Games with Incomplete Information Played by “Bayesian” Players

Abstract

The paper develops a new theory for the analysis of games with incomplete information where the players are uncertain about some important parameters of the game situation, such as the payoff functions, the strategies available to various players, the information other players have about the game, etc How-ever, each player has a subjective probability distribution over the alternative possibibties In most of the paper it is assumed that these probability distributions enter-tained by the different players are mutually "consistent", in the sense that they can be regarded as conditional probability distributions derived from a certain "basic probability distribution" over the parameters unknown to the various players But later the theory is extended also to cases where the different players' subjective probability distributions fail to satisfy this consistency assumption In cases where the consistencj assumption holds, the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probablity distribution, and the outcome of this lottery will decide Tvhich particular subgame will be played, l e , what the actual values of the rele-vant parameters will be in the game Yet, each player will receive only partial information about the outcome of tbe lottery, and about the values of these parameters However, every player will know the "basic probability distribu-tion" govermng the lottery Thus, technically, the resulting game will be a game with complete information It is called the Bayes-equivalent of the original game Part I of the paper describes the basic model and discusses vari-ous intuitive interpretations for the latter Part II shows that the Nash equi-librium points of the Bayes-equivalent game yield "Bayesian equilibrium points" for the original game Finally, Part III considers the main properties of the "basic probablity distribution"