Stability analysis of multiplayer games on adaptive simplicial complexes

TL;DR

This paper investigates how multiplayer interactions and network adaptation affect the stability of equilibrium points in evolutionary games on adaptive simplicial complexes, revealing that stability is robust despite complex adaptations.

Contribution

It introduces a low-dimensional differential equation model for adaptive multiplayer games on simplicial complexes and demonstrates stability robustness under adaptation.

Findings

Stability of fixed points remains unchanged despite network adaptation.

Derived low-dimensional differential equations accurately approximate system dynamics.

Higher-order interactions do not destabilize equilibrium points.

Abstract

We analyze the influence of multiplayer interactions and network adaptation on the stability of equilibrium points in evolutionary games. We consider the Snowdrift game with both two-player and three-player interactions on simplicial complexes. 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 interactions are taken into account.