Original article (link) posted: 22/09/2005
The following is the memo about Bayesian Games. All the sentences are quoted from Myerson (1991) "Game Theory" (Chapter 2.8 and 2.9).
A game with incomplete information is a game in which, at the first point in time when the players can begin to plan their moves in the game, some players already have private information about the game that other players do not know.
The initial private information that a player has at this point in time is called the type of the player.
Harsanyi (1967-68) argued that a generalization of the strategic form, called the Bayesian form, is needed to represent games with incomplete information.
Most of the Bayesian games that have been studied in applied game theory have beliefs that are consistent with a common prior. One reason for this tendency to use consistent models is that consistency simplifies the definition of the model. Furthermore, inconsistency often seems like a strikingly unnatural feature of a model. In a consistent model, differences in beliefs among players can be explained by differences in information, whereas inconsistent beliefs involve differences of opinion that cannot be derived from any differences in observations and must be simply assumed a priori.
Agreeing to disagree
In a sports match, suppose it is common knowledge among the coaches of two teams that each believes that his own team has a 2/3 probability of winning its next game against the other team, then the coaches' beliefs cannot be consistent with a common prior. In a consistent model, it can happen that each coach believes that his team has a 2/3 probability of winning, but this difference of beliefs cannot be common knowledge among the coaches. (see Aumann, 1976)
Bayesian Games are general enough?
To describe a situation in which many individuals have substantial uncertainty about one another's information and beliefs, we may have to develop a very complicated Bayesian-game model with large type sets and assume that this model is common knowledge among the players. This result begs the question; is it possible to construct a situation for which there are no sets of types large enough to contain all the private information that players are supposed to have, so that no Bayesian game could represent this situation?
Mertens and Zamir (1985) showed under some technical assumptions, that no such counterexample to the generality of the Bayesian game model can be constructed, because a universal belief space can be constructed that is always big enough to serve as the set of types for each player.
Although constructing an accurate model for any given situation may be extremely difficult, we can at least be confident that no one will ever be able to prove that some specific conflict situation cannot be described by any sufficient complicated Bayesian game.
Aumann (1976) "Agreeing to Disagree" Annals of Statistics, 4
Harsanyi (1967-68) "Games with Incomplete Information Played by 'Bayesian' Players" Management Science, 14
Mertens and Zamir (1985) "Formulation of Bayesian Analysis for Games with Incomplete Information" IJGT, 14