Evolutionary cooperation dynamics of combining imitation and super-rational aspiration induced strategy updating

In evolutionary game theory, strategy updating plays an important role in the evolution of cooperation, mainly including the Moran process, imitation, aspiration-driven updating, and super-rational aspiration induced strategy updating. Previous studies have focused on a single strategy updating but ignored the impact of environmental stochasticity and individual preference. In this paper, we study the evolutionary cooperation dynamics combining imitation and super-rational aspiration induced strategy updating in well-mixed finite populations. That is, individuals can no longer use a single update, but can choose to use imitation updating or super-rational aspiration induced strategy updating (i.e. mixed strategy updating). The closed-form expression of the fixation probability under arbitrary selection intensity is given, the approximate expression of the cooperation fixation probability is given, and the parameters favorable to cooperation fixation are given. The results show that cooperation is promoted when the cooperator is super-rational and inhibited when the defector is super-rational. This conclusion is verified by the approximate results of the mean-field theory and the simulation results in the structured population. In addition, the evolution of cooperation is given in two cases: the cooperator is completely super-rational but the defector is not super-rational, and the cooperator is not super-rational but the defector is completely super-rational. These results provide new perspectives on the evolution of cooperation.