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.     

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.     

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.     

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.     

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.     

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.     

Coevolutionary,coexisting learning and teaching agents model for prisoner’s dilemma games enhancing cooperation with assortative heterogeneous networks

Unlike other natural network systems, assortativity can be observed in most human social networks, although it has been reported that a social dilemma situation represented by the prisoner’s dilemma favors dissortativity to enhance cooperation. We established a new coevolutionary model for both agents’ strategy and network topology, where teaching and learning agents coexist. Remarkably, this model enables agents’ enhancing cooperation more than a learners-only model on a time-frozen scale-free network and produces an underlying assortative network with a fair degree of power-law distribution. The model may imply how and why assortative networks are adaptive in human society.     

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.     

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.     

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.     

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.     

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.     

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.     

Elimination mechanism promotes cooperation in coevolutionary prisoner’s dilemma games

We propose an elimination mechanism in the study of the evolutionary prisoner’s dilemma games on evolving networks. It assumes that after each round of playing, players whose payoffs are below a certain threshold will be eliminated from the game and the same number of new nodes will be added to the network to maintain the size of the network constant. Numerical results show that moderate values of elimination threshold can result in a maximum cooperation level in the evolutionary prisoner’s dilemma game. Moreover, the elimination mechanism can make the network structure evolve into a high heterogeneity in degree distribution, which is considered to be helpful in promoting cooperation in evolutionary games. The present study may provide new insight for understanding the evolution of cooperation in light of the law ‘survival of the fittest’ in nature.     

Properties of wealth distribution in multi-agent systems of a complex network

We present a simple model for examining the wealth distribution with agents playing evolutionary games (the Prisoners’ Dilemma and the Snowdrift Game) on complex networks. Pareto’s power law distribution of wealth (from 1897) is reproduced on a scale-free network, and the Gibbs or log-normal distribution for a low income population is reproduced on a random graph. The Pareto exponents of a scale-free network are in agreement with empirical observations. The Gini coefficient of an ER random graph shows a sudden increment with game parameters. We suggest that the social network of a high income group is scale-free, whereas it is more like a random graph for a low income group.     

Evolutionary prisoner’s dilemma game in flocks

We investigate an evolutionary prisoner’s dilemma game among self-driven agents, where collective motion of biological flocks is imitated through averaging directions of neighbors. Depending on the temptation to defect and the velocity at which agents move, we find that cooperation can not only be maintained in such a system but there exists an optimal size of interaction neighborhood, which can induce the maximum cooperation level. When compared with the case that all agents do not move, cooperation can even be enhanced by the mobility of individuals, provided that the velocity and the size of neighborhood are not too large. Besides, we find that the system exhibits aggregation behavior, and cooperators may coexist with defectors at equilibrium.     

Evolutionary game on a stochastic growth network

In some real complex systems the structures are difficult to map or changing over time. To explore the evolution of strategies on these complex systems, it is not realistic enough to specify their structures or topological properties in advance. In this paper, we address the evolutionary game on a stochastic growth network adopting the prisoner’s dilemma game. We introduce a growing rate q$q$ to control the ratio of network growth to strategy evolution. A large q$q$ denotes that the network grows faster than strategy evolution. Simulation results show that a fast growing rate is helpful to promote the average payoffs of both cooperators and defectors. Moreover, this parameter also significantly influences the cooperation frequency on the resulting networks. The coexisting mechanisms in this paper may provide a beneficial insight for understanding the emergence of complex topological structures and game behaviors in numerous real systems.     

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.     

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.     

Benefit community promotes evolution of cooperation in prisoners' dilemma game

Exploring the emergence and maintenance of cooperation in social dilemma is valuable and it arises considerable concerns of many researchers. In this paper, we propose a mechanism to promote cooperation, called benefit community,in which cooperators linking together form a common benefit community and all their payoffs obtained from game are divided coequally. The robustness of conclusions is tested for the PDG(prisoners' dilemma game) on square lattice and WS small world network. We find that cooperation can be promoted by this typical mechanism, especially, it can diffuse and prevail more easily and rapidly on the WS small world network than it on the square lattice, even if a big temptation to defect b. Our research provides a feasible direction to resolve the social dilemma.