Punishment Induces Secondary Cooperation within Structured Populations Facing Social Dilemmas

Abstract Interactions in the real world are not limited to pairwise relationships but often involve higher-order structures. However, the evolution of cooperation in the presence of both interaction types remains poorly understood, especially regarding which mechanisms can sustain cooperation in social dilemmas. Among these, punishment is widely recognized as a regulatory mechanism that can discourage non-cooperative behavior and potentially promote collective action. In this study, we propose a multi-strategy game framework on hypergraphs to investigate how punishment promotes cooperation in structured populations facing social dilemmas, where pairwise and higher-order interactions coexist. Within this framework, pairwise and higher-order interactions are modeled as two-player and multi-player games, respectively, and individuals participate concurrently across both types. Through stability analysis, we find that punishment can trigger a secondary cooperation effect. Remarkably, cooperation can be effectively promoted regardless of how severe the punishment is—even when its cost exceeds the net loss imposed on the punished individual. Extending the analysis to finite-size populations, we observe that small groups tend to suppress cooperation, whereas large groups promote higher overall cooperation levels. We further validate our findings via numerical simulations on three empirical network structures, all of which confirm that punishment leads to enhanced cooperation. Our results demonstrate that punishment serves as an effective and scalable driver of cooperation in systems with complex interaction structures.