Co-evolutionary Dynamics and Bayesian Interaction Games

In a Bayesian interaction game players have diverse preferences and are randomly matched according to an inhomogeneous random graph. A co-evolutionary process of networks and play gives a dynamic formalism for the joint evolution of the random graph and the actions the players use per match. Assuming that the players select actions and links according to log-linear functions taking as arguments the reward per match, we provide closed form solutions for the joint invariant distribution of the co-evolutionary process. We give sufficient conditions for the general selection of potential maximizers in the small noise limit, and also discuss concentration of the invariant distribution in the large population limit. Further, we present a general characterization theorem that a co-evolutionary process generates inhomogeneous random graphs, a large and important class of random graphs recently discussed in the economic and mathematical literature on random networks