Economics 109 Midterm Examination
Prof. Watson, Fall 2003
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. You may use the scratch paper that has been
distributed but, when you have
f
nished the examination, submit
only
your answer sheet.
1.
Write your name in the designated space on the answer sheet. In the space marked “version,”
write the following number: 1.
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 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.
For the normal-form game pictured below, answer the following questions. (You do not need
to show any work; these questions will be graded only on the basis of whether your
f
nal answers
are correct.)
3, 2
3, 4
5, 0
3, 1
w
y
ef
2
1
g
3, 2
0, 5
z
4, 5
4, 4
4, 2
(a)
Calculate the set
BR
2
(
μ
1
) for the belief
μ
1
=(0
,
3
/
8
,
5
/
8). (That is,
μ
1
puts zero probability
on w, 3
/
8ony
,and5
/
8onz
.)
(b)
How many of the nine strategy pro
f
les are e
ﬃ
cient? (Do not identify them; just give the
number.)
(c)
Is the set
X
=
{
w
,
y
}×{
e
,
f
}
weakly congruent?
(d)
How many (pure strategy) Nash equilibria does this game have?
(e)
Calculate and report the set of rationalizable strategies.