Population games on dynamic community networks

TL;DR

This paper introduces a new mathematical framework for evolutionary game dynamics on networks with evolving communities, coupling replicator equations with environmental feedback to analyze stability and convergence.

Contribution

It develops a coupled differential equation model for population games on dynamic networks and proves convergence to stable states for specific game classes.

Findings

Characterization of equilibria in co-evolving community networks

Proof of convergence to stable states in symmetric two-action games

Numerical simulations supporting theoretical results

Abstract

In this letter, we deal with evolutionary game theoretic learning processes for population games on networks with dynamically evolving communities. Specifically, we propose a novel mathematical framework in which a deterministic, continuous-time replicator equation on a community network is coupled with a closed dynamic flow process between communities that is governed by an environmental feedback mechanism, resulting in co-evolutionary dynamics. Through a rigorous analysis of the system of differential equations obtained, we characterize the equilibria of the coupled dynamical system. Moreover, for a class of population games with two actions and symmetric rewards a Lyapunov argument is employed to establish an evolutionary folk theorem that guarantees convergence to the evolutionary stable states of the game. Numerical simulations are provided to illustrate and corroborate our findings.