Phase transitions and volunteering in spatial public goods games

Evolution of cooperation by phenotypic similarity

The emergence of cooperation in populations of selfish individuals is a fascinating topic that has inspired much work in theoretical biology. Here, we study the evolution of cooperation in a model where individuals are characterized by phenotypic properties that are visible to others. The population is well mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies can depend on distance in phenotype space. We study the interaction of cooperators and defectors. In our model, cooperators cooperate with those who are similar and defect otherwise. Defectors always defect. Individuals mutate to nearby phenotypes, which generates a random walk of the population in phenotype space. Our analysis brings together ideas from coalescence theory and evolutionary game dynamics. We obtain a precise condition for natural selection to favor cooperators over defectors. Cooperation is favored when the phenotypic mutation rate is large and the strategy mutation rate is small. In the optimal case for cooperators, in a one-dimensional phenotype space and for large population size, the critical benefit-to-cost ratio is given by b/c = 1 + 2/square root(3). We also derive the fundamental condition for any two-strategy symmetric game and consider high-dimensional phenotype spaces.

Diversity-optimized cooperation on complex networks

We propose a strategy for achieving maximum cooperation in evolutionary games on complex networks. Each individual is assigned a weight that is proportional to the power of its degree, where the exponent alpha is an adjustable parameter that controls the level of diversity among individuals in the network. During the evolution, every individual chooses one of its neighbors as a reference with a probability proportional to the weight of the neighbor, and updates its strategy depending on their payoff difference. It is found that there exists an optimal value of alpha, for which the level of cooperation reaches maximum. This phenomenon indicates that, although high-degree individuals play a prominent role in maintaining the cooperation, too strong influences from the hubs may counterintuitively inhibit the diffusion of cooperation. Other pertinent quantities such as the payoff, the cooperator density as a function of the degree, and the payoff distribution are also investigated computationally and theoretically. Our results suggest that in order to achieve strong cooperation on a complex network, individuals should learn more frequently from neighbors with higher degrees, but only to a certain extent.

Evolutionary dynamics on graphs: Efficient method for weak selection

Investigating the evolutionary dynamics of game theoretical interactions in populations where individuals are arranged on a graph can be challenging in terms of computation time. Here, we propose an efficient method to study any type of game on arbitrary graph structures for weak selection. In this limit, evolutionary game dynamics represents a first-order correction to neutral evolution. Spatial correlations can be empirically determined under neutral evolution and provide the basis for formulating the game dynamics as a discrete Markov process by incorporating a detailed description of the microscopic dynamics based on the neutral correlations. This framework is then applied to one of the most intriguing questions in evolutionary biology: the evolution of cooperation. We demonstrate that the degree heterogeneity of a graph impedes cooperation and that the success of tit for tat depends not only on the number of rounds but also on the degree of the graph. Moreover, considering the mutation-selection equilibrium shows that the symmetry of the stationary distribution of states under weak selection is skewed in favor of defectors for larger selection strengths. In particular, degree heterogeneity--a prominent feature of scale-free networks--generally results in a more pronounced increase in the critical benefit-to-cost ratio required for evolution to favor cooperation as compared to regular graphs. This conclusion is corroborated by an analysis of the effects of population structures on the fixation probabilities of strategies in general 2 x 2 games for different types of graphs. Computer simulations confirm the predictive power of our method and illustrate the improved accuracy as compared to previous studies.

Partner switching stabilizes cooperation in coevolutionary prisoner's dilemma

Previous studies suggest that cooperation prevails when individuals can switch their interaction partners quickly. However, it is still unclear how quickly individuals should switch adverse partners to maximize cooperation. To address this issue, we propose a simple model of coevolutionary prisoner's dilemma in which individuals are allowed to either adjust their strategies or switch their defective partners. Interestingly, we find that, depending on the game parameter, there is an optimal tendency of switching adverse partnerships that maximizes the fraction of cooperators in the population. We confirm that the stabilization of cooperation by partner switching remains effective under some situations, where either normalized or accumulated payoff is used in strategy updating, and where either only cooperators or all individuals are privileged to sever disadvantageous partners. We also provide an extended pair approximation to study the coevolutionary dynamics. Our results may be helpful in understanding the role of partner switching in the stabilization of cooperation in the real world.

The outbreak of cooperation among success-driven individuals under noisy conditions

According to Thomas Hobbes' Leviathan [1651; 2008 (Touchstone, New York), English Ed], "the life of man [is] solitary, poor, nasty, brutish, and short," and it would need powerful social institutions to establish social order. In reality, however, social cooperation can also arise spontaneously, based on local interactions rather than centralized control. The self-organization of cooperative behavior is particularly puzzling for social dilemmas related to sharing natural resources or creating common goods. Such situations are often described by the prisoner's dilemma. Here, we report the sudden outbreak of predominant cooperation in a noisy world dominated by selfishness and defection, when individuals imitate superior strategies and show success-driven migration. In our model, individuals are unrelated, and do not inherit behavioral traits. They defect or cooperate selfishly when the opportunity arises, and they do not know how often they will interact or have interacted with someone else. Moreover, our individuals have no reputation mechanism to form friendship networks, nor do they have the option of voluntary interaction or costly punishment. Therefore, the outbreak of prevailing cooperation, when directed motion is integrated in a game-theoretical model, is remarkable, particularly when random strategy mutations and random relocations challenge the formation and survival of cooperative clusters. Our results suggest that mobility is significant for the evolution of social order, and essential for its stabilization and maintenance.

Exploration dynamics in evolutionary games

Evolutionary game theory describes systems where individual success is based on the interaction with others. We consider a system in which players unconditionally imitate more successful strategies but sometimes also explore the available strategies at random. Most research has focused on how strategies spread via genetic reproduction or cultural imitation, but random exploration of the available set of strategies has received less attention so far. In genetic settings, the latter corresponds to mutations in the DNA, whereas in cultural evolution, it describes individuals experimenting with new behaviors. Genetic mutations typically occur with very small probabilities, but random exploration of available strategies in behavioral experiments is common. We term this phenomenon "exploration dynamics" to contrast it with the traditional focus on imitation. As an illustrative example of the emerging evolutionary dynamics, we consider a public goods game with cooperators and defectors and add punishers and the option to abstain from the enterprise in further scenarios. For small mutation rates, cooperation (and punishment) is possible only if interactions are voluntary, whereas moderate mutation rates can lead to high levels of cooperation even in compulsory public goods games. This phenomenon is investigated through numerical simulations and analytical approximations.

Spatial three-player prisoners' dilemma

This paper extends traditional two-player prisoners' dilemma (PD) to three-player PD. We have studied spatial patterns of cooperation behaviors, growth patterns of cooperator clusters and defector clusters, and cooperation frequency of the players. It is found while three-player PD exhibits many properties similar to two-player PD, some new features arise. Specifically, (i) a new region appears, in which neither a 3x3 cooperator cluster nor a 3x3 defector cluster could grow; (ii) more growth patterns of cooperator clusters and defector clusters are identified; (iii) multiple cooperation frequencies exist in the region that exhibits dynamic chaos. Some theoretical analysis of these features is presented.

Evolutionary games on minimally structured populations

Population structure induced by both spatial embedding and more general networks of interaction, such as model social networks, have been shown to have a fundamental effect on the dynamics and outcome of evolutionary games. These effects have, however, proved to be sensitive to the details of the underlying topology and dynamics. Here we introduce a minimal population structure that is described by two distinct hierarchical levels of interaction, similar to the structured metapopulation concept of ecology and island models in population genetics. We believe this model is able to identify effects of spatial structure that do not depend on the details of the topology. While effects depending on such details clearly lie outside the scope of our approach, we expect that those we are able to reproduce should be generally applicable to a wide range of models. We derive the dynamics governing the evolution of a system starting from fundamental individual level stochastic processes through two successive mean-field approximations. In our model of population structure the topology of interactions is described by only two parameters: the effective population size at the local scale and the relative strength of local dynamics to global mixing. We demonstrate, for example, the existence of a continuous transition leading to the dominance of cooperation in populations with hierarchical levels of unstructured mixing as the benefit to cost ratio becomes smaller then the local population size. Applying our model of spatial structure to the repeated prisoner's dilemma we uncover a counterintuitive mechanism by which the constant influx of defectors sustains cooperation. Further exploring the phase space of the repeated prisoner's dilemma and also of the "rock-paper-scissor" game we find indications of rich structure and are able to reproduce several effects observed in other models with explicit spatial embedding, such as the maintenance of biodiversity and the emergence of global oscillations.

Social diversity promotes the emergence of cooperation in public goods games

Humans often cooperate in public goods games and situations ranging from family issues to global warming. However, evolutionary game theory predicts that the temptation to forgo the public good mostly wins over collective cooperative action, and this is often also seen in economic experiments. Here we show how social diversity provides an escape from this apparent paradox. Up to now, individuals have been treated as equivalent in all respects, in sharp contrast with real-life situations, where diversity is ubiquitous. We introduce social diversity by means of heterogeneous graphs and show that cooperation is promoted by the diversity associated with the number and size of the public goods game in which each individual participates and with the individual contribution to each such game. When social ties follow a scale-free distribution, cooperation is enhanced whenever all individuals are expected to contribute a fixed amount irrespective of the plethora of public goods games in which they engage. Our results may help to explain the emergence of cooperation in the absence of mechanisms based on individual reputation and punishment. Combining social diversity with reputation and punishment will provide instrumental clues on the self-organization of social communities and their economical implications.

Reputation-based partner choice promotes cooperation in social networks

We investigate the cooperation dynamics attributed to the interplay between the evolution of individual strategies and evolution of individual partnerships. We focus on the effect of reputation on an individual's partner-switching process. We assume that individuals can either change their strategies by imitating their partners or adjust their partnerships based on local information about reputations. We manipulate the partner switching in two ways; that is, individuals can switch from the lowest reputation partners, either to their partners' partners who have the highest reputation (i.e., ordering in partnership) or to others randomly chosen from the entire population (i.e., randomness in partnership). We show that when individuals are able to alter their behavioral strategies and their social interaction partnerships on the basis of reputation, cooperation can prevail. We find that the larger temptation to defect and the denser the partner network, the more frequently individuals need to shift their partnerships in order for cooperation to thrive. Furthermore, an increasing tendency of switching to partners' partners is more likely to lead to a higher level of cooperation. We show that when reputation is absent in such partner-switching processes, cooperation is much less favored than that of the reputation involved. Moreover, we investigate the effect of discounting an individual's reputation on the evolution of cooperation. Our results highlight the importance of the consideration of reputation (indirect reciprocity) on the promotion of cooperation when individuals can adjust their partnerships.

Phase transition and hysteresis loop in structured games with global updating

We present a global payoff-based strategy updating model for studying cooperative behavior of a networked population. We adopt the Prisoner's Dilemma game and the snowdrift game as paradigms for characterizing the interactions among individuals. We investigate the model on regular, small-world, and scale-free networks, and find multistable cooperation states depending on the initial cooperator density. In particular for the snowdrift game on small-world and scale-free networks, there exist a discontinuous phase transition and hysteresis loops of cooperator density. We explain the observed properties by theoretical predictions and simulation results of the average number of neighbors of cooperators and defectors, respectively. Our work indicates that individuals with more neighbors have a trend to preserve their initial strategies, which has strong impacts on the strategy updating of individuals with fewer neighbors; while the fact that individuals with few neighbors have to become cooperators to avoid gaining the lowest payoff plays significant roles in maintaining and spreading of cooperation strategy.

Emergence of synchronization induced by the interplay between two prisoner's dilemma games with volunteering in small-world networks

We studied synchronization between prisoner's dilemma games with voluntary participation in two Newman-Watts small-world networks. It was found that there are three kinds of synchronization: partial phase synchronization, total phase synchronization, and complete synchronization, for varied coupling factors. Besides, two games can reach complete synchronization for the large enough coupling factor. We also discussed the effect of the coupling factor on the amplitude of oscillation of cooperator density.

Cyclic dominance and biodiversity in well-mixed populations

Coevolutionary dynamics is investigated in chemical catalysis, biological evolution, social and economic systems. The dynamics of these systems can be analyzed within the unifying framework of evolutionary game theory. In this Letter, we show that even in well-mixed finite populations, where the dynamics is inherently stochastic, biodiversity is possible with three cyclic-dominant strategies. We show how the interplay of evolutionary dynamics, discreteness of the population, and the nature of the interactions influences the coexistence of strategies. We calculate a critical population size above which coexistence is likely.

Cyclic coevolution of cooperative behaviors and network structures

This paper aims at understanding coevolutionary dynamics of cooperative behaviors and network structures of interactions. We constructed an evolutionary model in which each individual not only has a strategy for prisoner's dilemma to play with its neighboring members on the network, but also has a strategy for changing its neighboring structure of the network. By conducting evolutionary experiments with various settings of the payoff matrix, we found that the coevolutionary cycles of cooperative behaviors of individuals and their network structures repeatedly occurred when both the temptation to defect and the cost for playing a game were moderate.

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k > 2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.

Spatial invasion of cooperation

The evolutionary puzzle of cooperation describes situations where cooperators provide a fitness benefit to other individuals at some cost to themselves. Under Darwinian selection, the evolution of cooperation is a conundrum, whereas non-cooperation (or defection) is not. In the absence of supporting mechanisms, cooperators perform poorly and decrease in abundance. Evolutionary game theory provides a powerful mathematical framework to address the problem of cooperation using the prisoner's dilemma. One well-studied possibility to maintain cooperation is to consider structured populations, where each individual interacts only with a limited subset of the population. This enables cooperators to form clusters such that they are more likely to interact with other cooperators instead of being exploited by defectors. Here we present a detailed analysis of how a few cooperators invade and expand in a world of defectors. If the invasion succeeds, the expansion process takes place in two stages: first, cooperators and defectors quickly establish a local equilibrium and then they uniformly expand in space. The second stage provides good estimates for the global equilibrium frequencies of cooperators and defectors. Under hospitable conditions, cooperators typically form a single, ever growing cluster interspersed with specks of defectors, whereas under more hostile conditions, cooperators form isolated, compact clusters that minimize exploitation by defectors. We provide the first quantitative assessment of the way cooperators arrange in space during invasion and find that the macroscopic properties and the emerging spatial patterns reveal information about the characteristics of the underlying microscopic interactions.

Repeated games and direct reciprocity under active linking

Direct reciprocity relies on repeated encounters between the same two individuals. Here we examine the evolution of cooperation under direct reciprocity in dynamically structured populations. Individuals occupy the vertices of a graph, undergoing repeated interactions with their partners via the edges of the graph. Unlike the traditional approach to evolutionary game theory, where individuals meet at random and have no control over the frequency or duration of interactions, we consider a model in which individuals differ in the rate at which they seek new interactions. Moreover, once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. Whenever the active dynamics of links is sufficiently fast, population structure leads to a simple transformation of the payoff matrix, effectively changing the game under consideration, and hence paving the way for reciprocators to dominate defectors. We derive analytical conditions for evolutionary stability.

Effects of inhomogeneous activity of players and noise on cooperation in spatial public goods games

We study the public goods game in the noisy case by considering the players with inhomogeneous activity of teaching on a square lattice. It is shown that the introduction of the inhomogeneous activity of teaching 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 rhoC with the noise parameter kappa displays several different behaviors: rhoC monotonically increases (decreases) with kappa; rhoC first increases (decreases) with kappa and then it decreases (increases) monotonically after reaching its maximum (minimum) value, which depends on the amount of the multiplication factor r, on whether the system is homogeneous or inhomogeneous, and on whether the adopted updating is synchronous or asynchronous. These results imply that the noise plays an important and nontrivial role in the evolution of cooperation.

Participation costs dismiss the advantage of heterogeneous networks in evolution of cooperation

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 pay-off 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 pay-offs from being positively biased. In replicator-type evolutionary dynamics, it is shown that even a relatively small participation cost extinguishes the merit of heterogeneous networks in terms of cooperation. In this situation, more connected players earn less so that they are no longer spreaders of cooperation. Instead, those with fewer contacts win and guide the evolution. The participation cost, or the baseline pay-off, is irrelevant in homogeneous populations, but is essential for evolutionary games on heterogeneous networks.

Impact of fraud on the mean-field dynamics of cooperative social systems

The evolution of costly cooperation between selfish individuals seems to contradict Darwinian selection, as it reduces the fitness of a cooperating individual. However, several mechanisms such as repeated interactions or spatial structure can lead to the evolution of cooperation. One such mechanism for the evolution of cooperation, in particular among humans, is indirect reciprocity, in which individuals base their decision to cooperate on the reputation of the potential receiver, which has been established in previous interactions. Cooperation can evolve in these systems if individuals preferably cooperate with those that have shown to be cooperative in the past. We analyze the impact of fake reputations or fraud on the dynamics of reputation and on the success of the reputation system itself, using a mean-field description for evolutionary games given by the replicator equation. This allows us to classify the qualitative dynamics of our model analytically. Our results show that cooperation based on indirect reciprocity is robust with respect to fake reputations and can even be enhanced by them. We conclude that fraud per se does not necessarily have a detrimental effect on social systems.

Evolutionary graph theory: breaking the symmetry between interaction and replacement

We study evolutionary dynamics in a population whose structure is given by two graphs: the interaction graph determines who plays with whom in an evolutionary game; the replacement graph specifies the geometry of evolutionary competition and updating. First, we calculate the fixation probabilities of frequency dependent selection between two strategies or phenotypes. We consider three different update mechanisms: birth-death, death-birth and imitation. Then, as a particular example, we explore the evolution of cooperation. Suppose the interaction graph is a regular graph of degree h, the replacement graph is a regular graph of degree g and the overlap between the two graphs is a regular graph of degree l. We show that cooperation is favored by natural selection if b/c>hg/l. Here, b and c denote the benefit and cost of the altruistic act. This result holds for death-birth updating, weak-selection and large population size. Note that the optimum population structure for cooperators is given by maximum overlap between the interaction and the replacement graph (g=h=l), which means that the two graphs are identical. We also prove that a modified replicator equation can describe how the expected values of the frequencies of an arbitrary number of strategies change on replacement and interaction graphs: the two graphs induce a transformation of the payoff matrix.

Randomness enhances cooperation: a resonance-type phenomenon in evolutionary games

We investigate the effect of randomness in both relationships and decisions on the evolution of cooperation. Simulation results show, in such randomness' presence, the system evolves more frequently to a cooperative state than in its absence. Specifically, there is an optimal amount of randomness, which can induce the highest level of cooperation. The mechanism of randomness promoting cooperation resembles a resonancelike fashion, which could be of particular interest in evolutionary game dynamics in economic, biological, and social systems.

Direct reciprocity on graphs

Direct reciprocity is a mechanism for the evolution of cooperation based on the idea of repeated encounters between the same two individuals. Here we examine direct reciprocity in structured populations, where individuals occupy the vertices of a graph. The edges denote who interacts with whom. The graph represents spatial structure or a social network. For birth-death or pairwise comparison updating, we find that evolutionary stability of direct reciprocity is more restrictive on a graph than in a well-mixed population, but the condition for reciprocators to be advantageous is less restrictive on a graph. For death-birth and imitation updating, in contrast, both conditions are easier to fulfill on a graph. Moreover, for all four update mechanisms, reciprocators can dominate defectors on a graph, which is never possible in a well-mixed population. We also study the effect of an error rate, which increases with the number of links per individual; interacting with more people simultaneously enhances the probability of making mistakes. We provide analytic derivations for all results.

Transition from Gaussian to Levy distributions of stochastic payoff variations in the spatial prisoner's dilemma game

We study the impact of stochastic payoff variations with different distributions on the evolution of cooperation in the spatial prisoner's dilemma game. We find that Gaussian-distributed payoff variations are most successful in promoting cooperation irrespective of the temptation to defect. In particular, the facilitative effect of noise on the evolution of cooperation decreases steadily as the frequency of rare events increases. Findings are explained via an analysis of local payoff ranking violations. The relevance of results for economics and sociology is discussed.

Stochastic payoff evaluation increases the temperature of selection

We study stochastic evolutionary game dynamics in populations of finite size. Moreover, each individual has a randomly distributed number of interactions with other individuals. Therefore, the payoff of two individuals using the same strategy can be different. The resulting "payoff stochasticity" reduces the intensity of selection and therefore increases the temperature of selection. A simple mean-field approximation is derived that captures the average effect of the payoff stochasticity. Correction terms to the mean-field theory are computed and discussed.

Networks with dispersed degrees save stable coexistence of species in cyclic competition

Coexistence of individuals with different species or phenotypes is often found in nature in spite of competition between them. Stable coexistence of multiple types of individuals have implications for maintenance of ecological biodiversity and emergence of altruism in society, to name a few. Various mechanisms of coexistence including spatial structure of populations, heterogeneous individuals, and heterogeneous environments, have been proposed. In reality, individuals disperse and interact on complex networks. We examine how heterogeneous degree distributions of networks influence coexistence, focusing on models of cyclically competing species. We show analytically and numerically that heterogeneity in degree distributions promotes stable coexistence.

The replicator equation on graphs

We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called 'birth-death', 'death-birth' and 'imitation'. A fourth update rule, 'pairwise comparison', is shown to be equivalent to birth-death updating in our model. We use pair approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a well-mixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner's Dilemma, the Snow-Drift game, a coordination game and the Rock-Scissors-Paper game.

Memory-based snowdrift game on networks

We present a memory-based snowdrift game (MBSG) taking place on networks. We found that, when a lattice is taken to be the underlying structure, the transition of spatial patterns at some critical values of the payoff parameter is observable for both four- and eight-neighbor lattices. The transition points as well as the styles of spatial patterns can be explained by local stability analysis. In sharp contrast to previously reported results, cooperation is promoted by the spatial structure in the MBSG. Interestingly, we found that the frequency of cooperation of the MBSG on a scale-free network peaks at a specific value of the payoff parameter. This phenomenon indicates that properly encouraging selfish behaviors can optimally enhance the cooperation. The memory effects of individuals are discussed in detail and some non-monotonous phenomena are observed on both lattices and scale-free networks. Our work may shed some new light on the study of evolutionary games over networks.

Time scales in evolutionary dynamics

Evolutionary game theory has traditionally assumed that all individuals in a population interact with each other between reproduction events. We show that eliminating this restriction by explicitly considering the time scales of interaction and selection leads to dramatic changes in the outcome of evolution. Examples include the selection of the inefficient strategy in the Harmony and Stag-Hunt games, and the disappearance of the coexistence state in the Snowdrift game. Our results hold for any population size and in more general situations with additional factors influencing fitness.

Evolutionary prisoner's dilemma game with dynamic preferential selection

We study a modified prisoner's dilemma game taking place on two-dimensional disordered square lattices. The players are pure strategists and can either cooperate or defect with their immediate neighbors. In the generations each player updates its strategy by following one of the neighboring strategies with a probability dependent on the payoff difference. The neighbor selection obeys a dynamic preferential rule, i.e., the more frequently a neighbor's strategy was adopted by the focal player in the previous rounds, the larger probability it will be chosen to refer to in the subsequent rounds. It is found that cooperation is substantially promoted due to this simple selection mechanism. Corresponding analysis is provided by the investigation of the distribution of the players' impact weights, persistence, and correlation function.

Stochasticity and evolutionary stability

In stochastic dynamical systems, different concepts of stability can be obtained in different limits. A particularly interesting example is evolutionary game theory, which is traditionally based on infinite populations, where strict Nash equilibria correspond to stable fixed points that are always evolutionarily stable. However, in finite populations stochastic effects can drive the system away from strict Nash equilibria, which gives rise to a new concept for evolutionary stability. The conventional and the new stability concepts may apparently contradict each other leading to conflicting predictions in large yet finite populations. We show that the two concepts can be derived from the frequency dependent Moran process in different limits. Our results help to determine the appropriate stability concept in large finite populations. The general validity of our findings is demonstrated showing that the same results are valid employing vastly different co-evolutionary processes.

Active linking in evolutionary games

In the traditional approach to evolutionary game theory, the individuals of a population meet each other at random, and they have no control over the frequency or duration of interactions. Here we remove these simplifying assumptions. We introduce a new model, where individuals differ in the rate at which they seek new interactions. Once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. In a limiting case, the linking dynamics introduces a simple transformation of the payoff matrix. We outline conditions for evolutionary stability. As a specific example, we study the interaction between cooperators and defectors. We find a simple relationship that characterizes those linking dynamics which allow natural selection to favour cooperation over defection.

Coevolutionary dynamics in large, but finite populations

Coevolving and competing species or game-theoretic strategies exhibit rich and complex dynamics for which a general theoretical framework based on finite populations is still lacking. Recently, an explicit mean-field description in the form of a Fokker-Planck equation was derived for frequency-dependent selection with two strategies in finite populations based on microscopic processes [A. Traulsen, J. C. Claussen, and C. Hauert, Phys. Rev. Lett. 95, 238701 (2005)]. Here we generalize this approach in a twofold way: First, we extend the framework to an arbitrary number of strategies and second, we allow for mutations in the evolutionary process. The deterministic limit of infinite population size of the frequency-dependent Moran process yields the adjusted replicator-mutator equation, which describes the combined effect of selection and mutation. For finite populations, we provide an extension taking random drift into account. In the limit of neutral selection, i.e., whenever the process is determined by random drift and mutations, the stationary strategy distribution is derived. This distribution forms the background for the coevolutionary process. In particular, a critical mutation rate uc is obtained separating two scenarios: above uc the population predominantly consists of a mixture of strategies whereas below uc the population tends to be in homogeneous states. For one of the fundamental problems in evolutionary biology, the evolution of cooperation under Darwinian selection, we demonstrate that the analytical framework provides excellent approximations to individual based simulations even for rather small population sizes. This approach complements simulation results and provides a deeper, systematic understanding of coevolutionary dynamics.

Dynamic-persistence of cooperation in public good games when group size is dynamic

The evolution of cooperation is possible with a simple model of a population of agents that can move between groups. The agents play public good games within their group. The relative fitness of individuals within the whole population affects their number of offspring. Groups of cooperators evolve but over time are invaded by defectors which eventually results in the group's extinction. However, for small levels of migration and mutation, high levels of cooperation evolve at the population level. Thus, evolution of cooperation based on individual fitness without kin selection, indirect or direct reciprocity is possible. We provide an analysis of the parameters that affect cooperation, and describe the dynamics and distribution of population sizes over time.

Multi-state epidemic processes on complex networks

Infectious diseases are practically represented by models with multiple states and complex transition rules corresponding to, for example, birth, death, infection, recovery, disease progression, and quarantine. In addition, networks underlying infection events are often much more complex than described by meanfield equations or regular lattices. In models with simple transition rules such as the SIS and SIR models, heterogeneous contact rates are known to decrease epidemic thresholds. We analyse steady states of various multi-state disease propagation models with heterogeneous contact rates. In many models, heterogeneity simply decreases epidemic thresholds. However, in models with competing pathogens and mutation, coexistence of different pathogens for small infection rates requires network-independent conditions in addition to heterogeneity in contact rates. Furthermore, models without spontaneous neighbor-independent state transitions, such as cyclically competing species, do not show heterogeneity effects.

Spatial prisoner's dilemma game with volunteering in Newman-Watts small-world networks

A modified spatial prisoner's dilemma game with voluntary participation in Newman-Watts small-world networks is studied. Some reasonable ingredients are introduced to the game evolutionary dynamics: each agent in the network is a pure strategist and can only take one of three strategies (cooperator, defector, and loner); its strategical transformation is associated with both the number of strategical states and the magnitude of average profits, which are adopted and acquired by its coplayers in the previous round of play; a stochastic strategy mutation is applied when it gets into the trouble of local commons that the agent and its neighbors are in the same state and get the same average payoffs. In the case of very low temptation to defect, it is found that agents are willing to participate in the game in typical small-world region and intensive collective oscillations arise in more random region.

Non-Gaussian fluctuations arising from finite populations: Exact results for the evolutionary Moran process

The appropriate description of fluctuations within the framework of evolutionary game theory is a fundamental unsolved problem in the case of finite populations. The Moran process recently introduced into this context in Nowak, [Nature (London) 428, 646 (2004)] defines a promising standard model of evolutionary game theory in finite populations for which analytical results are accessible. In this paper, we derive the stationary distribution of the Moran process population dynamics for arbitrary 2 x 2 games for the finite-size case. We show that a nonvanishing background fitness can be transformed to the vanishing case by rescaling the payoff matrix. In contrast to the common approach to mimic finite-size fluctuations by Gaussian distributed noise, the finite-size fluctuations can deviate significantly from a Gaussian distribution.

Effects of neighbourhood size and connectivity on the spatial Continuous Prisoner's Dilemma

The Prisoner's Dilemma, a two-person game in which the players can either cooperate or defect, is a common paradigm for studying the evolution of cooperation. In real situations cooperation is almost never all or nothing. This observation is the motivation for the Continuous Prisoner's Dilemma, in which individuals exhibit variable degrees of cooperation. It is known that in the presence of spatial structure, when individuals "play against" (i.e. interact with) their neighbours, and "compare to" ("learn from") them, cooperative investments can evolve to considerable levels. Here, we examine the effect of increasing the neighbourhood size: we find that the mean-field limit of no cooperation is reached for a critical neighbourhood size of about five neighbours on each side in a Moore neighbourhood, which does not depend on the size of the spatial lattice. We also find the related result that in a network of players, the critical average degree (number of neighbours) of nodes for which defection is the final state does not depend on network size, but only on the network topology. This critical average degree is considerably (about 10 times) higher for clustered (social) networks, than for distributed random networks. This result strengthens the argument that clustering is the mechanism which makes the development and maintenance of the cooperation possible. In the lattice topology, it is observed that when the neighbourhood sizes for "interacting" and "learning" differ by more than 0.5, cooperation is not sustainable, even for neighbourhood sizes that are below the mean-field limit of defection. We also study the evolution of neighbourhood sizes, as well as investment level. Here, we observe that the series of the interaction and learning neighbourhoods converge, and a final cooperative state with considerable levels of average investment is achieved.

Similarity-based cooperation and spatial segregation

We analyze a cooperative game, where the cooperative act is not based on the previous behavior of the coplayer, but on the similarity between the players. This system has been studied in a mean-field description recently [Phys. Rev. E 68, 046129 (2003)]]. Here, the spatial extension to a two-dimensional lattice is studied, where each player interacts with eight players in a Moore neighborhood. The system shows a strong segregation independent of parameters. The introduction of a local conversion mechanism towards tolerance allows for four-state cycles and the emergence of spiral waves in the spatial game. In the case of asymmetric costs of cooperation a rich variety of complex behavior is observed depending on both cooperation costs. Finally, we study the stabilization of a cooperative fixed point of a forecast rule in the symmetric game, which corresponds to cooperation across segregation borders. This fixed point becomes unstable for high cooperation costs, but can be stabilized by a linear feedback mechanism.

Spatial structure often inhibits the evolution of cooperation in the snowdrift game

Understanding the emergence of cooperation is a fundamental problem in evolutionary biology. Evolutionary game theory has become a powerful framework with which to investigate this problem. Two simple games have attracted most attention in theoretical and experimental studies: the Prisoner's Dilemma and the snowdrift game (also known as the hawk-dove or chicken game). In the Prisoner's Dilemma, the non-cooperative state is evolutionarily stable, which has inspired numerous investigations of suitable extensions that enable cooperative behaviour to persist. In particular, on the basis of spatial extensions of the Prisoner's Dilemma, it is widely accepted that spatial structure promotes the evolution of cooperation. Here we show that no such general predictions can be made for the effects of spatial structure in the snowdrift game. In unstructured snowdrift games, intermediate levels of cooperation persist. Unexpectedly, spatial structure reduces the proportion of cooperators for a wide range of parameters. In particular, spatial structure eliminates cooperation if the cost-to-benefit ratio of cooperation is high. Our results caution against the common belief that spatial structure is necessarily beneficial for cooperative behaviour.

Cooperation for volunteering and partially random partnerships

Competition among cooperative, defective, and loner strategies is studied by considering an evolutionary prisoner's dilemma game for different partnerships. In this game each player can adopt one of its coplayer's strategy with a probability depending on the difference of payoffs coming from games with the corresponding coplayers. Our attention is focused on the effects of annealed and quenched randomness in the partnership for fixed number of coplayers. It is shown that only the loners survive if the four coplayers are chosen randomly (mean-field limit). On the contrary, on the square lattice all the three strategies are maintained by the cyclic invasions resulting in a self-organizing spatial pattern. If the fixed partnership is described by a regular small-world structure then a homogeneous oscillation occurs in the population dynamics when the measure of quenched randomness exceeds a threshold value. Similar behavior with higher sensitivity to the randomness is found if temporary partners are substituted for the standard ones with some probability at each step of iteration.

Minimal model for tag-based cooperation

Recently, Riolo et al. [Nature (London) 414, 441 (2001)] showed by computer simulations that cooperation can arise without reciprocity when agents donate only to partners who are sufficiently similar to themselves. One striking outcome of their simulations was the observation that the number of tolerant agents that support a wide range of players was not constant in time, but showed characteristic fluctuations. The cause and robustness of these tides of tolerance remained to be explored. Here we clarify the situation by solving a minimal version of the model of Riolo et al. It allows us to identify a net surplus of random changes from intolerant to tolerant agents as a necessary mechanism that produces these oscillations of tolerance, which segregate different agents in time. This provides a new mechanism for maintaining different agents, i.e., for creating biodiversity. In our model the transition to the oscillating state is caused by a saddle node bifurcation. The frequency of the oscillations increases linearly with the transition rate from tolerant to intolerant agents.

The dynamics of multidimensional secession: fixed points and ideological condensation

We explore a generalized, stochastic seceder model of societal dynamics with variable size polling groups and higher-dimensional opinion vectors, revealing its essential modes of self-organized segregation. Renormalizing to a discrete, deterministic version, we pin down the upper critical size of the sampling group and analytically uncover a self-similar hierarchy of dynamically stable, multiple-branch fixed points. In d>/=3, the evolving, coarsening population suffers collapse to a 2D ideological plane.

Evolutionary prisoner's dilemma games with voluntary participation

Voluntary participation in public good games has recently been demonstrated to be a simple yet effective mechanism to avoid deadlocks in states of mutual defection and to promote persistent cooperative behavior. Apart from cooperators and defectors a third strategical type is considered: the risk averse loners who are unwilling to participate in the social enterprise and rather rely on small but fixed earnings. This results in a rock-scissors-paper type of cyclic dominance of the three strategies. In the prisoner's dilemma, the effects of voluntary participation crucially depend on the underlying population structure. While leading to homogeneous states of all loners in well-mixed populations, we demonstrate that cyclic dominance produces self-organizing patterns on square lattices but leads to different types of oscillatory behavior on random regular graphs: the temptation to defect determines whether damped, periodic, or increasing oscillations occur. These Monte Carlo simulations are complemented by predictions from pair approximation reproducing the results for random regular graphs particularly well.