A study on emergence of alternating reciprocity in a 2 x 2 game with 2-length memory strategy

It is recognized that bilateral cooperation (C), a reward-state in other words, emergently comes up in a 2 × 2 prisoner's dilemma game, if you assume a strategy set with a memory concept. Also observed is a mixture state of cooperation (C) and defect (D), saint- and temptation-state in other words, to obtain a higher payoff than R (R reciprocity) in a hero or leader game that is a chicken-type dilemma game; this phenomenon is called alternating reciprocity (AR) or ST reciprocity. Observing a holistic 2 × 2 game world including trivial games and various dilemma games, where 2-length memory and infinite interactions are assumed, the paper reports on the specific mechanism of AR. It is observed there are three different phases relating to AR, which can be explained by the stress of the dilemma.     

Evolution of Conformity in Social Dilemmas

People often deviate from their individual Nash equilibrium strategy in game experiments based on the prisoner’s dilemma (PD) game and the public goods game (PGG), whereas conditional cooperation, or conformity, is supported by the data from these experiments. In a complicated environment with no obvious “dominant” strategy, conformists who choose the average strategy of the other players in their group could be able to avoid risk by guaranteeing their income will be close to the group average. In this paper, we study the repeated PD game and the repeated m-person PGG, where individuals’ strategies are restricted to the set of conforming strategies. We define a conforming strategy by two parameters, initial action in the game and the influence of the other players’ choices in the previous round. We are particularly interested in the tit-for-tat (TFT) strategy, which is the well-known conforming strategy in theoretical and empirical studies. In both the PD game and the PGG, TFT can prevent the invasion of non-cooperative strategy if the expected number of rounds exceeds a critical value. The stability analysis of adaptive dynamics shows that conformity in general promotes the evolution of cooperation, and that a regime of cooperation can be established in an AllD population through TFT-like strategies. These results provide insight into the emergence of cooperation in social dilemma games.     

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.     

Direct Reciprocity in Spatial Populations Enhances R-Reciprocity As Well As ST-Reciprocity

As is well-known, spatial reciprocity plays an important role in facilitating the emergence of cooperative traits, and the effect of direct reciprocity is also obvious for explaining the cooperation dynamics. However, how the combination of these two scenarios influences cooperation is still unclear. In the present work, we study the evolution of cooperation in 2×2 games via considering both spatial structured populations and direct reciprocity driven by the strategy with 1-memory length. Our results show that cooperation can be significantly facilitated on the whole parameter plane. For prisoner''s dilemma game, cooperation dominates the system even at strong dilemma, where maximal social payoff is still realized. In this sense, R-reciprocity forms and it is robust to the extremely strong dilemma. Interestingly, when turning to chicken game, we find that ST-reciprocity is also guaranteed, through which social average payoff and cooperation is greatly enhanced. This reciprocity mechanism is supported by mean-field analysis and different interaction topologies. Thus, our study indicates that direct reciprocity in structured populations can be regarded as a more powerful factor for the sustainability of cooperation.     

A study of indirect reciprocity involving a reputation system or a simple tag system in a one-shot, multi-player game

The possibility of the evolution of cooperation backed by indirect reciprocity (IDR) in a one-shot, multi-player game is investigated focusing on two mechanisms. First of all, the reputation system with image score (RS with IS), as proposed by Nowak and Sigmund [Nowak, M.A., Sigmund, K., 1998. Evolution of indirect reciprocity by image scoring. Nature 393, 573–577], is investigated in various game structures. A numerical experiment demonstrates that the RS with IS is a robust mechanism for the support of IDR in various dilemma games, but whose effectiveness decreases with an increase in the number of players in a game. It is fair to say that the RS is an information mapping function to relate between player's cooperative fraction on his action and IS. As the second mechanism, a simple tag system which could be applicable to animals having no cognitive complexity is considered to support IDR. Computer simulations of the tag system's strategy for invading a population initially consisting of AllD, AllC, and Random strategies suggest several novel facts. The Tag strategy with plausible crossover and mutation probabilities can only invade to settle down if the game structure is not Trivial and contains a moderate dilemma. The Tag strategy's evolutionary competition takes place mainly in the presence of the AllD strategy. During the competition, the Tag strategy frequently metamorphoses to shake off the AllD strategy, but stops after winning in order to avoid a shrinking payoff due to fragmentation.     

What initially brought about communications?

This paper reports an intelligent agent equipped with two-layer finite state machines (FSMs) that can communicate by turning lighting on and off, leading to social cooperation that solves the dilemma situation, modeled by a one-shot 2x2 game. This communication between two gaming agents can be observed in hero- and leader-type dilemma games, where alternating reciprocity, repeating cooperation (C)-defeat (D) after D-C, is the equal pareto optimum instead of a sequence of mutual cooperation that is the equal pareto optimum for a prisoner's dilemma (PD) game.     

Strategy Selection in Evolutionary Game Dynamics on Group Interaction Networks

Evolutionary game theory provides an appropriate tool for investigating the competition and diffusion of behavioral traits in biological or social populations. A core challenge in evolutionary game theory is the strategy selection problem: Given two strategies, which one is favored by the population? Recent studies suggest that the answer depends not only on the payoff functions of strategies but also on the interaction structure of the population. Group interactions are one of the fundamental interactive modes within populations. This work aims to investigate the strategy selection problem in evolutionary game dynamics on group interaction networks. In detail, the strategy selection conditions are obtained for some typical networks with group interactions. Furthermore, the obtained conditions are applied to investigate selection between cooperation and defection in populations. The conditions for evolution of cooperation are derived for both the public goods game and volunteer’s dilemma game. Numerical experiments validate the above analytical results.     

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.     

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.     

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.     

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.     

A simple scaling of the effectiveness of supporting mutual cooperation in donor-recipient games by various reciprocity mechanisms

In donor-recipient games (DRG), one of the sub-classes of Prisoner's Dilemma (PD), it is well-known that a game structure is described by two parameters benefit (b) and cost (c) of cooperation. By means of a series of numerical experiments, we proved that the effectiveness of supporting mutual cooperation in DRG by various reciprocity mechanisms can be expressed in a single game structural parameter, b/c. This also implies that the dilemma strength in various donor-recipient games with various reciprocity mechanisms can be evaluated only by b/c, which is consistent with the previous novel finding by Nowak. It was also discussed whether this kind of parameterization idea can be extended to general games in PD game class.     

Second‐order cooperation: Cooperative offspring as a living public good arising from second‐order selection on non‐cooperative individuals

Switching rate between cooperating and non‐cooperating genotypes is a crucial social evolution factor, often neglected by game theory‐inspired theoretical and experimental frameworks. We show that the evolution of alleles increasing the mutation or phenotypic switching rates toward cooperation is in itself a social dilemma. Although cooperative offspring are often unlikely to reproduce, due to high cost of cooperation, they can be seen both as a living public good and a part of the extended parental phenotype. The competition between individuals that generate cooperators and ones that do not is often more relevant than the competition between cooperators and non‐cooperators. The dilemma of second‐order cooperation we describe relates directly to eusociality, but can be also interpreted as a division of labor or a soma‐germline distinction. The results of our simulations shine a new light on what Darwin had already termed a “special difficulty” of evolutionary theory and describe a novel type of cooperation dynamics.     

Evolution of Cooperation in Spatial Traveler's Dilemma Game

Traveler''s dilemma (TD) is one of social dilemmas which has been well studied in the economics community, but it is attracted little attention in the physics community. The TD game is a two-person game. Each player can select an integer value between and () as a pure strategy. If both of them select the same value, the payoff to them will be that value. If the players select different values, say and (), then the payoff to the player who chooses the small value will be and the payoff to the other player will be . We term the player who selects a large value as the cooperator, and the one who chooses a small value as the defector. The reason is that if both of them select large values, it will result in a large total payoff. The Nash equilibrium of the TD game is to choose the smallest value . However, in previous behavioral studies, players in TD game typically select values that are much larger than , and the average selected value exhibits an inverse relationship with . To explain such anomalous behavior, in this paper, we study the evolution of cooperation in spatial traveler''s dilemma game where the players are located on a square lattice and each player plays TD games with his neighbors. Players in our model can adopt their neighbors'' strategies following two standard models of spatial game dynamics. Monte-Carlo simulation is applied to our model, and the results show that the cooperation level of the system, which is proportional to the average value of the strategies, decreases with increasing until is greater than the critical value where cooperation vanishes. Our findings indicate that spatial reciprocity promotes the evolution of cooperation in TD game and the spatial TD game model can interpret the anomalous behavior observed in previous behavioral experiments.     

ASSORTMENT AND THE EVOLUTION OF GENERALIZED RECIPROCITY

Reciprocity is often invoked to explain cooperation. Reciprocity is cognitively demanding, and both direct and indirect reciprocity require that individuals store information about the propensity of their partners to cooperate. By contrast, generalized reciprocity, wherein individuals help on the condition that they received help previously, only relies on whether an individual received help in a previous encounter. Such anonymous information makes generalized reciprocity hard to evolve in a well‐mixed population, as the strategy will lose out to pure defectors. Here we analyze a model for the evolution of generalized reciprocity, incorporating assortment of encounters, to investigate the conditions under which it will evolve. We show that, in a well‐mixed population, generalized reciprocity cannot evolve. However, incorporating assortment of encounters can favor the evolution of generalized reciprocity in which indiscriminate cooperation and defection are both unstable. We show that generalized reciprocity can evolve under both the prisoner's dilemma and the snowdrift game.     

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.     

Coveting thy neighbors fitness as a means to resolve social dilemmas

In spatial evolutionary games the fitness of each individual is traditionally determined by the payoffs it obtains upon playing the game with its neighbors. Since defection yields the highest individual benefits, the outlook for cooperators is gloomy. While network reciprocity promotes collaborative efforts, chances of averting the impending social decline are slim if the temptation to defect is strong. It is, therefore, of interest to identify viable mechanisms that provide additional support for the evolution of cooperation. Inspired by the fact that the environment may be just as important as inheritance for individual development, we introduce a simple switch that allows a player to either keep its original payoff or use the average payoff of all its neighbors. Depending on which payoff is higher, the influence of either option can be tuned by means of a single parameter. We show that, in general, taking into account the environment promotes cooperation. Yet coveting the fitness of one's neighbors too strongly is not optimal. In fact, cooperation thrives best only if the influence of payoffs obtained in the traditional way is equal to that of the average payoff of the neighborhood. We present results for the prisoner's dilemma and the snowdrift game, for different levels of uncertainty governing the strategy adoption process, and for different neighborhood sizes. Our approach outlines a viable route to increased levels of cooperative behavior in structured populations, but one that requires a thoughtful implementation.     

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.     

The iterated continuous prisoner's dilemma game cannot explain the evolution of interspecific mutualism in unstructured populations

The evolutionionary origin of inter- and intra-specific cooperation among non-related individuals has been a great challenge for biologists for decades. Recently, the continuous prisoner's dilemma game has been introduced to study this problem. In function of previous payoffs, individuals can change their cooperative investment iteratively in this model system. Killingback and Doebeli (Am. Nat. 160 (2002) 421-438) have shown analytically that intra-specific cooperation can emerge in this model system from originally non-cooperating individuals living in a non-structured population. However, it is also known from an earlier numerical work that inter-specific cooperation (mutualism) cannot evolve in a very similar model. The only difference here is that cooperation occurs among individuals of different species. Based on the model framework used by Killingback and Doebeli (2002), this Note proves analytically that mutualism indeed cannot emerge in this model system. Since numerical results have revealed that mutualism can evolve in this model system if individuals interact in a spatially structured manner, our work emphasizes indirectly the role of spatial structure of populations in the origin of mutualism.     

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.