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.

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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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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