Random Fixed Points of M U Ltifu N Ctio N S in Games an D Dynam Ic Programming

Recently several authors demonstrated random fixed point theorems for various classes of multifunctions ([7], [8], [2], [3], [12], [10]). On the other hand we do not know any work on applications of these theorems. In this paper we apply to games and dynamic programming a random analogue of the Fan-Kakutani fixed point theorem. We consider a zero-sum two-person game depending on a random parameter, and present sufficient conditions for the existence of a measurable solution. Then we study the existence of measurable stationary optimal programs in discounted dynamic programming with a random parameter. 1. Preliminaries. Let X, Y be non-empty sets. A multifunction Aę?(-*i)+( 1/ 1) (*) is quasi-concave, and the sets O ) != {y e<p{x):v (x) = u (x , y)} are convex (possibly empty). Received December 10, 1980. AMS (MOS) subject classification (1980). Primary 60H25. Secondary 58C30. * Instytut Matematyki Uniwersytetu Śląskiego, Katowice, ul. Bankowa 14, Poland.