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.     

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.     

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.     

Evolutionary public goods games on scale-free networks with unequal payoff allocation mechanism

In this paper, we bring an unequal payoff allocation mechanism into evolutionary public goods game on scale-free networks and focus on the cooperative behavior of the system. The unequal mechanism can be tuned by one parameter α: if α>0, the hub nodes can use its degree advantage to collect more payoff; if α<0, numerous non-hub nodes will obtain more payoff in a single round game. Simulation results show that the cooperation level has a non-trivial dependence on α. For the small enhancement factor r, the cooperator frequency can be promoted by both negative and positive α. For large r, there exists an optimal α that can obtain the highest cooperation level. Our results may sharpen the understanding of the emergence of cooperation induced by the unequal payoff allocation mechanism.     

Effect of imitation in evolutionary minority game on small-world networks

The system performance in an evolutionary minority game with imitation on small-world networks is studied. Numerical results show that system performance positively correlates with the clustering coefficients. The domain structure of the agents’ strategies can be used to give a qualitative explanation for it. We also find that the time series of the reduced variance σ²/N could have a phasic evolution from a metastable state (two crowds are formed but the distribution of their probabilities does not peak at p≈0 and p≈1) to a steadystate (the two crowds evolve into a crowd and an anticrowd with the distribution of their probabilities peaking at p≈0 and p≈1).     

Heterogeneity Improves Cooperation in Continuous Snowdrift Game

We study the networking effects on the population divergence and the increased level of cooperation in the continuous snowdrift game （CSG）. In the regular world, limited interaction inhibits the occurrence of evolutionary branching. The formation of clusters defends the intermediate-investors from intruding by high- or low-investors. In a rewired network, the collective behaviour is related to the rewiring rules. A linear relationship I = aσ ＋ 5 between the average investment and the standard deviation of the degree distribution is found.     

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.     

Cooperative dynamics of snowdrift game on spatial distance-dependent small-world networks

We investigate the evolution of cooperative behaviors of small-world networking agents in a snowdrift game mode, where two agents (nodes) are connected with probability depending on their spatial Euclidean lattice distance in the power-law form controlled by an exponent α. Extensive numerical simulations indicate that the game dynamics crucially depends on the spatial topological structure of underlying networks with different values of the exponent α. Especially, in the distance-independent case of α=0, the small-world connectivity pattern contributes to an enhancement of cooperation compared with that in regular lattices, even for the case of having a high cost-to-benefit ratio r. However, with the increment of α>0, when r≥0.4, the spatial distance-dependent small-world (SDSW) structure tends to inhibit the evolution of cooperation in the snowdrift game.     

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.     

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.     

Evolutionary prisoner’s dilemma game on weighted scale-free networks

This paper investigates the evolutionary prisoner’s dilemma on weighted scale-free networks. The weighted networks are generated by adopting Barabási-Albert scale-free network and assigning link weight with w_ij=(k_i×k_j)^β. Simulation results show that the cooperation frequency has a strong dependence on β. The value of β which is associated with the maximal cooperation frequency has been sought out. Moreover, Gini coefficient and Pareto exponent of the system’s wealth distribution are investigated. The inequality of wealth distribution is minimized at β≈−1.     

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.     

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.     

Prisoner's Dilemma on Co-Evolving Networks without Strategy Update

We investigate the game theory in a structured population with the assumption that the evolution of network structure is far faster than that of strategy update. We find that the degree distribution for the finM network consists of two distinct parts： the low degree part which is contributed to by defectors and a broadband in the regime with high degree which is formed by cooperators. The structure of the final network and the final strategy pattern have also been numerically proved to be independent of the game parameters.     

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.     

Cooperative behavior in evolutionary snowdrift game with bounded rationality

An evolutionary snowdrift game (SG) that incorporates bounded rationality and limited information in the evolutionary process is proposed and studied. Based on SG in a well-mixed population and defining the winning action at a turn to be the one that gets a higher payoff, the most recent m winning actions can be used as a public information based on which the competing agents decide their next actions. This defines a strategy pool from which each agent picks a number of strategies as their tool in adapting to the competing environment. The payoff parameter r in SG serves to set the maximum number of winners per turn. Due to the bounded rationality and limited information, the cooperative frequency shows steps and plateaux as a function of r and these features tend to be smoothed out as m increases. These features are results of an interplay between a restricted subset of m-bit histories that the system can visit at a value of r and the limited capacity that agents can adapt. The standard deviation in the number of agents taking the cooperative action is also studied. For general values of r, our model generates a realization of the binary-agent-resource model. The idea of introducing bounded rationality into a two-person game to realize the minority game or binary-agent-resource model could be a useful tool for future research.     

Uncertainty of cooperation in random scale-free networks

We study the spatial prisoner’s dilemma game where the players are located on the nodes of a random scale-free network. The prisoner’s dilemma game is a powerful tool and has been used for the study of mutual trust and cooperation among individuals in structured populations. We vary the structure of the network and the payoff values for the game, and show that the specific conditions can greatly influence the outcome of the game. A variety of behaviors are reproduced and the percentage of cooperating agents fluctuates significantly, even in the absence of irrational behavior. For example, the steady state of the game may be a configuration where either cooperators or defectors dominate, while in many cases the solution fluctuates between these two limiting behaviors.     

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.     

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.     

Self-organized cooperative behavior and critical penalty in an evolving population

The emergence of cooperation and the effectiveness of penalties among competing agents are studied via a model of evolutionary game incorporating adaptive behavior and penalties for illegal acts. For initially identical agents, a phase diagram is obtained via an analytic approach, with results in good agreement with numerical simulations. The results show that there exists a critical penalty for achieving a completely honest population and a sufficiently well-behaved initial population requires no penalty. Self-organized segregation to extreme actions emerges in the dynamics for a system with uniformly distributed initial tendencies for cooperation. After training, the penalty can be relaxed without ruining the adapted cooperative behavior. Results of our model in a population taking on the form of a 2D square lattice are also reported.