Semicontinuous nonstationary stochastic games
A zero-sum discrete-time infinite horizon stochastic game with metric state spaces and compact (separable) metric action spaces for player II (I). The payoff for player I is a lower semicontinuous function on the space of all plays of the game. The transition law is nonstationary and consists of a sequence of weakly continuous transition probabilities associating with each n-stage history of the game a probability distribution of the \((n+1)st\) state. Such a stochastic game is a generalization of semicontinuous discounted and positive Markov games studied for the first time by \textit{A. Maitra} and \textit{T. Parthasarathy} [J. Optimization Theory Appl. 5, 289-300 (1970; Zbl 0181.232), and ibid. 8, 154-160 (1971; Zbl 0206.492)] and subsequently by \textit{T. Parthasarathy} [Int. J. Game Theory 2, 25-37 (1973; Zbl 0368.90139)], \textit{P. R. Kumar} and \textit{T. H. Shiau} [SIAM J. Control Optimization 19, 617-634 (1981; Zbl 0474.93053)] and \textit{U. Rieder} [On semicontinuous dynamic games, Preprint, University of Karlsruhe (1978)]. It is shown that such a game has a value and player II has an optimal strategy. This generalizes an earlier result of the author [Proc. Am. Math. Soc. 92, 418-424 (1984; Zbl 0559.90103)] where the state and action spaces are assumed to be countable sets. An alternative approach to zero-sum nonstationary stochastic games, where the payoff function (transition law) is semicontinuous (strongly continuous), but with respect to the actions of player II only, is taken in a recent report of the author [''A Borel space framework for zero-sum nonstationary stochastic games, Preprint, Wrocław Tech. Univ., Poland (1986)].
zero-sum discrete-time infinite horizon stochastic game
scientific article; zbMATH DE number 3956858
scientific article