Indirect reciprocity is one of the major mechanisms for the evolution of cooperation in human societies. There are two types of indirect reciprocity: upstream and downstream. Cooperation in downstream reciprocity follows the pattern, 'You helped someone, and I will help you'. The direction of cooperation is reversed in upstream reciprocity, which instead follows the pattern, 'You helped me, and I will help someone else'. In reality, these two types of indirect reciprocity often occur in combination. However, upstream and downstream reciprocity have mostly been studied theoretically in isolation. Here, we propose a new model that integrates both types. We apply the standard giving-game framework of indirect reciprocity and analyze the model by means of evolutionary game theory. We show that the model can result in the stable coexistence of altruistic reciprocators and free riders in well-mixed populations. We also found that considering inattention in the assessment rule can strengthen the stability of this mixed equilibrium, even resulting in a global attractor. Our results indicate that the cycles of forwarding help and rewarding help need to be established for creating and maintaining diversity and inclusion in a society.
The success of modern civilization is built upon widespread cooperation in human society, deciphering the mechanisms behind has being a major goal for centuries. A crucial fact is, however, largely missing in most prior studies that games in the real world are typically played simultaneously and interactively rather than separately as assumed. Here we introduce the idea of interacting games that different games coevolve and influence each other's decision-making. We show that as the game-game interaction becomes important, the cooperation phase transition dramatically improves, a fairly high l…
Indirect reciprocity in which players cooperate with unacquainted other players having good reputations is a mechanism for cooperation in relatively large populations subjected to social dilemma situations. When the population has group structure, as is often found in social networks, players in experiments are considered to show behavior that deviates from existing theoretical models of indirect reciprocity. First, players often show ingroup favoritism (i.e., cooperation only within the group) rather than full cooperation (i.e., cooperation within and across groups), even though the latter is…
Recent studies suggest that the emergence of cooperative behavior can be explained by generalized reciprocity, a behavioral mechanism based on the principle of "help anyone if helped by someone". In complex systems, the cooperative dynamics is largely determined by the network structure which dictates the interactions among neighboring individuals. These interactions often exhibit multidimensional features, either as relationships of different types or temporal dynamics, both of which may be modeled as a "multiplex" network. Against this background, here we advance the research on cooperation …
Although various norms for reciprocity-based cooperation have been suggested that are evolutionarily stable against invasion from free riders, the process of alternation of norms and the role of diversified norms remain unclear in the evolution of cooperation. We clarify the co-evolutionary dynamics of norms and cooperation in indirect reciprocity and also identify the indispensable norms for the evolution of cooperation. Inspired by the gene knockout method, a genetic engineering technique, we developed the norm knockout method and clarified the norms necessary for the establishment of cooper…
In multi-agent systems, cooperative behavior is largely determined by the network structure which dictates the interactions among neighboring agents. These interactions often exhibit multidimensional features, either as relationships of different types or temporal dynamics, both of which may be modeled as a "multiplex" network. Against this background, here we advance the research on cooperation models inspired by generalized reciprocity, a simple pay-it-forward behavioral mechanism, by considering a multidimensional networked society. Our results reveal that a multiplex network structure can …
The emergence of structure in cooperative relation is studied in a game theoretical model. It is proved that specific types of reciprocity norm lead individuals to split into two groups. The condition for the evolutionary stability of the norms is also revealed. This result suggests a connection between group formation and a specific type of reciprocity norm in our society.
Indirect reciprocity is a foundational mechanism of human cooperation. Existing models of indirect reciprocity fail to robustly support social cooperation: image scoring models fail to provide robust incentives, while social standing models are not informationally robust. Here we provide a new model of indirect reciprocity based on simple, decentralized records: each individual's record depends on their own past behavior alone, and not on their partners' past behavior or their partners' partners' past behavior. When social dilemmas exhibit a coordination motive (or strategic complementarity), …
We introduce a framework for studying social dilemmas in networked societies where individuals follow a simple state-based behavioral mechanism based on generalized reciprocity, which is rooted in the principle "help anyone if helped by someone". Within this general framework, which applies to a wide range of social dilemmas including, among others, public goods, donation and snowdrift games, we study the cooperation dynamics on a variety of complex network examples. By interpreting the studied model through the lenses of nonlinear dynamical systems, we show that cooperation through generalize…
Many biological and social systems show significant levels of collective action. Several cooperation mechanisms have been proposed, yet they have been mostly studied independently. Among these, direct reciprocity supports cooperation on the basis of repeated interactions among individuals. Signals and quorum dynamics may also drive cooperation. Here, we resort to an evolutionary game theoretical model to jointly analyse these two mechanisms and study the conditions in which evolution selects for direct reciprocity, signalling, or their combination. We show that signalling alone leads to higher…
We study the evolution of two mutually interacting games with both pairwise games as well as the public goods game on different topologies. On 2d square lattices, we reveal that the game-game interaction can promote the cooperation prevalence in all cases, and the cooperation-defection phase transitions even become absent and fairly high cooperation is expected when the interaction goes to be very strong. A mean-field theory is developed that points out new dynamical routes arising therein. Detailed analysis shows indeed that there are rich categories of interactions in either individual or bu…
In the study of the evolution of cooperation, many mechanisms have been proposed to help overcome the self-interested cheating that is individually optimal in the Prisoners' Dilemma game. These mechanisms include assortative or networked social interactions, other-regarding preferences considering the payoffs of others, reciprocity rules to establish cooperation as a social norm, and multilevel selection involving simultaneous competition between individuals favoring cheaters and competition between groups favoring cooperators. In this paper, we build on recent work studying PDE replicator equ…
Indirect reciprocity is a major mechanism in the maintenance of cooperation among unrelated individuals. Indirect reciprocity leads to conditional cooperation according to social norms that discriminate the good (those who deserve to be rewarded with help) and the bad (those who should be punished by refusal of help). Despite intensive research, however, there is no definitive consensus on what social norms best promote cooperation through indirect reciprocity, and it remains unclear even how those who refuse to help the bad should be assessed. Here, we propose a new simple norm called "Stayin…
Reputation is a powerful mechanism to enforce cooperation among unrelated individuals through indirect reciprocity, but it suffers from disagreement originating from private assessment, noise, and incomplete information. In this work, we investigate stability of cooperation in the donation game by regarding each player's reputation and behaviour as continuous variables. Through perturbative calculation, we derive a condition that a social norm should satisfy to give penalties to its close variants, provided that everyone initially cooperates with a good reputation, and this result is supported…
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 n…
Incomplete Reputation Information and Punishment in Indirect Reciprocity
Pukyong National University（釜庆国立大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Science (iTHEMS)（理化学研究所交叉理论数学科学中心（iTHEMS））；RIKEN Center for Computational Science（理化学研究所计算科学中心）
Recent research has identified the heterogeneity as crucial for the evolution of cooperation in spatial population. However, the influence of heterogeneous noise is still lack. Inspired by this interesting question, in this work, we try to incorporate heterogeneous noise into the evaluation of utility, where only a proportion of population possesses noise, whose range can also be tuned. We find that increasing heterogeneous noise monotonously promotes cooperation and even translates the full defection phase (of the homogeneous version) into the complete cooperation phase. Moreover, the promoti…
How cooperation emerges in human societies is both an evolutionary enigma, and a practical problem with tangible implications for societal health. Population structure has long been recognized as a catalyst for cooperation because local interactions enable reciprocity. Analysis of this phenomenon typically assumes bi-directional social interactions, even though real-world interactions are often uni-directional. Uni-directional interactions -- where one individual has the opportunity to contribute altruistically to another, but not conversely -- arise in real-world populations as the result of …
Reputation is one of key mechanisms to maintain human cooperation, but its analysis gets complicated if we consider the possibility that reputation does not reach consensus because of erroneous assessment. The difficulty is alleviated if we assume that reputation and cooperation do not take binary values but have continuous spectra so that disagreement over reputation can be analysed in a perturbative way. In this work, we carry out the analysis by expanding the dynamics of reputation to the second order of perturbation under the assumption that everyone initially cooperates with good reputati…
University of Pennsylvania（宾夕法尼亚大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Science (iTHEMS)（理化学研究所交叉理论与数学科学中心(iTHEMS)）；RIKEN Center for Computational Science（理化学研究所计算科学中心）；Saitama University（埼玉大学）
Besides the structure of interactions within networks, also the interactions between networks are of the outmost importance. We therefore study the outcome of the public goods game on two interdependent networks that are connected by means of a utility function, which determines how payoffs on both networks jointly influence the success of players in each individual network. We show that an unbiased coupling allows the spontaneous emergence of interdependent network reciprocity, which is capable to maintain healthy levels of public cooperation even in extremely adverse conditions. The mechanis…
Stern-judging is one of the best-known assessment rules in indirect reciprocity. Indirect reciprocity is a fundamental mechanism for the evolution of cooperation. It relies on mutual monitoring and assessments, i.e., individuals judge, following their own assessment rules, whether other individuals are "good" or "bad" according to information on their past behaviors. Among many assessment rules, stern-judging is known to provide stable cooperation in a population, as observed when all members in the population know all about others' behaviors (public information case) and when the members neve…
Direct reciprocity is a mechanism for the evolution of cooperation based on repeated interactions. When individuals meet repeatedly, they can use conditional strategies to enforce cooperative outcomes that would not be feasible in one-shot social dilemmas. Direct reciprocity requires that individuals keep track of their past interactions and find the right response. However, there are natural bounds on strategic complexity: Humans find it difficult to remember past interactions accurately, especially over long timespans. Given these limitations, it is natural to ask how complex strategies need…
Indirect reciprocity is a reputation-based mechanism for cooperation in social dilemma situations when individuals do not repeatedly meet. The conditions under which cooperation based on indirect reciprocity occurs have been examined in great details. Most previous theoretical analysis assumed for mathematical tractability that an individual possesses a binary reputation value, i.e., good or bad, which depends on their past actions and other factors. However, in real situations, reputations of individuals may be multiple valued. Another puzzling discrepancy between the theory and experiments i…
Evolution of cooperation is a widely studied problem in biology, social science, economics, and artificial intelligence. Most of the existing approaches that explain cooperation rely on some notion of direct or indirect reciprocity. These reciprocity based models assume agents recognize their partner and know their previous interactions, which requires advanced cognitive abilities. In this paper we are interested in developing a model that produces cooperation without requiring any explicit memory of previous game plays. Our model is based on the notion of, a concept introduced within behavior…
According to the evolutionary game theory principle, a strategy representing a higher payoff can spread among competitors. But there are cases when a player consistently overestimates or underestimates her own payoff, which undermines proper comparison. Interestingly, both underconfident and overconfident individuals are capable of elevating the cooperation level significantly. While former players stimulate a local coordination of strategies, the presence of overconfident individuals enhances the spatial reciprocity mechanism. In both cases the propagations of competing strategies are influen…
People tend to have their social interactions with members of their own community. Such group-structured interactions can have a profound impact on the behaviors that evolve. Group structure affects the way people cooperate, and how they reciprocate each other's cooperative actions. Past work has shown that population structure and reciprocity can both promote the evolution of cooperation. Yet the impact of these mechanisms has been typically studied in isolation. In this work, we study how the two mechanisms interact. Using a game-theoretic model, we explore how people engage in reciprocal co…
Indirect reciprocity is a mechanism for cooperation in social dilemma situations, in which an individual is motivated to help another to acquire a good reputation and receive help from others afterwards. Ingroup favoritism is another aspect of human cooperation, whereby individuals help members in their own group more often than those in other groups. Ingroup favoritism is a puzzle for the theory of cooperation because it is not easily evolutionarily stable. In the context of indirect reciprocity, ingroup favoritism has been shown to be a consequence of employing a double standard when assigni…
Indirect reciprocity is one of the main mechanisms to explain the emergence and sustainment of altruism in societies. The standard approach to indirect reciprocity are reputation models. These are games in which players base their decisions on their opponent's reputation gained in past interactions with other players (moral assessment). The combination of actions and moral assessment leads to a large diversity of strategies, thus determining the stability of any of them against invasions by all the others is a difficult task. We use a variant of a previously introduced reputation-based model t…
Direct reciprocity is one of the key mechanisms accounting for cooperation in our social life. According to recent understanding, most of classical strategies for direct reciprocity fall into one of two classes, `partners' or `rivals'. A `partner' is a generous strategy achieving mutual cooperation, and a `rival' never lets the co-player become better off. They have different working conditions: For example, partners show good performance in a large population, whereas rivals do in head-to-head matches. By means of exhaustive enumeration, we demonstrate the existence of strategies that act as …
Indirect reciprocity is a key mechanism that promotes cooperation in social dilemmas by means of reputation. Although it has been a common practice to represent reputations by binary values, either `good' or `bad', such a dichotomy is a crude approximation considering the complexity of reality. In this work, we studied norms with three different reputations, i.e., `good', `neutral', and `bad'. Through massive supercomputing for handling more than thirty billion possibilities, we fully identified which norms achieve cooperation and possess evolutionary stability against behavioural mutants. By …
Economic experiments reveal that humans value cooperation and fairness. Punishing unfair behavior is therefore common, and according to the theory of strong reciprocity, it is also directly related to rewarding cooperative behavior. However, empirical data fail to confirm that positive and negative reciprocity are correlated. Inspired by this disagreement, we determine whether the combined application of reward and punishment is evolutionary advantageous. We study a spatial public goods game, where in addition to the three elementary strategies of defection, rewarding and punishment, a fourth …
Many real systems are strongly characterized by collective cooperative phenomena whose existence and properties still need a satisfactory explanation. Coherently with their collective nature, they call for new and more accurate descriptions going beyond pairwise models, such as graphs, in which all the interactions are considered as involving only two individuals at a time. Hypergraphs respond to this need, providing a mathematical representation of a system allowing from pairs to larger groups. In this work, through the use of different hypergraphs, we study how group interactions influence t…
Reciprocity is a vital feature of social networks, but relatively little is known about its temporal structure or the mechanisms underlying its persistence in real world behavior. In pursuit of these two questions, we study the stationary and dynamical signals of reciprocity in a network of manioc beer (Spanish: chicha; Tsimane': shocdye') drinking events in a Tsimane' village in lowland Bolivia. At the stationary level, our analysis reveals that social exchange within the community is heterogeneously patterned according to kinship and spatial proximity. A positive relationship between the fre…
Spatial structure is one of the simplest and most studied ecological factors that affect the evolution of cooperation. It has been shown that spatial reciprocity promotes cooperation due to the formation of cooperative clusters, which provide mutual support against defectors. The usual assumption is that of constant population size, where no density-related effect is possible. Here, we extend the investigation of density-related effects on the evolution of cooperation. We integrate evolutionary game theory to the Lattice Lotka-Volterra Model. In our model, the birth rate depends on the local d…
