The characteristics of average abundance function of multi-player threshold public goods evolutionary game model under redistribution mechanism

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. The characteristics of average abundance function based on multi-player threshold public goods evolutionary game model under redistribution mechanism have been explored by analytical analysis and numerical simulation in this article. The main research findings contain four aspects. Firstly, we deduce the concrete expression of expected payoff function. In addition, we obtain the intuitive expression of average abundance function by taking the detailed balance condition as the point of penetration. Secondly, we obtain the approximate expression of average abundance function when selection intensity is sufficient small. In this case, average abundance function can be simplified from composite function to linear function. In addition, this conclusion will play a significant role when analyzing the results of the numerical simulation. Thridly, we deduce the approximate expression of average abundance function when selection intensity is large enough. Because of this approximation expression, the range of summation will be reduced, the number of operations for average abundance function will be reduced, and the operating efficiency for numerical simulation will be improved. Fourthly, we explore the influences of parameters (the size of group d, multiplication factor r, cost c, aspiration level α and the proportion of income redistribution τ) on the average abundance function through numerical simulation. Also the corresponding results have been explained based on the expected payoff function and function h(i, ω). It can be concluded that when selection intensity ω is small, the effects of parameters (d, r, c, α and τ) on average abundance function is slight. When selection intensity ω is large, there will be five conditions. (1) Average abundance function will decrease with d regardless of whether threshold m is small or large. (2) Average abundance function will decrease at first and then increase with r when threshold m is small. Average abundance function will increase with r when threshold m is large. (3) Average abundance function will basically remain unchanged with c regardless of whether threshold m is small or large. (4) Average abundance function will remain stable at first and then increase with α when threshold m is small. Average abundance function will remain stable at first and then decrease with α when threshold m is large. It should be noted that average abundance function will get close to 1/2 when α is large enough. (5) Average abundance function will increase with τ regardless of whether threshold m is small or large.