An Adaptive Strategy via Reinforcement Learning for the Prisoner＇s Dilemma Game

The iterated prisoner's dilemma(IPD) is an ideal model for analyzing interactions between agents in complex networks. It has attracted wide interest in the development of novel strategies since the success of tit-for-tat in Axelrod's tournament. This paper studies a new adaptive strategy of IPD in different complex networks, where agents can learn and adapt their strategies through reinforcement learning method. A temporal difference learning method is applied for designing the adaptive strategy to optimize the decision making process of the agents. Previous studies indicated that mutual cooperation is hard to emerge in the IPD. Therefore, three examples which based on square lattice network and scale-free network are provided to show two features of the adaptive strategy. First, the mutual cooperation can be achieved by the group with adaptive agents under scale-free network, and once evolution has converged mutual cooperation, it is unlikely to shift. Secondly, the adaptive strategy can earn a better payoff compared with other strategies in the square network. The analytical properties are discussed for verifying evolutionary stability of the adaptive strategy.

THE OPTIMAL CONTROL FOR PROMOTING THE COOPERATION IN EVOLUTION GAME GENERATED BY PRISONER'S DILEMMA

Natural selection opposes the evolution of cooperation unless specific mechanisms are at work in Prisoner＇s Dilemma. By taking advantage of the modern control theory, the controller design is discussed and the optimal control is designed for promoting cooperation based on the recent advances in mechanisms for the evolution of cooperation. Two con- trol strategies are proposed： compensation control strategy for the cooperator when playing against a defector and reward control strategy for cooperator when playing against a coop- erator. The feasibility and effectiveness of these control strategies for promoting cooperation in different stages are analyzed. The reward for cooperation can＇t prevent defection from being evolutionary stable strategy （ESS）. On the other hand, compensation for the coopera- tor can＇t prevent defection from emerging and sustaining. By considering the effect and the cost, an optimal control scheme with constraint on the admissible control set is put forward. By analyzing the special nonlinear system of replicator dynamics, the exact analytic solution of the optimal control scheme is obtained based on the maximum principle. Finally, the effectiveness of the proposed method is illustrated by examples.

Evolutionary prisoner's dilemma game on highly clustered community networks

This paper studies the evolutionary prisoner＇s dilemma game on a highly clustered community network in which the clustering coefficient and the community size can be tuned. It finds that the clustering coefficient in such a degree-homogeneous network inhibits the emergence of cooperation for the entire range of the payoff parameter. Moreover, it finds that the community size can also have a marked influence on the evolution of cooperation, with a larger community size leading to not only a lower cooperation level but also a smaller threshold of the payoff parameter above which cooperators become extinct.

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.

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.

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.

Effects of Dimers on Cooperation in the Spatial Prisoner's Dilemma Game

We investigate the evolutionary prisoner’s dilemma game in structured populations by introducing dimers,which are defined as that two players in each dimer always hold a same strategy.We find that influences of dimerson cooperation depend on the type of dimers and the population structure.For those dimers in which players interactwith each other,the cooperation level increases with the number of dimers though the cooperation improvement leveldepends on the type of network structures.On the other hand,the dimers,in which there are not mutual interactions,will not do any good to the cooperation level in a single community,but interestingly,will improve the cooperationlevel in a population with two communities.We explore the relationship between dimers and self-interactions and findthat the effects of dimers are similar to that of self-interactions.Also,we find that the dimers,which are establishedover two communities in a multi-community network,act as one type of interaction through which information betweencommunities is communicated by the requirement that two players in a dimer hold a same strategy.

Effects of Topological Randomness on Cooperation in a Deterministic Prisoner's Dilemma Game

In this work, we consider an evolutionary prisoner's dilemma game on a homogeneous random network with the richest-following strategy adoption rule. By constructing homogeneous random networks from a regular ring graph, we investigate the effects of topological randomness on cooperation. In contrast to the ordinary view that the presence of small amount of shortcuts in ring graphs favors cooperation, we find the cooperation inhibition by weak topological randomness. The explanations on the observations are

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.

Effect of Strategy-Homogeneity on the Prisoner’s Dilemma Game in a Square Lattice

We investigate the effect of strategy-homogeneity on the prisoner’s dilemma game in a square lattice. Strategy-homogeneity means that the population contains at least one connected group in which individuals maintain the same strategy at each iteration and may update according to updating rule at next iteration. The simulation results show that the introduction of strategy-homogeneity increases the cooperation in the evolutionary stable state. For any value of temptation to defect, the density of cooperators in equilibrium state increases firstly and then decreases as the level of strategy-homogeneity increases constantly, and there exists an appropriate level of strategy-homogeneity, maximizing the density of cooperators. The results may be favorable for comprehending cooperative behaviors in societies composed of connected groups with coherent strategy.

Defection or Cooperation—Richard Power’s Moral Insights in Prisoner’s Dilemma

Prisoner’s Dilemma is a master trope for relaying the permanent impasse or dilemma of cooperation versus defection.In Prisoner’s Dilemma Richard Powers narrates a multiform of dilemma mainly narrating Eddie Hobson’s traumatic history,along with Eddie’s attempt to carry out a strategy—Eddie’s ideal Hobstown which is supposed to cope with the confused dilemma.Pow⁃ers’strategic moral insights are revealed by assuming that each player feels sympathy for the other and a moral cooperative instead of antagonistic solution to the Prisoner’s Dilemma will be found.

Richard Power’s Philosophical Thoughts in Prisoner’s Dilemma

The“prisoner’s dilemma”illustrates that everyone is rationally thinking about maximizing their own interests and taking their own best strategies,but the end result is counterproductive.In Prisoner’s Dilemma Richard Powers describes multiformareas of dilemma;to lose balance;to make choices;and ask each other:what to do next?It is a master trope for relaying the permanent impasse or dilemma of cooperation versus defection.In most cases,the prisoner's dilemma is more about the number of defeats in the game,and even a major loss of society.Individual rationality is likely to be a collective disregard,and if everyone is acting on the principle of self-interest,it is often the result that everyone is lost.Powers expresses his philosophical insight:Until a cooperative instead of antagonistic solution is collectively agreed upon,we still remain imprisoned in an interminable and oppressive struggle and conflict decided by self-interest and rational control,which is also a moral response to historical consciousness of postmodernism.

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 c~ and is remarkably promoted when c~ 〉 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.

PREFERENCE AND EVOLUTION IN THE ITERATED PRISONER'S DILEMMA

Game theory is extensively used to study strategy-making and actions of play- ers. The authors proposed an analysis method for study the evolutionary outcome and behaviors of players with preference in iterated prisoner＇s dilemma. In this article, a preference parameter k was introduced in the payoff matrix, wherein the value of k denotes the player＇s degree of egoism and altruism （preference）. Then, a game-theoretic dynamical model was formulated using Birth-and-Death process. The authors studied how preference influences the evolutionary equilibrium and behaviors of players. The authors get the general results： egoism leads to defection, and altruism can make players build trust and maintain cooperation, and so, the hope of the Pareto optimal solution. In the end, the simulation experiments proved the efficiency of the method.

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.

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 ac- cording 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 play- ers, 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.

Strategic games on a hierarchical network model

Among complex network models,the hierarchical network model is the one most close to such real networks as world trade web,metabolic network,WWW,actor network,and so on.It has not only the property of power-law degree distribution,but also the scaling clustering coefficient property which Barabási-Albert(BA)model does not have.BA model is a model of network growth based on growth and preferential attachment,showing the scale-free degree distribution property.In this paper,we study the evolution of cooperation on a hierarchical network model,adopting the prisoner's dilemma(PD)game and snowdrift game(SG)as metaphors of the interplay between connected nodes.BA model provides a unifying framework for the emergence of cooperation.But interestingly,we found that on hierarchical model,there is no sign of cooperation for PD game,while the fre-quency of cooperation decreases as the common benefit decreases for SG.By comparing the scaling clustering coefficient prop-erties of the hierarchical network model with that of BA model,we found that the former amplifies the effect of hubs.Considering different performances of PD game and SG on complex network,we also found that common benefit leads to cooperation in the evolution.Thus our study may shed light on the emergence of cooperation in both natural and social environments.

异质Erd?s-Rényi随机超网络上的演化博弈

提出了一种异质Erd9s-Rényi (ER)随机超网络.与传统同质ER随机超网络相比,异质ER随机超网络具有两个重要优点:能更有效地描述现实世界中的多层级结构和复杂关系,避免了孤立点的出现.同时,改进了传统同质ER随机超网络的构造方法,以避免出现孤立点.研究了在两种新的ER随机超网络上的演化博弈,采用了两种策略更新规则和一种博弈机制.实验揭示了以下结果:断连机制显著地促进了两种ER随机超网络的合作水平;基于两种策略更新规则,演化雪堆博弈的合作水平会随着成本效益比的增加而减小;虽然两种ER随机超网络的结构和性质是不同的,但是在其他参数相同的情况下,两种演化博弈的合作水平在两种网络上是相似的.

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, w. 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 w. 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.

基于网络演化博弈的信息传播行为分析

演化博弈论将博弈理论与动态演化过程相结合,在演化的过程中追求动态平衡,该方法较好地实现了对在线社交网络信息传播演化趋势的分析。为了进行在线社交网络中的信息传播演化动力学研究,同时分析到达某一均衡状态的原因和方式,首先结合在线社交网络特性构建改进聚集的无标度网络,用户间的交互采用单参数囚徒困境博弈模型,并采用费米规则更新策略。其次,通过数据仿真对影响网络中合作者比例演化的因素进行分析,得到网络聚集性、博弈收益权重会促进网络中的合作行为,而背叛诱惑与噪声因子则对其有抑制作用的结论。最后,结合新浪微博的真实舆论传播数据,对现实网络进行信息传播分析。研究表明,合作者比例是由博弈动力学和网络拓扑结构共同作用的结果,需采取适当的激励机制,以解决在线社交网络中信息传播出现的合作困境。