Games with Incomplete Information Played by ÜBayesianÝ Players, I--III: Part I. The Basic Model&

(This article originally appeared in Management Science, November 1967, Volume 14, Number 3, pp. 159--182, published by The Institute of Management Sciences.) 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. However, each player has a subjective probability distribution over the alternative possibilities. In most of the paper it is assumed that these probability distributions entertained by the different players are mutually Üconsistent,Ý in the sense that they can be regarded as conditional probability distributions derived from a certain Übasic probability distributionÝ 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 consistency assumption holds, the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probability distribution, and the outcome of this lottery will decide which particular subgame will be played, i.e., what the actual values of the relevant parameters will be in the game. Yet, each player will receive only partial information about the outcome of the lottery, and about the values of these parameters. However, every player will know the Übasic probability distributionÝ governing 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 various intuitive interpretations for the latter. Part II shows that the Nash equilibrium points of the Bayes-equivalent game yield ÜBayesian equilibrium pointsÝ for the original game. Finally, Part III considers the main properties of the Übasic probability distribution.Ý