Crosstalk in concurrent repeated games impedes direct reciprocity and requires stronger levels of forgiveness

Efficiency and resilience of cooperation in asymmetric social dilemmas

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.

Personal sustained cooperation based on networked evolutionary game theory

Evolutionary game theory on complex networks provides an effective theoretical tool to explain the emergence of sustained cooperative behavior. Human society has formed various organizational networks. The network structure and individual behavior take on a variety of forms. This diversity provides the basis for choice, so it is crucial for the emergence of cooperation. This article provides a dynamic algorithm for individual network evolution, and calculates the importance of different nodes in the network evolution process. In the dynamic evolution simulation, the probability of the cooperation strategy and betrayal strategy is described. In the individual interaction network, cooperative behavior will promote the continuous evolution of individual relationships and form a better aggregative interpersonal network. The interpersonal network of betrayal has been in a relatively loose state, and its continuity must rely on the participation of new nodes, but there will be certain "weak links" in the existing nodes of the network.

Adaptive dynamics of memory-one strategies in the repeated donation game

Human interactions can take the form of social dilemmas: collectively, people fare best if all cooperate but each individual is tempted to free ride. Social dilemmas can be resolved when individuals interact repeatedly. Repetition allows them to adopt reciprocal strategies which incentivize cooperation. The most basic model for direct reciprocity is the repeated donation game, a variant of the prisoner's dilemma. Two players interact over many rounds; in each round they decide whether to cooperate or to defect. Strategies take into account the history of the play. Memory-one strategies depend only on the previous round. Even though they are among the most elementary strategies of direct reciprocity, their evolutionary dynamics has been difficult to study analytically. As a result, much previous work has relied on simulations. Here, we derive and analyze their adaptive dynamics. We show that the four-dimensional space of memory-one strategies has an invariant three-dimensional subspace, generated by the memory-one counting strategies. Counting strategies record how many players cooperated in the previous round, without considering who cooperated. We give a partial characterization of adaptive dynamics for memory-one strategies and a full characterization for memory-one counting strategies.

Mutation enhances cooperation in direct reciprocity

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.

Evolution of cooperation through cumulative reciprocity

Reciprocity is a simple principle for cooperation that explains many of the patterns of how humans seek and receive help from each other. To capture reciprocity, traditional models often assume that individuals use simple strategies with restricted memory. These memory-1 strategies are mathematically convenient, but they miss important aspects of human reciprocity, where defections can have lasting effects. Here we instead propose a strategy of cumulative reciprocity. Cumulative reciprocators count the imbalance of cooperation across their previous interactions with their opponent. They cooperate as long as this imbalance is sufficiently small. Using analytical and computational methods, we show that this strategy can sustain cooperation in the presence of errors, that it enforces fair outcomes and that it evolves in hostile environments. Using an economic experiment, we confirm that cumulative reciprocity is more predictive of human behaviour than several classical strategies. The basic principle of cumulative reciprocity is versatile and can be extended to a range of social dilemmas.

Indirect exclusion can promote cooperation in repeated group interactions

Social exclusion has been regarded as one of the most effective measures to promote the evolution of cooperation. In real society, the way in which social exclusion works can be direct or indirect. However, thus far there is no related work to explore how indirect exclusion influences the evolution of cooperation from a theoretical perspective. Here, we introduce indirect exclusion into the repeated public goods game where the game organizer probabilistically selects cooperators after the first game round to participate in the following possible game interactions. We then investigate the evolutionary dynamics of cooperation both in infinite and finite well-mixed populations. Through theoretical analysis and numerical calculations, we find that the introduction of indirect exclusion can induce the stable coexistence of cooperators and defectors or the dominance of cooperators, which thus effectively promotes the evolution of cooperation. Besides, we show that the identifying probability of the organizer has a nonlinear effect on public cooperation when its value is lower than an intermediate value, while the higher identifying probability can maintain a high level of cooperation. Furthermore, our results show that increasing the average rounds of game interactions can effectively promote the evolution of cooperation.

Direct reciprocity between individuals that use different strategy spaces

In repeated interactions, players can use strategies that respond to the outcome of previous rounds. Much of the existing literature on direct reciprocity assumes that all competing individuals use the same strategy space. Here, we study both learning and evolutionary dynamics of players that differ in the strategy space they explore. We focus on the infinitely repeated donation game and compare three natural strategy spaces: memory-1 strategies, which consider the last moves of both players, reactive strategies, which respond to the last move of the co-player, and unconditional strategies. These three strategy spaces differ in the memory capacity that is needed. We compute the long term average payoff that is achieved in a pairwise learning process. We find that smaller strategy spaces can dominate larger ones. For weak selection, unconditional players dominate both reactive and memory-1 players. For intermediate selection, reactive players dominate memory-1 players. Only for strong selection and low cost-to-benefit ratio, memory-1 players dominate the others. We observe that the supergame between strategy spaces can be a social dilemma: maximum payoff is achieved if both players explore a larger strategy space, but smaller strategy spaces dominate.

Cooperation in alternating interactions with memory constraints

In repeated social interactions, individuals often employ reciprocal strategies to maintain cooperation. To explore the emergence of reciprocity, many theoretical models assume synchronized decision making. In each round, individuals decide simultaneously whether to cooperate or not. Yet many manifestations of reciprocity in nature are asynchronous. Individuals provide help at one time and receive help at another. Here, we explore such alternating games in which players take turns. We mathematically characterize all Nash equilibria among memory-one strategies. Moreover, we use evolutionary simulations to explore various model extensions, exploring the effect of discounted games, irregular alternation patterns, and higher memory. In all cases, we observe that mutual cooperation still evolves for a wide range of parameter values. However, compared to simultaneous games, alternating games require different strategies to maintain cooperation in noisy environments. Moreover, none of the respective strategies are evolutionarily stable.

In many instances of reciprocity, individuals cooperate in turns. Here, the authors analyze the equilibria and the dynamics of such alternating games, and in particular describe all strategies with one-round memory that maintain cooperation.

A unified framework of direct and indirect reciprocity

Direct and indirect reciprocity are key mechanisms for the evolution of cooperation. Direct reciprocity means that individuals use their own experience to decide whether to cooperate with another person. Indirect reciprocity means that they also consider the experiences of others. Although these two mechanisms are intertwined, they are typically studied in isolation. Here, we introduce a mathematical framework that allows us to explore both kinds of reciprocity simultaneously. We show that the well-known 'generous tit-for-tat' strategy of direct reciprocity has a natural analogue in indirect reciprocity, which we call 'generous scoring'. Using an equilibrium analysis, we characterize under which conditions either of the two strategies can maintain cooperation. With simulations, we additionally explore which kind of reciprocity evolves when members of a population engage in social learning to adapt to their environment. Our results draw unexpected connections between direct and indirect reciprocity while highlighting important differences regarding their evolvability.

Evolving cooperation in multichannel games

Humans routinely engage in many distinct interactions in parallel. Team members collaborate on several concurrent projects, and even whole nations interact with each other across a variety of issues, including trade, climate change and security. Yet the existing theory of direct reciprocity studies isolated repeated games. Such models cannot account for strategic attempts to use the vested interests in one game as a leverage to enforce cooperation in another. Here we introduce a general framework of multichannel games. Individuals interact with each other over multiple channels; each channel is a repeated game. Strategic choices in one channel can affect decisions in another. With analytical equilibrium calculations for the donation game and evolutionary simulations for several other games we show that such linkage facilitates cooperation. Our results suggest that previous studies tend to underestimate the human potential for reciprocity. When several interactions occur in parallel, people often learn to coordinate their behavior across games to maximize cooperation in each of them.

Helping, fast and slow: Exploring intuitive cooperation in early ontogeny

Cooperative behavior is central to human societies. Human adults who reach their cooperative decisions more rapidly and independently of cognitive control display greater levels of prosocial behavior. This is taken to show that cooperation is guided by intuitive processes rather than by active control of selfish impulses. The current study investigated the emergence of intuitive cooperation in early human ontogeny. We measured helping behavior (latency and frequency) in a longitudinal sample of infants at ages 14 and 18 months. Between 14 and 18 months, the frequency of helping significantly increased and latency to help significantly decreased, suggesting advances in helping behavior during this period of development. Moreover, at 18 months and to some extent, even at 14 months, infants who helped more rapidly (as indexed by a shorter latency) acted more prosocially (as indexed by a greater frequency of helping) than infants who were slower to help. This link between latency and frequency of prosocial behavior was independent of infants' ability for inhibitory control and general sociability levels. Prosocial behavior thus begins to be governed by intuitive processes that operate independently of cognitive control early in human ontogeny. This informs our understanding of the nature and emergence of cooperative behavior by supporting accounts that assign a central role to intuition in the evolution of human cooperation.

Social dilemmas among unequals

Direct reciprocity is a powerful mechanism for the evolution of cooperation on the basis of repeated interactions^1-4. It requires that interacting individuals are sufficiently equal, such that everyone faces similar consequences when they cooperate or defect. Yet inequality is ubiquitous among humans^5,6 and is generally considered to undermine cooperation and welfare^7-10. Most previous models of reciprocity do not include inequality^11-15. These models assume that individuals are the same in all relevant aspects. Here we introduce a general framework to study direct reciprocity among unequal individuals. Our model allows for multiple sources of inequality. Subjects can differ in their endowments, their productivities and in how much they benefit from public goods. We find that extreme inequality prevents cooperation. But if subjects differ in productivity, some endowment inequality can be necessary for cooperation to prevail. Our mathematical predictions are supported by a behavioural experiment in which we vary the endowments and productivities of the subjects. We observe that overall welfare is maximized when the two sources of heterogeneity are aligned, such that more productive individuals receive higher endowments. By contrast, when endowments and productivities are misaligned, cooperation quickly breaks down. Our findings have implications for policy-makers concerned with equity, efficiency and the provisioning of public goods.

Indirect reciprocity with private, noisy, and incomplete information

Indirect reciprocity is a mechanism for cooperation based on shared moral systems and individual reputations. It assumes that members of a community routinely observe and assess each other and that they use this information to decide who is good or bad, and who deserves cooperation. When information is transmitted publicly, such that all community members agree on each other's reputation, previous research has highlighted eight crucial moral systems. These "leading-eight" strategies can maintain cooperation and resist invasion by defectors. However, in real populations individuals often hold their own private views of others. Once two individuals disagree about their opinion of some third party, they may also see its subsequent actions in a different light. Their opinions may further diverge over time. Herein, we explore indirect reciprocity when information transmission is private and noisy. We find that in the presence of perception errors, most leading-eight strategies cease to be stable. Even if a leading-eight strategy evolves, cooperation rates may drop considerably when errors are common. Our research highlights the role of reliable information and synchronized reputations to maintain stable moral systems.

Evolution of cooperation in stochastic games

Social dilemmas occur when incentives for individuals are misaligned with group interests^1-7. According to the 'tragedy of the commons', these misalignments can lead to overexploitation and collapse of public resources. The resulting behaviours can be analysed with the tools of game theory⁸. The theory of direct reciprocity^9-15 suggests that repeated interactions can alleviate such dilemmas, but previous work has assumed that the public resource remains constant over time. Here we introduce the idea that the public resource is instead changeable and depends on the strategic choices of individuals. An intuitive scenario is that cooperation increases the public resource, whereas defection decreases it. Thus, cooperation allows the possibility of playing a more valuable game with higher payoffs, whereas defection leads to a less valuable game. We analyse this idea using the theory of stochastic games^16-19 and evolutionary game theory. We find that the dependence of the public resource on previous interactions can greatly enhance the propensity for cooperation. For these results, the interaction between reciprocity and payoff feedback is crucial: neither repeated interactions in a constant environment nor single interactions in a changing environment yield similar cooperation rates. Our framework shows which feedbacks between exploitation and environment-either naturally occurring or designed-help to overcome social dilemmas.

Spillover modes in multiplex games: double-edged effects on cooperation and their coevolution

In recent years, there has been growing interest in studying games on multiplex networks that account for interactions across linked social contexts. However, little is known about how potential cross-context interference, or spillover, of individual behavioural strategy impact overall cooperation. We consider three plausible spillover modes, quantifying and comparing their effects on the evolution of cooperation. In our model, social interactions take place on two network layers: repeated interactions with close neighbours in a lattice, and one-shot interactions with random individuals. Spillover can occur during the learning process with accidental cross-layer strategy transfer, or during social interactions with errors in implementation. Our analytical results, using extended pair approximation, are in good agreement with extensive simulations. We find double-edged effects of spillover: increasing the intensity of spillover can promote cooperation provided cooperation is favoured in one layer, but too much spillover is detrimental. We also discover a bistability phenomenon: spillover hinders or promotes cooperation depending on initial frequencies of cooperation in each layer. Furthermore, comparing strategy combinations emerging in each spillover mode provides good indication of their co-evolutionary dynamics with cooperation. Our results make testable predictions that inspire future research, and sheds light on human cooperation across social domains.