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.     

Evolution of direct and indirect reciprocity

Indirect reciprocity (IR) occurs when individuals help those who help others. It is important as a potential explanation for why people might develop cooperative reputations. However, previous models of IR are based on the assumption that individuals never meet again. Yet humans and other animals often interact repeatedly within groups, thereby violating the fundamental basis of these models. Whenever re-meeting can occur, discriminating reciprocators can decide whether to help those who helped others (IR) or those who helped them (direct reciprocity, DR). Here I used simulation models to investigate the conditions in which we can expect the different forms of reciprocity to predominate. I show that IR through image scoring becomes unstable with respect to DR by experience scoring as the probability of re-meeting increases. However, using the standing strategy, which takes into account the context of observed defections, IR can be stable with respect to DR even when individuals interact with few partners many times. The findings are important in showing that IR cannot explain a concern for reputation in typical societies unless reputations provide as reliable a guide to cooperative behaviour as does experience.     

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.     

Participation costs can suppress the evolution of upstream reciprocity

Indirect reciprocity, one of the many mechanisms proposed to explain the evolution of cooperation, is the idea that altruistic actions can be rewarded by third parties. Upstream or generalized reciprocity is one type of indirect reciprocity in which individuals help someone if they have been helped by somebody else in the past. Although empirically found to be at work in humans, the evolution of upstream reciprocity is difficult to explain from a theoretical point of view. A recent model of upstream reciprocity, first proposed by Nowak and Roch (2007) and further analyzed by Iwagami and Masuda (2010), shows that while upstream reciprocity alone does not lead to the evolution of cooperation, it can act in tandem with mechanisms such as network reciprocity and increase the total level of cooperativity in the population. We argue, however, that Nowak and Roch's model systematically leads to non-uniform interaction rates, where more cooperative individuals take part in more games than less cooperative ones. As a result, the critical benefit-to-cost ratios derived under this model in previous studies are not invariant with respect to the addition of participation costs. We show that accounting for these costs can hinder and even suppress the evolution of upstream reciprocity, both for populations with non-random encounters and graph-structured populations.     

Reactive strategies in indirect reciprocity

Evolution of reactive strategy of indirect reciprocity is discussed, where individuals interact with others through the one-shot Prisoner's Dilemma game, changing their partners in every round. We investigate all of the reactive strategies that are stochastic, including deterministic ones as special cases. First we study adaptive dynamics of reactive strategies by assuming monomorphic population. Results are very similar to the corresponding evolutionary dynamics of direct reciprocity. The discriminating strategy, which prescribes cooperation only with those who cooperated in the previous round, cannot be an outcome of the evolution. Next we examine the case where the population includes a diversity of strategies. We find that only the mean 'discriminatoriness' in the population is the parameter that affects the evolutionary dynamics. The discriminating strategy works as a promoter of cooperation there. However, it is again not the end point of the evolution. This is because retaliatory defection, which was prescribed by the discriminating strategy, is regarded as another defection toward the society. These results caution that we have to reconsider the role of retaliatory defection much more carefully.     

Evolution of indirect reciprocity in groups of various sizes and comparison with direct reciprocity

Recently many studies have investigated the evolution of indirect reciprocity through which cooperative action is returned by a third individual, e.g. individual A helped B and then receives help from C. Most studies on indirect reciprocity have presumed that only two individuals take part in a single interaction (group), e.g. A helps B and C helps A. In this paper, we investigate the evolution of indirect reciprocity when more than two individuals take part in a single group, and compare the result with direct reciprocity through which cooperative action is directly returned by the recipient. Our analyses show the following. In the population with discriminating cooperators and unconditional defectors, whether implementation error is included or not, (i) both strategies are evolutionarily stable and the evolution of indirect reciprocity becomes more difficult as group size increases, and (ii) the condition for the evolution of indirect reciprocity under standing reputation criterion where the third individuals distinguish between justified and unjustified defections is more relaxed than that under image scoring reputation criterion in which the third individuals do not distinguish with. Furthermore, in the population that also includes unconditional cooperators, (iii) in the presence of errors in implementation, the discriminating strategy is evolutionarily stable not only under standing but also under image scoring if group size is larger than two. Finally, (iv) in the absence of errors in implementation, the condition for the evolution of direct reciprocity is equivalent to that for the evolution of indirect reciprocity under standing, and, in the presence of errors, the condition for the evolution of direct reciprocity is very close to that for the evolution of indirect reciprocity under image scoring.     

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.     

Altruism may arise from individual selection

The fact that humans cooperate with non-kin in large groups, or with people they will never meet again, is a long-standing evolutionary puzzle. Altruism, the capacity to perform costly acts that confer benefits on others, is at the core of cooperative behavior. Behavioral experiments show that humans have a predisposition to cooperate with others and to punish non-cooperators at personal cost (so-called strong reciprocity) which, according to standard evolutionary game theory arguments, cannot arise from selection acting on individuals. This has led to the suggestion of group and cultural selection as the only mechanisms that can explain the evolutionary origin of human altruism. We introduce an agent-based model inspired on the Ultimatum Game, that allows us to go beyond the limitations of standard evolutionary game theory and show that individual selection can indeed give rise to strong reciprocity. Our results are consistent with the existence of neural correlates of fairness and in good agreement with observations on humans and monkeys.     

ASSORTMENT AND THE EVOLUTION OF GENERALIZED RECIPROCITY

Reciprocity is often invoked to explain cooperation. Reciprocity is cognitively demanding, and both direct and indirect reciprocity require that individuals store information about the propensity of their partners to cooperate. By contrast, generalized reciprocity, wherein individuals help on the condition that they received help previously, only relies on whether an individual received help in a previous encounter. Such anonymous information makes generalized reciprocity hard to evolve in a well‐mixed population, as the strategy will lose out to pure defectors. Here we analyze a model for the evolution of generalized reciprocity, incorporating assortment of encounters, to investigate the conditions under which it will evolve. We show that, in a well‐mixed population, generalized reciprocity cannot evolve. However, incorporating assortment of encounters can favor the evolution of generalized reciprocity in which indiscriminate cooperation and defection are both unstable. We show that generalized reciprocity can evolve under both the prisoner's dilemma and the snowdrift game.     

Revisiting “The Evolution of Reciprocity in Sizable Groups”: Continuous reciprocity in the repeated n-person prisoner's dilemma

For many years in evolutionary science, the consensus view has been that while reciprocal altruism can evolve in dyadic interactions, it is unlikely to evolve in sizable groups. This view had been based on studies which have assumed cooperation to be discrete rather than continuous (i.e., individuals can either fully cooperate or else fully defect, but they cannot continuously vary their level of cooperation). In real world cooperation, however, cooperation is often continuous. In this paper, we re-examine the evolution of reciprocity in sizable groups by presenting a model of the n-person prisoner's dilemma that assumes continuous rather than discrete cooperation. This model shows that continuous reciprocity has a dramatically wider basin of attraction than discrete reciprocity, and that this basin's size increases with efficiency of cooperation (marginal per capita return). Further, we find that assortative interaction interacts synergistically with continuous reciprocity to a much greater extent than it does with discrete reciprocity. These results suggest that previous models may have underestimated reciprocity's adaptiveness in groups. However, we also find that the invasion of continuous reciprocators into a population of unconditional defectors becomes realistic only within a narrow parameter space in which the efficiency of cooperation is close to its maximum bound. Therefore our model suggests that continuous reciprocity can evolve in large groups more easily than discrete reciprocity only under unusual circumstances.     

Marmoset monkeys evaluate third-party reciprocity

Many non-human primates have been observed to reciprocate and to understand reciprocity in one-to-one social exchanges. A recent study demonstrated that capuchin monkeys are sensitive to both third-party reciprocity and violation of reciprocity; however, whether this sensitivity is a function of general intelligence, evidenced by their larger brain size relative to other primates, remains unclear. We hypothesized that highly pro-social primates, even with a relatively smaller brain, would be sensitive to others'' reciprocity. Here, we show that common marmosets discriminated between human actors who reciprocated in social exchanges with others and those who did not. Monkeys accepted rewards less frequently from non-reciprocators than they did from reciprocators when the non-reciprocators had retained all food items, but they accepted rewards from both actors equally when they had observed reciprocal exchange between the actors. These results suggest that mechanisms to detect unfair reciprocity in third-party social exchanges do not require domain-general higher cognitive ability based on proportionally larger brains, but rather emerge from the cooperative and pro-social tendencies of species, and thereby suggest this ability evolved in multiple primate lineages.     

Indirect reciprocity can overcome free-rider problems on costly moral assessment

Indirect reciprocity is one of the major mechanisms of the evolution of cooperation. Because constant monitoring and accurate evaluation in moral assessments tend to be costly, indirect reciprocity can be exploited by cost evaders. A recent study crucially showed that a cooperative state achieved by indirect reciprocators is easily destabilized by cost evaders in the case with no supportive mechanism. Here, we present a simple and widely applicable solution that considers pre-assessment of cost evaders. In the pre-assessment, those who fail to pay for costly assessment systems are assigned a nasty image that leads to them being rejected by discriminators. We demonstrate that considering the pre-assessment can crucially stabilize reciprocal cooperation for a broad range of indirect reciprocity models. In particular for the most leading social norms, we analyse the conditions under which a prosocial state becomes locally stable.     

The concepts of asymmetric and symmetric power can help resolve the puzzle of altruistic and cooperative behaviour

Evolutionary theory predicts competition in nature yet altruistic and cooperative behaviour appears to reduce the ability to compete in order to help others compete better. This evolutionary puzzle is usually explained by kin selection where close relatives perform altruistic and cooperative acts to help each other and by reciprocity theory (i.e. direct, indirect and generalized reciprocity) among non‐kin. Here, it is proposed that the concepts of asymmetry and symmetry in power and dominance are critical if we are ever to resolve the puzzle of altruism and cooperation towards non‐kin. Asymmetry in power and dominance is likely to emerge under competition in nature as individuals strive to gain greater access to the scarce resources needed to survive and reproduce successfully. Yet asymmetric power presents serious problems for reciprocity theory in that a dominant individual faces a temptation to cheat in interactions with subordinates that is likely to far outweigh any individual selective benefits gained through reciprocal mechanisms. Furthermore, action taken by subordinates to deter non‐reciprocation by dominants is likely to prove prohibitively costly to their fitness, making successful enforcement of reciprocal mechanisms unlikely. It is also argued here that many apparently puzzling forms of cooperation observed in nature (e.g. cooperative breeding in which unrelated subordinates help dominants to breed) might be best explained by asymmetry in power and dominance. Once it is recognized that individuals in these cooperative interactions are subject to the constraints and opportunities imposed on them by asymmetric power then they can be seen as pursuing a ‘least bad’ strategy to promote individual fitness – one that is nevertheless consistent with evolutionary theory. The concept of symmetric power also provides important insights. It can inhibit reciprocal mechanisms in the sense that symmetric power makes it easier for a cheat to appropriate common resources while incurring fewer penalties. Nevertheless under certain restrictive conditions, symmetric power is seen as likely to promote direct reciprocity through ‘tit for tat’.     

Three-person game facilitates indirect reciprocity under image scoring

Reputation building plays an important role in the evolution of reciprocal altruism when the same individuals do not interact repeatedly because, by referring to reputation, a reciprocator can know which partners are cooperative and can reciprocate with a cooperator. This reciprocity based on reputation is called indirect reciprocity. Previous studies of indirect reciprocity have focused only on two-person games in which only two individuals participate in a single interaction, and have claimed that indirectly reciprocal cooperation cannot be established under image scoring reputation criterion where the reputation of an individual who has cooperated (defected) becomes good (bad). In this study, we specifically examine three-person games, and reveal that indirectly reciprocal cooperation can be formed and maintained stably, even under image scoring, by a nucleus shield mechanism. In the nucleus shield, reciprocators are a shield that keeps out unconditional defectors, whereas unconditional cooperators are the backbone of cooperation that retains a good reputation among the population.     

Transforming the dilemma

How does natural selection lead to cooperation between competing individuals? The Prisoner's Dilemma captures the essence of this problem. Two players can either cooperate or defect. The payoff for mutual cooperation, R, is greater than the payoff for mutual defection, P. But a defector versus a cooperator receives the highest payoff, T, where as the cooperator obtains the lowest payoff, S. Hence, the Prisoner's Dilemma is defined by the payoff ranking T > R > P > S . In a well‐mixed population, defectors always have a higher expected payoff than cooperators, and therefore natural selection favors defectors. The evolution of cooperation requires specific mechanisms. Here we discuss five mechanisms for the evolution of cooperation: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity (or graph selection). Each mechanism leads to a transformation of the Prisoner's Dilemma payoff matrix. From the transformed matrices, we derive the fundamental conditions for the evolution of cooperation. The transformed matrices can be used in standard frameworks of evolutionary dynamics such as the replicator equation or stochastic processes of game dynamics in finite populations.     

Personal experience and reputation interact in human decisions to help reciprocally

There is ample evidence that human cooperative behaviour towards other individuals is often conditioned on information about previous interactions. This information derives both from personal experience (direct reciprocity) and from experience of others (i.e. reputation; indirect reciprocity). Direct and indirect reciprocity have been studied separately, but humans often have access to both types of information. Here, we experimentally investigate information use in a repeated helping game. When acting as donor, subjects can condition their decisions to help recipients with both types of information at a small cost to access such information. We find that information from direct interactions weighs more heavily in decisions to help, and participants tend to react less forgivingly to negative personal experience than to negative reputation. Moreover, effects of personal experience and reputation interact in decisions to help. If a recipient''s reputation is positive, the personal experience of the donor has a weak effect on the decision to help, and vice versa. Yet if the two types of information indicate conflicting signatures of helpfulness, most decisions to help follow personal experience. To understand the roles of direct and indirect reciprocity in human cooperation, they should be studied in concert, not in isolation.     

The good, the bad and the discriminator--errors in direct and indirect reciprocity

This paper presents, in a series of simple diagrams, concise results about the replicator dynamics of direct and indirect reciprocity. We consider repeated interactions between donors and recipients, and analyse the relationship between three basic strategies for the donor: unconditional cooperation, all-out defection, and conditional cooperation. In other words, we investigate the competition of discriminating and indiscriminating altruists with defectors. Discriminators and defectors form a bistable community, and hence a population of discriminators cannot be invaded by defectors. But unconditional altruists can invade a discriminating population and 'soften it up' for a subsequent invasion by defectors. The resulting dynamics exhibits various forms of rock-paper-scissors cycles and depends in subtle ways on noise, in the form of errors in implementation. The probability for another round (in the case of direct reciprocity), and information about the co-player (in the case of indirect reciprocity), add further elements to the ecology of reciprocation.     

Strong reciprocity or strong ferocity? A population genetic view of the evolution of altruistic punishment

Strong reciprocity, defined as a predisposition to help others and to punish those that are not helping, has been proposed as a potent force leading to the evolution of cooperation and altruism. However, the conditions under which strong reciprocity might be favored are not clear. Here we investigate the selective pressure on strong reciprocity by letting both limited dispersal (i.e., spatial structure) and recombination between helping and punishment jointly determine the evolutionary dynamics of strong reciprocity. Our analytical model suggests that when helping and punishment are perfectly linked traits (no recombination occurring between them), strong reciprocity can spread even when the initial frequency of strong reciprocators is close to 0 in the population (i.e., a rare mutant can invade). By contrast, our results indicate that when recombination can occur between helping and punishment (i.e., both traits coevolve) and is stronger than selection, punishment is likely to invade a population of defectors only when it gives a direct fitness benefit to the actor. Overall, our results delineate the conditions under which strong reciprocity is selected for in a spatially structured population and highlight that the forces behind its evolution involves kinship (be it genetic or cultural).     

Indirect reciprocity among imperfect individuals

Reciprocal cooperation occurs when the overall benefits of receiving help exceed the costs of donating help (Q. Rev. Biol. 46 (197) 35). That is, individuals in good condition--for whom the pertinent costs are relatively small; donate help in order to secure reciprocity in their hour of need--when the benefits of receiving a donation are large. Consequently, reciprocity occurs among individuals who occasionally need help. In particular, such individuals will be unable to help others, no matter how deserving, when in need of help themselves--involuntary defection. This paper deals with the effects of involuntary defection in the context of a specific model of indirect reciprocity (i.e. reciprocal altruism that is directed toward all the cooperative members of the community) due to Nowak and Sigmund (J. Theor. Biol.194 (1998b) 561: Sections 2-4). In that model, the authors formulate the decision rules for conditional cooperation in the context of indirect reciprocity, and demonstrate that these decision rules can account for a long-term persistence of cooperation. Here we show that addition of involuntary defection to the decision rules formulated by Nowak and Sigmund results in indirect reciprocity that is evolutionary stable under appropriate conditions. Moreover, for a wide range of parameter values, evolutionary stability of cooperation requires a mixture of conditional- and unconditional-altruist behaviors. To recollect, unconditional altruist strategy can be viewed as conditional altruist strategy sans the ability to decide when the help-soliciting individual should be refused help. That is, given involuntary defection, stability of cooperation requires an occasional forgiveness, if only by default, of a failure to donate help. Thus, we see that evolutionary stable indirect reciprocity does not require perfection in either the ability to assess the merits of the help-soliciting individuals, or the ability to donate help when it is merited. On the contrary, we are forced to conclude that reciprocity, at least in the current case, is stable only among imperfect individuals.     

Good predictors for the fixation probability on complex networks of multi-player games using territorial interactions

In 2012 Broom and Rychtar developed a new framework to consider the evolution of a population over a non-homogeneous underlying structure, where fitness depends upon multiplayer interactions amongst the individuals within the population played in groups of various sizes (including one). This included the independent model, and as a special case the territorial raider model, which has been considered in a series of subsequent papers. Here individuals are based upon the vertex of a graph but move to interact with their neighbours, sometimes meeting in large groups. The most important single property of such populations is the fixation probability, the probability of a single mutant completely replacing the existing population. In a recent paper we considered the fixation probability for the Birth Death Birth (BDB) dynamics for three games, a Public Goods game, the Hawk–Dove game and for fixed fitnesses for a large number of randomly generated graphs, in particular seeing if important underlying graph properties could be used as predictors. We found two good predictors, temperature and mean group size, but some interesting and unusual features for one type of graph, Barabasi–Albert graphs. In this paper we use a regression analysis to investigate (the usual) three alternative evolutionary dynamics (BDD, DBB, DBD) in addition to the original BDB. In particular, we find that the dynamics split into two pairs, BDB/DBD and BDD/DBB, each of which give essentially the same results and found a good fit to the data using a quadratic regression involving the above two variables. Further we find that temperature is the most important predictor for the Hawk–Dove game, whilst for the Public Goods game the group size also plays a key role, and is more important than the temperature for the BDD/DBB dynamics.