N–person Stochastic Games: Extensions of the Finite State Space Case and Correlation

In this chapter, we present a framework for m-person stochastic games with an infinite state space. Our main purpose is to present a correlated equilibrium theorem proved by Nowak and Raghavan [42] for discounted stochastic games with a measurable state space, where the correlation of the different players’ strategies employs only “public signals” [16]. We will also provide a detailed survey of the literature containing related results, some approximation theorems for general state space stochastic games (the existence of ε-equilibria), and the existence of equilibria in some classes of countable state space stochastic games with applications to queueing models. We consider m-person non-zero-sum stochastic games for which: (i) S is a nonempty Borel state space. (ii) Xi is a nonempty compact metric space of actions for player i. We put X = X1 ×X2 × · · · ×Xm. (iii) Ai(s) is a nonempty compact subset of Xi and represents the set of actions available to player i in state s. We assume that {(s, a) : s ∈ S and a ∈ Ai(s)} is a Borel subset of S ×Xi. Put