Universally measurable strategies in zero-sum stochastic games. (English) Zbl 0592.90106

Nowak, Andrzej S.

Ann. Probab. 13, 269-287 (1985).

This paper deals with zero-sum discrete-time stationary models of stochastic games with Borel state and action spaces. It is known that the value function of a Borel space stochastic game which satisfies neither semi-continuity nor compactness conditions need not be universally measurable [U. Rieder, On semi-continuous dynamic games, Tech. Rep., Univ. Karlsruhe (1978)]. The main purpose of this paper is to give some semi-continuity and compactness conditions that ensure the existence and universal measurability of the value function of the game as well as the existence of universally measurable optimal strategies for both players. The fundamental result in this paper is a minimax selection theorem extending a selection theorem of L. D. Brown and R. Purves [Ann. Statist. 1, 902-912 (1973; Zbl 0265.28003)]. As applications of this basic result, some new theorems on absorbing discounted, and positive stochastic games are proved.

MSC:

91A15	Stochastic games, stochastic differential games
91A05	2-person games
91A60	Probabilistic games; gambling
28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets

Keywords:

optimal stationary strategies; zero-sum discrete-time stationary models of stochastic games; Borel state and action spaces; semi-continuity and compactness conditions; existence and universal measurability of the value function; universally measurable optimal strategies; minimax selection theorem; absorbing discounted, and positive stochastic games

Citations:

Zbl 0265.28003