Stability analysis of multiplayer games on adaptive simplicial complexes

We analyze the influence of multiplayer interactions and network adaptation on the stability of equilibrium points in evolutionary games. We consider the Snowdrift game on simplicial complexes. In particular, we consider as a starting point the extension from only two-player interactions to coexistence of two- and three-player interactions. The state of the system and the topology of the interactions are both adaptive through best-response strategies of nodes and rewiring strategies of edges, respectively. We derive a closed set of low-dimensional differential equations using pairwise moment closure, which yields an approximation of the lower moments of the system. We numerically confirm the validity of these moment equations. Moreover, we demonstrate that the stability of the fixed points remains unchanged for the considered adaption process. This stability result indicates that rational best-response strategies in games are very difficult to destabilize, even if higher-order multiplayer interactions are taken into account.