Evolution of interactions and cooperation in the spatial prisoner's dilemma game

We study the evolution of cooperation in the spatial prisoner's dilemma game where players are allowed to establish new interactions with others. By employing a simple coevolutionary rule entailing only two crucial parameters, we find that different selection criteria for the new interaction partners as well as their number vitally affect the outcome of the game. The resolution of the social dilemma is most probable if the selection favors more successful players and if their maximally attainable number is restricted. While the preferential selection of the best players promotes cooperation irrespective of game parametrization, the optimal number of new interactions depends somewhat on the temptation to defect. Our findings reveal that the "making of new friends" may be an important activity for the successful evolution of cooperation, but also that partners must be selected carefully and their number limited.     

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.     

Resolution of the stochastic strategy spatial prisoner's dilemma by means of particle swarm optimization

We study the evolution of cooperation among selfish individuals in the stochastic strategy spatial prisoner's dilemma game. We equip players with the particle swarm optimization technique, and find that it may lead to highly cooperative states even if the temptations to defect are strong. The concept of particle swarm optimization was originally introduced within a simple model of social dynamics that can describe the formation of a swarm, i.e., analogous to a swarm of bees searching for a food source. Essentially, particle swarm optimization foresees changes in the velocity profile of each player, such that the best locations are targeted and eventually occupied. In our case, each player keeps track of the highest payoff attained within a local topological neighborhood and its individual highest payoff. Thus, players make use of their own memory that keeps score of the most profitable strategy in previous actions, as well as use of the knowledge gained by the swarm as a whole, to find the best available strategy for themselves and the society. Following extensive simulations of this setup, we find a significant increase in the level of cooperation for a wide range of parameters, and also a full resolution of the prisoner's dilemma. We also demonstrate extreme efficiency of the optimization algorithm when dealing with environments that strongly favor the proliferation of defection, which in turn suggests that swarming could be an important phenomenon by means of which cooperation can be sustained even under highly unfavorable conditions. We thus present an alternative way of understanding the evolution of cooperative behavior and its ubiquitous presence in nature, and we hope that this study will be inspirational for future efforts aimed in this direction.     

Moving away from nasty encounters enhances cooperation in ecological prisoner's dilemma game

We study the role of migration in the evolution of cooperation. Individuals spatially located on a square lattice play the prisoner's dilemma game. Dissatisfied players, who have been exploited by defectors, tend to terminate interaction with selfish partners by leaving the current habitats, and explore unknown physical niches available surrounding them. The time scale ratio of game interaction to natural selection governs how many game rounds occur before individuals experience strategy updating. Under local migration and strong selection, simulation results demonstrate that cooperation can be stabilized for a wide range of model parameters, and the slower the natural selection, the more favorable for the emergence of cooperation. Besides, how the selection intensity affects cooperators' evolutionary fate is also investigated. We find that increasing it weakens cooperators' viability at different speeds for different time scale ratios. However, cooperation is greatly improved provided that individuals are offered with enough chance to agglomerate, while cooperation can always establish under weak selection but vanishes under very strong selection whenever individuals have less odds to migrate. Whenever the migration range restriction is removed, the parameter area responsible for the emergence of cooperation is, albeit somewhat compressed, still remarkable, validating the effectiveness of collectively migrating in promoting cooperation.     

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.     

The shared reward dilemma

One of the most direct human mechanisms of promoting cooperation is rewarding it. We study the effect of sharing a reward among cooperators in the most stringent form of social dilemma, namely the prisoner's dilemma (PD). Specifically, for a group of players that collect payoffs by playing a pairwise PD game with their partners, we consider an external entity that distributes a fixed reward equally among all cooperators. Thus, individuals confront a new dilemma: on the one hand, they may be inclined to choose the shared reward despite the possibility of being exploited by defectors; on the other hand, if too many players do that, cooperators will obtain a poor reward and defectors will outperform them. By appropriately tuning the amount to be shared a vast variety of scenarios arises, including the traditional ones in the study of cooperation as well as more complex situations where unexpected behavior can occur. We provide a complete classification of the equilibria of the n-player game as well as of its evolutionary dynamics.     

Friendship, cliquishness, and the emergence of cooperation

The evolution of cooperation is a central problem in biology and the social sciences. While theoretical work using the iterated prisoner's dilemma (IPD) has shown that cooperation among non-kin can be sustained among reciprocal strategies (i.e. tit-for-tat), these results are sensitive to errors in strategy execution, cyclical invasions by free riders, and the specific ecology of strategies. Moreover, the IPD assumes that a strategy's probability of playing the PD game with other individuals is independent of the decisions made by others. Here, we remove the assumption of independent pairing by studying a more plausible cooperative dilemma in which players can preferentially interact with a limited set of known partners and also deploy longer-term accounting strategies that can counteract the effects of random errors. We show that cooperative strategies readily emerge and persist in a range of noisy environments, with successful cooperative strategies (henceforth, cliquers) maintaining medium-term memories for partners and low thresholds for acceptable cooperation (i.e. forgiveness). The success of these strategies relies on their cliquishness-a propensity to defect with strangers if they already have an adequate number of partners. Notably, this combination of medium-term accounting, forgiveness, and cliquishness fits with empirical studies of friendship and other long-term relationships among humans.     

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.     

The evolution of punishment and apology: an iterated prisoner's dilemma model

Recently, behaviors that seem to function as punishment or apology have been reported among non-human primates as well as humans. Such behaviors appear to play an important role in maintaining cooperation between individuals. Therefore, the evolution of these behaviors should be examined from the viewpoint of the evolution of cooperation. The iterated prisoner's dilemma (IPD) game is generally considered to be a standard model for the evolution of cooperation. In the present study, strategies accompanied by punishment-like attacks or apology-like behavior were introduced into the common IPD simulation. Punishment and apology were represented by the P signal and the AS signal given immediately after defection. A strategy with the P and AS signals, named the pPAS strategy, was proved to be an evolutionarily stable strategy under certain conditions. Numerical simulations were carried out according to different assigned values of the costs of punishment and apology. The simulations showed that pPAS could dominate the population (1) when the cost of giving P is relatively small, (2) when the cost of receiving P is relatively large, or (3) when the cost of giving AS is relatively large. The relative cost of giving AS had the clearest effect on the success of pPAS. pPAS can dominate the population even when a dominance asymmetry of the costs between two players was introduced. The present results suggest the possible evolution of social behaviors like punishment or apology as a means of maintaining cooperation. This revised version was published online in November 2006 with corrections to the Cover Date.     

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.     

Replicator dynamics for optional public good games

The public goods game represents a straightforward generalization of the prisoner's dilemma to an arbitrary number of players. Since the dominant strategy is to defect, both classical and evolutionary game theory predict the asocial outcome that no player contributes to the public goods. In contrast to the compulsory public goods game, optional participation provides a natural way to avoid deadlocks in the state of mutual defection. The three resulting strategies--collaboration or defection in the public goods game, as well as not joining at all--are studied by means of a replicator dynamics, which can be completely analysed in spite of the fact that the payoff terms are nonlinear. If cooperation is valuable enough, the dynamics exhibits a rock-scissors-paper type of cycling between the three strategies, leading to sizeable average levels of cooperation in the population. Thus, voluntary participation makes cooperation feasible. But for each strategy, the average payoff value remains equal to the earnings of those not participating in the public goods game.     

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   

Invasion and expansion of cooperators in lattice populations: Prisoner's dilemma vs. snowdrift games

The evolution of cooperation is an enduring conundrum in biology and the social sciences. Two social dilemmas, the prisoner's dilemma and the snowdrift game have emerged as the most promising mathematical metaphors to study cooperation. Spatial structure with limited local interactions has long been identified as a potent promoter of cooperation in the prisoner's dilemma but in the spatial snowdrift game, space may actually enhance or inhibit cooperation. Here we investigate and link the microscopic interaction between individuals to the characteristics of the emerging macroscopic patterns generated by the spatial invasion process of cooperators in a world of defectors. In our simulations, individuals are located on a square lattice with Moore neighborhood and update their strategies by probabilistically imitating the strategies of better performing neighbors. Under sufficiently benign conditions, cooperators can survive in both games. After rapid local equilibration, cooperators expand quadratically until global saturation is reached. Under favorable conditions, cooperators expand as a large contiguous cluster in both games with minor differences concerning the shape of embedded defectors. Under less favorable conditions, however, distinct differences arise. In the prisoner's dilemma, cooperators break up into isolated, compact clusters. The compact clustering reduces exploitation and leads to positive assortment, such that cooperators interact more frequently with other cooperators than with defectors. In contrast, in the snowdrift game, cooperators form small, dendritic clusters, which results in negative assortment and cooperators interact more frequently with defectors than with other cooperators. In order to characterize and quantify the emerging spatial patterns, we introduce a measure for the cluster shape and demonstrate that the macroscopic patterns can be used to determine the characteristics of the underlying microscopic interactions.     

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.     

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.     

Coevolution of trustful buyers and cooperative sellers in the trust game

Many online marketplaces enjoy great success. Buyers and sellers in successful markets carry out cooperative transactions even if they do not know each other in advance and a moral hazard exists. An indispensable component that enables cooperation in such social dilemma situations is the reputation system. Under the reputation system, a buyer can avoid transacting with a seller with a bad reputation. A transaction in online marketplaces is better modeled by the trust game than other social dilemma games, including the donation game and the prisoner's dilemma. In addition, most individuals participate mostly as buyers or sellers; each individual does not play the two roles with equal probability. Although the reputation mechanism is known to be able to remove the moral hazard in games with asymmetric roles, competition between different strategies and population dynamics of such a game are not sufficiently understood. On the other hand, existing models of reputation-based cooperation, also known as indirect reciprocity, are based on the symmetric donation game. We analyze the trust game with two fixed roles, where trustees (i.e., sellers) but not investors (i.e., buyers) possess reputation scores. We study the equilibria and the replicator dynamics of the game. We show that the reputation mechanism enables cooperation between unacquainted buyers and sellers under fairly generous conditions, even when such a cooperative equilibrium coexists with an asocial equilibrium in which buyers do not buy and sellers cheat. In addition, we show that not many buyers may care about the seller's reputation under cooperative equilibrium. Buyers' trusting behavior and sellers' reputation-driven cooperative behavior coevolve to alleviate the social dilemma.     

Eco-evolutionary feedback and the invasion of cooperation in prisoner's dilemma games

Unveiling the origin and forms of cooperation in nature poses profound challenges in evolutionary ecology. The prisoner's dilemma game is an important metaphor for studying the evolution of cooperation. We here classified potential mechanisms for cooperation evolution into schemes of frequency- and density-dependent selection, and focused on the density-dependent selection in the ecological prisoner's dilemma games. We found that, although assortative encounter is still the necessary condition in ecological games for cooperation evolution, a harsh environment, indicated by a high mortality, can foster the invasion of cooperation. The Hamilton rule provides a fundamental condition for the evolution of cooperation by ensuring an enhanced relatedness between players in low-density populations. Incorporating ecological dynamics into evolutionary games opens up a much wider window for the evolution of cooperation, and exhibits a variety of complex behaviors of dynamics, such as limit and heteroclinic cycles. An alternative evolutionary, or rather succession, sequence was proposed that cooperation first appears in harsh environments, followed by the invasion of defection, which leads to a common catastrophe. The rise of cooperation (and altruism), thus, could be much easier in the density-dependent ecological games than in the classic frequency-dependent evolutionary games.     

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.     

The evolution of altruism: game theory in multilevel selection and inclusive fitness

Although the prisoner's dilemma (PD) has been used extensively to study reciprocal altruism, here we show that the n-player prisoner's dilemma (NPD) is also central to two other prominent theories of the evolution of altruism: inclusive fitness and multilevel selection. An NPD model captures the essential factors for the evolution of altruism directly in its parameters and integrates important aspects of these two theories such as Hamilton's rule, Simpson's paradox, and the Price covariance equation. The model also suggests a simple interpretation of the Price selection decomposition and an alternative decomposition that is symmetrical and complementary to it. In some situations this alternative shows the temporal changes in within- and between-group selection more clearly than the Price equation. In addition, we provide a new perspective on strong vs. weak altruism by identifying their different underlying game structures (based on absolute fitness) and showing how their evolutionary dynamics are nevertheless similar under selection (based on relative fitness). In contrast to conventional wisdom, the model shows that both strong and weak altruism can evolve in periodically formed random groups of non-conditional strategies if groups are multigenerational. An integrative approach based on the NPD helps unify different perspectives on the evolution of altruism.     

The economic basis of cooperation: tradeoffs between selfishness and generosity

The current study examined the economics of cooperation in controlled-payoffgames by using captive blue jays, Cyanocitta cristata. Thisinvestigation used a special feeding apparatus to test for thestability of cooperative choice in a series of iterated games.The jays experienced experimentally determined game theoreticalpayoff matrices, which determined the distribution of food tothemselves and their opponent, depending on their decision tocooperate or defect. The experiment tested four game matrices,called the cooperate only, defect only, prisoner's dilemma,and opponent control treatments. This study found little cooperationin the defect only and prisoner's dilemma treatments. Cooperationoccurred significantly more often in the opponent control treatment.These findings suggest that the jays attend to short-term consequences;they do not cooperate in the absence of an immediate benefit(defect only), even if a long-term benefit may exist (prisoner'sdilemma). The opponent control treatment suggests that cooperationcan occur when an individual's benefits depend completely onthe actions of others; therefore, generosity is cheap. Thisstudy, therefore, agrees with recent studies in proposing alternativemodels of cooperation.