A Mean-Field Approach for Ergodic Nonzero-Sum Stochastic Games in a System of Interacting Objects with Additive Costs

This paper studies a mean-field approach for a nonzero-sum stochastic discrete-time game where the dynamic system is comprised of a large number of objects that interact with each other according to an observable—but unknown—law for the players. The central agents act under the ergodic cost criterion with Borel state and control spaces, bounded and additive costs, and compact action space. We depart from characterizing the Borel state space with an additive one-step transition function. Then, we prove the existence of stationary Nash equilibria for the discounted criterion, and, using an Abelian theorem, we study the existence of average cost optimal stationary equilibria. We also present a mean-field (deterministic) approximation of the original model to overcome the problem arising from the large dimensionality of the stochastic game. The paper concludes by analyzing the performance of the mean-field limit saddle points in the original model.