A Bayesian game is one where the players have incomplete information about the game. This is most often structured as saying that a player may have one of several (or even infinite) "types", and that the type of any player is known to that player, but unknown to others.

The type of a player determines the payoffs that player receives from any outcome of the game.

The common equilibrium notion for such games is Bayesian Nash Equilibrium (BNE). In a BNE, each player picks a strategy function, rather than a simple strategy. The strategy function then selects a particular strategy for the player's type.

Beyond that, the idea is just like like a normal Nash equilibrium. A BNE is a profile of strategy functions such that no single player can improve their expected utility by changing their function.

BNE is the solution concept most often applied to auctions. Bayesian auction games can be implemented with bidding experiments. Bayesian matrix games can be implemented as extensiveform games with the Extensive Form Game software at Carnegie Mellon University.
