Direct reciprocity is a mechanism for the evolution of cooperation in repeated social interactions. According to the literature, individuals naturally learn to adopt conditionally cooperative strategies if they have multiple encounters with their partner. Corresponding models have greatly facilitated our understanding of cooperation, yet they often make strong assumptions on how individuals remember and process payoff information. For example, when strategies are updated through social learning, it is commonly assumed that individuals compare their average payoffs. This would require them to compute (or remember) their payoffs against everyone else in the population. To understand how more realistic constraints influence direct reciprocity, we consider the evolution of conditional behaviours when individuals learn based on more recent experiences. Even in the most extreme case that they only take into account their very last interaction, we find that cooperation can still evolve. However, such individuals adopt less generous strategies, and they cooperate less often than in the classical setup with average payoffs. Interestingly, once individuals remember the payoffs of two or three recent interactions, cooperation rates quickly approach the classical limit. These findings contribute to a literature that explores which kind of cognitive capabilities are required for reciprocal cooperation. While our results suggest that some rudimentary form of payoff memory is necessary, it suffices to remember a few interactions.

Direct reciprocity is a powerful mechanism for cooperation in social dilemmas. The very logic of reciprocity, however, seems to require that individuals are symmetric, and that everyone has the same means to influence each others\' payoffs. Yet in many applications, individuals are asymmetric. Herein, we study the effect of asymmetry in linear public good games. Individuals may differ in their endowments (their ability to contribute to a public good) and in their productivities (how effective their contributions are). Given the individuals\' productivities, we ask which allocation of endowments is optimal for cooperation. To this end, we consider two notions of optimality. The first notion focuses on the resilience of cooperation. The respective endowment distribution ensures that full cooperation is feasible even under the most adverse conditions. The second notion focuses on efficiency. The corresponding endowment distribution maximizes group welfare. Using analytical methods, we fully characterize these two endowment distributions. This analysis reveals that both optimality notions favor some endowment inequality: More productive players ought to get higher endowments. Yet the two notions disagree on how unequal endowments are supposed to be. A focus on resilience results in less inequality. With additional simulations, we show that the optimal endowment allocation needs to account for both the resilience and the efficiency of cooperation.

Many evolutionary processes occur in phenotype spaces which are continuous. It is therefore of interest to explore how selection operates in continuous spaces. One approach is adaptive dynamics, which assumes that mutants are local. Here we study a different process which also allows non-local mutants. We assume that a resident population is challenged by an invader who uses a strategy chosen from a random distribution on the space of all strategies. We study the repeated donation game of direct reciprocity. We consider reactive strategies given by two probabilities, denoting respectively the probability to cooperate after the co-player has cooperated or defected. The strategy space is the unit square. We derive analytic formulae for the stationary distribution of evolutionary dynamics and for the average cooperation rate as function of the cost-to-benefit ratio. For positive reactive strategies, we prove that cooperation is more abundant than defection if the area of the cooperative region is greater than 1/2 which is equivalent to benefit, b, divided by cost, c, exceeding [Formula: see text]. We introduce the concept of strategies that are stable with probability one. We also study an extended process and discuss other games.

Wild-type Norway rats reciprocate help received in a well-replicated experimental food-giving task, but the criteria to appraise the received help\'s value are unclear. We tested whether quality or quantity of received help is more important when deciding to return help, and whether partner familiarity and own current need affect this evaluation. We experimentally varied recipients of help\'s hunger state, and familiar or unfamiliar partners provided either higher caloric food (enhanced quantity; carrots) or food higher in protein and fat (enhanced quality; cheese). Reciprocation of received help was our criterion for the rats\' value assessment. Familiarity, food type and hunger state interacted and affected help returned by rats. Rats returned less help to familiar partners than to unfamiliar partners. With unfamiliar partners, rats returned more help to partners that had donated preferred food (cheese) than to partners that had donated less preferred food (carrots), and they returned help earlier if they were satiated and had received cheese. With familiar partners, food-deprived rats that had received cheese returned more help than satiated rats that had received carrots. Our results suggest that Norway rats assess the received help\'s value based on its quality, their current need and partner familiarity before reciprocating received help.

Direct reciprocity is a powerful mechanism for the evolution of cooperation based on repeated interactions between the same individuals. But high levels of cooperation evolve only if the benefit-to-cost ratio exceeds a certain threshold that depends on memory length. For the best-explored case of one-round memory, that threshold is two. Here, we report that intermediate mutation rates lead to high levels of cooperation, even if the benefit-to-cost ratio is only marginally above one, and even if individuals only use a minimum of past information. This surprising observation is caused by two effects. First, mutation generates diversity which undermines the evolutionary stability of defectors. Second, mutation leads to diverse communities of cooperators that are more resilient than homogeneous ones. This finding is relevant because many real-world opportunities for cooperation have small benefit-to-cost ratios, which are between one and two, and we describe how direct reciprocity can attain cooperation in such settings. Our result can be interpreted as showing that diversity, rather than uniformity, promotes evolution of cooperation.

Direct reciprocity requires the ability to recognize and memorize social partners, and to remember their previous actions. \'Insufficient cognitive abilities\' have been assumed to potentially impair the ability to cooperate by direct reciprocity. Here we compare the propensity of rats to use direct reciprocity with their ability to memorize and recognize sensory cues in a non-social task. Female rats enriched in one of three sensory modalities (visual, olfactory or auditory) performed better in a learning task when they were tested with the specific sensory modality in which they have been enriched. For the cooperation test, during three subsequent reciprocity experiments the rats could provide two partners differing in their previous helpfulness with food. Individuals performing better in the non-social learning task that involved olfactory cues applied direct reciprocity more successfully in one experiment. However, in the experiment preventing visual cues and physical contact, rats applied direct reciprocity rules irrespective of their performance in the learning task with olfactory cues. This indicates that an enhanced olfactory recognition ability, despite being beneficial, is not a prerequisite for the rats\' ability to cooperate by direct reciprocity. This might suggest that when rats have all types of information about their social partner, individuals may apply other criteria than the reciprocity decision rule when determining how much help to provide, as for instance coercion. Interestingly, when all individuals are constrained to mostly rely on olfactory memory, individuals apply direct reciprocity independently of their ability to memorize olfactory cues in a non-social context. \'Insufficient cognitive abilities\' may thus not be the true reason when direct reciprocity is not observed.

Several studies have confirmed the existence of zero-determinant (ZD) strategies in repeated social dilemmas since Press and Dyson\'s ingenious discovery of ZD strategies in iterated prisoner\'s dilemmas. However, less research studies evolutionary performance of multiplayer ZD strategies, especially from a theoretical perspective. Here, we use a state-clustering method to theoretically analyze evolutionary dynamics of two representative ZD strategies: generous ZD strategies and extortionate ZD strategies. We consider two new settings for multiplayer ZD strategies: competitions with all ZD strategies and competitions with all memory-one strategies, apart from the competitions between these strategies and some classical ones. Moreover, we investigate the influence of the level of generosity and extortion on evolutionary dynamics of generous and extortionate ZD strategies, which was commonly ignored in previous studies. Theoretical results show that players with limited generosity are at an advantageous place and extortioners extorting more severely hold their ground more readily. Our results may provide new insights into better understanding evolutionary dynamics of ZD strategies in repeated multiplayer games.

Direct and indirect reciprocity are two fundamental mechanisms that promote prosocial behavior within groups and across societies. Here, we review recent work that illustrates how a (direct and indirect) reciprocity framework can illuminate our understanding of several factors related to prosocial behavior - namely group membership, gossip, and third-party punishment. We propose that each of these factors can promote prosocial behavior via proximate psychological mechanisms related to direct and indirect reciprocity: reputational concern, expectations, and anticipation of future interaction. Finally, we discuss the implications of adopting such a framework and highlight a number of avenues for future research.

Recent research suggests that children\'s sharing behavior is affected by experiencing or observing others\' sharing. These effects have been studied within research on the development of reciprocity and the effects of social modeling. In the current study, direct and indirect types of reciprocity and social modeling were assessed in 3- to 6-year-old children in three experiments (overall N = 382). In each experiment, we explored whether negative and positive social behavior were similarly paid back and forward in each of the different types of reciprocity. Moreover, we assessed the extent to which children reciprocated toward the protagonist who had performed the actual behavior and toward a neutral other. In Experiment 1, children experienced another\'s sharing behavior as recipients and could then allocate resources to this character and a neutral other. In Experiment 2, children observed another\'s sharing behavior and could then allocate resources to this character and a neutral other. In Experiment 3, children were asked to predict another protagonist\'s sharing in the same context as in Experiment 1. Overall, children treated the protagonist and the neutral other similarly and predicted others to do the same. Yet, they were more likely to reciprocate negative acts in indirect types of reciprocity. The results are interpreted with respect to the impact of observational learning and representational development on children\'s social behavior.

Egg trading-whereby simultaneous hermaphrodites exchange each other\'s eggs for fertilization-constitutes one of the few rigorously documented and most widely cited examples of direct reciprocity among unrelated individuals. Yet how egg trading may initially invade a population of nontrading simultaneous hermaphrodites is still unresolved. Here, we address this question with an analytical model that considers mate encounter rates and costs of egg production in a population that may include traders (who provide eggs for fertilization only if their partners also have eggs to reciprocate), providers (who provide eggs regardless of whether their partners have eggs to reciprocate), and withholders (cheaters who mate only in the male role and just use their eggs to elicit egg release from traders). Our results indicate that a combination of intermediate mate encounter rates, sufficiently high costs of egg production, and a sufficiently high probability that traders detect withholders (in which case eggs are not provided) is conducive to the evolution of egg trading. Under these conditions, traders can invade-and resist invasion from-providers and withholders alike. The prediction that egg trading evolves only under these specific conditions is consistent with the rare occurrence of this mating system among simultaneous hermaphrodites.