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 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.

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.

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.

Evolution of Cooperation in Continuous Prisoner's Dilemma Games on Barabasi-Albert Networks with Degree-Dependent Guilt Mechanism

This paper studies the continuous prisoner's dilemma games (CPDG) on Barabasi-Albert (BA) networks.In the model,each agent on a vertex of the networks makes an investment and interacts with all of his neighboring agents.Making an investment is costly,but which benefits its neighboring agents,where benefit and cost depend on the level of investment made.The payoff of each agent is given by the sum of payoffs it receives in its interactions with all its neighbors.Not only payoff,individual's guilty emotion in the games has also been considered.The negative guilty emotion produced in comparing with its neighbors can reduce the utility of individuals directly.We assume that the reduction amount depends on the individual's degree and a baseline level parameter.The group's cooperative level is characterized by the average investment of the population.Each player makes his investment in the next step based on a convex combination of the investment of his best neighbors in the last step,his best history strategies in the latest steps which number is controlled by a memory length parameter,and a uniformly distributed random number.Simulation results show that this degree-dependent guilt mechanism can promote the evolution of cooperation dramatically comparing with degree-independent guilt or no guilt cases.Imitation,memory,uncertainty coefficients and network structure also play determinant roles in the cooperation level of the population.All our results may shed some new light on studying the evolution of cooperation based on network reciprocity mechanisms.

Effects of Inertia on Evolutionary Prisoner's Dilemma Game

Considering the inertia of individuals in real inertia of players is introduced into evolutionary prisoner＇s life, we propose a modified Fermi updating rule, where the dilemma game （PDG） on square lattices. We mainly focus on how the inertia affects the cooperative behavior of the system. Interestingly, we find that the cooperation level has a nonmonotonic dependence on the inertia： with small inertia, cooperators will soon be invaded by defectors; with large inertia, players are unwilling to change their strategies and the cooperation level remains the same as the initial state; while a moderate inertia can induce the highest cooperation level. Moreover, effects of environmental noise and individual inertia are studied. Our work may be helpful in understanding the emergence and persistence of cooperation in nature and society.

An Improved Fitness Evaluation Mechanism with Memory in Spatial Prisoner's Dilemma Game on Regular Lattices

To deeply understand the emergence of cooperation in natural,social and economical systems,we present an improved fitness evaluation mechanism with memory in spatial prisoner's dilemma game on regular lattices.In our model,the individual fitness is not only determined by the payoff in the current game round,but also by the payoffs in previous round bins.A tunable parameter,termed as the memory strength(μ),which lies between 0 and 1,is introduced into the model to regulate the ratio of payoffs of current and previous game rounds in the individual fitness calculation.When μ = 0,our model is reduced to the standard prisoner's dilemma game;while μ = 1 represents the case in which the payoff is totally determined by the initial strategies and thus it is far from the realistic ones.Extensive numerical simulations indicate that the memory effect can substantially promote the evolution of cooperation.For μ < 1,the stronger the memory effect,the higher the cooperation level,but μ = 1 leads to a pathological state of cooperation,but can partially enhance the cooperation in the very large temptation parameter.The current results are of great significance for us to account for the role of memory effect during the evolution of cooperation among selfish players.

Evolutionary prisoner＇s dilemma games with local interaction and best-response dynamics

This paper studies the long run behavior in evolutionary prisoner＇s dilemma games. All players are assumed to sit around a circle and to interact only with their neighbors. It is known that full-defection is the unique long run equilibrium as the probability of players＇ experimentation （or mutation） tends to zero in the best response dynamics. Here, it is shown that full-cooperation could emerge in the long run if one also cares for his neighbors in the best- response dynamics.

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.

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.

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.

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.

Conditions Where the Ruled Class Unites for the Revolution:Applicability of a Game Theory on Social Dilemmas

Revolutions,typical cases of crucial social transformations,cannot be realized successfully without a large number of activists.Therefore,creating conditions favorable for acquiring enough participants should be an important topic of Marxist social science.In particular,this problem includes the “free-ride,” because the benefits of revolutionaries’ activities are gained not only by the activists but also by all other members.The paper analyzes problems such as this one,applying non-cooperative game theory to social dilemma problems.This leads to some interesting results.In this research,the problem of the workers’ choice between unity or freeride is first defined using numerical examples of the gain structure.It is defined again in a more generalized form using other parameters.In so doing,we express both the cost of participating in the movement and the gains from the concession of the ruling class.Because this analysis focuses on the importance of the number of participants,the concession of the ruling class is framed as a function of the number of participants.The results of this analysis revealed that the economic base and superstructure accurately correspond in some game structures but not in others.In other words,the social dilemma presents either as a case of prisoners’ dilemma or as a chicken game.Furthermore,this paper analyzes the influence of group size,and it was revealed that groups with a large number of members,such as a ruled class,find it particularly difficult to unite.This phenomenon is called the “large group dilemma.” In these ways,this research shows that the aforementioned type of game theory can be used to analyze the difficulties and possibilities of social movements.

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.

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.

Inference of interactions between players based on asynchronously updated evolutionary game data

The interactions between players of the prisoner's dilemma game are inferred using observed game data.All participants play the game with their counterparts and gain corresponding rewards during each round of the game.The strategies of each player are updated asynchronously during the game.Two inference methods of the interactions between players are derived with naive mean-field(n MF)approximation and maximum log-likelihood estimation(MLE),respectively.Two methods are tested numerically also for fully connected asymmetric Sherrington-Kirkpatrick models,varying the data length,asymmetric degree,payoff,and system noise(coupling strength).We find that the mean square error of reconstruction for the MLE method is inversely proportional to the data length and typically half(benefit from the extra information of update times)of that by n MF.Both methods are robust to the asymmetric degree but work better for large payoffs.Compared with MLE,n MF is more sensitive to the strength of couplings and prefers weak couplings.

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.