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W3211 Spring 2009
Professor Vogel
Review for Midterm #1: Solutions
Review Questions and solutions:
1. Consider the following payo/ matrix:
L
R
T
x
1
;y
1
x
2
;y
2
B
x
3
;y
3
x
4
;y
4
In the following questions, suppose this is a simultaneous move game:
(a) What are the weakest conditions under which
(
T;L
)
is a strict dominant strategy
equilibrium?
Solution:
x
1
> x
3
,
x
2
> x
4
,
y
1
> y
2
,
y
3
> y
4
(b) What are the weakest conditions under which
(
T;R
)
is a dominant strategy equi
librium?
Solution:
x
1
x
3
,
x
2
x
4
,
y
2
y
1
,
y
4
y
3
(c) What are the weakest conditions under which
(
B;R
)
is a pure strategy Nash
equilibrium?
Solution:
(B, R) is a pure strategy Nash equilibrium if no player has an incentive
to deviate, given the strategy of the other player.
)
y
4
y
3
and
x
4
x
2
(d) Suppose
(
b
;r
)
is a strict mixed strategy Nash equilibrium, where
b
is the equi
librium probability Player
1
plays
B
and
r
is the equilibrium probability Player
2
plays
R
. Solve for
b
and
r
as functions of the
x
y
of
b
and
r
for a strict mixed strategy Nash equilibrium to exist?
Solution:
If
(
b
;r
)
is a strict mixed strategy Nash Equilibrium, then both
players must be indi/erent between playing their respective pure strategies given
(1
±
r
)
x
1
+
r
x
2
= (1
±
r
)
x
3
+
r
x
4
(1)
(1
±
b
)
y
1
+
b
y
3
= (1
±
b
)
y
2
+
b
y
4
(2)
We can rearrange (1) and (2) to get:
r
=
x
1
±
x
3
x
1
+
x
4
±
x
2
±
x
3
(3)
b
=
y
1
±
y
2
y
4
+
y
1
±
y
2
±
y
3
(4)
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View Full DocumentReview for Midterm 1
2
There is a strict mixed strategy Nash Equilibrium if both
0
< b
<
1
and
0
<
r
<
1
hold.
(e) What are the weakest conditions under which both
(
T;L
)
and
(
B;R
)
are pure
strategy Nash equilibria?
Solution:
From (c), (B, R) is a pure strategy Nash equilibrium if
y
4
y
3
and
x
4
x
2
. Similarly, (T, L) is a pure strategy Nash equilibrium if
y
1
y
2
and
x
1
x
3
. For both (T, L) and (B, R) to be pure strategy Nash equilibria, these 4
conditions should hold simultaneously.
(f) If
(
B;L
)
is a pure strategy Nash Equilibrium, can there exist a strict mixed
strategy Nash equilibrium?
Solution:
Yes. If (B, L) is a pure strategy Nash Equilibrium
)
x
3
x
1
and
y
1
y
2
. If
y
4
> y
3
and
x
2
> x
4
, then conditions in part (d) still hold and
therefore, we can have a strict mixed strategy Nash Equilibrium.
(g) If
(
B;L
)
is a strict dominant strategy equilibrium, can there exist a strict mixed
strategy Nash equilibrium?
Solution:
No, because in order for the players to randomize strictly, they need
to be indi/erent between their pure strategies.
Now suppose this is a sequential game in which Player
1
2
moves second after observing Player
1
±s move
(h) Write out the payo/ matrix for the sequential move game.
Solution:
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 Spring '08
 Govel

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