Optimal bias of utility function between two-layer network for the evolution of prosocial behavior in two-order game and higher-order game

Cooperation is an important research object in economics, sociology, and biology, and the evolution of cooperation in structured populations is a interesting research topic. We mainly focus on the evolution of cooperation with two-order and higher-order game in two-layer network. We introduce a bias coefficient of utility function and study the influence of bias coefficient on the evolution of cooperation in two-layer network. We firstly provide theoretical analysis of fixation probabilities of two-order and higher-order game under weak selection in two-layer network.Secondly,based on the expression of fixation probability, we obtain the critical value of the two different games by comparing the size relationship of fixation probability under weak selection condition and neutral selection condition. Finally, by comparing the relationship between the critical value of single-layer and two-layer network in two-order game and higher-order game, when the nonlinear factor satisfies certain conditions, it is concluded that when the optimal bias coefficient tends towards 0 is met, some two-layer networks promote the evolution of cooperative behavior more than some single-layer networks.