# Bayesian Game - University of Illinois at Chicago

Bayesian Games
Yasuhiro Kirihata
University 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
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 (N, {A i }iN , {u i }iN ) is given by three
objects specified by
(1)the set of players: N  {1, 2, ..., n}
(2)the action(strategy)space of players: Ai, i  1, 2, ..., n
(3)the payoff functions of players:
u i : Ai  R, i  1, 2, ..., n
- Note that ui is determined by the outcome (strategy profile).
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 Ai, (A  i  N Ai)
- A type set Θi, (Θ  i  N Θi)
- A probability function, pi : i  (  i)
- A payoff function, ui : A    R
- The function pi is what player i believes about the types of the other players
- Payoff is determined by outcome A and type 
Bayesian Game
Definition
Bayesian game ( N ,{ Ai }iN ,{i }iN ,{ pi }iN ,{ui }iN ) is finite
if N , Ai , and  i are all finite
Definition(pure strategy, mixed strategy)
Given a Bayesian Game ( N ,{ Ai }iN ,{i }iN ,{ pi }iN ,{ui }iN ),
A pure strategy for player i is a function which maps player
i’s type into its action set
ai : i  Ai
A mixed strategy for player i is
 i : i  ( Ai ) : i   i (. | i )
Bayesian Equilibrium
Definition(Bayesian Equilibrium)
A Bayesian equilibrium of a Bayesian game is a mixed
strategy profile   ( i ) iN , such that for every player i∈N and
every type i  i , we have
 i (. | i )  arg max
  ( Ai )
p (


i  i
i
i
| i ){
aA

j
(a j |  j )} (ai )ui (a,  )
jN \{i}
- 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
Battle of Sexes with incomplete information
(1) N={1,2}: player1 and player2(wife and husband)
(2) A1=A2={B,S} (Ballet and Soccer)
(3) 1  { x }, 2  { l , h }
- 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) p1 (l | x)  p1 (h | x)  1/ 2, p2 ( x | l )  p2 ( x | h)  1
(5)u1 and u2 are given in the game matrix on the next slide
Examples ー Battle of Sexes
Game matrixes of BoS
S
B
B
2,1
0,0
S
0,0
1,2
type l
B
B
2,0
S
0,2
S
0,1
1,0
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 1
- Action B:  2 ( B | l )  1
EP  p2 ( x | l ){1 ( B) 2 ( B | l )u2 ( B, B, l )  1 (S ) 2 ( B | l )u2 (S , B, l )}
 1 ( B)
- Action S:  2 (S | l )  1
EP  p2 ( x | l ){1 ( B) 2 (S | l )u2 ( B, S , l )  1 (S ) 2 (S | l )u2 (S , S , l )}
 2(1  1 ( B))
Best response is B if 1 ( B)  2 / 3 , S if 1 ( B)  2 / 3
Examples ー Battle of Sexes
Player 2 of type h: Given player 1’s strategy 1
- Action B:  2 ( B | h)  1
EP  p2 ( x | h){1 ( B) 2 ( B | h)u2 ( B, B, h)  1 (S ) 2 ( B | h)u2 (S , B, h)}
 (1  1 ( B))
- Action S:  2 (S | h)  1
EP  p2 ( x | h){1 ( B) 2 (S | h)u2 ( B, S , h)  1 (S ) 2 (S | h)u2 (S , S , h)}
 21 ( B)
Best response is B if 1 ( B)  1 / 3, S if 1 ( B)  1 / 3
Examples ー Battle of Sexes
Player 1: Given player 2’s strategy  2 (. | l ) and  2 (. | h)
-Action B: 1 ( B)  1
EP  p1 (l | x){1 ( B) 2 ( B | l )u1 ( B, B, l )  1 ( B) 2 (S | l )u1 ( B, S , l )}
 p1 (h | x){1 ( B) 2 ( B | h)u1 ( B, B, h)  1 ( B) 2 (S | h)u1 ( B, S , h)}
  2 ( B | l )   2 ( B | h)
-Action S: 1 (S )  1
EP  p1 (l | x){1 (S ) 2 ( B | l )u1 (S , B, l )  1 (S ) 2 (S | l )u1 (S , S , l )}
 p1 (h | x){1 (S ) 2 ( B | h)u1 (S , B, h)  1 (S ) 2 (S | h)u1 (S , S , h)}
 2 ( B | l )   2 ( B | h)
 1
2
Best response is B if  2 ( B | l )   2 ( B | h)  2 / 3
Best response is S if  2 ( B | l )   2 ( B | h)  2 / 3
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.
( 2 ( B | l ),  2 ( B | h))  (1,1), (1, 0), (0,1), (0, 0)
- Condition of Bayesian equilibrium is not satisfied by:
( 2 ( B | l ),  2 ( B | h))  (1,1), (0,1), (0, 0)
- Bayesian equilibrium for pure strategy is given by:
(1 ( B | x),  2 ( B | l ),  2 ( B | h))  (1,1, 0)
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, 1 ( B) should be 2/3 and
1/3 in the equilibrium. This is contradiction!)
- Suppose only type l mixes. Then, 1 ( B)  2 / 3. This implies that
strategy of player 1 mixes, i.e.  2 ( B | l )   2 ( B | h)  2 / 3.
- Type h of player 2 does not mix and  2 ( B | l )  0.  2 ( B | h)  0.
- Bayesian equilibrium is given by:
(1 ( B | x),  2 ( B | l ),  2 ( B | h))  (2 / 3, 2 / 3, 0)
- Similarly, Bayesian equilibrium when type h mixes is given by:
(1 ( B | x),  2 ( B | l ),  2 ( B | h))  (1/ 3, 0, 2 / 3)
Examples ー Cournot Duopoly
Cournot Duopoly model
Players (2 firms): N  {1, 2}
Action set (outcome of firms): qi  R  , (i  1, 2)
Type set: 1  {1}, 2  {3 / 4, 5 / 4}
Probability function:
p(2  3 / 4 | 1 )  1/ 2, p(2  5 / 4 | 1 )  1/ 2
(5) Profit function:
(1)
(2)
(3)
(4)
u1 (q1 , q2 ,1 , 2 )  q1 (1  q1  q2 )
u2 (q1 , q2 ,1 , 2 )  q2 ( 2  q1  q2 )
Examples ー Cournot Duopoly
Bayesian equilibrium for pure strategy
- The Bayesian equilibrium is a maximal point of expected
payoff of firm 2, EP2:
EP2 * *
(q1 , q2 )   2  q1*  2q2*  0
EP2  u2
q2
q2* ( 2 )  ( 2  q1* ) / 2, ( 2  3 / 4,5 / 4)
- The expected payoff of player 1, EP1, is given as follows:
1
1
EP1  q1 (1  q1  q2 (3 / 4))  q1 (1  q1  q2 (5 / 4))
2
2
Examples ー Cournot Duopoly
Bayesian equilibrium is also the maximal point of expected
payoff EP1:
EP1 * *
1 *
*
(q1 , q2 )  1  2q1  {q2 (3 / 4)  q2* (5 / 4)}  0
q1
2
2  q2* (3 / 4) q 2* (5 / 4)
q 
4
*
1
Solving above equations, we can get Bayesian equilibrium
as follows:
1
11 *
5
q1*  , q2* (3 / 4) 
, q2 (5 / 4) 
.
3
24
24