Average abundance function of multi-player threshold public goods without initial endowment evolutionary game model under differential aspiration levels and redistribution mechanism

Abstract

Abstract The average abundance function reflects the level of cooperation in the population.So it is important to analyze how to increase the average abundance function in order to facilitate the proliferation of cooperative behavior.We explore the characteristics of average abundance function X A ( ω ) based on threshold public goods evolutionary game model without initial endowment under differential aspiration levels and redistribution mechanism by analytical analysis and numerical simulation. The main work contains four aspects. (1) We deduce the concrete expression of expected payoff function. We also obtain the intuitive expression of average abundance function on the basis of detailed balance condition. (2) We have deduced the approximate expressions of average abundance function when selection intensity is sufficient small. (3) We have deduced the approximate expressions of average abundance function when selection intensity is large enough. The range of summation for average abundance function will be reduced because of this approximation expression. (4) We analyze the influence of the size of group d , multiplication factor r , and cost c on average abundance function through numerical simulation. On one hand, when selection intensity is small, the influence of parameters on average abundance function is slight. On the other hand, when selection intensity is large, average abundance function will decrease with d . The average abundance function will decrease at first, and then increase quickly with the increase of r if threshold m = 4 . The average abundance function will decrease slowly at first, and then increase with the increase of r if threshold m = 9 . The average abundance function will decrease with the increase of c if threshold m = 4 . The average abundance function will remain unchanged at first, then increase, at last it will remain stable with the increase of c if threshold m = 9 . Furthermore, these conclusions have been explained based on expected payoff function π ( · ) and function h ( i , ω ) .