Bayesian game

In game theory, a Bayesian game is a game in which the players do not have complete information on the other players (e.g. on their available strategies or payoffs), but, they have beliefs with known probability distribution.

A Bayesian game can be converted into a game of complete but imperfect information under the "common prior assumption". John C. Harsanyi describes a Bayesian game in the following way. In addition to the actual players in the game, there is a special player called Nature. Nature assigns a random variable to each player which could take values of types for each player and associating probabilities or a probability mass function with those types (in the course of the game, Nature randomly chooses a type for each player according to the probability distribution across each player's type space). Harsanyi's approach to modeling a Bayesian game in such a way allows games of incomplete information to become games of imperfect information (in which the history of the game is not available to all players). The type of a player determines that player's payoff function. The probability associated with a type is the probability that the player, for whom the type is specified, is that type. In a Bayesian game, the incompleteness of information means that at least one player is unsure of the type (and so the payoff function) of another player.

Such games are called Bayesian because of the probabilistic analysis inherent in the game. Players have initial beliefs about the type of each player (where a belief is a probability distribution over the possible types for a player) and can update their beliefs according to Bayes' rule as play takes place in the game, i.e. the belief a player holds about another player's type might change on the basis of the actions they have played. The lack of information held by players and modeling of beliefs mean that such games are also used to analyse imperfect information scenarios.

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