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In games of complete information, the players know not only their own preferences, but also the preferences of other players as well. In many situations of economic interest, however, there is likely to be considerable uncertainty on the part of each player regarding the preferences of his or her opponents.
To incorporate this possibility, we introduce Bayesian Games, or games of incomplete information. Complete games are characterized by three things: players, actions and payoffs. Bayesian Games build on the static games of complete information.
In Bayesian Games, each player can be one of a certain number of types. Each player knows his or her own type, but cannot observe the type of the other players directly. From the perspective of a given player the others types are random draws from some given prior distribution.
The type of a player influences that player in two ways. First, the players payoff in the game can depend on both the actions chosen, and on the types of all players. Thus, the combination of types influences each players payoff. Second, each type of each player has its own set of possible actions in the game, but we frequently assume that the set of possible actions does not depend on the type of player. Given his type, each player in the game selects an action from those available.
Similar to dynamic games, in Bayesian Games a strategy for each player is a plan which specifies an action for each possible type. Knowing another players strategy in a Bayesian Game does not yet allow a player to predict the others action for certain, because other players type is unknown.
However, given the other players strategy and the players prior distribution each player can figure out his or her expected payoff from the actions that are available to him or her.
Of the actions available, each player will choose some particular action. Therefore, given his or her type, the strategy of each player and the prior distribution, each player can figure out his or her expected payoff from the actions that are available to him or her. Of the actions available, each player can therefore figure out a best possible action or actions for each of his or her own types. The best possible action for each of his or her types is a players best responses to each other then the strategies constitute a Bayesian Nash Equilibrium.
An Example of Bayesian Game: Take It Or Leave It Offer
As in the standard Take it or leave it offer game, there are two players: a buyer and a seller. Each players type is his value for the good, vB; vS.

Both values are independent draws form a uniform distribution on [0; 1] ; so that Pr (vi x) = x. The buyer is given the opportunity to make a take it or leave it offer to the seller of a price p at which he or she will be willing to buy the good. If the price exceeds the sellers value, then the seller accepts, if it isnt the seller rejects the proposal. What price will the buyer choose? Given his or her value vB the buyer selects her price to maximize her expected payoff

Max Pr (vS > p) (vB – p) =
p

max p (vB – p)
p

p* = vB/2

Thus the buyer “shades” her bid by half in order to trade o the chance that her offer is
rejected with her payoff if it is accepted. This leads to a outcomes that can be bad socially.

If the sellers value were known, whenever the buyers value was larger the parties would transact; thus the party who values the good most would always receive it and the collective payoff is as large as possible.

When the sellers value is unknown, the buyer shades his bid, so in order for trade to occur, the buyers value has to be larger than twice the sellers value, which occurs less often. Thus, because a value is private, it is more difficult to trade and the collective good is harmed.

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