LAUSR

The average abundance function under aspiration rule has received much attention in recent years due to the fact that it reflects the level of cooperation of the population. Under aspiration rule, individuals will adjust their strategies based on a comparison of their payoffs from the evolutionary game to a value they aspire, called the aspiration level. This means it is important to analyze how to increase the average abundance function in order to facilitate the proliferation of cooperative behavior. However, further analytical insights have been lacking so far. In addition, the theoretical basis of the corresponding data simulation results is still lacking sufficient understanding. Therefore, this article analyses the characteristics of average abundance function $${X_A}(\omega )$$ of multi-player threshold public goods evolutionary game model under aspiration rule through analytical analysis and numerical simulation. In general, the main work of this article contains three aspects. (1) The concrete expression of expected payoff function and the intuitive expression of average abundance function has been obtained. (2) The approximate expressions of average abundance function when selection intensity is large enough has been deduced. The range of summation for average abundance function will be reduced because of this approximation expression. (3) We analyze the influence of the size of group d, multiplication factor r, cost (and initial endowment) c, aspiration level $$\alpha $$ on average abundance function through numerical simulation. On the one hand, the influence of parameters on average abundance function when selection intensity is small is slight. On the other hand, average abundance function will decrease with d. It will show an U-shaped trend or an upward trend when $${X_A}(\omega )$$ changes with r. The $${X_A}(\omega )$$ basically remains stable with the increase of c. It will show a stair-like trend when $${X_A}(\omega )$$ changes with $$\alpha $$ . Furthermore, these conclusions have been explained on the basis of expected payoff function $$\pi \left( \centerdot \right) $$ and function $$h(i,\omega )$$ .