Upload
ellen-adams
View
222
Download
0
Embed Size (px)
DESCRIPTION
A static Bayesian game Equivalent representation: What if the uninformed gets to observe the informed choice? What if the informed gets to observe the uninformed choice? 26.04.2017 Daniel Spiro, ECON3200
Citation preview
03.05.23 Daniel Spiro, ECON3200 1
Incomplete information: Perfect Bayesian equilibrium
Lectures in Game TheoryFall 2015, Lecture 6
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?
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:
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
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?
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?
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.
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
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.
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.
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.
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.
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
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