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Theorem 4. In the uniform market sharing game, a pure-strategy NE can be found after at most \( {m}^{2}n \) steps, where \( m \) is the total number of markets and \( n \) is the number of players. | The idea behind proving Theorem 4 is to iteratively add and exchange markets to the action sets of players until a pure-strategy NE is achieved. This process is based on several rounds. The first stage of each round corresponds to adding only one new market to the action set of one player. The round then continues by a... | Yes |
Theorem 5. The PoA in the market sharing game is bounded above by 2 . | In fact, the result of Theorem 5 can be viewed as a special case of PoA bounds for so-called valid utility games (Vetta 2002). Valid utility games are a class of games where the utility functions satisfy certain conditions with sub-modular social welfare function. \( {}^{6} \) In general it is known that the PoA of val... | No |
Theorem 9. Let \( {d}_{m} \) denote the minimum degree of the graph in the CSR game with \( n \) players. Then, the least best response dynamics will converge to a pure-strategy NE in no longer than \( T \) steps, where\n\n\[ T = \\left\\{ \\begin{array}{ll} {n}^{2} & \\text{ if }{d}_{m} \\geq \\left| \\mathcal{O}\\rig... | The idea of establishing the above bounds is to show that once players start to play their best responses based on Algorithm 1, those who have radii smaller than the current updating player will still play their best responses. As a result, after at most every constant number of stages, one more player will play his be... | No |
Theorem 10. Given a real number \( \epsilon > 1 \), the \( \epsilon \) -best response algorithm terminates after at most \( \mathcal{O}\left( {{n}^{2}{D}^{{\log }_{\epsilon }n}}\right) \) steps with an allocation profile \( \widehat{a} \) which is an \( \epsilon \) - approximation of a pure-strategy \( {NE} \), i.e., a... | Using the equivalence between reduction in cost and increase in the radius, and by the definition of the \( \epsilon \) -best response dynamics, it is not hard to see that if the \( \epsilon \) - best response algorithm terminates, it must end at an action profile where no player can further increase its cost by a fact... | Yes |
Theorem 12. Given a graph \( \mathcal{G} \) and \( m \geq 2 \) players, deciding whether the deterministic diffusion game admits a pure-strategy \( {NE} \) on \( \mathcal{G} \) is \( {NP} \) complete. | The idea behind establishing the result of Theorem 12 is to reduce the existence of a pure-strategy NE in diffusion game to a solution of the 3-partitioning problem which is known to be an NP-complete problem. In the 3-partitioning problem, we are given integers \( {\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{3m} \) ... | Yes |
Before we get into analysis of the dynamics of the iterative coordination-based diffusion game, we first note that the single-stage game is an exact potential game. This is because for a given state of the game \( \mathbf{a} \), if we define\n\n\[ \n{W}_{AA}\left( \mathbf{a}\right) \mathrel{\text{:=}} \mathop{\sum }\li... | Now in order to study the action adoption dynamics over the course of the time \( t = 1,2,\ldots \), let us assume that each player updates his action at random times following a Poisson arrival process. Hence, without loss of generality, we may assume that each player on the average updates once per unit time. These u... | Yes |
Theorem 18. Starting from a tree network, the SG game is a generalized ordinal potential game. In particular, any sequence of better responses by players will converge after at most \( O\left( {n}^{3}\right) \) steps to a pure-strategy \( {NE} \) . | First, we note that starting from a tree \( T \) as the initial network, after each swap action by a player, the resulting network will still be a tree. Now, let us consider an arbitrary player \( i \) and assume that he swaps one of his edges from \( \{ i, j\} \) to \( \{ i, k\} \) . Let \( {T}^{\prime } \) and \( {T}... | Yes |
Theorem 1. A set of controls \( \left\{ {\left\lbrack {{\psi }_{1}^{\alpha }\left( {t, x}\right) ,{\psi }_{2}^{\alpha }\left( {t, x}\right) ,\ldots ,{\psi }_{n}^{\alpha }\left( {t, x}\right) }\right\rbrack \text{, for}t \in \left\lbrack {0, T}\right\rbrack }\right\} \) provides an optimal solution to the control proble... | Proof. The result follows directly from dynamic programming.\n\nSubstituting \( \left\lbrack {{\psi }_{1}^{\alpha }\left( {t, x}\right) ,{\psi }_{2}^{\alpha }\left( {t, x}\right) ,\ldots ,{\psi }_{n}^{\alpha }\left( {t, x}\right) }\right\rbrack \) for \( t \in \left\lbrack {0, T}\right\rbrack \) into (14.2) yields the ... | Yes |
Theorem 2. If there exist continuously differentiable functions \( {W}^{\left( \alpha \right) i}\left( {t, x}\right) : \left\lbrack {0, T}\right\rbrack \times {R}^{m} \rightarrow R \), for \( i \in N \), satisfying \[ - {W}_{t}^{\left( \alpha \right) i}\left( {t, x}\right) = {g}^{i}\left\lbrack {t, x,{\psi }_{1}^{\alph... | ## Proof. See Yeung (2004). | No |
Proposition 1. The value function of the non cooperative payoff of player \( i \) in the game (14.13) and (14.14) is:\n\n\[ \n{V}^{i}\left( {t, x}\right) = {e}^{-{rt}}\left\lbrack {{A}_{i}\left( t\right) x + {B}_{i}\left( t\right) }\right\rbrack ,\text{ for }i \in \{ 1,2\} \text{ and }t \in \left\lbrack {\tau, T}\right... | Proof. Upon substitution of \( {\phi }_{i}^{ * }\left( {t, x}\right) \) from (14.16) into (14.15) yields a set of partial differential equations. One can readily verify that (14.17) is a solution to the set of equations (14.15). | Yes |
Corollary 1. A set of controls \( \left\{ {\left\lbrack {{\psi }_{1}^{\alpha }\left( {t, x}\right) ,{\psi }_{2}^{\alpha }\left( {t, x}\right) }\right\rbrack \text{, for}t \in \left\lbrack {0, T}\right\rbrack }\right. \) \} provides an optimal solution to the control problem (14.13) and (14.18), if there exists a contin... | Performing the indicated maximization in Corollary 1 yields:\n\n\[ {\psi }_{1}^{\alpha }\left( {t, x}\right) = \frac{\left\lbrack {{\alpha }^{1}{h}_{1} - {W}_{x}^{\alpha }\left( {t, x}\right) {e}^{rt}}\right\rbrack x}{2{\alpha }^{2}{c}_{1}},\text{ and } \]\n\n\[ {\psi }_{2}^{\alpha }\left( {t, x}\right) = \frac{\left\l... | Yes |
Proposition 2. The maximized value function of the optimal control problem (14.13) and (14.18) is:\n\n\\[ \n{W}^{\alpha }\\left( {t, x}\\right) = \\exp \\left\\lbrack {-r\\left( {t - {t}_{0}}\\right) }\\right\\rbrack \\left\\lbrack {{A}^{\alpha }\\left( t\\right) x + {B}^{\alpha }\\left( t\\right) }\\right\\rbrack ,\n\... | Proof. Upon substitution of \\( {\\psi }_{1}^{\alpha }\\left( {t, x}\\right) \\) and \\( {\\psi }_{2}^{\alpha }\\left( {t, x}\\right) \\) from (14.20) into (14.19) yields a partial differential equation. One can readily verify that (14.21) is a solution to equation (14.19).\n\nSubstituting the partial derivatives \\( {... | Yes |
Proposition 3. The function \( {W}^{\left( \alpha \right) 1}\left( {t, x}\right) \) satisfying (14.25) and (14.26) can be solved as:\n\n\[ \n{W}^{\left( \alpha \right) 1}\left( {t, x}\right) = {e}^{-{rt}}\left\lbrack {{A}_{1}^{\alpha }\left( t\right) x + {B}_{1}^{\alpha }\left( t\right) }\right\rbrack , \n\] \n\n(14.27... | Proof. Upon calculating the derivatives \( {W}_{t}^{\left( \alpha \right) 1}\left( {t, x}\right) \) and \( {W}_{x}^{\left( \alpha \right) 1}\left( {t, x}\right) \) from (14.27) and then substituting them into (14.25)-(14.26) yield Proposition 3. | Yes |
Proposition 4. The function \( {W}^{\left( \alpha \right) 2}\left( {t, x}\right) \) can be solved as:\n\n\[ \n{W}^{\left( \alpha \right) 2}\left( {t, x}\right) = {e}^{-{rt}}\left\lbrack {{A}_{2}^{\alpha }\left( t\right) x + {B}_{2}^{\alpha }\left( t\right) }\right\rbrack ,\n\]\n\n(14.28)\n\nwhere \( {A}_{2}^{\alpha }\l... | Proof. The proof follows that of Proposition 3. | No |
Theorem 3. A set of strategies \( \left\{ {{\psi }_{k}^{\left( \alpha \right) i}\left( x\right) }\right. \), for \( \left. {k \in \kappa \text{and}i \in N}\right\} \) provides an optimal solution to the problem (14.51) and (14.53) if there exist functions \( {W}^{\left( \alpha \right) }\left( {k, x}\right) \), for \( k... | Proof. The conditions in (14.54)-(14.55) follow directly from dynamic programming. | No |
Theorem 4. The payoff of player \( i \) at stage \( k \) can be characterized as the value function \( {W}^{\left( \alpha \right) i}\left( {k, x}\right) \) satisfying the following recursive system of equations: | \[ {W}^{\left( \alpha \right) i}\left( {T + 1, x}\right) = {q}^{i}\left( {x}_{T + 1}\right) ,\] \[ {W}^{\left( \alpha \right) i}\left( {k, x}\right) = {g}_{k}^{i}\left\lbrack {{x}_{k}^{\left( \alpha \right) },{\psi }_{k}^{\left( \alpha \right) 1}\left( x\right) ,{\psi }_{k}^{\left( \alpha \right) 2}\left( x\right) ,\ld... | Yes |
Theorem 5. A set of payoff weights \( \left\{ {{\widehat{\alpha }}_{k} = \left( {{\widehat{\alpha }}_{k}^{1},{\widehat{\alpha }}_{k}^{2},\ldots ,{\widehat{\alpha }}_{k}^{n}}\right) \text{, for}k \in \kappa }\right\} \) and a set of strategies \( \left\{ {{\psi }_{k}^{\left( {\widehat{\alpha }}_{k}\right) i}\left( x\rig... | Proof. See the exposition from equation (14.62) to equation (14.80) in Sects. 6.2.1 and 6.2.2. | No |
Proposition 5. The value function which represents the payoff of agent \( i \) can be obtained as:\n\n\[ \n{V}^{i}\left( {t, x}\right) = \left\lbrack {{A}_{t}^{i}x + {C}_{t}^{i}}\right\rbrack {\left( 1 + r\right) }^{-\left( {t - 1}\right) }, \]\n\n(14.89)\n\nfor \( i \in \{ 1,2\} \) and \( t \in \{ 1,2,3\} \), where\n\... | Proof. Using (14.88) and (14.89) to evaluate the system (14.86) and (14.87) yields the results in (14.89) and (14.90). | Yes |
Proposition 6. The value function can be obtained as:\n\n\[ \n{W}^{\left( {\alpha }_{3}\right) }\left( {3, x}\right) = \left\lbrack {{A}_{3}^{\left( {\alpha }_{3}\right) }x + {C}_{3}^{\left( {\alpha }_{3}\right) }}\right\rbrack {\left( 1 + r\right) }^{-2}, \]\n\n(14.93)\n\nwhere\n\n\[ \n{A}_{3}^{\left( {\alpha }_{3}\ri... | Proof. Substituting the cooperative strategies from (14.92) into (14.91) yields the function \( {W}^{\left( {\alpha }_{3}\right) }\left( {3, x}\right) \) in (14.93). | Yes |
Proposition 7. The value function \( {W}^{\left( {\alpha }_{3}\right) i}\left( {3, x}\right) \) in (14.95) can be obtained as:\n\n\[ \n{W}^{\left( {\alpha }_{3}\right) i}\left( {3, x}\right) = \left\lbrack {{A}_{3}^{\left( {\alpha }_{3}\right) i}x + {C}_{3}^{\left( {\alpha }_{3}\right) i}}\right\rbrack {\left( 1 + r\ri... | Proof. The right-hand side of the equation (14.95) is a linear function with coefficients \( {A}_{3}^{\left( {\alpha }_{3}\right) i} \) and \( {C}_{3}^{\left( {\alpha }_{3}\right) i} \) in (14.97). Hence Proposition 7 follows. | Yes |
Consider a two-period economy with preferences given by\n\n\[ u\left( {c}_{1}\right) + \delta \left( {u\left( {c}_{2}\right) + {\gamma u}\left( g\right) }\right) \]\n\nwith linear production, full depreciation, and initial capital \( {k}_{0} > 0 \) . Then the economy’s resource constraint is \( {c}_{1} + k = {k}_{0} \)... | \[ {u}^{\prime }\left( {{k}_{0} - k}\right) = {u}^{\prime }\left( {k - g}\right) = {u}^{\prime }\left( g\right) ,\]\n\nwhich gives \( {2g} = k = \frac{2}{3}{k}_{0} \) with \( {c}_{1} = {c}_{2} = g = \frac{1}{3}{k}_{0} \) . | Yes |
Lemma 1. If \( W \) is self-generating, then \( W \subset {V}^{ * } \) . | Self-generation is an extension of the basic principle of optimality from dynamic programming. Let \( W \) be some self-generating set. Then, if \( w \in W \), by self-generation, \( w \in B\left( W\right) \) . Consequently, we may find a sequence of functions \( {\left( {v}^{t}\right) }_{t = 1}^{\infty } \) such that ... | Yes |
Theorem 1. Suppose that Assumption 1 holds. Then, (i) \( \mathop{\bigcap }\limits_{{t = 1}}^{\infty }{B}^{t}\left( V\right) \neq \varnothing \) , (ii) \( \mathop{\bigcap }\limits_{{t = 1}}^{\infty }{B}^{t}\left( V\right) \) is the greatest fixed point of \( B \) , (iii) \( \mathop{\bigcap }\limits_{{t = 1}}^{\infty }{B... | To see (i) of the aforementioned theorem, observe that \( V \) is a nonempty compact set. By Assumption 1 (ii),(iv) and (v), we may conclude that \( B\left( V\right) \) is nonempty compact set and consequently that each \( {B}^{t}\left( V\right) \) is nonempty and compact. Obviously, \( B \) is an increasing operator, ... | Yes |
How sensitive are a firm's profits to deviations from equilibrium advertising strategies? | After having characterized an OLNE, goodwill dynamics were estimated using data for a prescription drug manufactured by two firms in the US pharmaceutical industry. Then estimates of equilibrium advertising paths can be made. Numerical simulations suggested, for a quite wide range of parameter values, that profits are ... | Yes |
Example 5. Breton et al. (1997) studied a Stackelberg differential game. Let the leader and follower be represented by subscripts \( l \) and \( f \), respectively, such that \( i \in \{ l, f\} \) . The dynamics are (cf. Dockner and Jørgensen 1988a)\n\n\[ \n{\\dot{Y}}_{i}\\left( t\\right) = \\left\\lbrack {{\\alpha }_{... | Cost experience effects are included and the end result is that the model is quite complex. An FSE is not analytically tractable, but numerical simulations suggest that prices decline over time and the firm with the lowest cost charges the lowest price. The shape of price trajectories seems to be driven mainly by cost ... | No |
In problems where an incumbent firm faces potential entry of competitors, the latter should be viewed as rational opponents and not as exogenous decision-makers who implement predetermined actions. | A first attempt to do this in a dynamic setup was Lieber and Barnea (1977) in which an incumbent firm sets the price of its product while competitors invest in productive capacity. The authors viewed, however, the problem as an optimal control problem of the incumbent, supposing that potential entrants believe that the... | No |
Jørgensen and Zaccour (2003a) considered a game where aggregate retailer promotions have negative effects on goodwill, cf. Jørgensen et al. (2003), but in contrast to the latter reference, retailer promotions now have carry-over effects. Retailers - who are symmetric - determine local promotional efforts, while the man... | Technically, to find an equilibrium of the incentive game, retailers determine their optimal promotion rates under the belief that the manufacturer will act in accordance with (20.4). This provides \( P\left( {\phi \left( t\right) }\right) \) and the manufacturer selects the incentive coefficient \( \phi \left( t\right... | No |
In Jørgensen et al. (2006) a manufacturer is a Stackelberg leader who wishes to induce a retailer to increase her local advertising. This can be accomplished by implementing a promotion allowance scheme such that the manufacturer pays an amount, denoted \( D\left( P\right) \), per unit of retailer promotional effort \(... | \[ D\left( P\right) \left( t\right) = {\theta P}\left( t\right) \] where \( \theta \) is a constant to be determined by the manufacturer before the start of the game. Two scenarios were considered. In the first, the manufacturer wishes the retailer to behave in accordance with the full cooperation outcome. In the secon... | Yes |
In Jørgensen and Zaccour (2003b) a manufacturer controls the transfer price \( {p}_{M}\left( t\right) \) and national advertising effort \( {a}_{M}\left( t\right) \), while the retailer controls the consumer price \( {p}_{R}\left( t\right) \) and local advertising effort \( {a}_{R}\left( t\right) \). As we have seen, a... | \[ {\gamma }_{M}\left( {a}_{R}\right) \left( t\right) = {a}_{M}^{d}\left( t\right) + {\mu }_{M}\left( t\right) \left\lbrack {{a}_{R}\left( t\right) - {a}_{R}^{d}\left( t\right) }\right\rbrack \] \[ {\gamma }_{R}\left( {a}_{M}\right) \left( t\right) = {a}_{R}^{d}\left( t\right) + {\mu }_{R}\left( t\right) \left\lbrack {... | Yes |
Example 14. Sigué (2002) studied a system with a franchisor and two franchisees. The former takes care of national advertising efforts, while franchisees are in charge of local advertising and service provision. Franchisees benefit from a high level of system goodwill, built up through national advertising and local se... | It turns out that cooperation among franchisees leads to higher local advertising efforts (which is expected). If franchisees do not cooperate, free riding will lead to an outcome in which less service than desirable is provided. | No |
Who should do promotional and brand-image advertising, respectively, if franchisor and franchisees wish to maximize their individual profits? | The problem was cast as a two-stage game. First, the franchisor selects one out of three different advertising arrangements, each characterized by its degree of centralization of advertising decisions. Second, given the franchisor's choice, franchisees decide if they should cooperate or not. | No |
A seminal paper by Thépot (1983) studied pricing, advertising goodwill creation, and productive capacity expansion. Capacity dynamics follow the standard capital accumulation equations\n\n\[ \n{\\dot{K}}_{i}\\left( t\\right) = {u}_{i}\\left( t\\right) - \\delta {K}_{i}\\left( t\\right) \n\]\n\nwhere \( {K}_{i}\\left( t... | The author identified a series of advertising-investment-pricing regimes that may occur as OLNE. | No |
In Mukhopadhyay and Kouvelis (1997), firm \( i \) controls the rate of change of the design quality, denoted \( {u}_{i}\left( t\right) \), of its product as well as the product price \( {p}_{i}\left( t\right) \) . Sales dynamics are given by a modified V-W model: | \[ {\dot{S}}_{i}\left( t\right) = {\alpha }_{i}\left\lbrack {m\left( t\right) - {S}_{i}\left( t\right) }\right\rbrack - {\delta }_{i}{S}_{i}\left( t\right) + {\gamma }_{i}\dot{m}\left( t\right) \] in which \( m\left( t\right) \) is the market potential and the term \( {\gamma }_{i}\dot{m}\left( t\right) \) is the fract... | No |
Theorem 1. Consider game \( \left( {\mathcal{P}, x, u}\right) \) with individuals’ utilities defined by (21.8) and social graph \( \mathcal{G} \) satisfying Assumption 2. If there are no stubborn individuals, i.e., \( {K}_{i} = 0 \) for every \( {\mathcal{P}}_{i} \in \mathcal{P} \), then the best response dynamics (21.... | \[ \mathop{\lim }\limits_{{t \rightarrow \infty }}{x}_{i}\left( t\right) = \frac{1}{2\left| \mathcal{E}\right| }\mathop{\sum }\limits_{{j = 1}}^{n}{d}_{j}{x}_{j}\left( 0\right) ,\forall {\mathcal{P}}_{i} \in \mathcal{P}. \] (21.13) According to Theorem 1, each individual's impact on the agreement equilibrium is proport... | Yes |
Theorem 2. Consider game \( \left( {\mathcal{P}, x, u}\right) \) with individuals’ utilities defined as (21.8) and social graph \( \mathcal{G} \) satisfying Assumption 2. If there are no stubborn individuals, i.e., \( {K}_{i} = 0 \) for every \( {\mathcal{P}}_{i} \in \mathcal{P} \), under best response strategies we ha... | Based on Theorem 2, one can now conclude that:\n\n\[ \left( {\frac{1}{1 - {\rho }_{2}} - 1}\right) \log \left( \frac{\parallel e\left( 0\right) {\parallel }_{\pi }}{v}\right) \leq \tau \left( v\right) \leq \frac{1}{1 - {\rho }_{2}}\log \left( \frac{\parallel e\left( 0\right) {\parallel }_{\pi }}{v}\right) . \]\n\nIn pa... | Yes |
Theorem 3. Consider game \( \left( {\mathcal{P}, x, u}\right) \) with individuals’ utilities defined as in (21.8) and let social graph \( \mathcal{G} \) be connected. If there is at least one stubborn individual, the best response dynamics (21.10) asymptotically converges to the following equilibrium: | One proves Theorem 3 by simply taking (21.27) into account and using relation \( \mathop{\sum }\limits_{{s = 0}}^{\infty }{A}^{s} = {\left( I - A\right) }^{-1} \) . Notice that \( \mathop{\sum }\limits_{{s = 0}}^{\infty }{A}^{s} \) converges since each eigenvalue \( {\lambda }_{i} \) of \( A \) satisfies \( \begin{Vmat... | Yes |
Lemma 2. For the asynchronous game \( \left( {\mathcal{P}, x, u}\right) \), where the action set for every player is \( {\mathbb{R}}^{d} \) and utilities are defined as in (21.40), under strategy (21.38), if player \( i \) is selected to update at time \( t \geq 0 \), we have:\n\n\[ \phi \left( {{x}_{i}\left( {t + 1}\r... | \[ \mathbb{E}\{ \phi \left( {x\left( {t + 1}\right) }\right) - \phi \left( {x\left( t\right) }\right) \}\n\n\geq \frac{2}{n}\mathop{\sum }\limits_{{j = 1}}^{n}{n}_{i}\mathbb{E}\left\{ {\begin{Vmatrix}{x}_{i}\left( t + 1\right) - {x}_{i}\left( t\right) \end{Vmatrix}}^{2}\right\}\n\n\geq \frac{2}{n}\mathop{\sum }\limits_... | No |
Theorem 5. For the asynchronous game \( \left( {\mathcal{P}, x, u}\right) \), where the action set for each player is \( {\mathbb{R}}^{d} \) and utilities are defined as in (21.40), under strategy (21.38), we have:\n\n(i) The game converges to a Nash equilibrium, where the set of all Nash equilibria of the game is char... | To summarize, in this subsection, by employing game theory, we have concluded that for the given asynchronous HK dynamics (21.40), \( \mathop{\lim }\limits_{{t \rightarrow \infty }}x\left( t\right) \) exists, which means that clustering occurs. The steady-state opinions of those in the same cluster are equal by definit... | Yes |
Theorem 6 (Young 2001). In a potential game where players adopt the SAP (21.48) as their strategies, stationary distribution \( \mu : \mathcal{A} \rightarrow \left\lbrack {0,1}\right\rbrack \) of the joint action profiles is given by:\n\n\[ \mu \left( x\right) = \frac{\exp \{ {\beta \phi }\left( x\right) \} }{\mathop{\... | In other words, as \( t \rightarrow \infty, x\left( t\right) \) approaches \( x \) with probability \( \mu \left( x\right) \) for any \( x \in \mathcal{A} \) . Consequently, if \( \beta \) is sufficiently large, \( \mu \left( x\right) \) is arbitrarily small for those joint action profiles which do not maximize the pot... | Yes |
Theorem 7. In a potential game where players adopt the RSAP (21.50) and (21.51) as their strategies, stationary distribution \( \mu : \mathcal{A} \rightarrow \left\lbrack {0,1}\right\rbrack \) of the joint action profiles is given by (21.49). | Hence, similar to the discussion on the SAP, given \( \mathcal{G} \) is connected and \( \beta \) is sufficiently large, an agreement is achieved with an arbitrarily high probability. | No |
Theorem 1 Under the class of HQSF latency functions, \( \operatorname{NCF}\left( {N, r,\alpha }\right) \) is an optimal Stackelberg strategy for the Stackelberg instance \( \left( {N, r,\alpha }\right) \) . | We give a proof of Theorem 1 in Sect. 4. We will also show that for the class of HQSF latency functions, the best Nash equilibria can be computed in polynomial time in the size \( N \) of the network, and as a consequence, the NCF strategy can also be computed in polynomial time. This stands in contrast to previous res... | No |
Proposition 2 Let \( \left( {x, m}\right) \in \mathrm{{NE}}\left( {N, r}\right) \) be a Nash equilibrium. Then \( k \in \operatorname{supp}\left( x\right) \Rightarrow \) \( \forall n < k \), link \( n \) is congested. | Proof By contradiction, if \( {m}_{n} = 0 \), then \( {\ell }_{n}\left( {{x}_{n},{m}_{n}}\right) = {a}_{n} < {a}_{k} \leq {\ell }_{k}\left( {{x}_{k},{m}_{k}}\right) \) , which contradicts Definition 2 of a Nash equilibrium. | Yes |
Corollary 1 (Support of a Nash equilibrium) Let \( \left( {\mathbf{x},\mathbf{m}}\right) \in \mathrm{{NE}}\left( {N, r}\right) \) be a Nash equilibrium and \( k = \max \operatorname{supp}\left( \mathbf{x}\right) \) be the last link in the support of \( \mathbf{x} \) (i.e., the one with the largest free-flow latency). T... | Proof Since \( k \in \operatorname{supp}\left( \mathbf{x}\right) \), we have by Proposition 2 that \( \forall n < k \), link \( n \) is congested, thus \( n \in \operatorname{supp}\left( \mathbf{x}\right) \) (by definition, a congested link cannot be empty). | Yes |
Proposition 3 (Single-link-free-flow equilibria) \( \left( {x, m}\right) \) is a single-link-free-flow equilibrium if and only if \( \exists k \in \{ 1,\ldots, N\} \) such that \( 0 < r - \mathop{\sum }\limits_{{n = 1}}^{{k - 1}}{\widehat{x}}_{n}\left( k\right) \leq {x}_{k}^{\max } \) , and | \[ \mathbf{x} \triangleq \left( {{\widehat{x}}_{1}\left( k\right) ,\ldots ,{\widehat{x}}_{k - 1}\left( k\right), r - \mathop{\sum }\limits_{{n = 1}}^{{k - 1}}{\widehat{x}}_{n}\left( k\right) ,0,\ldots ,0}\right) \] (25.9) \[ \mathbf{m} \triangleq \left( \begin{matrix} \overset{k}{\mathbf{h}} \\ 1,\ldots ,1,0,\ldots ,0 ... | Yes |
Corollary 2 The maximum demand \( r \) such that the set of Nash equilibria \( \mathrm{{NE}}\left( {N, r}\right) \) is nonempty is \( {r}^{\mathrm{{NE}}}\left( N\right) \) . | Proof By the previous Lemma, \( {r}^{\mathrm{{NE}}}\left( N\right) \) is a lower bound on the maximum demand. To show that it is also an upper bound, suppose that \( \operatorname{NE}\left( {N, r}\right) \) is nonempty and let \( \left( {\mathbf{x},\mathbf{m}}\right) \in \operatorname{NE}\left( {N, r}\right) \) and \( ... | Yes |
Proposition 4 (An upper bound on the number of equilibria) Consider a routing game instance \( \left( {N, r}\right) \) . For any given \( k \in \{ 1,\ldots, N\} \), there is at most one single-link-free-flow equilibrium and one congested equilibrium with support \( \{ 1,\ldots, k\} \) . As a consequence, by Corollary 1... | Proof We prove the result for single-link-free-flow equilibria, the proof for congested equilibria is similar. Let \( k \in \{ 1,\ldots, N\} \), and assume \( \left( {\mathbf{x},\mathbf{m}}\right) \) and \( \left( {{\mathbf{x}}^{\prime },{\mathbf{m}}^{\prime }}\right) \) are single-link-free-flow equilibria such that \... | Yes |
Lemma 2 (Best Nash Equilibrium) For a routing game instance \( \left( {N, r}\right), r \leq \) \( {r}^{\mathrm{{NE}}}\left( N\right) \), the unique best Nash equilibrium is the single-link-free-flow equilibrium that has the smallest support | \[ \operatorname{BNE}\left( {N, r}\right) = \underset{\left( {\mathbf{x},\mathbf{m}}\right) \in {\mathrm{{NE}}}_{\mathrm{f}}\left( {N, r}\right) }{\arg \min }\{ \max \operatorname{supp}\left( \mathbf{x}\right) \} . \] Proof We first show that a congested equilibrium cannot be a best Nash equilibrium. Let \( \left( {\ma... | Yes |
Lemma 3 Let \( k = \max \operatorname{supp}\left( t\right) \) and \( \bar{k} = \max \operatorname{supp}\left( \bar{t}\right) \) . Then \( k \geq \bar{k} \) . | Proof We first note that \( \left( {s + t\left( s\right), m}\right) \) restricted to supp \( \left( {t\left( s\right) }\right) \) is a Nash equilibrium. Then since link \( k \) is in free-flow, we have \( {\ell }_{k}\left( {{s}_{k} + {t}_{k}\left( s\right) ,{m}_{k}}\right) = {a}_{k} \), and since \( k \in \operatorname... | Yes |
Lemma 4 Under \( \left( {\mathbf{x},\mathbf{m}}\right) \), the links \( \{ 1,\ldots, l - 1\} \) have greater (or equal) latency and hold less (or equal) flow than under \( \left( {\bar{x},\bar{m}}\right) \), i.e., \( \forall n \in \{ 1,\ldots, l - 1\} ,{\ell }_{n}\left( {{x}_{n},{m}_{n}}\right) \geq \) \( {\ell }_{n}\l... | Proof Since \( k \in \operatorname{supp}\left( t\right) \), we have by definition of a Stackelberg strategy and its induced equilibrium that \( \forall n \in \{ 1,\ldots, k - 1\} ,{\ell }_{n}\left( {{x}_{n},{m}_{n}}\right) \geq {\ell }_{k}\left( {{x}_{k},{m}_{k}}\right) \geq {a}_{k} \), see Eq. (25.5). We also have by ... | Yes |
Lemma 5 Consider a Stackelberg strategy \( s \) of the form \( s = \bar{s} + \varepsilon \), where\n\n\[ \mathbf{\varepsilon } = \left( {{\varepsilon }_{1},{\varepsilon }_{2},\ldots ,{\varepsilon }_{\bar{k} - 1}, - \mathop{\sum }\limits_{{n = 1}}^{{\bar{k} - 1}}{\varepsilon }_{n},0,\ldots ,0}\right) \]\n\n(25.18)\n\nan... | Proof We show that \( s = \bar{s} + \varepsilon \) is a feasible assignment of the compliant flow \( {\alpha r} \) and that the induced equilibrium of the followers is \( \left( {\mathbf{t}\left( s\right) ,\mathbf{m}\left( s\right) }\right) = \left( {\overline{\mathbf{t}} - \mathbf{\varepsilon },\overline{\mathbf{m}}}\... | Yes |
Proposition 5 \( \left( {{x}^{ \star },{m}^{ \star }}\right) \) is optimal for (SO) only if \( \forall n \in \{ 1,\ldots, N\} ,{m}_{n}^{ \star } = 0 \) . | Proof This follows immediately from the fact the latency on a link in congestion is always greater than the latency of the link in free-flow \( {\ell }_{n}\left( {{x}_{n},1}\right) > {\ell }_{n}\left( {{x}_{n},0}\right) \) \( \forall {x}_{n} \in \left( {0,{x}_{n}^{\max }}\right) \) . | Yes |
Theorem 3. There exists an \( \operatorname{MFE}\left( {\rho ,{\widehat{\theta }}_{\rho }}\right) \) such that\n\n\[ \rho \left( x\right) = \gamma \left( x\right) \triangleq {\Pi }_{\rho }\left( {{\widehat{\theta }}_{\rho }^{-1}\left\lbrack {0, x}\right\rbrack }\right) ,\forall x \in {\mathbb{R}}^{ + }.\] | In order to prove the above theorem, we need to show that \( \mathcal{F} \) has a fixed point, i.e., \( \mathcal{F}\left( \rho \right) = \rho \) . | No |
Theorem 4 (Schauder Fixed Point Theorem). Suppose \( \mathcal{F}\left( \mathcal{P}\right) \subset \mathcal{P} \) . If \( \mathcal{F}\left( \cdot \right) \) is continuous and \( \mathcal{F}\left( \mathcal{P}\right) \) is contained in a convex and compact subset of \( \mathcal{P} \), then \( \mathcal{F}\left( \cdot \righ... | We will show that the mapping \( \mathcal{F} \) satisfies the conditions of the above theorem, and hence it has a fixed point. Note that \( \mathcal{P} \) is a convex set. Therefore, we only need to verify the other two conditions.\n\nTo prove the continuity of mapping \( \mathcal{F} \), we first show that \( {\theta }... | No |
For any bid distribution \( \rho \in \mathcal{P} \) and for any stationary policy \( \theta \in \Theta \) , the Markov chain described by the transition probabilities given in (26.82) has a unique invariant distribution \( {\Pi }_{\rho ,\theta }\left( \cdot \right) \) . Also \( {\Pi }_{\rho ,\theta } \) and \( {\Upsilo... | Proof. \( {\Upsilon }_{\rho ,\theta }^{\left( k\right) }\left( {B \mid q}\right) \) is the queue length distribution assuming no regeneration has happened yet, and the regeneration event occurs with probability \( \beta \) independently of the rest of the system. It is then easy to find \( {\Pi }_{\rho ,\theta }\left( ... | Yes |
Theorem 6. The mapping \( {\Pi }^{ * } : \mathcal{P} \mapsto \Omega \) is continuous. | Proof. By Portmanteau theorem, (Billingsley 2009), we only need to show that for any sequence \( {\rho }_{n} \rightarrow \rho \) in \( w \) -norm and any open set \( B \) , \( \mathop{\liminf }\limits_{{n \rightarrow \infty }}{\Pi }_{{\rho }_{n}}\left( B\right) \geq \) \( {\Pi }_{\rho }\left( B\right) \) . By Fatou’s l... | Yes |
Lemma 5. For any \( \rho \in \mathcal{P} \), let \( \gamma \left( w\right) = \left( {\mathcal{F}\left( \rho \right) }\right) \left( w\right) = {\Pi }_{\rho }\left( {{\widehat{\theta }}_{\rho }^{-1}\left( \left\lbrack {0, w}\right\rbrack \right) }\right), w \in {\mathbb{R}}^{ + } \) . Then, \( \gamma \in \mathcal{P} \) ... | Proof. From the definition of \( {\Pi }_{\rho } \), it is easy to see that \( \gamma \) is a distribution function. Since \( {\widehat{\theta }}_{\rho } \) is continuous and strictly increasing function as shown in Lemma 2, \( {\widehat{\theta }}_{\rho }^{-1}\left( {\{ w\} }\right) \) is either empty or a singleton. Th... | Yes |
Theorem 7. The mapping \( \mathcal{F} : \mathcal{P} \mapsto \mathcal{P} \) given by \( \left( {\mathcal{F}\left( \rho \right) }\right) \left( w\right) = {\Pi }_{\rho }\left( {{\widehat{\theta }}_{\rho }^{-1}\left( \left\lbrack {0, w}\right\rbrack \right) }\right) \) is continuous. | Proof. Let \( {\rho }_{n} \rightarrow \rho \) in uniform norm. From previous steps, we have \( {\widehat{\theta }}_{{\rho }_{n}} \rightarrow {\widehat{\theta }}_{\rho } \) in \( \mu \) -norm and \( {\Pi }_{{\rho }_{n}} \Rightarrow {\Pi }_{\rho } \) . Then, using Theorem 5.5 of Billingsley (2009), one can show that the ... | Yes |
Lemma 6. For any interval \( \left\lbrack {a, b}\right\rbrack ,{\Pi }_{\rho }\left( \left\lbrack {a, b}\right\rbrack \right) < c \cdot \left( {b - a}\right) \), for some large enough \( c \) . | Proof. The proof follows easily from our characterization of \( {\Pi }_{\rho } \) in terms of \( {\Upsilon }_{\rho }^{\left( k\right) }.▱ | No |
Lemma 7. \( \mathcal{F}\left( \mathcal{P}\right) \) is uniformly tight, i.e., for any \( \epsilon > 0 \) and any \( f \in \mathcal{F}\left( \mathcal{P}\right) \), there exists an \( {x}_{\epsilon } \in \mathbb{R} \) such that \( 1 - \epsilon \leq f\left( {x}_{\epsilon }\right) \leq 1 \) . | Proof. From Lemma 5, we have \( \mathcal{F}\left( \mathcal{P}\right) \subseteq \mathcal{P} \) . Hence, the expectation of the bid distributions in \( \mathcal{F}\left( \mathcal{P}\right) \) is bounded uniformly. An application of Markov inequality will give uniform tightness. | No |
Theorem 1.2 (Debreu 1952; Glicksberg 1952; Fan 1952) Consider a strategic-form game whose strategy spaces \( {S}_{i} \) are nonempty compact convex subsets of an Euclidean space. If the payoff functions \( {u}_{i} \) are continuous in \( s \) and quasi-concave in \( {s}_{i} \), there exists a pure-strategy Nash equilib... | Proof The proof is very similar to that of Nash's theorem: We verify that continuous payoffs imply nonempty, closed-graph reaction correspondences, and that quasi-concavity in players' own actions implies that the reaction correspondences are convex-valued. | No |
Theorem 1.3 (Glicksberg 1952) Consider a strategic-form game whose strategy spaces \( {S}_{i} \) are nonempty compact subsets of a metric space. If the payoff functions \( {u}_{i} \) are continuous then there exists a Nash equilibrium in mixed strategies. | Here the mixed strategies are the (Borel) probability measures over the pure strategies, which we endow with the topology of weak convergence. \( {}^{23} \) Once more, the proof applies a fixed-point theorem to the reaction correspondences. As we remarked above, the introduction of mixed strategies again makes the stra... | Yes |
Theorem 2.1 (Bernheim 1984; Pearce 1984) The set of rationalizable strategies is nonempty and contains at least one pure strategy for each player. Further, each \( {\sigma }_{i} \in {R}_{i} \) is (in \( {\sum }_{i} \) ) a best response to an element of \( { \times }_{j \neq i} \) convex hull \( \left( {R}_{j}\right) \)... | Sketch of Proof The proof shows inductively that the \( {\widetilde{\sum }}_{i}^{n} \) in the definition of rationalizability are closed, nonempty, and nested and that they contain a pure strategy. Their infinite intersection is thus nonempty and contains a pure strategy. The existence of an element of \( { \times }_{j... | No |
Theorem 2.2 (Pearce 1984) Rationalizability and iterated strict dominance coincide in two-player games. | Proof Let \( {S}^{n} \) denote the set of pure strategies remaining after \( n \) rounds of the deletion of strictly dominated strategies, let \( {\sum }^{n} \) be the corresponding mixed strategies, and let \( {\widetilde{\mathbf{\sum }}}^{n} \) be the set of mixed strategies that survive \( n \) rounds of the iterati... | Yes |
Theorem 3.2 (Zermelo 1913; Kuhn 1953) A finite game of perfect information has a pure-strategy Nash equilibrium. | The proof of this theorem constructs the equilibrium strategies using \ | No |
Theorem 4.1 (one-stage-deviation principle for finite-horizon games) In a finite multi-stage game with observed actions, strategy profile \( s \) is subgame perfect if and only if it satisfies the one-stage-deviation condition that no player \( i \) can gain by deviating from \( s \) in a single stage and conforming to... | Proof The necessity of the one-stage-deviation condition (\ | No |
Theorem 4.2 (one-stage deviation principle for infinite-horizon games) In an infinite-horizon multi-stage game with observed actions that is continuous at infinity, profile \( s \) is subgame perfect if and only if there is no player \( i \) and strategy \( {\widehat{s}}_{i} \) that agrees with \( {s}_{i} \) except at ... | Proof The proof of the last theorem establishes necessity, and also shows that if \( s \) satisfies the one-stage-deviation condition then it cannot be improved by any finite sequence of deviations in any subgame. Suppose to the contrary that \( s \) were not subgame perfect. Then there would be a stage \( t \) and a h... | Yes |
Theorem 4.3 In a finite- or infinite-horizon game of perfect information, no subgame-perfect strategy profile is removed by iterated conditional dominance. | ## Proof Proving this theorem is exercise 4.7. | No |
Theorem 4.4 (Fudenberg and Levine 1983) Consider an infinite-horizon finite-action game \( {G}^{\infty } \) whose payoffs are continuous at infinity. Then\n\n(i) \( {\sigma }^{ * } \) is a subgame-perfect equilibrium of \( {G}^{\infty } \) if and only if it is the limit (in the product topology) of a sequence \( {\sigm... | Proof First note that if \( {\sigma }^{n} \rightarrow \sigma \) in the product topology, then the continuation payoffs \( u\left( {{\sigma }^{n} \mid {h}^{t}}\right) \) under \( {\sigma }^{n} \) in the subgame starting with \( {h}^{t} \) converge to the payoffs \( u\left( {\sigma \mid {h}^{t}}\right) \) under \( \sigma... | Yes |
Theorem 5.1 (folk theorem) \( {}^{9} \) For every feasible payoff vector \( v \) with \( {v}_{i} > {\underline{v}}_{i} \) for all players \( i \), there exists a \( \underline{\delta } < 1 \) such that for all \( \delta \in \left( {\underline{\delta },1}\right) \) there is a Nash equilibrium of \( G\left( \delta \right... | Proof Assume first that there is a pure action profile \( a \) such that \( g\left( a\right) = v \) , and consider the following strategies for each player \( i \) : \ | No |
Theorem 5.2 (Friedman 1971) Let \( {\alpha }^{ * } \) be a static equilibrium (an equilibrium of the stage game) with payoffs \( e \) . Then for any \( v \in V \) with \( {v}_{i} > {e}_{i} \) for all players \( i \), there is a \( \underline{\delta } \) such that for all \( \delta > \underline{\delta } \) there is a su... | Proof Assume that there is an \( \widehat{a} \) with \( g\left( \widehat{a}\right) = v \), and consider the following strategy profile: In period 0 each player \( i \) plays \( {a}_{i} \) . Each player \( i \) continues to play \( {\widehat{a}}_{l} \) so long as the realized actions were \( \widehat{a} \) in all previo... | Yes |
Theorem 5.3 (Aumann and Shapley 1976) If players evaluate sequences of stage-game utilities by the time-average criterion, then for any \( v \in V \) with \( {v}_{i} > {\underline{v}}_{i} \) for all players \( i \), there is a subgame-perfect equilibrium with payoffs \( v \) . | Proof Consider the following strategies: \ | No |
Theorem 5.4 (Fudenberg and Maskin 1986a) Assume that the dimension of the set \( V \) of feasible payoffs equals the number of players. Then, for any \( v \in V \) with \( {v}_{i} > {\underline{v}}_{i} \) for all \( i \), there is a discount factor \( \underline{\delta } < 1 \) such that for all \( \delta \in \left( {\... | (i) For simplicity, suppose that there is a pure action profile \( a \) with \( g\left( a\right) = v \) . The proof for the general case follows essentially the same lines. Assume first that the minmax profile \( {m}_{-i}^{i} \) against each player \( i \) is in pure strategies, so that deviations from this profile are... | Yes |
Theorem 5.8 (Benoit and Krishna 1987) Assume that for each player \( i \) there is a static equilibrium \( {\alpha }^{ * }\left( i\right) \) with \( {g}_{i}\left( {{\alpha }^{ * }\left( i\right) }\right) > {\underline{v}}_{i} \) . Then the set of Nash-equilibrium payoffs of the \( T \) -period game with time averaging ... | Proof The key idea of the proof is to first construct a \ | No |
Theorem 5.9 (Benoit and Krishna 1985) Assume that for each player \( i \) there are static equilibria \( {\alpha }^{ * }\left( i\right) \) and \( \widehat{\alpha }\left( i\right) \) with \( {g}_{i}\left( {{\alpha }^{ * }\left( i\right) }\right) > {g}_{i}\left( {\widehat{\alpha }\left( i\right) }\right) \), and that the... | ## Proof Omitted. | No |
Suppose that a single long-run firm faces a sequence of short-run consumers, each of whom plays only once but is informed of all previous play when choosing his actions. Each period, the consumer moves first, and chooses whether or not to purchase a good from the firm. If the consumer does not purchase, then both playe... | The following strategies are a subgame-perfect equilibrium of this game when the firm is sufficiently patient: The firm starts out producing high quality every time a consumer purchases, and continues to do so as long as it has never produced low quality in the past. If ever the firm produces low quality, it produces l... | Yes |
Theorem 5.10 (Fudenberg, Kreps, and Maskin 1990; Fudenberg and Levine 1990) Assume that the dimension of \( V \) is equal to \( \ell \), the number of long-run players. Then for every \( v \in \mathcal{V} \) with \( {\underline{v}}_{i} < {v}_{i} < {\bar{v}}_{i} \) for all \( i = 1,\ldots ,\ell \) , there is a \( \under... | Proof Omitted. | No |
Theorem 5.11 (Fudenberg, Levine, and Maskin 1990) If (i) the individual full-rank condition is satisfied at every pure strategy \( a \) ,(ii) for each pair \( i, j \) of players there is a profile that satisfies pairwise full rank for \( i \) and \( j \), and (iii) the feasible set \( V \) has dimension equal to the nu... | Outline of Proof Approximate the set of feasible individually rational payoffs by a smooth convex set \( W \) . Condition i implies that the minmax profile against player \( i \) and the best profile for player \( i \) can both be enforced on hyperplanes where player \( i \) ’s payoff is constant. Condition ii implies ... | Yes |
Theorem 7.1 (necessity) A piecewise \( {C}^{1} \) decision function \( x\left( \cdot \right) \) is implementable only if\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{n}\frac{\partial }{\partial \theta }\left( \frac{\partial {u}_{1}/\partial {x}_{k}}{\partial {u}_{1}/\partial t}\right) \frac{d{x}_{k}}{d\theta } \geq 0 \]\n\n(... | Proof Type \( \theta \) chooses an announcement \( \widehat{\theta } \) so as to maximize \( \Phi \left( {\widehat{\theta },\theta }\right) \equiv \) \( {u}_{1}\left( {x\left( \theta \right), t\left( \theta \right) ,\theta }\right) \) . Because \( {u}_{1} \) is \( {C}^{2} \) and \( x \) is piecewise \( {C}^{1} \), any ... | Yes |
Theorem 7.2 (monotonicity) Assume that the decision space is single dimensional and that \( {\mathrm{{CS}}}^{ + } \) holds. A necessary condition for \( x\left( \cdot \right) \) to be implementable is that it be nondecreasing: \( {\theta }_{2} > {\theta }_{1} \Rightarrow x\left( {\theta }_{2}\right) \geq x\left( {\thet... | Of course, if \( {\mathrm{{CS}}}^{ - } \) held, the necessary condition would be that \( x\left( \cdot \right) \) be nonincreasing. Note that, whereas theorem 7.1 implies monotonicity at points of differentiability, the proof of theorem 7.2 relies on the simple revealed-preference argument outlined in the discussion of... | No |
Theorem 7.3 Under assumptions \( \mathrm{{Al}}\left( {\mathrm{{CS}}}^{ + }\right) \) and \( \mathrm{A}2 \), any piecewise \( {C}^{1} \) decision function \( x\left( \cdot \right) \) satisfying \( d{x}_{k}/{d\theta } \geq 0 \) for all \( k \) is implementable. That is, there exists \( t\left( \cdot \right) \) such that ... | Proof From the agent’s first-order condition (equation 7.7), \( t\left( \cdot \right) \) must satisfy\n\n\[ \frac{dt}{d\theta } = - \mathop{\sum }\limits_{{k = 1}}^{n}\left( \frac{\partial {u}_{1}/\partial {x}_{k}}{\partial {u}_{1}/\partial t}\right) \frac{d{x}_{k}}{d\theta }. \]\n\n(7.9)\n\nAssumption A2 guarantees th... | No |
Lemma 7.1 The expectation of the virtual valuation is equal to the lower bound of the interval. | Proof Integrating by parts,\n\n\[ \n{\int }_{{\underline{\theta }}_{i}}^{{\bar{\theta }}_{i}}\left( {{\theta }_{i} - \frac{1 - {P}_{i}\left( {\theta }_{i}\right) }{{p}_{i}\left( {\theta }_{i}\right) }}\right) {p}_{i}\left( {\theta }_{i}\right) d{\theta }_{i} \n\]\n\n\[ \n= {\int }_{{\underline{\theta }}_{i}}^{{\bar{\th... | Yes |
Theorem 8.3 (Fudenberg and Tirole 1991) For multi-stage games of incomplete information with independent types, condition \( {\mathrm{B}}^{ * } \) implies condition \( \mathrm{C} \), and any assessment satisfying \( \mathrm{C} \) can be extended to a generalized assessment satisfying \( {\mathrm{B}}^{ * } \) . Therefor... | The idea of the proof of these two results (that condition B for two types or two periods, or more generally that condition \( {\mathrm{B}}^{ * } \) implies condition C) is as follows: Suppose one has built trembles up to date \( t \) that yield strictly positive beliefs at the beginning of date \( t \) and converge to... | No |
Theorem 8.4 The three definitions of perfect equilibrium (8.5A-8.5C) are equivalent. | Proof We show that definition A implies definition \( \mathrm{C} \), which implies definition B, which in turn implies definition A. First, by construction, the sequence \( {\sigma }^{\ell } \) defined in definition \( \mathrm{A} \) is an \( \varepsilon \) -perfect equilibrium, so that \( {\sigma }^{\ell } \) satisfies... | Yes |
Theorem 8.5 In finite games, at least one perfect equilibrium exists (Selten 1975). | A perfect equilibrium is sequential, but the converse is not true; however, for generic games the two concepts coincide (Kreps and Wilson 1982a).\n\nThe perfect-equilibrium correspondence need not be upper hemi-continuous in the payoffs. Figure 8.14 depicts a small perturbation of the\n\n<table><thead><tr><th></th><th>... | Yes |
Theorem 8.6 (Myerson 1978) All finite strategic-form games have proper equilibria. | Proof We first prove the existence of \( \varepsilon \) -proper equilibria. Let\n\n\[ \n{\widetilde{\sum }}_{i} = \left\{ {{\sigma }_{i} \in {\sum }_{i}^{0} \mid {\sigma }_{i}\left( {s}_{i}\right) \geq \frac{{\varepsilon }^{m}}{m}\text{ for all }{s}_{i}\text{ in }{S}_{i}}\right\} , \n\] \n\nwhere \( m \equiv \mathop{\m... | Yes |
Theorem 9.1 (Fudenberg and Levine 1991) Suppose that the long-run player’s choice of \( {a}_{1} \) is revealed at the end of each period. Then for all \( {\theta }_{0} \) with \( p\left( {\theta }_{0}\right) > 0 \), and all \( \lambda > 0 \), there is a \( \underline{\delta } < 1 \) such that, for all \( \delta \in \le... | \[ \left( {1 - \lambda }\right) {g}_{1}^{ * }\left( {p,{\theta }_{0}}\right) + \lambda \mathop{\min }\limits_{\alpha }{g}_{1}\left( {{\alpha }_{1},{\alpha }_{2},{\theta }_{0}}\right) \leq \underline{N}\left( {\delta, p,{\theta }_{0}}\right) \] (9.1a) and \[ \bar{N}\left( {\delta, p,{\theta }_{0}}\right) \leq \left( {1 ... | Yes |
Theorem 9.2 (Fudenberg and Maskin 1986) For any \( v = \left( {{v}_{1},{v}_{2}}\right) \in {V}^{ * } \) and any \( \varepsilon > 0 \), there exists a \( \underline{T} \) such that, for all \( T > \underline{T} \), there exists a \( T \) -period game such that each player \( i \) has probability \( 1 - \varepsilon \) of... | Partial Proof We will prove only the weaker theorem that any payoffs that Pareto dominate those of a static equilibrium can be approximated. Let \( e \) be a static-equilibrium profile with payoffs \( y = \left( {{y}_{1},{y}_{2}}\right) \), and let \( v \) be a payoff vector that Pareto dominates \( y \) . To avoid a d... | No |
Theorem 9.3 (Aumann and Sorin 1989) Let the stage game \( g \) be a game of common interests, and let \( z \) be its unique Pareto-optimal payoff vector. Fix a recall length \( \ell \), and let \( {p}^{m} \) be a sequence of \( \ell \) -perturbations that support the associated discounted repeated game \( G\left( \delt... | Idea of Proof We give a partial intuition for the convergence of equilibrium payoffs for the case in which \( \delta \) goes to 1 much faster than \( m \) goes to \( \infty \) (the theorem holds uniformly over sequences \( \left( {\delta, m}\right) \) ). Suppose more strongly that the game is symmetric and that a symme... | No |
Lemma 10.1 (skimming or cutoff-rule property) Suppose that the buyer accepts price \( {m}^{t} \) at date \( t \) when he has valuation \( v \) . Then he accepts price \( {m}^{t} \) with probability 1 when he has valuation \( {v}^{\prime } > v \) . | Proof Let \( {h}^{t} = \left( {{m}^{0},\ldots ,{m}^{t - 1}}\right) \) denote the history at date \( t \) (where the fact that the buyer has rejected all offers is implicit). Type \( v \) accepts \( {m}^{t} \) only if\n\n\[ v - {m}^{t} \geq \delta {U}_{\mathrm{b}}\left( {v,\left( {{h}^{t},{m}^{t}}\right) }\right) \]\n\n... | Yes |
Theorem 10.1 (Fudenberg, Levine, and Tirole 1985; Gul, Sonnenschein, and Wilson 1986) Suppose that the distribution of the buyer's type satisfies conditions \( G \) and \( R \). Then (i) a perfect Bayesian equilibrium exists and is generically unique, (ii) the equilibrium satisfies the Coase conjecture as \( \delta \ri... | Instead of proving this theorem, we give the flavor of the argument by analyzing the two-type case. Suppose that \( v = \bar{v} \) with probability \( \bar{p} \), and \( \underline{v} \) with probability \( \underline{p} \). Let \( {\bar{\mu }}^{t} \) denote the date- \( t \) posterior probability of \( \bar{v} \), con... | No |
Theorem 10.3 (Ausubel and Deneckere 1989a) \( {}^{20} \) Assume NG and that there exists \( L > M > 0 \) such that, for all \( v \in \left\lbrack {0,\bar{v}}\right\rbrack ,{Lv} \leq P\left( v\right) \leq {Mv} \) . Let \( {U}_{s}^{ * } \equiv \mathop{\sup }\limits_{m}\left\lbrack {m\left( {1 - P\left( m\right) }\right) ... | 19. A sketch of the argument follows: Fix a real time \( \varepsilon > 0 \), and suppose that there exists a sequence \( \Delta \rightarrow 0 \) such that the price at date \( \varepsilon \) does not converge to \( 0 : {m}^{\alpha /\Delta } \geq \bar{m} \geq 0 \) . The proof that this is impossible has four steps: (1) ... | No |
Theorem 10.4 (no price discrimination) Suppose that the buyer has \( n \) possible valuations \( 0 < {v}_{1} = \underline{v} < {v}_{2} < \cdots < {v}_{n} = \bar{v} \) . And assume \( \delta > \frac{1}{2} \) and \( T < + \infty \) . | - Let \( n = 2 \) . There exist \( {T}_{0} \) and \( {T}_{1} \) such that, for any \( T \geq {T}_{0} \) and any perfect Bayesian equilibrium of the rental game, the seller charges \( {r}^{t} = \underline{v} \) for all \( t = 0,1,\ldots, T - {T}_{1} \) . (Hart and Tirole 1988)\n\n- Let \( n \geq 2 \) . There exist \( {T... | Yes |
Theorem 10.5 (Hart and Tirole 1988) Suppose that the buyer has valuation \( \underline{v} \) or \( \bar{v} \) . Then the outcome of the rental model with (possibly renegotiated) long-term contracts coincides with that of the durable-good model. | To obtain some intuition about this result, \( {}^{30} \) it is useful to recall the basic conflict between efficiency and rent extraction in mechanism design (see section 7.3 and the price-discrimination example in subsection 7.1.1). Let \( {\underline{x}}^{t} \) and \( {\bar{x}}^{t} \) denote the probabilities of con... | Yes |
Theorem 10.6 (Ausubel and Deneckere 1989b) Assume \( \underline{v} = 0 \) . A feasible mechanism \( \{ M\left( \cdot \right), X\left( \cdot \right) \} \) is implementable by a perfect Bayesian equilibrium of the alternating-offer bargaining game with one-sided asymmetric information (in the sense that for any \( \varep... | \[ \bar{v}X\left( \bar{v}\right) - M\left( \bar{v}\right) \geq \bar{v}/2 \] That is, any feasible mechanism is an equilibrium outcome as long as the highest-valuation buyer obtains at least his full information payoff for \( \Delta \) close to 0 (which is approximately the Nash bargaining solution \( \bar{v}/2 \) ; see... | Yes |
Theorem 11.1 (Kreps and Wilson 1982) In a fixed tree, for generic assignments of payoffs to terminal nodes, the set of Nash-equilibrium probability distributions over terminal nodes is finite. | Since the distribution over terminal nodes is a continuous function of the strategy profile, when there is only a finite number of equilibrium distributions, every equilibrium in the same connected component \( {}^{3} \) must have the same probability distribution over endpoints, and hence the same play at every inform... | No |
Theorem 11.2 (Kohlberg and Mertens 1986) There exists a stable set that is contained in a single connected component of the set of Nash equilibria, and every tree with generic payoffs has a stable payoff (i.e., a payoff that obtains for every equilibrium in a stable set). A stable set contains a stable set of any game ... | This last property, called \ | No |
Theorem 11.3 (Cho and Sobel 1990) Suppose that a signaling game satisfies the following conditions:\n\n(i) (Monotonicity) If \( {a}_{2}^{\prime } > {a}_{2} \), then all types \( \theta \) prefer \( {a}_{2}^{\prime } \) to \( {a}_{2} \) .\n\n(ii) For each \( \mu \in \Delta \left( \Theta \right) ,\operatorname{MBR}\left(... | Since Cho and Sobel require player 1's action space to be bounded above, one possible equilibrium configuration has a set of types pooling at the highest possible action. Cho and Sobel show that this is the only possible kind of pooling, so if no type chooses to send the highest action then the equilibrium must be full... | Yes |
Lemma 11.2 Under the hypotheses of theorem 11.3, if type \( {\theta }^{\prime \prime } \) chooses action \( {a}_{1}^{\prime } \) with positive probability in equilibrium, then D1 implies that \( \mu \left( {{\theta }^{\prime } \mid {a}_{1}^{\prime \prime }}\right) = 0 \) if \( {\theta }^{\prime \prime } > {\theta }^{\p... | Proof Fix an equilibrium \( \left( {{\sigma }_{1}^{ * },{\sigma }_{2}^{ * }}\right) \) such that type \( {\theta }^{\prime \prime } \) chooses \( {a}_{1}^{\prime } \) with positive probability. Let \( {a}_{2}^{ * }\left( {a}_{1}\right) \) be the action prescribed by \( {\sigma }_{2}^{ * }\left( {\cdot \mid {a}_{1}}\rig... | No |
Theorem 11.4 (Fudenberg, Kreps, and Levine 1988) \( {}^{18} \) A pure-strategy profile \( s \) of an extensive-form game \( E \) with payoffs \( u \) is near-strict with respect to personal types if and only if there is a sequence \( {u}^{k} \rightarrow u \) such that \( s \) is c-perfect in the corresponding strategic... | Theorem 11.4 implies that any c-perfect pure-strategy \( {}^{19} \) equilibrium is near-strict; the set of near-strict equilibria also includes equilibria that are c-perfect in games where the payoffs are slightly different. That is, to obtain an \ | No |
Theorem 12.2 (Wu and Jiang 1962) Almost all finite strategic-form games are essential. | The proof of this theorem relies on the essential fixed-point theorem of Fort (1950). Consider a compact metric space \( \sum \) with distance \( d \) . A fixed point \( \sigma \) of a continuous mapping \( f \) from \( \sum \) into itself is essential if for any \( \varepsilon > 0 \) there exists \( \eta > 0 \) such t... | Yes |
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