Game Theory. Uncertainty about Payoffs: Bayesian Games. Manar Mohaisen Department of EEC Engineering

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1 Game Theory Uncertainty about Payoffs: Bayesian Games Manar Mohaisen Department of EEC Engineering Korea University of Technology and Education (KUT)

2 Content Bayesian Games Bayesian Nash Equilibrium How to compute Bayes-Nash Equilibrium 2

3 Auction Uncertainty about payoffs Players don t know the value of the auctioned items to other players 3

4 Bayesian Games So far, agents know the game being played number of players; available actions; and payoffs Bayesian Games All possible games have the same number of agents and the same strategy space for each agent; they differ only in their payoffs. The beliefs of the different agents are posteriors, obtained by conditioning a common prior on individual private signals. 4

5 Bayesian Games All possible games have the same number of agents and the same strategy space for each agent; they differ only in their payoffs. Example o Player 1 does not know whether P2 has 2 or three strategies. P2:A P2:B P1: A 1, 1 1, 3 P1: B 0, 5 1, 13 P2:A P2:C P2:B P1: A 1, 1 0, 2 1, 3 P1: B 0, 5 2, 8 1, 13 o The uncertainty in number of actions is transformed into uncertainty in payoffs by padding P2:A P2:C P2:B P1: A 1, 1 0, , 3 P1: B 0, 5 2, , 13 5

6 Bayesian Games: First Definition Definition I: Information Sets A Bayesian game is a set of games that differ only in their payoffs, a common prior defined over them, and a partition structure over games for each agent. A Bayesian Games is a tuple (N, G, P, I) where 6

7 Bayesian Games: First Definition Example (Normal-form) Known: Players, Games, associated probability for each game, and partition structure for each player 7

8 Bayesian Games: Second Definition Definition II: Extensive Form with Chance Moves Nature randomizes the games in a commonly known way o Information sets indicate that players make their choices without knowing taken actions by other players 8

9 Bayesian Games: Third Definition Definition III: Epistemic Type User s type is used to define the uncertainty A Bayesian game is a tuple (N, A,, p, u) 9

10 Bayesian Games: Third Definition Example {, } 1 1,1 1,2 {, } 2 2,1 2,2 p(, ) 0.3 1,1 2,1 p(, ) 0.1 1,1 2,2 p(, ) 0.2 1,2 2,1 p(, ) 0.4 1,2 2,2 p( ) 0.6 1,1 2,1 p( ) 0.2 1,1 2,2 p( ) 0.4 1,2 2,1 p( ) 0.8 1,2 2,2 10

11 Bayesian Nash Equilibrium How to define agent s strategy space in Bayesian game? Pure strategy s A : i i i o Is a mapping from each type agent i could have to the action he would play if he had that type. Mixed Strategy s : ( A) i i i o Is a mapping from each type agent i could have to a mixed action he would play if he had that type. Probability under mixed strategy sa ( ) i i i o The probability under mixed strategy s i that player i played action a i given that her type is I 11

12 Bayesian Nash Equilibrium Three Settings of Expected Utility Ex-ante o An gent does not know anybody's type Ex-interim o Each agent knows her own type but not those of other agents Ex-post o Each agent knows all agents types 12

13 Bayesian Nash Equilibrium Ex-post Expected Utility Agent i s ex-post expected utility in a Bayesian game (N, A,, p, u), where the agent s strategies are given by s and the agent s types are given by, is defined as o The uncertainty concerns the other agents mixed strategies; o In a Bayesian game, of course, no agent will know other agents types! o This definition is useful to define other expected utilities 13

14 Bayesian Nash Equilibrium Ex-interim Expected Utility Agent i s ex-interim expected utility in a Bayesian game (N, A,, p, u), where i s type is i, and where the agents strategies are given by the mixed strategy profile s, is defined as o Equivalently written as 14

15 Bayesian Nash Equilibrium Ex-ante Expected Utility Agent i s ex-ante expected utility in a Bayesian game (N, A,, p, u), where the agents strategies are given by the mixed strategy profile s, is defined as 15

16 Bayesian Nash Equilibrium Best Response in a Bayesian Game The set of agent i s best responses to mixed strategy profile s -i are given by Bayes-Nash Equilibrium A Bayes-Nash equilibrium is a mixed-strategy profile s that satisfies i s i BR i (s -i ) 16

17 Computing Equilibria Example Each player has 4 pure strategies o UU, UD, DU and DD for P1 o RR, RL, LR, and LL for P2 The normal-form game is 4x4 o The payoffs are the expected payoffs 17

18 Computing Equilibria Example contd. The ex-ante expected utility under the strategy profile (UU, LL) is given by o And so forth all the entries are computed. 18

19 Computing Equilibria Example contd. Ex-ante Ex-interim P1 observes 1,1 19

20 Summary Bayesian Games Bayesian Nash Equilibrium How to compute Bayes-Nash Equilibrium 20