Repeated thinking promotes cooperation in spatial prisoner's dilemma game

Inspired by the realistic process of taking decisions in social life, we have proposed a repeated thinking mechanism in the evolutionary spatial prisoner's dilemma game where players are denoted by the vertices and play games with their direct neighbors. Under our mechanism, a player i will randomly select a neighbor j and then deliberate for M times before strategy updating. It will remain unchanged if not all M considerations suggest it to learn the strategy of j. We mainly focus on the evolution of cooperation in the systems. Interestingly, we find that the cooperation level f _C is remarkably promoted and f _C has a monotonic dependence on the caution parameter M, indicating that being cautious facilitates the emergence and persistence of cooperation. We give a simple but clear explanation for this cooperation promotion via detecting the cooperatordefector transition process. Moreover, the robustness of this mechanism is also examined on different noise levels and game models.