Stochastic Evolutionary Game Dynamics In Finite Populations

Competition among individuals is ubiquitous in human society,as well as in biological systems,but the phenomenon of group cooperation emerges frequently.Theoretical and empirical studies try to explain the phenomenon of cooperation in a fiercely competitive world from the perspective of biological evolution or by the use of analogies with evolution,but,whether the perspective of evolutionary biology or social science is used,the intrinsic mechanism of cooperation is still an unsolved problem.Evolutionary game theory,which combines game theory with evolutionary dynamics,has become a systematic and effective analytical tool for solving such problems.Traditional evolutionary game theory studies evolutionary dynamics and the equilibria of infinite populations by means of a single or group of replication dynamic equations,in which random effects are neglected.In reality,however,most populations are limited.The games of individuals in finite populations are affected by various uncertainty factors in the environment,so the evolutionary dynamics are significantly different from those of infinite populations.Considering the influence of random factors,a stochastic evolutionary game dynamic established by a stochastic process can describe the evolutionary dynamics of finite populations more accurately and realistically.A stochastic evolutionary dynamic considers systems as discrete processes and determines the directions of system evolution with certain probabilities.The updating rules that describe how individuals change their strategies as they acquire new knowledge or information play very important roles in the stochastic evolutionary dynamics of finite populations.Therefore,stochastic evolutionary games of finite populations must specify their updating rules,so that we could obtain different evolutionary game dynamics that result from different updating rules.The Moran process,the Wright–Fisher process,imitation updating,and aspiration-driven updating are four common rules used to update the strategies in finite populations.This paper first summarizes the research on stochastic evolutionary dynamics,and then discusses the four common strategy update rules.The main content and conclusions of this study are described below.Firstly,with the assumption that individuals who choose different strategies have different selection intensities,evolutionary game dynamics based on the Moran process and Wright–Fisher process with selection differences were established.The fixation probabilities of cooperative strategies in two evolutionary game dynamics with weak selection were solved.The fixation probabilities are related not only to the game matrix and population size but also to the selection intensities.By comparing the fixation probabilities with weak selection to those with neutral selection,the necessary conditions required for natural selection to favor one strategy that is the evolutionary stable strategy were analyzed.For the prisoner’s dilemma game,coexistence game,and coordination game,the relationships between the fixation probabilities of cooperative strategies and population size,as well as the selection intensities,were obtained by numerical analysis.The relationships between the fixation times of cooperative strategies and selection intensities were obtained by simulation analysis.Numerical analysis showed that,in the prisoner’s dilemma game,the fixation probability decreased as the selection intensities of the cooperation and betrayal strategies increased.However,in the coexistence and coordination games,the fixation probabilities increased as the selection intensity of the cooperative strategy increased but decreased as that of the betrayal strategy increased.Moreover,in the three games,the fixation probabilities decrease monotonously as population size increases.A simulation analysis showed that,in the prisoner’s dilemma game,the fixation time decreased as the cooperative strategy’s selection intensity increased but remained unchanged as the betrayal strategy’s selection intensity increased.In the coexistence and coordination games,the fixation time increased as the cooperative strategy’s selection intensity increased.In the coexistence game,the fixation time remained almost unchanged as the betrayal strategy’s selection intensity increased,whereas,in the coordination game,the fixation time decreased as the betrayal strategy’s selection intensity increased.In addition,the dynamic stochastic game model of Moran process was applied to solving the problems of third-party logistic companies participating in supply chain finance and strategy selection.Secondly,it was assumed that individuals could adopt two updating rules to change strategies.A hybrid evolutionary game dynamic that integrated the Moran process and imitation updating was established to study how population evolutionary dynamics were affected by individuals’ changing their strategies according to two updating rules.The fixation probability and time of the cooperative strategy were obtained by analysis.The fixation probability was independent of the probability of an individual’s adoption of imitation updating,while the fixation time is related to the probability of an individual’s adoption of imitation updating.The relationships between the fixation time and selection intensities,and the probability of an individual’s adopting imitation updating in the prisoner’s dilemma,coexistence,and coordination games were obtained by numerical analysis.In the prisoner’s dilemma and coordination games,the fixation time decreased as the selection intensity increased,whereas,in the coexistence game,it increased as the selection intensity increased.In all three games,the fixation time increased with the probability of imitation updating’s adoption,but the relationship was not linear.Thirdly,it was also assumed that individuals could adopt two updating rules to change strategies.A hybrid evolutionary game dynamic model that integrated both imitation and aspiration-driven updating was established.We conducted a numerical analysis of the unstructured population and a simulation analysis of the structured population to construct a random matching model.Then,we obtained the change in the relationships between the mean proportion of the cooperators(MFC),and the selection intensity and the probability of an individual’s adoption of aspiration-driven updating.In the prisoner’s dilemma and coexistence games,aspiration-driven updating promotes cooperation in an unstructured population but suppresses it in a structured one and a random matching model.In the coordination game,aspiration-driven updating promotes cooperation in a structured population and a stochastic matching model but suppresses it in an unstructured population.Moreover,MFC in different populations and random matching models varies with the aspiration level.The evolutionary characteristics of MFC in different structural populations and a random matching model remain robust.Finally,previous studies on evolutionary games always assumed that an individual’s payoff could be represented by exact numbers.However,because of various uncertain factors in the environment,the payoff would not have an accurate value but should be represented by a fuzzy number.Therefore,we studied a 2×2symmetric game with fuzzy payoffs to explore the fuzzy evolutionary game dynamics of the Moran process in a finite population.Individuals’ payoffs were represented by trapezoid and normal fuzzy numbers.The fuzzy fixation probabilities of cooperative strategies were obtained with weak selection.The conditions for natural selection to favor one strategy to fixate and become the fuzzy evolutionary stable strategy were analyzed.The stochastic evolution game models with the trapezoid and normal fuzzy numbers were applied to solving the problems of third-party logistic companies participating in supply chain finance and strategy selection,respectively,among pollution-producing enterprises.