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,… Click to show full 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, 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.
               
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