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7
Rationalizability and Iterated Dominance
1.
(a)
R
=
{
U, M, D
}×{
L, R
}
.
(b) Here there is a dominant strategy. So we can iteratively delete dom
inated strategies. U dominates D. When D is ruled out, R dominates C.
Thus,
R
=
{
U, M
L, R
}
.
(c)
R
=
{
(U, L)
}
.
(d)
R
=
{
A, B
X, Y
}
.
(e)
R
=
{
A, B
X, Y
}
.
(f)
R
=
{
A, B
X, Y
}
.
(g)
R
=
{
(D, Y)
}
.
2.
Chapter 2, problem 1(a) (the normal form is found in Chapter 4, problem
2):
R
=
{
(Ea, aa
I
)
,
(Ea, an
I
)
}
. Chapter 5:
R
=
{
U, M, D
L, C, R
}
.
3.
No. This is because 1/2 A 1/2 B dominates C.
4.
For “give in” to be rationalizable, it must be that
x
≤
0. The man
ager must believe that the probability that the employee plays “settle” is
(weakly) greater than 1/2.
5.
R
=
{
(
w,c
)
}
. The order does not matter because if a strategy is domi
nated (not a best response) relative to some set of strategies of the other
player, then this strategy will also be dominated relative to a smaller set
of strategies for the other player.
6.
R
=
{
(7:00, 6:00, 6:00)
}
.
86
7 RATIONALIZABILITY AND ITERATED DOMINANCE
87
7.
Yes. If
s
1
is rationalizable, then
s
2
is a best response to a strategy of
player 1 that may rationally be played. Thus, player 2 can rationalize
strategy
s
2
.
8.
No. It may be that
s
1
is rationalizable because it is a best response to some
other rationalizable strategy of player 2, say ˆ
s
2
, and just also happens to
be a best response to
s
2
.
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View Full Document 9
Congruous Strategies and Nash Equilibrium
1.
(a) The Nash equilibria are (B, CF) and (B, DF).
(b) The Nash equilibria are (IU, I), (OU, O) and (OD, O).
(c) The Nash equilibria are (UE, BD), (UF, BD), (DE, AC), and (DE,
BC).
(d) There is no Nash equilibrium.
2.
(a) The set of Nash equilibria is
{
(B, L)
}
=
R
.
(b
)Th
es
e
to
fNa
shequ
i
l
ib
r
iai
s
{
(U, L),(M, C)
}
.
R
=
{
U, M, D
}×
{
L, C
}
.
(c) The set of Nash equilibria is
{
(U, X)
}
=
R
.
(d) The set of Nash equilibria is
{
(U, L), (D, R)
}
.
R
=
{
U, D
}×{
L, R
}
.
3.
Figure 7.1: The Nash equilibrium is (B,Z).
Figure 7.3: The Nash equilibrium is (M,R).
Figure 7.4: The Nash equilibria are (stag,stag) and (hare,hare).
Exercise 1: (a) No Nash equilibrium. (b) The Nash equilibria are (U,R)
and (M,L). (c) The Nash equilibrium is (U,L). (d) The Nash equilibria are
(A,X) and (B,Y). (e) The Nash equilibria are (A,X) and (B,Y). (f) The
Nash equilibria are (A,X) and (B,Y). (g) The Nash equilibrium is (D,Y).
Chapter 4, Exercise 2: The Nash equilibria are (Ea,aa
I
) and (Ea,an
I
).
Chapter 5, Exercise 1: The Nash equilibrium is (D,R).
Exercise 3: No Nash equilibrium.
4.
Only at (1/2, 1/2) would no player wish to unilaterally deviate. Thus,
the Nash equilibrium is (1/2, 1/2).
92
9 CONGRUOUS STRATEGIES AND NASH EQUILIBRIUM
93
5.
Player 1 solves max
s
1
3
s
1
−
2
s
1
s
2
−
2
s
2
1
.Tak
ing
s
2
as given and di
f
erentiat
ing with respect to
s
1
yields the
f
rst order condition 3
−
2
s
2
−
4
s
1
=0
.Re
arranging, we obtain player 1’s best response function:
s
1
(
s
2
)=3
/
4
−
s
2
/
2.
player 2 solves max
s
2
s
2
+2
s
1
s
2
−
2
s
2
2
. This yields the best response function
s
2
(
s
1
)=1
/
4+
s
1
/
2. The Nash equilibrium is found by
f
nding the strategy
pro
f
le that satis
f
es both of these equations. Substituting player 2’s best
response function into player 1’s, we have
s
1
=3
/
4
−
1
/
2[1
/
s
1
/
2]. This
implies that the Nash equilibrium is (1/2, 1/2).
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This note was uploaded on 08/20/2011 for the course ECON 101 taught by Professor Etw during the Spring '11 term at UniversitÃ di Bologna.
 Spring '11
 etw

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