Role of strategy update rules in the spatial memory-based mixed strategy games

Based on a continuous mixed strategy game model considering the memory mechanism and the uncertainty of individual decision-making, we explore the influence of several different update rules on the spatial cooperation behavior. Here, we mainly investigate the role of the Moran-like process, Fermi rule and replicator dynamics in the evolution of cooperation. Meanwhile, we further compare the evolution of cooperation when the memory mechanism and individual uncertainty are taken into account. In particular, memory length M and strategy adjustment factor $$\delta $$ δ can promote the cooperation behavior in different ways. Extensive Monte Carlo simulations indicate that the Moran-like process can generally drive the expansion of cooperative clusters faster and finally achieve the highest frequency of cooperation although the Fermi update rule performs better under the same condition. Moreover, the replicator dynamics create the worst scenario as far as the evolution of cooperation is concerned. Therefore, the level of cooperation strongly depends on the strategy update rules; these findings may be helpful to understand and analyze the evolutionary process of cooperation in many real-world natural and social systems.