Imitation Dynamics in Population Games on Community Networks

In this article, we study the asymptotic behavior of deterministic, continuous-time imitation dynamics for population games over networks. The basic assumption of this learning mechanism—encompassing the replicator dynamics—is that players belonging to a single population exchange information through pairwise interactions, where they become aware of the actions played by the other players and the corresponding rewards. Using this information, they can revise their current action, imitating one of the players they interact with. The pattern of interactions regulating the learning process is determined by a community structure. First, the set of equilibrium points of such network imitation dynamics is characterized. Second, for the class of potential games and for undirected and connected community networks, global asymptotic convergence is proved. In particular, our results guarantee convergence to a Nash equilibrium from every fully supported initial population state in the special case when the Nash equilibria are isolated and fully supported. Examples and numerical simulations are offered to validate the theoretical results, and counterexamples are discussed for scenarios when the assumptions on the community structure are not verified.