The Multi-player Evolutionary Game Model And Average Abundance Function Under Aspiration Dynamic

Evolutionary game is developed from biological Moran process,it provides new ideas and research tools for explaining how cooperation is generated and maintained,so as an important mathematical model,it is widely used in various social and economic issues.Under the premise of inheriting the basic ideas of game theory,evolutionary game has continuously expanded new research categories,the main research contents of evolutionary game include four:game model,population structure,strategy updating rule and method for evaluating evolutionary result.Studying game model can help us understand how individuals participate in the game and the corresponding game characteristics and ways,and also help us establish new research model according to new game phenomenon in social development.Population structure is divided into finite population and infinite population.Different population structure will lead to differences in individual encounters and game occurrences,for example,for infinite well-mixed population,individuals in the population meet randomly,while individuals in finite structured population can only meet neighboring individuals and make game happen,so we can understand the interaction behavior of individuals by studying population structure.Strategy updating rule refers to individuals who participate in the game will update strategy according to certain rules after the game,currently,the main strategy updating rules include fitness rule,aspiration rule and imitation rule,and the combination among them,different strategy updating rules will lead to different evolutionary dynamics.Method for evaluating evolutionary result mainly include two types,fixation probability and average abundance.This paper mainly studies the evolution model of the multi-player game in finite population(three classic multi-player social dilemma game models as background)in the aspiration evolution dynamics(referred to aspiration rule).The research of this paper is different from other researches of evolutionary games in the following two aspects:first,this paper focuses on the multi-player game in the finite population rather than the pairwise symmetric game;second,this paper focuses on the average abundance of evolutionary game under aspiration evolutionary dynamic of different multi-player game rather than fixation probability.Average abundance function is one of the effective tools for judging the results of evolutionary games.It characterizes the average number of individuals taking a certain type in population in the evolutionary outcome.Therefore,it has statistical significance,which is easy to understand and cognize the judgment results,and is especially suitable for the analysis of multi-player game evolution results.Taking the multi-player snowdrift,linear public goods,and threshold public goods as the research background,driven by the aspiration rule,the representation and characteristics of the average abundance function of the three evolutionary game model are systematically studied from the theoretical and simulation aspects for the case of no mutation and mutation.Firstly,the behavior of multi-person evolutionary game is discussed,the corresponding income matrix is analyzed,using probability theory and mathematical statistics,it reveals that the income function can be expressed as the weighted sum of the product of the income matrix element and the corresponding extraction probability,thus a general expression of income function is obtained.Based on the general expression of income function and combining the three classical game models,the mathematical expressions of the snowdrift game,linear public goods game,and threshold public goods game income function are given by mathematical and physical analysis method,and the relationship between the average abundance function and the corresponding parameters can be analyzed intuitively and quantitatively,which provides the theoretical basis for further study of the average abundance function,in addition,according to the expression of the income function,the curves about the income function changes with the evolution state i are drawn,more intuitively reveals the impact of the corresponding parameters on the income function and more deeply explains the trend of the income function changes with i.Through the operation analysis of the income function,the progressive characteristics and the relative size relationship of the income functions between different strategies can be further grasped,which lays a foundation for subsequent analysis.Based on the traditional research combined with the detail balance formula,the visual expression of the average abundance function of the multi-player evolutionary game model driven by aspiration rule is given by the function h(i,ω)with clear and intuitive meaning,the expression forms the average abundance function through a plurality of compound operations through functions h(i,ω)and related parameters.Secondly,the basic characteristics of the average abundance function of the above three multi-player evolutionary game models are studied in depth.In the case of weak selection strength,the expression of the first-order Taylor expansion of the average abundance function is given for the case of no mutation and mutation,and the simulation results of the average abundance function are performed by the first-order Taylor expansion,expands the range of applications for the first-order Taylor expansion.For the case of strong selection strength,the truncation analysis method is used to prove that the average abundance function can be simplified to the sum of partial population in the case of no dominant(the proportion of A stratege is less than 1/2).On this basis,according to the difference of the income function of different strategies tends to be fixed when i(the number of individuals who adopt A stratege,ie,evolutionary state)is large enough,the characteristic range of the truncation approximation formula is discussed,and then the range of parameters that makes the truncation approximation significant in the three models is given.Further,a limited range of multiplication coefficients for the multi-player linear public goods evolutionary game model is given.The research on the evolutionary game model of multi-player threshold public goods shows that the threshold m and i must meet the corresponding inequality together,and the limited range of threshold m and i is obtained through corresponding mathematical calculations.The truncation approximation theory is verified and further analyzed by numerical calculation.We simulate the influence of parameters change because the average abundance function is regarded as a multi-parameter function.Since not only the relationship between the parameters is complicated but also the parameters contain continuous variables and discrete variables,the data simulation is used to analyze the influence of multi-parameters on the average abundance function of the finite well-mixed population based on aspiration rule.The multi-player snowdrift game model,the linear public goods game model and the threshold public goods game model are combined to simulate and analyze the influence of parameters,such as group size,aspiration level and cost,on the average abundance function under different selection strengths.Some regular conclusions have been obtained,revealing the relationship between the average abundance function and related parameters from both qualitative and quantitative aspects.The multi-person snowdrift evolutionary game model is applied to the environmental pollution control decision-making problem,and the traditional model has been improved to make it closer to the actual situation.Thirdly,it analyzes the multi-player evolutionary game behavior under the combination of aspiration rule and mutation,and gives the intuitive expression of the average abundance function with the characteristic of mutation.Approximate analysis of the average abundance function with the characteristic of mutation is performed by power series expansion.Based on the different characteristics of the income functionπ(i)and function h(i,ω)of three multi-player evolutionary game models,the approximation of the average abundance function and the applicable range of selection strength corresponding to different multi-player evolutionary game models are given.For the multi-player snowdrift evolutionary game model,for the two cases of relatively small and relatively large selection intensity,the empirical formula for of the average abundance function with significant approximation effect is obtained within the corresponding selection intensity range.For the linear public goods evolutionary game model,the remarkable approximation of the average abundance function is given under the condition of relatively small multiplication factor,and it is proved that the approximation effect is not remarkable when the multiplication factor is relatively large.For the threshold public goods evolutionary game model,the ranges for threshold and multiplication factor are classified in the dominant and non-dominant cases,and the approximate representation of the corresponding range average abundance function is given.Finally,based on three multi-player evolutionary game models,the influence of multi-parameters on the average abundance function with the characteristic of mutation is calculated and analyzed by data simulation method.By comparing the average abundance function with the characteristic of mutation with the average abundance function without the characteristic of mutation,it is found that when the selection intensity is relatively small,the average abundance function is faintly affected and interfered by the parameters when mutation is relatively small;when the selection intensity is relatively large,even if mutation is relatively small,the average abundance function is fiercely affected and interfered by the parameters.At the same time,combined with the influence of corresponding parameters and mutation 8 on the income function π(i)and function h(i,ω),the reason for the corresponding change trend is explained,and the law for the average abundance function with the characteristic of mutation changes with relevant parameters is mastered.