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.     

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.     

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.     

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.     

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.     

Evolutionary Prisoner＇s Dilemma on a Weighted Scale-Free Network

We investigate the evolutionary prisoner＇s dilemma on a weighted scale-free network where the diversity of teaching ability is introduced to the network in the form of weight. Though the diversity of teaching ability is not a sufficient condition for the enhancement of cooperation, we find that the degree-dependent teaching ability plays an active role in the evolution of cooperation. A new phenomenon is found when the degree-dependent teaching ability is used： the distribution of the cooperator frequency displays a two-peak structure for a certain parameter range. We also investigate the effects of the degree-degree correlation of the network on the evolution of cooperation in the presence of the diversity of the teaching ability.     

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.     

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.     

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.     

Evolutionary Games on Weighted Newman-Watts Small-World Networks

We investigate the evolutionary prisoner＇s dilemma game （PDG） on weighted Newman-Watts （NW） networks. In weighted NW networks, the link weight ωij is assigned to the link between the nodes i and j as： ωij = （ki · kj）^β, where ki（kj） is the degree of node i（j） and β represents the strength of the correlations. Obviously, the link weight can be tuned by only one parameter β. We focus on the cooperative behavior and wealth distribution in the system. Simulation results show that the cooperator frequency is promoted by a large range of β and there is a minimal cooperation frequency around β = -1. Moreover, we also employ the Gini coefficient to study the wealth distribution in the population. Numerical results show that the Gini coefficient reaches its minimum when β≈ -1. Our work may be helpful in understanding the emergence of cooperation and unequal wealth distribution in society.     

Clustering Effect on the Evolution of Cooperation in a Herding Snowdrift Game

We study the evolution of cooperation for two cluster breaking mechanisms in a herding snowdrift game. The cooperative behavior is observed to be related to the duster size. A negative dependence of the payoff parameter r on cooperative behavior is discovered. For a low r, herding helps promote the cooperation, whereas for a high r, herding tends to prevent cooperative behavior.     

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.     

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.     

Local Minority Game with Evolutionary Strategies

We study a model of local minority game in the random Kauffman network with evolutionary strategies and propose three methods to update the strategy of poor agents, with lower points in a given generation： namely to update either the Boolean function of their strategies randomly, or their local information of randomly adjacent m agents, or the number m of randomly chosen adjacent agents. The results of extended numerical simulations show that the behaviour of strategies in the three methods may enhance significantly the entire coordination of agents in the system. It is also found that a poor agent tends to use both small m strategies and correlated strategies, and the strategies of agents will finally self-organize into a steady-state distribution for a long time playing of the game.     

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.     

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.     

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.     

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.     

Modelling Moran Process with Network Dynamics for the Evolution of Cooperation

We introduce a simple model based on the Moran process with network dynamics. Using pair approximation, the cooperation frequencies at equilibrium states are deduced for general interactions. Three usual social dilemmas are discussed in the framework of our model. It is found that they all have a phase transition at the same value of cost-to-benefit ratio. For the prisoner＇s dilemma game, notably it is exactly the simple rule reported in the literature [Nature 441 （2006） 502]. In our model, the simple rule results from the parent-offspring link. Thus the basic mechanism for cooperation enhancement in network reciprocity is in line with the Hamilton rule of kin selection. Our simulations verify the analysis obtained from pair approximation.     

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).