Efficiency in evolutionary games: Darwin, Nash and the secret handshake

This paper considers any evolutionary game possessing several evolutionarily stable strategies, or ESSs, with differing payoffs. A mutant is introduced which will "destroy" any ESS which yields a lower payoff than another. This mutant possesses a costless signal and also conditions on the presence of this signal in each opponent. The mutant then can protect itself against a population playing an inefficient ESS by matching this against these non-signalers. At the same time, the mutants can achieve the more efficient ESS against the signaling mutant population itself. This construction is illustrated by means of the simplest possible example, a co-ordination game. The one-shot prisoner's dilemma is used to illustrate how a superior outcome which is not induced by an ESS may be temporarily but not permanently attained. In the case of the repeated prisoner's dilemma, the present argument seems to render the "evolution of co-operation" ultimately inevitable.     

An escape from ‘the prisoner's dilemma‘

Conventional escapes from the paradox that noncooperation between two organisms may be rational, even when cooperation would yield a higher reward to each, are based on the mechanism of reciprocity; but an analytical model of foraging among oviposition sites reveals a more immediate rationale, namely, equivalence of selfishness and altruism. The resulting game is unconditionally the prisoner's dilemma if the players have perfect recognition; however, in the absence thereof and for three different parameter regimes, it yields either the prisoner's dilemma, or two evolutionarily stable strategies, or a unique cooperative ESS. Thus unrecognition can favor cooperation; and environments can exist in which cooperation persists, or even invades, without reciprocity. The results suggest that different mechanisms for cooperation may operate at different levels of neural complexity.     

Are there really no evolutionarily stable strategies in the iterated prisoner's dilemma?

The evolutionary form of the iterated prisoner's dilemma (IPD) is a repeated game where players strategically choose whether to cooperate with or exploit opponents and reproduce in proportion to game success. It has been widely used to study the evolution of cooperation among selfish agents. In the past 15 years, researchers proved over a series of papers that there is no evolutionarily stable strategy (ESS) in the IPD when players maintain long-term relationships. This makes it difficult to make predictions about what strategies can actually persist as prevalent in a population over time. Here, we show that this no ESS finding may be a mathematical technicality, relying on implausible players who are "too perfect" in that their probability of cooperating on any move is arbitrarily close to either 0 or 1. Specifically, in the no ESS proof, all strategies were allowed, meaning that after a strategy X experiences any history H, X cooperates with an unrestricted probability p (X, H) where 0< or =p (X, H)< or =1. Here, we restrict strategies to the set S in which X is a member of S [corrected] if after any H, X cooperates with a restricted probability p (X, H) where e< or =p (X, H)< or =1-e and 0相似文献   

Evolutionary escape from the prisoner's dilemma

The classic prisoner's dilemma model of game theory is modified by introducing occasional variations on the options available to players. Mutation and selection of game options reliably change the game matrix, gradually, from a prisoner's dilemma game into a byproduct mutualism one, in which cooperation is stable, and "temptation to defect" is replaced by temptation to cooperate. This result suggests that when there are many different potential ways of interacting, exploring those possibilities may make escape from prisoner's dilemmas a common outcome in the world. A consequence is that persistent prisoner's dilemma structures may be less common than one might otherwise expect.     

Evolutionary dynamics of the continuous iterated prisoner's dilemma

The iterated prisoner's dilemma (IPD) has been widely used in the biological and social sciences to model dyadic cooperation. While most of this work has focused on the discrete prisoner's dilemma, in which actors choose between cooperation and defection, there has been some analysis of the continuous IPD, in which actors can choose any level of cooperation from zero to one. Here, we analyse a model of the continuous IPD with a limited strategy set, and show that a generous strategy achieves the maximum possible payoff against its own type. While this strategy is stable in a neighborhood of the equilibrium point, the equilibrium point itself is always vulnerable to invasion by uncooperative strategies, and hence subject to eventual destabilization. The presence of noise or errors has no effect on this result. Instead, generosity is favored because of its role in increasing contributions to the most efficient level, rather than in counteracting the corrosiveness of noise. Computer simulation using a single-locus infinite alleles Gaussian mutation model suggest that outcomes ranging from a stable cooperative polymorphism to complete collapse of cooperation are possible depending on the magnitude of the mutational variance. Also, making the cost of helping a convex function of the amount of help provided makes it more difficult for cooperative strategies to invade a non-cooperative equilibrium, and for the cooperative equilibrium to resist destabilization by non-cooperative strategies. Finally, we demonstrate that a much greater degree of assortment is required to destabilize a non-cooperative equilibrium in the continuous IPD than in the discrete IPD. The continuous model outlined here suggests that incremental amounts of cooperation lead to rapid decay of cooperation and thus even a large degree of assortment will not be sufficient to allow cooperation to increase when cooperators are rare. The extreme degree of assortment required to destabilize the non-cooperative equilibrium, as well as the instability of the cooperative equilibrium, may help explain why cooperation in Prisoner's Dilemmas is so rare in nature.     

The evolution of reciprocity in sizable groups

Recently, several authors have investigated the evolution of reciprocal altruism using the repeated prisoner's dilemma game. These models suggest that natural selection is likely to favor behavioral strategies leading to reciprocal cooperation when pairs of individuals interact repeatedly in potentially cooperative situations. Using the repeated n-person prisoner's dilemma game, we consider whether reciprocal altruism is also likely to evolve when social interactions involve more individuals. We show that the conditions that allow the evolution of reciprocal cooperation become extremely restrictive as group size increases.     

Win-stay-lose-learn promotes cooperation in the spatial prisoner's dilemma game

Holding on to one's strategy is natural and common if the later warrants success and satisfaction. This goes against widespread simulation practices of evolutionary games, where players frequently consider changing their strategy even though their payoffs may be marginally different than those of the other players. Inspired by this observation, we introduce an aspiration-based win-stay-lose-learn strategy updating rule into the spatial prisoner's dilemma game. The rule is simple and intuitive, foreseeing strategy changes only by dissatisfied players, who then attempt to adopt the strategy of one of their nearest neighbors, while the strategies of satisfied players are not subject to change. We find that the proposed win-stay-lose-learn rule promotes the evolution of cooperation, and it does so very robustly and independently of the initial conditions. In fact, we show that even a minute initial fraction of cooperators may be sufficient to eventually secure a highly cooperative final state. In addition to extensive simulation results that support our conclusions, we also present results obtained by means of the pair approximation of the studied game. Our findings continue the success story of related win-stay strategy updating rules, and by doing so reveal new ways of resolving the prisoner's dilemma.     

Evolutionary Stability in the Asymmetric Volunteer's Dilemma

It is often assumed that in public goods games, contributors are either strong or weak players and each individual has an equal probability of exhibiting cooperation. It is difficult to explain why the public good is produced by strong individuals in some cooperation systems, and by weak individuals in others. Viewing the asymmetric volunteer''s dilemma game as an evolutionary game, we find that whether the strong or the weak players produce the public good depends on the initial condition (i.e., phenotype or initial strategy of individuals). These different evolutionarily stable strategies (ESS) associated with different initial conditions, can be interpreted as the production modes of public goods of different cooperation systems. A further analysis revealed that the strong player adopts a pure strategy but mixed strategies for the weak players to produce the public good, and that the probability of volunteering by weak players decreases with increasing group size or decreasing cost-benefit ratio. Our model shows that the defection probability of a “strong” player is greater than the “weak” players in the model of Diekmann (1993). This contradicts Selten''s (1980) model that public goods can only be produced by a strong player, is not an evolutionarily stable strategy, and will therefore disappear over evolutionary time. Our public good model with ESS has thus extended previous interpretations that the public good can only be produced by strong players in an asymmetric game.     

Cheating as a mixed strategy in a simple model of aggressive communication

The possibility that frequency-dependent cheating can persist in an evolutionarily stable communication system has frequently been proposed. Although there is empirical evidence for this idea, however, it has not been investigated in terms of game theory. In the present paper I show for a simple symmetric game that cheating can be part of a mixed evolutionarily stable strategy (ESS). Furthermore, despite the widespread assumption that cheaters must be rare, I show that most of the population can be cheaters, while the signalling system remains evolutionarily stable. Consequences for signalling theory and experiments to detect such mixed ESS are discussed. Copyright 2000 The Association for the Study of Animal Behaviour.     

The continuous prisoner's dilemma and the evolution of cooperation through reciprocal altruism with variable investment

Understanding the evolutionary origin and persistence of cooperative behavior is a fundamental biological problem. The standard "prisoner's dilemma," which is the most widely adopted framework for studying the evolution of cooperation through reciprocal altruism between unrelated individuals, does not allow for varying degrees of cooperation. Here we study the continuous iterated prisoner's dilemma, in which cooperative investments can vary continuously in each round. This game has been previously considered for a class of reactive strategies in which current investments are based on the partner's previous investment. In the standard iterated prisoner's dilemma, such strategies are inferior to strategies that take into account both players' previous moves, as is exemplified by the evolutionary dominance of "Pavlov" over "tit for tat." Consequently, we extend the analysis of the continuous prisoner's dilemma to a class of strategies in which current investments depend on previous payoffs and, hence, on both players' previous investments. We show, both analytically and by simulation, that payoff-based strategies, which embody the intuitively appealing idea that individuals invest more in cooperative interactions when they profit from these interactions, provide a natural explanation for the gradual evolution of cooperation from an initially noncooperative state and for the maintenance of cooperation thereafter.     

Prisoner's dilemma posed by fitness-associated recombination strategies

Genetic recombination is a central and repeated topic of study in the evolution of life. However, along with the influence of recombination on evolution, we understand surprisingly little of how selection shapes the nature of recombination. One explanation for recombination is that it allows organisms to escape from perilous situations where they experience very low fitness. As a corollary, it has been suggested that selection should favor recombination at low fitness and not at high fitness (fitness-associated recombination, FAR), and theory suggests that such strategies can indeed be selected. Here we develop models to further investigate the evolution of FAR. Consistent with previous works, we find that FAR can invade and dominate over a strategy of uniform recombination that is independent of fitness. However, our simulation results suggest that extreme FAR strategies, known as group-elitism, are not necessarily superior to other FAR strategies. Moreover, we argue that FAR domination will often occur with a net loss of mean population fitness. Interestingly, this suggests that the strategy of not recombining at high fitness will sometimes be analogous to a defector strategy from the famous "prisoner's dilemma" game: a selfish strategy that is selected but leads to a loss of mean fitness for all players.     

Revisiting matrix games: the concept of neighborhood invader strategies

We extend the concept of neighborhood invader strategy (NIS) to finite-dimensional matrix games and compare this concept to the evolutionarily stable strategy (ESS) concept. We show that these two concepts are not equivalent in general. Just as ESS's may not be unique, NIS's may also not be unique. However, if there is an ESS and a NIS then these strategies must be the same. We show that an ESNIS (an ESS and NIS) for any matrix game is unique and that a mixed ESS with full support is a NIS. Thus a mixed ESS with full support is not invadable by any pure or mixed strategy and it can invade any pure or mixed strategy. An ESS which is an ESNIS, therefore, has better chance of being established evolutionarily through dynamic selection.     

Resolving the iterated prisoner's dilemma: theory and reality

Pairs of unrelated individuals face a prisoner's dilemma if cooperation is the best mutual outcome, but each player does best to defect regardless of his partner's behaviour. Although mutual defection is the only evolutionarily stable strategy in one-shot games, cooperative solutions based on reciprocity can emerge in iterated games. Among the most prominent theoretical solutions are the so-called bookkeeping strategies, such as tit-for-tat, where individuals copy their partner's behaviour in the previous round. However, the lack of empirical data conforming to predicted strategies has prompted the suggestion that the iterated prisoner's dilemma (IPD) is neither a useful nor realistic basis for investigating cooperation. Here, we discuss several recent studies where authors have used the IPD framework to interpret their data. We evaluate the validity of their approach and highlight the diversity of proposed solutions. Strategies based on precise accounting are relatively uncommon, perhaps because the full set of assumptions of the IPD model are rarely satisfied. Instead, animals use a diverse array of strategies that apparently promote cooperation, despite the temptation to cheat. These include both positive and negative reciprocity, as well as long-term mutual investments based on 'friendships'. Although there are various gaps in these studies that remain to be filled, we argue that in most cases, individuals could theoretically benefit from cheating and that cooperation cannot therefore be explained with the concept of positive pseudo-reciprocity. We suggest that by incorporating empirical data into the theoretical framework, we may gain fundamental new insights into the evolution of mutual reciprocal investment in nature.     

The structure of mutations and the evolution of cooperation

Evolutionary game dynamics in finite populations assumes that all mutations are equally likely, i.e., if there are n strategies a single mutation can result in any strategy with probability 1/n. However, in biological systems it seems natural that not all mutations can arise from a given state. Certain mutations may be far away, or even be unreachable given the current composition of an evolving population. These distances between strategies (or genotypes) define a topology of mutations that so far has been neglected in evolutionary game theory. In this paper we re-evaluate classic results in the evolution of cooperation departing from the assumption of uniform mutations. We examine two cases: the evolution of reciprocal strategies in a repeated prisoner's dilemma, and the evolution of altruistic punishment in a public goods game. In both cases, alternative but reasonable mutation kernels shift known results in the direction of less cooperation. We therefore show that assuming uniform mutations has a substantial impact on the fate of an evolving population. Our results call for a reassessment of the "model-less" approach to mutations in evolutionary dynamics.     

Dear Enemies and the Prisoner's Dilemma: Why Should Territorial Neighbors Form Defensive Coalitions?

Game-theoretic arguments are used to derive two new hypothesesto explain why territorial residents so consistently defeatpotential usurpers. Both hypotheses are based on help from established,familiar neighbors. The first hypothesis follows simply fromKrebs' (1982) assertion that the value of a territory to a usurpermust be decremented by the costs of negotiating dear-enemy relationshipswith the remaining neighbors. An implication is that the remainingneighbors will also have to pay these renegotiation costs ifthe usurper succeeds. The first hypothesis is that it may benefita territorial animal to help its established neighbors defendso it can avoid having to renegotiate territorial boundarieswith a new, unfamiliar neighbor. This hypothesis assumes netpositive benefits to helping without requiring reciprocation. The second hypothesis requires reciprocation to compensate forimmediate net costs of helping. An animal should help its neighborsfight off usurpers only if the neighbors will reciprocate. Thishypothesis is based on the prisoner's dilemma game and buildson Axelrod's (1984) work. Cooperative defense (reciprocal help)can be an evolutionarily stable strategy (ESS) if several conditionsare met. One critical condition is that the relationship betweenneighbors is relatively stable. Cooperative defense should helpestablished neighbors retain their territories, and should thereforebe a cause, as well as a consequence of stability. It is suggestedthat the necessary conditions are not very restrictive, thatthey are often met in nature, and that shared defense is observedbut not recognized as such.     

The evolution of payoff matrices: providing incentives to cooperate

Most of the work in evolutionary game theory starts with a model of a social situation that gives rise to a particular payoff matrix and analyses how behaviour evolves through natural selection. Here, we invert this approach and ask, given a model of how individuals behave, how the payoff matrix will evolve through natural selection. In particular, we ask whether a prisoner's dilemma game is stable against invasions by mutant genotypes that alter the payoffs. To answer this question, we develop a two-tiered framework with goal-oriented dynamics at the behavioural time scale and a diploid population genetic model at the evolutionary time scale. Our results are two-fold: first, we show that the prisoner's dilemma is subject to invasions by mutants that provide incentives for cooperation to their partners, and that the resulting game is a coordination game similar to the hawk-dove game. Second, we find that for a large class of mutants and symmetric games, a stable genetic polymorphism will exist in the locus determining the payoff matrix, resulting in a complex pattern of behavioural diversity in the population. Our results highlight the importance of considering the evolution of payoff matrices to understand the evolution of animal social systems.     

Teaching and learning in a probabilistic prisoner's dilemma

The prisoner's dilemma is much studied in social psychology and decision-making because it models many real-world conflicts. In everyday terms, the choice to 'cooperate' (maximize reward for the group) or 'defect' (maximize reward for the individual) is often attributed to altruistic or selfish motives. Alternatively, behavior during a dilemma may be understood as a function of reinforcement and punishment. Human participants played a prisoner's-dilemma-type game (for points exchangeable for money) with a computer that employed either a teaching strategy (a probabilistic version of tit-for-tat), in which the computer reinforced or punished participants' cooperation or defection, or a learning strategy (a probabilistic version of Pavlov), in which the computer's responses were reinforced and punished by participants' cooperation and defection. Participants learned to cooperate against both computer strategies. However, in a second experiment which varied the context of the game, they learned to cooperate only against one or other strategy; participants did not learn to cooperate against tit-for-tat when they believed that they were playing against another person; participants did not learn to cooperate against Pavlov when the computer's cooperation probability was signaled by a spinner. The results are consistent with the notion that people are biased not only to cooperate or defect on individual social choices, but also to employ one or other strategy of interaction in a pattern across social choices.     

The dependency game: Multiperson reciprocal sharing leads to stable cooperation which can evolve into group formation

In the standard model for reciprocal collaboration, the repeated prisoner's dilemma (PD), it has proved difficult to establish collaboration in larger groups, necessitating the introduction of additional mechanisms such as reputation or assortedness. The problem is corroborated because current multiperson PDs model simultaneous player action, known as a common goods situation, whereas multiperson collaboration could be easier to obtain in a PD with alternate player action, a private goods situation. Here we present such a game, called a dependency game, and show that stable collaboration can be obtained in a 255 player simulation if only players are allowed to remember three previous benefactors, so they can play advanced tit-for-tat. Furthermore, we show that such a freely collaborating population is threatened by assorted strategies, which define groups that parasitize on independent tit-for-tat players. By excluding others, these groups engage in indirect reciprocal behaviour. Our model therefore combines many hitherto separate collaboration-enhancing concepts into one game, and suggests that group formation and collaboration are two separate social phenomena.     

Simple adaptive strategy wins the prisoner's dilemma

The prisoner's dilemma has become the leading paradigm to explain the evolution of cooperation among selfish individuals. Here, we present an adaptive strategy that implements new mechanisms to process information about past encounters. The history of moves is summarized in an internal state which then determines the subsequent move. This enables the strategy to adjust its decisions to the character of the current opponent and to adapt the most promising strategic behavior. For this reason, we call such strategies Adaptor. Through evolutionary simulations, we demonstrate that the concept of Adaptor leads to strategic patterns that are (a) highly cooperative when playing against kin, (b) stable in a sense that goes far beyond the concept of evolutionary stability, (c) robust to environmental changes, i.e. variations of the parameter values and finally (d) superior in performance to the most prominent strategies in the literature.     

Whence tit-for-tat?

Summary In theoretical and empirical studies of the evolution of cooperation, the tit-for-tat strategy (i.e. cooperate unless your partner did not cooperate in the previous interaction) is widely considered to be of central importance. Nevertheless, surprisingly little is known about the conditions in which tit-for-tat appears and disappears across generations in a population of interacting individuals. Here, we apply a newly developed classifier-system model (EvA) in addressing this issue when the key features of interactions are caricatured using the iterated prisoner's dilemma game. Our simple representation of behavioural strategies as algorithms composed of two interacting rules allowed us to determine conditions in which tit-for-tat can replace noncooperative strategies and vice versa. Using direct game-theoretic analysis and simulations with the EvA model, we determined that no strategy is evolutionarily stable, but larger population sizes and longer sequences of interactions between individuals can yield transient dominance by tit-for-tat. Genetic drift among behaviourally equivalent strategies is the key mechanism underlying this dominance. Our analysis suggests that tit-for-tat could be important in nature for cognitively simple organisms of limited memory capacity, in strongly kin-selected or group-selected populations, when interaction sequences between individuals are relatively short, in moderate-sized populations of widely interacting individuals and when defectors appear in the population with moderate frequency.