Spatial effects in social dilemmas

Social dilemmas and the evolutionary conundrum of cooperation are traditionally studied through various kinds of game theoretical models such as the prisoner's dilemma, public goods games, snowdrift games or by-product mutualism. All of them exemplify situations which are characterized by different degrees of conflicting interests between the individuals and the community. In groups of interacting individuals, cooperators produce a common good benefitting the entire group at some cost to themselves, whereas defectors attempt to exploit the resource by avoiding the costly contributions. Based on synergistic or discounted accumulation of cooperative benefits a unifying theoretical framework was recently introduced that encompasses all games that have traditionally been studied separately (Hauert, Michor, Nowak, Doebeli, 2005. Synergy and discounting of cooperation in social dilemmas. J. Theor. Biol., in press.). Within this framework we investigate the effects of spatial structure with limited local interactions on the evolutionary fate of cooperators and defectors. The quantitative effects of space turn out to be quite sensitive to the underlying microscopic update mechanisms but, more general, we demonstrate that in prisoner's dilemma type interactions spatial structure benefits cooperation-although the parameter range is quite limited-whereas in snowdrift type interactions spatial structure may be beneficial too, but often turns out to be detrimental to cooperation.     

Ecological public goods games: cooperation and bifurcation

The Public Goods Game is one of the most popular models for studying the origin and maintenance of cooperation. In its simplest form, this evolutionary game has two regimes: defection goes to fixation if the multiplication factor r is smaller than the interaction group size N, whereas cooperation goes to fixation if the multiplication factor r is larger than the interaction group size N. Hauert et al. [Hauert, C., Holmes, M., Doebeli, M., 2006a. Evolutionary games and population dynamics: Maintenance of cooperation in public goods games. Proc. R. Soc. Lond. B 273, 2565-2570] have introduced the Ecological Public Goods Game by viewing the payoffs from the evolutionary game as birth rates in a population dynamic model. This results in a feedback between ecological and evolutionary dynamics: if defectors are prevalent, birth rates are low and population densities decline, which leads to smaller interaction groups for the Public Goods game, and hence to dominance of cooperators, with a concomitant increase in birth rates and population densities. This feedback can lead to stable co-existence between cooperators and defectors. Here we provide a detailed analysis of the dynamics of the Ecological Public Goods Game, showing that the model exhibits various types of bifurcations, including supercritical Hopf bifurcations, which result in stable limit cycles, and hence in oscillatory co-existence of cooperators and defectors. These results show that including population dynamics in evolutionary games can have important consequences for the evolutionary dynamics of cooperation.     

Evolution of cooperation under -person snowdrift games

In the animal world, performing a given task which is beneficial to an entire group requires the cooperation of several individuals of that group who often share the workload required to perform the task. The mathematical framework to study the dynamics of collective action is game theory. Here we study the evolutionary dynamics of cooperators and defectors in a population in which groups of individuals engage in N-person, non-excludable public goods games. We explore an N-person generalization of the well-known two-person snowdrift game. We discuss both the case of infinite and finite populations, taking explicitly into consideration the possible existence of a threshold above which collective action is materialized. Whereas in infinite populations, an N-person snowdrift game (NSG) leads to a stable coexistence between cooperators and defectors, the introduction of a threshold leads to the appearance of a new interior fixed point associated with a coordination threshold. The fingerprints of the stable and unstable interior fixed points still affect the evolutionary dynamics in finite populations, despite evolution leading the population inexorably to a monomorphic end-state. However, when the group size and population size become comparable, we find that spite sets in, rendering cooperation unfeasible.     

Evolutionary games and population dynamics: maintenance of cooperation in public goods games

The emergence and abundance of cooperation in nature poses a tenacious and challenging puzzle to evolutionary biology. Cooperative behaviour seems to contradict Darwinian evolution because altruistic individuals increase the fitness of other members of the population at a cost to themselves. Thus, in the absence of supporting mechanisms, cooperation should decrease and vanish, as predicted by classical models for cooperation in evolutionary game theory, such as the Prisoner's Dilemma and public goods games. Traditional approaches to studying the problem of cooperation assume constant population sizes and thus neglect the ecology of the interacting individuals. Here, we incorporate ecological dynamics into evolutionary games and reveal a new mechanism for maintaining cooperation. In public goods games, cooperation can gain a foothold if the population density depends on the average population payoff. Decreasing population densities, due to defection leading to small payoffs, results in smaller interaction group sizes in which cooperation can be favoured. This feedback between ecological dynamics and game dynamics can generate stable coexistence of cooperators and defectors in public goods games. However, this mechanism fails for pairwise Prisoner's Dilemma interactions and the population is driven to extinction. Our model represents natural extension of replicator dynamics to populations of varying densities.     

Coexistence of cooperation and defection in public goods games

The production of public goods by the contribution of individual volunteers is a social dilemma because an individual that does not volunteer can benefit from the public good produced by the contributions of others. Therefore it is generally believed that public goods can be produced only in the presence of repeated interactions (which allow reciprocation, reputation effects and punishment) or relatedness (kin selection). Cooperation, however, often occurs in the absence of iterations and relatedness. We show that when the production of a public good is a Volunteer's Dilemma, in which a fixed number of cooperators is necessary to produce the public good, cooperators and defectors persist in a mixed equilibrium, without iterations and without relatedness. This mixed equilibrium is absent in the N-person Prisoner's Dilemma, in which the public good is a linear function of the individual contributions. We also show that the Prisoner's Dilemma and the Volunteer's Dilemma are the two opposite extremes of a general public goods game, and that all intermediate cases can have a mixed equilibrium like the Volunteer's Dilemma. The coexistence of cooperators and defectors, therefore, is a typical outcome of most social dilemmas, which requires neither relatedness nor iterations.     

Replicator dynamics of reward & reputation in public goods games

Public goods games have become the mathematical metaphor for game theoretical investigations of cooperative behavior in groups of interacting individuals. Cooperation is a conundrum because cooperators make a sacrifice to benefit others at some cost to themselves. Exploiters or defectors reap the benefits and forgo costs. Despite the fact that groups of cooperators outperform groups of defectors, Darwinian selection or utilitarian principles based on rational choice should favor defectors. In order to overcome this social dilemma, much effort has been expended for investigations pertaining to punishment and sanctioning measures against defectors. Interestingly, the complementary approach to create positive incentives and to reward cooperation has received considerably less attention—despite being heavily advocated in education and social sciences for increasing productivity or preventing conflicts. Here we show that rewards can indeed stimulate cooperation in interaction groups of arbitrary size but, in contrast to punishment, fail to stabilize it. In both cases, however, reputation is essential. The combination of reward and reputation result in complex dynamics dominated by unpredictable oscillations.     

Synergy and discounting of cooperation in social dilemmas

The emergence and maintenance of cooperation by natural selection is an enduring conundrum in evolutionary biology, which has been studied using a variety of game theoretical models inspired by different biological situations. The most widely studied games are the Prisoner's Dilemma, the Snowdrift game and by-product mutualism for pairwise interactions, as well as Public Goods games in larger groups of interacting individuals. Here, we present a general framework for cooperation in social dilemmas in which all the traditional scenarios can be recovered as special cases. In social dilemmas, cooperators provide a benefit to the group at some cost, while defectors exploit the group by reaping the benefits without bearing the costs of cooperation. Using the concepts of discounting and synergy for describing how benefits accumulate when more than one cooperator is present in a group of interacting individuals, we recover the four basic scenarios of evolutionary dynamics given by (i) dominating defection, (ii) coexistence of defectors and cooperators, (iii) dominating cooperation and (iv) bi-stability, in which cooperators and defectors cannot invade each other. Generically, for groups of three or more interacting individuals further, more complex, dynamics can occur. Our framework provides the first unifying approach to model cooperation in different kinds of social dilemmas.     

The Increased Risk of Joint Venture Promotes Social Cooperation

The joint venture of many members is common both in animal world and human society. In these public enterprizes, highly cooperative groups are more likely to while low cooperative groups are still possible but not probable to succeed. Existent literature mostly focuses on the traditional public goods game, in which cooperators create public wealth unconditionally and benefit all group members unbiasedly. We here institute a model addressing this public goods dilemma with incorporating the public resource foraging failure risk. Risk-averse individuals tend to lead a autarkic life, while risk-preferential ones tend to participate in the risky public goods game. For participants, group''s success relies on its cooperativeness, with increasing contribution leading to increasing success likelihood. We introduce a function with one tunable parameter to describe the risk removal pattern and study in detail three representative classes. Analytical results show that the widely replicated population dynamics of cyclical dominance of loner, cooperator and defector disappear, while most of the time loners act as savors while eventually they also disappear. Depending on the way that group''s success relies on its cooperativeness, either cooperators pervade the entire population or they coexist with defectors. Even in the later case, cooperators still hold salient superiority in number as some defectors also survive by parasitizing. The harder the joint venture succeeds, the higher level of cooperation once cooperators can win the evolutionary race. Our work may enrich the literature concerning the risky public goods games.     

The durability of public goods changes the dynamics and nature of social dilemmas

An implicit assumption underpins basic models of the evolution of cooperation, mutualism and altruism: The benefits (or pay-offs) of cooperation and defection are defined by the current frequency or distribution of cooperators. In social dilemmas involving durable public goods (group resources that can persist in the environment-ubiquitous from microbes to humans) this assumption is violated. Here, we examine the consequences of relaxing this assumption, allowing pay-offs to depend on both current and past numbers of cooperators. We explicitly trace the dynamic of a public good created by cooperators, and define pay-offs in terms of the current public good. By raising the importance of cooperative history in determining the current fate of cooperators, durable public goods cause novel dynamics (e.g., transient increases in cooperation in Prisoner's Dilemmas, oscillations in Snowdrift Games, or shifts in invasion thresholds in Stag-hunt Games), while changes in durability can transform one game into another, by moving invasion thresholds for cooperation or conditions for coexistence with defectors. This enlarged view challenges our understanding of social cheats. For instance, groups of cooperators can do worse than groups of defectors, if they inherit fewer public goods, while a rise in defectors no longer entails a loss of social benefits, at least not in the present moment (as highlighted by concerns over environmental lags). Wherever durable public goods have yet to reach a steady state (for instance due to external perturbations), the history of cooperation will define the ongoing dynamics of cooperators.     

Economic game theory for mutualism and cooperation

We review recent work at the interface of economic game theory and evolutionary biology that provides new insights into the evolution of partner choice, host sanctions, partner fidelity feedback and public goods. (1) The theory of games with asymmetrical information shows that the right incentives allow hosts to screen-out parasites and screen-in mutualists, explaining successful partner choice in the absence of signalling. Applications range from ant-plants to microbiomes. (2) Contract theory distinguishes two longstanding but weakly differentiated explanations of host response to defectors: host sanctions and partner fidelity feedback. Host traits that selectively punish misbehaving symbionts are parsimoniously interpreted as pre-adaptations. Yucca-moth and legume-rhizobia mutualisms are argued to be examples of partner fidelity feedback. (3) The theory of public goods shows that cooperation in multi-player interactions can evolve in the absence of assortment, in one-shot social dilemmas among non-kin. Applications include alarm calls in vertebrates and exoenzymes in microbes.     

Public goods games with reward in finite populations

Public goods games paraphrase the problem of cooperation in game theoretical terms. Cooperators contribute to a public good and thereby increase the welfare of others at a cost to themselves. Defectors consume the public good but do not pay its cost and therefore outperform cooperators. Hence, according to genetic or cultural evolution, defectors should be favored and the public good disappear – despite the fact that groups of cooperators are better off than groups of defectors. The maximization of short term individual profits causes the demise of the common resource to the detriment of all. This outcome can be averted by introducing incentives to cooperate. Negative incentives based on the punishment of defectors efficiently stabilize cooperation once established but cannot initiate cooperation. Here we consider the complementary case of positive incentives created by allowing individuals to reward those that contribute to the public good. The finite-population stochastic dynamics of the public goods game with reward demonstrate that reward initiates cooperation by providing an escape hatch out of states of mutual defection. However, in contrast to punishment, reward is unable to stabilize cooperation but, instead, gives rise to a persistent minority of cooperators.     

Adaptive dynamics of cooperation may prevent the coexistence of defectors and cooperators and even cause extinction

It has recently been demonstrated that ecological feedback mechanisms can facilitate the emergence and maintenance of cooperation in public goods interactions: the replicator dynamics of defectors and cooperators can result, for example, in the ecological coexistence of cooperators and defectors. Here we show that these results change dramatically if cooperation strategy is not fixed but instead is a continuously varying trait under natural selection. For low values of the factor with which the value of resources is multiplied before they are shared among all participants, evolution will always favour lower cooperation strategies until the population falls below an Allee threshold and goes extinct, thus evolutionary suicide occurs. For higher values of the factor, there exists a unique evolutionarily singular strategy, which is convergence stable. Because the fitness function is linear with respect to the strategy of the mutant, this singular strategy is neutral against mutant invasions. This neutrality disappears if a nonlinear functional response in receiving benefits is assumed. For strictly concave functional responses, singular strategies become uninvadable. Evolutionary branching, which could result in the evolutionary emergence of cooperators and defectors, can occur only with locally convex functional responses, but we illustrate that it can also result in coevolutionary extinction.     

Adaptive Dynamics of Altruistic Cooperation in a Metapopulation: Evolutionary Emergence of Cooperators and Defectors or Evolutionary Suicide?

We investigate the evolution of public goods cooperation in a metapopulation model with small local populations, where altruistic cooperation can evolve due to assortment and kin selection, and the evolutionary emergence of cooperators and defectors via evolutionary branching is possible. Although evolutionary branching of cooperation has recently been demonstrated in the continuous snowdrift game and in another model of public goods cooperation, the required conditions on the cost and benefit functions are rather restrictive, e.g., altruistic cooperation cannot evolve in a defector population. We also observe selection for too low cooperation, such that the whole metapopulation goes extinct and evolutionary suicide occurs. We observed intuitive effects of various parameters on the numerical value of the monomorphic singular strategy. Their effect on the final coexisting cooperator–defector pair is more complex: changes expected to increase cooperation decrease the strategy value of the cooperator. However, at the same time the population size of the cooperator increases enough such that the average strategy does increase. We also extend the theory of structured metapopulation models by presenting a method to calculate the fitness gradient in a general class of metapopulation models, and try to make a connection with the kin selection approach.     

DILEMMAS OF PARTIAL COOPERATION

Related to the often applied cooperation models of social dilemmas, we deal with scenarios in which defection dominates cooperation, but an intermediate fraction of cooperators, that is, “partial cooperation,” would maximize the overall performance of a group of individuals. Of course, such a solution comes at the expense of cooperators that do not profit from the overall maximum. However, because there are mechanisms accounting for mutual benefits after repeated interactions or through evolutionary mechanisms, such situations can constitute “dilemmas” of partial cooperation. Among the 12 ordinally distinct, symmetrical 2 × 2 games, three (barely considered) variants are correspondents of such dilemmas. Whereas some previous studies investigated particular instances of such games, we here provide the unifying framework and concisely relate it to the broad literature on cooperation in social dilemmas. Complementing our argumentation, we study the evolution of partial cooperation by deriving the respective conditions under which coexistence of cooperators and defectors, that is, partial cooperation, can be a stable outcome of evolutionary dynamics in these scenarios. Finally, we discuss the relevance of such models for research on the large biodiversity and variation in cooperative efforts both in biological and social systems.     

The Effect of Ostracism and Optional Participation on the Evolution of Cooperation in the Voluntary Public Goods Game

Not only animals, plants and microbes but also humans cooperate in groups. The evolution of cooperation in a group is an evolutionary puzzle, because defectors always obtain a higher benefit than cooperators. When people participate in a group, they evaluate group member’s reputations and then decide whether to participate in it. In some groups, membership is open to all who are willing to participate in the group. In other groups, a candidate is excluded from membership if group members regard the candidate’s reputation as bad. We developed an evolutionary game model and investigated how participation in groups and ostracism influence the evolution of cooperation in groups when group members play the voluntary public goods game, by means of computer simulation. When group membership is open to all candidates and those candidates can decide whether to participate in a group, cooperation cannot be sustainable. However, cooperation is sustainable when a candidate cannot be a member unless all group members admit them to membership. Therefore, it is not participation in a group but rather ostracism, which functions as costless punishment on defectors, that is essential to sustain cooperation in the voluntary public goods game.     

Models of cooperation based on the Prisoner's Dilemma and the Snowdrift game

Understanding the mechanisms that can lead to the evolution of cooperation through natural selection is a core problem in biology. Among the various attempts at constructing a theory of cooperation, game theory has played a central role. Here, we review models of cooperation that are based on two simple games: the Prisoner's Dilemma, and the Snowdrift game. Both games are two‐person games with two strategies, to cooperate and to defect, and both games are social dilemmas. In social dilemmas, cooperation is prone to exploitation by defectors, and the average payoff in populations at evolutionary equilibrium is lower than it would be in populations consisting of only cooperators. The difference between the games is that cooperation is not maintained in the Prisoner's Dilemma, but persists in the Snowdrift game at an intermediate frequency. As a consequence, insights gained from studying extensions of the two games differ substantially. We review the most salient results obtained from extensions such as iteration, spatial structure, continuously variable cooperative investments, and multi‐person interactions. Bridging the gap between theoretical and empirical research is one of the main challenges for future studies of cooperation, and we conclude by pointing out a number of promising natural systems in which the theory can be tested experimentally.     

Probabilistic participation in public goods games

Voluntary participation in public goods games (PGGs) has turned out to be a simple but effective mechanism for promoting cooperation under full anonymity. Voluntary participation allows individuals to adopt a risk-aversion strategy, termed loner. A loner refuses to participate in unpromising public enterprises and instead relies on a small but fixed pay-off. This system leads to a cyclic dominance of three pure strategies, cooperators, defectors and loners, but at the same time, there remain two considerable restrictions: the addition of loners cannot stabilize the dynamics and the time average pay-off for each strategy remains equal to the pay-off of loners. Here, we introduce probabilistic participation in PGGs from the standpoint of diversification of risk, namely simple mixed strategies with loners, and prove the existence of a dynamical regime in which the restrictions ono longer hold. Considering two kinds of mixed strategies associated with participants (cooperators or defectors) and non-participants (loners), we can recover all basic evolutionary dynamics of the two strategies: dominance; coexistence; bistability; and neutrality, as special cases depending on pairs of probabilities. Of special interest is that the expected pay-off of each mixed strategy exceeds the pay-off of loners at some interior equilibrium in the coexistence region.     

Evolutionary game dynamics with non-uniform interaction rates

The classical setting of evolutionary game theory, the replicator equation, assumes uniform interaction rates. The rate at which individuals meet and interact is independent of their strategies. Here we extend this framework by allowing the interaction rates to depend on the strategies. This extension leads to non-linear fitness functions. We show that a strict Nash equilibrium remains uninvadable for non-uniform interaction rates, but the conditions for evolutionary stability need to be modified. We analyze all games between two strategies. If the two strategies coexist or exclude each other, then the evolutionary dynamics do not change qualitatively, only the location of the equilibrium point changes. If, however, one strategy dominates the other in the classical setting, then the introduction of non-uniform interaction rates can lead to a pair of interior equilibria. For the Prisoner's Dilemma, non-uniform interaction rates allow the coexistence between cooperators and defectors. For the snowdrift game, non-uniform interaction rates change the equilibrium frequency of cooperators.     

For whom is it more beneficial to stop interactions with defectors: Cooperators or defectors?

Cooperation is a mysterious evolutionary phenomenon and its mechanisms require elucidation. When cooperators can stop interactions with defectors, the evolution of cooperation becomes possible; this is one mechanism that facilitates the evolution of cooperation. Here, stopping interactions with defectors is beneficial not only for cooperators but also for defectors. The question then arises, for whom is stopping interactions with defectors more beneficial: cooperators or defectors? By utilizing evolutionary game theory, I addressed this question using a two-player game involving four strategies: (1) cooperators who stop the interaction if the current partner is a defector, (2) cooperators who attempt to maintain a relationship with anyone, (3) defectors who stop the interaction if the current partner is a defector, and (4) defectors who attempt to maintain a relationship with anyone. Our results show that, at equilibrium, the ratio of cooperators who stop the interaction if the current partner is a defector to cooperators who attempt to maintain a relationship with anyone is larger than the ratio of defectors who stop the interaction if the current partner is a defector to defectors who attempt to maintain a relationship with anyone. Thus, cooperators rather than defectors are more likely to stop interactions with defectors at equilibrium. This result is consistent with a previous experimental study in which a positive correlation was detected between the degree of individuals’ cooperativeness and how accurately the individuals recognize whether other individuals are cooperators or defectors. Thus, the theoretical work presented in this study provides relevant insights into the natural phenomena of cooperation and recognition.     

Pattern formation and chaos in spatial ecological public goodsgames

Cooperators and defectors can coexist in ecological public goods games. When the game is played in two-dimensional continuous space, a reaction diffusion model produces highly irregular dynamics, in which cooperators and defectors survive in ever-changing configurations (Wakano et al., 2009. Spatial dynamics of ecological public goods. Proc. Natl. Acad. Sci. 106, 7910-7914). The dynamics is related to the formation of Turing patterns, but the origin of the irregular dynamics is not well understood. In this paper, we present a classification of the spatio-temporal dynamics based on the dispersion relation, which reveals that the spontaneous pattern formation can be attributed to the dynamical interplay between two linearly unstable modes: temporal instability arising from a Hopf-bifurcation and spatial instability arising from a Turing-bifurcation. Moreover, we provide a detailed analysis of the highly irregular dynamics through Fourier analysis, the break-down of symmetry, the maximum Lyapunov exponent, and the excitability of the reaction-term dynamics. All results clearly support that the observed irregular dynamics qualifies as spatio-temporal chaos. A particularly interesting type of chaotic dynamics, which we call intermittent bursts, clearly demonstrates the effects of the two unstable modes where (local) periods of stasis alternate with rapid changes that may induce local extinction.