Effects of average degree on cooperation in networked evolutionary game

We study effects of average degree on cooperation in the networked prisoner's dilemma game. Typical structures are considered, including random networks, small-world networks and scale-free networks. Simulation results show that the average degree plays a universal role in cooperation occurring on all these networks, that is the density of cooperators peaks at some specific values of the average degree. Moreover, we investigated the average payoff of players through numerical simulations together with theoretical predictions and found that simulation results agree with the predictions. Our work may be helpful in understanding network effects on the evolutionary games.     

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.     

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.     

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.     

Effect of community structure on coevolutionary dynamics with dynamical linking

We investigate the effect of community structure on the evolution of cooperation in the prisoner’s dilemma and the snowdrift game with dynamical linking. We show both analytically and numerically that cooperators are generally more favorable on community networks than on networks without community structure, and in particular, there exists an optimal intermediate value of the model parameter leading to the easiest fixation of cooperators. We show that our results are robust with respect to the initial number of cooperators and are valid for a wide range of the ratio of time scales associated with linking and strategy dynamics. Since community structure is ubiquitous in real social networks, our results may provide new insights into the evolution of cooperation in real world.     

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.     

Diversity of contribution promotes cooperation in public goods games

In most previous studies of public goods game, individuals conventionally donate their contributions equally to the games they participate in. We develop an extended public goods game model, in which individuals distribute their contributions based on the groups’ qualities. Namely, the individuals are allowed to increase their investment to the superior groups at the expense of the nasty ones. The quality of a group is positively correlated with its cooperation level. In numerical simulations, synchronized stochastic strategy updating rule based on pairwise comparison for a fixed noise level is adopted. The results show that the high-quality group preference mechanism can greatly improve cooperation, compared with conventional models. Besides, the system with stronger preference toward high-quality groups performs better. Investigation of wealth distribution at equilibrium reveals that cooperators’ wealth appreciates with the increase of preference degree when cooperators take up the same fraction of the population.     

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.     

Role of recommendation in spatial public goods games

We study the role of recommendation in a co-evolutionary public goods game in which groups can recommend their members for establishment of new relationships with individuals outside the current group according to group quality. Intriguingly, for square lattices and ER graphs there exists optimal group quality for recommendation that induces positive feedback between cooperation and recommendation. Snapshots of spatial patterns of cooperators, defectors, recommended cooperators and recommended defectors show that if group quality is appropriate for recommendation, cooperation and recommendation can simultaneously emerge. Moreover, we find that local recommendation improves cooperation more than global recommendation. As an extension, we also present results for Barabási–Albert networks. The positive effect of recommendation on cooperation for Barabási–Albert networks is independent of group quality. Our results provide an insight into the evolution of cooperation in real social systems.     

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.     

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.     

The roles of heterogeneous investment mechanism in the public goods game on scale-free networks

The public goods game is an important theoretical model for investigating the emergence of cooperation in the multi-player social dilemma. It has been proven that scale-free networks can significantly promote cooperation, but fail to sustain cooperation when the player obtains the normalized payoff. In this paper, we introduce heterogeneous investment mechanism into the public goods game on scale-free networks, and study the evolution of cooperation in both cases of accumulated and normalized payoff. Our research reveals that the heterogeneous investment mechanism can obviously facilitate cooperation as the adjusted parameter α increases. The increase of α allows cooperators to emerge under lower values of r. In the case of accumulated payoff, cooperators always firmly occupy the hubs, and the population keeps high cooperation level. In the case of normalized payoff, the increase of α changes the situation that the hubs are easily invaded by defectors, and inhibits the spread of defectors.     

Diversity of reproduction rate supports cooperation in the prisoner's dilemma game on complex networks

In human societies the probability of strategy adoption from a given person may be affected by the personal features. Now we investigate how an artificially imposed restricted ability to reproduce, overruling ones fitness, affects an evolutionary process. For this purpose we employ the evolutionary prisoner's dilemma game on different complex graphs. Reproduction restrictions can have a facilitative effect on the evolution of cooperation that sets in irrespective of particularities of the interaction network. Indeed, an appropriate fraction of less fertile individuals may lead to full supremacy of cooperators where otherwise defection would be widespread. By studying cooperation levels within the group of individuals having full reproduction capabilities, we reveal that the recent mechanism for the promotion of cooperation is conceptually similar to the one reported previously for scale-free networks. Our results suggest that the diversity in the reproduction capability, related to inherently different attitudes of individuals, can enforce the emergence of cooperative behavior among selfish competitors.     

Effect of habitat destruction on cooperation in public goods games

We explore the evolution of cooperation in a public goods game where the habitat destruction is taken into account. A model of ordinary differential equations is proposed, in which incorporate the habitat destroyed. And we focus on the impact of different levels of habitat destruction on cooperation. Our results show that the fraction of cooperation receives the biggest boost for moderate habitat destruction, and decreases to zero very quickly for a higher degree of habitat destruction. Similarly, our results suggest that low or moderate habitat degradation can promote the evolution of cooperators, and once habitat degradation is too severe, the fraction of cooperators will decline dramatically. Furthermore, we investigate the cooperation level with different multiplication factor, and results demonstrate that the cooperation increases monotonously with the increasing of the multiplication factor. Our findings may provide many more insights for understanding the emergence and maintenance of cooperation in the complex biological and social systems.     

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.     

Individual’s strategy characterized by local topology conditions in prisoner’s dilemma on scale-free networks

We introduce a “gradient” to find out the defectors, and further a “topology potential” to characterize the individual’s strategy preference in the prisoner’s dilemma on scale-free networks. It is shown that the cooperators typically locate on the nodes with high topology potential and the defectors are mainly found on the nodes with small topology potential. A critical topology potential is found for the nodes where cooperators are nip and tuck with defectors. So the information of node’s degree, gradient and topology potential together can predict individual’s strategy decision in the prisoner’s dilemma on the complex networks.     

Prisoner’s Dilemma Game with Cooperation-Defection Dominance Strategies on Correlational Multilayer Networks

As multilayer networks are widely applied in modern society, numerous studies have shown the impact of a multilayer network structure and the network nature on the proportion of cooperators in the network. In this paper, we use Barabási–Albert scale-free networks (BA) and Watts and Strogatz networks (WS) to build a multilayer network structure, and we propose a new strategy-updating rule called “cooperation-defection dominance”, which can be likened to dominant and recessive traits in biogenetics. With the newly constructed multilayer network structure and the strategy-updating rules, based on the simulation results, we find that in the BA-BA network, the cooperation dominance strategy can make the networks with different rs show a cooperative trend, while the defection dominance strategy only has an obvious effect on the network cooperation with a larger r. When the BA network is connected to the WS network, we find that the effect of strategy on the proportion of cooperators in the network decreases, and the main influencing factor is the structure of the network. In the three-layer network, the cooperation dominance strategy has a greater impact on the BA network, and the proportion of the cooperators is enhanced more than under the natural evolution strategy, but the promotion effect is still smaller than that of the two-layer BA network because of the WS network. Under the defection dominance strategy, the WS layer appears different from the first two strategies, and we conclude through simulation that when the payoff parameter is at the middle level, its cooperator proportion will be suppressed, and we deduce that the proportion of cooperators and defectors, as well as the payoff, play an important role.     

Evolutionary Prisoner's Dilemma Game Based on Pursuing Higher Average Payoff

We investigate the prisoner＇s dilemma game based on a new rule： players will change their current strategies to opposite strategies with some probability if their neighbours＇ average payoffs are higher than theirs. Compared with the cases on regular lattices （RL） and Newman-Watts small world network （NW）, cooperation can be best enhanced on the scale-free Barabasi-Albert network （BA）. It is found that cooperators are dispersive on RL network, which is different from previously reported results that cooperators will form large clusters to resist the invasion of defectors. Cooperative behaviours on the BA network are discussed in detail. It is found that large-degree individuals have lower cooperation level and gain higher average payoffs than that of small-degree individuals. In addition, we find that small-degree individuals more frequently change strategies than do large- degree individuals.     

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.