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In the classical version of the Prisoner’s Dilemma, two agents can choose either cooperation or defection. Hence, there are four payoffs associated with each pairwise interaction between the two agents. In consonance with common practice |
, payoffs are characterized by the reward for mutual cooperation ( [MATH] ), punishment for mutual defection ( [MATH] ), sucker’s payoff ( [MATH] ) and temptation to defect ( [MATH] , where [MATH] ). Note that this parametrization refers to the weak version of the Prisoner’s Dilemma game, where [MATH] can be equal to [... |
The extended version of the PD game presented in this paper includes the concept of abstention, in which agents can not only cooperate ( [MATH] ) or defect [MATH] ) but can also choose to abstain ( [MATH] ) from a game interaction, obtaining the loner’s payoff ( [MATH] ) which is awarded to both players if one or both ... |
abstainers receive a payoff greater than [MATH] and less than [MATH] (i.e., [MATH] ). Thus, considering the normalized payoff matrix adopted, [MATH] . The payoff matrix and the associated values are illustrated in Table |
In these experiments, the following parameters are used: a [MATH] [MATH] ) regular lattice grid with periodic boundary conditions is created and fully populated with agents, which can play with their eight immediate neighbours (Moore neighbourhood). We adopt an unbiased environment (in which initially each agent is des... |
Monte Carlo methods are used to perform the Coevolutionary Optional Prisoner’s Dilemma game. In one Monte Carlo (MC) step, each player is selected once on average. This means that one MC step comprises [MATH] inner steps where the following calculations and updates occur: |
Select an agent ( [MATH] ) at random from the population. Calculate the utility [MATH] of each interaction of [MATH] with its eight neighbours (each neighbour represented as agent [MATH] ) as follows: |
[EQUATION] where [MATH] is the edge weight between agents [MATH] and [MATH] , and [MATH] corresponds to the payoff obtained by agent [MATH] on playing the game with agent [MATH] |
Calculate [MATH] the accumulated utility of [MATH] , that is: [EQUATION] where [MATH] denotes the set of neighbours of the agent [MATH] |
In order to update the link weights, [MATH] , between agents, compare the values of [MATH] and the average accumulated utility (i.e., [MATH] ) as follows: |
[EQUATION] where [MATH] is a constant such that [MATH] In line with previous research [MATH] is adjusted to be within the range [EQUATION] |
where [MATH] [MATH] ) defines the weight heterogeneity. Note that when [MATH] or [MATH] are equal to [MATH] , the link weight keeps constant ( [MATH] ), which results in the traditional scenario where only the strategies evolve. |
In order to update the strategy of [MATH] , the accumulated utility [MATH] is recalculated (based on the new link weights) and compared with the accumulated utility of one randomly selected neighbour ( [MATH] ). If [MATH] , agent [MATH] will copy the strategy of agent [MATH] with a probability proportional to the utili... |
[EQUATION] where [MATH] is the temptation to defect and [MATH] is the punishment for mutual defection. This equation has been considered previously by Huang et al. |
Simulations are run for [MATH] MC steps and the fraction of cooperation is determined by calculating the average of the final [MATH] MC steps. To alleviate the effect of randomness in the approach, the final results are obtained by averaging [MATH] independent runs. The following scenarios are investigated: |
The benefits of coevolution and abstention. Presence of abstainers in the coevolutionary model. Inspecting the coevolutionary environment. |
Investigating the properties of the parameters [MATH] and [MATH] Varying the number of states. Investigating the relationship between [MATH] [MATH] and [MATH] |
Investigating the robustness of cooperation in a biased environment. The Benefits of Coevolution and Abstention This section presents some of the main differences between the outcomes obtained by the proposed Coevolutionary Optional Prisoner’s Dilemma (COPD) game and other models which do not adopt the concept of coevo... |
4.1 Presence of Abstainers in the Coevolutionary Model In order to provide a means to effectively explore the impact of our coevolutionary model, i.e., the Coevolutionary Prisoner’s Dilemma (COPD) game, in the emergence of cooperation, we start by investigating the performance of some of the existing models. Namely, th... |
As shown in Figure , it can be observed that for both PD and CPD games, when the defector’s payoff is very high (i.e., [MATH] ) defectors spread quickly and dominate the environment. On the other hand, when abstainers are present in a static and unweighted network, i.e., playing the OPD game, we end up with abstainers ... |
Surprisingly, as shown in Figure , when considering the Coevolutionary Optional Prisoner’s Dilemma (COPD) game for the same environmental settings of Figure (i.e., [MATH] |
[MATH] and [MATH] ), with the temptation of defection almost at its peak (i.e., [MATH] ), it is possible to reach high levels of cooperation. |
Figure shows a typical phase diagram for both CPD and COPD games for a fixed value of [MATH] and [MATH] (on the COPD game). It can be observed that if a given environmental setting (i.e, [MATH] [MATH] and |
[MATH] ) produces a stable population of cooperators in the CPD game, then the presence of abstainers will not change it. In other words, the COPD game does not affect the outcome of scenarios in which cooperation is stable in the absence of abstainers. Thus, the main changes occur in scenarios in which defection domin... |
To summarize, despite the fact that the Coevolutionary Prisoner’s Dilemma (CPD) game succeeds in the promotion of cooperation in a wide range of scenarios, it is still not able to avoid the invasion by defectors in cases where [MATH] , which does not happen when the abstainers are present (i.e., COPD game). |
4.2 Inspecting the Coevolutionary Environment In order to further explain the results witnessed in the previous experiments, we investigate how the population evolves over time for the Coevolutionary Optional Prisoner’s Dilemma game. Figure features the time course of cooperation for three different values of [MATH] , ... |
[MATH] . Based on these results, in Figure we show snapshots for the Monte Carlo steps [MATH] [MATH] [MATH] and [MATH] for the three scenarios shown in Figure |
We see from Figure that for the traditional case (i.e., [MATH] ), abstainers spread quickly and reach a stable state in which single defectors are completely isolated by abstainers. In this way, as the payoffs obtained by a defector and an abstainer are the same, neither will ever change their strategy. In fact, even i... |
[MATH] (COPD in Figure ). When [MATH] , it is possible to observe some sort of equilibrium between the three strategies. They reach a state of cyclic competition in which abstainers invade defectors, defectors invade cooperators and cooperators invade abstainers. |
This behaviour, of balancing the three possible outcomes, is very common in nature where species with different reproductive strategies remain in equilibrium in the environment. For instance, the same scenario was observed as being responsible for preserving biodiversity in the neighbourhoods of the |
Escherichia coli , which is a bacteria commonly found in the lower intestine of warm-blooded organisms. According to Fisher studies were performed with three natural populations mixed together, in which one population produces a natural antibiotic but is immune to its effects; a second population is sensitive to the an... |
Because of this balance, they observed that each population ends up establishing its own territory in the environment, as the first population could kill off any other bacteria sensitive to the antibiotic, the second population could use their faster growth rate to displace the bacteria which are resistant to the antib... |
Another interesting behaviour is noticed for [MATH] . In this scenario, defectors are dominated by abstainers, allowing a few clusters of cooperators to survive. As a result of the absence of defectors, cooperators invade abstainers and dominate the environment. |
Exploring the Coevolutionary Optional Prisoner’s Dilemma game In this section, we present some of the relevant experimental results of the Monte Carlo simulations of the Coevolutionary Optional Prisoner’s Dilemma game in an unbiased environment. That is, a well-mixed initial population with a balanced amount of coopera... |
5.1 Investigating the properties of [MATH] and [MATH] This section aims to investigate the properties of the presented model (Sect. ) in regard to the parameters [MATH] and |
[MATH] . These parameters play a key role in the evolutionary dynamics of this model because they define the number of possible link weights that an agent is allowed to have (i.e., they define the number of states). |
Despite the fact that the number of states is discrete, the act of counting them is not straightforward. For instance, when counting the number of states between [MATH] and [MATH] for [MATH] and [MATH] , we could incorrectly state that there are four possible states for this scenario (i.e., [MATH] ). However, consideri... |
In order to better understand the relationship between [MATH] and [MATH] we plot [MATH] [MATH] and [MATH] as a function of the number of states (numerically counted) for a number of different values of both parameters (Figure ). It was observed that given the pairs |
[MATH] and [MATH] , if [MATH] is equal to [MATH] , then the number of states of both settings is the same. Figure shows the ratio [MATH] as a function of the number of states. As we can see, although the function is non-linear and non-monotonic, in general, higher values of [MATH] have less states. |
5.2 Varying the Number of States Figure shows the impact of the coevolutionary model on the emergence of cooperation when the ratio [MATH] varies for a range of fixed values of the loner’s payoff ( [MATH] ), temptation to defect ( [MATH] ) and |
[MATH] . In this experiment, we observe that when [MATH] , the outcomes of the Coevolutionary Optional Prisoner’s Dilemma (COPD) game are very similar to those observed by Huang et al. |
for the Coevolutionary Prisoner’s Dilemma (CPD) game. This result can be explained by the normalized payoff matrix adopted in this work (Table ). Clearly, when [MATH] , there is no advantage in abstaining from playing the game, thus agents choose the option to cooperate or defect. |
Results indicate that, in cases where the temptation to defect is very low (e.g, [MATH] ), the level of cooperation does not seem to be affected by the increment of the loner’s payoff, except when the advantage of abstaining is very high (e.g, [MATH] ). However, these results highlight that the presence of the abstaine... |
[MATH] ) for [MATH] and all other values of [MATH] is strong evidence that our coevolutionary model is very advantageous to the promotion of cooperative behaviour. |
Namely, when [MATH] , in the traditional case with a static and unweighted network ( [MATH] ), the cooperators have no chance of surviving; except, of course, when [MATH] is very close to the reward for mutual cooperation |
[MATH] , where it is possible to observe scenarios of quasi-stable states of the three strategies or between cooperators and defectors. In fact, in the traditional OPD ( [MATH] ), when [MATH] and [MATH] , abstainers are always the dominant strategy. However, when the coevolutionary rules are used, cooperators do much b... |
It is noteworthy that the curves in Figure are usually non-linear and/or non-monotonic because of the properties of the ratio [MATH] in regard to the number of states of each combination of [MATH] and [MATH] (Sect. 5.1 ). |
5.3 Investigating the Relationship between [MATH] [MATH] and [MATH] To investigate the outcomes in other scenarios, we explore a wider range of settings by varying the values of the temptation to defect ( [MATH] ), the loner’s payoff ( [MATH] ) and the ratio [MATH] for a fixed value of [MATH] |
As shown in Figure , cooperation is the dominant strategy in the majority of cases. Note that in the traditional case, with an unweighted and static network, i.e., [MATH] , abstainers dominate in all scenarios illustrated in this ternary diagram. In addition, it is also possible to observe that certain combinations of ... |
Another observation is that defectors are attacked more efficiently by abstainers as we increase the loner’s payoff ( [MATH] ). Simulations reveal that, for any scenario, if the loner’s payoff is greater than [MATH] [MATH] ), defectors have no chance of surviving. |
However, the drawback of increasing the value of [MATH] is that it makes it difficult for cooperators to dominate abstainers, which might produce a quasi-stable population of cooperators and abstainers. It is noteworthy that it is a counter-intuitive result from the COPD game, since the loner’s payoff is always less th... |
In fact, it is still expected that, in the COPD game, cooperators dominate abstainers, but depending on the value of the loner’s payoff, or the amount of abstainers in the population at this stage, it might take several Monte Carlo steps to reach a stable state, which is usually a state of cooperation fully dominating ... |
An interesting behaviour is noticed when [MATH] and [MATH] . In this scenario, abstainers quickly dominate the population, making a clear division between two states: before this range (defectors hardly die off) and after this range (defectors hardly survive). In this way, a loner’s payoff value greater than [MATH] [MA... |
Although the combinations shown in Figure for higher values of b ( [MATH] ) are just a small subset of an infinite number of possible values, it is clearly shown that a reasonable fraction of cooperators can survive even in an extremely adverse situation where the advantage of defecting is very high. Indeed, our result... |
Investigating the Robustness of Cooperation in a Biased Environment The previous experiments revealed that the presence of abstainers together with simple coevolutionary rules (i.e., the COPD game) act as a powerful mechanism to avoid the spread of defectors, which also allows the dominance of cooperation in a wide ran... |
However, the distribution of the strategies in the initial population used in all of the previous experiments was uniform. That is, we have explored cases in which the initial population contained a balanced amount of cooperators, defectors and abstainers. Thus, in order to explore the robustness of these outcomes in r... |
Figure features the fraction of each strategy in the population (i.e., cooperators, defectors and abstainers) over time for fixed values of [MATH] [MATH] and [MATH] . In this experiment, several independent simulations were performed, in which the loner’s payoff ( [MATH] ) and the number of abstainers in the initial po... |
and from [MATH] to [MATH] , respectively. Other special cases were also analyzed, such as placing only one abstainer into a balanced population of cooperators and defectors, and placing only one defector and one cooperator in a population of abstainers. For the sake of simplicity, we report only the values of [MATH] fo... |
Note that, for all these simulations, the initial population of cooperators and defectors remained in balance. For instance, an initial population with [MATH] |
of abstainers, will consequently have [MATH] of cooperators and [MATH] of defectors. Experiments reveal that the COPD game is actually extremely robust to radical changes in the initial population of abstainers. It has been shown that if the loner’s payoff is greater than [MATH] [MATH] ), then one abstainer might alone... |
a, b and c). However, this outcome is only possible if the single abstainer is in the middle of a big cluster of defectors. This outcome can happen because the payoff obtained by the abstainers is always greater than the one obtained by pairs of defectors (i.e., [MATH] ). Thus, in a cluster of defectors, abstention is ... |
In this way, as the loner’s payoff is the only parameter that directly affects the evolutionary dynamics of the abstainers, intuition might lead one to expect to see a clear and perhaps linear relationship between the loner’s payoff and the initial number of abstainers in the population. That is, given the same set of ... |
As discussed in Section 5.3 , populations of cooperators and abstainers tend to converge to cooperation. In this way, the scenario showed in Figure for [MATH] will probably end up with cooperators dominating the population, but as the loner’s payoff is close to the reward for mutual cooperation, the case in Figure i wi... |
Another very counter-intuitive behaviour occurs in the range [MATH] (this range may shift a little bit depending on the value of [MATH] ), where the outcome is usually of abstainers quickly dominating the population (Sect. 5.3 ). In this scenario, we would expect that changes in the initial population of abstainers wou... |
In summary, results show that an initial population with [MATH] of abstainers is usually enough to make reasonable changes in the outcome, increasing the chances of cooperators surviving or dominating the population. |
Conclusions and Future Work This paper studies the impact of a simple coevolutionary model in which not only the agents’ strategies but also the network evolves over time. The model consists of placing agents playing the Optional Prisoner’s Dilemma game in a dynamic spatial environment, which in turn, defines the Coevo... |
In summary, based on the results of several Monte Carlo simulations, it was shown that the COPD game allows for the emergence of cooperation in a wider range of scenarios than the Coevolutionary Prisoner’s Dilemma (CPD) game (i.e., the same coevolutionary model in populations which do not have the option to abstain fro... |
CPD: [MATH] OPD: [MATH] (or [MATH] ). PD: [MATH] and [MATH] (or [MATH] ). Also, it was possible to observe that abstention acts as an important mechanism to avoid the dominance of defectors. For instance, in adverse scenarios such as when the defector’s payoff is very high (i.e., [MATH] ), for both PD and CPD games, de... |
Furthermore, simulations showed that defectors die off when the loner’s payoff is greater than [MATH] [MATH] ). However, it was observed that increasing the loner’s payoff makes it difficult for cooperators to dominate abstainers, which is a counter-intuitive result, since the loner’s payoff is always less than the rew... |
Results revealed that the COPD game also allows scenarios of cyclic dominance between the three strategies (i.e., cooperation, defection and abstention), indicating that, for some parameter settings, the COPD game is intransitive. That is, the population remains balanced in such a way that cooperators invade abstainers... |
We also explored the robustness of these outcomes in regard to the initial amount of abstainers in the population (biased population). In summary, it was shown that, in some of the scenarios, even one abstainer might alone be enough to protect cooperators from the invasion of defectors, which in turn increases the chan... |
Although recent research has considered coevolving game strategy (with optional games) and link weights , this work presents a more complete analysis. We conclude that the combination of both of these trends in evolutionary game theory may shed additional light on gaining an in-depth understanding of the emergence of c... |
Future work will consider the exploration of different topologies and the influence of a wider range of scenarios, where, for example, agents could rewire their links, which, in turn, adds another level of complexity to the model. Future work will also involve applying our studies and results to realistic scenarios, su... |
# Source: arxiv 1706.03762 # Title: Attention Is All You Need # Sections: all # Downloaded: 2026-03-02T07:39:05.319657+00:00 Provided proper attribution is provided, Google hereby grants permission to reproduce the tables and figures in this paper solely for use in journalistic or scholarly works. |
Attention Is All You Need \AND Ashish Vaswani Google Brain avaswani@google.com &Noam Shazeer footnotemark: Google Brain noam@google.com |
&Niki Parmar footnotemark: Google Research nikip@google.com &Jakob Uszkoreit footnotemark: Google Research usz@google.com &Llion Jones footnotemark: |
Google Research llion@google.com &Aidan N. Gomez footnotemark: University of Toronto aidan@cs.toronto.edu &Łukasz Kaiser footnotemark: |
Google Brain lukaszkaiser@google.com &Illia Polosukhin footnotemark: illia.polosukhin@gmail.com Equal contribution. Listing order is random. Jakob proposed replacing RNNs with self-attention and started the effort to evaluate this idea. Ashish, with Illia, designed and implemented the first Transformer models and has b... |
Abstract The dominant sequence transduction models are based on complex recurrent or convolutional neural networks that include an encoder and a decoder. The best performing models also connect the encoder and decoder through an attention mechanism. We propose a new simple network architecture, the Transformer, based s... |
Introduction Recurrent neural networks, long short-term memory and gated recurrent neural networks in particular, have been firmly established as state of the art approaches in sequence modeling and transduction problems such as language modeling and machine translation |
. Numerous efforts have since continued to push the boundaries of recurrent language models and encoder-decoder architectures Recurrent models typically factor computation along the symbol positions of the input and output sequences. Aligning the positions to steps in computation time, they generate a sequence of hidde... |
and conditional computation , while also improving model performance in case of the latter. The fundamental constraint of sequential computation, however, remains. |
Attention mechanisms have become an integral part of compelling sequence modeling and transduction models in various tasks, allowing modeling of dependencies without regard to their distance in the input or output sequences |
. In all but a few cases , however, such attention mechanisms are used in conjunction with a recurrent network. In this work we propose the Transformer, a model architecture eschewing recurrence and instead relying entirely on an attention mechanism to draw global dependencies between input and output. The Transformer ... |
Background The goal of reducing sequential computation also forms the foundation of the Extended Neural GPU , ByteNet and ConvS2S |
, all of which use convolutional neural networks as basic building block, computing hidden representations in parallel for all input and output positions. In these models, the number of operations required to relate signals from two arbitrary input or output positions grows in the distance between positions, linearly f... |
. In the Transformer this is reduced to a constant number of operations, albeit at the cost of reduced effective resolution due to averaging attention-weighted positions, an effect we counteract with Multi-Head Attention as described in section 3.2 |
Self-attention, sometimes called intra-attention is an attention mechanism relating different positions of a single sequence in order to compute a representation of the sequence. Self-attention has been used successfully in a variety of tasks including reading comprehension, abstractive summarization, textual entailmen... |
End-to-end memory networks are based on a recurrent attention mechanism instead of sequence-aligned recurrence and have been shown to perform well on simple-language question answering and language modeling tasks |
To the best of our knowledge, however, the Transformer is the first transduction model relying entirely on self-attention to compute representations of its input and output without using sequence-aligned RNNs or convolution. In the following sections, we will describe the Transformer, motivate self-attention and discus... |
and Model Architecture Most competitive neural sequence transduction models have an encoder-decoder structure . Here, the encoder maps an input sequence of symbol representations subscript subscript (x_{1},...,x_{n}) to a sequence of continuous representations subscript subscript \mathbf{z}=(z_{1},...,z_{n}) . Given \m... |
, consuming the previously generated symbols as additional input when generating the next. The Transformer follows this overall architecture using stacked self-attention and point-wise, fully connected layers for both the encoder and decoder, shown in the left and right halves of Figure , respectively. |
3.1 Encoder and Decoder Stacks Encoder: The encoder is composed of a stack of N=6 identical layers. Each layer has two sub-layers. The first is a multi-head self-attention mechanism, and the second is a simple, position-wise fully connected feed-forward network. We employ a residual connection |
around each of the two sub-layers, followed by layer normalization . That is, the output of each sub-layer is LayerNorm Sublayer LayerNorm Sublayer \mathrm{LayerNorm}(x+\mathrm{Sublayer}(x)) , where Sublayer Sublayer \mathrm{Sublayer}(x) is the function implemented by the sub-layer itself. To facilitate these residual ... |
Decoder: The decoder is also composed of a stack of N=6 identical layers. In addition to the two sub-layers in each encoder layer, the decoder inserts a third sub-layer, which performs multi-head attention over the output of the encoder stack. Similar to the encoder, we employ residual connections around each of the su... |
3.2 Attention An attention function can be described as mapping a query and a set of key-value pairs to an output, where the query, keys, values, and output are all vectors. The output is computed as a weighted sum of the values, where the weight assigned to each value is computed by a compatibility function of the que... |
3.2.1 Scaled Dot-Product Attention We call our particular attention "Scaled Dot-Product Attention" (Figure ). The input consists of queries and keys of dimension subscript d_{k} , and values of dimension subscript d_{v} . We compute the dot products of the query with all keys, divide each by subscript \sqrt{d_{k}} , an... |
In practice, we compute the attention function on a set of queries simultaneously, packed together into a matrix . The keys and values are also packed together into matrices and . We compute the matrix of outputs as: |
(1) The two most commonly used attention functions are additive attention , and dot-product (multiplicative) attention. Dot-product attention is identical to our algorithm, except for the scaling factor of subscript \frac{1}{\sqrt{d_{k}}} . Additive attention computes the compatibility function using a feed-forward net... |
While for small values of subscript d_{k} the two mechanisms perform similarly, additive attention outperforms dot product attention without scaling for larger values of subscript d_{k} |
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