if and only if
C
(
0
)
=
C
e
(
0
)
=
X
{\displaystyle C(0)=C^{e}{(0)=X}}
for all e>0.
Example
Let the system be an n dimensional discrete-time-invariant system from the formula:
ϕ
(
n
,
0
,
0
,
w
)
=
∑
i
=
1
n
A
i
−
1
B
w
(
n
−
1
)
{\displaystyle \phi (n,0,0,w)=\sum \limits _{i=1}^{n}A^{i-1}Bw(n-1)}
(Where
ϕ
{\displaystyle \phi }
(final time, initial time, state variable, restrictions) is defined as the transition matrix of a state variable x from an initial time 0 to a final time n with some restrictions w).
It follows that the future state is in
R
k
(
0
)
{\displaystyle R^{k}{(0)}}
if and only if it is in
I
m
(
R
)
{\displaystyle \mathrm {Im} (R)}
, the image of the linear map
R
{\displaystyle R}
, defined as:
R
(
A
,
B
)
≜
[
B
A
B
.
.
.
.
A
n
−
1
B
]
{\displaystyle R(A,B)\triangleq [B\ AB....A^{n-1}B]}
,
which maps,
u
n
↦
X
{\displaystyle u^{n}\mapsto X}
When
u
=
K
m
{\displaystyle u=K^{m}}
and
X
=
K
n
{\displaystyle X=K^{n}}
we identify
R
(
A
,
B
)
{\displaystyle R(A,B)}
with a
n
×
n
m
{\displaystyle n\times nm}
matrix whose columns are
B
,
A
B
,
.
.
.
.
,
A
n
−
1
B
{\displaystyle B,\ AB,....,A^{n-1}B}
in that order. If the system is controllable the rank of
[
B
A
B
.
.
.
.
A
n
−
1
B
]
{\displaystyle [B\ AB....A^{n-1}B]}
is
n
{\displaystyle n}
. If this is true, the image of the linear map
R
{\displaystyle R}
is all of
X
{\displaystyle X}
. Based on that, we have:
R
(
0
)
=
R
k
(
0
)
=
X
{\displaystyle R(0)=R^{k}{(0)=X}}
with
X
∈
R
n
{\displaystyle X\in \mathbb {R} ^{n}}
.
== See also ==
Observability
Hautus lemma
== Notes ==
== References ==
== External links ==
MATLAB function for checking controllability of a system Archived 2012-02-10 at the Wayback Machine
Mathematica function for checking controllability of a system | Wikipedia/Controllability |
In linear algebra, the nonnegative rank of a nonnegative matrix is a concept similar to the usual linear rank of a real matrix, but adding the requirement that certain coefficients and entries of vectors/matrices have to be nonnegative.
For example, the linear rank of a matrix is the smallest number of vectors, such that every column of the matrix can be written as a linear combination of those vectors. For the nonnegative rank, it is required that the vectors must have nonnegative entries, and also that the coefficients in the linear combinations are nonnegative.
== Formal definition ==
There are several equivalent definitions, all modifying the definition of the linear rank slightly. Apart from the definition given above, there is the following: The nonnegative rank of a nonnegative m×n-matrix A is equal to the smallest number q such there exists a nonnegative m×q-matrix B and a nonnegative q×n-matrix C such that A = BC (the usual matrix product). To obtain the linear rank, drop the condition that B and C must be nonnegative.
Further, the nonnegative rank is the smallest number of nonnegative rank-one matrices into which the matrix can be decomposed additively:
where Rj ≥ 0 stands for "Rj is nonnegative". (To obtain the usual linear rank, drop the condition that the Rj have to be nonnegative.)
Given a nonnegative
m
×
n
{\displaystyle m\times n}
matrix A the nonnegative rank
r
a
n
k
+
(
A
)
{\displaystyle rank_{+}(A)}
of A satisfies
=== A Fallacy ===
The rank of the matrix A is the largest number of columns which are linearly independent, i.e., none of the selected columns can be written as a linear combination of the other selected columns. It is not true that adding nonnegativity to this characterization gives the nonnegative rank: The nonnegative rank is in general less than or equal to the largest number of columns such that no selected column can be written as a nonnegative linear combination of the other selected columns.
== Connection with the linear rank ==
It is always true that rank(A) ≤ rank+(A). In fact rank+(A) = rank(A) holds whenever rank(A) ≤ 2.
In the case rank(A) ≥ 3, however, rank(A) < rank+(A) is possible. For example, the matrix
A
=
[
1
1
0
0
1
0
1
0
0
1
0
1
0
0
1
1
]
,
{\displaystyle \mathbf {A} ={\begin{bmatrix}1&1&0&0\\1&0&1&0\\0&1&0&1\\0&0&1&1\end{bmatrix}},}
satisfies rank(A) = 3 < 4 = rank+(A).
These two results (including the 4×4 matrix example above) were first provided by Thomas in a response to a question posed in 1973 by Berman and Plemmons.
== Computing the nonnegative rank ==
The nonnegative rank of a matrix can be determined algorithmically.
It has been proved that determining whether
rank
+
(
A
)
=
rank
(
A
)
{\displaystyle {{\text{rank}}_{+}}(A)={\text{rank}}(A)}
is NP-hard.
Obvious questions concerning the complexity of nonnegative rank computation remain unanswered to date. For example, the complexity of determining the nonnegative rank of matrices of fixed rank k is unknown for k > 2.
== Ancillary facts ==
Nonnegative rank has important applications in Combinatorial optimization: The minimum number of facets of an extension of a polyhedron P is equal to the nonnegative rank of its so-called slack matrix.
== References == | Wikipedia/Nonnegative_rank_(linear_algebra) |
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), and general relativity (stress–energy tensor, curvature tensor, ...). In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others – as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.
== Definition ==
Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction.
=== As multidimensional arrays ===
A tensor may be represented as a (potentially multidimensional) array. Just as a vector in an n-dimensional space is represented by a one-dimensional array with n components with respect to a given basis, any tensor with respect to a basis is represented by a multidimensional array. For example, a linear operator is represented in a basis as a two-dimensional square n × n array. The numbers in the multidimensional array are known as the components of the tensor. They are denoted by indices giving their position in the array, as subscripts and superscripts, following the symbolic name of the tensor. For example, the components of an order-2 tensor T could be denoted Tij , where i and j are indices running from 1 to n, or also by T ij. Whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. Thus while Tij and T ij can both be expressed as n-by-n matrices, and are numerically related via index juggling, the difference in their transformation laws indicates it would be improper to add them together.
The total number of indices (m) required to identify each component uniquely is equal to the dimension or the number of ways of an array, which is why a tensor is sometimes referred to as an m-dimensional array or an m-way array. The total number of indices is also called the order, degree or rank of a tensor, although the term "rank" generally has another meaning in the context of matrices and tensors.
Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis. The components of a vector can respond in two distinct ways to a change of basis (see Covariance and contravariance of vectors), where the new basis vectors
e
^
i
{\displaystyle \mathbf {\hat {e}} _{i}}
are expressed in terms of the old basis vectors
e
j
{\displaystyle \mathbf {e} _{j}}
as,
e
^
i
=
∑
j
=
1
n
e
j
R
i
j
=
e
j
R
i
j
.
{\displaystyle \mathbf {\hat {e}} _{i}=\sum _{j=1}^{n}\mathbf {e} _{j}R_{i}^{j}=\mathbf {e} _{j}R_{i}^{j}.}
Here R ji are the entries of the change of basis matrix, and in the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article. The components vi of a column vector v transform with the inverse of the matrix R,
v
^
i
=
(
R
−
1
)
j
i
v
j
,
{\displaystyle {\hat {v}}^{i}=\left(R^{-1}\right)_{j}^{i}v^{j},}
where the hat denotes the components in the new basis. This is called a contravariant transformation law, because the vector components transform by the inverse of the change of basis. In contrast, the components, wi, of a covector (or row vector), w, transform with the matrix R itself,
w
^
i
=
w
j
R
i
j
.
{\displaystyle {\hat {w}}_{i}=w_{j}R_{i}^{j}.}
This is called a covariant transformation law, because the covector components transform by the same matrix as the change of basis matrix. The components of a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called contravariant and is conventionally denoted with an upper index (superscript). If the transformation matrix of an index is the basis transformation itself, then the index is called covariant and is denoted with a lower index (subscript).
As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array
T
{\displaystyle T}
that transforms under a change of basis matrix
R
=
(
R
i
j
)
{\displaystyle R=\left(R_{i}^{j}\right)}
by
T
^
=
R
−
1
T
R
{\displaystyle {\hat {T}}=R^{-1}TR}
. For the individual matrix entries, this transformation law has the form
T
^
j
′
i
′
=
(
R
−
1
)
i
i
′
T
j
i
R
j
′
j
{\displaystyle {\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}}
so the tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1).
Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above:
v
=
v
^
i
e
^
i
=
(
(
R
−
1
)
j
i
v
j
)
(
e
k
R
i
k
)
=
(
(
R
−
1
)
j
i
R
i
k
)
v
j
e
k
=
δ
j
k
v
j
e
k
=
v
k
e
k
=
v
i
e
i
{\displaystyle \mathbf {v} ={\hat {v}}^{i}\,\mathbf {\hat {e}} _{i}=\left(\left(R^{-1}\right)_{j}^{i}{v}^{j}\right)\left(\mathbf {e} _{k}R_{i}^{k}\right)=\left(\left(R^{-1}\right)_{j}^{i}R_{i}^{k}\right){v}^{j}\mathbf {e} _{k}=\delta _{j}^{k}{v}^{j}\mathbf {e} _{k}={v}^{k}\,\mathbf {e} _{k}={v}^{i}\,\mathbf {e} _{i}}
,
where
δ
j
k
{\displaystyle \delta _{j}^{k}}
is the Kronecker delta, which functions similarly to the identity matrix, and has the effect of renaming indices (j into k in this example). This shows several features of the component notation: the ability to re-arrange terms at will (commutativity), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of the transformation matrix and its inverse cancel, so that expressions like
v
i
e
i
{\displaystyle {v}^{i}\,\mathbf {e} _{i}}
can immediately be seen to be geometrically identical in all coordinate systems.
Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector. The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations. That is, the components
(
T
v
)
i
{\displaystyle (Tv)^{i}}
are given by
(
T
v
)
i
=
T
j
i
v
j
{\displaystyle (Tv)^{i}=T_{j}^{i}v^{j}}
. These components transform contravariantly, since
(
T
v
^
)
i
′
=
T
^
j
′
i
′
v
^
j
′
=
[
(
R
−
1
)
i
i
′
T
j
i
R
j
′
j
]
[
(
R
−
1
)
k
j
′
v
k
]
=
(
R
−
1
)
i
i
′
(
T
v
)
i
.
{\displaystyle \left({\widehat {Tv}}\right)^{i'}={\hat {T}}_{j'}^{i'}{\hat {v}}^{j'}=\left[\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}\right]\left[\left(R^{-1}\right)_{k}^{j'}v^{k}\right]=\left(R^{-1}\right)_{i}^{i'}(Tv)^{i}.}
The transformation law for an order p + q tensor with p contravariant indices and q covariant indices is thus given as,
T
^
j
1
′
,
…
,
j
q
′
i
1
′
,
…
,
i
p
′
=
(
R
−
1
)
i
1
i
1
′
⋯
(
R
−
1
)
i
p
i
p
′
{\displaystyle {\hat {T}}_{j'_{1},\ldots ,j'_{q}}^{i'_{1},\ldots ,i'_{p}}=\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}}
T
j
1
,
…
,
j
q
i
1
,
…
,
i
p
{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}}
R
j
1
′
j
1
⋯
R
j
q
′
j
q
.
{\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}
Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. Such a tensor is said to be of order or type (p, q). The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. Here, the term "order" or "total order" will be used for the total dimension of the array (or its generalization in other definitions), p + q in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices. A tensor of type (p, q) is also called a (p, q)-tensor for short.
This discussion motivates the following formal definition:
Definition. A tensor of type (p, q) is an assignment of a multidimensional array
T
j
1
…
j
q
i
1
…
i
p
[
f
]
{\displaystyle T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}[\mathbf {f} ]}
to each basis f = (e1, ..., en) of an n-dimensional vector space such that, if we apply the change of basis
f
↦
f
⋅
R
=
(
e
i
R
1
i
,
…
,
e
i
R
n
i
)
{\displaystyle \mathbf {f} \mapsto \mathbf {f} \cdot R=\left(\mathbf {e} _{i}R_{1}^{i},\dots ,\mathbf {e} _{i}R_{n}^{i}\right)}
then the multidimensional array obeys the transformation law
T
j
1
′
…
j
q
′
i
1
′
…
i
p
′
[
f
⋅
R
]
=
(
R
−
1
)
i
1
i
1
′
⋯
(
R
−
1
)
i
p
i
p
′
{\displaystyle T_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}[\mathbf {f} \cdot R]=\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}}
T
j
1
,
…
,
j
q
i
1
,
…
,
i
p
[
f
]
{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]}
R
j
1
′
j
1
⋯
R
j
q
′
j
q
.
{\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}
The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.
An equivalent definition of a tensor uses the representations of the general linear group. There is an action of the general linear group on the set of all ordered bases of an n-dimensional vector space. If
f
=
(
f
1
,
…
,
f
n
)
{\displaystyle \mathbf {f} =(\mathbf {f} _{1},\dots ,\mathbf {f} _{n})}
is an ordered basis, and
R
=
(
R
j
i
)
{\displaystyle R=\left(R_{j}^{i}\right)}
is an invertible
n
×
n
{\displaystyle n\times n}
matrix, then the action is given by
f
R
=
(
f
i
R
1
i
,
…
,
f
i
R
n
i
)
.
{\displaystyle \mathbf {f} R=\left(\mathbf {f} _{i}R_{1}^{i},\dots ,\mathbf {f} _{i}R_{n}^{i}\right).}
Let F be the set of all ordered bases. Then F is a principal homogeneous space for GL(n). Let W be a vector space and let
ρ
{\displaystyle \rho }
be a representation of GL(n) on W (that is, a group homomorphism
ρ
:
GL
(
n
)
→
GL
(
W
)
{\displaystyle \rho :{\text{GL}}(n)\to {\text{GL}}(W)}
). Then a tensor of type
ρ
{\displaystyle \rho }
is an equivariant map
T
:
F
→
W
{\displaystyle T:F\to W}
. Equivariance here means that
T
(
F
R
)
=
ρ
(
R
−
1
)
T
(
F
)
.
{\displaystyle T(FR)=\rho \left(R^{-1}\right)T(F).}
When
ρ
{\displaystyle \rho }
is a tensor representation of the general linear group, this gives the usual definition of tensors as multidimensional arrays. This definition is often used to describe tensors on manifolds, and readily generalizes to other groups.
=== As multilinear maps ===
A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach that is common in differential geometry is to define tensors relative to a fixed (finite-dimensional) vector space V, which is usually taken to be a particular vector space of some geometrical significance like the tangent space to a manifold. In this approach, a type (p, q) tensor T is defined as a multilinear map,
T
:
V
∗
×
⋯
×
V
∗
⏟
p
copies
×
V
×
⋯
×
V
⏟
q
copies
→
R
,
{\displaystyle T:\underbrace {V^{*}\times \dots \times V^{*}} _{p{\text{ copies}}}\times \underbrace {V\times \dots \times V} _{q{\text{ copies}}}\rightarrow \mathbf {R} ,}
where V∗ is the corresponding dual space of covectors, which is linear in each of its arguments. The above assumes V is a vector space over the real numbers,
R
{\displaystyle \mathbb {R} }
. More generally, V can be taken over any field F (e.g. the complex numbers), with F replacing
R
{\displaystyle \mathbb {R} }
as the codomain of the multilinear maps.
By applying a multilinear map T of type (p, q) to a basis {ej} for V and a canonical cobasis {εi} for V∗,
T
j
1
…
j
q
i
1
…
i
p
≡
T
(
ε
i
1
,
…
,
ε
i
p
,
e
j
1
,
…
,
e
j
q
)
,
{\displaystyle T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\equiv T\left({\boldsymbol {\varepsilon }}^{i_{1}},\ldots ,{\boldsymbol {\varepsilon }}^{i_{p}},\mathbf {e} _{j_{1}},\ldots ,\mathbf {e} _{j_{q}}\right),}
a (p + q)-dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of T thus form a tensor according to that definition. Moreover, such an array can be realized as the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.
In viewing a tensor as a multilinear map, it is conventional to identify the double dual V∗∗ of the vector space V, i.e., the space of linear functionals on the dual vector space V∗, with the vector space V. There is always a natural linear map from V to its double dual, given by evaluating a linear form in V∗ against a vector in V. This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify V with its double dual.
=== Using tensor products ===
For some mathematical applications, a more abstract approach is sometimes useful. This can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property as explained here and here.
A type (p, q) tensor is defined in this context as an element of the tensor product of vector spaces,
T
∈
V
⊗
⋯
⊗
V
⏟
p
copies
⊗
V
∗
⊗
⋯
⊗
V
∗
⏟
q
copies
.
{\displaystyle T\in \underbrace {V\otimes \dots \otimes V} _{p{\text{ copies}}}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{q{\text{ copies}}}.}
A basis vi of V and basis wj of W naturally induce a basis vi ⊗ wj of the tensor product V ⊗ W. The components of a tensor T are the coefficients of the tensor with respect to the basis obtained from a basis {ei} for V and its dual basis {εj}, i.e.
T
=
T
j
1
…
j
q
i
1
…
i
p
e
i
1
⊗
⋯
⊗
e
i
p
⊗
ε
j
1
⊗
⋯
⊗
ε
j
q
.
{\displaystyle T=T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\;\mathbf {e} _{i_{1}}\otimes \cdots \otimes \mathbf {e} _{i_{p}}\otimes {\boldsymbol {\varepsilon }}^{j_{1}}\otimes \cdots \otimes {\boldsymbol {\varepsilon }}^{j_{q}}.}
Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (p, q) tensor. Moreover, the universal property of the tensor product gives a one-to-one correspondence between tensors defined in this way and tensors defined as multilinear maps.
This 1 to 1 correspondence can be achieved in the following way, because in the finite-dimensional case there exists a canonical isomorphism between a vector space and its double dual:
U
⊗
V
≅
(
U
∗
∗
)
⊗
(
V
∗
∗
)
≅
(
U
∗
⊗
V
∗
)
∗
≅
Hom
2
(
U
∗
×
V
∗
;
F
)
{\displaystyle U\otimes V\cong \left(U^{**}\right)\otimes \left(V^{**}\right)\cong \left(U^{*}\otimes V^{*}\right)^{*}\cong \operatorname {Hom} ^{2}\left(U^{*}\times V^{*};\mathbb {F} \right)}
The last line is using the universal property of the tensor product, that there is a 1 to 1 correspondence between maps from
Hom
2
(
U
∗
×
V
∗
;
F
)
{\displaystyle \operatorname {Hom} ^{2}\left(U^{*}\times V^{*};\mathbb {F} \right)}
and
Hom
(
U
∗
⊗
V
∗
;
F
)
{\displaystyle \operatorname {Hom} \left(U^{*}\otimes V^{*};\mathbb {F} \right)}
.
Tensor products can be defined in great generality – for example, involving arbitrary modules over a ring. In principle, one could define a "tensor" simply to be an element of any tensor product. However, the mathematics literature usually reserves the term tensor for an element of a tensor product of any number of copies of a single vector space V and its dual, as above.
=== Tensors in infinite dimensions ===
This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are naturally isomorphic. Constructions of spaces of tensors based on the tensor product and multilinear mappings can be generalized, essentially without modification, to vector bundles or coherent sheaves. For infinite-dimensional vector spaces, inequivalent topologies lead to inequivalent notions of tensor, and these various isomorphisms may or may not hold depending on what exactly is meant by a tensor (see topological tensor product). In some applications, it is the tensor product of Hilbert spaces that is intended, whose properties are the most similar to the finite-dimensional case. A more modern view is that it is the tensors' structure as a symmetric monoidal category that encodes their most important properties, rather than the specific models of those categories.
=== Tensor fields ===
In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space. This was the setting of Ricci's original work. In modern mathematical terminology such an object is called a tensor field, often referred to simply as a tensor.
In this context, a coordinate basis is often chosen for the tangent vector space. The transformation law may then be expressed in terms of partial derivatives of the coordinate functions,
x
¯
i
(
x
1
,
…
,
x
n
)
,
{\displaystyle {\bar {x}}^{i}\left(x^{1},\ldots ,x^{n}\right),}
defining a coordinate transformation,
T
^
j
1
′
…
j
q
′
i
1
′
…
i
p
′
(
x
¯
1
,
…
,
x
¯
n
)
=
∂
x
¯
i
1
′
∂
x
i
1
⋯
∂
x
¯
i
p
′
∂
x
i
p
∂
x
j
1
∂
x
¯
j
1
′
⋯
∂
x
j
q
∂
x
¯
j
q
′
T
j
1
…
j
q
i
1
…
i
p
(
x
1
,
…
,
x
n
)
.
{\displaystyle {\hat {T}}_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}\left({\bar {x}}^{1},\ldots ,{\bar {x}}^{n}\right)={\frac {\partial {\bar {x}}^{i'_{1}}}{\partial x^{i_{1}}}}\cdots {\frac {\partial {\bar {x}}^{i'_{p}}}{\partial x^{i_{p}}}}{\frac {\partial x^{j_{1}}}{\partial {\bar {x}}^{j'_{1}}}}\cdots {\frac {\partial x^{j_{q}}}{\partial {\bar {x}}^{j'_{q}}}}T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\left(x^{1},\ldots ,x^{n}\right).}
== History ==
The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century. The word "tensor" itself was introduced in 1846 by William Rowan Hamilton to describe something different from what is now meant by a tensor. Gibbs introduced dyadics and polyadic algebra, which are also tensors in the modern sense. The contemporary usage was introduced by Woldemar Voigt in 1898.
Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented in 1892. It was made accessible to many mathematicians by the publication of Ricci-Curbastro and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications). In Ricci's notation, he refers to "systems" with covariant and contravariant components, which are known as tensor fields in the modern sense.
In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Albert Einstein's theory of general relativity, around 1915. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann. Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect:
I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.
Tensors and tensor fields were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors, and the Riemann curvature tensor. The exterior algebra of Hermann Grassmann, from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of differential forms, as naturally unified with tensor calculus. The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics, and Hassler Whitney popularized the tensor product.
From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem). Correspondingly there are types of tensors at work in many branches of abstract algebra, particularly in homological algebra and representation theory. Multilinear algebra can be developed in greater generality than for scalars coming from a field. For example, scalars can come from a ring. But the theory is then less geometric and computations more technical and less algorithmic. Tensors are generalized within category theory by means of the concept of monoidal category, from the 1960s.
== Examples ==
An elementary example of a mapping describable as a tensor is the dot product, which maps two vectors to a scalar. A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The cross product, where two vectors are mapped to a third one, is strictly speaking not a tensor because it changes its sign under those transformations that change the orientation of the coordinate system. The totally anti-symmetric symbol
ε
i
j
k
{\displaystyle \varepsilon _{ijk}}
nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems.
This table shows important examples of tensors on vector spaces and tensor fields on manifolds. The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner products. In the (0, M)-entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to select that dimension to get a maximally covariant antisymmetric tensor.
Raising an index on an (n, m)-tensor produces an (n + 1, m − 1)-tensor; this corresponds to moving diagonally down and to the left on the table. Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. Contraction of an upper with a lower index of an (n, m)-tensor produces an (n − 1, m − 1)-tensor; this corresponds to moving diagonally up and to the left on the table.
== Properties ==
Assuming a basis of a real vector space, e.g., a coordinate frame in the ambient space, a tensor can be represented as an organized multidimensional array of numerical values with respect to this specific basis. Changing the basis transforms the values in the array in a characteristic way that allows to define tensors as objects adhering to this transformational behavior. For example, there are invariants of tensors that must be preserved under any change of the basis, thereby making only certain multidimensional arrays of numbers a tensor. Compare this to the array representing
ε
i
j
k
{\displaystyle \varepsilon _{ijk}}
not being a tensor, for the sign change under transformations changing the orientation.
Because the components of vectors and their duals transform differently under the change of their dual bases, there is a covariant and/or contravariant transformation law that relates the arrays, which represent the tensor with respect to one basis and that with respect to the other one. The numbers of, respectively, vectors: n (contravariant indices) and dual vectors: m (covariant indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers (n, m), which determine the precise form of the transformation law. The order of a tensor is the sum of these two numbers.
The order (also degree or rank) of a tensor is thus the sum of the orders of its arguments plus the order of the resulting tensor. This is also the dimensionality of the array of numbers needed to represent the tensor with respect to a specific basis, or equivalently, the number of indices needed to label each component in that array. For example, in a fixed basis, a standard linear map that maps a vector to a vector, is represented by a matrix (a 2-dimensional array), and therefore is a 2nd-order tensor. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. Scalars are simple numbers and are thus 0th-order tensors. This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order 2 + 0 = 2, the same as the stress tensor, taking one vector and returning another 1 + 1 = 2. The
ε
i
j
k
{\displaystyle \varepsilon _{ijk}}
-symbol, mapping two vectors to one vector, would have order 2 + 1 = 3.
The collection of tensors on a vector space and its dual forms a tensor algebra, which allows products of arbitrary tensors. Simple applications of tensors of order 2, which can be represented as a square matrix, can be solved by clever arrangement of transposed vectors and by applying the rules of matrix multiplication, but the tensor product should not be confused with this.
== Notation ==
There are several notational systems that are used to describe tensors and perform calculations involving them.
=== Ricci calculus ===
Ricci calculus is the modern formalism and notation for tensor indices: indicating inner and outer products, covariance and contravariance, summations of tensor components, symmetry and antisymmetry, and partial and covariant derivatives.
=== Einstein summation convention ===
The Einstein summation convention dispenses with writing summation signs, leaving the summation implicit. Any repeated index symbol is summed over: if the index i is used twice in a given term of a tensor expression, it means that the term is to be summed for all i. Several distinct pairs of indices may be summed this way.
=== Penrose graphical notation ===
Penrose graphical notation is a diagrammatic notation which replaces the symbols for tensors with shapes, and their indices by lines and curves. It is independent of basis elements, and requires no symbols for the indices.
=== Abstract index notation ===
The abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates. This notation captures the expressiveness of indices and the basis-independence of index-free notation.
=== Component-free notation ===
A component-free treatment of tensors uses notation that emphasises that tensors do not rely on any basis, and is defined in terms of the tensor product of vector spaces.
== Operations ==
There are several operations on tensors that again produce a tensor. The linear nature of tensor implies that two tensors of the same type may be added together, and that tensors may be multiplied by a scalar with results analogous to the scaling of a vector. On components, these operations are simply performed component-wise. These operations do not change the type of the tensor; but there are also operations that produce a tensor of different type.
=== Tensor product ===
The tensor product takes two tensors, S and T, and produces a new tensor, S ⊗ T, whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.,
(
S
⊗
T
)
(
v
1
,
…
,
v
n
,
v
n
+
1
,
…
,
v
n
+
m
)
=
S
(
v
1
,
…
,
v
n
)
T
(
v
n
+
1
,
…
,
v
n
+
m
)
,
{\displaystyle (S\otimes T)(v_{1},\ldots ,v_{n},v_{n+1},\ldots ,v_{n+m})=S(v_{1},\ldots ,v_{n})T(v_{n+1},\ldots ,v_{n+m}),}
which again produces a map that is linear in all its arguments. On components, the effect is to multiply the components of the two input tensors pairwise, i.e.,
(
S
⊗
T
)
j
1
…
j
k
j
k
+
1
…
j
k
+
m
i
1
…
i
l
i
l
+
1
…
i
l
+
n
=
S
j
1
…
j
k
i
1
…
i
l
T
j
k
+
1
…
j
k
+
m
i
l
+
1
…
i
l
+
n
.
{\displaystyle (S\otimes T)_{j_{1}\ldots j_{k}j_{k+1}\ldots j_{k+m}}^{i_{1}\ldots i_{l}i_{l+1}\ldots i_{l+n}}=S_{j_{1}\ldots j_{k}}^{i_{1}\ldots i_{l}}T_{j_{k+1}\ldots j_{k+m}}^{i_{l+1}\ldots i_{l+n}}.}
If S is of type (l, k) and T is of type (n, m), then the tensor product S ⊗ T has type (l + n, k + m).
=== Contraction ===
Tensor contraction is an operation that reduces a type (n, m) tensor to a type (n − 1, m − 1) tensor, of which the trace is a special case. It thereby reduces the total order of a tensor by two. The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component. Components for which those two indices are different are discarded. For example, a (1, 1)-tensor
T
i
j
{\displaystyle T_{i}^{j}}
can be contracted to a scalar through
T
i
i
{\displaystyle T_{i}^{i}}
, where the summation is again implied. When the (1, 1)-tensor is interpreted as a linear map, this operation is known as the trace.
The contraction is often used in conjunction with the tensor product to contract an index from each tensor.
The contraction can also be understood using the definition of a tensor as an element of a tensor product of copies of the space V with the space V∗ by first decomposing the tensor into a linear combination of simple tensors, and then applying a factor from V∗ to a factor from V. For example, a tensor
T
∈
V
⊗
V
⊗
V
∗
{\displaystyle T\in V\otimes V\otimes V^{*}}
can be written as a linear combination
T
=
v
1
⊗
w
1
⊗
α
1
+
v
2
⊗
w
2
⊗
α
2
+
⋯
+
v
N
⊗
w
N
⊗
α
N
.
{\displaystyle T=v_{1}\otimes w_{1}\otimes \alpha _{1}+v_{2}\otimes w_{2}\otimes \alpha _{2}+\cdots +v_{N}\otimes w_{N}\otimes \alpha _{N}.}
The contraction of T on the first and last slots is then the vector
α
1
(
v
1
)
w
1
+
α
2
(
v
2
)
w
2
+
⋯
+
α
N
(
v
N
)
w
N
.
{\displaystyle \alpha _{1}(v_{1})w_{1}+\alpha _{2}(v_{2})w_{2}+\cdots +\alpha _{N}(v_{N})w_{N}.}
In a vector space with an inner product (also known as a metric) g, the term contraction is used for removing two contravariant or two covariant indices by forming a trace with the metric tensor or its inverse. For example, a (2, 0)-tensor
T
i
j
{\displaystyle T^{ij}}
can be contracted to a scalar through
T
i
j
g
i
j
{\displaystyle T^{ij}g_{ij}}
(yet again assuming the summation convention).
=== Raising or lowering an index ===
When a vector space is equipped with a nondegenerate bilinear form (or metric tensor as it is often called in this context), operations can be defined that convert a contravariant (upper) index into a covariant (lower) index and vice versa. A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index. This operation is quite graphically known as lowering an index.
Conversely, the inverse operation can be defined, and is called raising an index. This is equivalent to a similar contraction on the product with a (2, 0)-tensor. This inverse metric tensor has components that are the matrix inverse of those of the metric tensor.
== Applications ==
=== Continuum mechanics ===
Important examples are provided by continuum mechanics. The stresses inside a solid body or fluid are described by a tensor field. The stress tensor and strain tensor are both second-order tensor fields, and are related in a general linear elastic material by a fourth-order elasticity tensor field. In detail, the tensor quantifying stress in a 3-dimensional solid object has components that can be conveniently represented as a 3 × 3 array. The three faces of a cube-shaped infinitesimal volume segment of the solid are each subject to some given force. The force's vector components are also three in number. Thus, 3 × 3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment. Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Thus, a second-order tensor is needed.
If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type (2, 0), in linear elasticity, or more precisely by a tensor field of type (2, 0), since the stresses may vary from point to point.
=== Other examples from physics ===
Common applications include:
Electromagnetic tensor (or Faraday tensor) in electromagnetism
Finite deformation tensors for describing deformations and strain tensor for strain in continuum mechanics
Permittivity and electric susceptibility are tensors in anisotropic media
Four-tensors in general relativity (e.g. stress–energy tensor), used to represent momentum fluxes
Spherical tensor operators are the eigenfunctions of the quantum angular momentum operator in spherical coordinates
Diffusion tensors, the basis of diffusion tensor imaging, represent rates of diffusion in biological environments
Quantum mechanics and quantum computing utilize tensor products for combination of quantum states
=== Computer vision and optics ===
The concept of a tensor of order two is often conflated with that of a matrix. Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop. This happens, for instance, in the field of computer vision, with the trifocal tensor generalizing the fundamental matrix.
The field of nonlinear optics studies the changes to material polarization density under extreme electric fields. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:
P
i
ε
0
=
∑
j
χ
i
j
(
1
)
E
j
+
∑
j
k
χ
i
j
k
(
2
)
E
j
E
k
+
∑
j
k
ℓ
χ
i
j
k
ℓ
(
3
)
E
j
E
k
E
ℓ
+
⋯
.
{\displaystyle {\frac {P_{i}}{\varepsilon _{0}}}=\sum _{j}\chi _{ij}^{(1)}E_{j}+\sum _{jk}\chi _{ijk}^{(2)}E_{j}E_{k}+\sum _{jk\ell }\chi _{ijk\ell }^{(3)}E_{j}E_{k}E_{\ell }+\cdots .\!}
Here
χ
(
1
)
{\displaystyle \chi ^{(1)}}
is the linear susceptibility,
χ
(
2
)
{\displaystyle \chi ^{(2)}}
gives the Pockels effect and second harmonic generation, and
χ
(
3
)
{\displaystyle \chi ^{(3)}}
gives the Kerr effect. This expansion shows the way higher-order tensors arise naturally in the subject matter.
=== Machine learning ===
The properties of tensors, especially tensor decomposition, have enabled their use in machine learning to embed higher dimensional data in artificial neural networks. This notion of tensor differs significantly from that in other areas of mathematics and physics, in the sense that a tensor is usually regarded as a numerical quantity in a fixed basis, and the dimension of the spaces along the different axes of the tensor need not be the same.
== Generalizations ==
=== Tensor products of vector spaces ===
The vector spaces of a tensor product need not be the same, and sometimes the elements of such a more general tensor product are called "tensors". For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense, and an order-d tensor may likewise be defined as an element of a tensor product of d different vector spaces. A type (n, m) tensor, in the sense defined previously, is also a tensor of order n + m in this more general sense. The concept of tensor product can be extended to arbitrary modules over a ring.
=== Tensors in infinite dimensions ===
The notion of a tensor can be generalized in a variety of ways to infinite dimensions. One, for instance, is via the tensor product of Hilbert spaces. Another way of generalizing the idea of tensor, common in nonlinear analysis, is via the multilinear maps definition where instead of using finite-dimensional vector spaces and their algebraic duals, one uses infinite-dimensional Banach spaces and their continuous dual. Tensors thus live naturally on Banach manifolds and Fréchet manifolds.
=== Tensor densities ===
Suppose that a homogeneous medium fills R3, so that the density of the medium is described by a single scalar value ρ in kg⋅m−3. The mass, in kg, of a region Ω is obtained by multiplying ρ by the volume of the region Ω, or equivalently integrating the constant ρ over the region:
m
=
∫
Ω
ρ
d
x
d
y
d
z
,
{\displaystyle m=\int _{\Omega }\rho \,dx\,dy\,dz,}
where the Cartesian coordinates x, y, z are measured in m. If the units of length are changed into cm, then the numerical values of the coordinate functions must be rescaled by a factor of 100:
x
′
=
100
x
,
y
′
=
100
y
,
z
′
=
100
z
.
{\displaystyle x'=100x,\quad y'=100y,\quad z'=100z.}
The numerical value of the density ρ must then also transform by 100−3 m3/cm3 to compensate, so that the numerical value of the mass in kg is still given by integral of
ρ
d
x
d
y
d
z
{\displaystyle \rho \,dx\,dy\,dz}
. Thus
ρ
′
=
100
−
3
ρ
{\displaystyle \rho '=100^{-3}\rho }
(in units of kg⋅cm−3).
More generally, if the Cartesian coordinates x, y, z undergo a linear transformation, then the numerical value of the density ρ must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula for integration. Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a scalar density. To model a non-constant density, ρ is a function of the variables x, y, z (a scalar field), and under a curvilinear change of coordinates, it transforms by the reciprocal of the Jacobian of the coordinate change. For more on the intrinsic meaning, see Density on a manifold.
A tensor density transforms like a tensor under a coordinate change, except that it in addition picks up a factor of the absolute value of the determinant of the coordinate transition:
T
j
1
′
…
j
q
′
i
1
′
…
i
p
′
[
f
⋅
R
]
=
|
det
R
|
−
w
(
R
−
1
)
i
1
i
1
′
⋯
(
R
−
1
)
i
p
i
p
′
T
j
1
,
…
,
j
q
i
1
,
…
,
i
p
[
f
]
R
j
1
′
j
1
⋯
R
j
q
′
j
q
.
{\displaystyle T_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}[\mathbf {f} \cdot R]=\left|\det R\right|^{-w}\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}
Here w is called the weight. In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor. An example of a tensor density is the current density of electromagnetism.
Under an affine transformation of the coordinates, a tensor transforms by the linear part of the transformation itself (or its inverse) on each index. These come from the rational representations of the general linear group. But this is not quite the most general linear transformation law that such an object may have: tensor densities are non-rational, but are still semisimple representations. A further class of transformations come from the logarithmic representation of the general linear group, a reducible but not semisimple representation, consisting of an (x, y) ∈ R2 with the transformation law
(
x
,
y
)
↦
(
x
+
y
log
|
det
R
|
,
y
)
.
{\displaystyle (x,y)\mapsto (x+y\log \left|\det R\right|,y).}
=== Geometric objects ===
The transformation law for a tensor behaves as a functor on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as local diffeomorphisms). This makes a tensor a special case of a geometrical object, in the technical sense that it is a function of the coordinate system transforming functorially under coordinate changes. Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles.
=== Spinors ===
When changing from one orthonormal basis (called a frame) to another by a rotation, the components of a tensor transform by that same rotation. This transformation does not depend on the path taken through the space of frames. However, the space of frames is not simply connected (see orientation entanglement and plate trick): there are continuous paths in the space of frames with the same beginning and ending configurations that are not deformable one into the other. It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1. A spinor is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.
Spinors are elements of the spin representation of the rotation group, while tensors are elements of its tensor representations. Other classical groups have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well.
== See also ==
The dictionary definition of tensor at Wiktionary
Array data type, for tensor storage and manipulation
Bitensor
=== Foundational ===
=== Applications ===
== Explanatory notes ==
== References ==
=== Specific ===
=== General ===
This article incorporates material from tensor on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
== External links == | Wikipedia/Tensors |
The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst Steinitz. The result is often called the Steinitz–Mac Lane exchange lemma, also recognizing the generalization
by Saunders Mac Lane
of Steinitz's lemma to matroids.
== Statement ==
Let
U
{\displaystyle U}
and
W
{\displaystyle W}
be finite subsets of a vector space
V
{\displaystyle V}
. If
U
{\displaystyle U}
is a set of linearly independent vectors, and
W
{\displaystyle W}
spans
V
{\displaystyle V}
, then:
1.
|
U
|
≤
|
W
|
{\displaystyle |U|\leq |W|}
;
2. There is a set
W
′
⊆
W
{\displaystyle W'\subseteq W}
with
|
W
′
|
=
|
W
|
−
|
U
|
{\displaystyle |W'|=|W|-|U|}
such that
U
∪
W
′
{\displaystyle U\cup W'}
spans
V
{\displaystyle V}
.
== Proof ==
Suppose
U
=
{
u
1
,
…
,
u
m
}
{\displaystyle U=\{u_{1},\dots ,u_{m}\}}
and
W
=
{
w
1
,
…
,
w
n
}
{\displaystyle W=\{w_{1},\dots ,w_{n}\}}
. We wish to show that
m
≤
n
{\displaystyle m\leq n}
, and that after rearranging the
w
j
{\displaystyle w_{j}}
if necessary, the set
{
u
1
,
…
,
u
m
,
w
m
+
1
,
…
,
w
n
}
{\displaystyle \{u_{1},\dotsc ,u_{m},w_{m+1},\dotsc ,w_{n}\}}
spans
V
{\displaystyle V}
. We proceed by induction on
m
{\displaystyle m}
.
For the base case, suppose
m
{\displaystyle m}
is zero.
In this case, the claim holds because there are no vectors
u
i
{\displaystyle u_{i}}
, and the set
{
w
1
,
…
,
w
n
}
{\displaystyle \{w_{1},\dotsc ,w_{n}\}}
spans
V
{\displaystyle V}
by hypothesis.
For the inductive step, assume the proposition is true for
m
−
1
{\displaystyle m-1}
. By the inductive hypothesis we may reorder the
w
i
{\displaystyle w_{i}}
so that
{
u
1
,
…
,
u
m
−
1
,
w
m
,
…
,
w
n
}
{\displaystyle \{u_{1},\ldots ,u_{m-1},w_{m},\ldots ,w_{n}\}}
spans
V
{\displaystyle V}
. Since
u
m
∈
V
{\displaystyle u_{m}\in V}
, there exist coefficients
μ
1
,
…
,
μ
n
{\displaystyle \mu _{1},\ldots ,\mu _{n}}
such that
u
m
=
∑
i
=
1
m
−
1
μ
i
u
i
+
∑
j
=
m
n
μ
j
w
j
{\displaystyle u_{m}=\sum _{i=1}^{m-1}\mu _{i}u_{i}+\sum _{j=m}^{n}\mu _{j}w_{j}}
.
At least one of the
μ
j
{\displaystyle \mu _{j}}
for
j
≥
m
{\displaystyle j\geq m}
must be non-zero, since otherwise this equality would contradict the linear independence of
{
u
1
,
…
,
u
m
}
{\displaystyle \{u_{1},\ldots ,u_{m}\}}
; this also shows that indeed
m
≤
n
.
{\displaystyle m\leq n.}
By reordering
μ
m
w
m
,
…
,
μ
n
w
n
{\displaystyle \mu _{m}w_{m},\ldots ,\mu _{n}w_{n}}
if necessary, we may assume that
μ
m
{\displaystyle \mu _{m}}
is nonzero. Therefore, we have
w
m
=
1
μ
m
(
u
m
−
∑
j
=
1
m
−
1
μ
j
u
j
−
∑
j
=
m
+
1
n
μ
j
w
j
)
{\displaystyle w_{m}={\frac {1}{\mu _{m}}}\left(u_{m}-\sum _{j=1}^{m-1}\mu _{j}u_{j}-\sum _{j=m+1}^{n}\mu _{j}w_{j}\right)}
.
In other words,
w
m
{\displaystyle w_{m}}
is in the span of
{
u
1
,
…
,
u
m
,
w
m
+
1
,
…
,
w
n
}
{\displaystyle \{u_{1},\ldots ,u_{m},w_{m+1},\ldots ,w_{n}\}}
. Since this span contains each of the vectors
u
1
,
…
,
u
m
−
1
,
w
m
,
w
m
+
1
,
…
,
w
n
{\displaystyle u_{1},\ldots ,u_{m-1},w_{m},w_{m+1},\ldots ,w_{n}}
, by the inductive hypothesis it contains
V
{\displaystyle V}
.
== Applications ==
The Steinitz exchange lemma is a basic result in computational mathematics, especially in linear algebra and in combinatorial algorithms.
== References ==
Julio R. Bastida, Field extensions and Galois Theory, Addison–Wesley Publishing Company (1984).
== External links ==
Mizar system proof: http://mizar.org/version/current/html/vectsp_9.html#T19 | Wikipedia/Steinitz_exchange_lemma |
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
In differential geometry, an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used extensively in abstract algebra and homological algebra, where tensors arise naturally.
== Definition via tensor products of vector spaces ==
Given a finite set {V1, ..., Vn} of vector spaces over a common field F, one may form their tensor product V1 ⊗ ... ⊗ Vn, an element of which is termed a tensor.
A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:
V
⊗
⋯
⊗
V
⊗
V
∗
⊗
⋯
⊗
V
∗
{\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}}
where V∗ is the dual space of V.
If there are m copies of V and n copies of V∗ in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant of order n and of total order m + n. The tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V∗ (for this reason, the elements of the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m, n) is denoted
T
n
m
(
V
)
=
V
⊗
⋯
⊗
V
⏟
m
⊗
V
∗
⊗
⋯
⊗
V
∗
⏟
n
.
{\displaystyle T_{n}^{m}(V)=\underbrace {V\otimes \dots \otimes V} _{m}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{n}.}
Example 1. The space of type (1, 1) tensors,
T
1
1
(
V
)
=
V
⊗
V
∗
,
{\displaystyle T_{1}^{1}(V)=V\otimes V^{*},}
is isomorphic in a natural way to the space of linear transformations from V to V.
Example 2. A bilinear form on a real vector space V,
V
×
V
→
F
,
{\displaystyle V\times V\to F,}
corresponds in a natural way to a type (0, 2) tensor in
T
2
0
(
V
)
=
V
∗
⊗
V
∗
.
{\displaystyle T_{2}^{0}(V)=V^{*}\otimes V^{*}.}
An example of such a bilinear form may be defined, termed the associated metric tensor, and is usually denoted g.
== Tensor rank ==
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor) is a tensor that can be written as a product of tensors of the form
T
=
a
⊗
b
⊗
⋯
⊗
d
{\displaystyle T=a\otimes b\otimes \cdots \otimes d}
where a, b, ..., d are nonzero and in V or V∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors. The rank of a tensor T is the minimum number of simple tensors that sum to T.
The zero tensor has rank zero. A nonzero order 0 or 1 tensor always has rank 1. The rank of a non-zero order 2 or higher tensor is less than or equal to the product of the dimensions of all but the highest-dimensioned vectors in (a sum of products of) which the tensor can be expressed, which is dn−1 when each product is of n vectors from a finite-dimensional vector space of dimension d.
The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an outer product of two nonzero vectors:
A
=
v
w
T
.
{\displaystyle A=vw^{\mathrm {T} }.}
The rank of a matrix A is the smallest number of such outer products that can be summed to produce it:
A
=
v
1
w
1
T
+
⋯
+
v
k
w
k
T
.
{\displaystyle A=v_{1}w_{1}^{\mathrm {T} }+\cdots +v_{k}w_{k}^{\mathrm {T} }.}
In indices, a tensor of rank 1 is a tensor of the form
T
i
j
…
k
ℓ
…
=
a
i
b
j
⋯
c
k
d
ℓ
⋯
.
{\displaystyle T_{ij\dots }^{k\ell \dots }=a_{i}b_{j}\cdots c^{k}d^{\ell }\cdots .}
The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix, and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often very difficult to determine, and low rank decompositions of tensors are sometimes of great practical interest. In fact, the problem of finding the rank of an order 3 tensor over any finite field is NP-Complete, and over the rationals, is NP-Hard. Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of bilinear forms
z
k
=
∑
i
j
T
i
j
k
x
i
y
j
{\displaystyle z_{k}=\sum _{ij}T_{ijk}x_{i}y_{j}}
for given inputs xi and yj. If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known.
== Universal property ==
The space
T
n
m
(
V
)
{\displaystyle T_{n}^{m}(V)}
can be characterized by a universal property in terms of multilinear mappings. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for free modules, and the "universal" approach carries over more easily to more general situations.
A scalar-valued function on a Cartesian product (or direct sum) of vector spaces
f
:
V
1
×
⋯
×
V
N
→
F
{\displaystyle f:V_{1}\times \cdots \times V_{N}\to F}
is multilinear if it is linear in each argument. The space of all multilinear mappings from V1 × ... × VN to W is denoted LN(V1, ..., VN; W). When N = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from V to W is denoted L(V; W).
The universal characterization of the tensor product implies that, for each multilinear function
f
∈
L
m
+
n
(
V
∗
,
…
,
V
∗
⏟
m
,
V
,
…
,
V
⏟
n
;
W
)
{\displaystyle f\in L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};W)}
(where W can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function
T
f
∈
L
(
V
∗
⊗
⋯
⊗
V
∗
⏟
m
⊗
V
⊗
⋯
⊗
V
⏟
n
;
W
)
{\displaystyle T_{f}\in L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};W)}
such that
f
(
α
1
,
…
,
α
m
,
v
1
,
…
,
v
n
)
=
T
f
(
α
1
⊗
⋯
⊗
α
m
⊗
v
1
⊗
⋯
⊗
v
n
)
{\displaystyle f(\alpha _{1},\ldots ,\alpha _{m},v_{1},\ldots ,v_{n})=T_{f}(\alpha _{1}\otimes \cdots \otimes \alpha _{m}\otimes v_{1}\otimes \cdots \otimes v_{n})}
for all vi in V and αi in V∗.
Using the universal property, it follows, when V is finite dimensional, that the space of (m, n)-tensors admits a natural isomorphism
T
n
m
(
V
)
≅
L
(
V
∗
⊗
⋯
⊗
V
∗
⏟
m
⊗
V
⊗
⋯
⊗
V
⏟
n
;
F
)
≅
L
m
+
n
(
V
∗
,
…
,
V
∗
⏟
m
,
V
,
…
,
V
⏟
n
;
F
)
.
{\displaystyle T_{n}^{m}(V)\cong L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};F)\cong L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};F).}
Each V in the definition of the tensor corresponds to a V∗ inside the argument of the linear maps, and vice versa. (Note that in the former case, there are m copies of V and n copies of V∗, and in the latter case vice versa). In particular, one has
T
0
1
(
V
)
≅
L
(
V
∗
;
F
)
≅
V
,
T
1
0
(
V
)
≅
L
(
V
;
F
)
=
V
∗
,
T
1
1
(
V
)
≅
L
(
V
;
V
)
.
{\displaystyle {\begin{aligned}T_{0}^{1}(V)&\cong L(V^{*};F)\cong V,\\T_{1}^{0}(V)&\cong L(V;F)=V^{*},\\T_{1}^{1}(V)&\cong L(V;V).\end{aligned}}}
== Tensor fields ==
Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term tensor is sometimes used as a shorthand for tensor field. A tensor field expresses the concept of a tensor that varies from point to point on the manifold.
== References ==
Abraham, Ralph; Marsden, Jerrold E. (1985), Foundations of Mechanics (2nd ed.), Reading, Massachusetts: Addison-Wesley, ISBN 0-201-40840-6.
Bourbaki, Nicolas (1989), Elements of Mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9.
de Groote, H. F. (1987), Lectures on the Complexity of Bilinear Problems, Lecture Notes in Computer Science, vol. 245, Springer, ISBN 3-540-17205-X.
Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4.
Håstad, Johan (November 15, 1989), "Tensor Rank Is NP-Complete", Journal of Algorithms, 11 (4): 644–654, doi:10.1016/0196-6774(90)90014-6.
Jeevanjee, Nadir (2011), "An Introduction to Tensors and Group Theory for Physicists", Physics Today, 65 (4): 64, Bibcode:2012PhT....65d..64P, doi:10.1063/PT.3.1523, ISBN 978-0-8176-4714-8.
Knuth, Donald E. (1998) [1969], The Art of Computer Programming, vol. 2 (3rd ed.), Addison-Wesley, pp. 145–146, ISBN 978-0-201-89684-8.
Hackbusch, Wolfgang (2012), Tensor Spaces and Numerical Tensor Calculus, Springer, p. 4, ISBN 978-3-642-28027-6. | Wikipedia/Simple_tensor |
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
In differential geometry, an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used extensively in abstract algebra and homological algebra, where tensors arise naturally.
== Definition via tensor products of vector spaces ==
Given a finite set {V1, ..., Vn} of vector spaces over a common field F, one may form their tensor product V1 ⊗ ... ⊗ Vn, an element of which is termed a tensor.
A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:
V
⊗
⋯
⊗
V
⊗
V
∗
⊗
⋯
⊗
V
∗
{\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}}
where V∗ is the dual space of V.
If there are m copies of V and n copies of V∗ in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant of order n and of total order m + n. The tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V∗ (for this reason, the elements of the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m, n) is denoted
T
n
m
(
V
)
=
V
⊗
⋯
⊗
V
⏟
m
⊗
V
∗
⊗
⋯
⊗
V
∗
⏟
n
.
{\displaystyle T_{n}^{m}(V)=\underbrace {V\otimes \dots \otimes V} _{m}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{n}.}
Example 1. The space of type (1, 1) tensors,
T
1
1
(
V
)
=
V
⊗
V
∗
,
{\displaystyle T_{1}^{1}(V)=V\otimes V^{*},}
is isomorphic in a natural way to the space of linear transformations from V to V.
Example 2. A bilinear form on a real vector space V,
V
×
V
→
F
,
{\displaystyle V\times V\to F,}
corresponds in a natural way to a type (0, 2) tensor in
T
2
0
(
V
)
=
V
∗
⊗
V
∗
.
{\displaystyle T_{2}^{0}(V)=V^{*}\otimes V^{*}.}
An example of such a bilinear form may be defined, termed the associated metric tensor, and is usually denoted g.
== Tensor rank ==
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor) is a tensor that can be written as a product of tensors of the form
T
=
a
⊗
b
⊗
⋯
⊗
d
{\displaystyle T=a\otimes b\otimes \cdots \otimes d}
where a, b, ..., d are nonzero and in V or V∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors. The rank of a tensor T is the minimum number of simple tensors that sum to T.
The zero tensor has rank zero. A nonzero order 0 or 1 tensor always has rank 1. The rank of a non-zero order 2 or higher tensor is less than or equal to the product of the dimensions of all but the highest-dimensioned vectors in (a sum of products of) which the tensor can be expressed, which is dn−1 when each product is of n vectors from a finite-dimensional vector space of dimension d.
The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an outer product of two nonzero vectors:
A
=
v
w
T
.
{\displaystyle A=vw^{\mathrm {T} }.}
The rank of a matrix A is the smallest number of such outer products that can be summed to produce it:
A
=
v
1
w
1
T
+
⋯
+
v
k
w
k
T
.
{\displaystyle A=v_{1}w_{1}^{\mathrm {T} }+\cdots +v_{k}w_{k}^{\mathrm {T} }.}
In indices, a tensor of rank 1 is a tensor of the form
T
i
j
…
k
ℓ
…
=
a
i
b
j
⋯
c
k
d
ℓ
⋯
.
{\displaystyle T_{ij\dots }^{k\ell \dots }=a_{i}b_{j}\cdots c^{k}d^{\ell }\cdots .}
The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix, and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often very difficult to determine, and low rank decompositions of tensors are sometimes of great practical interest. In fact, the problem of finding the rank of an order 3 tensor over any finite field is NP-Complete, and over the rationals, is NP-Hard. Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of bilinear forms
z
k
=
∑
i
j
T
i
j
k
x
i
y
j
{\displaystyle z_{k}=\sum _{ij}T_{ijk}x_{i}y_{j}}
for given inputs xi and yj. If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known.
== Universal property ==
The space
T
n
m
(
V
)
{\displaystyle T_{n}^{m}(V)}
can be characterized by a universal property in terms of multilinear mappings. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for free modules, and the "universal" approach carries over more easily to more general situations.
A scalar-valued function on a Cartesian product (or direct sum) of vector spaces
f
:
V
1
×
⋯
×
V
N
→
F
{\displaystyle f:V_{1}\times \cdots \times V_{N}\to F}
is multilinear if it is linear in each argument. The space of all multilinear mappings from V1 × ... × VN to W is denoted LN(V1, ..., VN; W). When N = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from V to W is denoted L(V; W).
The universal characterization of the tensor product implies that, for each multilinear function
f
∈
L
m
+
n
(
V
∗
,
…
,
V
∗
⏟
m
,
V
,
…
,
V
⏟
n
;
W
)
{\displaystyle f\in L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};W)}
(where W can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function
T
f
∈
L
(
V
∗
⊗
⋯
⊗
V
∗
⏟
m
⊗
V
⊗
⋯
⊗
V
⏟
n
;
W
)
{\displaystyle T_{f}\in L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};W)}
such that
f
(
α
1
,
…
,
α
m
,
v
1
,
…
,
v
n
)
=
T
f
(
α
1
⊗
⋯
⊗
α
m
⊗
v
1
⊗
⋯
⊗
v
n
)
{\displaystyle f(\alpha _{1},\ldots ,\alpha _{m},v_{1},\ldots ,v_{n})=T_{f}(\alpha _{1}\otimes \cdots \otimes \alpha _{m}\otimes v_{1}\otimes \cdots \otimes v_{n})}
for all vi in V and αi in V∗.
Using the universal property, it follows, when V is finite dimensional, that the space of (m, n)-tensors admits a natural isomorphism
T
n
m
(
V
)
≅
L
(
V
∗
⊗
⋯
⊗
V
∗
⏟
m
⊗
V
⊗
⋯
⊗
V
⏟
n
;
F
)
≅
L
m
+
n
(
V
∗
,
…
,
V
∗
⏟
m
,
V
,
…
,
V
⏟
n
;
F
)
.
{\displaystyle T_{n}^{m}(V)\cong L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};F)\cong L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};F).}
Each V in the definition of the tensor corresponds to a V∗ inside the argument of the linear maps, and vice versa. (Note that in the former case, there are m copies of V and n copies of V∗, and in the latter case vice versa). In particular, one has
T
0
1
(
V
)
≅
L
(
V
∗
;
F
)
≅
V
,
T
1
0
(
V
)
≅
L
(
V
;
F
)
=
V
∗
,
T
1
1
(
V
)
≅
L
(
V
;
V
)
.
{\displaystyle {\begin{aligned}T_{0}^{1}(V)&\cong L(V^{*};F)\cong V,\\T_{1}^{0}(V)&\cong L(V;F)=V^{*},\\T_{1}^{1}(V)&\cong L(V;V).\end{aligned}}}
== Tensor fields ==
Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term tensor is sometimes used as a shorthand for tensor field. A tensor field expresses the concept of a tensor that varies from point to point on the manifold.
== References ==
Abraham, Ralph; Marsden, Jerrold E. (1985), Foundations of Mechanics (2nd ed.), Reading, Massachusetts: Addison-Wesley, ISBN 0-201-40840-6.
Bourbaki, Nicolas (1989), Elements of Mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9.
de Groote, H. F. (1987), Lectures on the Complexity of Bilinear Problems, Lecture Notes in Computer Science, vol. 245, Springer, ISBN 3-540-17205-X.
Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4.
Håstad, Johan (November 15, 1989), "Tensor Rank Is NP-Complete", Journal of Algorithms, 11 (4): 644–654, doi:10.1016/0196-6774(90)90014-6.
Jeevanjee, Nadir (2011), "An Introduction to Tensors and Group Theory for Physicists", Physics Today, 65 (4): 64, Bibcode:2012PhT....65d..64P, doi:10.1063/PT.3.1523, ISBN 978-0-8176-4714-8.
Knuth, Donald E. (1998) [1969], The Art of Computer Programming, vol. 2 (3rd ed.), Addison-Wesley, pp. 145–146, ISBN 978-0-201-89684-8.
Hackbusch, Wolfgang (2012), Tensor Spaces and Numerical Tensor Calculus, Springer, p. 4, ISBN 978-3-642-28027-6. | Wikipedia/Tensor_rank |
In mathematics, specifically linear algebra, a degenerate bilinear form f (x, y ) on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V ) given by v ↦ (x ↦ f (x, v )) is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero x in V such that
f
(
x
,
y
)
=
0
{\displaystyle f(x,y)=0\,}
for all
y
∈
V
.
{\displaystyle \,y\in V.}
== Nondegenerate forms ==
A nondegenerate or nonsingular form is a bilinear form that is not degenerate, meaning that
v
↦
(
x
↦
f
(
x
,
v
)
)
{\displaystyle v\mapsto (x\mapsto f(x,v))}
is an isomorphism, or equivalently in finite dimensions, if and only if
f
(
x
,
y
)
=
0
{\displaystyle f(x,y)=0}
for all
y
∈
V
{\displaystyle y\in V}
implies that
x
=
0
{\displaystyle x=0}
.
== Using the determinant ==
If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero – if and only if the matrix is singular, and accordingly degenerate forms are also called singular forms. Likewise, a nondegenerate form is one for which the associated matrix is non-singular, and accordingly nondegenerate forms are also referred to as non-singular forms. These statements are independent of the chosen basis.
== Related notions ==
If for a quadratic form Q there is a non-zero vector v ∈ V such that Q(v) = 0, then Q is an isotropic quadratic form. If Q has the same sign for all non-zero vectors, it is a definite quadratic form or an anisotropic quadratic form.
There is the closely related notion of a unimodular form and a perfect pairing; these agree over fields but not over general rings.
== Examples ==
The study of real, quadratic algebras shows the distinction between types of quadratic forms. The product zz* is a quadratic form for each of the complex numbers, split-complex numbers, and dual numbers. For z = x + ε y, the dual number form is x2 which is a degenerate quadratic form. The split-complex case is an isotropic form, and the complex case is a definite form.
The most important examples of nondegenerate forms are inner products and symplectic forms. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map
V
→
V
∗
{\displaystyle V\to V^{*}}
be an isomorphism, not positivity. For example, a manifold with an inner product structure on its tangent spaces is a Riemannian manifold, while relaxing this to a symmetric nondegenerate form yields a pseudo-Riemannian manifold.
== Infinite dimensions ==
Note that in an infinite-dimensional space, we can have a bilinear form ƒ for which
v
↦
(
x
↦
f
(
x
,
v
)
)
{\displaystyle v\mapsto (x\mapsto f(x,v))}
is injective but not surjective. For example, on the space of continuous functions on a closed bounded interval, the form
f
(
ϕ
,
ψ
)
=
∫
ψ
(
x
)
ϕ
(
x
)
d
x
{\displaystyle f(\phi ,\psi )=\int \psi (x)\phi (x)\,dx}
is not surjective: for instance, the Dirac delta functional is in the dual space but not of the required form. On the other hand, this bilinear form satisfies
f
(
ϕ
,
ψ
)
=
0
{\displaystyle f(\phi ,\psi )=0}
for all
ϕ
{\displaystyle \phi }
implies that
ψ
=
0.
{\displaystyle \psi =0.\,}
In such a case where ƒ satisfies injectivity (but not necessarily surjectivity), ƒ is said to be weakly nondegenerate.
== Terminology ==
If f vanishes identically on all vectors it is said to be totally degenerate. Given any bilinear form f on V the set of vectors
{
x
∈
V
∣
f
(
x
,
y
)
=
0
for all
y
∈
V
}
{\displaystyle \{x\in V\mid f(x,y)=0{\mbox{ for all }}y\in V\}}
forms a totally degenerate subspace of V. The map f is nondegenerate if and only if this subspace is trivial.
Geometrically, an isotropic line of the quadratic form corresponds to a point of the associated quadric hypersurface in projective space. Such a line is additionally isotropic for the bilinear form if and only if the corresponding point is a singularity. Hence, over an algebraically closed field, Hilbert's Nullstellensatz guarantees that the quadratic form always has isotropic lines, while the bilinear form has them if and only if the surface is singular.
== See also ==
Indefinite inner product space – generalization of Hilbert space with indefinite signaturePages displaying wikidata descriptions as a fallback
Dual system
Linear form – Linear map from a vector space to its field of scalars
== References == | Wikipedia/Degenerate_form |
A monograph is generally a long-form work on one (usually scholarly) subject, or one aspect of a subject, typically created by a single author or artist (or, sometimes, by two or more authors). Traditionally it is in written form and published as a book, but it may be an artwork, audiovisual work, or exhibition made up of visual artworks. In library cataloguing, the word has a specific and broader meaning, while in the United States, the Food and Drug Administration uses the term to mean a set of published standards.
== Written works ==
=== Academic works ===
The English term monograph is derived from modern Latin monographia, which has its root in Greek. In the English word, mono- means 'single' and -graph means 'something written'.
Unlike a textbook, which surveys the state of knowledge in a field, the main purpose of a monograph is to present primary research and original scholarship. This research is presented at length, distinguishing a monograph from an article. For these reasons, publication of a monograph is commonly regarded as vital for career progression in many academic disciplines. Intended for other researchers and bought primarily by libraries, monographs are generally published as individual volumes in a short print run. In Britain and the U.S., what differentiates a scholarly monograph from an academic trade title varies by publisher, though generally it is the assumption that the readership has not only specialised or sophisticated knowledge but also professional interest in the subject of the work.
A written monograph is usually a specialist book on one topic, although the term is sometimes used loosely, with its meaning being broadened to include any works which are not reference works and which may be written by one or more authors, or an edited collection.
This broadened use of the term, however, does not affect the essential difference in academic publishing and assessment between an authored academic book (i.e., a traditional academic monograph) and an edited volume (i.e., a non-authored book). In the case of an academic monograph, it is a "a focused work of scholarship pitched at a relatively high level of intellectual sophistication", whose author (or authors) has carried out the research and written the text of the book. By contrast, the editor of an edited volume owns the copyright to the concept, structure and organization of the book, as well as any text he or she has authored, while the authors of the individual chapters retain the copyright to the text and content of the chapters they authored.
=== Library definition ===
In library cataloguing, monograph has a broader meaning: a non-serial publication complete in one volume (book) or a definite number of books. Thus it differs from a serial or periodical publication such as a magazine, academic journal, or newspaper. In this context only, books such as novels are considered monographs.
== Types of monographs ==
=== Biology ===
In biological taxonomy, a monograph is a comprehensive treatment of a taxon in written form. Monographs typically review all known species within a group, add any newly discovered species, and collect and synthesize available information on the ecological associations, geographic distributions, and morphological variations within the group.
The first-ever monograph of a plant taxon was Robert Morison's 1672 Plantarum Umbelliferarum Distributio Nova, a treatment of the Apiaceae.
=== Art ===
Book publishers use the term "artist monograph" or "art monograph" to indicate books dealing with a single artist, as opposed to broader surveys of art subjects.
=== Film and multimedia ===
The term monograph is also used for audiovisual or film documentary-type representations of a subject, often creatively expressed. The term "monographic film" has also been used for short fiction or animated films.
Video or film essays on a single topic are also referred to as monographs.
IndyVinyl, by Scottish film academic Ian Garwood, is a monographic research project focused on "vinyl records in American independent cinema between 1987 and 2018". It includes an 8,000-word peer-reviewed academic book chapter; video compilations; "critical montages"; and a series of social media posts, all curated on a website. Garwood has written that his project is "an attempt to produce a research output equivalent to an academic monograph, but incorporating video-based forms of criticism that have been popularised through online film culture".
== FDA usage ==
In the context of Food and Drug Administration regulation, monographs represent published standards by which the use of one or more substances is automatically authorized. For example, the following is an excerpt from the Federal Register: "The Food and Drug Administration (FDA) is issuing a final rule in the form of a final monograph establishing conditions under which over-the-counter (OTC) sunscreen drug products are generally recognized as safe and effective and not misbranded as part of FDA's ongoing review of OTC drug products." Such usage has given rise to the use of the word monograph as a verb, as in "this substance has been monographed by the FDA".
== See also ==
Compendium
Compilation thesis
Documentation
Open access monograph
Treatise
== References == | Wikipedia/Monograph |
In mathematics, an algebraic equation or polynomial equation is an equation of the form
P
=
0
{\displaystyle P=0}
, where P is a polynomial with coefficients in some field, often the field of the rational numbers.
For example,
x
5
−
3
x
+
1
=
0
{\displaystyle x^{5}-3x+1=0}
is an algebraic equation with integer coefficients and
y
4
+
x
y
2
−
x
3
3
+
x
y
2
+
y
2
+
1
7
=
0
{\displaystyle y^{4}+{\frac {xy}{2}}-{\frac {x^{3}}{3}}+xy^{2}+y^{2}+{\frac {1}{7}}=0}
is a multivariate polynomial equation over the rationals.
For many authors, the term algebraic equation refers only to the univariate case, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables (the multivariate case), in which case the term polynomial equation is usually preferred.
Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations, not all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).
== Terminology ==
The term "algebraic equation" dates from the time when the main problem of algebra was to solve univariate polynomial equations. This problem was completely solved during the 19th century; see Fundamental theorem of algebra, Abel–Ruffini theorem and Galois theory.
Since then, the scope of algebra has been dramatically enlarged. In particular, it includes the study of equations that involve nth roots and, more generally, algebraic expressions. This makes the term algebraic equation ambiguous outside the context of the old problem. So the term polynomial equation is generally preferred when this ambiguity may occur, specially when considering multivariate equations.
== History ==
The study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000 BC could solve some kinds of quadratic equations (displayed on Old Babylonian clay tablets).
Univariate algebraic equations over the rationals (i.e., with rational coefficients) have a very long history. Ancient mathematicians wanted the solutions in the form of radical expressions, like
x
=
1
+
5
2
{\displaystyle x={\frac {1+{\sqrt {5}}}{2}}}
for the positive solution of
x
2
−
x
−
1
=
0
{\displaystyle x^{2}-x-1=0}
. The ancient Egyptians knew how to solve equations of degree 2 in this manner. The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. In the 9th century Muhammad ibn Musa al-Khwarizmi and other Islamic mathematicians derived the quadratic formula, the general solution of equations of degree 2, and recognized the importance of the discriminant. During the Renaissance in 1545, Gerolamo Cardano published the solution of Scipione del Ferro and Niccolò Fontana Tartaglia to equations of degree 3 and that of Lodovico Ferrari for equations of degree 4. Finally Niels Henrik Abel proved, in 1824, that equations of degree 5 and higher do not have general solutions using radicals. Galois theory, named after Évariste Galois, showed that some equations of at least degree 5 do not even have an idiosyncratic solution in radicals, and gave criteria for deciding if an equation is in fact solvable using radicals.
== Areas of study ==
The algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is the study of (univariate) algebraic equations over the rationals (that is, with rational coefficients). Galois theory was introduced by Évariste Galois to specify criteria for deciding if an algebraic equation may be solved in terms of radicals. In field theory, an algebraic extension is an extension such that every element is a root of an algebraic equation over the base field. Transcendental number theory is the study of the real numbers which are not solutions to an algebraic equation over the rationals. A Diophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. Algebraic geometry is the study of the solutions in an algebraically closed field of multivariate polynomial equations.
Two equations are equivalent if they have the same set of solutions. In particular the equation
P
=
Q
{\displaystyle P=Q}
is equivalent to
P
−
Q
=
0
{\displaystyle P-Q=0}
. It follows that the study of algebraic equations is equivalent to the study of polynomials.
A polynomial equation over the rationals can always be converted to an equivalent one in which the coefficients are integers. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the previously mentioned polynomial equation
y
4
+
x
y
2
=
x
3
3
−
x
y
2
+
y
2
−
1
7
{\displaystyle y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}
becomes
42
y
4
+
21
x
y
−
14
x
3
+
42
x
y
2
−
42
y
2
+
6
=
0.
{\displaystyle 42y^{4}+21xy-14x^{3}+42xy^{2}-42y^{2}+6=0.}
Because sine, exponentiation, and 1/T are not polynomial functions,
e
T
x
2
+
1
T
x
y
+
sin
(
T
)
z
−
2
=
0
{\displaystyle e^{T}x^{2}+{\frac {1}{T}}xy+\sin(T)z-2=0}
is not a polynomial equation in the four variables x, y, z, and T over the rational numbers. However, it is a polynomial equation in the three variables x, y, and z over the field of the elementary functions in the variable T.
== Theory ==
=== Polynomials ===
Given an equation in unknown x
(
E
)
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
=
0
{\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}
,
with coefficients in a field K, one can equivalently say that the solutions of (E) in K are the roots in K of the polynomial
P
=
a
n
X
n
+
a
n
−
1
X
n
−
1
+
⋯
+
a
1
X
+
a
0
∈
K
[
X
]
{\displaystyle P=a_{n}X^{n}+a_{n-1}X^{n-1}+\dots +a_{1}X+a_{0}\quad \in K[X]}
.
It can be shown that a polynomial of degree n in a field has at most n roots. The equation (E) therefore has at most n solutions.
If K' is a field extension of K, one may consider (E) to be an equation with coefficients in K and the solutions of (E) in K are also solutions in K' (the converse does not hold in general). It is always possible to find a field extension of K known as the rupture field of the polynomial P, in which (E) has at least one solution.
=== Existence of solutions to real and complex equations ===
The fundamental theorem of algebra states that the field of the complex numbers is closed algebraically, that is, all polynomial equations with complex coefficients and degree at least one have a solution.
It follows that all polynomial equations of degree 1 or more with real coefficients have a complex solution. On the other hand, an equation such as
x
2
+
1
=
0
{\displaystyle x^{2}+1=0}
does not have a solution in
R
{\displaystyle \mathbb {R} }
(the solutions are the imaginary units i and −i).
While the real solutions of real equations are intuitive (they are the x-coordinates of the points where the curve y = P(x) intersects the x-axis), the existence of complex solutions to real equations can be surprising and less easy to visualize.
However, a monic polynomial of odd degree must necessarily have a real root. The associated polynomial function in x is continuous, and it approaches
−
∞
{\displaystyle -\infty }
as x approaches
−
∞
{\displaystyle -\infty }
and
+
∞
{\displaystyle +\infty }
as x approaches
+
∞
{\displaystyle +\infty }
. By the intermediate value theorem, it must therefore assume the value zero at some real x, which is then a solution of the polynomial equation.
=== Connection to Galois theory ===
There exist formulas giving the solutions of real or complex polynomials of degree less than or equal to four as a function of their coefficients. Abel showed that it is not possible to find such a formula in general (using only the four arithmetic operations and taking roots) for equations of degree five or higher. Galois theory provides a criterion which allows one to determine whether the solution to a given polynomial equation can be expressed using radicals.
== Explicit solution of numerical equations ==
=== Approach ===
The explicit solution of a real or complex equation of degree 1 is trivial. Solving an equation of higher degree n reduces to factoring the associated polynomial, that is, rewriting (E) in the form
a
n
(
x
−
z
1
)
…
(
x
−
z
n
)
=
0
{\displaystyle a_{n}(x-z_{1})\dots (x-z_{n})=0}
,
where the solutions are then the
z
1
,
…
,
z
n
{\displaystyle z_{1},\dots ,z_{n}}
. The problem is then to express the
z
i
{\displaystyle z_{i}}
in terms of the
a
i
{\displaystyle a_{i}}
.
This approach applies more generally if the coefficients and solutions belong to an integral domain.
=== General techniques ===
==== Factoring ====
If an equation P(x) = 0 of degree n has a rational root α, the associated polynomial can be factored to give the form P(X) = (X − α)Q(X) (by dividing P(X) by X − α or by writing P(X) − P(α) as a linear combination of terms of the form Xk − αk, and factoring out X − α. Solving P(x) = 0 thus reduces to solving the degree n − 1 equation Q(x) = 0. See for example the case n = 3.
==== Elimination of the sub-dominant term ====
To solve an equation of degree n,
(
E
)
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
=
0
{\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}
,
a common preliminary step is to eliminate the degree-n - 1 term: by setting
x
=
y
−
a
n
−
1
n
a
n
{\displaystyle x=y-{\frac {a_{n-1}}{n\,a_{n}}}}
, equation (E) becomes
a
n
y
n
+
b
n
−
2
y
n
−
2
+
⋯
+
b
1
y
+
b
0
=
0
{\displaystyle a_{n}y^{n}+b_{n-2}y^{n-2}+\dots +b_{1}y+b_{0}=0}
.
Leonhard Euler developed this technique for the case n = 3 but it is also applicable to the case n = 4, for example.
=== Quadratic equations ===
To solve a quadratic equation of the form
a
x
2
+
b
x
+
c
=
0
{\displaystyle ax^{2}+bx+c=0}
one calculates the discriminant Δ defined by
Δ
=
b
2
−
4
a
c
{\displaystyle \Delta =b^{2}-4ac}
.
If the polynomial has real coefficients, it has:
two distinct real roots if
Δ
>
0
{\displaystyle \Delta >0}
;
one real double root if
Δ
=
0
{\displaystyle \Delta =0}
;
no real root if
Δ
<
0
{\displaystyle \Delta <0}
, but two complex conjugate roots.
=== Cubic equations ===
The best-known method for solving cubic equations, by writing roots in terms of radicals, is Cardano's formula.
=== Quartic equations ===
For detailed discussions of some solution methods see:
Tschirnhaus transformation (general method, not guaranteed to succeed);
Bezout method (general method, not guaranteed to succeed);
Ferrari method (solutions for degree 4);
Euler method (solutions for degree 4);
Lagrange method (solutions for degree 4);
Descartes method (solutions for degree 2 or 4);
A quartic equation
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
=
0
{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}
with
a
≠
0
{\displaystyle a\neq 0}
may be reduced to a quadratic equation by a change of variable provided it is either biquadratic (b = d = 0) or quasi-palindromic (e = a, d = b).
Some cubic and quartic equations can be solved using trigonometry or hyperbolic functions.
=== Higher-degree equations ===
Évariste Galois and Niels Henrik Abel showed independently that in general a polynomial of degree 5 or higher is not solvable using radicals. Some particular equations do have solutions, such as those associated with the cyclotomic polynomials of degrees 5 and 17.
Charles Hermite, on the other hand, showed that polynomials of degree 5 are solvable using elliptical functions.
Otherwise, one may find numerical approximations to the roots using root-finding algorithms, such as Newton's method.
== See also ==
Algebraic function
Algebraic number
Root finding
Linear equation (degree = 1)
Quadratic equation (degree = 2)
Cubic equation (degree = 3)
Quartic equation (degree = 4)
Quintic equation (degree = 5)
Sextic equation (degree = 6)
Septic equation (degree = 7)
System of linear equations
System of polynomial equations
Linear Diophantine equation
Linear equation over a ring
Cramer's theorem (algebraic curves), on the number of points usually sufficient to determine a bivariate n-th degree curve
== References ==
"Algebraic equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Algebraic Equation". MathWorld. | Wikipedia/Algebraic_equations |
Natural science or empirical science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and reproducibility of findings are used to try to ensure the validity of scientific advances.
Natural science can be divided into two main branches: life science and physical science. Life science is alternatively known as biology. Physical science is subdivided into branches: physics, astronomy, Earth science and chemistry. These branches of natural science may be further divided into more specialized branches (also known as fields). As empirical sciences, natural sciences use tools from the formal sciences, such as mathematics and logic, converting information about nature into measurements that can be explained as clear statements of the "laws of nature".
Modern natural science succeeded more classical approaches to natural philosophy. Galileo, Kepler, Descartes, Bacon, and Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way. Still, philosophical perspectives, conjectures, and presuppositions, often overlooked, remain necessary in natural science. Systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, animals, minerals, and so on. Today, "natural history" suggests observational descriptions aimed at popular audiences.
== Criteria ==
Philosophers of science have suggested several criteria, including Karl Popper's controversial falsifiability criterion, to help them differentiate scientific endeavors from non-scientific ones. Validity, accuracy, and quality control, such as peer review and reproducibility of findings, are amongst the most respected criteria in today's global scientific community.
In natural science, impossibility assertions come to be widely accepted as overwhelmingly probable rather than considered proven to the point of being unchallengeable. The basis for this strong acceptance is a combination of extensive evidence of something not occurring, combined with an underlying theory, very successful in making predictions, whose assumptions lead logically to the conclusion that something is impossible. While an impossibility assertion in natural science can never be proved, it could be refuted by the observation of a single counterexample. Such a counterexample would require that the assumptions underlying the theory that implied the impossibility be re-examined.
== Branches of natural science ==
=== Biology ===
This field encompasses a diverse set of disciplines that examine phenomena related to living organisms. The scale of study can range from sub-component biophysics up to complex ecologies. Biology is concerned with the characteristics, classification and behaviors of organisms, as well as how species were formed and their interactions with each other and the environment.
The biological fields of botany, zoology, and medicine date back to early periods of civilization, while microbiology was introduced in the 17th century with the invention of the microscope. However, it was not until the 19th century that biology became a unified science. Once scientists discovered commonalities between all living things, it was decided they were best studied as a whole.
Some key developments in biology were the discovery of genetics, evolution through natural selection, the germ theory of disease, and the application of the techniques of chemistry and physics at the level of the cell or organic molecule.
Modern biology is divided into subdisciplines by the type of organism and by the scale being studied. Molecular biology is the study of the fundamental chemistry of life, while cellular biology is the examination of the cell; the basic building block of all life. At a higher level, anatomy and physiology look at the internal structures, and their functions, of an organism, while ecology looks at how various organisms interrelate.
=== Earth science ===
Earth science (also known as geoscience) is an all-embracing term for the sciences related to the planet Earth, including geology, geography, geophysics, geochemistry, climatology, glaciology, hydrology, meteorology, and oceanography.
Although mining and precious stones have been human interests throughout the history of civilization, the development of the related sciences of economic geology and mineralogy did not occur until the 18th century. The study of the earth, particularly paleontology, blossomed in the 19th century. The growth of other disciplines, such as geophysics, in the 20th century led to the development of the theory of plate tectonics in the 1960s, which has had a similar effect on the Earth sciences as the theory of evolution had on biology. Earth sciences today are closely linked to petroleum and mineral resources, climate research, and to environmental assessment and remediation.
==== Atmospheric sciences ====
Although sometimes considered in conjunction with the earth sciences, due to the independent development of its concepts, techniques, and practices and also the fact of it having a wide range of sub-disciplines under its wing, atmospheric science is also considered a separate branch of natural science. This field studies the characteristics of different layers of the atmosphere from ground level to the edge of the space. The timescale of the study also varies from day to century. Sometimes, the field also includes the study of climatic patterns on planets other than Earth.
==== Oceanography ====
The serious study of oceans began in the early- to mid-20th century. As a field of natural science, it is relatively young, but stand-alone programs offer specializations in the subject. Though some controversies remain as to the categorization of the field under earth sciences, interdisciplinary sciences, or as a separate field in its own right, most modern workers in the field agree that it has matured to a state that it has its own paradigms and practices.
==== Planetary science ====
Planetary science or planetology, is the scientific study of planets, which include terrestrial planets like the Earth, and other types of planets, such as gas giants and ice giants. Planetary science also concerns other celestial bodies, such as dwarf planets moons, asteroids, and comets. This largely includes the Solar System, but recently has started to expand to exoplanets, particularly terrestrial exoplanets. It explores various objects, spanning from micrometeoroids to gas giants, to establish their composition, movements, genesis, interrelation, and past. Planetary science is an interdisciplinary domain, having originated from astronomy and Earth science, and currently encompassing a multitude of areas, such as planetary geology, cosmochemistry, atmospheric science, physics, oceanography, hydrology, theoretical planetology, glaciology, and exoplanetology. Related fields encompass space physics, which delves into the impact of the Sun on the bodies in the Solar System, and astrobiology.
Planetary science comprises interconnected observational and theoretical branches. Observational research entails a combination of space exploration, primarily through robotic spacecraft missions utilizing remote sensing, and comparative experimental work conducted in Earth-based laboratories. The theoretical aspect involves extensive mathematical modelling and computer simulation.
Typically, planetary scientists are situated within astronomy and physics or Earth sciences departments in universities or research centers. However, there are also dedicated planetary science institutes worldwide. Generally, individuals pursuing a career in planetary science undergo graduate-level studies in one of the Earth sciences, astronomy, astrophysics, geophysics, or physics. They then focus their research within the discipline of planetary science. Major conferences are held annually, and numerous peer reviewed journals cater to the diverse research interests in planetary science. Some planetary scientists are employed by private research centers and frequently engage in collaborative research initiatives.
=== Chemistry ===
Constituting the scientific study of matter at the atomic and molecular scale, chemistry deals primarily with collections of atoms, such as gases, molecules, crystals, and metals. The composition, statistical properties, transformations, and reactions of these materials are studied. Chemistry also involves understanding the properties and interactions of individual atoms and molecules for use in larger-scale applications.
Most chemical processes can be studied directly in a laboratory, using a series of (often well-tested) techniques for manipulating materials, as well as an understanding of the underlying processes. Chemistry is often called "the central science" because of its role in connecting the other natural sciences.
Early experiments in chemistry had their roots in the system of alchemy, a set of beliefs combining mysticism with physical experiments. The science of chemistry began to develop with the work of Robert Boyle, the discoverer of gases, and Antoine Lavoisier, who developed the theory of the conservation of mass.
The discovery of the chemical elements and atomic theory began to systematize this science, and researchers developed a fundamental understanding of states of matter, ions, chemical bonds and chemical reactions. The success of this science led to a complementary chemical industry that now plays a significant role in the world economy.
=== Physics ===
Physics embodies the study of the fundamental constituents of the universe, the forces and interactions they exert on one another, and the results produced by these interactions. Physics is generally regarded as foundational because all other natural sciences use and obey the field's principles and laws. Physics relies heavily on mathematics as the logical framework for formulating and quantifying principles.
The study of the principles of the universe has a long history and largely derives from direct observation and experimentation. The formulation of theories about the governing laws of the universe has been central to the study of physics from very early on, with philosophy gradually yielding to systematic, quantitative experimental testing and observation as the source of verification. Key historical developments in physics include Isaac Newton's theory of universal gravitation and classical mechanics, an understanding of electricity and its relation to magnetism, Einstein's theories of special and general relativity, the development of thermodynamics, and the quantum mechanical model of atomic and subatomic physics.
The field of physics is vast and can include such diverse studies as quantum mechanics and theoretical physics, applied physics and optics. Modern physics is becoming increasingly specialized, where researchers tend to focus on a particular area rather than being "universalists" like Isaac Newton, Albert Einstein, and Lev Landau, who worked in multiple areas.
=== Astronomy ===
Astronomy is a natural science that studies celestial objects and phenomena. Objects of interest include planets, moons, stars, nebulae, galaxies, and comets. Astronomy is the study of everything in the universe beyond Earth's atmosphere, including objects we can see with our naked eyes. It is one of the oldest sciences.
Astronomers of early civilizations performed methodical observations of the night sky, and astronomical artifacts have been found from much earlier periods. There are two types of astronomy: observational astronomy and theoretical astronomy. Observational astronomy is focused on acquiring and analyzing data, mainly using basic principles of physics. In contrast, Theoretical astronomy is oriented towards developing computer or analytical models to describe astronomical objects and phenomena.
This discipline is the science of celestial objects and phenomena that originate outside the Earth's atmosphere. It is concerned with the evolution, physics, chemistry, meteorology, geology, and motion of celestial objects, as well as the formation and development of the universe.
Astronomy includes examining, studying, and modeling stars, planets, and comets. Most of the information used by astronomers is gathered by remote observation. However, some laboratory reproduction of celestial phenomena has been performed (such as the molecular chemistry of the interstellar medium). There is considerable overlap with physics and in some areas of earth science. There are also interdisciplinary fields such as astrophysics, planetary sciences, and cosmology, along with allied disciplines such as space physics and astrochemistry.
While the study of celestial features and phenomena can be traced back to antiquity, the scientific methodology of this field began to develop in the middle of the 17th century. A key factor was Galileo's introduction of the telescope to examine the night sky in more detail.
The mathematical treatment of astronomy began with Newton's development of celestial mechanics and the laws of gravitation. However, it was triggered by earlier work of astronomers such as Kepler. By the 19th century, astronomy had developed into formal science, with the introduction of instruments such as the spectroscope and photography, along with much-improved telescopes and the creation of professional observatories.
== Interdisciplinary studies ==
The distinctions between the natural science disciplines are not always sharp, and they share many cross-discipline fields. Physics plays a significant role in the other natural sciences, as represented by astrophysics, geophysics, chemical physics and biophysics. Likewise chemistry is represented by such fields as biochemistry, physical chemistry, geochemistry and astrochemistry.
A particular example of a scientific discipline that draws upon multiple natural sciences is environmental science. This field studies the interactions of physical, chemical, geological, and biological components of the environment, with particular regard to the effect of human activities and the impact on biodiversity and sustainability. This science also draws upon expertise from other fields, such as economics, law, and social sciences.
A comparable discipline is oceanography, as it draws upon a similar breadth of scientific disciplines. Oceanography is sub-categorized into more specialized cross-disciplines, such as physical oceanography and marine biology. As the marine ecosystem is vast and diverse, marine biology is further divided into many subfields, including specializations in particular species.
There is also a subset of cross-disciplinary fields with strong currents that run counter to specialization by the nature of the problems they address. Put another way: In some fields of integrative application, specialists in more than one field are a key part of most scientific discourse. Such integrative fields, for example, include nanoscience, astrobiology, and complex system informatics.
=== Materials science ===
Materials science is a relatively new, interdisciplinary field that deals with the study of matter and its properties and the discovery and design of new materials. Originally developed through the field of metallurgy, the study of the properties of materials and solids has now expanded into all materials. The field covers the chemistry, physics, and engineering applications of materials, including metals, ceramics, artificial polymers, and many others. The field's core deals with relating the structure of materials with their properties.
Materials science is at the forefront of research in science and engineering. It is an essential part of forensic engineering (the investigation of materials, products, structures, or components that fail or do not operate or function as intended, causing personal injury or damage to property) and failure analysis, the latter being the key to understanding, for example, the cause of various aviation accidents. Many of the most pressing scientific problems that are faced today are due to the limitations of the materials that are available, and, as a result, breakthroughs in this field are likely to have a significant impact on the future of technology.
The basis of materials science involves studying the structure of materials and relating them to their properties. Understanding this structure-property correlation, material scientists can then go on to study the relative performance of a material in a particular application. The major determinants of the structure of a material and, thus, of its properties are its constituent chemical elements and how it has been processed into its final form. These characteristics, taken together and related through the laws of thermodynamics and kinetics, govern a material's microstructure and thus its properties.
== History ==
Some scholars trace the origins of natural science as far back as pre-literate human societies, where understanding the natural world was necessary for survival. People observed and built up knowledge about the behavior of animals and the usefulness of plants as food and medicine, which was passed down from generation to generation. These primitive understandings gave way to more formalized inquiry around 3500 to 3000 BC in the Mesopotamian and Ancient Egyptian cultures, which produced the first known written evidence of natural philosophy, the precursor of natural science. While the writings show an interest in astronomy, mathematics, and other aspects of the physical world, the ultimate aim of inquiry about nature's workings was, in all cases, religious or mythological, not scientific.
A tradition of scientific inquiry also emerged in Ancient China, where Taoist alchemists and philosophers experimented with elixirs to extend life and cure ailments. They focused on the yin and yang, or contrasting elements in nature; the yin was associated with femininity and coldness, while yang was associated with masculinity and warmth. The five phases – fire, earth, metal, wood, and water – described a cycle of transformations in nature. The water turned into wood, which turned into the fire when it burned. The ashes left by fire were earth. Using these principles, Chinese philosophers and doctors explored human anatomy, characterizing organs as predominantly yin or yang, and understood the relationship between the pulse, the heart, and the flow of blood in the body centuries before it became accepted in the West.
Little evidence survives of how Ancient Indian cultures around the Indus River understood nature, but some of their perspectives may be reflected in the Vedas, a set of sacred Hindu texts. They reveal a conception of the universe as ever-expanding and constantly being recycled and reformed. Surgeons in the Ayurvedic tradition saw health and illness as a combination of three humors: wind, bile and phlegm. A healthy life resulted from a balance among these humors. In Ayurvedic thought, the body consisted of five elements: earth, water, fire, wind, and space. Ayurvedic surgeons performed complex surgeries and developed a detailed understanding of human anatomy.
Pre-Socratic philosophers in Ancient Greek culture brought natural philosophy a step closer to direct inquiry about cause and effect in nature between 600 and 400 BC. However, an element of magic and mythology remained. Natural phenomena such as earthquakes and eclipses were explained increasingly in the context of nature itself instead of being attributed to angry gods. Thales of Miletus, an early philosopher who lived from 625 to 546 BC, explained earthquakes by theorizing that the world floated on water and that water was the fundamental element in nature. In the 5th century BC, Leucippus was an early exponent of atomism, the idea that the world is made up of fundamental indivisible particles. Pythagoras applied Greek innovations in mathematics to astronomy and suggested that the earth was spherical.
=== Aristotelian natural philosophy (400 BC–1100 AD) ===
Later Socratic and Platonic thought focused on ethics, morals, and art and did not attempt an investigation of the physical world; Plato criticized pre-Socratic thinkers as materialists and anti-religionists. Aristotle, however, a student of Plato who lived from 384 to 322 BC, paid closer attention to the natural world in his philosophy. In his History of Animals, he described the inner workings of 110 species, including the stingray, catfish and bee. He investigated chick embryos by breaking open eggs and observing them at various stages of development. Aristotle's works were influential through the 16th century, and he is considered to be the father of biology for his pioneering work in that science. He also presented philosophies about physics, nature, and astronomy using inductive reasoning in his works Physics and Meteorology.
While Aristotle considered natural philosophy more seriously than his predecessors, he approached it as a theoretical branch of science. Still, inspired by his work, Ancient Roman philosophers of the early 1st century AD, including Lucretius, Seneca and Pliny the Elder, wrote treatises that dealt with the rules of the natural world in varying degrees of depth. Many Ancient Roman Neoplatonists of the 3rd to the 6th centuries also adapted Aristotle's teachings on the physical world to a philosophy that emphasized spiritualism. Early medieval philosophers including Macrobius, Calcidius and Martianus Capella also examined the physical world, largely from a cosmological and cosmographical perspective, putting forth theories on the arrangement of celestial bodies and the heavens, which were posited as being composed of aether.
Aristotle's works on natural philosophy continued to be translated and studied amid the rise of the Byzantine Empire and Abbasid Caliphate.
In the Byzantine Empire, John Philoponus, an Alexandrian Aristotelian commentator and Christian theologian, was the first to question Aristotle's physics teaching. Unlike Aristotle, who based his physics on verbal argument, Philoponus instead relied on observation and argued for observation rather than resorting to a verbal argument. He introduced the theory of impetus. John Philoponus' criticism of Aristotelian principles of physics served as inspiration for Galileo Galilei during the Scientific Revolution.
A revival in mathematics and science took place during the time of the Abbasid Caliphate from the 9th century onward, when Muslim scholars expanded upon Greek and Indian natural philosophy. The words alcohol, algebra and zenith all have Arabic roots.
=== Medieval natural philosophy (1100–1600) ===
Aristotle's works and other Greek natural philosophy did not reach the West until about the middle of the 12th century, when works were translated from Greek and Arabic into Latin. The development of European civilization later in the Middle Ages brought with it further advances in natural philosophy. European inventions such as the horseshoe, horse collar and crop rotation allowed for rapid population growth, eventually giving way to urbanization and the foundation of schools connected to monasteries and cathedrals in modern-day France and England. Aided by the schools, an approach to Christian theology developed that sought to answer questions about nature and other subjects using logic. This approach, however, was seen by some detractors as heresy.
By the 12th century, Western European scholars and philosophers came into contact with a body of knowledge of which they had previously been ignorant: a large corpus of works in Greek and Arabic that were preserved by Islamic scholars. Through translation into Latin, Western Europe was introduced to Aristotle and his natural philosophy. These works were taught at new universities in Paris and Oxford by the early 13th century, although the practice was frowned upon by the Catholic church. A 1210 decree from the Synod of Paris ordered that "no lectures are to be held in Paris either publicly or privately using Aristotle's books on natural philosophy or the commentaries, and we forbid all this under pain of ex-communication."
In the late Middle Ages, Spanish philosopher Dominicus Gundissalinus translated a treatise by the earlier Persian scholar Al-Farabi called On the Sciences into Latin, calling the study of the mechanics of nature Scientia naturalis, or natural science. Gundissalinus also proposed his classification of the natural sciences in his 1150 work On the Division of Philosophy. This was the first detailed classification of the sciences based on Greek and Arab philosophy to reach Western Europe. Gundissalinus defined natural science as "the science considering only things unabstracted and with motion," as opposed to mathematics and sciences that rely on mathematics. Following Al-Farabi, he separated the sciences into eight parts, including: physics, cosmology, meteorology, minerals science, and plant and animal science.
Later, philosophers made their own classifications of the natural sciences. Robert Kilwardby wrote On the Order of the Sciences in the 13th century that classed medicine as a mechanical science, along with agriculture, hunting, and theater, while defining natural science as the science that deals with bodies in motion. Roger Bacon, an English friar and philosopher, wrote that natural science dealt with "a principle of motion and rest, as in the parts of the elements of fire, air, earth, and water, and in all inanimate things made from them." These sciences also covered plants, animals and celestial bodies.
Later in the 13th century, a Catholic priest and theologian Thomas Aquinas defined natural science as dealing with "mobile beings" and "things which depend on a matter not only for their existence but also for their definition." There was broad agreement among scholars in medieval times that natural science was about bodies in motion. However, there was division about including fields such as medicine, music, and perspective. Philosophers pondered questions including the existence of a vacuum, whether motion could produce heat, the colors of rainbows, the motion of the earth, whether elemental chemicals exist, and where in the atmosphere rain is formed.
In the centuries up through the end of the Middle Ages, natural science was often mingled with philosophies about magic and the occult. Natural philosophy appeared in various forms, from treatises to encyclopedias to commentaries on Aristotle. The interaction between natural philosophy and Christianity was complex during this period; some early theologians, including Tatian and Eusebius, considered natural philosophy an outcropping of pagan Greek science and were suspicious of it. Although some later Christian philosophers, including Aquinas, came to see natural science as a means of interpreting scripture, this suspicion persisted until the 12th and 13th centuries. The Condemnation of 1277, which forbade setting philosophy on a level equal with theology and the debate of religious constructs in a scientific context, showed the persistence with which Catholic leaders resisted the development of natural philosophy even from a theological perspective. Aquinas and Albertus Magnus, another Catholic theologian of the era, sought to distance theology from science in their works. "I don't see what one's interpretation of Aristotle has to do with the teaching of the faith," he wrote in 1271.
=== Newton and the scientific revolution (1600–1800) ===
By the 16th and 17th centuries, natural philosophy evolved beyond commentary on Aristotle as more early Greek philosophy was uncovered and translated. The invention of the printing press in the 15th century, the invention of the microscope and telescope, and the Protestant Reformation fundamentally altered the social context in which scientific inquiry evolved in the West. Christopher Columbus's discovery of a new world changed perceptions about the physical makeup of the world, while observations by Copernicus, Tyco Brahe and Galileo brought a more accurate picture of the solar system as heliocentric and proved many of Aristotle's theories about the heavenly bodies false. Several 17th-century philosophers, including René Descartes, Pierre Gassendi, Marin Mersenne, Nicolas Malebranche, Thomas Hobbes, John Locke and Francis Bacon, made a break from the past by rejecting Aristotle and his medieval followers outright, calling their approach to natural philosophy superficial.
The titles of Galileo's work Two New Sciences and Johannes Kepler's New Astronomy underscored the atmosphere of change that took hold in the 17th century as Aristotle was dismissed in favor of novel methods of inquiry into the natural world. Bacon was instrumental in popularizing this change; he argued that people should use the arts and sciences to gain dominion over nature. To achieve this, he wrote that "human life [must] be endowed with discoveries and powers." He defined natural philosophy as "the knowledge of Causes and secret motions of things; and enlarging the bounds of Human Empire, to the effecting of all things possible." Bacon proposed that scientific inquiry be supported by the state and fed by the collaborative research of scientists, a vision that was unprecedented in its scope, ambition, and forms at the time.
Natural philosophers came to view nature increasingly as a mechanism that could be taken apart and understood, much like a complex clock. Natural philosophers including Isaac Newton, Evangelista Torricelli and Francesco Redi, Edme Mariotte, Jean-Baptiste Denis and Jacques Rohault conducted experiments focusing on the flow of water, measuring atmospheric pressure using a barometer and disproving spontaneous generation. Scientific societies and scientific journals emerged and were spread widely through the printing press, touching off the scientific revolution. Newton in 1687 published his The Mathematical Principles of Natural Philosophy, or Principia Mathematica, which set the groundwork for physical laws that remained current until the 19th century.
Some modern scholars, including Andrew Cunningham, Perry Williams, and Floris Cohen, argue that natural philosophy is not properly called science and that genuine scientific inquiry began only with the scientific revolution. According to Cohen, "the emancipation of science from an overarching entity called 'natural philosophy is one defining characteristic of the Scientific Revolution." Other historians of science, including Edward Grant, contend that the scientific revolution that blossomed in the 17th, 18th, and 19th centuries occurred when principles learned in the exact sciences of optics, mechanics, and astronomy began to be applied to questions raised by natural philosophy. Grant argues that Newton attempted to expose the mathematical basis of nature – the immutable rules it obeyed – and, in doing so, joined natural philosophy and mathematics for the first time, producing an early work of modern physics.
The scientific revolution, which began to take hold in the 17th century, represented a sharp break from Aristotelian modes of inquiry. One of its principal advances was the use of the scientific method to investigate nature. Data was collected, and repeatable measurements were made in experiments. Scientists then formed hypotheses to explain the results of these experiments. The hypothesis was then tested using the principle of falsifiability to prove or disprove its accuracy. The natural sciences continued to be called natural philosophy, but the adoption of the scientific method took science beyond the realm of philosophical conjecture and introduced a more structured way of examining nature.
Newton, an English mathematician and physicist, was a seminal figure in the scientific revolution. Drawing on advances made in astronomy by Copernicus, Brahe, and Kepler, Newton derived the universal law of gravitation and laws of motion. These laws applied both on earth and in outer space, uniting two spheres of the physical world previously thought to function independently, according to separate physical rules. Newton, for example, showed that the tides were caused by the gravitational pull of the moon. Another of Newton's advances was to make mathematics a powerful explanatory tool for natural phenomena. While natural philosophers had long used mathematics as a means of measurement and analysis, its principles were not used as a means of understanding cause and effect in nature until Newton.
In the 18th century and 19th century, scientists including Charles-Augustin de Coulomb, Alessandro Volta, and Michael Faraday built upon Newtonian mechanics by exploring electromagnetism, or the interplay of forces with positive and negative charges on electrically charged particles. Faraday proposed that forces in nature operated in "fields" that filled space. The idea of fields contrasted with the Newtonian construct of gravitation as simply "action at a distance", or the attraction of objects with nothing in the space between them to intervene. James Clerk Maxwell in the 19th century unified these discoveries in a coherent theory of electrodynamics. Using mathematical equations and experimentation, Maxwell discovered that space was filled with charged particles that could act upon each other and were a medium for transmitting charged waves.
Significant advances in chemistry also took place during the scientific revolution. Antoine Lavoisier, a French chemist, refuted the phlogiston theory, which posited that things burned by releasing "phlogiston" into the air. Joseph Priestley had discovered oxygen in the 18th century, but Lavoisier discovered that combustion was the result of oxidation. He also constructed a table of 33 elements and invented modern chemical nomenclature. Formal biological science remained in its infancy in the 18th century, when the focus lay upon the classification and categorization of natural life. This growth in natural history was led by Carl Linnaeus, whose 1735 taxonomy of the natural world is still in use. Linnaeus, in the 1750s, introduced scientific names for all his species.
=== 19th-century developments (1800–1900) ===
By the 19th century, the study of science had come into the purview of professionals and institutions. In so doing, it gradually acquired the more modern name of natural science. The term scientist was coined by William Whewell in an 1834 review of Mary Somerville's On the Connexion of the Sciences. But the word did not enter general use until nearly the end of the same century.
=== Modern natural science (1900–present) ===
According to a famous 1923 textbook, Thermodynamics and the Free Energy of Chemical Substances, by the American chemist Gilbert N. Lewis and the American physical chemist Merle Randall, the natural sciences contain three great branches:
Aside from the logical and mathematical sciences, there are three great branches of natural science which stand apart by reason of the variety of far reaching deductions drawn from a small number of primary postulates — they are mechanics, electrodynamics, and thermodynamics.
Today, natural sciences are more commonly divided into life sciences, such as botany and zoology, and physical sciences, which include physics, chemistry, astronomy, and Earth sciences.
== See also ==
Branches of science
Empiricism
List of academic disciplines and sub-disciplines
Logology (science)
Natural history
Natural Sciences (Cambridge), for the Tripos at the University of Cambridge
== References ==
=== Bibliography ===
== Further reading ==
Defining Natural Sciences Ledoux, S. F., 2002: Defining Natural Sciences, Behaviorology Today, 5(1), 34–36.
Stokes, Donald E. (1997). Pasteur's Quadrant: Basic Science and Technological Innovation. Revised and translated by Albert V. Carozzi and Marguerite Carozzi. Washington, D.C.: Brookings Institution Press. ISBN 978-0-8157-8177-6.
The History of Recent Science and Technology
Natural Sciences Contains updated information on research in the Natural Sciences including biology, geography and the applied life and earth sciences.
Reviews of Books About Natural Science This site contains over 50 previously published reviews of books about natural science, plus selected essays on timely topics in natural science.
Scientific Grant Awards Database Contains details of over 2,000,000 scientific research projects conducted over the past 25 years.
E!Science Up-to-date science news aggregator from major sources including universities. | Wikipedia/Natural_sciences |
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. It is then called a simple algebra over this field.
Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple.
Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring over a field does not have any nontrivial two-sided ideals (since any ideal of
M
n
(
R
)
{\displaystyle M_{n}(R)}
is of the form
M
n
(
I
)
{\displaystyle M_{n}(I)}
with
I
{\displaystyle I}
an ideal of
R
{\displaystyle R}
), but it has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns).
An immediate example of a simple ring is a division ring, where every nonzero element has a multiplicative inverse, for instance, the quaternions. Also, for any
n
≥
1
{\displaystyle n\geq 1}
, the algebra of
n
×
n
{\displaystyle n\times n}
matrices with entries in a division ring is simple.
Joseph Wedderburn proved that if a ring
R
{\displaystyle R}
is a finite-dimensional simple algebra over a field
k
{\displaystyle k}
, it is isomorphic to a matrix algebra over some division algebra over
k
{\displaystyle k}
. In particular, the only simple rings that are finite-dimensional algebras over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions.
Wedderburn proved these results in 1907 in his doctoral thesis, On hypercomplex numbers, which appeared in the Proceedings of the London Mathematical Society. His thesis classified finite-dimensional simple and also semisimple algebras over fields. Simple algebras are building blocks of semisimple algebras: any finite-dimensional semisimple algebra is a Cartesian product, in the sense of algebras, of finite-dimensional simple algebras.
One must be careful of the terminology: not every simple ring is a semisimple ring, and not every simple algebra is a semisimple algebra. However, every finite-dimensional simple algebra is a semisimple algebra, and every simple ring that is left- or right-artinian is a semisimple ring.
Wedderburn's result was later generalized to semisimple rings in the Wedderburn–Artin theorem: this says that every semisimple ring is a finite product of matrix rings over division rings. As a consequence of this generalization, every simple ring that is left- or right-artinian is a matrix ring over a division ring.
== Examples ==
Let
R
{\displaystyle \mathbb {R} }
be the field of real numbers,
C
{\displaystyle \mathbb {C} }
be the field of complex numbers, and
H
{\displaystyle \mathbb {H} }
the quaternions.
A central simple algebra (sometimes called a Brauer algebra) is a simple finite-dimensional algebra over a field
F
{\displaystyle F}
whose center is
F
{\displaystyle F}
.
Every finite-dimensional simple algebra over
R
{\displaystyle \mathbb {R} }
is isomorphic to an algebra of
n
×
n
{\displaystyle n\times n}
matrices with entries in
R
{\displaystyle \mathbb {R} }
,
C
{\displaystyle \mathbb {C} }
, or
H
{\displaystyle \mathbb {H} }
. Every central simple algebra over
R
{\displaystyle \mathbb {R} }
is isomorphic to an algebra of
n
×
n
{\displaystyle n\times n}
matrices with entries
R
{\displaystyle \mathbb {R} }
or
H
{\displaystyle \mathbb {H} }
. These results follow from the Frobenius theorem.
Every finite-dimensional simple algebra over
C
{\displaystyle \mathbb {C} }
is a central simple algebra, and is isomorphic to a matrix ring over
C
{\displaystyle \mathbb {C} }
.
Every finite-dimensional central simple algebra over a finite field is isomorphic to a matrix ring over that field.
Over a field of characteristic zero, the Weyl algebra is simple but not semisimple, and in particular not a matrix algebra over a division algebra over its center; the Weyl algebra is infinite-dimensional, so Wedderburn's theorem does not apply to it.
== See also ==
Simple (algebra)
Simple algebra (universal algebra)
== References ==
Albert, A. A. (2003). Structure of Algebras. Colloquium publications. Vol. 24. American Mathematical Society. p. 37. ISBN 0-8218-1024-3.
Bourbaki, Nicolas (2012), Algèbre Ch. 8 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-35315-7
Nicholson, William K. (1993). "A short proof of the Wedderburn-Artin theorem" (PDF). New Zealand J. Math. 22: 83–86.
Henderson, D. W. (1965). "A short proof of Wedderburn's theorem". Amer. Math. Monthly. 72: 385–386. doi:10.2307/2313499.
Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4419-8616-0, ISBN 978-0-387-95325-0, MR 1838439
Lang, Serge (2002), Algebra (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0387953854
Jacobson, Nathan (1989), Basic Algebra II (2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5 | Wikipedia/Simple_algebra |
Abstrakt Algebra was a Swedish experimental metal band with influences from power metal and doom metal. It was founded by bassist Leif Edling in 1994, shortly after his main project Candlemass split up.
They made one album, but Edling had already started working on a second album with a different line-up. However, due to the commercial failure of Abstrakt Algebra, Edling reformed Candlemass while taking with him some of the ideas for that second album, as well as drummer Jejo Perkovic. There, materialised on the Dactylis Glomerata album. And as such Abstrakt Algebra was over. That second album, called Abstrakt Algebra II, was later included as a bonus disc in the 2006 re-release of Dactylis Glomerata.
Mats Levén later appeared as the singer of the band Krux, another band by Edling, which has a similar take on the experimentation Edling started with Abstrakt Algebra.
== Members ==
== Discography ==
Abstrakt Algebra (1995)
Abstrakt Algebra II (2008)
== References == | Wikipedia/Abstrakt_Algebra |
In quantum field theory, a force carrier is a type of particle that gives rise to forces between other particles. They serve as the quanta of a particular kind of physical field. Force carriers are also known as messenger particles, intermediate particles, or exchange particles.
== Particle and field viewpoints ==
Quantum field theories describe nature in terms of fields. Each field has a complementary description as the set of particles of a particular type. A force between two particles can be described either as the action of a force field generated by one particle on the other, or in terms of the exchange of virtual force-carrier particles between them.
The energy of a wave in a field (for example, an electromagnetic wave in the electromagnetic field) is quantized, and the quantum excitations of the field can be interpreted as particles. The Standard Model contains the following force-carrier particles, each of which is an excitation of a particular force field:
Gluons, excitations of the strong gauge field.
Photons, W bosons, and Z bosons, excitations of the electroweak gauge fields.
Higgs bosons, excitations of one component of the Higgs field, which gives mass to fundamental particles.
In addition, composite particles such as mesons, as well as quasiparticles, can be described as excitations of an effective field.
Gravity is not a part of the Standard Model, but it is thought that there may be particles called gravitons which are the excitations of gravitational waves. The status of this particle is still tentative, because the theory is incomplete and because the interactions of single gravitons may be too weak to be detected.
== Forces from the particle viewpoint ==
When one particle scatters off another, altering its trajectory, there are two ways to think about the process. In the field picture, we imagine that the field generated by one particle caused a force on the other. Alternatively, we can imagine one particle emitting a virtual particle which is absorbed by the other. The virtual particle transfers momentum from one particle to the other. This particle viewpoint is especially helpful when there are a large number of complicated quantum corrections to the calculation since these corrections can be visualized as Feynman diagrams containing additional virtual particles.
Another example involving virtual particles is beta decay where a virtual W boson is emitted by a nucleon and then decays to e± and (anti)neutrino.
The description of forces in terms of virtual particles is limited by the applicability of the perturbation theory from which it is derived. In certain situations, such as low-energy QCD and the description of bound states, perturbation theory breaks down.
== History ==
The concept of messenger particles dates back to the 18th century when the French physicist Charles Coulomb showed that the electrostatic force between electrically charged objects follows a law similar to Newton's Law of Gravitation. In time, this relationship became known as Coulomb's law. By 1862, Hermann von Helmholtz had described a ray of light as the "quickest of all the messengers". In 1905, Albert Einstein proposed the existence of a light-particle in answer to the question: "what are light quanta?"
In 1923, at the Washington University in St. Louis, Arthur Holly Compton demonstrated an effect now known as Compton scattering. This effect is only explainable if light can behave as a stream of particles, and it convinced the physics community of the existence of Einstein's light-particle. Lastly, in 1926, one year before the theory of quantum mechanics was published, Gilbert N. Lewis introduced the term "photon", which later became the name for Einstein's light particle. From there, the concept of messenger particles developed further, notably to massive force carriers (e.g. for the Yukawa potential).
== See also ==
Virtual particle
Fundamental interaction
Exciton
== References == | Wikipedia/Force_carrier |
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