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• The study aimed to quantify the cyclic dominance resembling rock-paper-scissors observed in the optional prisoner's dilemma, considering cooperators, defectors, and loners as the three competing strategies. • The research utilized theoretical and numerical analyses to explore the impact of environmental feedback on the replicator dynamics. • The rate of environmental replenishment and the payoff from draw games were identified as crucial factors influencing gameplay dynamics. • The study revealed that the loner's payoff and the strength of the dilemma played pivotal roles in determining the dominant strategies. This research investigation delves into the concept of cyclic dominance and its interplay with environmental feedback within the framework of the Optional Prisoner's Dilemma (OPD) game. A quantitative model has been meticulously constructed to encompass both the OPD and Rock-Paper-Scissors (RPS) games, wherein the outcomes are contingent upon environmental feedback. The intricate relationship between player populations and their surrounding environment is elucidated through the utilization of replicator dynamics. The empirical findings of this study shed light on the existence of distinct equilibria contingent upon the prevailing environmental circumstances. Under scenarios characterized by resource depletion, an equilibrium state emerges wherein only solitary individuals (referred to as "loners") persist. Conversely, under conditions of resource replenishment, an equilibrium is established where various strategic approaches coexist in equal proportions, akin to the dynamics of the Rock-Paper-Scissors game. Notably, cyclic dominance manifests itself during periods characterized by intermediate environmental conditions. Several pivotal factors have been identified as determinants of these dynamic phenomena, including the rate of environmental replenishment, the strength of the dilemma presented, the payoff for loners, and the reward for a draw. It has been empirically demonstrated that the presence of extreme values in the replenishment rate, irrespective of whether exceptionally high or exceptionally low, serves as an impediment to the emergence of cyclic dominance. This model underscores the substantial influence of environmental feedback on social dilemmas, particularly in the context of the OPD game, ultimately leading to the emergence of competitive cyclic dominance games reminiscent of the Rock-Paper-Scissors game. In doing so, it elucidates the intricate interplay between evolutionary game incentives and the dynamics of the surrounding environment. [ABSTRACT FROM AUTHOR]

Exploring the emergence of cooperation in the populations of self-interested individuals is of huge challenge. Evolutionary game theory has been utilized to investigate the evolution of cooperation. In recent years, many researchers have paid massive attention on the evolution of cooperative behavior from the perspective of multiple games, especially the transition of games. Nevertheless, not only are the current interactive actions crucial, but also the historical interactive experience is fateful. Motivated by this, the persistence-based game transition mechanism is proposed and introduced to investigate the evolution of cooperation, where both the current pairwise strategies and the persistence of the pairwise strategies determine the game played in the next time step. Through analyzing the evolutionary results of strategies and games, we find that the cooperation can be facilitated by the enhanced network reciprocity as the persistence-based game transition makes mutual cooperators play games with weak dilemma more persistently. • A persistence-based game transition is proposed in multi-games. • In our model, both pairwise interactive strategies and the persistence of pairwise strategies determine the game transition. • The persistence-based game transition can facilitate cooperation through enhancing network reciprocity. [ABSTRACT FROM AUTHOR]

As an important part of the evolution of cooperation, interaction diversity has attracted extensive attention. Traditional studies mainly focus exclusively on either time or space, and the comprehensive impact of these two dimensions on the evolution of cooperation deserves further exploration. In this paper, the interaction willingness is abstracted as the network weight, and an adaptive mechanism for interaction willingness based on the delay effect is proposed by combining the network weight and accumulated payoff. Specifically, individuals adjust their willingness to participate in the game with a delay, considering the change in accumulated payoffs within a certain period, rather than the change in immediate payoffs. The degree of delay is reflected in the length of individuals' memory of historical payoffs. Simulation results deviate from the intuition that more information facilitates cooperation, and in certain modes of interaction willingness adjustment, the correlation between cooperation level and memory–length is non–monotonic. Long–memory is conducive to promoting cooperation under the condition of a relatively large range adjustment of interaction willingness. Conversely, with a narrow adjustment range of interaction willingness, cooperation can be effectively established with short–memory. These results hold substantial implications for both theory and practice, as slight adjustment often corresponds to low loss and high operability. In addition, the underlying mechanism is furthermore explored, revealing that the 'isolation zones' composed of individuals with low interaction willingness play an important role in promoting cooperation. • The self-adaptive mechanism based on memory influences cooperation evolution. • The relationship between cooperation level and memory-length is non-monotonic. • Memory-length affect cooperation differently based on willingness adjustment mode. [ABSTRACT FROM AUTHOR]

In this work, we have studied the evolution of the cooperation and the global synchronization on the Kuramoto model upon random networks with two games, in which one is the evolutionary Kuramoto dilemma and the other is the weak prisoner's dilemma. Agents can gain their payoffs from these two games synchronously. In the evolutionary Kuramoto game, cooperators prefer the formation of local synchronization but pay less cooperative cost. Cooperators in weak prisoner's dilemma can form cooperator clusters because of the network reciprocity. Combining these two games shows a mutual promotion effect upon the order of the global synchronization but a competition effect on the cooperation level. The global synchronization begins to rise only when most agents choose cooperation. It is manifested by the fact that the optimal coupling strength for the highest cooperation level is smaller than that for the highest synchronization level. The high cooperation cost makes defection the most advantageous strategy when the coupling strength reaches rather high, which diminishes the effect of cooperator clusters, and ultimately results in the absence of both cooperators and global synchronization. When agents have more neighbors in the random network, it is observed that cooperation is consistently hindered unless the coupling strength is relatively small. Additionally, there is always an optimal average degree that allows for global synchronization. • A multi-game is proposed that combines the Kuramoto dilemma and the prisoner's dilemma. • The coevolutionary dynamic of synchronization and cooperation is investigated. • There shows a mutual promotion effect upon the global synchronization but a competition effect on the cooperation. [ABSTRACT FROM AUTHOR]

In a social dilemma situation, where individual and collective interests are in conflict, it sounds a reasonable assumption that the presence of super or smart players, who simultaneously punish defection and reward cooperation without allowing exploitation, could solve the basic problem. The behavior of such a multi-strategy system, however, is more subtle than it is firstly anticipated. When exploring the complete parameter space we find that the emergence of cyclic dominance among strategies is rather common, which results in several counter-intuitive phenomena. For example, the defection level can be lowered at higher temptation, or weaker punishment provides better conditions for smart players. Our study indicates that smart cooperators can unexpectedly thrive under high temptation, emphasizing the complexity of strategic interactions. This study suggests that the principles governing these interactions can be applied to other moral behaviors, such as truth-telling and honesty, providing valuable insights for future research in multi-agent systems. • Introduces a smart cooperator strategy in the prisoner's dilemma, rewarding cooperative neighbors and punishing defectors. • Reveals complex interactions and cyclic dominance among strategies, showing reduced defection at higher temptations. • Weaker punishments can favor smart cooperators, adding to the dynamics of multi-strategy systems. • Provides insights that could extend to other moral behaviors and offer new directions for future research in social dilemmas. [ABSTRACT FROM AUTHOR]

In contemporary society, cooperation is a crucial element for individuals pursuing shared interests and attaining triumph. However, conventional cooperative evolution approaches frequently disregard the intricacies amongst individuals. In order to gain improved comprehension and refine the cooperative evolution process, we suggest a pioneering deep cooperation strategy. The strategy of deep cooperation is grounded on the conventional Prisoner's dilemma model. It highlights that individuals incur additional costs in the game to attain more benefits while also maximizing mutual social advantages. By doing so, individuals and the collective attain a win-win situation. After research, we find that the system has the following five evolutionary results: only deep cooperation strategy, pure defection strategy, deep cooperation strategy and cooperation strategy coexistence, deep cooperation strategy and defection strategy retained, and three strategies exist simultaneously. The simulation validates that the expensive deep cooperation strategy can enhance the system's resilience to defection, prompt cooperation strategy in the populace to transition their approach to garner a greater payoff, and foster a favorable atmosphere for cooperation strategy. It provides a new perspective for understanding cooperation. • We propose a more advantageous deep cooperation strategy. • The deep cooperation strategy is conducive to relieving the social dilemma. • The model with a deep cooperation strategy is more practical than the traditional prisoner's dilemma model. [ABSTRACT FROM AUTHOR]

The relationships among individuals on the network are often dynamic and adaptive adjustment with the interactions, which have a strong effect on the cooperation behavior. In this paper, we propose a new spatial evolutionary game model to further investigate the emergence and persistence of cooperation, in which the link weights among individuals are dynamically and adaptively adjusted by comparing between their own reputation and the average one of their nearest neighbors. If the reputation of the focal player is greater than the average one of his/her nearest neighbors, the nearest neighbors around the focal player will follow him/her, thus the link weight between the nearest neighbors and the focal player will increase by one unit as a reward. In contrast, the corresponding link weight will be decreased by one unit as a punishment. In addition, we use another parameter ε to control the possible range of link weight. By conducting extensive Monte Carlo numerical simulations in the prisoner's dilemma and the snowdrift games, it is clearly verified that the adaptive adjustment of link weight can dramatically promote the evolution of cooperation, and the cooperation can be further elevated into a higher level when compared with the standard spatial dilemma game. The current results are helpful for us to better understand the influence of dynamic link weight on the collective cooperation behavior. [ABSTRACT FROM AUTHOR]

• A three-strategy social dilemma is studied where quasi-cooperators are also present. • A structured population may provide lower cooperation level than a wellmixed. • This unexpected behavior is related to the relation of different strategies. • The usage of myopic best response rule can restore the standard behavior. • Our model highlights the importance of strategy relations in a multistrategy system. Introducing strategy complexity into the basic conflict of cooperation and defection is a natural response to avoid the tragedy of the common state. As an intermediate approach, quasi-cooperators were recently suggested to address the original problem. In this study, we test its vitality in structured populations where players have fixed partners. Naively, the latter condition should support cooperation unambiguously via enhanced network reciprocity. However, the opposite is true because the spatial structure may provide a humbler cooperation level than a well-mixed population. This unexpected behavior can be understood if we consider that at a certain parameter interval the original prisoner's dilemma game is transformed into a snow-drift game. If we replace the original imitating strategy protocol by assuming myopic players, the spatial population becomes a friendly environment for cooperation. This observation is valid in a huge region of parameter space. This study highlights that spatial structure can reveal a new aspect of social dilemmas when strategy complexity is introduced. [ABSTRACT FROM AUTHOR]

The Iterated Prisoner's Dilemma (IPD) is a well studied framework for understanding direct reciprocity and cooperation in pairwise encounters. However, measuring the morality of various IPD strategies is still largely lacking. Here, we partially address this issue by proposing a suit of plausible morality metrics to quantify four aspects of justice. We focus our closed-form calculation on the class of reactive strategies because of their mathematical tractability and expressive power. We define reactive means as a tool for studying how actors in the IPD and Iterated Snowdrift Game (ISG) behave under typical circumstances. We compute reactive means for four functions intended to capture human intuitions about "goodness" and "fair play". Two of these functions are strongly anticorrelated with success in the IPD and ISG, and the other two are weakly anticorrelated with success. Our results will aid in evaluating and comparing powerful IPD strategies based on machine learning algorithms, using simple and intuitive morality metrics. [ABSTRACT FROM AUTHOR]

Abstract Understanding and explaining the widespread emergence of cooperation in social dilemmas remains a great challenge. Previous researches have shown that complex networks with high heterogeneity especially scale-free networks can remarkably facilitate cooperation in the case of accumulated payoff, but fail in the case of average payoff. In this paper, we investigate the role of particle swarm optimization (PSO) strategy update rules in the evolution of cooperation in the prisoner's dilemma (PD) on scale-free networks, when each player obtains the average payoff normalized with its degree. In the meantime, considering whether goods and resources are divisible, PSO strategy update rules based on continuous and mixed strategy systems are proposed respectively. The simulation results show that the PSO strategy update rules can promote cooperation in the continuous strategy system, but lead to the severe collapse of cooperation in the mixed strategy system. The nodes with low or medium degree play an important role to facilitate cooperative behaviors. [ABSTRACT FROM AUTHOR]

Abstract The problem of how cooperative behaviors emerge in real world has been widely investigated and it is found that cooperative behaviors are closely related to memory, which means a rational player can get a more reasonable strategy by comprehensively considering certain amount of historical information within its memory scope. On the other hand, due to the potential risk during pursuing high payoff in human society, selecting the most common strategy for a player inside the interaction range is relatively safer, at least its payoff is not much lower than average, which is known as conformity. Motivated by these facts, we here propose a mechanism of conformity with memory whose core lies in two facts: (i) Each player first applies a memory rule to compare its own historical payoffs with a limited memory length and to take the strategy corresponding to the maximal payoff as historical optimal strategy; (ii) Each player can get the virtual optimal strategy by selecting the most common strategy from its neighbors' historical optimal strategies according to the conformity rule. The asynchronous updating algorithm, together with the virtual strategy, is used to study the evolution of cooperation with different memory lengths on a regular lattice. Simulation results show that the proposed mechanism remarkably promotes cooperation in three classical evolutionary social dilemmas. More specifically, it is revealed that the cooperation level decreases gradually as the memory length increases in the prisoner's dilemma and snowdrift game, and that the cooperation level increases as so does the memory length in the stag-hunt game. [ABSTRACT FROM AUTHOR]

Abstract In realistic world, the role or influence of each individual is heterogeneous and usually varies according to surroundings. Inspired by this fact, here we study the emergence of cooperative behavior in weighted networks under the coevolution of game strategy and node weight, where the node weight is used to mimic the role or influence of subjects. In the prisoner's dilemma, if an individual's fitness exceeds the aspiration level, its weight becomes larger; otherwise weight decreases. While such an adjustment of weight is defined by the intensity parameter δ, it is interesting that there is an optimal range for δ guaranteeing the best evolution of cooperation. The facilitation of cooperative behavior mainly depends on the weight distribution of players, which is based on the formation of a cooperative cluster controlled by high-weight cooperators. These cooperators are able to prevail against defectors even when there is a large temptation to defect. Our research provides a viable route to resolve social dilemma and will inspire further applications. [ABSTRACT FROM AUTHOR]

Cooperation plays an essential role in the evolution of social species, chief among all in humans. In this paper, we study the effects of compassion on the evolution of cooperation in spatial social dilemmas by introducing a payoff redistribution mechanism. In particular, a player whose payoff is larger than the average in its neighborhood will share some it with its comparatively poor neighbors. We find that such a simple redistribution mechanism, which we interpret as a form of compassion, significantly promotes the evolution of cooperation. While traditional network reciprocity already supports the formation of compact cooperative clusters, an in-depth analysis of payoff transfer events between players reveals an enhanced form of this phenomenon through the reinforcement of payoffs of cooperators that reside along the borders of such clusters. This significantly enhances the resilience of cooperative clusters, who are in turn able to survive even at adverse conditions where traditional network reciprocity alone would long fail. We show that the observed positive effects of compassion on the evolution of cooperation are robust to changes of the interaction network and to changes in the type of the governing social dilemma. [ABSTRACT FROM AUTHOR]

Emergence of cooperation in social dilemmas is an essential phenomena for the functioning of numerous multilevel and complex systems. The evidence of cooperation ranges from the elementary biological organisms to the most sophisticated human societies. Although emergence of cooperation is widely experienced, but its evolution is not well explained, since natural selection typically promotes selfish behaviours which are often not socially optimal. In this paper the coevolution of network structure and emergence of cooperation is studied in four classes of social dilemmas, representing the prisoner's dilemma, Hawk–Dove, snowdrift and coordination classes of games, in structured populations defined by weighted complex networks. The strength of interaction between two individuals is represented by network (edge) weight, which changes according to individuals’ preference through the evolutionary network dynamics. Using evolutionary dynamic network based simulations of the games model on randomly weighted complete networks, we present a detailed study of the evolution of dynamic complex network through the evolution of the structural properties of a network, such as clustering coefficient, assortativity coefficient, entropy of degree distribution, average strength of interaction and the promotion of cooperative behaviour in all four classes of games. The effect of changing the cost-to-benefit ratio on these network properties and evolution of cooperation is also reported. [ABSTRACT FROM AUTHOR]

• A computational game model, in which individual can adaptively switch between two payoff matrices with different temptations based on aspiration to explore the cooperation dynamics in spatial prisoner's dilemma, is proposed. • We analyzed the dynamics of cooperation by investigating the switching rate between different payoff matrices as well as the average payoff of individuals at the boundary. • We find that the mechanism of adaptive switching between payoff matrices with different temptations can effectively promote the cooperation of spatial prisoner's dilemma, and the appropriate aspiration level of individual can provide the most favorable environment for the diffusion of cooperation. Under given social dilemma, different individuals perhaps have different perceptions for the dilemma intensity. Generally speaking, the degree of individual's perception should be closely related to its ability, the stronger the ability of individual, the weaker the perception for the dilemma intensity will be. On the other hand, the competent individuals may be more likely to carry out altruistic cooperation behavior than those who are relatively lacking in ability. Considering these facts, we introduce an evolutionary game model in which individual can adaptively switch between two payoff matrices with different temptations based on aspiration to explore the cooperation dynamics in spatial prisoner's dilemma. In this model, two different payoff matrices are denoted by M w and M s respectively, the temptation value of the former is small, and the one of the latter is large. For the individual who has adopted M w to interact, the higher the individual is satisfied with the current payoff, the lower the probability of switching to M s in the next round, while for the individual who has adopted M s to interact, the higher the individual is satisfied with the current payoff, the higher the probability of switching to M w in the next round. Experimental results show that this mechanism of adaptive switching between payoff matrices with different temptations can effectively promote the cooperation of spatial prisoner's dilemma, and the appropriate aspiration level of individual can provide the most favorable environment for the diffusion of cooperation. The difference in the perception for dilemma intensity will be a new perspective in the study of networked evolution game. [ABSTRACT FROM AUTHOR]

This study investigates how the user’s exchanges affect people’s trust in electronic commerce and empower them to interact with foreign customers. Online exchanges often occur between strangers who cannot rely on past behavior or the prospect of future interactions to establish mutual trust. Game theorists have formalized this problem as a prisoner’s dilemma and predict mutual noncooperation. In this paper, we introduce a social network based model to promote the cooperation in the prisoner’s dilemma game. For this study, we implemented an agent-based simulation framework, which models different types of behaviors in online auctions. Agents who represent buyers or sellers are located in the nodes of a small world graph. Each link weight between the agent and its neighbor symbolizes how much it trusts this neighbor. A link with trust inferior to some cut-link threshold will be removed from the graph. Our results show that the evolved structure of the graph induces considerable variation in the level of cooperation and the profit in the e-commerce system. This shows that the outcome of our co-evolutionary game, in terms of cooperative behavior, strongly depends on topology but also on the update rule used in the trust between users. Co-evolution of game dynamics in interconnected networks is also studied, where two networks are interconnected, and players have interactions not only with others in the same network, but also with players in the other network. We will demonstrate that the interdependence between networks promotes the cooperation in both networks. But, the degree of promotion changes as a function of interdependence degree. [ABSTRACT FROM AUTHOR]

• When only the role of punishing evil is considered, increasing punishment strength alone is not enough to weed out defectors. • When only the effect of promoting good is considered, under the larger temptation to defection, the increase in reward is not sufficient for cooperators to survive. • When considered simultaneously, there exists combinations of minimum values of punishment and reward that bring the system to a full cooperation state. In this paper, based on the social dilemma model, we introduce the punishing evil and promoting good strategy, whose basic strategy is to cooperate, while to obtain benefits at the cost of payment. Specifically, when the person who punishes evil and promotes good plays a game with the cooperator, his contributions will bring the same benefits to both parties, reflecting the promoting good. However, when the person who punishes evil and promotes good plays a game with the defector, making the benefit of the defector loss and become his own benefit, reflecting the punishing evil. The results show that when only the role of punishing evil is considered, increasing punishment strength can easily make those punishing evil and promoting good and ordinary cooperators appear. But only increasing the punishment is not enough to weed out the defectors. When only the effect of promoting good is considered, under a smaller temptation to defection, the increase of the reward strength can transform the system from a full defection state to a full cooperation state. Whereas when the temptation is greater, the increase in the reward is not enough for the cooperator to survive and only defectors exist in the system at this time. In addition, there exist combinations of minimum values of punishment and reward that bring the system to a defector-free state. In conclusion, our study finds effective intervals for promoting the development of those who punish evil and promote good and ordinary cooperators, respectively. [ABSTRACT FROM AUTHOR]

Repeated prisoner’s dilemma (RPD) is an evolutionary game, extensively utilised for modelling socio-economic dynamics. Such an instrument was used in order to investigate co-operation in different interesting aspects and strategies, e.g., gratuity (Bruni, 2008) [2], in terms of co-operation, that privileges a collective advantage. This work explores RPD extension to a spatial context (SRPD) by a stochastic distribution of agents with different strategies in the space of a Cellular Automaton (CA) according to the Nowak approach (Nowak and May, 1993) [22]. This approach coincides with SPD extension to SRPD. In SRPD, each cell is occupied by one agent, that relates with all other agents in neighbourhood cells in terms of RPD classic payoffs and symmetries. Results show that the SRPD convergence to total co-operation or defection is very rapid (more than RPD), but some values of payoff generate an interesting chaotic evolution of the system. Furthermore, introduction of new behaviours of agents through altering payoffs and symmetry characterises this our model. In such conditions, unexpected emergence of co-operative behaviours was detected for agents with complex strategies, in spite of their reduced conditions of increasing own payoff. [ABSTRACT FROM AUTHOR]

Although the evolutionary game theory provides a powerful framework to investigate various social dilemmas, the traveler’s dilemma game (TDG) is hardly concerned under this platform within the applied mathematical communities. We put forward an improved traveler’s dilemma game model on two coupled lattices to investigate the effect of coupling effect on the evolution of cooperation based on the traveler’s dilemma game, where the coupling effect between two lattices is added into the strategy imitation process. Large quantities of simulations indicate that the cooperation behavior can be greatly varied when compared to those obtained on the traditionally single lattices. When the model parameter R surpasses a specific threshold, the cooperation will be greatly enhanced, but the cooperation may be inhibited if R is smaller than this threshold. Meanwhile, we also explore the relationship between the critical value R c leading to the extinction of cooperation and the strategy range parameter radius in our model. Our results are surprisingly conducive to understanding the cooperation behavior of traveler’s dilemma game within many real-world systems, especially for coupled and interdependent networked systems. [ABSTRACT FROM AUTHOR]

• A new framework of multiple public goods games considering the persistence and transition of games is proposed. • The heterogeneous public resource of public goods game evolves with the persistence and transition of games. • Our new model can promote cooperation effectively, especially under the condition with scarce resource. 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. [ABSTRACT FROM AUTHOR]

• We propose a multilayer network model that consider the coupling effect of group selection and network reciprocity under spatial the prisoner's dilemma game, and find that it could promote cooperation greatly. • We explore the two kinds of updating rules which are deterministic updating rule and stochastic updating rule, finding different phenomena under these. • We explore two key parameters called evolution frequency and the imitation rate, both of which play key roles on the evolution process in our model. Group selection and network reciprocity are two well-known mechanisms for cooperation. And while they have separately received ample attention for their ability to promote cooperation in social dilemmas, their joint effects in this regard are much less explored. Here we propose a multilayer network model that takes into account that not only individuals are connected by means of networks, but that also groups can be connected in much the same way. The model thus couples together network reciprocity and group selection, and it allows us to study their joint effect on the evolution of cooperation. We use the prisoner's dilemma game as the paradigmatic social dilemma example, showing that the fine-tuning of evolution frequency and the imitation rate in the group network play key roles in determining the survival thresholds of cooperators. We also explore the importance of deterministic and stochastic updating rules, showing that the former provides more options to further promote cooperation. We discuss the importance of our findings for cooperation in structured interactions among groups and in higher-order networks. [ABSTRACT FROM AUTHOR]

• With the help of memory, we propose a stability-based strategy updating rule and study its effect on cooperative behaviors. • Under this updating rule, the probability of learning strategy is affected not only by the difference of average payoffs but also the stability. • Simulations show that compared with the Fermi rule, the introduction of stability enhances cooperative behaviors and average payoffs. • Compared with the memory-based rule which considers the difference of average payoffs, the stability-based rule shows these superiorities on cooperators when the payoff parameters are relatively large. The stability of payoffs, often depicted by variance, provides more information during the decision-making process. With the help of memory, we propose a stability-based strategy updating rule and study its effect on cooperative behaviors. Under this updating rule, the probability of learning strategy is affected not only by the difference of average payoffs but also the stability. Based on the prisoner's dilemma game and snowdrift game, simulations show that compared with the Fermi rule, the introduction of stability enhances cooperative behaviors and average payoffs. Moreover, cooperators in population own much higher and more stable payoffs than defectors when considering long memories. Compared with the memory-based rule which considers the difference of average payoffs, the stability-based rule shows these superiorities on cooperators when the payoff parameters are relatively large, which suggests that the proposed rule exhibits its dominant position when the fluctuation of payoffs is obvious. According to this conclusion, we obtain the application scope enhancing cooperators of memory-based and stability-based rules, which maybe helpful in understanding the influence of these two rules on cooperation under different circumstances. [ABSTRACT FROM AUTHOR]

• An asymmetric game model with the so-called carrot-stick strategy is established. • Asymmetric strategy setup can lower the dilemma of cooperation: the dominant players and recipient players might coexist through cycle dynamics. • Three basic evolutionary dynamics of systems: dominance; bistability; and coexistence, can be observed as each case is closely related with the cost-to-benefit, punishment-to-benefit, and reward-to-benefit. Explaining the evolution of cooperation represents one of the greatest challenges in both evolutionary biology and social science. However, asymmetrical interaction, as one typical characteristic of cooperative system, has not been sufficiently considered in the existing literature for the evolution of cooperation. Incorporating the asymmetry in the strategy sets, we here construct an asymmetric game model with the so-called carrot-stick strategy, which is a mixed strategy of reward and punishment. Based on mathematical analyses, it is unveiled that this asymmetric interaction can lower the dilemma of cooperation: the dominant players and recipient players might coexist through cycle frequency. Further analysis shows that the multi-equilibria are possible, which depend on the payoff parameters and the initial conditions. That is to say, by setting up different values of the cost-to-benefit, the punishment-to-benefit, and the reward-to-benefit, we can recover three basic evolutionary dynamics of systems: dominance; bistability and coexistence. These theoretical observations are consistent with existing empirical outcomes that asymmetric sanction or reward of host species might solve the conflicts between the actors in the fig-fig wasp mutualism or in the cleaner fish-client mutualism. It is thus suggested that this asymmetric strategy setup may shed new light into the solution of social dilemmas. [ABSTRACT FROM AUTHOR]

• The rate of failure for collective cooperation increases for larger groups, along with the arising of bystander effect and a decrease in average contributions. • Our experimental observations suggest that bystanders contribute less if others give less contributions. • More players choose to be bystanders who strategically contribute less when the collective goal is unlikely to achieve, intriguing the failure of collective cooperation. Collective cooperation is essential to human society, and it exists in many social dilemmas. In the scenario of a collective-risk social dilemma, a group of players have to collectively contribute to a public fund to prevent the tragedy of the commons, such as dangerous climate change, because everybody will lose all their remaining money when the damage happens with a certain probability if the group fails to reach a fixed fundraising target. Yet, it remains largely unclear how the group size affects the probability of reaching the collective target and the mechanism that drives different outcomes of the collective cooperation. Here, we contribute to the literature by exploring the role of group size in the collective-risk social dilemma and the potential underlying mechanism using both model simulations and human experiments. Through simulations we found that the rate of failure for collective cooperation increases for larger groups, along with the arising of bystander effect and a decrease in average contributions, which are confirmed by our experimental observations. We further analyze the patterns of investment behaviors in the experiment setting by categorizing players into cooperators, altruists, and free riders using both a clustering method and a golden standard. We found that altruists who tend to contribute more, rather than cooperators who prefer contributing a fair-share investment, play a crucial role in groups with success outcome in early and/or middle stages of the game. Our results indicate that bystanders are dynamic and their amount depends on the contribution of others. When others contribute less, bystanders also contribute less. If the collective goal is unlikely to achieve, more players choose to be bystanders who strategically contribute less, intriguing the failure of the collective goal. Our findings suggest a potentially effective way to solve the collective-risk social dilemma by reducing the bystander effect through the mechanism design of forming small groups. [ABSTRACT FROM AUTHOR]

We investigate the repeated prisoner's dilemma game where both players alternately use reinforcement learning to obtain their optimal memory-one strategies. We theoretically solve the simultaneous Bellman optimality equations of reinforcement learning. We find that the Win-stay Lose-shift strategy, the Grim strategy, and the strategy which always defects can form symmetric equilibrium of the mutual reinforcement learning process amongst all deterministic memory-one strategies. [ABSTRACT FROM AUTHOR]

In this paper we consider the Pure and Impure Prisoner's Dilemmas. Our purpose is to theoretically extend them when using non-Archimedean quantities and to work with them numerically, potentially on a computer. The recently introduced Sergeyev's Grossone Methodology proved to be effective in addressing our problem, because it is both a simple yet effective way to model non-Archimedean quantities and a framework which allows one to perform numerical computations between them. In addition, we could be able, in the future, to perform the same computations in hardware, resorting to the infinity computer patented by Sergeyev himself. After creating the theoretical model for Pure and Impure Prisoner's Dilemmas using Grossone Methodology, we have numerically reproduced the diagrams associated to our two new models, using a Matlab simulator of the Infinity Computer. Finally, we have proved some theoretical properties of the simulated diagrams. Our tool is thus ready to assist the modeler in all that problems for which a non-Archimedean Pure/Impure Prisoner's Dilemma model provides a good description of reality: energy market modeling, international trades modeling, political merging processes, etc. [ABSTRACT FROM AUTHOR]

• We study the emergence of cooperation in spatial social dilemmas with expulsion. • We show that expulsion can facilitate the emergence of cooperation regardless of dilemma type. • We analyze the expansion conditions of critical cooperative clusters. • We determine key changes in the spatial dynamics that ensure promotion of cooperation. In social dilemmas cooperators pay costs for other individuals to receive benefits, whereas defectors are the benefactors but do not provide anything in return. What is more, except in the snowdrift game, defective strategies are evolutionarily stable against invasion by cooperative strategies under strong selection. It is thus of interest to determine under which conditions the spontaneous emergence of cooperation is possible in a population consisting entirely of defectors. To that effect, we here want to concentrate on exploring how expulsion influences the emergent dynamics of cooperation in the context of the spatial social dilemmas. Interestingly, expulsion can indeed support the emergence of cooperators for all classes of social dilemma games. By analyzing the spatial expansion and invasion processes of cooperative individuals, we further reveal that merely a 1 × 2 (or 2 × 1) cluster of expulsive cooperators, introduced for example by means of mutations, is required for invading and completely dominating a spatial population of expulsive defectors provided that the cost of cooperation is no larger than a constant value. [ABSTRACT FROM AUTHOR]

Punishment and reward are effective ways to sustain cooperation among selfish groups. Researches show that the ultimate result of selflessness is self-interest. This paper considers that individuals take defensive measures to reduce losses in the face of unknown risks. Defensive cooperators need to pay the defensive cost. When they interact with unconditional defectors, the defensive cooperators gain benefit due to defensive behavior, while the unconditional defectors lose some benefit due to the defense of defensive cooperators. Results show that in the well-mixed population, reducing defensive cost and improving defensive benefit can promote cooperation, and maintain a higher cooperation level unchanged within a larger range of temptation. In the structured population, the three strategies perform the phenomenon of cyclic dominance, that is unconditional cooperators dominate defensive cooperators who dominate unconditional defectors who dominate unconditional cooperators. It is worth noting that both in the well-mixed population and structured population, three strategies have obvious phased advantages with regard to defensive benefit. Lower defensive benefit leads to an advantage for unconditional defectors. If defensive benefit is moderate, the weakened competitiveness of unconditional defectors leads defensive cooperators dominate. And excessive defensive benefit results in the second-order free-rider phenomenon of the unconditional cooperation strategy. In addition, structured population do not always support cooperation better than well-mixed population. When the defensive cost is small and defensive benefit is high, the spatial effect can enhance network reciprocity. More generally, our results prove that defensive self-interested behavior leads to the maintenance of cooperation, which provides a new perspective for understanding the cooperation evolutionary. [ABSTRACT FROM AUTHOR]

• Climate change dilemma is formulated as a public goods game with environmental feedback. • A dynamic model based on Particle Swarm Optimization algorithm is proposed with restricted exchange of information. • Theoretical analysis and simulation results both show that environmental feedback promotes cooperation. • Continuous contribution is more conducive to cooperation. • Population composed of individuals that make decisions replying more on historical information than social information can promote cooperation. This paper formulates the climate change dilemma as an adaption of public goods game. The Nash equilibrium of the climate change dilemma is analyzed in the cases of discrete contribution and continuous contribution. Analytic results show that environmental feedback promotes cooperation to a certain extent, but as the number of players increases, zero contribution becomes the only Nash equilibrium in most cases. A dynamic model based on Particle Swarm Optimization algorithm is then proposed for the climate change dilemma, where the information exchange is restricted by a network. Simulation results show that the proposed dynamic model can effectively promote cooperation, especially in the case of continuous contribution, and making decisions based on historical information can largely boost the average contribution. [ABSTRACT FROM AUTHOR]

• A fascinating spiral pattern is spontaneously formed, adding a new population always brings additional prey to cooperators in other populations, leading to competing spatial dynamics and pattern formation. • The inherent cyclic dominance of the strategies results in the self-organization of populations on the square lattice and ultimately effectively promotes cooperation. • The cooperation facilitation mechanism depends on the spontaneous emergence of the dynamically accelerating invasion processes. Through these processes, unlike defectors, significant evolutionary advantages are created for cooperators. The coexistence of different populations has recently been confirmed to be a simple yet effective mechanism to understand the stability of cooperation. Based on this intuition, we use the voluntary prisoner's dilemma (VPD) game as the mathematical model to explore the spatiotemporal dynamics of cooperation among different populations. By using parameter α as a key quantity that takes into account the strength of connections between populations, we observe that a fascinating spiral pattern is spontaneously formed, adding a new population always brings additional prey to cooperators in other populations, leading to competing spatial dynamics and pattern formation. Moreover, the system gradually changes from the C + D + L state to the C + D state and finally to the full C state. The inherent cyclic dominance of the strategies results in the self-organization of populations on the square lattice and ultimately effectively promotes cooperation. Our work emphasizes that the complexity of the evolutionary dynamics of structural populations is significantly increased by the simultaneous existence of different populations. [ABSTRACT FROM AUTHOR]

• The pure prisoner's dilemma games is adopted, where parameters are set as: T=b, R=1, P=0 and S=-0.1. • The alienation from defectors reinforces larger and more compact cooperative clusters. The number of cooperative clusters depend on the viscosity of network, where direct reciprocity works well. • The concepts of END (the enduring time) and EXP (the expanding time) are used to analyze the dynamic evolution of cooperative clusters. It is shown that large cooperative clusters play key role in shifting from END episode to EXP episode. • The parameters Dg and Dr are employed to depict the strength of the dilemmas. Thus, we verified what dilemmas are, where the mechanism could still enhance cooperation effectively. Individuals often move to distance themselves from defectors, or to seek better chances for higher payoffs, for example moving from rural to urban areas. Regardless of the reason, however, moving frequently also means alienation, which in turn means bearing costs for seeking new opportunities. With this motivation, we study a prisoner's dilemma game, where individuals with defectors in their communities either move or update their strategy. We find that the alienation from defectors reinforces larger and more compact cooperative clusters. However, the number of cooperative clusters depends on the viscosity of the interaction network, where network reciprocity still works well. And it is the fine-tuned interplay between the mobility to alienate from defectors and a still functioning network reciprocity that works best in promoting cooperation. Our results suggest that a limited mobility of minorities could spare public resources in social dilemma situations more effectively than reward and punishment. [ABSTRACT FROM AUTHOR]

• We have built a new spatial prisoner's dilemma model that enhances network reciprocity. • The key idea is the introduction of a third strategy: "fence-sitters" who cooperate and defect with equal probability. • Situations with larger Stag-Hunt-type and smaller Chicken-type dilemmas are specifically relaxed. We introduce a new spatial prisoner's dilemma (SPD) model in which the so-called network reciprocity is enhanced more than in the conventional model. In addition to the usual binary strategies—perfect cooperator and perfect defector—we introduce "fence-sitters", who either cooperate or defect with equal probability, as a third strategy. In regions with larger Stag-Hunt-type dilemmas but smaller Chicken-type dilemmas, the existence of the fence-sitters works as a buffer, hampering the exploitation of cooperators by neighboring defectors and significantly enhancing the cooperative phase. Conversely, in regions with larger Chicken-type dilemmas but smaller Stag-Hunt-type dilemmas, the existence of the fence-sitters expands the critical dilemma strength for the survival of cooperation, but it reduces the cooperating fraction more than in the conventional model. What we have found may justify the conclusion that the existence of people with neutral opinions—often regarded betwixt and between—absorbs the severe competition between two extreme groups, bringing a more accommodationist situation to our society, backed with greater cooperation. [ABSTRACT FROM AUTHOR]

• Prisoner's dilemma game played on two-layer dynamic multiplex networks is examined. • Link updating induces symmetry breaking in cooperation levels across two layers. • Broken symmetry disappears with sufficiently fast link updating. • Effects of link updating are more nuanced considering network multiplexity. Understanding the role of network structure in the evolution of cooperation is a key research goal at the intersection between physics and biology. Recent studies have particularly focused on multiplex networks given that multiple social domains are interrelated and cannot be represented by single-layer networks. However, the role of network multiplexity is not fully understood when combined with another important network characteristic: network dynamics. In the present study, we investigated evolutionary prisoner's dilemma games played on dynamic two-layer multiplex networks in which the payoff combined across the two layers determined strategy evolution. In addition, we introduced network dynamics where agents can sever links with defecting neighbors and construct new links. Our simulation showed that link updating enhances cooperation but the resultant states are far from those of full cooperation. This modest enhancement in cooperation was related to symmetry breaking whereby the cooperation frequency in one layer disproportionately increased while that in the other layer remained the same or even diminished. However, this broken symmetry disappeared with sufficiently fast link updating. Our results show that the introduction of network dynamics enhances cooperation in the prisoner's dilemma as previously reported, but this enhancement is accompanied by significant asymmetry once network multiplexity is considered. [ABSTRACT FROM AUTHOR]

• Two new reputation forming mechanisms using successive behaviors are proposed to encourage successive cooperation. • When encouraging continuous cooperation, the system is not sensitive to reputation's adjustment. • An ideal society where everyone chooses to cooperate can be achieved without total abandonment of personal interests. • Which mechanism is better under different parameters is shown by the heat-map. We propose a cooperative evolution model in which successive behaviors influence reputation. There are two mechanisms to form reputation. One is enhancing the reputation of players who cooperate insistently, and reducing it when they defect continuously. The other is increasing reputation when defection is transformed into cooperation, and decreasing when cooperation is turned into defection. These two reputation formation mechanisms are simulated on the weak prisoner's dilemma game. We use a weighted average Fermi equation to describe the probability of players learning the strategy of a randomly selected neighbor. The results show that, under these two mechanisms, both the reward for reputation and emphasis on it have an impact on the cooperation. We find that encouraging "C after C" is better for cooperation than encouraging "C after D", when people attach importance to reputation to a certain extent. Before reaching that threshold, however, encouraging "C after D" is better when reputation reward is large. [ABSTRACT FROM AUTHOR]

Choosing a suitable restricted strategic space is crucial for the novel features of quantum games. Based on the Eisert–Wilkens–Lewenstein quantization scheme, a quantum prisoner's dilemma is introduced using a simple restricted one-parameter strategic space. Our scheme is found to preserve the quantum advantages of the Eisert–Wilkens–Lewenstein scheme, and improve its Nash equilibrium characteristics. The simplicity of the calculations enables us to address some important aspects of quantum games which can be applied to many real systems. The robustness of Nash equilibria to corruption in the initial state is studied, and a measure of robustness is defined. The robustness of Nash equilibria is found to scale inversely with the entanglement. Our scheme is applied to various prisoner's dilemma games having different dilemma strength. Also, Some variant forms of restricted one-parameter strategic space are studied. [ABSTRACT FROM AUTHOR]

We present a computational model of bidirectional choice based on the length of information of previous payoff to study the cooperative dynamics of iterated prisoner's dilemma game on directed weighted lattices. For paired two individuals in network, they can attend game under the condition that they both successfully choose the other party with the intensity of unidirectional interaction, and the changing of the intensity of unidirectional interaction according to the relation between individual's current payoff and its previous payoff which are both gained from the other party. The simulation results show that this evolutionary rule can availably remove the stumbling blocks of cooperators for existence and hugely boost the cooperation in population. Interestingly, for a smaller or moderate cost-to-benefit ratio (0 < r < 0.235), the cooperation level ascends monotonically as the length of information increases; however, for a bigger cost-to-benefit ratio (0.235 ≤ r < 0.249), there are some moderate values of length of information, resulting in the status of full cooperation. Surprisingly, we find that there is an optimal length of information which not just leads to the ultimately full cooperation in population but benefits the learning of cooperation strategy for individual, regardless of the value of cost-to-benefit ratio (0 < r < 0.249). Our results may contribute to the understanding and the analysis of cooperative dynamics of bidirectional choice in network. [ABSTRACT FROM AUTHOR]