Existence of Value and Randomized Strategies in Zero-Sum Discrete-Time Stochastic Dynamic Games

Two players with conflicting objectives are simultaneously controlling a discrete-time stochastic system. The goal of this paper is to analyze such zero-sum, discrete-time, stochastic systems when the two players are allowed to use randomized strategies.Previous results have been restricted to systems with finite or compact state spaces. Such restrictions are usually untenable from the point of view of applications, since many applications frequently use either the integers or $\mathbb{R}^n $ as a state space. Our results are proved for complete, separable, metric spaces which are very useful for applications.All previously known results emerge as special cases of our results. In addition, a variety of conjectures and open problems are resolved regarding the existence of a value function, its properties such as Borel measurability or continuity, and the existence for either or both players of optimal or $\varepsilon $-optimal stationary strategies.