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.

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.

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.

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.

Influence of Strategy Continuity on Cooperation in Spatial Prisoner’s Dilemma Games with Migrating Players

The phenomenon of cooperation is prevalent in both nature and human society. In this paper a simulative model is developed to examine how the strategy continuity influences cooperation in the spatial prisoner＇s games in which the players migrate through the success-driven migration mechanism. Numerical simulations illustrate that the strategy continuity promotes cooperation at a low rate of migration, while impeding cooperation when the migration rate is higher. The influence of strategy continuity is also dependent on the game types. Through a more dynamic analysis, the different effects of the strategy continuity at low and high rates of migration are explained by the formation, expansion, and extinction of the self-assembled clusters of ＂partial-cooperators＂ within the gaming population.

A New Type of Evolutionary Strategy Based on a Multi-player Iterated Prisoner’s Dilemma Game

According to the philosophy of self-cultivation that"one should refine his personal virtue when in poverty,and help save the world when in success",a new type of evolutionary strategy,PoorCompetition-Rich-Cooperation(PCRC),is proposed.To discuss its superiority and inferiority,based on a multi-player iterated Prisoner’s Dilemma game,PCRC and other six kinds of strategies are played by using the roulette method in three different populations(a uniformly distributed population,a cooperation-preference population,a defection-preference population).The payoff characteristics for each strategy under different temptation coefficients and noise values are also analyzed.Simulation results indicate that PCRC has a sufficient robustness and its payoff presents a basically monotonic increasing tendency with the increment of noise.The superiority of PCRC becomes more obvious when the temptation coefficient becomes larger.Furthermore,a higher population preference for defection yields a more obvious advantage for PCRC.

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.

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 topologicaJ randomness on cooperation. In contrast to the ordinary view that the presence of smaJ1 amount of shortcuts in ring graphs favors cooperation, we find the cooperation inhibition by weak topological randomness. The explanations on the observations are presented.

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.

Spatial prisoner’s dilemma games with increasing size of the interaction neighborhood on regular lattices

We studied the evolution of cooperation in the prisoner's dilemma game on a square lattice where the size of the interaction neighborhood is considered. Firstly, the effects of noise and the cost-to-benefit ratio on the maintenance of cooperation were investigated. The results indicate that the cooperation frequency depends on the noise and cost-to-benefit ratio: cooperation reaches a climax as noise increases, but it monotonously decreases and even vanishes with the ratio increasing. Furthermore, we investigated how the size of the interaction neighborhood affects the emergence of cooperation in detail. Our study demonstrates that cooperation is remarkably enhanced by an increase in the size of the interaction neighborhood. However, cooperation died out when the size of the interaction neighborhood became too large since the system was similar to the mean-field system. On this basis, a cluster-forming mechanism acting among cooperators was also explored, and it showed that the moderate range of the neighborhood size is beneficial for forming larger cooperative clusters. Finally, large-scale Monte Carlo simulations were carried out to visualize and interpret these phenomena explicitly.

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.

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.

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.