Condition to Gain More Profits and Promote Cooperation by Tit-for-Tat Strategy in Finite and Repeated Prisoner＇s Dilemma

As a basic study to prevent accidents or concealment caused by violation of rules or regulations （which are regarded as uncooperative behavior）, an attempt was made to clarify the condition necessary for promoting cooperation when the tit-for-tat strategy is adopted in the finite and repeated prisoner＇s dilemma situations. A mathematical model, in which three different strategies （tit-for-tat, all defection （individualism）, and all cooperation （altruism）） exist, was constructed in order to demonstrate the condition that can promote cooperative behaviors. As a result of an agent-agent computer simulation, it was shown that the tit-for-tat strategy promoted more cooperation than other strategies when the number of agents adopting the tit-for-tat strategy was dominant in the population and the discount parameter was larger. Next, it was explored how the tit-for-tat strategy in the finite and repeated prisoner＇s dilemma promotes cooperation using a human-agent computer simulation. In other words, the condition under which cooperative behavior is encouraged was clarified. In the simulation experiment, the discount rate was controlled as an experimental variable. As well as the first experiment above, the dominant occupation of the tit-for-tat strategy was found to lead to the promoted cooperation. Concerning the effect of discount parameter on the cooperative behavior, the cooperation rate tended to increase with the increase of discount parameter only when the t-t-for-tat strategy is dominant. As a whole, the type of change of discount parameter did not affect the cooperation rate.

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.

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.

Effects of Probability of Revelation of Defection and Penalty to Defection on Cooperative Behavior in 2-Person Prisoner＇s Dilemma Game

In our society, it is a major issue to enhance cooperative behaviors. Without this, our society fall into social dilemma situations, and gets worse and worse. Such a situation in an organization leads to violation of social or organizational rules, and at the worst case it suffers from serious accidents or scandals. Therefore, it is important for organizational managers to make efforts and take measures to enhance cooperative behaviors. Although there seem to be many ways to constantly elicit cooperative behaviors, the punishment is one of the most effective measures for enhancing cooperation. This study focused on the effects of penalty and probability of the revelation of defection on the cooperation, and getting insight into how punishment strategy should be used to get rid of social dilemmas and enhance cooperation. This study conducted a simulation experiment to find the proper penal regulations condition that can suppress violations （defective behavior） in a 2-person prisoner＇s dilemma situation. The effects of probability of the revelation of defection and penalty to revelation on the cooperative behavior were identified with the interactive effect of both experimental factors. The defection （uncooperative behavior） decreased when the penalty to the defection was heavy and the probability of the revelation of defection was low than that when the penalty to the defection was light and the probability of the revelation of the defection was high.

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.

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.

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.

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 dimets on cooperation depend on the type of dimers and the population structure. For those dimers in which players interact with each other, the cooperation level increases with the number of dimers though the cooperation improvement level depends 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 cooperation level in a population with two communities. We explore the relationship between dimers and self-interactions and find that the effects of dimers are similar to that of self-interactions. Also, we find that the dimers, which are established over two communities in a multi-community network, act as one type of interaction through which information between communities is communicated by the requirement that two players in a dimer hold a same strategy.

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.

SPECTRUM SHARING IN ITERATED PRISONER'S DILEMMA GAME BASED ON EVOLUTIONARY STRATEGIES FOR COGNITIVE RADIOS

We study a spectrum sharing problem where multiple systems coexist and interfere with each other. First, an analysis is proposed for distributed spectrum sharing based on Prisoners' Dilemma (PD) in Cognitive Radios (CRs). In one-shot game, selfish and rational CRs greedily full spread their own spectrum space in order to maximize their own rates, which leads to Nash Equilibrium (N.E.). But with long term interaction, i.e., Iterated Prisoner's Dilemma (IPD), CRs can come to cooperate and acquire the social optimal point by using different evolutionary strategies such as Tit For Tat (TFT), Generous TFT (GTFT), etc. Also we compare the performances of the different evolutionary strategies in noise-free and noisy environments for two-player games. Finally, N-player IPD (N-IPD) is simulated to verify our conclusions that TFT is a good strategy for spectrum sharing in CRs.

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.

Zero-determinant strategies in iterated multi-strategy games

Self-serving,rational agents sometimes cooperate to their mutual benefit.The two-player iterated prisoner′s dilemma game is a model for including the emergence of cooperation.It is generally believed that there is no simple ultimatum strategy which a player can control the return of the other participants.The zero-determinant strategy in the iterated prisoner′s dilemma dramatically expands our understanding of the classic game by uncovering strategies that provide a unilateral advantage to sentient players pitted against unwitting opponents.However,strategies in the prisoner′s dilemma game are only two strategies.Are there these results for general multi-strategy games?To address this question,the paper develops a theory for zero-determinant strategies for multi-strategy games,with any number of strategies.The analytical results exhibit a similar yet different scenario to the case of two-strategy games.The results are also applied to the Snowdrift game,the Hawk-Dove game and the Chicken game.

论无限次重复囚徒困境博弈中的针锋相对策略

在有限次重复囚徒困境博弈中,理性的参与人在每个阶段博弈中选择不合作是唯一的子博弈精炼纳什均衡,因此不会出现合作行为。然而,在无限次重复囚徒困境博弈中,参与人采取某些特定的策略可能会导致合作结果的出现。冷酷触发策略是几乎所有博弈论著作都会讨论的一种促进合作的策略,但学者对于针锋相对策略却甚少涉及。研究严格论证了针锋相对策略构成纳什均衡及子博弈精炼纳什均衡的条件。结果表明,当参与人对未来足够重视时,只有当阶段博弈的支付满足一定条件,针锋相对策略才能构成纳什均衡,其对贴现率的要求要比冷酷触发策略更高。研究还表明针锋相对策略并非不能构成子博弈精炼纳什均衡,只是要求的条件非常严格,除了支付满足特定条件外,贴现率也只能取某一特定值。

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。

基于SPD的网络演化仿真研究

复杂网络结构演化研究中多探讨“如何形成”,而忽视了“为什么这样形成”的问题。基于合作演化的角度,利用空间囚徒困境理论,对社会网络中的个体进行了分类,并建立网络演化中个体选择的微观动力学机制,建立了社会网络的结构演化模型。使用多主体系统仿真工具Repast进行了仿真。利用度分布、聚集系数、平均最短路径及社会总收益作为演化判据,给出了网络演化的仿真结果。结果表明合作机制下的演化网络展现出明显的小世界特性,说明合作机制可以在一定程度上解释现实网络形成的原因。并且指出对于社会整体来说,即使在合作者较少的情况下,也能够通过社会关系的改善极大的提高社会的总体收益。