Evolutionary prisoner’s dilemma on Newman-Watts social networks with an asymmetric payoff distribution mechanism

In this paper, we introduce an asymmetric payoff distribution mechanism into the evolutionary prisoner's dilemma game (PDG) on Newman--Watts social networks, and study its effects on the evolution of cooperation. The asymmetric payoff distribution mechanism can be adjusted by the parameter α: if α>0, the rich will exploit the poor to get richer; if α<0, the rich are forced to offer part of their income to the poor. Numerical results show that the cooperator frequency monotonously increases with α and is remarkably promoted when α>0. The effects of updating order and self-interaction are also investigated. The co-action of random updating and self-interaction can induce the highest cooperation level. Moreover, we employ the Gini coefficient to investigate the effect of asymmetric payoff distribution on the the system's wealth distribution. This work may be helpful for understanding cooperative behaviour and wealth inequality in society.     

Payoff-based accumulative effect promotes cooperation in spatial prisoner’s dilemma

We study the effect of accumulative payoff on the evolution of cooperation in the evolutionary prisoner’s dilemma on a square lattice. We introduce a decaying factor for the accumulative payoff, which characterizes the extent that the historical payoff is accumulated. It is shown that for fixed values of the temptation to defect, the density of cooperators increases with the value of the decaying factor. This indicates that the more the historical payoff is involved, the more favourable cooperators become. In the critical region where the cooperator density converges to zero, cooperators vanish according to a power-law-like behaviour. The associated exponents agree approximately with the two-dimensional directed percolation and depend weakly on the value of the decaying factor.     

Cooperation influenced by the correlation degree of two-layered complex networks in evolutionary prisoner’s dilemma games

An evolutionary prisoner's dilemma game is investigated on two-layered complex networks respectively representing interaction and learning networks in one and two dimensions. A parameter q is introduced to denote the correlation degree between the two-layered networks. Using Monte Carlo simulations we studied the effects of the correlation degree on cooperative behaviour and found that the cooperator density nontrivially changes with q for different payoff parameter values depending on the detailed strategy updating and network dimension. An explanation for the obtained results is provided.     

Adaptive co-evolution of strategies and network leading to optimal cooperation level in spatial prisoner’s dilemma game

We study evolutionary prisoner's dilemma game on adaptive networks where a population of players co-evolves with their interaction networks. During the co-evolution process, interacted players with opposite strategies either rewire the link between them with probability $p$ or update their strategies with probability $1-p$ depending on their payoffs. Numerical simulation shows that the final network is either split into some disconnected communities whose players share the same strategy within each community or forms a single connected network in which all nodes are in the same strategy. Interestingly, the density of cooperators in the final state can be maximised in an intermediate range of $p$ via the competition between time scale of the network dynamics and that of the node dynamics. Finally, the mean-field analysis helps to understand the results of numerical simulation. Our results may provide some insight into understanding the emergence of cooperation in the real situation where the individuals' behaviour and their relationship adaptively co-evolve.     

Integrating the environmental factor into the strategy updating rule to promote cooperation in evolutionary games

Many previous studies have shown that the environment plays an important role for social individuals.In this paper,we integrate the environmental factor,which is defined as the average payoff of all a player’s neighbours,with the standard Fermi updating rule by introducing a tunable parameter,ω.It is found that the level of cooperation increases remarkably,and that the cooperators can better resist the invasion of defection with an increase in ω.This interesting phenomenon is then explained from a microscopic view.In addition,the universality of this mechanism is also proved with the help of the small-world network and the random regular graph.This work may be helpful in understanding cooperation behaviour in species from unicellular organisms up to human beings.     

A weight’s agglomerative method for detecting communities in weighted networks based on weight’s similarity

This paper proposes the new definition of the community structure of the weighted networks that groups of nodes in which the edge's weights distribute uniformly but at random between them. It can describe the steady connections between nodes or some similarity between nodes' functions effectively. In order to detect the community structure efficiently, a threshold coefficient κ to evaluate the equivalence of edges' weights and a new weighted modularity based on the weight's similarity are proposed. Then, constructing the weighted matrix and using the agglomerative mechanism, it presents a weight's agglomerative method based on optimizing the modularity to detect communities. For a network with n nodes, the algorithm can detect the community structure in time O(n²log₂ⁿ). Simulations on networks show that the algorithm has higher accuracy and precision than the existing techniques. Furthermore, with the change of κ the algorithm discovers a special hierarchical organization which can describe the various steady connections between nodes in groups.     

Zero-determinant strategy:An underway revolution in game theory

Repeated games describe situations where players interact with each other in a dynamic pattern and make decisions according to outcomes of previous stage games. Very recently, Press and Dyson have revealed a new class of zero-determinant(ZD) strategies for the repeated games, which can enforce a fixed linear relationship between expected payoffs of two players, indicating that a smart player can control her unwitting co-player’s payoff in a unilateral way [Proc. Acad. Natl. Sci.USA 109, 10409(2012)]. The theory of ZD strategies provides a novel viewpoint to depict interactions among players,and fundamentally changes the research paradigm of game theory. In this brief survey, we first introduce the mathematical framework of ZD strategies, and review the properties and constrains of two specifications of ZD strategies, called pinning strategies and extortion strategies. Then we review some representative research progresses, including robustness analysis,cooperative ZD strategy analysis, and evolutionary stability analysis. Finally, we discuss some significant extensions to ZD strategies, including the multi-player ZD strategies, and ZD strategies under noise. Challenges in related research fields are also listed.     

Importance of tie strengths in the prisoner?s dilemma game on social networks

Though numerous researches have shown that tie strengths play a key role in the formation of collective behavior in social networks, little work has been done to explore their impact on the outcome of evolutionary games. In this Letter, we studied the effect of tie strength in the dynamics of evolutionary prisoner?s dilemma games by using online social network datasets. The results show that the fraction of cooperators has a non-trivial dependence on tie strength. Weak ties, just like previous researches on epidemics and information diffusion have shown, play a key role by the maintenance of cooperators in evolutionary prisoner?s dilemma games.     

A 2-stage strategy updating rule promotes cooperation in the prisoner's dilemma game

In this study,we propose a spatial prisoner’s dilemma game model with a 2-stage strategy updating rule,and focus on the cooperation behavior of the system.In the first stage,i.e.,the pre-learning stage,a focal player decides whether to update his strategy according to the pre-learning factor β and the payoff difference between himself and the average of his neighbors.If the player makes up his mind to update,he enters into the second stage,i.e.,the learning stage,and adopts a strategy of a randomly selected neighbor according to the standard Fermi updating rule.The simulation results show that the cooperation level has a non-trivial dependence on the pre-learning factor.Generally,the cooperation frequency decreases as the pre-learning factor increases;but a high cooperation level can be obtained in the intermediate region of 3 < β < 1.We then give some explanations via studying the co-action of pre-learning and learning.Our results may sharpen the understanding of the influence of the strategy updating rule on evolutionary games.     

Reputation-based co-evolutionary model promotes cooperation in prisoner's dilemma game

Indirect reciprocity is a fascinating topic in the field of social cooperation. In this paper, we propose a novel updating strategy based on the critical reputation-aware calculation. The joint of reputation allows players to make decisions not only on current payoffs but also from a third party, which improves the status of cooperators in the prisoner's dilemma game and provides a possibility for surviving. Experiments show that the discrepancies in initial fitness caused by reputation will support cooperators in occupying a high proportion in communities. Interestingly, we find that the massive scale of reputation fluctuation helps to enhance the cooperative effect, and newly name this character as “quasi-time lag”. The simulations show that the promotion of our proposed mechanism is effective and robust on different network topologies. This work provides a new perspective for the study of the cooperative game.     

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.     

Non-factorizable joint probabilities and evolutionarily stable strategies in the quantum prisoner's dilemma game

The well-known refinement of the Nash Equilibrium (NE) called an Evolutionarily Stable Strategy (ESS) is investigated in the quantum Prisoner's Dilemma (PD) game that is played using an Einstein-Podolsky-Rosen type setting. Earlier results report that in this scheme the classical NE remains intact as the unique solution of the quantum PD game. In contrast, we show here that interestingly in this scheme a non-classical solution for the ESS emerges for the quantum PD.     

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.