Spatial Games Based on Pursuing the Highest Average Payoff

We propose a strategy updating mechanism based on pursuing the highest average payoff to investigate the prisoner＇s dilemma game and the snowdrift game. We apply the new rule to investigate cooperative behaviours on regular, small-world, scale-free networks, and find spatial structure can maintain cooperation for the prisoner＇s dilemma game. fn the snowdrift game, spatial structure can inhibit or promote cooperative behaviour which depends on payoff parameter. We further study cooperative behaviour on scale-free network in detail. Interestingly, non-monotonous behaviours observed on scale-free network with middle-degree individuals have the lowest cooperation level. We also find that large-degree individuals change their strategies more frequently for both games.     

Cooperation in N-person evolutionary snowdrift game in scale-free Barabási-Albert networks

Cooperation in the N-person evolutionary snowdrift game (NESG) is studied in scale-free Barabási-Albert (BA) networks. Due to the inhomogeneity of the network, two versions of NESG are proposed and studied. In a model where the size of the competing group varies from agent to agent, the fraction of cooperators drops as a function of the payoff parameter. The networking effect is studied via the fraction of cooperative agents for nodes with a particular degree. For small payoff parameters, it is found that the small-k agents are dominantly cooperators, while large-k agents are of non-cooperators. Studying the spatial correlation reveals that cooperative agents will avoid to be nearest neighbors and the correlation disappears beyond the next-nearest neighbors. The behavior can be explained in terms of the networking effect and payoffs. In another model with a fixed size of competing groups, the fraction of cooperators could show a non-monotonic behavior in the regime of small payoff parameters. This non-trivial behavior is found to be a combined effect of the many agents with the smallest degree in the BA network and the increasing fraction of cooperators among these agents with the payoff for small payoffs.     

Prisoner's Dilemma Game on Two Types of Positively Correlated Networks

We study the effects of degree correlations on the evolution of cooperation in the prisoner＇s dilemma game with individuals located on two types of positively correlated networks. It is shown that the positive degree correlation can either promote or inhibit the emergence of cooperation depending on network configurations. Furthermore, we investigate the probability to cooperate as a function of connectivity degree, and find that high-degree individuals generally have a higher tendency to cooperate. Finally, it is found that small-degree individuals usually change their strategy more frequently, and such change is shown to be unfavourable to cooperation for both kinds of networks.     

Other-regarding preference and the evolutionary prisoner’s dilemma on complex networks

Prevailing models of the evolutionary prisoner’s game on networks always assume that agents are pursuing their own profit maximization. But the results from experimental games show that many agents have other-regarding preference. In this paper, we study the emergence of cooperation from the prisoner’s dilemma game on complex networks while some agents exhibit other-regarding preference such as inequality aversion, envious and guilty emotions. Contrary to common ideas, the simulation results show that the existence of inequality aversion agents does not promote cooperation emergence on a BA (Barabási and Albert) scale-free network in most situations. If the defection attraction is big and agents exhibit strong preference for inequality aversion, the frequency of cooperators will be lower than in situations where no inequality aversion agents exist. In some cases, the existence of the inequality agents will even induce the frequency of cooperators to zero, a feature which is not observed in previous research on the prisoner’s dilemma game when the underlying interaction topology is a BA scale-free network. This means that if an agent cares about equality too much, it will be difficult for cooperation to emerge and the frequency of cooperators will be low on BA networks. The research on the effect of envy or guilty emotions on the emergence of cooperation in the prisoner’s dilemma game on BA networks obtains similar results, though some differences exist. However, simulation results on a WS (Watts and Strogatz) small-world network display another scenario. If agents care about the inequality of agents very much, the WS network favors cooperation emergence in the prisoners’ dilemma game when other-regarding agents exist. If the agent weight on other-regarding is lowered, the cooperation frequencies emerging on a WS network are not much different from those in situations without other-regarding agents, although the frequency of cooperators is lower than those of the situation without other-regarding preference agents sometimes. All the simulation results imply that inequality aversion and its variations can have important effects on cooperation emergence in the prisoner’s dilemma game, and different network topologies have different effects on cooperation emergence in the prisoner’s dilemma game played on complex networks.     

Cooperation and community structure in social networks

Situations of conflict giving rise to social dilemmas are widespread in society. One way of studying these important phenomena is by using simplified models of individual behavior under conflicting situations such as evolutionary game theory. Starting from the observation that individuals interact through networks of acquaintances, we study the evolution of cooperation on model and real social networks through well known paradigmatic games. Using a new payoff scheme which leaves replicator dynamics invariant, we find that cooperation is sustainable in such networks, even in the difficult case of the prisoner’s dilemma. The evolution and stability of cooperation implies the condensation of game strategies into the existing community structures of the social network in which clusters of cooperators survive thanks to their higher connectivity towards other fellow cooperators.     

Evolutionary Prisoner's Dilemma on heterogeneous Newman-Watts small-world network

We focus on the heterogeneity of social networks and its role to the emergence of prevailing cooperators and sustainable cooperation. The social networks are representative of the interaction relationships between players and their encounters in each round of games. We study an evolutionary Prisoner's Dilemma game on a variant of Newman-Watts small-world network, whose heterogeneity can be tuned by a parameter. It is found that optimal cooperation level exists at some intermediate topological heterogeneity for different temptations to defect. That is, frequency of cooperators peaks at a certain specific value of degree heterogeneity — neither the most heterogeneous case nor the most homogeneous one would favor the cooperators. Besides, the average degree of networks and the adopted update rule also affect the cooperation level.     

Coevolution of game and network structure with adjustable linking

Most papers about the evolutionary game on graph assume the statistic network structure. However, in the real world, social interaction could change the relationship among people. And the change of social structure will also affect people’s strategies. We build a coevolution model of prisoner’s dilemma game and network structure to study the dynamic interaction in the real world. Differing from other coevolution models, players rewire their network connections according to the density of cooperation and other players’ payoffs. We use a parameter α to control the effect of payoff in the process of rewiring. Based on the asynchronous update rule and Monte Carlo simulation, we find that, when players prefer to rewire their links to those who are richer, the temptation can increase the cooperation density.     

Evolutionary ultimatum game on complex networks under incomplete information

This paper studies the evolutionary ultimatum game on networks when agents have incomplete information about the strategies of their neighborhood agents. Our model assumes that agents may initially display low fairness behavior, and therefore, may have to learn and develop their own strategies in this unknown environment. The Genetic Algorithm Learning Classifier System (GALCS) is used in the model as the agent strategy learning rule. Aside from the Watts-Strogatz (WS) small-world network and its variations, the present paper also extends the spatial ultimatum game to the Barabási-Albert (BA) scale-free network. Simulation results show that the fairness level achieved is lower than in situations where agents have complete information about other agents’ strategies. The research results display that fairness behavior will always emerge regardless of the distribution of the initial strategies. If the strategies are randomly distributed on the network, then the long-term agent fairness levels achieved are very close given unchanged learning parameters. Neighborhood size also has little effect on the fairness level attained. The simulation results also imply that WS small-world and BA scale-free networks have different effects on the spatial ultimatum game. In ultimatum game on networks with incomplete information, the WS small-world network and its variations favor the emergence of fairness behavior slightly more than the BA network where agents are heterogeneously structured.     

Cooperation and charity in spatial public goods game under different strategy update rules

Human cooperation can be influenced by other human behaviors and recent years have witnessed the flourishing of studying the coevolution of cooperation and punishment, yet the common behavior of charity is seldom considered in game-theoretical models. In this article, we investigate the coevolution of altruistic cooperation and egalitarian charity in spatial public goods game, by considering charity as the behavior of reducing inter-individual payoff differences. Our model is that, in each generation of the evolution, individuals play games first and accumulate payoff benefits, and then each egalitarian makes a charity donation by payoff transfer in its neighborhood. To study the individual-level evolutionary dynamics, we adopt different strategy update rules and investigate their effects on charity and cooperation. These rules can be classified into two global rules: random selection rule in which individuals randomly update strategies, and threshold selection rule where only those with payoffs below a threshold update strategies. Simulation results show that random selection enhances the cooperation level, while threshold selection lowers the threshold of the multiplication factor to maintain cooperation. When charity is considered, it is incapable in promoting cooperation under random selection, whereas it promotes cooperation under threshold selection. Interestingly, the evolution of charity strongly depends on the dispersion of payoff acquisitions of the population, which agrees with previous results. Our work may shed light on understanding human egalitarianism.     

Prisoner's Dilemma Game on Clustered Scale-Free Networks under Different Initial Distributions

The evolutionary prisoner＇s dilemma game is investigated under different initial distributions for cooperators and defectors on scale-free networks with a tunable clustering coefficient. It is found that, on the one hand, cooperation can be enhanced with the increasing clustering coefficient when only the most connected nodes are occupied by cooperators initially. On the other hand, if cooperators just occupy the lowest-degree nodes at the beginning, then the higher the value of the clustering coefficient, the more unfavorable the environment for cooperators to survive for the increment of temptation to defect. Thereafter, we analytically argue these nontrivial phenomena by calculating the cooperation probability of the nodes with different degrees in the steady state, and obtain the critical values of initial frequency of cooperators below which cooperators would vanish finally for the two initial distributions.     

Pattern formation,social forces,and diffusion instability in games with success-driven motion

A local agglomeration of cooperators can support the survival or spreading of cooperation, even when cooperation is predicted to die out according to the replicator equation, which is often used in evolutionary game theory to study the spreading and disappearance of strategies. In this paper, it is shown that success-driven motion can trigger such local agglomeration and may, therefore, be used to supplement other mechanisms supporting cooperation, like reputation or punishment. Success-driven motion is formulated here as a function of the game-theoretical payoffs. It can change the outcome and dynamics of spatial games dramatically, in particular as it causes attractive or repulsive interaction forces. These forces act when the spatial distributions of strategies are inhomogeneous. However, even when starting with homogeneous initial conditions, small perturbations can trigger large inhomogeneities by a pattern-formation instability, when certain conditions are fulfilled. Here, these instability conditions are studied for the prisoner’s dilemma and the snowdrift game. Furthermore, it is demonstrated that asymmetrical diffusion can drive social, economic, and biological systems into the unstable regime, if these would be stable without diffusion.     

Selective investment promotes cooperation in public goods game

Most previous investigations on spatial Public Goods Game assume that individuals treat neighbors equivalently, which is in sharp contrast with realistic situations, where bias is ubiquitous. We construct a model to study how a selective investment mechanism affects the evolution of cooperation. Cooperators selectively contribute to just a fraction among their neighbors. According to the interaction result, the investment network can be adapted. On selecting investees, three patterns are considered. In the random pattern, cooperators choose their investees among the neighbors equiprobably. In the social-preference pattern, cooperators tend to invest to individuals possessing large social ties. In the wealth-preference pattern, cooperators are more likely to invest to neighbors with higher payoffs. Our result shows robustness of selective investment mechanism that boosts emergence and maintenance of cooperation. Cooperation is more or less hampered under the latter two patterns, and we prove the anti-social-preference or anti-wealth-preference pattern of selecting investees can accelerate cooperation to some extent. Furthermore, the theoretical analysis of our mechanism on double-star networks coincides with simulation results. We hope our finding could shed light on better understanding of the emergence of cooperation among adaptive populations.     

Mechanisms for similarity based cooperation

Cooperation based on similarity has been discussed since Richard Dawkins introduced the term “green beard” effect. In these models, individuals cooperate based on an aribtrary signal (or tag) such as the famous green beard. Here, two different models for such tag based cooperation are analysed. As neutral drift is important in both models, a finite population framework is applied. The first model, which we term “cooperative tags” considers a situation in which groups of cooperators are formed by some joint signal. Defectors adopting the signal and exploiting the group can lead to a breakdown of cooperation. In this case, conditions are derived under which the average abundance of the more cooperative strategy exceeds 50%. The second model considers a situation in which individuals start defecting towards others that are not similar to them. This situation is termed “defective tags”. It is shown that in this case, individuals using tags to cooperate exclusively with their own kind dominate over unconditional cooperators.     

Promoting cooperation in social dilemmas via simple coevolutionary rules

We study the evolution of cooperation in structured populations within popular models of social dilemmas, whereby simple coevolutionary rules are introduced that may enhance players abilities to enforce their strategy on the opponent. Coevolution thus here refers to an evolutionary process affecting the teaching activity of players that accompanies the evolution of their strategies. Particularly, we increase the teaching activity of a player after it has successfully reproduced, yet we do so depending on the disseminated strategy. We separately consider coevolution affecting either only the cooperators or only the defectors, and show that both options promote cooperation irrespective of the applied game. Opposite to intuitive reasoning, however, we reveal that the coevolutionary promotion of players spreading defection is, in the long run, more beneficial for cooperation than the likewise promotion of cooperators. We explain the contradictive impact of the two considered coevolutionary rules by examining the differences between resulting heterogeneities that segregate participating players, and furthermore, demonstrate that the influential individuals completely determine the final outcome of the games. Our findings are immune to changes defining the type of considered social dilemmas and highlight that the heterogeneity of players, resulting in a positive feedback mechanism, is a fundamental property promoting cooperation in groups of selfish individuals.     

Evolution of cooperation in well-mixed N-person snowdrift games

We study the time evolution of cooperation in a recently proposed N-person evolutionary snowdrift game, by focusing on the details of the evolutionary dynamics. It is found that the analytic solution for the equilibrium fraction of cooperators as given previously by the replicator dynamics stems from a balance between the terms: the cost to contribute to a common task and the risk in refusing to participate in a common task. Analytic expressions for these two terms are given, and their magnitudes are studied over the whole range of parameter space. Away from equilibrium, it is the imbalance between these terms that drives the system to equilibrium. A continuous time first-order differential equation for the degree of cooperation is derived, for arbitrary interacting group size N and cost-to-benefit ratio. Analytic solutions to the time evolution of cooperation for the cases of N=2 and N=3 are obtained, with results in good agreement with those obtained by numerical simulations. For arbitrary N, numerical solutions to the equation give the time evolution of cooperation, with the long time limit giving the equilibrium fraction of cooperators.     

Does the Compelled Cooperation Determine the Structure of a Complex Network？

Cooperation among individuals is considered to play an important role in the evolution of complex networked systems in physical, biological, economical and even epidemiological worlds, but its effects on the development of the systems is not so clear. We consider a specific kind of primal cooperation in a group of individuals, i.e., an individual never cooperates with others except when compelled to do so. The lowest level of compelled cooperation, in which cooperators share no message or resources, is investigated in the background of complex networks driven by the simple game rock-paper-scissors. Simulation results show that with the evolution of the systems, the cooperation will spread all over the networks, and finally results in systems with modular structures and a scale-free property.     

To cooperate or to defect? Altruism and reputation

The basic difficulty in cooperation theory is to justify the cooperation. Here we propose a new approach, where players are driven by their altruism to cooperate or not. The probability of cooperation depends also on the co-player’s reputation. We find that players with positive altruism cooperate and meet cooperation. In this approach, payoffs are not relevant.     

Promotion of cooperation induced by the interplay between structure and game dynamics

We consider the coupled dynamics of the adaption of network structure and the evolution of strategies played by individuals occupying the network vertices. We propose a computational model in which each agent plays a n-round Prisoner's Dilemma game with its immediate neighbors, after that, based upon self-interest, partial individuals may punish their defective neighbors by dismissing the social tie to the one who defects the most times, meanwhile seek for a new partner at random from the neighbors of the punished agent. It is found that the promotion of cooperation is attributed to the entangled evolution of individual strategy and network structure. Moreover, we show that the emerging social networks exhibit high heterogeneity and disassortative mixing pattern. For a given average connectivity of the population and the number of rounds, there is a critical value for the fraction of individuals adapting their social interactions, above which cooperators wipe out defectors. Besides, the effects of the average degree, the number of rounds, and the intensity of selection are investigated by extensive numerical simulations. Our results to some extent reflect the underlying mechanism promoting cooperation.     

Randomness Effect on Cooperation in Memory-Based Snowdrift Game

A memory-based snowdrift game (MBSG) on spatial small-world networks is investigated. It is found that cooperation rate versus temptation shows some step structures on small-world networks, similar to the case on regular lattices. With the increment of rewiring probability based on four-neighbourregular lattices, more steps are observable. Interestingly, it is observed that cooperation rate peaks at a specific value of temptation, which indicates that properly encouraging selfish actions may lead to better cooperative behaviours in the MBSG on small-world networks. Memory effects are also discussed for different rewiring probabilities. Furthermore, optimal regions arefound in the parameter planes. The strategy-related average degrees of individuals are helpful to understand the obtained results.     

Influence of different initial distributions on robust cooperation in scale-free networks: A comparative study

We study the evolutionary Prisoner's dilemma game on scale-free networks, focusing on the influence of different initial distributions for cooperators and defectors on the evolution of cooperation. To address this issue, we consider three types of initial distributions for defectors: uniform distribution at random, occupying the most connected nodes, and occupying the lowest-degree nodes, respectively. It is shown that initial configurations for defectors can crucially influence the cooperation level and the evolution speed of cooperation. Interestingly, the situation where defectors initially occupy the lowest-degree vertices can exhibit the most robust cooperation, compared with two other distributions. That is, the cooperation level is least affected by the initial percentage of defectors. Moreover, in this situation, the whole system evolves fastest to the prevalent cooperation. Besides, we obtain the critical values of initial frequency of defectors above which the extinction of cooperators occurs for the respective initial distributions. Our results might be helpful in explaining the maintenance of high cooperation in scale-free networks.