Incomplete information: Perfect Bayesian equilibrium
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03.05.23 Daniel Spiro, ECON3200 1
Incomplete information: Perfect Bayesian equilibrium
Lectures in Game TheoryFall 2015, Lecture 6
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03.05.23 Daniel Spiro, ECON3200 2
A static Bayesian game
U
D1 2
2 L
R
R
L
U
D1 2
2 L
R
R
L
Nature)(A 2
1
)( B 21 L
R2 1
1 U
D
D
U
L
R2 1
1 U
D
D
U
Nature)(A 2
1
)( B 21
Equivalent representation:
What if the uninformed gets to observe the informed choice?
What if the informed gets to observe the uninformed choice?
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03.05.23 Daniel Spiro, ECON3200 3
A dynamic Bayesian game: Screening
U
D
2
1
1R
D
U
U
D1
1
LD
UNature)(A 2
1
)( B 21
L
R2 1
1 U
D
D
U
L
R2 1
1 U
D
D
U
Nature)(A 2
1
)( B 21
Use subgame perfect Nash equilibrium!
The informed gets to ob-serve the uninformed choice.
Nature)(A 2
1
)( B 21
Equivalent representation:
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03.05.23 Daniel Spiro, ECON3200 4
U
D1 2
2 L
R
R
L
U
D1 2
2 L
R
R
L
Nature)(A 2
1
)( B 21
The uninformed gets to ob-serve the informed choice. A dynamic
Bayesian game: Signaling
1
1
Nature
)( B 21
L
R
2 L
R)(A 21
2L
R
R
L
Equivalent representation:
Requires a new equilibrium concept: Perfect Bayesian equilibrium
Why?
U D
U D
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03.05.23 Daniel Spiro, ECON3200 5
1
1
Nature
)( E 21
A
R
2 A
R)( F 21
FN FG
EN EG
Gift game, version 1
0 0,
0 0,
2 2,
0 2,-
0 2,
2- 2,-
EFGGA
EFNG
R1 2,
0 1,-1 1,
1- 2,-1
2
EFGNEFNN
0 1,
0 0,0 0,
1- 1,-
EFGGA
EFNG
R1 2,
0 1,-1 1,
1- 2,-1
2
EFGNEFNN
0 1,
0 0,0 0,
1- 1,-
Are both Nash equilibria reasonable?Note that subgame perfection does not help. Why?
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03.05.23 Daniel Spiro, ECON3200 6
Remedy 1: Conditional beliefs about types
1
1
Nature
)( E 21
A
R
2 A
R)( F 21
FN FG
EN EG
0 0,
0 0,
2 2,
0 2,-
0 2,
2- 2,-
)(q
)1( q
Let player 2 assign probabilities to the two types player 1: the belief of player 2.
Remedy 2: Sequential rationalityEach player chooses rationally at all information sets, given his belief and the opponent’s strategy.
What will the players choose?
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03.05.23 Daniel Spiro, ECON3200 7
1
1
Nature
)( E 21
A
R
2 A
R)( F 21
FN FG
EN EG
Gift game, version 2
0 0,
0 0,
2 2,
0 2,
2- 2,-
0 2,-
EFGGA
EFNG
R0 0,
0 1,1 1,
0 0,1
2
EFGNEFNN
1- 1,-
0 0,0 0,
0 1,-
EFGGA
EFNG
R0 0,
0 1,1 1,
0 0,1
2
EFGNEFNN
1- 1,-
0 0,0 0,
0 1,-
)(q
)1( q
If q ½, then player 2 will choose R.If so, the outcome is not a Nash equil. outcome.
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03.05.23 Daniel Spiro, ECON3200 8
Remedy 3: Consistency of beliefs
1
1
Nature
)( E 21
A
R
2 A
R)( F 21
FN FG
EN EG
0 0,
0 0,
2 2,
0 2,
2- 2,-
0 2,-
)1(
)0(
Player 2 should find his belief by means of Bayes’ rule, when-ever possible.
An example of a separating equilibrium.An equilibrium is separating if the types of a player behave differently.
]GPr[]FPr[]F|GPr[]G|FPr[ q
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03.05.23 Daniel Spiro, ECON3200 9
1
1
Nature
)( E 21
A
R
2 A
R)( F 21
FN FG
EN EG
Gift game, version 3
0 0,
0 0,
2 2,
0 2,-
2- 2,
0 2,-
EFGGA
EFNG
R0 2,
0 1,-1 1,
0 2,-1
2
EFGNEFNN
1- 1,
0 0,0 0,
0 1,-
EFGGA
EFNG
R0 2,
0 1,-1 1,
0 2,-1
2
EFGNEFNN
1- 1,
0 0,0 0,
0 1,-
)(q
)1( q
If q ½, then player 2 will choose R.Bayes’ rule cannot be used.
An example of a pooling equilibrium. An equi-librium is pooling if the types behave the same.
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03.05.23 Daniel Spiro, ECON3200 10
Perfect Bayesian equilibrium Definition: Consider a strategy profile for the players, as well as beliefs over the nodes at
all information sets. These are called a perfect Bayesian equilibrium (PBE) if:
(a) each player’s strategy specifies optimal actions given his beliefs and the strategies of the other players.
(b) the beliefs are consistent with Bayes’ rule whenever possible.
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03.05.23 Daniel Spiro, ECON3200 11
Algorithm for finding perfect Bayesian equilibria in a signaling game: posit a strategy for player 1 (either
pooling or separating), calculate restrictions on conditional
beliefs, calculate optimal actions for player 2
given his beliefs, check whether player 1’s strategy is a
best response to player 2’s strategy.
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03.05.23 Daniel Spiro, ECON3200 12
Applying the algorithm in a signaling game
1
1
Nature
)( B 21
U
D
2 U
D)(A 21
2U
D
D
U
L R
L R
)(q
)1( q)1( r
)(r
3 1,
0 4,
4 2,
1 0,
1 2,
0 0,
0 1,
2 1,
Player 1 has four pure strategies.
PBE w/(LL)?YES]2/1 ,),U(D),L[(L rq
.3/2 where q
PBE w/(RR)?NO
PBE w/(LR)?NO
PBE w/(RL)?YES]0,1),U(U),L[(R rq
]2/1 ,),U(D),L[(L rq
]0,1),U(U),L[(R rq is a separating equilibrium.
is a pooling equilibrium.
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Are all perfect Bayesian equilibria reasonable?
1
1
Nature
)( B 21
U
D
2 U
D)(A 21
2U
D
D
U
L R
L R
)(q
)1( q)1( r
)(r
2 3,
0 2,
0 1,
1 1,
0 1,
1 0,
1 2,
0 0,
Player 1 has four pure strategies.
PBE w/(LL)?YES]2/1,),U(D),L[(L rq
.2/1 where q
PBE w/(RR)?NO
PBE w/(RL)?NO
PBE w/(LR)?YES]1,0),U(U),R[(L rq ]2/1,),U(D),L[(L rq
Choosing R is dominated for 1A. is an un-reasonable equilibrium, because it requires 2 to have
.2/1q
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Beer – Quiche game
1
1
Nature
)(W 101
2 N
F)( S 109
2N'
F'
Q B
Q B
)(q
)1( q)1( r
)(r
0 2,
1- 0,
0 3,
1 1,
0 3,
1- 1,
0 2,
1 0,
Player 1 has four pure strategies.
PBE w/(QQ)?YES]10/9,),(FN'),Q[(Q rq
.2/1 where q
PBE w/(BQ)?NO
PBE w/(QB)?NO
PBE w/(BB)?YES] ,10/9),(NF'),B[(B rq
]10/9,),(FN'),Q[(Q rqIs a reasonable equilibrium?
Only 1S has possibly something to gain by choosing B. But
.2/1 where r
.2/1q
N'
F'
N
F