The role of environmental feedback mechanisms in the evolution of cooperation is attracting increasing attention within the academic community. However, existing research has rarely explored how these mechanisms influence the emergence of multi-player games and the evolution of cooperation within higher-order networks. In reality, higher-order relationships within a group merely provide the structural foundation for multi-player games, whilst the ultimate formation of such games depends on the internal gaming environment of the group. On this basis, this study, grounded in simplicial complex structures, constructs an evolutionary game model with an environmental feedback mechanism to reveal the impact of the mechanism on the evolution of cooperation within higher-order networks. The results indicate that the environmental feedback mechanism in this model effectively promotes the evolution of cooperation within simplicial complex structures, and that stricter environmental feedback is more conducive to enhancing the level of cooperation. Furthermore, in many cases, as the proportion of 2-simplices p increases, the cooperation rate exhibits a non-monotonic trend that first increases and then decreases, reaching a peak at a relatively small value of p. This highlights the significant positive role of sparse 2-simplices in the structure for the evolution of cooperation. Moreover, our findings also reveal a shift: cooperation is facilitated by a preference for multi-player games' payoffs at small p, but by a preference for pairwise games' payoffs at large p. Overall, these findings reveal the rich dynamic characteristics of the evolution of cooperation in higher-order networks with environmental feedback, providing new insights into how cooperation arises and flourishes within complex interactive structures.

The role of indirect reciprocity in sustaining cooperation stems from individuals’ strategic investment in establishing a favourable reputation. Gossip contributes to the construction of reputational mechanisms that are universally agreed upon. However, past gossip models typically assumed (1) the game process is observed by an arbitrary third party, (2) gossip propagation is random and infinite. In reality, individuals can typically only observe the actions of those they know personally, and gossip propagation is influenced by social relationships, often ceasing as the social distance from the focal individual increases. These real-world factors underscore the necessity of considering population structure, while limited observation and gossip propagation also raise concerns about the potential failure to achieve reputational consensus. To address this, we propose a network-based gossip model with local observation and limited propagation. Here, an individual’s behaviour is observable only to neighbours connected by direct edges, while their reputation propagates along the edges of the social network, constrained by social distance. We examine how factors such as the benefit-to-cost ratio of cooperation, gossip propagation frequency, network structure, error rates, and social norms influence group cooperation levels and opinion consistency. Our findings reveal the detrimental effect of restricted gossip propagation on indirect reciprocity efficacy, the pivotal role of social structure in achieving reputational consensus, and the key insight that reward-based social norms can counteract limited consensus.

Indirect reciprocity sustaining large-scale human cooperation relies upon reputation consensus grounded in social norms. Whilst unified social norms may foster consensus, real-world communities often feature coexisting diverse norms - such as in multi-ethnic neighbourhoods and multicultural settings where individuals frequently judge others' reputations by differing standards. Particularly when reputations are privately assessed and disseminated through limited gossip within structured groups, resulting evaluative divergence may exacerbate group polarisation, thereby threatening cooperative stability. To address this, this study constructs a two-layer network model with limited gossip propagation and donation games, wherein individuals' reputations are judged according to differing norms, and gossip propagates along social network edges with finite reach. Results indicate that punitive norms readily induce group opinion polarisation and oscillatory shifts in reputation matrices under information scarcity, whereas introducing relatively simple norms fosters stable cooperation. Furthermore, we distinguish how gossip carriers-transmitting either reputation or behavioural information about focal individuals-amplify opinion divergence, finding that behavioural information propagation undermines indirect reciprocity mechanisms. Finally, results show that network segregation can help build local consensus and rescue cooperation when gossip propagation is infrequent, as modelled in segregated networks constructed via Markov Chain Monte Carlo sampling of bilateral exchanges. This study provides insights into indirect reciprocity mechanisms within structured societies featuring heterogeneous norms and constrained information propagation, aiding explanations for the coexistence of polarisation and cooperation in the real world.

Although the influence of social learning on collective decision-making has been extensively studied, the role of the resulting social pressure in shaping decision dynamics is often overlooked. This pressure not only prompts individuals to follow the majority, thereby reinforcing group inertia and stifling innovation; more significantly, it drives social learners to adaptively adjust their learning strategies, thereby reshaping the co-evolutionary dynamics between social learning and collective decision-making. To this end, we have constructed an evolutionary game model based on collective action, analysing the impact of the relative advantages of options, learning costs, and social pressure. Theoretical analysis demonstrates that social learning patterns dominate phase transitions in system equilibria, revealing the necessary conditions for bistable emergence. Spatially structured Monte Carlo simulations further reveal that when the benefit-to-cost ratio of collective action exceeds a critical threshold, moderate social pressure can spontaneously evolve into an effective coordination mechanism - even in the absence of explicit coordination - ensuring social learners achieve higher long-term decision accuracy than individual learners.

Maximizing payoff is individuals’ pursuit during games, and thus payoff information profoundly affects individuals’ strategic choices. However, the privatization of payoff information provides individuals with opportunities for pretense and deception, and it is commonplace for individuals to induce their opponents to cooperate with them by declaring false payoff information in order to obtain a future advantage. This begs the question of how false payoff information disseminated by individuals for self-interested purposes affects humans’ strategy selection, and whether it inevitably has a negative impact on human cooperation. In view of this, this paper focuses on the impact of different deceivers, i.e., cooperative deceiver, defective deceiver, and dual-mode deceiver, on the evolution of cooperation, in the context that players can choose to declare false payoff information depending on their game situation. The results showed that all three types of deceivers can positively influence the evolution of cooperation, despite the self-interested initial intent of their deception, especially the dual-mode deceivers who most notably drove group cooperation by their flexibility in deceiving. In addition, compared to cooperative deceivers, defective deceivers perform better, i.e., defectors understate their payoff is more favorable to group cooperation than cooperators overstate their payoff. Moreover, in some of the tougher dilemmas, the defective deceivers can drive the group to achieve a reversal in the cooperation rate, which triggers a series of interesting and counterintuitive phenomena.

Previous experience suggests that prior agreement (a mechanism to test how strongly peers are committed to a common goal) and posterior compensation (a way to compensate cooperators by punishing defectors who violate their commitments) together contribute to the flourishing of cooperation. Far less is known, however, how the determination of the optimal threshold for prior agreement in force and the posterior compensation affect collaboration, and the relevance of different strategies. Here, in the context of public goods games and using an evolutionary game theory model, we investigate the evolution of cooperation and the relevance of different strategies by introducing prior agreements and posterior compensation in structured populations. Analytical results show that posterior compensation does not always promote cooperation and that there are sudden increases and decreases in the first-order phase transition. Interestingly, prior agreements without posterior compensation can also effectively restrain defectors, with cooperation abundance remaining at a high level. Furthermore, the threshold of effective agreement has nonlinear effects on cooperation, with cooperation abundance peaking at intermediate thresholds under specific parameters.

The interaction radius is crucial in spatial evolutionary games, but previous studies have overlooked the dynamic nature of individuals adjusting their interaction circles based on their states. This manuscript investigates the impact of dynamic changes in interaction radius on cooperation, specifically examining the influence of emotion-induced heterogeneity. Individuals are categorized into non-competitive and competitive groups, highlighting their differing emotional responses to game outcomes. By affecting the dynamics of their interaction radius, emotions play a role in cooperation. Specifically, adjusting the interaction radius based on emotions effectively enhances cooperation by enabling individuals to secure survival opportunities through distinct evolutionary pathways. The study also explores the influence of unit emotional value size and unit game initiation cost on cooperation evolution, shedding light on their effects. Furthermore, the research examines the non-linear relationship between the maximum interaction radius and cooperation. By considering the interplay of emotions and the dynamics of interaction radius, this research advances the understanding of cooperation in evolutionary games and emphasizes the significance of individual-level adjustments in interaction radius.

Identifying the most influential individuals in structured populations is an important research topic throughout network science. Previous studies, whether identifying the initial cooperators in a network that can lead to the spread of cooperation or assessing the chances that an intrusive defector may lead to the collapse of cooperation, have assumed that all individual attributes are homogeneous. However, heterogeneity among humans is widespread and can sometimes affect cooperation and welfare. For example, the layout of non-cooperative individuals in cooperative individuals inhibits the evolution of cooperation. Therefore, specifying the heterogeneity of individuals and studying the effect of their initial position on cooperation is the focus of our attention. Here we hypothesize that individuals in the network are divided into cooperative and non-cooperative individuals according to the heterogeneity of social value orientations, and we investigate the effects of the initial position of non-cooperative individuals on cooperation by adding them through different ranking rules to the network composed of cooperative individuals. We investigate separately the effects of the proportion of non-cooperative individuals, the proportion of initial cooperators of different types of individuals and different game scenarios on cooperation under different ranking rules. The results show that the addition of non-cooperative individuals to cooperative populations can effectively hinder the spread of cooperation. We also find that the proportion of initial cooperators for different types of individuals and the effects of different game types on population evolution also differ significantly under different ranking rules.

In real life, people often participate in regional public goods games and face the social dilemma caused by the non-exclusivity of public goods. Intervening policies are expected to alleviate the social dilemma and promote intra-regional cooperation. However, owing to the existence of the costs of policy, the marginal benefits of establishing intervening policies may be diminishing, which raises questions as to whether it is necessary to establish intervening policies on a global scale; whether partial intervention is a more economical solution. In view of this, this paper investigates the evolution of cooperation in regional public goods game in the context of a central institution selecting some regions to set up an endogenous reward and punishment policy, focusing on the effects of the proportion of selection and the methods of selection on global cooperation and total social welfare. The results show that the partial intervention is superior to the whole intervention in terms of total social welfare when the coefficient of the costs of the policy is large or when strict reward and punishment are imposed and that the partial intervention can achieve the effect of the whole intervention in terms of cooperation in many cases. For all three selection methods, i.e., random selection, degree centrality selection, and size selection, size selection is optimal in most cases. Furthermore, allowing groups in non-intervened regions to spontaneously establish or abolish the policy can greatly facilitate cooperation under all three methods. Finally, we find that the central institution’s prolonged and sustained intervention is essential, and its termination will lead to serious consequences.

Central sanctioning institution controlled by third party is a double-edged sword for the evolution of cooperation. On the one hand, the institution can protect cooperators from the exploitation of free-riders; on the other hand, a corrupt third party can cause the institution to fail, thus resulting in the collapse of cooperation. However, previous relevant studies have always explored that third parties are bribed by free-riders, and therefore forgo sanctioning free-riders. Although bribery is an important component of corruption, it is not comprehensive. In this paper, we consider third parties who embezzle public funds, which is characterized by the fact that the deterrence of sanctions remains even if corruption occurs. There is a new dilemma here, where cooperators are exploited by third parties when they are in the institution and by free-riders when out of the institution. An evolutionary game model is established to represent this dilemma, and a complex network model is constructed to represent interactions between third parties and players. We discuss the evolutionary dynamics of cooperation, trust and corrupt behaviours both in finite and infinite population, while find that players trust corrupt third parties and maintain cooperation as long as the sanction institution is sufficiently effective and the corruption rate is not very high. This work presents a novel framework for representing interactions between players and third parties, and suggests that exploitation by free-riders is less acceptable to cooperators than exploitation by third parties.

Public resource redistribution is an important means of social income regulation, which maintains social income equality by transferring payoffs to low-income groups, usually in the form of taxes and social services. Originally studied as a means of maintaining social equality, redistribution has also been used to address social dilemmas through evolutionary game theory in recent years. However, the definition of low-income groups in previous studies is not comprehensive. They are used to studying the inequality generated by participants during the game but ignore the inequality caused by initial endowment differences. At the same time, is it beneficial to the whole that redistribution requires the rich to subsidize the poor, but leads the rich to choose defection (tax evasion, etc.)? How should the redistribution mechanism work to reduce inequality while facilitating the spread of altruistic behavior? To explore whether the redistribution mechanism can reach a balance between promoting cooperation and maintaining equality of payoffs, we develop a generic public resource redistribution framework based on heterogeneous populations by a spatial public goods game model. We control redistribution preferences for willingness to cooperate and initial endowments by parameters α and β. With a heterogeneous population, we find that helping the poor through redistribution mechanisms does not have significant positive effects and may even result in exploitation of the rich; focusing on participants' willingness to cooperate rather than total investment can promote cooperation, enhance overall payoffs and reduce inequality.

“Friends of friends are friends, and enemies of friends are enemies” is a well-known friendship transmission effect. In interpersonal interactions, friendship transmission may prompt individuals to imitate one partner’s attitude toward another. Real human interactive networks often show a high degree of clustering and are filled with triangles. In this context, friendship-transmission-based imitation (FI) may be very frequent and play a crucial role in the flourishing of cooperation. Here, we develop an evolutionary pairwise game model and conduct simulations on small-world networks with the aim of revealing exactly how FI affects cooperation. Specifically, we add four forms of FI to a strategy update framework with interactive diversity, using two sets of parameters to control the frequency of each update method. The results show that the different FIs play very different roles. Two of them even conditionally outperform the targeted imitation with reciprocal features. Furthermore, appropriate coexistence of multiple FIs may catalyze more cooperation. This work captures potential factors influencing the evolution of cooperation in real human societies, and contributes to understanding the drivers and the barriers to human cooperation in highly clustering interactions.

Collective action, like public goods game, often requires mechanism intervention to avoid exploitation by free-riders. Reward and punishment are two widely studied solutions, but the balance of budgets and diverse behavioral motivations need further emphasis. Here, we introduce a heterogeneous distribution system based on peer rating (PRHDS), a disguised reward and punishment mechanism under budget balancing, and consider a comprehensive range of behavioral motivations. Preliminary studies show that PRHDS has the potential to promote cooperation, but is too sensitive to random exploration. This vulnerability is attributed to the exploitation of prosocial cooperators by egoistic cooperators. Afterward, we propose three punishments for minority raters and find that two monetary punishments can compensate for the above limitation. A notable finding is that the cooperation potential at medium multipliers seems to exceed that at high multipliers. In addition, our further analysis suggests that PRHDS may perform better when strategy differentiation is considered. This study reveals the possible limitations of PRHDS as a solution to collective social dilemma, and may have some enlightening significances for seeking the countermeasures.

In repeated games, historical interactions lead to varying environment. Among existing studies, the variation of environment or game is basically considered from the perspective of nodes, while in pairwise game, it is more reasonable to consider the interactive environment from the perspective of edges. In addition, since historical interactive experience of pairwise players has an impact on the current interactive environment, the duration of pairwise strategies produces differentiated results. Therefore, we consider the persistence-dependent dynamic interactive environment where the persistence of pairwise strategy affects the weight between two interacting players, and the game resource of interactive environment is influenced by varying weight. The results suggest that the introduction of persistence-dependent dynamic interactive environment can effectively enhance cooperation, and a larger weight increment of CD link generates a higher cooperation level as it indicates that the reward for cooperative behavior is significant, although a single cooperator for the pairwise interacting players.

Individual selection, as an effective mechanism, is often used in spatial evolutionary games to promote cooperation. Previous research assumes that, individual selection usually occurs with people who fail to meet a certain criterion. However, individual selection is usually inevitable, regardless of whether players in population cooperate or defect. This paper studies the effects of wealth-based rule in costly public goods games when individual selection is inevitable. Specifically, we assume that only the top V individuals with relatively high cumulative payoffs in each group can be selected for costly PGG. The results show that when V is large, the increase of participation cost has slight inhibitory effects on the evolution of cooperation, but it alleviates the polarization of individuals. However, when V is small, the increase of participation cost within a certain range promotes cooperation prosperity, but it also causes an increase in the proportion of polarized individuals and a widening of the wealth gap between rich and poor individuals. (c) 2021 Elsevier Inc. All rights reserved.

Many various projects may exist in an enterprise in parallel, which leads to distinct group interactions. Team members cooperate in their respective projects, and sometimes it is possible to engage in several concurrent projects. Owing to practice makes perfect, the longer the duration of members participating in an identical project is, the higher benefits will be under same contribution. However, in a fiercely competitive environment, a huge conflict of interest between groups and individuals which results in social dilemma makes projects difficult to proceed smoothly. The experience of direct reciprocity shows that repeated interactions can alleviate this problem, but most multi-person dilemma studies consider a single game scenario, and public resources remain constant. Here we introduce a general framework of multiple public goods games. Taking the public goods game with threshold into consideration, public resource evolves with the persistence and transition of games. Our results suggest although collaborators are at a disadvantage in different game scenarios, our model can still effectively promote the overall cooperation in multiple public goods games. Especially, it performs better when public resource is relative scarce. (c) 2021 Elsevier Inc. All rights reserved.

Emotion emerges along with individuals' interactions, and numerous experimental studies have shown that emotion plays an important role in individual decision making. However, how emotion affects the evolution of cooperation in structured population is largely unknown. Here we introduce emotion into network reciprocity, where emotion is considered quantifiable and cumulative, and then divide individuals into two types, namely noncompetitive individuals whose social value orientation is mutually beneficial and competitive individuals whose social value orientation is maximizing outcomes for the self. Here, we explore the effects of the proportion of non-competitive individuals and emotional cumulative length on the evolution of cooperation. Simulation results show that the existence of non-competitive individuals promotes cooperation in the system. Furthermore, we also find that emotional cumulative length plays a key role in promoting overall cooperation, but surprisingly, the cooperation rate of competitive individuals peaks at an intermediate value of emotional cumulative length. (C) 2021 Elsevier Inc. All rights reserved.

Affected by geographical or social factors, human communities are organized with changeless groups for teamwork. Within groups, defection dominates because the payoffs of defectors are often higher than those of cooperators; among groups, higher cooperation level makes individuals wealthier. For the previous mechanism of evolutionary dynamics leading individuals to imitate the rich in their groups, defective strategy evolves. However, individuals who belong to different groups and have no direct interaction of interests actually could influence each other. Due to the similarity of environment, the strategies among groups have the channel and necessity of dissemination. To this end, we propose an evolutionary game model with inter-group selection of strategy. By simulations, the effects of groups’ selection preference, group size, multiplier factor, imitation intensity and competition among groups on cooperative evolution in populations are studied. We find that when the group makes selection without preference, the intensity of the choice positively affects the level of cooperation; the strategy selection with preference can effectively increase the group cooperation rate.

It is known that the vaccination is essential for suppressing the periodic prevalence of an infectious disease, while the vaccination decision is influenced by external factors. Whether or not to take vaccination depends not only on the expected benefit of individual, but also on the behavior of others. Nowadays, the speed of information transmission on the social media is fast with wide range, which extremely expands the scope of imitation, but it is unknown whether it is beneficial to vaccination. Inspired by this situation, a vaccination game model on two-layer network which increases the intensity of information sharing is proposed. The vaccination game layer and the information sharing layer are established to study the interaction between individual vaccination decisions and strategic exchanges. We find that although information sharing has inhibited individual vaccination decisions to some extent, the overall scale of disease spreading in society has decreased with increasing information sharing.