Breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs

Evolution of individual versus social learning on social networks

A number of studies have investigated the roles played by individual and social learning in cultural phenomena and the relative advantages of the two learning strategies in variable environments. Because social learning involves the acquisition of behaviours from others, its utility depends on the availability of 'cultural models' exhibiting adaptive behaviours. This indicates that social networks play an essential role in the evolution of learning. However, possible effects of social structure on the evolution of learning have not been fully explored. Here, we develop a mathematical model to explore the evolutionary dynamics of learning strategies on social networks. We first derive the condition under which social learners (SLs) are selectively favoured over individual learners in a broad range of social network. We then obtain an analytical approximation of the long-term average frequency of SLs in homogeneous networks, from which we specify the condition, in terms of three relatedness measures, for social structure to facilitate the long-term evolution of social learning. Finally, we evaluate our approximation by Monte Carlo simulations in complete graphs, regular random graphs and scale-free networks. We formally show that whether social structure favours the evolution of social learning is determined by the relative magnitudes of two effects of social structure: localization in competition, by which competition between learning strategies is evaded, and localization in cultural transmission, which slows down the spread of adaptive traits. In addition, our estimates of the relatedness measures suggest that social structure disfavours the evolution of social learning when selection is weak.

Boosting cooperation by involving extortion in spatial prisoner's dilemma games

We study the evolution of cooperation in spatial prisoner's dilemma games with and without extortion by adopting the aspiration-driven strategy updating rule. We focus explicitly on how the strategy updating manner (whether synchronous or asynchronous) and also the introduction of extortion strategy affect the collective outcome of the games. By means of Monte Carlo simulations as well as dynamical cluster techniques, we find that the involvement of extortioners facilitates the boom of cooperators in the population (and whom can always dominate the population if the temptation to defect is not too large) for both synchronous and asynchronous strategy updating, in stark contrast to the other case, where cooperation is promoted for an intermediate aspiration level with synchronous strategy updating, but is remarkably inhibited if the strategy updating is implemented asynchronously. We explain the results by configurational analysis and find that the presence of extortion leads to the checkerboard-like ordering of cooperators and extortioners, which enable cooperators to prevail in the population with both strategy updating manners. Moreover, extortion itself is evolutionary stable, and therefore acts as the incubator for the evolution of cooperation.

Different perceptions of social dilemmas: evolutionary multigames in structured populations

Motivated by the fact that the same social dilemma can be perceived differently by different players, we here study evolutionary multigames in structured populations. While the core game is the weak prisoner's dilemma, a fraction of the population adopts either a positive or a negative value of the sucker's payoff, thus playing either the traditional prisoner's dilemma or the snowdrift game. We show that the higher the fraction of the population adopting a different payoff matrix the more the evolution of cooperation is promoted. The microscopic mechanism responsible for this outcome is unique to structured populations, and it is due to the payoff heterogeneity, which spontaneously introduces strong cooperative leaders that give rise to an asymmetric strategy imitation flow in favor of cooperation. We demonstrate that the reported evolutionary outcomes are robust against variations of the interaction network, and they also remain valid if players are allowed to vary which game they play over time. These results corroborate existing evidence in favor of heterogeneity-enhanced network reciprocity, and they reveal how different perceptions of social dilemmas may contribute to their resolution.

Degree mixing in multilayer networks impedes the evolution of cooperation

Traditionally, the evolution of cooperation has been studied on single, isolated networks. Yet a player, especially in human societies, will typically be a member of many different networks, and those networks will play different roles in the evolutionary process. Multilayer networks are therefore rapidly gaining on popularity as the more apt description of a networked society. With this motivation, we here consider two-layer scale-free networks with all possible combinations of degree mixing, wherein one network layer is used for the accumulation of payoffs and the other is used for strategy updating. We find that breaking the symmetry through assortative mixing in one layer and/or disassortative mixing in the other layer, as well as preserving the symmetry by means of assortative mixing in both layers, impedes the evolution of cooperation. We use degree-dependent distributions of strategies and cluster-size analysis to explain these results, which highlight the importance of hubs and the preservation of symmetry between multilayer networks for the successful resolution of social dilemmas.

Biased imitation in coupled evolutionary games in interdependent networks

We explore the evolutionary dynamics of two games—the Prisoner's Dilemma and the Snowdrift Game—played within distinct networks (layers) of interdependent networks. In these networks imitation and interaction between individuals of opposite layers is established through interlinks. We explore an update rule in which revision of strategies is a biased imitation process: individuals imitate neighbors from the same layer with probability p, and neighbors from the second layer with complementary probability 1 − p. We demonstrate that a small decrease of p from p = 1 (which corresponds to forbidding strategy transfer between layers) is sufficient to promote cooperation in the Prisoner's Dilemma subpopulation. This, on the other hand, is detrimental for cooperation in the Snowdrift Game subpopulation. We provide results of extensive computer simulations for the case in which layers are modelled as regular random networks, and support this study with analytical results for coupled well-mixed populations.

Cooperation in networks where the learning environment differs from the interaction environment

We study the evolution of cooperation in a structured population, combining insights from evolutionary game theory and the study of interaction networks. In earlier studies it has been shown that cooperation is difficult to achieve in homogeneous networks, but that cooperation can get established relatively easily when individuals differ largely concerning the number of their interaction partners, such as in scale-free networks. Most of these studies do, however, assume that individuals change their behaviour in response to information they receive on the payoffs of their interaction partners. In real-world situations, subjects do not only learn from their interaction partners, but also from other individuals (e.g. teachers, parents, or friends). Here we investigate the implications of such incongruences between the ‘interaction network’ and the ‘learning network’ for the evolution of cooperation in two paradigm examples, the Prisoner's Dilemma game (PDG) and the Snowdrift game (SDG). Individual-based simulations and an analysis based on pair approximation both reveal that cooperation will be severely inhibited if the learning network is very different from the interaction network. If the two networks overlap, however, cooperation can get established even in case of considerable incongruence between the networks. The simulations confirm that cooperation gets established much more easily if the interaction network is scale-free rather than random-regular. The structure of the learning network has a similar but much weaker effect. Overall we conclude that the distinction between interaction and learning networks deserves more attention since incongruences between these networks can strongly affect both the course and outcome of the evolution of cooperation.

Spontaneous symmetry breaking in interdependent networked game

Spatial evolution game has traditionally assumed that players interact with direct neighbors on a single network, which is isolated and not influenced by other systems. However, this is not fully consistent with recent research identification that interactions between networks play a crucial rule for the outcome of evolutionary games taking place on them. In this work, we introduce the simple game model into the interdependent networks composed of two networks. By means of imitation dynamics, we display that when the interdependent factor α is smaller than a threshold value α(C), the symmetry of cooperation can be guaranteed. Interestingly, as interdependent factor exceeds α(C), spontaneous symmetry breaking of fraction of cooperators presents itself between different networks. With respect to the breakage of symmetry, it is induced by asynchronous expansion between heterogeneous strategy couples of both networks, which further enriches the content of spatial reciprocity. Moreover, our results can be well predicted by the strategy-couple pair approximation method.

Social dilemma alleviated by sharing the gains with immediate neighbors

We study the evolution of cooperation in the evolutionary spatial prisoner's dilemma game (PDG) and snowdrift game (SG), within which a fraction α of the payoffs of each player gained from direct game interactions is shared equally by the immediate neighbors. The magnitude of the parameter α therefore characterizes the degree of the relatedness among the neighboring players. By means of extensive Monte Carlo simulations as well as an extended mean-field approximation method, we trace the frequency of cooperation in the stationary state. We find that plugging into relatedness can significantly promote the evolution of cooperation in the context of both studied games. Unexpectedly, cooperation can be more readily established in the spatial PDG than that in the spatial SG, given that the degree of relatedness and the cost-to-benefit ratio of mutual cooperation are properly formulated. The relevance of our model with the stakeholder theory is also briefly discussed.

Impact of social punishment on cooperative behavior in complex networks

Social punishment is a mechanism by which cooperative individuals spend part of their resources to penalize defectors. In this paper, we study the evolution of cooperation in 2-person evolutionary games on networks when a mechanism for social punishment is introduced. Specifically, we introduce a new kind of role, punisher, which is aimed at reducing the earnings of defectors by applying to them a social fee. Results from numerical simulations show that different equilibria allowing the three strategies to coexist are possible as well as that social punishment further enhance the robustness of cooperation. Our results are confirmed for different network topologies and two evolutionary games. In addition, we analyze the microscopic mechanisms that give rise to the observed macroscopic behaviors in both homogeneous and heterogeneous networks. Our conclusions might provide additional insights for understanding the roots of cooperation in social systems.

Insight into the so-called spatial reciprocity

Up to now, there have been a great number of studies that demonstrate the effect of spatial topology on the promotion of cooperation dynamics (namely, the so-called "spatial reciprocity"). However, most researchers probably attribute it to the positive assortment of strategies supported by spatial arrangement. In this paper, we analyze the time course of cooperation evolution under different evolution rules. Interestingly, a typical evolution process can be divided into two evident periods: the enduring (END) period and the expanding (EXP) period where the former features that cooperators try to endure defectors' invasion and the latter shows that perfect C clusters fast expand their area. We find that the final cooperation level relies on two key factors: the formation of the perfect C cluster at the end of the END period and the expanding fashion of the perfect C cluster during the EXP period. For deterministic rule, the smooth expansion of C cluster boundaries enables cooperators to reach a dominant state, whereas, the rough boundaries for stochastic rule cannot provide a sufficient beneficial environment for the evolution of cooperation. Moreover, we show that expansion of the perfect C cluster is closely related to the cluster coefficient of interaction topology. To some extent, we present a viable method for understanding the spatial reciprocity mechanism in nature and hope that it will inspire further studies to resolve social dilemmas.

Adaptive dynamics with interaction structure

Evolutionary dynamics depend critically on a population's interaction structure-the pattern of which individuals interact with which others, depending on the state of the population and the environment. Previous research has shown, for example, that cooperative behaviors disfavored in well-mixed populations can be favored when interactions occur only between spatial neighbors or group members. Combining the adaptive dynamics approach with recent advances in evolutionary game theory, we here introduce a general mathematical framework for analyzing the long-term evolution of continuous game strategies for a broad class of evolutionary models, encompassing many varieties of interaction structure. Our main result, the canonical equation of adaptive dynamics with interaction structure, characterizes expected evolutionary trajectories resulting from any such model, thereby generalizing a central tool of adaptive dynamics theory. Interestingly, the effects of different interaction structures and update rules on evolutionary trajectories are fully captured by just two real numbers associated with each model, which are independent of the considered game. The first, a structure coefficient, quantifies the effects on selection pressures and thus on the shapes of expected evolutionary trajectories. The second, an effective population size, quantifies the effects on selection responses and thus on the expected rates of adaptation. Applying our results to two social dilemmas, we show how the range of evolutionarily stable cooperative behaviors systematically varies with a model's structure coefficient.

Decelerated invasion and waning-moon patterns in public goods games with delayed distribution

We study the evolution of cooperation in the spatial public goods game, focusing on the effects that are brought about by the delayed distribution of goods that accumulate in groups due to the continuous investments of cooperators. We find that intermediate delays enhance network reciprocity because of a decelerated invasion of defectors, who are unable to reap the same high short-term benefits as they do in the absence of delayed distribution. Long delays, however, introduce a risk because the large accumulated wealth might fall into the wrong hands. Indeed, as soon as the curvature of a cooperative cluster turns negative, the engulfed defectors can collect the heritage of many generations of cooperators and by doing so start a waning-moon pattern that nullifies the benefits of decelerated invasion. Accidental meeting points of growing cooperative clusters may also act as triggers for the waning-moon effect, thus linking the success of cooperators with their propensity to fail in a rather bizarre way. Our results highlight that "investing in the future" is a good idea only if that future is sufficiently near and not likely to be burdened by inflation.

Global migration can lead to stronger spatial selection than local migration

The outcome of evolutionary processes depends on population structure. It is well known that mobility plays an important role in affecting evolutionary dynamics in group structured populations. But it is largely unknown whether global or local migration leads to stronger spatial selection and would therefore favor to a larger extent the evolution of cooperation. To address this issue, we quantify the impacts of these two migration patterns on the evolutionary competition of two strategies in a finite island model. Global migration means that individuals can migrate from any one island to any other island. Local migration means that individuals can only migrate between islands that are nearest neighbors; we study a simple geometry where islands are arranged on a one-dimensional, regular cycle. We derive general results for weak selection and large population size. Our key parameters are: the number of islands, the migration rate and the mutation rate. Surprisingly, our comparative analysis reveals that global migration can lead to stronger spatial selection than local migration for a wide range of parameter conditions. Our work provides useful insights into understanding how different mobility patterns affect evolutionary processes.

Adaptive tag switching reinforces the coevolution of contingent cooperation and tag diversity

Most of the previous studies concerning the similarity-based interaction have assumed that the change of tags just happens in the imitation stage. Individuals actually can adjust their tags whenever the environments related to these tags grow nasty. We institute a spatial model to investigate the effect of the coevolution of tag and strategy on the evolution of cooperation in the context of the Prisoner's Dilemma game. Interactions just happen between tag-identical neighbors. Individuals exploited by defectors change their current tags at a certain cost. The time-scale ratio controls how fast interaction happens relatively to selection. Results show that whenever individuals have enough chance to adapt to the environment, cooperation is greatly improved even for quite large temptation to defect. Intensive exploration reveals that both little and large costs of tag switching can further favor the establishment of cooperation. Our work may add more into the literature concerning games on adaptive networks.

Evolution of emotions on networks leads to the evolution of cooperation in social dilemmas

We show that the resolution of social dilemmas in 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 to be the two emotions that determine the strategy of each player in any given interaction, and we define them as the probabilities of cooperating with players having a lower and a 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 that is eventually adopted by all players. However, this emotional profile depends not only on the payoffs but also on the heterogeneity of the interaction network. Homogeneous networks, such as lattices and regular random graphs, lead to fixations that are characterized by high sympathy and high envy, while heterogeneous networks lead to low or modest sympathy but also low envy. Our results thus suggest that public emotions and the propensity to cooperate at large depend, and are in fact determined by, the properties of the interaction network.

Consolidating birth-death and death-birth processes in structured populations

Network models extend evolutionary game theory to settings with spatial or social structure and have provided key insights on the mechanisms underlying the evolution of cooperation. However, network models have also proven sensitive to seemingly small details of the model architecture. Here we investigate two popular biologically motivated models of evolution in finite populations: Death-Birth (DB) and Birth-Death (BD) processes. In both cases reproduction is proportional to fitness and death is random; the only difference is the order of the two events at each time step. Although superficially similar, under DB cooperation may be favoured in structured populations, while under BD it never is. This is especially troubling as natural populations do not follow a strict one birth then one death regimen (or vice versa); such constraints are introduced to make models more tractable. Whether structure can promote the evolution of cooperation should not hinge on a simplifying assumption. Here, we propose a mixed rule where in each time step DB is used with probability and BD is used with probability . We derive the conditions for selection favouring cooperation under the mixed rule for all social dilemmas. We find that the only qualitatively different outcome occurs when using just BD (). This case admits a natural interpretation in terms of kin competition counterbalancing the effect of kin selection. Finally we show that, for any mixed BD-DB update and under weak selection, cooperation is never inhibited by population structure for any social dilemma, including the Snowdrift Game.

Evolution in a changing environment

We propose a simple model for genetic adaptation to a changing environment, describing a fitness landscape characterized by two maxima. One is associated with "specialist" individuals that are adapted to the environment; this maximum moves over time as the environment changes. The other maximum is static, and represents "generalist" individuals not affected by environmental changes. The rest of the landscape is occupied by "maladapted" individuals. Our analysis considers the evolution of these three subpopulations. Our main result is that, in presence of a sufficiently stable environmental feature, as in the case of an unchanging aspect of a physical habitat, specialists can dominate the population. By contrast, rapidly changing environmental features, such as language or cultural habits, are a moving target for the genes; here, generalists dominate, because the best evolutionary strategy is to adopt neutral alleles not specialized for any specific environment. The model we propose is based on simple assumptions about evolutionary dynamics and describes all possible scenarios in a non-trivial phase diagram. The approach provides a general framework to address such fundamental issues as the Baldwin effect, the biological basis for language, or the ecological consequences of a rapid climate change.

Evolutionary shift dynamics on a cycle

We present a new model of evolutionary dynamics in one-dimensional space. Individuals are arranged on a cycle. When a new offspring is born, another individual dies and the rest shift around the cycle to make room. This rule, which is inspired by spatial evolution in somatic tissue and microbial colonies, has the remarkable property that, in the limit of large population size, evolution acts to maximize the payoff of the whole population. Therefore, social dilemmas, in which some individuals benefit at the expense of others, are resolved. We demonstrate this principle for both discrete and continuous games. We also discuss extensions of our model to other one-dimensional spatial configurations. We conclude that shift dynamics in one dimension is an unusually strong promoter of cooperative behavior.

From local to global dilemmas in social networks

Social networks affect in such a fundamental way the dynamics of the population they support that the global, population-wide behavior that one observes often bears no relation to the individual processes it stems from. Up to now, linking the global networked dynamics to such individual mechanisms has remained elusive. Here we study the evolution of cooperation in networked populations and let individuals interact via a 2-person Prisoner's Dilemma – a characteristic defection dominant social dilemma of cooperation. We show how homogeneous networks transform a Prisoner's Dilemma into a population-wide evolutionary dynamics that promotes the coexistence between cooperators and defectors, while heterogeneous networks promote their coordination. To this end, we define a dynamic variable that allows us to track the self-organization of cooperators when co-evolving with defectors in networked populations. Using the same variable, we show how the global dynamics — and effective dilemma — co-evolves with the motifs of cooperators in the population, the overall emergence of cooperation depending sensitively on this co-evolution.

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.

Multi-player games on the cycle

In multi-player games n individuals interact in any one encounter and derive a payoff from that interaction. We assume that individuals adopt one of two strategies, and we consider symmetric games, which means the payoff depends only on the number of players using either strategy, but not on any particular configuration of the encounter. On the cycle we assume that any string of n neighbouring players interacts. We study fixation probabilities of stochastic evolutionary dynamics. We derive analytical results on the cycle both for linear and exponential fitness for any intensity of selection, and compare those to results for the well-mixed population. As particular examples we study multi-player public goods games, stag hunt games and snowdrift games.

Group-size effects on the evolution of cooperation in the spatial public goods game

We study the evolution of cooperation in public goods games on the square lattice, focusing on the effects that are brought about by different sizes of groups where individuals collect their payoffs and search for potential strategy donors. We find that increasing the group size does not necessarily lead to mean-field behavior, as is traditionally observed for games governed by pairwise interactions, but rather that public cooperation may be additionally promoted by means of enhanced spatial reciprocity that sets in for very large groups. Our results highlight that the promotion of cooperation due to spatial interactions is not rooted solely in having restricted connections among players, but also in individuals having the opportunity to collect payoffs separately from their direct opponents. Moreover, in large groups the presence of a small number of defectors is bearable, which makes the mixed-phase region expand with increasing group size. Having a chance of exploiting distant players, however, offers defectors a different way to break the phalanx of cooperators and even to resurrect from small numbers to eventually completely invade the population.

Effects of encounter in a population of spatial prisoner's dilemma players

We study the evolution of cooperation in spatial prisoner's dilemma games, whereby each player extends its interaction scope by trying to interact with a certain number of encounters randomly chosen from its non-neighbors, in addition to its permanently linked nearest neighbors. Furthermore, the non-neighbors treat the initiative interactions in two scenarios: definitely accepting that from the cooperators, whereas guardedly interacting with defectors with an acceptance probability which may take arbitrary value in [0,1]. Importantly, our results reveal that the proposed encounter mechanism is a potent extrinsic factor that is able to boost cooperation when appropriately adjusting the values of the encounter number and acceptance probability, though rational players would always defect in one-shot encounters, regardless of the action from the counterparts. We hope our studies may help understand that the proposed encounter mechanism is also an important ingredient of a flourishing cooperative society.

Crossover between structured and well-mixed networks in an evolutionary prisoner's dilemma game

In a spatial evolutionary prisoner's dilemma game (PDG), individuals interact with their neighbors and update their strategies according to some rules. As is well known, cooperators are destined to become extinct in a well-mixed population, whereas they could emerge and be sustained on a structured network. In this work, we introduce a simple model to investigate the crossover between a structured network and a well-mixed one in an evolutionary PDG. In the model, each link j is designated a rewiring parameter τ(j), which defines the time interval between two successive rewiring events for link j. By adjusting the rewiring parameter τ (the mean time interval for any link in the network), we could change a structured network into a well-mixed one. For the link rewiring events, three situations are considered: one synchronous situation and two asynchronous situations. Simulation results show that there are three regimes of τ: large τ where the density of cooperators ρ(c) rises to ρ(c,∞) (the value of ρ(c) for the case without link rewiring), small τ where the mean-field description for a well-mixed network is applicable, and moderate τ where the crossover between a structured network and a well-mixed one happens.

Mesoscopic effects in an agent-based bargaining model in regular lattices

The effect of spatial structure has been proved very relevant in repeated games. In this work we propose an agent based model where a fixed finite population of tagged agents play iteratively the Nash demand game in a regular lattice. The model extends the multiagent bargaining model by Axtell, Epstein and Young [1] modifying the assumption of global interaction. Each agent is endowed with a memory and plays the best reply against the opponent's most frequent demand. We focus our analysis on the transient dynamics of the system, studying by computer simulation the set of states in which the system spends a considerable fraction of the time. The results show that all the possible persistent regimes in the global interaction model can also be observed in this spatial version. We also find that the mesoscopic properties of the interaction networks that the spatial distribution induces in the model have a significant impact on the diffusion of strategies, and can lead to new persistent regimes different from those found in previous research. In particular, community structure in the intratype interaction networks may cause that communities reach different persistent regimes as a consequence of the hindering diffusion effect of fluctuating agents at their borders.

Effects of asynchronism on evolutionary games

We analyze the influence of the update dynamics on symmetric 2-player evolutionary games, which are among the most used tools to study the emergence of cooperation in populations of interacting agents. A synchronous dynamics means that, at each time step, all the agents of the population update their strategies simultaneously. An extreme case of asynchronism is sequential dynamics, in which only one agent is updated each time. We first show that these two opposite update dynamics can lead to very different outcomes and that sequential dynamics is detrimental to the emergence of cooperation only when the probability of imitating the most successful neighbors is high. In this sense, we can say that, when the update dynamics has some influence, in general asynchronism is beneficial to the emergence of cooperation. We then explore the consequences of using intermediate levels of asynchronism, where only a fraction of the agents update their behavior each time. In general, the level of cooperation changes smoothly and monotonically as we gradually go from synchronous to sequential dynamics. However, there are some exceptions that should be taken into account. In addition, the results show that the possibility of agents taking irrational decisions has a key role in the sensitivity of these models to changes in the update dynamics. Explanations for the observed behaviors are advanced.

How wealth accumulation can promote cooperation

Explaining the emergence and stability of cooperation has been a central challenge in biology, economics and sociology. Unfortunately, the mechanisms known to promote it either require elaborate strategies or hold only under restrictive conditions. Here, we report the emergence, survival, and frequent domination of cooperation in a world characterized by selfishness and a strong temptation to defect, when individuals can accumulate wealth. In particular, we study games with local adaptation such as the prisoner's dilemma, to which we add heterogeneity in payoffs. In our model, agents accumulate wealth and invest some of it in their interactions. The larger the investment, the more can potentially be gained or lost, so that present gains affect future payoffs. We find that cooperation survives for a far wider range of parameters than without wealth accumulation and, even more strikingly, that it often dominates defection. This is in stark contrast to the traditional evolutionary prisoner's dilemma in particular, in which cooperation rarely survives and almost never thrives. With the inequality we introduce, on the contrary, cooperators do better than defectors, even without any strategic behavior or exogenously imposed strategies. These results have important consequences for our understanding of the type of social and economic arrangements that are optimal and efficient.

Universality of weak selection

Weak selection, which means a phenotype is slightly advantageous over another, is an important limiting case in evolutionary biology. Recently, it has been introduced into evolutionary game theory. In evolutionary game dynamics, the probability to be imitated or to reproduce depends on the performance in a game. The influence of the game on the stochastic dynamics in finite populations is governed by the intensity of selection. In many models of both unstructured and structured populations, a key assumption allowing analytical calculations is weak selection, which means that all individuals perform approximately equally well. In the weak selection limit many different microscopic evolutionary models have the same or similar properties. How universal is weak selection for those microscopic evolutionary processes? We answer this question by investigating the fixation probability and the average fixation time not only up to linear but also up to higher orders in selection intensity. We find universal higher order expansions, which allow a rescaling of the selection intensity. With this, we can identify specific models which violate (linear) weak selection results, such as the one-third rule of coordination games in finite but large populations.

Early appraisal of the fixation probability in directed networks

In evolutionary dynamics, the probability that a mutation spreads through the whole population, having arisen from a single individual, is known as the fixation probability. In general, it is not possible to find the fixation probability analytically given the mutant's fitness and the topological constraints that govern the spread of the mutation, so one resorts to simulations instead. Depending on the topology in use, a great number of evolutionary steps may be needed in each of the simulation events, particularly in those that end with the population containing mutants only. We introduce two techniques to accelerate the determination of the fixation probability. The first one skips all evolutionary steps in which the number of mutants does not change and thereby reduces the number of steps per simulation event considerably. This technique is computationally advantageous for some of the so-called layered networks. The second technique, which is not restricted to layered networks, consists of aborting any simulation event in which the number of mutants has grown beyond a certain threshold value and counting that event as having led to a total spread of the mutation. For advantageous mutations in large populations and regardless of the network's topology, we demonstrate, both analytically and by means of simulations, that using a threshold of about [N/(r-1)](1/4) mutants, where N is the number of simulation events and r is the ratio of the mutants' fitness to that of the remainder of the population, leads to an estimate of the fixation probability that deviates in no significant way from that obtained from the full-fledged simulations. We have observed speedups of two orders of magnitude for layered networks with 10,000 nodes.

Effects of heterogeneous wealth distribution on public cooperation with collective risk

The distribution of wealth among individuals in real society can be well described by the Pareto principle or "80-20 rule." How does such heterogeneity in initial wealth distribution affect the emergence of public cooperation, when individuals, the rich and the poor, engage in a collective-risk enterprise, not to gain a profit but to avoid a potential loss? Here we address this issue by studying a simple but effective model based on threshold public goods games. We analyze the evolutionary dynamics for two distinct scenarios, respectively: one with fair sharers versus defectors and the other with altruists versus defectors. For both scenarios, particularly, we in detail study the dynamics of the population with dichotomic initial wealth-the rich versus the poor. Moreover, we demonstrate the possible steady compositions of the population and provide the conditions for stability of these steady states. We prove that in a population with heterogeneous wealth distribution, richer individuals are more likely to cooperate than poorer ones. Participants with lower initial wealth may choose to cooperate only if all players richer than them are cooperators. The emergence of pubic cooperation largely relies on rich individuals. Furthermore, whenever the wealth gap between the rich and the poor is sufficiently large, cooperation of a few rich individuals can substantially elevate the overall level of social cooperation, which is in line with the well-known Pareto principle. Our work may offer an insight into the emergence of cooperative behavior in real social situations where heterogeneous distribution of wealth among individual is omnipresent.

Evolutionary dynamics of public goods games with diverse contributions in finite populations

The public goods game is a powerful metaphor for exploring the maintenance of social cooperative behavior in a group of interactional selfish players. Here we study the emergence of cooperation in the public goods games with diverse contributions in finite populations. The theory of stochastic process is innovatively adopted to investigate the evolutionary dynamics of the public goods games involving a diversity of contributions. In the limit of rare mutations, the general stationary distribution of this stochastic process can be analytically approximated by means of diffusion theory. Moreover, we demonstrate that increasing the diversity of contributions greatly reduces the probability of finding the population in a homogeneous state full of defectors. This increase also raises the expectation of the total contribution in the entire population and thus promotes social cooperation. Furthermore, by investigating the evolutionary dynamics of optional public goods games with diverse contributions, we find that nonparticipation can assist players who contribute more in resisting invasion and taking over individuals who contribute less. In addition, numerical simulations are performed to confirm our analytical results. Our results may provide insight into the effect of diverse contributions on cooperative behaviors in the real world.

Phase transitions to cooperation in the prisoner's dilemma

Game theory formalizes certain interactions between physical particles or between living beings in biology, sociology, and economics and quantifies the outcomes by payoffs. The prisoner's dilemma (PD) describes situations in which it is profitable if everybody cooperates rather than defects (free rides or cheats), but as cooperation is risky and defection is tempting, the expected outcome is defection. Nevertheless, some biological and social mechanisms can support cooperation by effectively transforming the payoffs. Here, we study the related phase transitions, which can be of first order (discontinuous) or of second order (continuous), implying a variety of different routes to cooperation. After classifying the transitions into cases of equilibrium displacement, equilibrium selection, and equilibrium creation, we show that a transition to cooperation may take place even if the stationary states and the eigenvalues of the replicator equation for the PD stay unchanged. Our example is based on adaptive group pressure, which makes the payoffs dependent on the endogenous dynamics in the population. The resulting bistability can invert the expected outcome in favor of cooperation.

Evolutionary dynamics of populations with conflicting interactions: classification and analytical treatment considering asymmetry and power

Evolutionary game theory has been successfully used to investigate the dynamics of systems, in which many entities have competitive interactions. From a physics point of view, it is interesting to study conditions under which a coordination or cooperation of interacting entities will occur, be it spins, particles, bacteria, animals, or humans. Here, we analyze the case, where the entities are heterogeneous, particularly the case of two populations with conflicting interactions and two possible states. For such systems, explicit mathematical formulas will be determined for the stationary solutions and the associated eigenvalues, which determine their stability. In this way, four different types of system dynamics can be classified and the various kinds of phase transitions between them will be discussed. While these results are interesting from a physics point of view, they are also relevant for social, economic, and biological systems, as they allow one to understand conditions for (1) the breakdown of cooperation, (2) the coexistence of different behaviors ("subcultures"), (3) the evolution of commonly shared behaviors ("norms"), and (4) the occurrence of polarization or conflict. We point out that norms have a similar function in social systems that forces have in physics.

Population structure induces a symmetry breaking favoring the emergence of cooperation

The evolution of cooperation described in terms of simple two-person interactions has received considerable attention in recent years, where several key results were obtained. Among those, it is now well established that the web of social interaction networks promotes the emergence of cooperation when modeled in terms of symmetric two-person games. Up until now, however, the impacts of the heterogeneity of social interactions into the emergence of cooperation have not been fully explored, as other aspects remain to be investigated. Here we carry out a study employing the simplest example of a prisoner's dilemma game in which the benefits collected by the participants may be proportional to the costs expended. We show that the heterogeneous nature of the social network naturally induces a symmetry breaking of the game, as contributions made by cooperators may become contingent on the social context in which the individual is embedded. A new, numerical, mean-field analysis reveals that prisoner's dilemmas on networks no longer constitute a defector dominance dilemma—instead, individuals engage effectively in a general coordination game. We find that the symmetry breaking induced by population structure profoundly affects the evolutionary dynamics of cooperation, dramatically enhancing the feasibility of cooperators: cooperation blooms when each cooperator contributes the same cost, equally shared among the plethora of games in which she participates. This work provides clear evidence that, while individual rational reasoning may hinder cooperative actions, the intricate nature of social interactions may effectively transform a local dilemma of cooperation into a global coordination problem.

Humans contribute to a broad range of cooperative endeavors. In many of them, the amount or effort contributed often depends on the social context of each individual. Recent evidence has shown how modern societies are grounded in complex and heterogeneous networks of exchange and cooperation, in which some individuals play radically different roles and/or interact more than others. We show that such social heterogeneity drastically affects the behavioral dynamics and promotes cooperative behavior, whenever the social dilemma perceived by each individual is contingent on her/his social context. The multiplicity of roles and contributions induced by realistic population structures is shown to transform an initial defection dominance dilemma into a coordination challenge or even a cooperator dominance game. While locally defection may seem inescapable, globally there is an emergent new dilemma in which cooperation often prevails, illustrating how collective cooperative action may emerge from myopic individual selfishness.

Calculating evolutionary dynamics in structured populations

Evolution is shaping the world around us. At the core of every evolutionary process is a population of reproducing individuals. The outcome of an evolutionary process depends on population structure. Here we provide a general formula for calculating evolutionary dynamics in a wide class of structured populations. This class includes the recently introduced "games in phenotype space" and "evolutionary set theory." There can be local interactions for determining the relative fitness of individuals, but we require global updating, which means all individuals compete uniformly for reproduction. We study the competition of two strategies in the context of an evolutionary game and determine which strategy is favored in the limit of weak selection. We derive an intuitive formula for the structure coefficient, sigma, and provide a method for efficient numerical calculation.

Selection of noise level in strategy adoption for spatial social dilemmas

We studied spatial Prisoner's Dilemma and Stag Hunt games where both the strategy distribution and the players' individual noise level could evolve to reach higher individual payoff. Players are located on the sites of different two-dimensional lattices and gain their payoff from games with their neighbors by choosing unconditional cooperation or defection. The way of strategy adoption can be characterized by a single K (temperaturelike) parameter describing how strongly adoptions depend on the payoff difference. If we start the system from a random strategy distribution with many different player specific K parameters, the simultaneous evolution of strategies and K parameters drives the system to a final stationary state where only one K value remains. In the coexistence phase of cooperator and defector strategies the surviving K parameter is in good agreement with the noise level that ensures the highest cooperation level if uniform K is supposed for all players. In this paper we give a thorough overview about the properties of this evolutionary process.

Cluster approximations for infection dynamics on random networks

In this paper, we consider a simple stochastic epidemic model on large regular random graphs and the stochastic process that corresponds to this dynamics in the standard pair approximation. Using the fact that the nodes of a pair are unlikely to share neighbors, we derive the master equation for this process and obtain from the system size expansion the power spectrum of the fluctuations in the quasistationary state. We show that whenever the pair approximation deterministic equations give an accurate description of the behavior of the system in the thermodynamic limit, the power spectrum of the fluctuations measured in long simulations is well approximated by the analytical power spectrum. If this assumption breaks down, then the cluster approximation must be carried out beyond the level of pairs. We construct an uncorrelated triplet approximation that captures the behavior of the system in a region of parameter space where the pair approximation fails to give a good quantitative or even qualitative agreement. For these parameter values, the power spectrum of the fluctuations in finite systems can be computed analytically from the master equation of the corresponding stochastic process.

Coevolutionary games--a mini review

Prevalence of cooperation within groups of selfish individuals is puzzling in that it contradicts with the basic premise of natural selection. Favoring players with higher fitness, the latter is key for understanding the challenges faced by cooperators when competing with defectors. Evolutionary game theory provides a competent theoretical framework for addressing the subtleties of cooperation in such situations, which are known as social dilemmas. Recent advances point towards the fact that the evolution of strategies alone may be insufficient to fully exploit the benefits offered by cooperative behavior. Indeed, while spatial structure and heterogeneity, for example, have been recognized as potent promoters of cooperation, coevolutionary rules can extend the potentials of such entities further, and even more importantly, lead to the understanding of their emergence. The introduction of coevolutionary rules to evolutionary games implies, that besides the evolution of strategies, another property may simultaneously be subject to evolution as well. Coevolutionary rules may affect the interaction network, the reproduction capability of players, their reputation, mobility or age. Here we review recent works on evolutionary games incorporating coevolutionary rules, as well as give a didactic description of potential pitfalls and misconceptions associated with the subject. In addition, we briefly outline directions for future research that we feel are promising, thereby particularly focusing on dynamical effects of coevolutionary rules on the evolution of cooperation, which are still widely open to research and thus hold promise of exciting new discoveries.

Cooperation enhanced by moderate tolerance ranges in myopically selective interactions

We present a mode of myopically selective interaction to study the evolutionary prisoner's dilemma game in scale-free networks. Each individual has a reputation-based tolerance range and only tends to interact with the neighbors whose reputation is within its tolerance range. Moreover, its reputation is assessed in response to the interactions in the neighborhood. Interestingly, we show that moderate values of tolerance range can result in the best promotion of cooperation due to the emergence of group selection mechanism. Furthermore, we study the effects of weighting factor in the assessment rule of reputation on the evolution of cooperation. We also show how cooperation evolves in some extended situations, where an interaction stimulus payment is considered for individuals, and where the strategy and reputation of individuals can spread simultaneously. Our results may enhance the understanding of evolutionary dynamics in graph-structured populations where individuals conditionally play with their neighbors according to some myopic selection criteria.

Partner selections in public goods games with constant group size

Most of previous studies concerning the public goods game assume either participation is unconditional or the number of actual participants in a competitive group changes over time. How the fixed group size, prescribed by social institutions, affects the evolution of cooperation is still unclear. We propose a model where individuals with heterogeneous social ties might well engage in differing numbers of public goods games, yet with each public goods game being constant size during the course of evolution. To do this, we assume that each focal individual unidirectionally selects a constant number of interaction partners from his immediate neighbors with probabilities proportional to the degrees or the reputations of these neighbors, corresponding to degree-based partner selection or reputation-based partner selection, respectively. Because of the stochasticity the group formation is dynamical. In both selection regimes, monotonical dependence of the stationary density of cooperators on the group size was found, the latter over the whole range but the former over a restricted range of the renormalized enhancement factor. Moreover, the reputation-based regime can substantially improve cooperation. To interpret these differences, the microscopic characteristics of individuals are probed. We later extend the degree-based partner selection to general cases where focal individuals have preferences toward their neighbors of varying social ties to form groups. As a comparison, we as well investigate the situation where individuals locating on the degree regular graphs choose their coplayers at random. Our results may give some insights into better understanding the widespread teamwork and cooperation in the real world.

Impact of aging on the evolution of cooperation in the spatial prisoner's dilemma game

Aging is always present, tailoring our interactions with others, and postulating a finite lifespan during which we are able to exercise them. We consider the prisoner's dilemma game on a square lattice and examine how quenched age distributions and different aging protocols influence the evolution of cooperation when taking the life experience and knowledge accumulation into account as time passes. In agreement with previous studies, we find that a quenched assignment of age to players, introducing heterogeneity to the game, substantially promotes cooperative behavior. Introduction of aging and subsequent death as a coevolutionary process may act detrimental on cooperation but enhances it efficiently if the offspring of individuals that have successfully passed their strategy is considered newborn. We study resulting age distributions of players and show that the heterogeneity is vital-yet insufficient-for explaining the observed differences in cooperator abundance on the spatial grid. The unexpected increment of cooperation levels can be explained by a dynamical effect that has a highly selective impact on the propagation of cooperator and defector states.

Effects of strategy-migration direction and noise in the evolutionary spatial prisoner's dilemma

Spatial games are crucial for understanding patterns of cooperation in nature (and to some extent society). They are known to be more sensitive to local symmetries than, e.g., spin models. This paper concerns the evolution of the prisoner's dilemma game on regular lattices with three different types of neighborhoods--the von Neumann, Moore, and kagomé types. We investigate two kinds of dynamics for the players to update their strategies (that can be unconditional cooperator or defector). Depending on the payoff difference, an individual can adopt the strategy of a random neighbor [a voter-model-like dynamics (VMLD)] or impose its strategy on a random neighbor, i.e., invasion-process-like dynamics (IPLD). In particular, we focus on the effects of noise, in combination with the strategy dynamics, on the evolution of cooperation. We find that VMLD, compared to IPLD, better supports the spreading and sustaining of cooperation. We see that noise has nontrivial effects on the evolution of cooperation: maximum cooperation density can be realized either at a medium noise level, in the limit of zero noise or in both these regions. The temptation to defect and the local interaction structure determine the outcome. Especially, in the low noise limit, the local interaction plays a crucial role in determining the fate of cooperators. We elucidate these both by numerical simulations and mean-field cluster approximation methods.

Network growth for enhanced natural selection

Natural selection and random drift are competing phenomena for explaining the evolution of populations. Combining a highly fit mutant with a population structure that improves the odds that the mutation spreads through the whole population tips the balance in favor of natural selection. The probability that the spread occurs, known as the fixation probability, depends heavily on how the population is structured. Certain topologies, albeit highly artificially contrived, have been shown to exist that favor fixation. We present a randomized mechanism for network growth that is loosely inspired in some of these topologies' key properties and demonstrate, through simulations, that it is capable of giving rise to structured populations for which the fixation probability significantly surpasses that of an unstructured population. This discovery provides important support to the notion that natural selection can be enhanced over random drift in naturally occurring population structures.

Emergence of social cooperation in threshold public goods games with collective risk

In real situations, people are often faced with the option of voluntary contribution to achieve a collective goal, for example, building a dam or a fence, in order to avoid an unfavorable loss. Those who do not donate, however, can free ride on others' sacrifices. As a result, cooperation is difficult to maintain, leading to an enduring collective-risk social dilemma. To address this issue, here we propose a simple yet effective theoretical model of threshold public goods game with collective risk and focus on the effect of risk on the emergence of social cooperation. To do this, we consider the population dynamics represented by replicator equation for two simplifying scenarios, respectively: one with fair sharers, who contribute the minimum average amount versus defectors and the other with altruists contributing more than average versus defectors. For both cases, we find that the dilemma is relieved in high-risk situations where cooperation is likely to persist and dominate defection in the population. Large initial endowment to individuals also encourages the risk-averse action, which means that, as compared to poor players (with small initial endowment), wealthy individuals (with large initial endowment) are more likely to cooperate in order to protect their private accounts. In addition, we show that small donation amount and small threshold (collective target) can encourage and sustain cooperation. Furthermore, for other parameters fixed, the impacts of group size act differently on the two scenarios because of distinct mechanisms: in the former case where the cost of cooperation depends on the group size, large size of group readily results in defection, while easily maintains cooperation in the latter case where the cost of cooperation is fixed irrespective of the group size. Our theoretical results of the replicator dynamics are in excellent agreement with the individual based simulation results.

Deterministic evolutionary game dynamics in finite populations

Evolutionary game dynamics describes the spreading of successful strategies in a population of reproducing individuals. Typically, the microscopic definition of strategy spreading is stochastic such that the dynamics becomes deterministic only in infinitely large populations. Here, we present a microscopic birth-death process that has a fully deterministic strong selection limit in well-mixed populations of any size. Additionally, under weak selection, from this process the frequency-dependent Moran process is recovered. This makes it a natural extension of the usual evolutionary dynamics under weak selection. We find simple expressions for the fixation probabilities and average fixation times of the process in evolutionary games with two players and two strategies. For cyclic games with two players and three strategies, we show that the resulting deterministic dynamics crucially depends on the initial condition in a nontrivial way.