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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 .• 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/
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
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
ep p
e p
When low type does not have incentive to deviate, so does high type.
QED
L
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.
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 0
High 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
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
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)
Topic Two: Kreps-Cho Intuitive Criterion
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
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.”
LM
R
2,2
1,3 0,0 0,0 5,1
22
1
L R L R
Beer or Quiche (Slide deleted)
1,0 0,13,0
c
strong weak1 1
Q Q
B B2
2N N
NN
F F
FF
0,01,11,0
1,1
3,1
0.9 0.1
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 weak
1 1
Q Q
B B2
2N N
NN
F F
FF
0,01,11,0
1,1
3,1
0.9 0.1
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.
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.
The most efficient separating equilibrium
e
w
eHeL=0
H type equilibrium payoff
L type equilibrium payoff
wH
wL
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
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!
Topic Three:Cheap Talk Game
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 M
a 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 a
a
Perfect Information Transmission?
• An equilibrium in which each type will report honestly does not exist unless b=0.
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.
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.
Some information transmission
1
2
1 1
2
2 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 xb
2 1
1 1 2
2
4
each segment exceeds the preceding segment by 4b
k k
k k k k
x xb
x x x x b
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 d
Kd d b d K
,1]
Remark: *( ) is increasing in
b
K b b
Final Remark:
• Relationship among different equilibrium concepts:
• Sequential equilibrium satisfying Cho-kreps => sequential equilibrium => Perfect Bayesian equilibrium => subgame perfect equilibrium => Nash equilibrium