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Extensive Game with Imperfect Information III

Extensive Game with Imperfect Information III

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Extensive Game with Imperfect Information III. Topic One: Costly Signaling Game. Spence’s education game. Players: worker (1) and firm (2) 1 has two types: high ability  H with probability p H and low ability  L with probability p L . - PowerPoint PPT Presentation

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Page 1: Extensive Game with Imperfect Information III

Extensive Game with Imperfect Information III

Page 2: Extensive Game with Imperfect Information III

Topic One:Costly Signaling Game

Page 3: Extensive Game with Imperfect Information III

Spence’s education game• Players: worker (1) and firm (2)• 1 has two types: high ability H with probability p

H and low ability L with probability p L .• The two types of worker choose education level

e H and e L (messages).• The firm also choose a wage w equal to the

expectation of the ability• The worker’s payoff is w – e/

Page 4: Extensive Game with Imperfect Information III

Pooling equilibrium

• e H = e L = e* L pH (H - L) • w* = pHH + pLL

• Belief: he who chooses a different e is thought with probability one as a low type

• Then no type will find it beneficial to deviate.• Hence, a continuum of perfect Bayesian

equilibria

Page 5: Extensive Game with Imperfect Information III

Proof

The "best" deviation is to choose no education.

In this case, the worker gets a wage ' .The low type worker does not have incentive to deviate

*' *

*

* 1

*

L

L

L H H L LL

H H L LL

L H H

w

ew w

ep p

e p p

e p

When low type does not have incentive to deviate, so does high type.QED

L

Page 6: Extensive Game with Imperfect Information III

Separating equilibrium• e L = 0 H (H - L) ≥ e H ≥ L (H - L) • w H = H and w L = L

• Belief: he who chooses a different e is thought with probability one as a low type

• Again, a continuum of perfect Bayesian equilibria

• Remark: all these (pooling and separating) perfect Bayesian equilibria are sequential equilibria as well.

Page 7: Extensive Game with Imperfect Information III

Proof

Low type does not have incentive to deviate to any

Low type does not have incentive to deviate to

High type does not have incentive to deviate to any 0High type does not

H

H

HH L

L

L H L H

e e

e

ew w

w w e

e

have incentive to deviate to 0

QED

HL H

H

H H H L

e

ew w

e w w

Page 8: Extensive Game with Imperfect Information III

The most efficient separating equilibrium

LL

ew w

e

w

eH

Increase in payoff

eL=0

H type equilibrium payoff

H type payoff by choosing e=0

L type equilibrium payoff

HH

H H

e ew w

wH

wL

LH

ew w

Page 9: Extensive Game with Imperfect Information III

When does signaling work?

• The signal is costly• Single crossing condition holds (i.e., signal is

more costly for the low-type than for the high-type)

Page 10: Extensive Game with Imperfect Information III

Topic Two: Kreps-Cho Intuitive Criterion

Page 11: Extensive Game with Imperfect Information III

Refinement of sequential equilibrium

• There are too many sequential equilibria in the education game. Are some more appealing than others?

• Cho-Kreps intuitive criterion– A refinement of sequential equilibrium—

not every sequential equilibrium satisfies this criterion

Page 12: Extensive Game with Imperfect Information III

An example where a sequential equilibrium is unreasonable

(slided deleted)• Two sequential equilibria with outcomes: (R,R) and (L,L), respectively

• (L,L) is supported by belief that, in case 2’s information set is reached, with high probability 1 chose M.

• If 2’s information set is reached, 2 may think “since M is strictly dominated by L, it is not rational for 1 to choose M and hence 1 must have chosen R.”

L MR

2,2

1,3 0,0 0,0 5,1

22

1

L R L R

Page 13: Extensive Game with Imperfect Information III

Beer or Quiche (Slide deleted)

1,0 0,13,0

cstrong weak

1 1Q Q

B B2

2N N

NN

F F

FF

0,0 1,11,0

1,1

3,1

0.9 0.1

Page 14: Extensive Game with Imperfect Information III

Why the second equilibrium is not reasonable? (slide deleted)• If player 1 is weak she should

realize that the choice for B is worse for her than following the equilibrium, whatever the response of player 2.

• If player 1 is strong and if player 2 correctly concludes from player 1 choosing B that she is strong and hence chooses N, then player 1 is indeed better than she is in the equilibrium.

• Hence player 2’s belief is unreasonable and the equilibrium is not appealing under scrutiny.

1,0 0,13,0

cstrong weak1 1

Q Q

B B2

2N N

NN

F F

FF

0,0 1,11,0

1,1

3,1

0.9 0.1

Page 15: Extensive Game with Imperfect Information III

Cho-Kreps Intuitive Criterion• Consider a signaling game. Consider a sequential

equilibrium (β,μ). We call an action that will not reach in equilibrium as an out-of-equilibrium action (denoted by a).

• (β,μ) is said to violate the Cho-Kreps Intuitive Criterion if:– there exists some out-of-equilibrium action a so that one type,

say θ*, can gain by deviating to this action when the receiver interprets her type correctly, while every other type cannot gain by deviating to this action even if the receiver interprets her as type θ*.

• (β,μ) is said to satisfy the Cho-Kreps Intuitive Criterion if it does not violate it.

Page 16: Extensive Game with Imperfect Information III

Spence’s education game• Only one separating equilibrium survives the Cho-

Kreps Intuitive criterion, namely: e L = 0 and e H = L (H - L)

• Any separating equilibrium where e L = 0 and e H > L (H - L) does not satisfy Cho-Kreps intuitive criterion.

• A high type worker after choosing an e slightly smaller will benefit from it if she is correctly construed as a high type.

• A low type worker cannot benefit from it however.• Hence, this separating equilibrium does not survive

Cho-Kreps intuitive criterion.

Page 17: Extensive Game with Imperfect Information III

The most efficient separating equilibrium

e

w

eHeL=0

H type equilibrium payoff

L type equilibrium payoff

wH

wL

Page 18: Extensive Game with Imperfect Information III

Inefficient separating equilibrium

e

w

eH’eL=0

H type equilibrium payoff

L type equilibrium payoff

wH’

wL

eHe#

H type better off by deviating to e# if believed to be High type

L type worse off by deviating to e# if believed to be High type

Page 19: Extensive Game with Imperfect Information III

Spence’s education game• All the pooling equilibria are eliminated by the Cho-Kreps

intuitive criterion.• Let e satisfy w* – e*/ L > H – e/ L and w* – e*/ H > H –

e/ L (such a value of e clearly exists.)• If a high type work deviates and chooses e and is

correctly viewed as a good type, then she is better off than under the pooling equilibrium

• If a low type work deviates and successfully convinces the firm that she is a high type, still she is worse off than under the pooling equilibrium.

• Hence, according to the intuitive criterion, the firm’s belief upon such a deviation should be such that the deviator is a high type rather than a low type.

• The pooling equilibrium break down!

Page 20: Extensive Game with Imperfect Information III

Topic Three:Cheap Talk Game

Page 21: Extensive Game with Imperfect Information III

Cheap Talk Model

Two players: S (sender) and R (receiver)S's type is uniformly distributed in the unit interval [0,1]In the first stage: S chooses a message [0,1]In the second stage: R chooses an action [0,1

m Ma A

2

2

]

( , ) ( ( )) where 0

( , ) ( )If S is not allowed to send message, R will choose 1/ 2.Does cheap talk matter?

S

R

u a a b b

u a aa

Page 22: Extensive Game with Imperfect Information III

Perfect Information Transmission?

• An equilibrium in which each type will report honestly does not exist unless b=0.

Page 23: Extensive Game with Imperfect Information III

No information transmission

• There always exists an equilibrium in which no useful information is transmitted.

• The receiver regards every message from the sender as useless, uninformative.

• The sender simply utters uninformative messages.

Page 24: Extensive Game with Imperfect Information III

Some information transmission

1 1 2 1 1

There exists a perfect Bayesian equilibrium as follows:

Partition of the unit interval: [0, ),[ , ),...,[ , ),...,[ , 1]Types of S who are in the same segment give the same message (wlog, m

k k K Kx x x x x x x

1 1

=segment's lower bound)Types in different segments give different messages

R always choose an action ( ) / 2 when message is received

R holds the belief that: S has a type equally likely in the

k k kx x x

segment in which the message is in.

Page 25: Extensive Game with Imperfect Information III

Some information transmission

1

21 1

22 1

1

Consider the incentive of the type

If she reports , she gets ( ) / 2 .

If she reports a message in the preceding segment,

she gets ( ) / 2 .

The two must be the same

2

k

k k k

k k

k k

x

x x x b

x x b

x x b

2 1

1 1 2

24

each segment exceeds the preceding segment by 4b

k k

k k k k

x x b

x x x x b

Page 26: Extensive Game with Imperfect Information III

Some Information Transmission

( 4 ) ... ( 4( 1) ) 2 ( 1) 1

Let *( ) be the largest integer satisfying 1- 2 ( -1) 0

1- 2 ( -1)For any *( ), determine a unique number

and a unique partition [0, ),[ ,4 ),...,[ (4 -1)

d d b d K b Kd K K b

K b K K b

K K bK K b dK

d d b d K

,1]

Remark: *( ) is increasing in

b

K b b

Page 27: Extensive Game with Imperfect Information III

Final Remark:

• Relationship among different equilibrium concepts:

• Sequential equilibrium satisfying Cho-kreps => sequential equilibrium => Perfect Bayesian equilibrium => subgame perfect equilibrium => Nash equilibrium