1
problem set 13
see last slide
Last Slide
Extensive form Games with
Incomplete Information
Information set of player 1
Belief
p1
p2
p3
1
1
1
p1 + p2 + p3 = 1, pi ≥ 0
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Extensive form Games with
Incomplete Information
Information set of player 1
p1
p2
p3
1
1
1
player 2 mixes
1- α1
What is player 1’s
belief at this
information set ??
2
2
2
α1
α2
α3
1
1
1
4
p1
p2
p3
1
1
1
player 2 mixes
What
playerLaw:
1’s
UsingisBayes’
belief at this
information set ??
p11
p11  p2 2  p3 3
2
α1
1
2
2
α2
α3
1
1
p2 2
p11  p2 2  p3 3
p3 3
p11  p2 2  p3 3
5
p1
p2
p3
1
1
1
2
Updating the belief is
consistent with the
strategy profile
(whenever possible)
α1
1
2
2
α2
α3
1
1
p11
p2 2
p11  p2 2  p3 3 > 0 p11  p2 2  p3 3
p3 3
p11  p2 2  p3 3
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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.
p1
p2
p3
1
1
1
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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)
p1
p2
p3
1
1
1
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Signalling Games
The sender, a player who has complete information
(about the state of nature, or his own type) sends a
signal to the other player, the receiver.
The receiver observes the signal and takes an action.
The payoffs depend on the state of nature, the signal
and the action takren
Michael Spence
Nobel Prtize, 2001
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Education as a signal
The worker, has skills H or L with probability q, 1-q,
resp. He knows his own productivity.
He chooses a level of education
e which costs him
c(η,e) where η is his type.
His productivity is y(η,e).
The firm observes the signal, and pays the worker a
wage rate w which equals the productivity that it
believes he has.
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Education as a signal
The payoff to the worker is:
The payoff to the firm is:
w - c(η,e)
y(η,e) - w
dc
dc
assume :
 L,e  > (H,e) > 0
de
de
IL
The indifference curves of type η
I η : w - c  η,e  = Const.
IH
Single Crossing property
e
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Education as a signal
assume :
y  H,e  > y(L,e),
dy
0
de
y  H,e 
y  L,e 
y  H,e  ???
e
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Education as a signal
In complete information , the worker maximizes
y  η,e  - c(η,e).
Iη
y  η,e 
e*  η
e
two cases:
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Education as a signal
IH
No Envy
y  H,e 
IL
y  L,e 
e*  L
e*  H 
e
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Education as a signal
Envy
IH
IL
e*  L e*  H 
y  H,e 
y  L,e 
e
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Pooling Equilibrium
Both types choose education level: ep
Observing e p , the firm believes that the worker's type is :
 = qH +  1 - q  L.
Observing e  e p , the firm believes that the worker's type is :
L.
The wage rate :
w  e p  = w p = qy  H,e p  +  1 - q  y  L,e p 
w  e  = y  L,e  , e  e p
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Pooling Equilibrium
IH
IL
e*  L e p
y  H,e  y  η,e 
y  L,e 
e
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Separating Equilibrium
Type L chooses : e*  L
Payoffs for education
Type H chooses : e*S
Envy
y  L,e 
IL
e*  L e*  H 
Beliefs:
y  H,e 
IH
L
e
e
*
S
H
L
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Can the firm have these beliefs???
Separating Equilibrium
It is
a strictly
dominated
Type L
chooses
: e*
 L (inferior) strategy for type L to
send a signal in this interval
Type H chooses : e*S


Even if he is identfied as H he is better
off sending
e*(L).
I
y H,e
H
y  L,e 
IL
e*  L e*  H 
e
*
S
e
Beliefs:
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L
L
(An H is better off in this interval if H
he is identified as H. )
accept this argument then
Separating Equilibrium Ifthewefirm
‘s belief in this interval
Type L chooses : e*  L
Type H chooses : e*S
should be H.
The only separating equilibrium
is when eS* is at the left of this
IH
y H,e
interval




y L,e
This argument is known as TheIIntuitive
Criterion
L
of In-Koo Cho & David Kreps
e*  L e*  H 
Beliefs:
L
e
e
*
S
H
L
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1. Find All sepaprating Equilibria of the Spence Model
2. Find Hybrid Equilibria, in which one type mixes, and
the other plays a pure strategy
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