Bayesian Games Yasuhiro Kirihata University of Illinois at Chicago.
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Transcript of Bayesian Games Yasuhiro Kirihata University of Illinois at Chicago.
Bayesian Games
Yasuhiro KirihataUniversity of Illinois at Chicago
Outline
- Game in strategic form- Bayesian Game- Bayesian Equilibrium- Examples - Battle of Sexes with incomplete information - Cournot Duopoly with incomplete information
Game in strategic form
Game in strategic form is given by threeobjects 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).
n ..., 2, 1,i ,Ai
)}{u ,}{A (N, NiiNii
n ..., 2, 1,i ,A:u ii R
n} ..., 2, 1,{N
What is the game? …
- It is regarded as a multiagent decision problem - Several players has several strategies. - Performs strategies to maximize its payoff funciton
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
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
RA :ui
)( :p iii )Θ(Θ ,Θ i Nii
)A(A ,A i Nii
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
)}{,}{,}{,}{,( NiiNiiNiiNii upAN
iii Aa :
)|(.:)(: iiiiii A
)}{,}{,}{,}{,( NiiNiiNiiNii upAN
N iiA
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 e∈very type , we have
ii Nii )(
),()(})|({)|(maxarg)|(.}\{
)(
auaap ii
Aa iNjjjjiii
Aii
iii
- Bayesian equilibrium is one of the mixed strategy profiles
which maximize the each players’ expected payoffs for each type.
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
Examples ー Battle of Sexes
(1) N={1,2}: player1 and player2(wife and husband)(2) 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
},{},{ 21 hlx
1)|()|(,2/1)|()|( 2211 hxplxpxhpxlp
Battle of Sexes with incomplete information
Examples ー Battle of Sexes
2,1 0,0
0,0 1,2
2,0 0,2
0,1 1,0
Game matrixes of BoS
B
B
B
B
S
SS
S
type l type h
- 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
Examples ー Battle of Sexes
Calculate the Bayesian Equilibrium
Player 2 of type l: Given player 1’s strategy - Action B:
- Action S:
1
)(1 B
3/2)(1 B 3/2)(1 B
))(1(2 1 B
1)|(2 lB
1)|(2 lS
)},,()|()(),,()|()(){|(EP 2212212 lBSulBSlBBulBBlxp
)},,()|()(),,()|()(){|(EP 2212212 lSSulSSlSBulSBlxp
Best response is B if , S if
Examples ー Battle of Sexes
Player 2 of type h: Given player 1’s strategy
- Action B:
- Action S:
1
3/1)(1 B3/1)(1 B
))(1( 1 B
)(2 1 B
)},,()|()(),,()|()(){|(EP 2212212 hBSuhBShBBuhBBhxp 1)|(2 hB
)},,()|()(),,()|()(){|(EP 2212212 hSSuhSShSBuhSBhxp 1)|(2 hS
Best response is B if , S if
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
)|(.2 l )|(.2 h
)},,()|()(),,()|()(){|(EP 1211211 lSBulSBlBBulBBxlp
)|()|( 22 hBlB )},,()|()(),,()|()(){|( 1211211 hSBuhSBhBBuhBBxhp
)},,()|()(),,()|()(){|(EP 1211211 lSSulSSlBSulBSxlp )},,()|()(),,()|()(){|( 1211211 hSSuhSShBSuhBSxhp
2
)|()|(1 22 hBlB
3/2)|()|( 22 hBlB 3/2)|()|( 22 hBlB
1)(1 B
1)(1 S
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:
)0,0(),1,0(),0,1(),1,1())|(),|(( 22 hBlB
)0,1,1())|(),|(),|(( 221 hBlBxB
)0,0(),1,0(),1,1())|(),|(( 22 hBlB
- 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
)(1 B
3/2)(1 B3/2)|()|( 22 hBlB
0)|(2 lB 0)|(2 hB
)0,3/2,3/2())|(),|(),|(( 221 hBlBxB
)3/2,0,3/1())|(),|(),|(( 221 hBlBxB
Examples ー Cournot Duopoly
)(),,,(
)(),,,(
212221212
211121211
qqqqqu
qqqqqu
Cournot Duopoly model
}2,1{N(1) Players (2 firms):(2) Action set (outcome of firms):(3) Type set:(4) Probability function:
(5) Profit function:
}4/5,4/3{},1{ 21 )2,1(, iqi R
2/1)|4/5(,2/1)|4/3( 1212 pp
Examples ー Cournot Duopoly
Bayesian equilibrium for pure strategy
- The Bayesian equilibrium is a maximal point of expected payoff of firm 2, EP2:
02),( *2
*12
*2
*1
2
2
qqqqq
EP
)4/5,4/3(,2/)()( 2*122
*2 qq
- The expected payoff of player 1, EP1, is given as follows:
))4/5((2
1))4/3((
2
1211121111 qqqqqqEP
22 uEP
Examples ー Cournot Duopoly
0)}4/5()4/3({2
121),( *
2*2
*1
*2
*1
1
1
qqqqqq
EP
4
)4/5()4/3(2 *2
*2*
1
qqq
Bayesian equilibrium is also the maximal point of expected payoff EP1:
.24
5)4/5(,
24
11)4/3(,
3
1 *2
*2
*1 qqq
Solving above equations, we can get Bayesian equilibrium as follows:
