Evolutionary dynamics of cooperation driven by a mixed update rule in structured prisoner's dilemma games

How to understand the evolution of cooperation remains a scientific challenge. Individual strategy update rule plays an important role in the evolution of cooperation in a population. Previous works mainly assume that individuals adopt one single update rule during the evolutionary process. Indeed, individuals may adopt a mixed update rule influenced by different preferences such as payoff-driven and conformity-driven factors. It is still unclear how such mixed update rules influence the evolutionary dynamics of cooperation from a theoretical analysis perspective. In this work, in combination with the pairwise comparison rule and the conformity rule, we consider a mixed updating procedure into the evolutionary prisoner's dilemma game. We assume that individuals adopt the conformity rule for strategy updating with a certain probability in a structured population. By means of the pair approximation and mean-field approaches, we obtain the dynamical equations for the fraction of cooperators in the population. We prove that under weak selection, there exists one unique interior equilibrium point, which is stable, in the system. Accordingly, cooperators can survive with defectors under the mixed update rule in the structured population. In addition, we find that the stationary fraction of cooperators increases as the conformity strength increases, but is independent of the benefit parameter. Furthermore, we perform numerical calculations and computer simulations to confirm our theoretical predictions.