14.12 Game TheoryMidterm I
10/15/2003
Prof. Haluk Ergin
Instructions:
This is an open book exam, you can use any written material. You
have 1 hour and 20 minutes. Each question is 35 points where the bonus question 3(c)
accounts for the extra 5 points. Good luck!
1. Consider the following two player game where Player 1 chooses one of the three
rows and Player 2 chooses one of the three columns:
C
1
C
2
C
3
R
1
2,1
4,2
2,0
R
2
3,3
0,0
1,1
R
3
1,2
2,8
5,1
(a) What are the strategies that survive IESDS?
Solution:
Strategy
C
3
is strictly dominated by mixture
1
2
C
1
+
1
2
C
2
;once
C
3
is out,
R
3
is strictly dominated by
R
1
.
The answer is
R
1
,R
2
,C
1
2
.
(b) At each step of the elimination what were your rationality and knowledge
assumptions?
Solution:
At the
f
rst step we assume that player 2 is rational; at the
second step we assume that player 1 is rational and knows that player 2
is rational.
(c) Find all Nash equilibria, including the mixed one.
Solution:
By inspection,
(
R
1
2
)
and
(
R
2
1
)
arepurestrategyNashequi
libria. To
f
nd the mixed one, assume that player 1 plays
αR
1
+(1
−
α
)
R
2
.
Playing
C
1
gives player 2 the payo
f
of
−
α
+3(1
−
α
)=3
−
4
α,
while
playing
C
2
gives him
2
α.
He will be willing to mix if
α
=
1
2
.
Likewise,
assume that player 2 plays
βC
1
−
β
)
C
2
.
Playing
R
1
gives player 1
the payo
f
of
2
β
+4(1
−
β
)=4
−
2
β
, while playing
R
2
gives her
3
β.
She
will be willing to randomize if
β
=
4
5
.
So the mixed Nash equilibrium is
(
1
2
R
1
+
1
2
R
2
,
4
5
C
1
+
1
5
C
2
)
.
2. Consider the following extensive form game with perfect information:
(a) Find out the backwards induction outcome.
Solution:
i.
in the last stage of the game, player 1 chooses to play a.
ii.
in the previous stage, player 2 chooses to play L.
1
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(0,1)
(1,0)
1
2
2
A
B
Ll
r
1
a
b
R
Figure 1:
iii.
if we’d been in right hand side part of the game (following a move to the
right B in the
f
rst stage), player 2 would have chosen r.
iv.
in the
f
rst stage of the game player 1 choses A.
So the backward induction outcome of this game is "player 1 plays A, player
2 responds by playing L; if player 1 had played B in the
f
rst place, player
2 would have responded by playing r; if A and R had been played, player 1
would have responded by playing a".
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 Spring '05
 MuhammadYildiz
 Game Theory, Nash, C1 C2 C3

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