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 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 u i is determined by the outcome (strategy profile). What is the game? … - It is regarded as a multiagent decision problem - Several players has several strategies. - Performs strategies to maximize its payoff funciton

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 p i 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 (1)N={1,2}: player1 and player2(wife and husband) (2)A 1 =A 2 ={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)u 1 and u 2 are given in the game matrix on the next slide Battle of Sexes with incomplete information

10. Examples ー Battle of Sexes 2,10,0 1,2 2,00,2 0,11,0 Game matrixes of BoS B B B B S S S S type l type h - Since player1 has only type x, we omit the parameter x in the payoff functions u i, i = 1,2. - These matrixes define the payoff functions: u 1 (B,B,l) = 2, u 2 (B,B,l) = 1, u 1 (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. - 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: Examples ー Battle of Sexes Bayesian equilibrium for mixed strategy

16. Examples ー Cournot Duopoly Cournot Duopoly model (1)Players (2 firms): (2)Action set (outcome of firms): (3)Type set: (4)Probability function: (5)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, EP 2 : - The expected payoff of player 1, EP 1, is given as follows:

18. Examples ー Cournot Duopoly Bayesian equilibrium is also the maximal point of expected payoff EP 1 : Solving above equations, we can get Bayesian equilibrium as follows: