Games with Self-Generating Distributions

Abstract

In this paper I develop an N-person stochastic game in which each player views himself as facing a Markov decision process. Specifically, every player is assumed to choose his strategy as a policy which is optimal with respect to the process in question. It is clear that such a strategy need not be a best reply to the strategy choices of the other players, so a Nash equilibrium might not be appropriate for this game. Consider the following alternative equilibrium concept. Every player compares the subjective transition probabilities from his Markov decision process with the objective frequencies which he encounters during the course of the game. If no player receives disconfirming evidence about the subjective probabilities he is using, then the game is said to be in informational equilibrium. In this case the subjective probabilities are termed self-generating. Clearly, an informational equilibrium falls into the class of fulfilled expectations or rational equilibria. The main result of the paper is that an informational equilibrium exists for the game under consideration. The motivation for studying this particular class of games derives from the microeconomic treatment of household and firm intertemporal decisions under uncertainty. In economic theory it is common to treat these decision problems within a dynamic programming Markov decision process framework.' Here Markov transistion probabilities are used to describe the random prices faced by households and the random demand functions faced by firms. A question which would seem to be of considerable interest involves the problem of how one might model a complete economy in which each agent treats his own decision problem as a Markov decision process. Choosing an appropriate equilibrium concept would naturally be of paramount importance. This is the essential issue with which the present paper is concerned. It is important though to emphasize that the underlying approach does not really depend upon the specification of any particular economy. Therefore the general framework is described in terms of an