Extensive Game with Imperfect Information III

1
Extensive Game with Imperfect Information III
2
Topic OneCostly Signaling Game
3
Spences 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 workers payoff is w e/?

4
Pooling equilibrium

• e H e L e ? ?L pH (?H - ?L)
• w pH?H pL?L
• 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

5
Proof
6
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.

7
Proof
8
The most efficient separating equilibrium
9
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)

10
Topic Two Kreps-Cho Intuitive Criterion
11
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 equilibriumnot every
  sequential equilibrium satisfies this criterion

12
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 2s
  information set is reached, with high probability
  1 chose M.
• If 2s 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.

13
Beer or Quiche (Slide deleted)
14
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 2s belief is unreasonable and the
  equilibrium is not appealing under scrutiny.

15
Cho-Kreps Intuitive Criterion

• Consider a signaling game. Consider a sequential
  equilibrium (ß,µ). We call an action that will
  not reach in equilibrium as an out-of-equilibrium
  action (denoted by a).
• (ß,µ) is said to violate the Cho-Kreps Intuitive
  Criterion if
• there exists some out-of-equilibrium action a so
  that one type, say ?, 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 ?.
• (ß,µ) is said to satisfy the Cho-Kreps Intuitive
  Criterion if it does not violate it.

16
Spences 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 gt ?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.

17
The most efficient separating equilibrium
18
Inefficient separating equilibrium
L type worse off by deviating to e if believed
to be High type
w
H type better off by deviating to e if believed
to be High type
L type equilibrium payoff
H type equilibrium payoff
wH
e
wL
eH
e
eL0
eH
19
Spences education game

• All the pooling equilibria are eliminated by the
  Cho-Kreps intuitive criterion.
• Let e satisfy w e/ ?L gt ?H e/ ?L and w
  e/ ?H gt ?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
  firms 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!

20
Topic ThreeCheap Talk Game
21
Cheap Talk Model
22
Perfect Information Transmission?

• An equilibrium in which each type will report
  honestly does not exist unless b0.

23
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.

24
Some information transmission
25
Some information transmission
26
Some Information Transmission
27
Final Remark

• Relationship among different equilibrium
  concepts
• Sequential equilibrium satisfying Cho-kreps gt
  sequential equilibrium gt Perfect Bayesian
  equilibrium gt subgame perfect equilibrium gt
  Nash equilibrium