Evolutionary dynamics of cooperation with tolerance and historical payoff memory in the prisoner’s dilemma

The emergence and maintenance of cooperative behavior is a fundamental issue in evolutionary game theory. In this paper, within the framework of the prisoner’s dilemma game, we construct an evolutionary game model of agent behavior that simultaneously incorporates an individual tolerance mechanism and historical payoff memory to investigate the evolution of cooperation on complex networks. The results show that, as the temptation to defect increases, the overall level of cooperation in the system declines significantly. In addition, increasing the historical discount factor strengthens the influence of historical payoffs, thereby promoting the emergence and persistence of cooperative behavior. In contrast, the memory length has a relatively limited impact on the cooperation level. Further analysis reveals that the tolerance threshold plays different roles under different network structures, exhibiting a non-monotonic effect in scale-free networks. Meanwhile, the coevolution of strategy dynamics and network structure reconstruction facilitates the formation of local cooperative clusters, leading to a gradual stabilization of the network clustering structure. These findings highlight the important roles of historical memory and tolerance mechanisms in the evolution of cooperation on complex networks and provide new theoretical insights into understanding cooperative behavior in real-world social systems.