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Department of Economics
The Ohio State University
Final Exam Answers—Econ 805
Prof. Peck
Winter 2008
1. (40 points)
Consider the following two player, zero sum game between an attacker and
a defender.
The attacker must choose which one of three targets to attack,
and the defender must choose which one of the three targets to defend. If the
defender defends the target that the attacker attacks, both sides receive a payo
f
of zero.
If the defender does not defend the target that the attacker attacks,
then the payo
f
s depend on which target was attacked, as follows: if target 1
is successfully attacked, the payo
f
s to the attacker and defender are
(2
,
−
2)
;i
f
target 2 is successfully attacked, the payo
f
s to the attacker and defender are
(1
,
−
1)
; and if target 3 is successfully attacked, the payo
f
s to the attacker and
defender are
(
v,
−
v
)
, where we have
v<
1
.
(a) (10 points) Prove that this game does not have a Nash equilibrium in
pure strategies.
(b) (20 points) Assuming that there is a mixed strategy Nash equilibrium
in which all three targets are attacked with positive probability,
f
nd the equilib
rium mixing probabilities for the attacker
(
p
1
,p
2
3
)
and the equilibrium mixing
probabilities for the defender
(
q
1
,q
2
3
)
. Your answer should be a function of
the parameter,
v
.
(c) (10 points) For what values of
v
w
i
l
ltherebeam
ixeds
tra
tegyNash
equilibrium in which all three targets are attacked with positive probability?
Answer:
(a) Suppose there is a NE in pure strategies. If the attacker is
choosing the same target that the defender is defending, then the attacker can
dev
iatetoad
i
f
erent target and increase his payo
f
. If the attacker is choosing
ad
i
f
erent target from the one that the defender is defending, then the defender
can deviate to the attacker’s target and increase her payo
f
.T
h
u
s
,
t
h
e
s
t
r
a
t
e
g
y
pro
f
le cannot be a NE.
(b) If the attacker is attacking each target with positive probability, he must
be indi
f
erent between all three actions. Therefore, we have
2(1
−
q
1
)=1
−
q
2
=
v
(1
−
q
3
)
and
q
1
+
q
2
+
q
3
=1
.
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This note was uploaded on 07/17/2008 for the course ECON 805 taught by Professor Peck during the Spring '08 term at Ohio State.
 Spring '08
 PECK
 Economics

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