The conflict between individual and collective interests is in the heart of every social dilemmas established by evolutionary game theory. We cannot avoid these conflicts but sometimes we may choose which interaction framework to use as a battlefield. For instance some people like to be part of a larger group while other persons prefer to interact in a more personalized, individual way. Both attitudes can be formulated via appropriately chosen traditional games. In particular, the prisoner's dilemma game is based on pair interaction while the public goods game represents multi-point interactions of group members. To reveal the possible advantage of a certain attitude we extend these models by allowing players not simply to change their strategies but also let them to vary their attitudes for a higher individual income. We show that both attitudes could be the winner at a specific parameter value. Interestingly, however, the subtle interplay between different states may result in a counterintuitive evolutionary outcome where the increase of the multiplication factor of public goods game drives the population to a fully defector state. We point out that the accompanying pattern formation can only be understood via the multipoint or multi-player interactions of different microscopic states where the vicinity of a particular state may influence the relation of two other competitors.
In an iterated non-cooperative game, if all the players act to maximize their individual accumulated payoff, the system as a whole usually converges to a Nash equilibrium that poorly benefits any player. Here we show that such an undesirable destiny is avoidable in an iterated Rock-Paper-Scissors (RPS) game involving two players X and Y. Player X has the option of proactively adopting a cooperation-trap strategy, which enforces complete cooperation from the rational player Y and leads to a highly beneficial as well as maximally fair situation to both players. That maximal degree of cooperation…
We study the evolution of cooperation within the spatial prisoner's dilemma game on a square lattice where a fraction of players $μ$ can spread their strategy more easily than the rest due to a predetermined larger teaching capability. In addition, players characterized with the larger teaching capability are allowed to temporarily link with distant opponents of the same kind with probability $p$, thus introducing shortcut connections among the distinguished. We show that these additional temporary connections are able to sustain cooperation throughout the whole range of the temptation to defe…
Spatial reciprocity is a well known tour de force of cooperation promotion. A thorough understanding of the effects of different population densities is therefore crucial. Here we study the evolution of cooperation in social dilemmas on different interaction graphs with a certain fraction of vacant nodes. We find that sparsity may favor the resolution of social dilemmas, especially if the population density is close to the percolation threshold of the underlying graph. Regardless of the type of the governing social dilemma as well as particularities of the interaction graph, we show that under…
This online supplement mainly presents the details of empirical investigations of so-called "cooperation sharing" distributions in 12 real world cooperation-competition systems, which will be used for building an evolution model that will be reported elsewhere. All the distributions obey a function form P(ω)(ω+α)^(gamma). The heterogeneities of the cooperation sharing distributions in different systems can be described by two distribution parameters,αand γ. The empirically determined system evolution durations are also presented.
Inspired by the fact that people have diverse propensities to punish wrongdoers, we study a spatial public goods game with defectors and different types of punishing cooperators. During the game, cooperators punish defectors with class-specific probabilities and subsequently share the associated costs of sanctioning. We show that in the presence of different punishing cooperators the highest level of public cooperation is always attainable through a selection mechanism. Interestingly, the selection not necessarily favors the evolution of punishers who would be able to prevail on their own agai…
According to the evolutionary death-birth protocol, a player is chosen randomly to die and neighbors compete for the available position proportional to their fitness. Hence, the status of the focal player is completely ignored and has no impact on the strategy update. In this work, we revisit and generalize this rule by introducing a weight factor to compare the payoff values of the focal and invading neighbors. By means of evolutionary graph theory, we analyze the model on joint transitive graphs to explore the possible consequences of the presence of a weight factor. We find that focal weigh…
We study a spatial Prisoner's dilemma game with two types (A and B) of players located on a square lattice. Players following either cooperator or defector strategies play Prisoner's Dilemma games with their 24 nearest neighbors. The players are allowed to adopt one of their neighbor's strategy with a probability dependent on the payoff difference and type of the given neighbor. Players A and B have different efficiency in the transfer of their own strategy therefore the strategy adoption probability is reduced by a multiplicative factor (w < 1) from the players of type B. We report that the m…
Systems with simultaneous cooperation and competition among the elements are ubiquitous. In spite of their practical importance, knowledge on the evolution mechanism of this class of complex system is still very limit. In this work, by conducting extensive empirical survey to a large number of cooperation-competition systems which cover wide categories and contain the information of network topology, cooperation-competition gain, and the evolution time, we try to get some insights to the universal mechanism of their evolutions. Empirical investigations show that the distributions of the cooper…
According to the evolutionary game theory principle, a strategy representing a higher payoff can spread among competitors. But there are cases when a player consistently overestimates or underestimates her own payoff, which undermines proper comparison. Interestingly, both underconfident and overconfident individuals are capable of elevating the cooperation level significantly. While former players stimulate a local coordination of strategies, the presence of overconfident individuals enhances the spatial reciprocity mechanism. In both cases the propagations of competing strategies are influen…
We study the evolution of cooperation in structured populations within popular models of social dilemmas, whereby simple coevolutionary rules are introduced that may enhance players abilities to enforce their strategy on the opponent. Coevolution thus here refers to an evolutionary process affecting the teaching activity of players that accompanies the evolution of their strategies. Particularly, we increase the teaching activity of a player after it has successfully reproduced, yet we do so depending on the disseminated strategy. We separately consider coevolution affecting either only the co…
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 ir…
The hypothesis that living systems can benefit from operating at the vicinity of critical points has gained momentum in recent years. Criticality may confer an optimal balance between exceedingly ordered and too noisy states. We here present a model, based on information theory and statistical mechanics, illustrating how and why a community of agents aimed at understanding and communicating with each other converges to a globally coherent state in which all individuals are close to an internal critical state, i.e. at the borderline between order and disorder. We study --both analytically and c…
Recent work highlights that cooperation in the one-shot Prisoner's dilemma (PD) is primarily driven by moral preferences for doing the right thing, rather than social preferences for equity or efficiency. By contrast, little is known on what motivates cooperation in the Stag-Hunt Game (SHG). Cooperation in the SHG fundamentally differs from cooperation in the PD in that it is not costly, but risky: players have no temptation to deviate from the cooperative outcome, but cooperation only pays off if the other player cooperates. Here, we provide data from a large (N=436), pre-registered, experime…
Indirect reciprocity in which players cooperate with unacquainted other players having good reputations is a mechanism for cooperation in relatively large populations subjected to social dilemma situations. When the population has group structure, as is often found in social networks, players in experiments are considered to show behavior that deviates from existing theoretical models of indirect reciprocity. First, players often show ingroup favoritism (i.e., cooperation only within the group) rather than full cooperation (i.e., cooperation within and across groups), even though the latter is…
We show that the heterogeneity index, which was proposed by Hu and Wang (Physica A 2008 387 3769), can be used to describe the disparity of the cooperation sharing or competition gain distributions that is very important for the cooperation/competition system dynamic understanding. An analytical relation between the distribution parameters and the heterogeneity index is derived, which is in a good agreement with the empirical results. Our theoretical and empirical analyses also show that the relation between the distribution parameters can be analytically derived from so-called Zhang-Chang mod…
In the traditional setup of public goods game all players are involved in every available groups and the mutual benefit is shared among competing cooperator and defector strategies. But in real life situations the group formation of players could be more sophisticated because not all players are attractive enough for others to participate in a joint venture. What if only those players can initiate a group formation and establish a game who are successful enough to the neighbors? To elaborate this idea we employ a modified protocol and demonstrate that a carefully chosen threshold to establish …
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 $R$ and $M$ ($R < M$) 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 $i$ and $j$ ($R \le i < j \le M$), then the payoff to the player who chooses the small value will be $i+R$ and the payoff to the other player will be $i-R$. We term the player who selec…
Interactions among individuals in natural populations often occur in a dynamically changing environment. Understanding the role of environmental variation in population dynamics has long been a central topic in theoretical ecology and population biology. However, the key question of how individuals, in the middle of challenging social dilemmas (e.g., the "tragedy of the commons"), modulate their behaviors to adapt to the fluctuation of the environment has not yet been addressed satisfactorily. Utilizing evolutionary game theory and stochastic games, we develop a game-theoretical framework that…
In previous studies of spatial public goods game, each player is able to establish a group. However, in real life, some players cannot successfully organize groups for various reasons. In this paper, we propose a mechanism of reputation-driven group formation, in which groups can only be organized by players whose reputation reaches or exceeds a threshold. We define a player's reputation as the frequency of cooperation in the last $T$ time steps. We find that the highest cooperation level can be obtained when groups are only established by pure cooperators who always cooperate in the last $T$ …
Evolution of cooperation in the prisoner's dilemma game is studied where initially all players are linked via a regular graph, having four neighbors each. Simultaneously with the strategy evolution, players are allowed to make new connections and thus permanently extend their neighborhoods, provided they have been successful in passing their strategy to the opponents. We show that this simple coevolutionary rule shifts the survival barrier of cooperators towards high temptations to defect and results in highly heterogeneous interaction networks with an exponential fit best characterizing their…
We study the public goods game in the noisy case by considering the players with inhomogeneous activity teaching on a square lattice. It is shown that the introduction of the inhomogeneous activity of teaching of the players can remarkably promote cooperation. By investigating the effects of noise on cooperative behavior in detail, we find that the variation of cooperator density $ρ_C$ with the noise parameter $κ$ displays several different behaviors: $ρ_C$ monotonically increases (decreases) with $κ$; $ρ_C$ firstly increases (decreases) with $κ$ and then it decreases (increases) monotonically…
To be the fittest is central to proliferation in evolutionary games. Individuals thus adopt the strategies of better performing players in the hope of successful reproduction. In structured populations the array of those that are eligible to act as strategy sources is bounded to the immediate neighbors of each individual. But which one of these strategy sources should potentially be copied? Previous research dealt with this question either by selecting the fittest or by selecting one player uniformly at random. Here we introduce a parameter $u$ that interpolates between these two extreme optio…
In the last few decades, numerous experiments have shown that humans do not always behave so as to maximize their material payoff. Cooperative behavior when non-cooperation is a dominant strategy (with respect to the material payoffs) is particularly puzzling. Here we propose a novel approach to explain cooperation, assuming what Halpern and Pass (2013) call "translucent players". Typically, players are assumed to be "opaque", in the sense that a deviation by one player does not affect the strategies used by other players. But a player may believe that if he switches from one strategy to anoth…
We study a spatial two-strategy (cooperation and defection) Prisoner's Dilemma game with two types ($A$ and $B$) of players located on the sites of a square lattice. The evolution of strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff difference and also on the type of the neighbor. The strategy adoption probability is reduced by a pre-factor ($w < 1$) from the players of type $B$. We consider the competition between two opposite effects when increasing the number of neighbors ($k=4$, 8, and 24). Within a …
We study a condition of favoring cooperation in a Prisoner's Dilemma game on complex networks. There are two kinds of players: cooperators and defectors. Cooperators pay a benefit b to their neighbors at a cost c, whereas defectors only receive a benefit. The game is a death-birth process with weak selection. Although it has been widely thought that b/c> <k> is a condition of favoring cooperation, we find that b/c> <knn> is the condition. We also show that among three representative networks, namely, regular, random, and scale-free, a regular network favors cooperation the most, whereas a scal…
The basic problem in the cooperation theory is to justify the cooperation. Here we propose a new approach, where players are driven by their altruism to cooperate or not. The probability of cooperation depends also on the co-player's reputation. We find that players with positive altruism cooperate and met cooperation. In this approach, payoffs are not relevant. The mechanism is most efficient in the fully connected network.
Strategy changes are an essential part of evolutionary games. Here we introduce a simple rule that, depending on the value of a single parameter $w$, influences the selection of players that are considered as potential sources of the new strategy. For positive $w$ players with high payoffs will be considered more likely, while for negative $w$ the opposite holds. Setting $w$ equal to zero returns the frequently adopted random selection of the opponent. We find that increasing the probability of adopting the strategy from the fittest player within reach, i.e. setting $w$ positive, promotes the …
This paper investigates the evolution of strategic play where players drawn from a finite well-mixed population are offered the opportunity to play in a public goods game. All players accept the offer. However, due to the possibility of unforeseen circumstances, each player has a fixed probability of being unable to participate in the game, unlike similar models which assume voluntary participation. We first study how prescribed stochastic opting-out affects cooperation in finite populations. Moreover, in the model, cooperation is favored by natural selection over both neutral drift and defect…
In recent years, there has been growing interest in studying evolutionary games with environmental feedback. Previous studies exclusively focus on two-player games. However, extension to multi-player game is needed to study problems such as microbial cooperation and crowdsourcing collaborations. Here, we study coevolutionary public goods games where strategies coevolve with the multiplication factors of group cooperation. Asymmetry can arise in such environmental feedback, where games organized by focal cooperators may have a different efficiency than the ones by defectors. Our analysis shows …
In this paper, we empirically analyze the spatial distribution of Chinese cities using a method based on triangle transition. This method uses a regular triangle mapping from the observed cities and its three neighboring cities to analyze their distribution of mapping positions. We find that obvious center-gathering tendency for the relationship between cities and its nearest three cities, indicating the spatial competition between cities. Moreover, we observed the competitive trends between neighboring cities with similar economic volume, and the remarkable cooperative tendency between neighb…
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 o…
Evolution of cooperation in the prisoner's dilemma and the public goods game is studied, where initially players belong to two independent structured populations. Simultaneously with the strategy evolution, players whose current utility exceeds a threshold are rewarded by an external link to a player belonging to the other population. Yet as soon as the utility drops below the threshold, the external link is terminated. The rewarding of current evolutionary fitness thus introduces a time-varying interdependence between the two populations. We show that, regardless of the details of the evoluti…
Real social interactions occur on networks in which each individual is connected to some, but not all, of others. In social dilemma games with a fixed population size, heterogeneity in the number of contacts per player is known to promote evolution of cooperation. Under a common assumption of positively biased payoff structure, well-connected players earn much by playing frequently, and cooperation once adopted by well-connected players is unbeatable and spreads to others. However, maintaining a social contact can be costly, which would prevent local payoffs from being positively biased. In re…
The diversity in wealth and social status is present not only among humans, but throughout the animal world. We account for this observation by generating random variables that determ ine the social diversity of players engaging in the prisoner's dilemma game. Here the term social diversity is used to address extrinsic factors that determine the mapping of game pay offs to individual fitness. These factors may increase or decrease the fitness of a player depending on its location on the spatial grid. We consider different distributions of extrin sic factors that determine the social diversity …
A network effect is introduced taking into account competition, cooperation and mixed-type interaction amongst agents along a generalized Verhulst-Lotka-Volterra model. It is also argued that the presence of a market capacity enforces an indubious limit on the agent's size growth. The state stability of triadic agents, i.e., the most basic network plaquette, is investigated analytically for possible scenarios, through a fixed point analysis. It is discovered that: (i) \market" demand is only satisfied for full competition when one agent monopolizes the market; (ii) growth of agent size is enco…
Properties of cooperation's probability function in Prisoner`s Dilemma have impact on evolution of game. Basic model defines that probability of cooperation depends linearly, both on the player's altruism and the co-player's reputation. I propose modification of activation function to smooth one (hyperbolic tangent with scaling parameter a, which corresponds to its shape) and observe three phases for different range of a. (1) For small a, strategies seem to randomly change in time and situation of mixed choices (one cooperates and second defects) dominate. (2) For medium a, players choose only…
Social hierarchy is important that can not be ignored in human socioeconomic activities and in the animal world. Here we incorporate this factor into the evolutionary game to see what impact it could have on the cooperation outcome. The probabilistic strategy adoption between two players is then not only determined by their payoffs, but also by their hierarchy difference -- players in the high rank are more likely to reproduce their strategies than the peers in the low rank. Through simulating the evolution of Prisoners' dilemma game with three hierarchical distributions, we find that the leve…
In the study of the evolution of cooperation, many mechanisms have been proposed to help overcome the self-interested cheating that is individually optimal in the Prisoners' Dilemma game. These mechanisms include assortative or networked social interactions, other-regarding preferences considering the payoffs of others, reciprocity rules to establish cooperation as a social norm, and multilevel selection involving simultaneous competition between individuals favoring cheaters and competition between groups favoring cooperators. In this paper, we build on recent work studying PDE replicator equ…
We study the evolution of cooperation in the spatial prisoner's dilemma game, where besides unconditional cooperation and defection, tit-for-tat, win-stay-lose-shift and extortion are the five competing strategies. While pairwise imitation fails to sustain unconditional cooperation and extortion regardless of game parametrization, myopic updating gives rise to the coexistence of all five strategies if the temptation to defect is sufficiently large or if the degree distribution of the interaction network is heterogeneous. This counterintuitive evolutionary outcome emerges as a result of an unex…
The emergence and maintenance of cooperation within sizable groups of unrelated humans offer many challenges for our understanding. We propose that the humans' capacity of communication, such as how many and how far away the fellows can build up mutual communications, may affect the evolution of cooperation. We study this issue by means of the public goods game (PGG) with a two-layered network of contacts. Players obtain payoffs from five-person public goods interactions on a square lattice (the interaction layer). Also, they update strategies after communicating with neighbours in learning la…
We show that the resolution of social dilemmas on random graphs and scale-free networks is facilitated by imitating not the strategy of better performing players but rather their emotions. We assume sympathy and envy as the two emotions that determine the strategy of each player by any given interaction, and we define them as probabilities to cooperate with players having a lower and higher payoff, respectively. Starting with a population where all possible combinations of the two emotions are available, the evolutionary process leads to a spontaneous fixation to a single emotional profile tha…
Indirect reciprocity based on reputation is a leading mechanism driving human cooperation, where monitoring of behaviour and sharing reputation-related information are crucial. Because collecting information is costly, a tragedy of the commons can arise, with some individuals free-riding on information supplied by others. This can be overcome by organising monitors that aggregate information, supported by fees from their information users. We analyse a co-evolutionary model of individuals playing a social dilemma game and monitors watching them; monitors provide information and players vote fo…
Productive societies feature high levels of cooperation and strong connections between individuals. Public Goods Games (PGGs) are frequently used to study the development of social connections and cooperative behavior in model societies. In such games, contributions to the public good are made only by cooperators, while all players, including defectors, can reap public goods benefits. Classic results of game theory show that mutual defection, as opposed to cooperation, is the Nash Equilibrium of PGGs in well-mixed populations, where each player interacts with all others. In this paper, we expl…
In the context of Evolutionary Game Theory, one of the most noteworthy mechanisms to support cooperation is spatial reciprocity, usually accomplished by distributing players in a spatial structure allowing cooperators to cluster together and avoid exploitation. This raises an important question: how is the survival of cooperation affected by different topologies? Here, to address this question, we explore the Focal Public Goods (FPGG) and classic Public Goods Games (PGG), and the Prisoner's Dilemma (PD) on several regular lattices: honeycomb, square (with von Neumann and Moore neighborhoods), …
We investigate the evolutionary prisoner's dilemma game in structured populations by introducing dimers, which are defined as that two players in each dimer always hold a same strategy. We find that influences of dimers on cooperation depend on the type of dimers and the population structure. For those dimers in which players interact with each other, the cooperation level increases with the number of dimers though the cooperation improvement level depends on the type of network structures. On the other hand, the dimers, in which there are not mutual interactions, will not do any good to the c…
Evolutionary games are studied where the teaching activity of players can evolve in time. Initially all players following either the cooperative or defecting strategy are distributed on a square lattice. The rate of strategy adoption is determined by the payoff difference and a teaching activity characterizing the donor's capability to enforce its strategy on the opponent. Each successful strategy adoption process is accompanied with an increase in the donor's teaching activity. By applying an optimum value of the increment this simple mechanism spontaneously creates relevant inhomogeneities i…
An active participation of players in evolutionary games depends on several factors, ranging from personal stakes to the properties of the interaction network. Diverse activity patterns thus have to be taken into account when studying the evolution of cooperation in social dilemmas. Here we study the weak prisoner's dilemma game, where the activity of each player is determined in a probabilistic manner either by its degree or by its payoff. While degree-correlated activity introduces cascading failures of cooperation that are particularly severe on scale-free networks with frequently inactive …
Although cooperation is central to the organisation of many social systems, relatively little is known about cooperation in situations of collective emergency. When groups of people flee from a danger such as a burning building or a terrorist attack, the collective benefit of cooperation is important, but the cost of helping is high and the temptation to defect is strong. To explore the degree of cooperation in emergencies, we develop a new social game, the help-or-escape social dilemma. Under time and monetary pressure, players decide how much risk they are willing to take in order to help ot…
Individuals might abstain from participating in an instance of an evolutionary game for various reasons, ranging from lack of interest to risk aversion. In order to understand the consequences of such diverse activity patterns on the evolution of cooperation, we study a weak prisoner's dilemma where each player's participation is probabilistic rather than certain. Players that do not participate get a null payoff and are unable to replicate. We show that inactivity introduces cascading failures of cooperation, which are particularly severe on scale-free networks with frequently inactive hubs. …
Predicting how cooperative behavior arises in the thermodynamic limit is one of the outstanding problems in evolutionary game theory. For two player games, cooperation is seldom the Nash equilibrium. However, in the thermodynamic limit cooperation is the natural recourse regardless of whether we are dealing with humans or animals. In this work, we use the analogy with the Ising model to predict how cooperation arises in the thermodynamic limit.
Dynamics of evolutionary games strongly depend on underlying networks. We study the coevolutionary prisoner's dilemma in which players change their local networks as well as strategies (i.e., cooperate or defect). This topic has been increasingly explored by many researchers. On the basis of active linking dynamics [J. M. Pacheco et al., J. Theor. Biol. 243, 437 (2006), J. M. Pacheco et al., Phys. Rev. Lett. 97, 258103 (2006)], we show that cooperation is enhanced fairly robustly. In particular, cooperation evolves when the payoff of the player is normalized by the number of neighbors; this is…
Motion is a typical reaction among animals and humans trying to reach better conditions in a changing world. This aspect has been studied intensively in social dilemmas where competing players' individual and collective interests are in conflict. Starting from the traditional public goods game model, where players are locally fixed and unconditional cooperators or defectors are present, we introduce two additional strategies through which agents can change their positions of dependence on the local cooperation level. More importantly, these so-called sophisticated players should bear an extra …
