Cooperative Games in a Stochastic Environment

Abstract

In this chapter we face the task of learning to win a game characterized by highly complex aspects. Our player, Bob, has no information about the ability of his opponent, Nature; moreover the effects of his actions on the outcome of the single contest are neither known nor predictable, since the game mechanism is quite complex and uncertain. Bob can base his strategy only on a monotonicity property: the more Bob increases the value of a given parameter of the game (the one representing his strength), the more his winning ability improves. This class of games exploits the results of game theory [Nash, 1951] only partially since the payoffs associated to each strategy are random variables with unknown distribution. A similar lack of information can be overcome by considering this problem as a dynamic process in a computational learning context. In particular, we refer to the online learning paradigm [Angluin, 1988, Ben-David et al., 1997, Blum, 1996, Littlestone, 1988] along with the statistical inferential framework called algorithmic inference introduced in this book [Apolloni et al., 2002a]. In this framework we evaluate the actions through the construction of confidence intervals for the losing probability, thus abandoning the common goal of optimizing the expected value of a utility function [Blackwell and Ghirshick, 1979].