1. problem set 14 from osborne’s introd. to g.t. p.332 ex. 332.1
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problem set 14
from Osborne’sIntrod. To G.T.
p.332 Ex. 332.1
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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 educationThe firm’s action plan
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The structure of equilibrium
Type chooses : L e* L
Type chooses : *SH e
The firm must believe :
after observing :
after observing : *S
L e* L
H e
after observing anything else it can believe anything.
Specify what it believes after observing any signal.
The firm chooses an optimal action given its belief
, are 's optimal signals given the firm's
action plan
*se* L e L,H
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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
Whatever action the firm takes, (even if it pays him as if
he were H) he is better off sending e*(L).
*Se
L(An H is better off in this interval if he is identified as H. )Any signal above e*s is dominated for both types, so the
firm is allowed to believe something.
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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
L
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Beer or Quiche
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Beer or Quiche
Nature chooses the type of player 1 : Weak, Strong with probability r, 1-r
The Weak type likes Quiche for his breakfast, the Strong likes Beer. Player 1 chooses his breakfast
Player 2 does not know the type of player 1 but observes what he had for breakfast. He then decides whether to pick up a fight with player 1.
Player 1 gets 1 point if he had his favorite meal, and gets additional 2 points if there was no fight.
Player 2 gets 1 point if he fought the Weak type or if he did not fight the Strong
all other payoffs are 0
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Beer or Quiche
N
S
W
r
1-r
beer
beer
quiche
quiche
fight
fight
I II
fight
fight
II
0 , 1
3 , 1
1 , 0
2 , 0
1 , 1
2 , 1
0 , 0
3 , 0
Is there a separating equilibrium ?
pq
1-p1-q
10
Beer or Quiche
N
S
W
r
1-r
beer
beer
quiche
quiche
fight
fight
I II
fight
fight
II
0 , 1
1 , 0
1 , 1
2 , 1
3 , 0
Is there a separating equilibrium ?
01
10 3 , 1
not an equilibrium
2 , 0
0 , 0
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Beer or Quiche
N
S
W
r
1-r
beer
beer
quiche
quiche
fight
fight
I II
fight
fight
II
2 , 0
1 , 1
0 , 0
Is there a separating equilibrium ?
10
01 3 , 1
0 , 1
2 , 1
not an equilibrium
3 , 0
1 , 0
12
Beer or Quiche
N
S
W
r
1-r
beer
beer
quiche
quiche
fight
fight
I IIfight
fight
II
2 , 0
1 , 1
0 , 0
3 , 0
Is there a pooling equilibrium ?
rq
1-r1-q 3 , 1
0 , 1
2 , 1
If r > ½ then fight
1 , 0
not an equilibrium
13
Beer or Quiche
N
S
W
r
1-r
beer
beer
quiche
quiche
fight
fight
I IIfight
fight
II
2 , 03 , 0
Is there a pooling equilibrium ?
pr
1-p1-r 3 , 12 , 1
If r > ½ then fight
r > ½
0 , 0
1 , 1 0 , 1
1 , 0
not an equilibrium
14
Beer or Quiche
N
S
W
r
1-r
beer
beer
quiche
quiche
fight
fight
I IIfight
fight
II
3 , 0
Is there a pooling equilibrium ?
rq
1-r1-q
0 , 1
2 , 1
If r < ½ then not fight
1 , 0
r < ½
2 , 0
3 , 1
If q > ½ then fight
0 , 0
1 , 1
an equilibrium
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Beer or Quiche
N
S
W
r
1-r
beer
beer
quiche
quiche
fight
fight
I IIfight
fight
II
2 , 0
Is there a pooling equilibrium ?
pr
1-p1-r 3 , 1
If r < ½ then not fight
r < ½
0 , 0
1 , 1
If p > ½ then fight
0 , 1
1 , 0
p > ½
an equilibrium
3 , 0
2 , 1
but …..
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Beer or Quiche
N
S
W
r
1-r
beer
beer
quiche
quiche
fight
fight
I IIfight
fight
II
2 , 0
Is there a pooling equilibrium ?
pr
1-p1-r 3 , 1
r < ½
0 , 0
1 , 1 0 , 1
1 , 0
p > ½
3 , 0
2 , 1
Is it reasonable for player 2 to believe that W will drink beer when all his possible payoffs (after beer) are less than 3 ?
00 , 1
22 , 033 , 0