An Abelian Theorem for a Markov Decision Process in a System of Interacting Objects with Unknown Random Disturbance Law

This paper studies a mean-field approach for Markov decision processes in a class of systems of a large number of objects that interact with each other according to an observable -but unknown- law for the central controller. The central controller acts under the ergodic cost criterion with Borei state and control spaces, bounded costs, and compact action space. We depart from the characterization of the discounted optimal strategies, and then, by means of an Abelian theorem, we study the existence of average cost optimal stationary policies in the original model. We also analyze the performance of the mean-field limit optimal policies in the original model.