Bayesian game

In game theory, a Bayesian game is a strategic decision-making model which assumes players have incomplete information. Players hold private information relevant to the game, meaning that the payoffs are not common knowledge. Bayesian games model the outcome of player interactions using aspects of Bayesian probability. They are notable because they allowed, for the first time in game theory, for the specification of the solutions to games with incomplete information. Hungarian economist John C. Harsanyi introduced the concept of Bayesian games in three papers from 1967 and 1968: He was awarded the Nobel Memorial Prize in Economic Sciences for these and other contributions to game theory in 1994. Roughly speaking, Harsanyi defined Bayesian games in the following way: players are assigned by nature at the start of the game a set of characteristics. By mapping probability distributions to these characteristics and by calculating the outcome of the game using Bayesian probability, the result is a game whose solution is, for technical reasons, far easier to calculate than a similar game in a non-Bayesian context. For those technical reasons, see the Specification of games section in this article. A Bayesian game is defined by (N,A,T,p,u), where it consists of the following elements: Set of players, N: The set of players within the game Action sets, ai: The set of actions available to Player i. An action profile a = (a1, . . . , aN) is a list of types, one for each player Type sets, ti: The set of types of players i. "Types" capture the private information a player can have. A type profile t = (t1, . . . , tN) is a list of types, one for each player Payoff functions, u: Assign a payoff to a player given their type and the action profiles. A payoff function, u= (u1, . . . , uN) denotes the utilities of player i Prior, p: A probability distribution over all possible type profiles, where p(t) = p(t1, . . . ,tN) is the probability that Player 1 has type t1 and Player N has type tN.