In Harsanyi games with incomplete information, also known as Bayesian games, each player has a type. The type of the player describes all that he knows and believes about the situation he faces: who are the players, what are his and their available actions, what are his and their utility functions, and what are the beliefs of the other players about the situation.

Since the player’s type describes his knowledge and beliefs, a player always knows his own type. But a player need not know the other players’ types. Indeed, a chess player knows his own abilities, but he may not the level of his opponent: he may ascribe probability 1/3 to the event that his opponent is familiar with the Benko gambit, and probability 2/3 to the event that the opponent is not familiar with this opening.

In a Bayesian game, a chance move selects a vector of types, one type for each player, according to a known probability distribution *p* at the outset of the game. Each player learns his own type, but he does not know the types chosen for the other players. He does have a belief about the other players’ types, which is the conditional distribution of *p* given his own type.

The Bayesian game is an auxiliary construction. In reality there is no chance move that selects the player’s types: the knowledge and beliefs each player is equipped with determine his or her type. Bayesian games are merely a way to model the incomplete information each player has on the other players’ types. Thus, the true situation the players face is the situation after the vector of types was selected, which is called the *interim stage*. The situation before the vector of types is chosen, which is called* ex ante* *stage*, is the mathematical way that Harsanyi found to model the game.

Consider now the following Bayesian game, that depends on a real number *a* (which is in the unit interval; below, all additions and subtractions are modulo 1). There are two players; the type space of each player is the unit interval [0,1]. The types of the players are correlated: if player 1 has type x, then he believes that player 2′s type is either x of x+a (each with probability 1/2); if player 2 has type x, then he believes that player 1′s type is either x of x-a (each with probability 1/2). This belief structure can be described by a common prior distribution: the types of the two players are chosen according to the uniform distribution over the following set T (this is a variation of an example of Ehud Lehrer and Dov Samet):

If player 1′s type is *x*, then he believes that player 2 may be of type *x* or *x+a*. It follows that player 2 may believes that player 1′s type is *x-a*, *x*, or *x+a*. So player 2 may believe that player 1 believes that player 2′s type is *x-a*, *x*, *x+a* or *x+2a*. When the situation is a game, to decide how to play, player 1 needs to take all types of player 2 (and of himself) of the form {*x+na*, *n* is an integer}. This set of finite if *a* is a rational number, and countable if *a* is an irrational number. Denote by Zx the set of all pairs of type {(*x+na,x+na*), *n* is an integer} union with {(*x+na,x+(n+1)a*), *n* is an integer}. The set Zx is called the *minimal belief subspace* of player 1. In the interim stage, after his type was selected and told to him, player 1 knows that the type vector is in Zx, that only type vectors in Zx appear in the belief hierarchy of player 2, and therefore he can think about the situation as if the Bayesian game is restricted to Zx: a type vector in Zx was chosen according to the conditional distribution over Zx. To determine how to play, player 1 should find an equilibrium in the game restricted to Z.

The uniform distribution of the set T that appears in the figure above induces a probability distribution over Zx. When Zx is finite (= *a* is a rational number), this is the uniform distribution over a finite set. Alas, when Zx is countable (*a* is irrational) there is no uniform distribution over Zx. In particular, the interim stage is not well defined! Thus, even though the interim stage is the actual situation the players face, and even though they can describe their beliefs using a Harsanyi game with a larger type space, the situation they face cannot be described as a Harsanyi game if they take into account only the types that are possible according to their information.

It is interesting to note that one can find a Bayesian equilibrium in the game restricted to Zx, for every x. However, when one tries to “glue” these equilibria together, one might find out that the resulting pair of strategies over [0,1] is not measurable, and in particular an equilibrium in the Harsanyi game (over T) need not exist. This finding was first noted by Bob Simon.

Since indeed the true situation the players face is the interim stage, and the ex ante stage is merely an auxiliary construction, how come the ex ante stage does not define a proper game in the interim stage? If this is the case, is the auxiliary construction of Harsanyi game over T the correct one?

## 4 comments

April 8, 2012 at 2:24 am

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April 9, 2012 at 7:20 am

drbuddyHi, just a naive question (I don’t know anything about Harsanyi games). You say that “the interim stage is not well defined” when Zx is infinite, because “there is no uniform distribution over Zx.” This latter is true if one insists (as usual) that distributions are countably additive; but there is a perfectly good finitely-additive “uniform” distribution on Zx that is, in a finitely-additive sense, induced from the uniform distribution on T. I tend to think that examples like this indicate that assuming countable additivity is sometimes a mistake. So my question is, is countable additivity really important for making these games work? Is it easy to see why? Thanks!

April 11, 2012 at 2:36 pm

EilonFrankly, I do not know the answer. I do not know which parts of calculus for countably additive measures carries to finitely additive measures, and if we can develop the theory of Harsanyi games for finitely additive priors. A non-negligeable effort was invested in constructing the universal belief space for countable additive priors, see, for example, the following paper by Martin Meier (http://arxiv.org/pdf/math/0602656.pdf).

April 11, 2012 at 9:51 pm

Jonathan WeinsteinWell allowing finitely additive measures in game theory can lead to paradoxes like the following: Consider a symmetric 2-player game with action spaces Z+, the object is to pick the larger integer. I randomize according to a “uniform” measure which puts 0 mass on each integer. Now you have probability 0 of winning no matter what integer you pick, but the game is symmetric so you could pick the same mixture as me and have probability 1/2…and this makes no sense! Technically I believe this is a failure of the Fubini Theorem.