The evolution of cooperation with environmental feedback on higher-order networks

The role of environmental feedback mechanisms in the evolution of cooperation is attracting increasing attention within the academic community. However, existing research has rarely explored how these mechanisms influence the emergence of multi-player games and the evolution of cooperation within higher-order networks. In reality, higher-order relationships within a group merely provide the structural foundation for multi-player games, whilst the ultimate formation of such games depends on the internal gaming environment of the group. On this basis, this study, grounded in simplicial complex structures, constructs an evolutionary game model with an environmental feedback mechanism to reveal the impact of the mechanism on the evolution of cooperation within higher-order networks. The results indicate that the environmental feedback mechanism in this model effectively promotes the evolution of cooperation within simplicial complex structures, and that stricter environmental feedback is more conducive to enhancing the level of cooperation. Furthermore, in many cases, as the proportion of 2-simplices ρ increases, the cooperation rate exhibits a non-monotonic trend that first increases and then decreases, reaching a peak at a relatively small value of ρ. This highlights the significant positive role of sparse 2-simplices in the structure for the evolution of cooperation. Moreover, our findings also reveal a shift: cooperation is facilitated by a preference for multi-player games’ payoffs at small ρ, but by a preference for pairwise games’ payoffs at large ρ. Overall, these findings reveal the rich dynamic characteristics of the evolution of cooperation in higher-order networks with environmental feedback, providing new insights into how cooperation arises and flourishes within complex interactive structures.