Population games on dynamic community networks

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, in turn governed by an environmental feedback mechanism. When such a mechanism is independent of the game-theoretic learning process, a closed-loop system of differential equations is obtained. Through a direct analysis of the system, we study its asymptotic behavior. Specifically, we prove that, if the learning process converges, it converges to a (possibly restricted) Nash equilibrium of the game, even when the dynamic flow process does not converge. Moreover, for a class of population games-two-strategy matrix games- a Lyapunov argument is employed to establish an evolutionary folk theorem that guarantees convergence to a subset of Nash equilibria, that is, the evolutionary stable states of the game. Numerical simulations are provided to illustrate and corroborate our findings.