Incomplete
p
information:
Perfect Bayesian equilibrium
Lectures in Game Theory
Spring 2011, Part 7
10.12.2010
G.B. Asheim, ECON4240-7
A static Bayesian game
U
1
A ( 12 )
Nature
B ( 12 )
D
U
1
D
Equivalent
representation:
2 L
R
2 L
R
2 L
R
2 L
1
L
2
A ( 12 )
Nature
B ( 12 )
R
L
2
R
R
1 U
D
U
1
D
1 U
D
1 U
D
What if the uninformed gets to What if the informed gets to
observe the informed choice? observe the uninformed choice?
10.12.2010
G.B. Asheim, ECON4240-7
2
1
A dynamic Bayesian
game: Screening
A ( 12 )
Nature
1
L B (2)
2
R A ( 12 )
Nature
B ( 12 )
Equivalent
representation:
The informed gets to observe the uninformed choice.
1 U
1 U
D
1 U
D
1 U
D
1 U
L
2
A ( 12 )
Nature
B ( 12 )
D
1 U
R
D
1 U
L
D
U
1 
2
R
D
D
Use subgame perfect Nash equilibrium!
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3
G.B. Asheim, ECON4240-7
The uninformed gets to observe the informed choice.
U
1
A ( 12 )
Nature
B ( 12 )
D
U
1
D
2 L
A dynamic Bayesian
game: Signaling
Equivalent representation:
R
2 L
L 2 U1
R
R
2 L
R
2 L
D 2 L
R
A ( 12 )
Nature
B ( 12 )
L
R
U 1
R
R
Requires a new equilibrium concept:
Perfect Bayesian equilibrium
10.12.2010
D
L
G.B. Asheim, ECON4240-7
Why?
4
2
Gift game, version 1
F1
G
0 0 N
0,
F ( 12 )
F
2 A
2, 2
R - 2, 0
Nature
1
2
E ( )
0, 0
NE 1 GE
1
2
G FG E
R
A
2,, 1 - 2,, - 1
G F N E 1, 1 - 1, 0
N FG E 1, 0 - 1, - 1
A
2, 0
N F N E 0, 0 0, 0
R - 2,
2 -2
Are both Nash equilibria reasonable?
Note that subgame perfection does not help. Why?
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5
G.B. Asheim, ECON4240-7
Remedy 1: Conditional beliefs about types
F1
G F 2 A 2, 2
0, 0 N
( q ) R - 2, 0
F ( 12 )
Nature
E ( 12 )
(1  q ) A 2, 0
0, 0 E
N 1 GE
R - 2, - 2
Let player 2 assign
probabilities to the
two types player 1:
the belief of player 2.
What will the players
choose?
R d 22: Sequential
Remedy
S
ti l rationality
ti lit
Each player chooses rationally at all information
sets, given his belief and the opponent’s strategy.
10.12.2010
G.B. Asheim, ECON4240-7
6
3
Gift game, version 2
2 A
F1
G
0 0 N
0,
(q) R
F ( 12 )
F
2
2, 2
R
A
G FG E 0,, 0 0,, 0
2, 0
G F N E 1, 1 1, 0
Nature
1
N FG E - 1, - 1 - 1, 0
1
2
E( )
(1  q ) A - 2, - 2 N F N E 0, 0 0, 0
0, 0 E
N 1 GE
R - 2,
2 0
If q ½, then player 2 will choose R.
If so, the outcome is not a Nash equil. outcome.
10.12.2010
G.B. Asheim, ECON4240-7
7
Remedy 3: Consistency of beliefs
F1
GF 2 A
0, 0 N
(1) R
F ( 12 )
Nature
E ( 12 )
0, 0
2, 2
2,, 0
Player 2 should find
his belief by means
of Bayes’ rule,
when-ever possible.
(0) A - 2, - 2
Pr[ G | F] Pr[ F]
q  Pr[ F | G ] 
NE 1 GE
R - 2, 0
Pr[ G ]
An example of a separating equilibrium.
An equilibrium is separating if
the types of a player behave differently.
10.12.2010
G.B. Asheim, ECON4240-7
8
4
Gift game, version 3
If q ½, then player 2 will choose R.
Bayes’ rule cannot be used.
2 A
2, 2
F1
G
0 0 N
0,
( q ) R - 2, 0
F ( 12 )
F
Nature
1
2
E( )
2
R
A
G FG E 2,, 0 - 2,, 0
1
G F N E 1, 1 - 1, 0
N FG E 1, - 1 - 1, 0
(1  q ) A 2, - 2 N F N E 0, 0 0, 0
0, 0 E
N 1 GE
R - 2,
2 0
An example of a pooling equilibrium. An equilibrium is pooling if the types behave the same.
10.12.2010
G.B. Asheim, ECON4240-7
9
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)
(b)
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each player’s strategy specifies optimal
actions given his beliefs and the strategies
of the other players
players.
the beliefs are consistent with Bayes’ rule
whenever possible.
G.B. Asheim, ECON4240-7
10
5
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,
beliefs
check whether player 1’s strategy is a best
response to player 2’s strategy.
10.12.2010
11
G.B. Asheim, ECON4240-7
Applying the algorithm in a signaling game
Player 1 has four
pure strategies.
1, 3 U 2 L 1
R 2 U 2, 1
(r )
(q ) D 0, 0
PBE w/(LL
w/(LL)?
)? YES 4,, 0 D
A ( 12 )
[(LL), (DU), q , r  1 / 2]
where q  2 / 3.
Nature
B ( 12 )
2, 4 U (1  r )
(1  q ) U 1, 0
L 1 R 
D 1, 2
0, 1 D
PBE w/(RR)? NO
PBE w/(LR
w/(LR)?
)? NO
[(RL), (UU), q  1, r  0]
PBE w/(RL)? YES
[(RL), (UU), q  1, r  0]
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is a
separating equilibrium.
[(LL), (DU), q , r  1 / 2 ] is a
pooling equilibrium.
G.B. Asheim, ECON4240-7
12
6