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.     

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.     

The continuous Prisoner's dilemma: II. Linear reactive strategies with noise.

We present a general model for the Continuous Prisoner's Dilemma and study the effect of errors. We find that cooperative strategies that can resist invasion by defectors are optimistic (make high initial offers), generous (always offer more cooperation than the partner did in the previous round) and uncompromising (offer full cooperation only if the partner does). A necessary condition for the emergence of cooperation in the continuous Prisoner's Dilemma with noise is b (1-p)>c, where b and c denote, respectively, the benefit and cost of cooperation, while p is the error rate. This relation can be reformulated as an error threshold: cooperation can only emerge if the probability of making a mistake is below a critical value. We note, however, that cooperation in the continuous Prisoner's Dilemma with noise does not seem to be evolutionarily stable: while it is possible to find cooperative strategies that resist invasion by defectors, such cooperators are generally invaded by more cooperative strategies which eventually yield to defectors. Thus, the long-term evolution of the continuous Prisoner's Dilemma is either characterized by unending cycles or by stable polymorphisms of cooperators and defectors.     

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.     

Optimality under noise: higher memory strategies for the alternating prisoner's dilemma.

The Alternating Prisoner's Dilemma is a variant of the iterated Prisoner's Dilemma in which the players alternate in the roles of actor and recipient. We searched for strategies which are "optimal" in the Alternating Prisoner's Dilemma with noise (a non-zero probability that a player's decision will be transmitted incorrectly). In order to achieve success against a variety of other strategies, a strategy must be "self-cooperating" (able to achieve mutual cooperation with its clone), "C-exploiting" (able to exploit unconditional cooperators), and "D-unexploitable" (able to resist exploitation by defectors). It must also have high evolutionary "dominance", a general measure of evolutionary performance which considers both resistance to invasion and the ability to invade other strategies. A strategy which meets these optimality criteria can evolve cooperation by invading a population of defectors and establishing a stable cooperative society. Most of the strategies commonly discussed in the Alternating Prisoner's Dilemma literature are low-memory strategies such as Tit For Tat, Pavlov, and Firm But Fair, but none of these strategies can simultaneously meet all of the optimality criteria. However, we discovered a class of higher memory "Firm Pavlov" strategies, which not only meet our stringent optimality criteria, but also achieve remarkable success in round-robin tournaments and evolutionary interactions. These higher memory strategies are friendly enough to cooperate with their clone, pragmatic enough to exploit unconditional cooperators, and wary enough to resist exploitation by defectors: they are truly "optimal under noise" in the Alternating Prisoner's Dilemma.     

Joint evolution of altruistic cooperation and dispersal in a metapopulation of small local populations

We investigate the joint evolution of public goods cooperation and dispersal in a metapopulation model with small local populations. Altruistic cooperation can evolve due to assortment and kin selection, and dispersal can evolve because of demographic stochasticity, catastrophes and kin selection. Metapopulation structures resulting in assortment have been shown to make selection for cooperation possible. But how does dispersal affect cooperation and vice versa, when both are allowed to evolve as continuous traits? We found four qualitatively different evolutionary outcomes. (1) Monomorphic evolution to full defection with positive dispersal. (2) Monomorphic evolution to an evolutionarily stable state with positive cooperation and dispersal. In this case, parameter changes selecting for increased cooperation typically also select for increased dispersal. (3) Evolutionary branching can result in the evolutionarily stable coexistence of defectors and cooperators. Although defectors could be expected to disperse more than cooperators, here we show that the opposite case is also possible: Defectors tend to disperse less than cooperators when the total amount of cooperation in the dimorphic population is low enough. (4) Selection for too low cooperation can cause the extinction of the evolving population. For moderate catastrophe rates dispersal needs to be initially very frequent for evolutionary suicide to occur. Although selection for less dispersal in principle could prevent such evolutionary suicide, in most cases this rescuing effect is not sufficient, because selection in the cooperation trait is typically much stronger. If the catastrophe rate is large enough, a part of the boundary of viability can be evolutionarily attracting with respect to both strategy components, in which case evolutionary suicide is expected from all initial conditions.     

Quorum sensing provides a molecular mechanism for evolution to tune and maintain investment in cooperation

As selection frequently favors noncooperating defectors in mixed populations with cooperators, mechanisms that promote cooperation stability clearly exist. One potential mechanism is bacterial cell-to-cell communication, quorum sensing (QS), which can allow cooperators to prevent invasion by defectors. However, the impact of QS on widespread maintenance of cooperation in well-mixed conditions has not been experimentally demonstrated over extended evolutionary timescales. Here, we use wild-type (WT) Vibrio campbellii that regulates cooperation with QS and an unconditional cooperating (UC) mutant to examine the evolutionary origins and dynamics of novel defectors during a long-term evolution experiment. We found that UC lineages were completely outcompeted by defectors, whereas functioning QS enabled the maintenance of cooperative variants in most WT populations. Sequencing evolved populations revealed multiple luxR mutations that swept the UC lineages. However, the evolution of mutant lineages with reduced levels of bioluminescence (dims) occurred in many WT lineages. These dim variants also decreased other cooperative phenotypes regulated by QS, including protease production, indicating they result from changes to QS regulation. This diminished investment phenotype optimizes a tradeoff between cooperative input and growth output and suggests that decreasing the cost of QS could be a favorable strategy for maintaining the cooperative behaviors it regulates.Subject terms: Bacterial evolution, Microbial ecology, Molecular evolution     

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.     

Emergence and resilience of cooperation in the spatial prisoner's dilemma via a reward mechanism

We study the problem of the emergence of cooperation in the spatial Prisoner's Dilemma. The pioneering work by Nowak and May [1992. Evolutionary games and spatial chaos. Nature 415, 424-426] showed that large initial populations of cooperators can survive and sustain cooperation in a square lattice with imitate-the-best evolutionary dynamics. We revisit this problem in a cost-benefit formulation suitable for a number of biological applications. We show that if a fixed-amount reward is established for cooperators to share, a single cooperator can invade a population of defectors and form structures that are resilient to re-invasion even if the reward mechanism is turned off. We discuss analytically the case of the invasion by a single cooperator and present agent-based simulations for small initial fractions of cooperators. Large cooperation levels, in the sustainability range, are found. In the conclusions we discuss possible applications of this model as well as its connections with other mechanisms proposed to promote the emergence of cooperation.     

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.     

Global analyses of evolutionary dynamics and exhaustive search for social norms that maintain cooperation by reputation

Reputation formation is a key to understanding indirect reciprocity. In particular, the way to assign reputation to each individual, namely a norm that describes who is good and who is bad, greatly affects the possibility of sustained cooperation in the population. Previously, we have exhaustively studied reputation dynamics that are able to maintain a high level of cooperation at the ESS. However, this analysis examined the stability of monomorphic population and did not investigate polymorphic population where several strategies coexist. Here, we study the evolutionary dynamics of multiple behavioral strategies by replicator dynamics. We exhaustively study all 16 possible norms under which the reputation of a player in the next round is determined by the action of the self and the reputation of the opponent. For each norm, we explore evolutionary dynamics of three strategies: unconditional cooperators, unconditional defectors, and conditional cooperators. We find that only three norms, simple-standing, Kandori, and shunning, can make conditional cooperation evolutionarily stable, hence, realize sustained cooperation. The other 13 norms, including scoring, ultimately lead to the invasion by defectors. Also, we study the model in which private reputation errors exist to a small extent. In this case, we find the stable coexistence of unconditional and conditional cooperators under the three norms.     

Collective Chasing Behavior between Cooperators and Defectors in the Spatial Prisoner’s Dilemma

Cooperation is one of the essential factors for all biological organisms in major evolutionary transitions. Recent studies have investigated the effect of migration for the evolution of cooperation. However, little is known about whether and how an individuals’ cooperativeness coevolves with mobility. One possibility is that mobility enhances cooperation by enabling cooperators to escape from defectors and form clusters; the other possibility is that mobility inhibits cooperation by helping the defectors to catch and exploit the groups of cooperators. In this study we investigate the coevolutionary dynamics by using the prisoner’s dilemma game model on a lattice structure. The computer simulations demonstrate that natural selection maintains cooperation in the form of evolutionary chasing between the cooperators and defectors. First, cooperative groups grow and collectively move in the same direction. Then, mutant defectors emerge and invade the cooperative groups, after which the defectors exploit the cooperators. Then other cooperative groups emerge due to mutation and the cycle is repeated. Here, it is worth noting that, as a result of natural selection, the mobility evolves towards directional migration, but not to random or completely fixed migration. Furthermore, with directional migration, the rate of global population extinction is lower when compared with other cases without the evolution of mobility (i.e., when mobility is preset to random or fixed). These findings illustrate the coevolutionary dynamics of cooperation and mobility through the directional chasing between cooperators and defectors.     

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.     

The joker effect: cooperation driven by destructive agents

Understanding the emergence of cooperation is a central issue in evolutionary game theory. The hardest setup for the attainment of cooperation in a population of individuals is the Public Goods game in which cooperative agents generate a common good at their own expenses, while defectors “free-ride” this good. Eventually this causes the exhaustion of the good, a situation which is bad for everybody. Previous results have shown that introducing reputation, allowing for volunteer participation, punishing defectors, rewarding cooperators or structuring agents, can enhance cooperation. Here we present a model which shows how the introduction of rare, malicious agents - that we term jokers - performing just destructive actions on the other agents induce bursts of cooperation. The appearance of jokers promotes a rock-paper-scissors dynamics, where jokers outbeat defectors and cooperators outperform jokers, which are subsequently invaded by defectors. Thus, paradoxically, the existence of destructive agents acting indiscriminately promotes cooperation.     

Invasion and expansion of cooperators in lattice populations: Prisoner's dilemma vs. snowdrift games

The evolution of cooperation is an enduring conundrum in biology and the social sciences. Two social dilemmas, the prisoner's dilemma and the snowdrift game have emerged as the most promising mathematical metaphors to study cooperation. Spatial structure with limited local interactions has long been identified as a potent promoter of cooperation in the prisoner's dilemma but in the spatial snowdrift game, space may actually enhance or inhibit cooperation. Here we investigate and link the microscopic interaction between individuals to the characteristics of the emerging macroscopic patterns generated by the spatial invasion process of cooperators in a world of defectors. In our simulations, individuals are located on a square lattice with Moore neighborhood and update their strategies by probabilistically imitating the strategies of better performing neighbors. Under sufficiently benign conditions, cooperators can survive in both games. After rapid local equilibration, cooperators expand quadratically until global saturation is reached. Under favorable conditions, cooperators expand as a large contiguous cluster in both games with minor differences concerning the shape of embedded defectors. Under less favorable conditions, however, distinct differences arise. In the prisoner's dilemma, cooperators break up into isolated, compact clusters. The compact clustering reduces exploitation and leads to positive assortment, such that cooperators interact more frequently with other cooperators than with defectors. In contrast, in the snowdrift game, cooperators form small, dendritic clusters, which results in negative assortment and cooperators interact more frequently with defectors than with other cooperators. In order to characterize and quantify the emerging spatial patterns, we introduce a measure for the cluster shape and demonstrate that the macroscopic patterns can be used to determine the characteristics of the underlying microscopic interactions.     

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.     

Defectors Can Create Conditions That Rescue Cooperation

Cooperation based on the production of costly common goods is observed throughout nature. This is puzzling, as cooperation is vulnerable to exploitation by defectors which enjoy a fitness advantage by consuming the common good without contributing fairly. Depletion of the common good can lead to population collapse and the destruction of cooperation. However, population collapse implies small population size, which, in a structured population, is known to favor cooperation. This happens because small population size increases variability in cooperator frequency across different locations. Since individuals in cooperator-dominated locations (which are most likely cooperators) will grow more than those in defector-dominated locations (which are most likely defectors), cooperators can outgrow defectors globally despite defectors outgrowing cooperators in each location. This raises the possibility that defectors can lead to conditions that sometimes rescue cooperation from defector-induced destruction. We demonstrate multiple mechanisms through which this can occur, using an individual-based approach to model stochastic birth, death, migration, and mutation events. First, during defector-induced population collapse, defectors occasionally go extinct before cooperators by chance, which allows cooperators to grow. Second, empty locations, either preexisting or created by defector-induced population extinction, can favor cooperation because they allow cooperator but not defector migrants to grow. These factors lead to the counterintuitive result that the initial presence of defectors sometimes allows better survival of cooperation compared to when defectors are initially absent. Finally, we find that resource limitation, inducible by defectors, can select for mutations adaptive to resource limitation. When these mutations are initially present at low levels or continuously generated at a moderate rate, they can favor cooperation by further reducing local population size. We predict that in a structured population, small population sizes precipitated by defectors provide a “built-in” mechanism for the persistence of cooperation.     

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.     

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.     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.