Stochastic dynamics of the prisoner's dilemma with cooperation facilitators

Hostility prevents the tragedy of the commons in metapopulation with asymmetric migration: A lesson from queenless ants

A colony of the queenless ant species, Pristomyrmex punctatus, can broadly be seen as consisting of small-body sized worker ants and relatively larger body-sized cheater ants. Hence, in the presence of intercolony migration, a set of constituent colonies act as a metapopulation exclusively composed of cooperators and defectors. Such a setup facilitates an evolutionary game-theoretic replication-selection model of population dynamics of the ants in a metapopulation. Using the model, we analytically probe the effects of territoriality induced hostility. Such hostility in the ant metapopulation proves to be crucial in preventing the tragedy of the commons, specifically, the workforce, a social good formed by cooperation. This mechanism applies to any metapopulation-not necessarily the ants-composed of cooperators and defectors where interpopulation migration occurs asymmetrically, i.e., cooperators and defectors migrate at different rates. Furthermore, our model validates that there is evolutionary benefit behind the queenless ants' behavior of showing more hostility towards the immigrants from nearby colonies than those from the far-off ones. In order to calibrate our model's parameters, we have extensively used the data available on the queenless ant species, P. punctatus.

Zealous cooperation does not always promote cooperation in public goods games

There is a conventional belief that prosocial behaviors cannot arise through selfish human nature, because defection always exploits cooperation to achieve a higher payoff at an individual level. Unyieldingly, some people hope to move society to cooperation through their zealous cooperation, regardless of payoffs. From the perspective of spatial evolutionary games, however, such zealous behavior is unnecessary because cooperation can emerge from selfish human nature by aggregating in evolution. Yet, to what extent can zealous cooperation induce others to cooperate? We assume a fraction of zealous agents in spatial public goods games who always cooperate. The results show that a moderate proportion of these zealous cooperators can diminish the cooperation level in the system, and cooperation is only promoted when zealots are many. Regarding spatial behaviors, the areas of zealous cooperation in a medium density can prevent evolutionary cooperation from passing through and aggregating. The phenomenon of zealous cooperation impeding cooperation becomes more pronounced when agents become less random and more selfish. This is because dotted zealous cooperation provides significant payoffs to neighboring defection, making them more solid in fitness. In this way, we also find that when zealous cooperators have low productivity, the neighbors receive fewer benefits by exploitation, thus allowing cooperation to spread. We also study replicator dynamics in unstructured populations where zealous cooperation always promotes cooperation, agreeing that zealous cooperation hindering cooperation is a spatial effect.

Evolution of cooperation under punishment

Punishment has been considered as an effective mechanism for promoting and sustaining cooperation. In most existing models, punishment always comes as a third strategy alongside cooperation and defection, and it is commonly assumed to be executed based on individual decision rules rather than collective decision rules. Differently from previous works, we employ a democratic procedure by which cooperators cast votes independently and simultaneously for whether to impose punishment on defectors, and we establish a relationship between the cooperators' willingness to punish defectors (WTPD) and whether the punishment is inflicted on defectors. The results illustrate that the population can evolve to full cooperation under consensual punishment. It is noteworthy that, compared with autonomous punishment, whether consensual punishment is more in favor of cooperation crucially depends on the minimum number of votes required for punishment execution as well as the cooperators' WTPD. Our findings highlight the importance of collective decision making in the evolution of cooperation and may provide a mathematical framework for explaining the prevalence of democracy in modern societies.

Reciprocity-based cooperative phalanx maintained by overconfident players

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 influenced in a biased way resulting in a cooperation supporting environment. These effects are strongly related to the nonlinear character of invasion probabilities which is a common and frequently observed feature of microscopic dynamics.

Punishment diminishes the benefits of network reciprocity in social dilemma experiments

Network reciprocity has been widely advertised in theoretical studies as one of the basic cooperation-promoting mechanisms, but experimental evidence favoring this type of reciprocity was published only recently. When organized in an unchanging network of social contacts, human subjects cooperate provided the following strict condition is satisfied: The benefit of cooperation must outweigh the total cost of cooperating with all neighbors. In an attempt to relax this condition, we perform social dilemma experiments wherein network reciprocity is aided with another theoretically hypothesized cooperation-promoting mechanism-costly punishment. The results reveal how networks promote and stabilize cooperation. This stabilizing effect is stronger in a smaller-size neighborhood, as expected from theory and experiments. Contrary to expectations, punishment diminishes the benefits of network reciprocity by lowering assortment, payoff per round, and award for cooperative behavior. This diminishing effect is stronger in a larger-size neighborhood. An immediate implication is that the psychological effects of enduring punishment override the rational response anticipated in quantitative models of cooperation in networks.

Heterogeneous out-of-equilibrium nonlinear q-voter model with zealotry

We study the dynamics of the out-of-equilibrium nonlinear q-voter model with two types of susceptible voters and zealots, introduced in Mellor et al. [Europhys. Lett. 113, 48001 (2016)EULEEJ0295-507510.1209/0295-5075/113/48001]. In this model, each individual supports one of two parties and is either a susceptible voter of type q_{1} or q_{2}, or is an inflexible zealot. At each time step, a q_{i}-susceptible voter (i=1,2) consults a group of q_{i} neighbors and adopts their opinion if all group members agree, while zealots are inflexible and never change their opinion. This model violates detailed balance whenever q_{1}≠q_{2} and is characterized by two distinct regimes of low and high density of zealotry. Here, by combining analytical and numerical methods, we investigate the nonequilibrium stationary state of the system in terms of its probability distribution, nonvanishing currents, and unequal-time two-point correlation functions. We also study the switching time properties of the model by exploiting an approximate mapping onto the model of Mobilia [Phys. Rev. E 92, 012803 (2015)PLEEE81539-375510.1103/PhysRevE.92.012803] that satisfies the detailed balance, and we outline some properties of the model near criticality.

Cooperation driven by success-driven group formation

In the traditional setup of the public goods game all players are involved in every available group and the mutual benefit is shared among competing cooperator and defector strategies. However, 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 who are successful enough to the neighbors can initiate a group formation and establish a game? To elaborate this idea we employ a modified protocol and demonstrate that a carefully chosen threshold to establish a joint venture could efficiently improve the cooperation level even if the synergy factor would suggest a full defector state otherwise. The microscopic mechanism that is responsible for this effect is based on the asymmetric consequences of competing strategies: while the success of a cooperator provides a long-time well-being for the neighborhood, the temporary advantage of defection cannot be maintained if the protocol is based on the success of leaders.

Fixation probabilities of evolutionary coordination games on two coupled populations

Evolutionary forces resulted from competitions between different populations are common, which change the evolutionary behavior of a single population. In an isolated population of coordination games of two strategies (e.g., s_{1} and s_{2}), the previous studies focused on determining the fixation probability that the system is occupied by only one strategy (s_{1}) and their expectation times, given an initial mixture of two strategies. In this work, we propose a model of two interdependent populations, disclosing the effects of the interaction strength on fixation probabilities. In the well-mixing limit, a detailed linear stability analysis is performed, which allows us to find and to classify the different equilibria, yielding a clear picture of the bifurcation patterns in phase space. We demonstrate that the interactions between populations crucially alter the dynamic behavior. More specifically, if the coupling strength is larger than some threshold value, the critical initial density of one strategy (s_{1}) that corresponds to fixation is significantly delayed. Instead, the two populations evolve to the opposite state of all (s_{2}) strategy, which are in favor of the red queen hypothesis. We delineate the extinction time of strategy (s_{1}) explicitly, which is an exponential form. These results are validated by systematic numerical simulations.

Aspiration promotes cooperation in the prisoner's dilemma game with the imitation rule

A model of stochastic evolutionary game dynamics with finite population of size N+M was built. Among these individuals, N individuals update strategies with aspiration updating, while the other M individuals update strategies with imitation updating. In the proposed model, we obtain the expression of the mean fraction of cooperators and analyze some concrete cases. Compared with the standard imitation dynamics, there is always a positive probability to support the formation of cooperation in the system with the aspiration and imitation rules. Moreover, the numerical results indicate that more aspiration-driven individuals lead to a higher mean fraction of imitation-driven cooperators, which means the invasion of the aspiration-driven individuals is conducive to promoting the cooperation of the imitation-driven individuals.

Nonlinear q-voter model with inflexible zealots

We study the dynamics of the nonlinear q-voter model with inflexible zealots in a finite well-mixed population. In this system, each individual supports one of two parties and is either a susceptible voter or an inflexible zealot. At each time step, a susceptible adopts the opinion of a neighbor if this belongs to a group of q≥2 neighbors all in the same state, whereas inflexible zealots never change their opinion. In the presence of zealots of both parties, the model is characterized by a fluctuating stationary state and, below a zealotry density threshold, the distribution of opinions is bimodal. After a characteristic time, most susceptibles become supporters of the party having more zealots and the opinion distribution is asymmetric. When the number of zealots of both parties is the same, the opinion distribution is symmetric and, in the long run, susceptibles endlessly swing from the state where they all support one party to the opposite state. Above the zealotry density threshold, when there is an unequal number of zealots of each type, the probability distribution is single-peaked and non-Gaussian. These properties are investigated analytically and with stochastic simulations. We also study the mean time to reach a consensus when zealots support only one party.

Universal scaling for the dilemma strength in evolutionary games

Why would natural selection favor the prevalence of cooperation within the groups of selfish individuals? A fruitful framework to address this question is evolutionary game theory, the essence of which is captured in the so-called social dilemmas. Such dilemmas have sparked the development of a variety of mathematical approaches to assess the conditions under which cooperation evolves. Furthermore, borrowing from statistical physics and network science, the research of the evolutionary game dynamics has been enriched with phenomena such as pattern formation, equilibrium selection, and self-organization. Numerous advances in understanding the evolution of cooperative behavior over the last few decades have recently been distilled into five reciprocity mechanisms: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity. However, when social viscosity is introduced into a population via any of the reciprocity mechanisms, the existing scaling parameters for the dilemma strength do not yield a unique answer as to how the evolutionary dynamics should unfold. Motivated by this problem, we review the developments that led to the present state of affairs, highlight the accompanying pitfalls, and propose new universal scaling parameters for the dilemma strength. We prove universality by showing that the conditions for an ESS and the expressions for the internal equilibriums in an infinite, well-mixed population subjected to any of the five reciprocity mechanisms depend only on the new scaling parameters. A similar result is shown to hold for the fixation probability of the different strategies in a finite, well-mixed population. Furthermore, by means of numerical simulations, the same scaling parameters are shown to be effective even if the evolution of cooperation is considered on the spatial networks (with the exception of highly heterogeneous setups). We close the discussion by suggesting promising directions for future research including (i) how to handle the dilemma strength in the context of co-evolution and (ii) where to seek opportunities for applying the game theoretical approach with meaningful impact.

Analytical description for the critical fixations of evolutionary coordination games on finite complex structured populations

Evolutionary game theory is crucial to capturing the characteristic interaction patterns among selfish individuals. In a population of coordination games of two strategies, one of the central problems is to determine the fixation probability that the system reaches a state of networkwide of only one strategy, and the corresponding expectation times. The deterministic replicator equations predict the critical value of initial density of one strategy, which separates the two absorbing states of the system. However, numerical estimations of this separatrix show large deviations from the theory in finite populations. Here we provide a stochastic treatment of this dynamic process on complex networks of finite sizes as Markov processes, showing the evolutionary time explicitly. We describe analytically the effects of network structures on the intermediate fixations as observed in numerical simulations. Our theoretical predictions are validated by various simulations on both random and scale free networks. Therefore, our stochastic framework can be helpful in dealing with other networked game dynamics.

Evolutionary prisoner's dilemma game on graphs and social networks with external constraint

A game-theoretical model is constructed to capture the effect of external constraint on the evolution of cooperation. External constraint describes the case where individuals are forced to cooperate with a given probability in a society. Mathematical analyses are conducted via pair approximation and diffusion approximation methods. The results show that the condition for cooperation to be favored on graphs with constraint is b¯/c¯>k/A¯ (A¯=1+kp/(1-p)), where b¯ and c¯ represent the altruistic benefit and cost, respectively, k is the average degree of the graph and p is the probability of compulsory cooperation by external enforcement. Moreover, numerical simulations are also performed on a repeated game with three strategies, always defect (ALLD), tit-for-tat (TFT) and always cooperate (ALLC). These simulations demonstrate that a slight enforcement of ALLC can only promote cooperation when there is weak network reciprocity, while the catalyst effect of TFT on cooperation is verified. In addition, the interesting phenomenon of stable coexistence of the three strategies can be observed. Our model can represent evolutionary dynamics on a network structure which is disturbed by a specified external constraint.

Facilitators on networks reveal optimal interplay between information exchange and reciprocity

Reciprocity is firmly established as an important mechanism that promotes cooperation. An efficient information exchange is likewise important, especially on structured populations, where interactions between players are limited. Motivated by these two facts, we explore the role of facilitators in social dilemmas on networks. Facilitators are here mirrors to their neighbors-they cooperate with cooperators and defect with defectors-but they do not participate in the exchange of strategies. As such, in addition to introducing direct reciprocity, they also obstruct information exchange. In well-mixed populations, facilitators favor the replacement and invasion of defection by cooperation as long as their number exceeds a critical value. In structured populations, on the other hand, there exists a delicate balance between the benefits of reciprocity and the deterioration of information exchange. Extensive Monte Carlo simulations of social dilemmas on various interaction networks reveal that there exists an optimal interplay between reciprocity and information exchange, which sets in only when a small number of facilitators occupy the main hubs of the scale-free network. The drawbacks of missing cooperative hubs are more than compensated for by reciprocity and, at the same time, the compromised information exchange is routed via the auxiliary hubs with only marginal losses in effectivity. These results indicate that it is not always optimal for the main hubs to become leaders of the masses, but rather to exploit their highly connected state to promote tit-for-tat-like behavior.

Evolutionary dynamics in finite populations with zealots

We investigate evolutionary dynamics of two-strategy matrix games with zealots in finite populations. Zealots are assumed to take either strategy regardless of the fitness. When the strategy selected by the zealots is the same, the fixation of the strategy selected by the zealots is a trivial outcome. We study fixation time in this scenario. We show that the fixation time is divided into three main regimes, in one of which the fixation time is short, and in the other two the fixation time is exponentially long in terms of the population size. Different from the case without zealots, there is a threshold selection intensity below which the fixation is fast for an arbitrary payoff matrix. We illustrate our results with examples of various social dilemma games.

Cooperation dilemma in finite populations under fluctuating environments

We present a novel approach allowing the study of rare events like fixation under fluctuating environments, modeled as extrinsic noise, in evolutionary processes characterized by the dominance of one species. Our treatment consists of mapping the system onto an auxiliary model, exhibiting metastable species coexistence, that can be analyzed semiclassically. This approach enables us to study the interplay between extrinsic and demographic noise on the statistics of interest. We illustrate our theory by considering the paradigmatic prisoner's dilemma game, whose evolution is described by the probability that cooperators fixate the population and replace all defectors. We analytically and numerically demonstrate that extrinsic noise may drastically enhance the cooperation fixation probability and even change its functional dependence on the population size. These results, which generalize earlier works in population genetics, indicate that extrinsic noise may help sustain and promote a much higher level of cooperation than static settings.

Comment on "Stochastic dynamics of the prisoner's dilemma with cooperation facilitators"

In a recent paper [Phys. Rev. E 86, 011134 (2012)], Mobilia introduces cooperation facilitators in the standard prisoner's dilemma game. He claims that natural selection favors the replacement of defection by cooperation in the weak-selection case if and only if their frequency satisfies a certain inequality. We show that this is not true, and we point out an error in the author's proof which follows from the improper handling of the large-population limit. In addition, we prove a stronger result that cooperation is favored for any selection strength if and only if the average payoff of cooperation is bigger than the average payoff of defection (which is a weaker condition than the author's inequality). We also show that, if we include self-interaction, then the presence of a fixed number of facilitators causes a rescaling of the payoff matrix, and for their certain frequency, cooperation becomes a dominant strategy, and the prisoner's dilemma simply disappears.

Reply to "Comment on 'Stochastic dynamics of the prisoner's dilemma with cooperation facilitators' "

The fixation properties of a simple prisoner's dilemma game in the presence of "cooperation facilitators" have recently been investigated in finite and well-mixed populations for various dynamics [Mobilia, Phys. Rev. E 86, 011134 (2012)]. In a Comment, Miękisz claims that, for cooperation to be favored by selection in the standard prisoner's dilemma games with facilitators, it suffices that f(C)>f(D) (where f(C/D) are the respective fitnesses of cooperators and defectors). In this Reply, we show that, in generic prisoner's dilemma games with ℓ cooperation facilitators, it is generally not sufficient that a single cooperator has a higher fitness than defectors to ensure that selection favors cooperation. In fact, it is also necessary that selection promotes the replacement of defection by cooperation in a population of size N, which requires that the fixation probability of a single cooperator exceeds (N-ℓ)(-1). This replacement condition is independent of f(C)>f(D) and, when the payoff for mutual defection is negative, it is shown to be more stringent than the invasion condition. Our results, illustrated by a series of examples, considerably generalize those reported in the paper [Phys. Rev. E 86, 011134 (2012)] and in the aforementioned Comment whose claims are demonstrated to be relevant only for a special subclass of prisoner's dilemma games.

Influence of opinion dynamics on the evolution of games

Under certain circumstances such as lack of information or bounded rationality, human players can take decisions on which strategy to choose in a game on the basis of simple opinions. These opinions can be modified after each round by observing own or others payoff results but can be also modified after interchanging impressions with other players. In this way, the update of the strategies can become a question that goes beyond simple evolutionary rules based on fitness and become a social issue. In this work, we explore this scenario by coupling a game with an opinion dynamics model. The opinion is represented by a continuous variable that corresponds to the certainty of the agents respect to which strategy is best. The opinions transform into actions by making the selection of an strategy a stochastic event with a probability regulated by the opinion. A certain regard for the previous round payoff is included but the main update rules of the opinion are given by a model inspired in social interchanges. We find that the fixed points of the dynamics of the coupled model are different from those of the evolutionary game or the opinion models alone. Furthermore, new features emerge such as the independence of the fraction of cooperators with respect to the topology of the social interaction network or the presence of a small fraction of extremist players.

Evolution of cooperation driven by zealots

Recent experimental results with humans involved in social dilemma games suggest that cooperation may be a contagious phenomenon and that the selection pressure operating on evolutionary dynamics (i.e., mimicry) is relatively weak. I propose an evolutionary dynamics model that links these experimental findings and evolution of cooperation. By assuming a small fraction of (imperfect) zealous cooperators, I show that a large fraction of cooperation emerges in evolutionary dynamics of social dilemma games. Even if defection is more lucrative than cooperation for most individuals, they often mimic cooperation of fellows unless the selection pressure is very strong. Then, zealous cooperators can transform the population to be even fully cooperative under standard evolutionary dynamics.