Chapter summary

In this chapter we study situations in which players do not have complete information on the environment they face. Due to the interactive nature of the game, modeling such situations involves not only the knowledge and beliefs of the players, but also the whole hierarchy of knowledge of each player, that is, knowledge of the knowledge of the other players, knowledge of the knowledge of the other players of the knowledge of other players, and so on. When the players have beliefs (i.e. probability distributions) on the unknown parameters that define the game, we similarly run into the need to consider infinite hierarchies of beliefs. The challenge of the theory was to incorporate these infinite hierarchies of knowledge and beliefs in a workable model.

We start by presenting the Aumann model of incomplete information, which models the knowledge of the players regarding the payoff-relevant parameters in the situation that they face. We define the knowledge operator, the concept of common knowledge, and characterize the collection of events that are common knowledge among the players.

We then add to the model the notion of belief and prove Aumann's agreement theorem: it cannot be common knowledge among the players that they disagree about the probability of a certain event.

An equivalent model to the Aumann model of incomplete information is a Harsanyi game with incomplete information. After presenting the game, we define two notions of equilibrium: the Nash equilibrium corresponding to the ex ante stage, before players receive information on the game they face, and the Bayesian equilibrium corresponding to the interim stage, after the players have received information.