Mutation promote cooperation in repeated games on structured populations

Cooperation is a fundamental phenomenon in human societies and multi-agent systems, yet it remains challenging to sustain among rational individuals. In realistic evolutionary processes, cooperative behavior is inevitably affected by behavioral variation and random perturbations; a common way to capture these effects is to introduce mutation, which allows strategies to occasionally change independently of payoffs. Previous studies have shown that introducing mutation into well-mixed populations can enhance strategic diversity and thereby promote the emergence of cooperation. However, interactions in real social and biological systems are constrained by spatial and social networks rather than being well-mixed, and such population structure can qualitatively change evolutionary outcomes through network reciprocity. Here, we investigate how mutation shapes the evolution of cooperation in repeated games on structured populations. Our simulations reveal that cooperation is difficult to sustain when mutation is extremely rare, whereas moderate mutation substantially promotes cooperation. We further identify parameter-dependent mutation regimes that maximize cooperation, and show that their locations depend on the payoff parameters and the considered strategy sets. Remarkably, mutation can facilitate the emergence of cooperation even when the benefit-cost ratio is only slightly larger than one. Further simulations demonstrate that this promoting effect persists in Erdős-Rényi random networks with different average degrees, indicating the robustness of our results with respect to variations in network structures.