Evolutionary cooperation under collective risk on hypergraphs

Understanding how cooperation emerges and persists in complex systems remains a central challenge in evolutionary game theory. While prior research has predominantly focused on pairwise interactions, real-world societies and ecosystems are often shaped by group interactions that are inherently higher-order. Here, we investigate the evolution of cooperation in a threshold public goods game with collective risk, implemented on uniform random hypergraph. In this setting, individuals interact within hyperedges, and cooperation yields benefits only when the collective contribution exceeds a critical threshold; otherwise, all participants face the risk of losing their endowments. Using numerical simulations and analytical approximations based on replicator dynamics, we show that collective risk substantially promotes cooperation. Notably, a moderate risk threshold leads to the most pronounced enhancement of cooperative behavior. By tracking the distribution of cooperators across all hyperedges, we find that collective risk increases the resilience of highly cooperative groups to defector exploitation, while simultaneously reducing the prevalence of low-cooperation groups. This structural shift underpins the mechanism by which collective risk fosters cooperation. Finally, we validate the robustness of this mechanism by varying key structural parameters of the hypergraph, further confirming the effectiveness of collective risk in sustaining cooperation under high-order interaction networks.