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Bayesian Games

Explore the concept of Bayesian games in strategic form, including Bayesian equilibrium and its application in games like Battle of Sexes and Cournot Duopoly, with incomplete information.

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Bayesian Games

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  1. Bayesian Games Yasuhiro Kirihata University of Illinois at Chicago

  2. Outline • Game in strategic form • Bayesian Game • Bayesian Equilibrium • Examples • - Battle of Sexes with incomplete information • - Cournot Duopoly with incomplete information

  3. Game in strategic form What is the game? … - It is regarded as a multiagent decision problem - Several players has several strategies. - Performs strategies to maximize its payoff funciton Game in strategic form is given by three objects specified by (1)the set of players: (2)the action(strategy)space of players: (3)the payoff functions of players: - Note that ui is determined by the outcome (strategy profile).

  4. What is Bayesian Game? • Game in strategic form • Complete information(each player has perfect information • regarding the element of the game) • Iterated deletion of dominated strategy, Nash equilibrium: • solutions of the game in strategic form • Bayesian Game • A game with incomplete information • Each player has initial private information, type. • - Bayesian equilibrium: solution of the Bayesian game

  5. Bayesian Game Definition (Bayesian Game) A Bayesian game is a strategic form game with incomplete information. It consists of: - A set of players N={1, …, n} for each i∈N - An action set - A type set - A probability function, - A payoff function, - The function pi is what player i believes about the types of the other players - Payoff is determined by outcome A and type

  6. Bayesian Game Definition Bayesian game is finite if , , and are all finite Definition(pure strategy, mixed strategy) Given a Bayesian Game , A pure strategy for player i is a function which maps player i’s type into its action set A mixed strategy for player i is

  7. Bayesian Equilibrium Definition(Bayesian Equilibrium) A Bayesian equilibrium of a Bayesian game is a mixed strategy profile , such that for every player i∈N and every type , we have - Bayesian equilibrium is one of the mixed strategy profiles which maximize the each players’ expected payoffs for each type.

  8. Bayesian Equilibrium • Remark) • This equilibrium is the solution of the Bayesian game. This equilibrium means the best response to each player’s belief about the other player’s mixed strategy. • In the definition of Bayesian equilibrium, we need to specify strategies for each type of a player, even if in the actual game that is played all but one of these types are non-exist

  9. Examples ー Battle of Sexes Battle of Sexes with incomplete information • N={1,2}: player1 and player2(wife and husband) • A1=A2={B,S} (Ballet and Soccer) • (3) • - Type x: player1 loves going out with player2 • - Type l : player2 loves going out with player1 • - Type h: player2 hates going out with player1 • (4) • (5)u1 and u2 are given in the game matrix on the next slide

  10. Examples ー Battle of Sexes Game matrixes of BoS S S B B B B S S type h type l • Since player1 has only type x, we omit the parameter x • in the payoff functions ui, i = 1,2. • These matrixes define the payoff functions: • u1(B,B,l) = 2, u2(B,B,l) = 1, u1(B,B,h) = 2, …and so on

  11. Examples ー Battle of Sexes Calculate the Bayesian Equilibrium Player 2 of type l: Given player 1’s strategy - Action B: - Action S: Best response is B if , S if

  12. Examples ー Battle of Sexes Player 2 of type h: Given player 1’s strategy - Action B: - Action S: Best response is B if , S if

  13. Examples ー Battle of Sexes Player 1: Given player 2’s strategy and -Action B: -Action S: Best response is B if Best response is S if

  14. Examples ー Battle of Sexes Bayesian equilibrium for pure strategy • Assume that both types of player 2’s strategies are pure strategy, • and check the all combination of strategies pair. • Condition of Bayesian equilibrium is not satisfied by: • Bayesian equilibrium for pure strategy is given by:

  15. Examples ー Battle of Sexes Bayesian equilibrium for mixed strategy • There is no equilibrium in which both types of player 2 mixes. (Because, if both type of player 2 mixes, should be 2/3 and 1/3 in the equilibrium. This is contradiction!) • Suppose only type l mixes. Then, . This implies that strategy of player 1 mixes, i.e. . • Type h of player 2 does not mix and . . • Bayesian equilibrium is given by: • Similarly, Bayesian equilibrium when type h mixes is given by:

  16. Examples ー Cournot Duopoly Cournot Duopoly model • Players (2 firms): • Action set (outcome of firms): • Type set: • Probability function: • Profit function:

  17. Examples ー Cournot Duopoly Bayesian equilibrium for pure strategy • The Bayesian equilibrium is a maximal point of expected • payoff of firm 2, EP2: - The expected payoff of player 1, EP1, is given as follows:

  18. Examples ー Cournot Duopoly Bayesian equilibrium is also the maximal point of expected payoff EP1: Solving above equations, we can get Bayesian equilibrium as follows:

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