A tutorial on Zero-sum Stochastic Games

TL;DR

This paper provides a comprehensive mathematical introduction to zero-sum stochastic games, covering basic models, key results, and recent developments in the theory of these competitive dynamic systems.

Contribution

It offers a clear, structured presentation of fundamental results and proofs in zero-sum stochastic games, including convergence properties and examples, with a brief overview of recent advances.

Findings

Existence and formulas for game values

Convergence of discounted and finite-horizon values

Counterexamples with no asymptotic value

Abstract

Zero-sum stochastic games generalize the notion of Markov Decision Processes (i.e. controlled Markov chains, or stochastic dynamic programming) to the 2-player competitive case : two players jointly control the evolution of a state variable, and have opposite interests. These notes constitute a short mathematical introduction to the theory of such games. Section 1 presents the basic model with finitely many states and actions. We give proofs of the standard results concerning : the existence and formulas for the values of the n-stage games, of the $\lambda$-discounted games, the convergence of these values when $\lambda$ goes to 0 (algebraic approach) and when n goes to +$\infty$, an important example called 'The Big Match' and the existence of the uniform value. Section 2 presents a short and subjective selection of related and more recent results : 1-player games (MDP) and the compact non expansive case, a simple compact continuous stochastic game with no asymptotic value, and the general equivalence between the uniform convergence of (v n) n and (v $\lambda$) $\lambda$. More references on the topic can be found for instance in the books by Mertens-Sorin