1

22

problem set 13

see last slide

Last Slide

3

Extensive form Games with Incomplete Information

111

Information set of player 1

BeliefBelief

p1 p2 p3

p1 + p2 + p3 = 1, pi ≥ 0

4

Extensive form Games with Incomplete Information

111

Information set of player 1

p1 p2 p3

222

α3α2α1

111

player 2 mixes

What is player 1’s belief at this information set ??

1- α1

5

111

p1 p2 p3

222

111

player 2 mixes

What is player 1’s belief at this information set ??

α3α2α1

Using Bayes’ Law:

1 1

1 1 2 2 3 3

p

p p p

2 2

1 1 2 2 3 3

p

p p p

3 3

1 1 2 2 3 3

p

p p p

6

111

p1 p2 p3

222

111

α3α2α1

1 1

1 1 2 2 3 3

p

p p p

2 2

1 1 2 2 3 3

p

p p p

3 3

1 1 2 2 3 3

p

p p p

Updating the belief is consistent with the strategy profile

(whenever possible)

> 0

7

111

p1 p2 p3

Once we have a beliefs for each information set, we can define the equivalent of subgame perfect equilibrium.

We require that each player’s strategy is optimal in that part of the game that follows an information set of this player, given the strategy profile and that player’s belief at the information set.

8

111

p1 p2 p3

Once we have a beliefs for each information set, we can define the equivalent of subgame perfect equilibrium.

We require that each player’s strategy is optimal in that part of the game that follows an information set of this player, given the strategy profile and that player’s belief at the information set. (Sequential Rationality)

9

Signalling Games

The sendersender, a player who has complete information (about the state of nature, or his own type) sends a signal to the other player, the receiver.receiver.

The payoffs depend on the state of nature, the signal and the action takren

The receiver observes the signal and takes an action.

Michael SpenceNobel Prtize, 2001

10

Education as a signal

The workerworker, has skills H or L with probability q, 1-q, resp. He knows his own productivity.

The firm firm observes the signal, and pays the workerworker a

wage rate w which equals the productivity that it believes he has.

He chooses a level of education e which costs him

c(η,e) where η is his type.

His productivity is y(η,e).

11

Education as a signal

The payoff to the workerworker is: w - c(η,e)The payoff to the firmfirm is: y(η,e) - w

:assumedc dc

L,e > (H,e) > 0de de

The indifference curves of type

ηI : w - c η,e = Const.

η

e

HI

LI

Single Crossing property

12

Education as a signal

:assume dy

y H,e > y(L,e), 0de

y H,e

e

???y H,e

y L,e

13

Education as a signal

In , the worker maximizescomplete information

y η,e - c(η,e).

y η,e

e

ηI

e* η

two cases:

14

Education as a signal

y L,e

e

HI

e* L

y H,e

e* H

No EnvyNo Envy

LI

15

Education as a signal

y L,e

e

HI

e* L

y H,e

e* H

EnvyEnvy

LI

16

Pooling Equilibrium

Both types choose education level: ep

,

The wage rate :

p p p p

p

w e = w = qy H,e + 1 - q y L,e

w e = y L,e e e

Observing the firm believes that the worker's type , is :

pe

= qH + 1 - q L.

Observing the firm believes that the worker's type is , : pe e

L.

17

Pooling Equilibrium

y L,e

e

HI

e* L

y H,e

pe

LI

y η,e

18

Separating Equilibrium

Type chooses : L e* L

Type chooses : *SH e

y L,e

e

HI

e* L

y H,e

e* H

EnvyEnvyLI

*Se

Beliefs:

H LL

Payoffs for education

19

Separating Equilibrium

Type chooses : L e* L

Type chooses : *SH e

y L,e

e

HI

e* L

y H,e

e* H

LI

*Se

Beliefs:

H LL

Can the firm have these beliefs???

It is a strictly dominated (inferior) strategy for type L to send a signal in this interval

Even if he is identfied as H he is better off sending e*(L).

(An H is better off in this interval if he is identified as H. )

*Se

20

Separating Equilibrium

Type chooses : L e* L

Type chooses : *SH e

y L,e

e

HI

e* L

y H,e

e* H

LI

*Se

Beliefs:

H LL

If we accept this argument then the firm ‘s belief in this interval should be H.

*Se

The only separating equilibrium is when eS* is at the left of this

interval

This argument is known as The Intuitive Criterion The Intuitive Criterion of In-Koo Cho & David Kreps

21

1. Find All sepaprating Equilibria of the Spence Model2. Find Hybrid Equilibria, in which one type mixes, and the

other plays a pure strategy

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