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