Evolutionary dynamics and non-trivial phenomena of the Multi-player Stag Hunt Game on uniform power-law hypergraphs

Cooperation is a fundamental mechanism in biological and social systems, the evolution of cooperation within higher-order interaction structures remains a frontier issue in evolutionary game theory. The Multi-player Stag Hunt Game captures the trade-off between collective benefit and individual risk in groups. This paper investigates the evolutionary dynamics of the Multi-player Stag Hunt Game on Uniform Power-law Hypergraphs. By integrating heterogeneous mean-field theory and replicator dynamics, we derive the system's evolutionary dynamics equation, demonstrating the existence of two stable equilibrium states and one unstable critical point—which signifies typical coordination game characteristics. Theoretical analysis reveals the influence of parameters such as hyperedge order, power-law exponent, and cooperation threshold on cooperative behavior. Furthermore, through simulation experiments, we explore the effects of different strategy update rules and structural parameters. The results indicate that enhanced hypergraph heterogeneity accelerates evolution via high-hyperdegree nodes. Increasing the cooperation threshold suppresses the overall cooperation level, yet it also creates conditions for specific strategy rules to exert their influence. The simulations also uncover several non-trivial phenomena, such as the "decrease-then-increase" trend in cooperation levels under the Profit-First Imitation (PFI) rule and the emergence of topological locking under the Hyperdegree-Status Oriented Imitation (HSOI) rule, which facilitates the long-term coexistence of cooperation and defection.