Homework 6 Answer
(Total 20 points)
Q1
(3pt.): Find all the purestrategy Nash equilibria of the game below.
B
A
l
m
r
T
4,

5
1,

3

1,

6
M
2,

1

2, 0
0, 1
B
7, 6
0, 3

4, 8
Answer
: (
T,m
) and (
M,r
).
Q2
: Consider the game of Battle of Sexes,
B
A
Boxing
Opera
Boxing
5, 3
0, 0
Opera
0, 0
3, 5
Find all the Nash equilibria. There are two purestrategy equilibria and one mixed
strategy equilibrium. (2pt. for each purestrategy equilibrium, and 4pt. for the
mixedstrategy equilibrium)
Answer
: Let
p
A
be the probability that A goes to Baxing, let
p
B
be the probability
that B goes to Boxing. A’s expected utility is given (
p
A
,p
B
) is
EU
A
(
p
A
,p
B
) = 5
p
A
p
B
+ 3(1

p
A
)(1

p
B
)
= (8
p
B

3)
p
A
+ 3

3
p
B
Hence, A’s best response is
BR
A
(
p
B
) =
{
0
}
,
when
p
B
<
3
8
entire [0
,
1]
,
when
p
B
=
3
8
{
1
}
,
when
p
B
>
3
8
.
B’s expected utility is given (
p
A
,p
B
) is
EU
B
(
p
A
,p
B
) = 3
p
A
p
B
+ 5(1

p
A
)(1

p
B
)
= (8
p
A

5)
p
B
+ 5

5
p
A
1