Indirect reciprocity maintains cooperation in stranger societies by mapping individual behaviors onto reputation signals via social norms. Existing theoretical frameworks assume static environments with constant resources and fixed payoff structures. However, in real-world systems, individuals' strategic behaviors not only shape their reputation but also induce collective-level resource changes in ecological, economic, or other external environments, which in turn reshape the incentives governing future individual actions. To overcome this limitation, we establish a co-evolutionary framework that couples moral assessment, strategy updating, and environmental dynamics, allowing the payoff structure to dynamically adjust in response to the ecological consequences of collective actions. We find that this environmental feedback mechanism helps lower the threshold for the emergence of cooperation, enabling the system to spontaneously transition from a low-cooperation state to a stable high-cooperation regime, thereby reducing the dependence on specific initial conditions. Furthermore, while lenient norms demonstrate adaptability in static environments, norms with strict discrimination are shown to be crucial for curbing opportunism and maintaining evolutionary resilience in dynamic settings. Our results reveal the evolutionary dynamics of coupled systems involving reputation institutions and environmental constraints, offering a new theoretical perspective for understanding collective cooperation and social governance in complex environments.
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…
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 key mechanism behind the evolution of cooperation. Oishi et al. analytically showed the formation of two exclusive groups under the KANDORI assessment rule in the case of perfect information and no implementation error, regardless of the population size $N$. Here, we numerically show the formation of many exclusive groups under the JUDGING assessment rule in the same case. Introducing degrees of exclusive groups, we numerically examine the stability of the group formation under imperfect information and implementation error.
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…
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…
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）
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…
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…
Here we investigate the dynamics of indirect reciprocity on networks, a type of social dynamics in which the attitude of individuals, either cooperative or antagonistic, toward other individuals changes over time by their actions and mutual monitoring. We observe an absorbing state phase transition as we change the network's link or edge density. When the edge density is either small or large enough, opinions quickly reach an absorbing state, from which opinions never change anymore once reached. In contrast, if the edge density is in the middle range the absorbing state is not reached and the…
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…
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…
Cooperation can be supported by indirect reciprocity via reputation.Thanks to gossip, reputations are built and circulated and humans can identify defectors and ostracise them. However, the evolutionary stability of gossip is allegedly undermined by the fact that it is more error-prone that direct observation, whereas ostracism could be ineffective if the partner selection mechanism is not robust. The aim of this work is to investigate the conditions under which the combination of gossip and ostracism might support cooperation in groups of different sizes. We are also interested in exploring t…
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）
Evaluation relationships are pivotal for maintaining a cooperative society. A formation of the evaluation relationships has been discussed in terms of indirect reciprocity, by modeling dynamics of good or bad reputations among individuals. Recently, a situation that individuals independently evaluate others with errors (i.e., noisy and private reputation) is considered, where the reputation structure (from what proportion of individuals in the population each receives good reputations, defined as goodness here) becomes complex, and thus has been studied mainly with numerical simulations. The p…
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…
Indirect reciprocity based on reputation is a leading mechanism driving human cooperation, where monitoring of behaviour and sharing reputation-related information are crucial. Because collecting information is costly, a tragedy of the commons can arise, with some individuals free-riding on information supplied by others. This can be overcome by organising monitors that aggregate information, supported by fees from their information users. We analyse a co-evolutionary model of individuals playing a social dilemma game and monitors watching them; monitors provide information and players vote fo…
The self-organization in cooperative regimes in a simple mean-field version of a model based on "selfish" agents which play the Prisoner's Dilemma (PD) game is studied. The agents have no memory and use strategies not based on direct reciprocity nor 'tags'. Two variables are assigned to each agent $i$ at time $t$, measuring its capital $C(i;t)$ and its probability of cooperation $p(i;t)$. At each time step $t$ a pair of agents interact by playing the PD game. These 2 agents update their probability of cooperation $p(i)$ as follows: they compare the profits they made in this interaction $δC(i;t…
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 …
Upstream reciprocity (also called generalized reciprocity) is a putative mechanism for cooperation in social dilemma situations with which players help others when they are helped by somebody else. It is a type of indirect reciprocity. Although upstream reciprocity is often observed in experiments, most theories suggest that it is operative only when players form short cycles such as triangles, implying a small population size, or when it is combined with other mechanisms that promote cooperation on their own. An expectation is that real social networks, which are known to be full of triangles…
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 …
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…
We analyze an agent-based model to estimate how the costs and benefits of users in an online social network (OSN) impact the robustness of the OSN. Benefits are measured in terms of relative reputation that users receives from their followers. They can be increased by direct and indirect reciprocity in following each other, which leads to a core-periphery structure of the OSN. Costs relate to the effort to login, to maintain the profile, etc. and are assumed as constant for all users. The robustness of the OSN depends on the entry and exit of users over time. Intuitively, one would expect that…
The annual occurrence of many infectious diseases remains a constant burden to public health systems. The seasonal patterns in respiratory disease incidence observed in temperate regions have been attributed to the impact of environmental conditions on pathogen survival. A model describing the transmission of an infectious disease by means of a pathogenic state capable of surviving in an environmental reservoir outside of its host organism is presented in this paper. The ratio of pathogen lifespan to the duration of the infectious disease state is found to be a critical parameter in determinin…
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…
In this paper three computational models for the study of the evolution of cooperation under cultural propagation are studied: Kin Selection, Direct Reciprocity and Indirect Reciprocity. Two analyzes are reported, one comparing their behavior between them and a second one identifying the impact that different parameters have in the model dynamics. The results of these analyzes illustrate how game transitions may occur depending of some parameters within the models and also explain how agents adapt to these transitions by individually choosing their attachment to a cooperative attitude. These p…
Learning to cooperate in a changing world: How caring about the future promotes cooperation across scales
Beihang University（北京航空航天大学）；Zhongguancun Laboratory（中关村实验室）；Hangzhou International Innovation Institute, Beihang University（北京航空航天大学杭州创新研究院）；Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Beihang University（北京航空航天大学数学、信息学与行为语义重点实验室）；Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University（北京航空航天大学未来区块链与隐私计算高级创新中心）；Beijing Academy of Blockchain and Edge Computing（北京区块链与边缘计算研究院）
In dyadic models of indirect reciprocity, the receivers' history of giving has a significant impact on the donor's decision. When the interaction involves more than two agents things become more complicated, and in large groups cooperation can hardly emerge. In this work we use a Public Goods Game to investigate whether publicly available reputation scores may support the evolution of cooperation and whether this is affected by the kind of network structure adopted. Moreover, if agents interact on a bipartite graph with partner selection cooperation can thrive in large groups and in a small am…
When a population is engaged in successive prisoner's dilemmas, indirect reciprocity through reputation fosters cooperation through the emergence of moral and action rules. A simplified model has recently been proposed where individuals choose between helping or not others, and are judged good or bad for it by the rest of the population. The reputation so acquired will condition future actions. In this model, eight strategies (referred to as 'leading eight') enforce a high level of cooperation, generate high payoffs and are therefore resistant to invasions by other strategies. Here we show tha…
Our wellbeing depends as much on our personal success, as it does on the success of our society. The realization of this fact makes cooperation a very much needed trait. Experiments have shown that rewards can elevate our readiness to cooperate, but since giving a reward inevitably entails paying a cost for it, the emergence and stability of such behavior remain elusive. Here we show that allowing for the act of rewarding to self-organize in dependence on the success of cooperation creates several evolutionary advantages that instill new ways through which collaborative efforts are promoted. R…
We consider the problem of explaining the emergence and evolution of cooperation in dynamic network-structured populations. Building on seminal work by Poncela et al, which shows how cooperation (in one-shot prisoner's dilemma) is supported in growing populations by an evolutionary preferential attachment (EPA) model, we investigate the effect of fluctuations in the population size. We find that the fluctuating model is more robust than Poncela et al's in that cooperation flourishes for a wide variety of initial conditions. In terms of both the temptation to defect, and the types of strategies…
