Economics 109 Midterm Examination
Prof. Watson, Fall 2002
You have one hour and twenty minutes to complete this examination. You may
not
use your
notes or any books during the examination. Write your answers in the spaces provided on the
answer sheet that has been distributed separately. When you have
f
nished the examination,
submit
only
your answer sheet.
Basic Questions
1.
Write your name in the designated space on the answer sheet. In the space marked “version,”
write the following number: 3.
2.
Consider the following strategic setting. There are three players, numbered 1, 2, and 3. At
the beginning of the game, players 1 and 2 simultaneously make decisions, each choosing between
“X” and “Y.” If they both choose “X” then the game ends and the payo
f
vector is (1
,
0
,
0); that
is, player 1 gets 1, player 2 gets 0, and player 3 gets 0. If they both choose “Y” then the game
ends and the payo
f
vector is (0
,
1
,
0); that is, player 2 gets 1 and the other players get 0.
If one player chooses “X” while the other chooses “Y,” then player 3 must guess which of the
players selected “X.” That is, player 3 chooses between “1” and “2.” Player 3 makes his selection
knowing only that the game did not end following the choices of players 1 and 2. If player 3
guesses correctly, then both he and the player who selected “X” each obtains a payo
f
of 2, and
the player who selected “Y” gets 0. If player 3 guesses incorrectly, then everyone gets a payo
f
of
0.
Represent this game in the extensive form (draw the game tree).
3.
Convert the following extensive form game into the normal form (draw and label the matrix):
1
L
M
Q
P
1, 2
2
4, 1
8, 0
0, 3
0, 2
X
Y
X
Y
2
4.
For each of the normal form games pictured on your answer sheet,
f
nd the sets of rationalizable
strategies and the Nash equilibria. Designate the Nash equilibria by circling the cells corresponding
to equilibrium strategies. Designate the rationalizable sets by striking out strategies that are
iteratively dominated and by describing the sets with the proper notation. If you need a mixed
strategy to dominate a pure strategy, specify which mixed strategy you use.
1
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5.
Consider an asymmetric Cournot duopoly game, where the two
f
rms have di
f
erent costs of
production. Firm 1 selects quantity
q
1
at a production cost of 2
q
1
. Firm 2 selects quantity
q
2
and
pays the production cost 4
q
2
. The market price is given by
p
=12
−
q
1
−
q
2
.
(a) What are the payo
f
(pro
f
t) functions of the two
f
rms?
(b) Calculate the
f
rms’ bestresponse functions
BR
1
(
q
2
)and
2
(
q
1
) and graph them.
(c) Calculate the Nash equilibrium of this game and designate it on your graph.
6.
Consider the normal form game pictured here:
2, 0
1, 3
5, 4
1, 3
A
B
X
Y
2
1
Z
5,
x
6, 2
All of the payo
f
numbers are speci
f
ed, with the exception of that denoted by
x
.F
indanumb
e
r
for
x
such that the following three statements are
all
true: (B, X) is a Nash equilibrium, (A, Z) is
an e
ﬃ
cient strategy pro
f
le
,and
,forthebe
l
ie
f
μ
1
=(
1
2
,
1
2
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 Fall '08
 WATSON
 Economics, Game Theory, ........., Nash

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