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Homework 2 Solutions
David Easley and Jon Kleinberg
February 13, 2008
Grading: Part A is out of 10, Part B is out of 14.
1.a
[2 points]
D
is a dominant strategy for player
A
and
R
is a dominant
strategy for player
B
. Thus the only Nash equilibrium is (
D,R
).
1.b
[2 points]
U
is a dominant strategy for player
A
and
L
is player
B
’s
unique best response to
U
. Thus the only Nash equilibrium is (
U,L
).
1.c
[2 points] This game has two pure strategy Nash equilibria: (
D,L
)
and (
U,R
). It also has a Nash equilibrium in which both players use mixed
strategies. Let
Pr
A
(
U
) =
p
and
Pr
B
(
L
) =
q
. To have a Nash equilibrium
p,q
, with both
p
and
q
greater than 0 and less than 1,
A
’s expected payoFs
from
U
and
D
must be equal; and
B
’s expected payoFs from
L
and
R
must
be equal. So we have
q
+ 4(1

q
) = 3
q
+ 2(1

q
)
and
p
+ 3(1

p
) = 2
p
+ 2(1

p
)
.
Solving we see that
p
=
q
= 1
/
2.
2.a
[ 2 points] This game has two pure strategy Nash equilibria (
U,L
)
and (
D,R
). Strategy
L
is a weakly dominated strategy for player
B
. So the
equilibrium (
U,L
) uses a weakly dominated strategy.
2.b
[2 points] Here we are just looking for a discussion of why a player
might or might not use a weakly dominated strategy. There are at least two
equally good answers and an argument along either line received full credit
for this part:
•
One way to reason about the game is that
B
knows that if he has an
opportunity to make a move that matters, then
A
must have chosen
D
. As
R
is the unique best response to
D
, player
B
should chose
R
.
•
A second possibility is to note that, no matter what
A
does,
B
has
nothing to lose from choosing
R
rather than
L
. So he should chose
R
.
In either of these cases, if this is how
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 Spring '08
 EASLEY/KLEINBERG

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