1
Quiz 1 Solutions
ECOS3012 - Strategic Behavior, Semester 1, 2011
Note.
Here is the sketch of solutions to Quiz 1. I will put up the explanation for Q7 a bit later – it is easiest to explain with
a couple of pictures and I am not able to draw those on this computer.
Begin Quiz
Q1.
Assume that there are two states of the world and a typical action is denoted by a vector (
x,y
) where
x
≥
0 and
y
≥
0 denotes monetary reward in the two states. Consider the utility functions
U
(
x,y
) =
x
α
y
1
/
2
,
V
(
x,y
) =
α
log(
x
) +
β
log(
y
) where 0
< α < β <
1.
A decision maker is necessarily consistent with the expected utility hypothesis if her preferences are give by the utility
(a)
Either U or V.
(b)
Neither U nor V.
(c)
Only U.
(d)
Only V.
Solution
: See solution to Q1, Problem Set 1 to note that the preferences of a DM that are described by
U
will be
the as thsoe of a DM whose utility function is given by
U
0
, where
U
0
(
x,y
) =
α
α
+ 1
/
2
log(
x
) +
1
/
2
α
+ 1
/
2
log(
y
)
Similarly both
V
and
V
0
both describe the same preferences where
V
0
(
x,y
) =
α
α
+
β
log(
x
) +
β
α
+
β
log(
y
)
.
Both
U
0
and
V
0
are utility functions that would conform to a DM who satisﬁes Expected Utiltiy Hypothesis.
±
Q2.
Suppose there are three states of the world and a typical action is given by a triple (
x,y,z
), denoting the monetary
reward in the three states of the world. Consider two actions
A
= (
x,y,z
) and
B
= (ˆ
x,
ˆ
y,z
). ( Both actions give the
same reward in state 3). Now consider any other actions
C
= (
x,y,z
*
) and
D
= (ˆ
x,
ˆ
y,z
*
) obtained from A and B
by changing the reward in state 3. Suppose a Decision Maker (DM) is known to satisfy Expected Utility Hypothesis
and it is known that she strictly prefers A to B.
(a)
DM must strictly prefer C to D.
(b)
DM may strictly prefer C to D but depends on the value of
z
*
.
(c)
DM may prefer C to D but this depends on payoﬀs in the ﬁrst two states
(d)
None of the above.
Solution
: This is straightforward. If the Expected utility hypothesis holds, then there must be a probability
distribution (
p
1
,p
2
,p
3
, and a utility function
u
such that
U
(
A
)
=
p
1
u
(
x
) +
p
2
u
(
y
) +
p
3
u
(
z
)
(1)
U
(
B
)
=
p
1
u
(
x
0
) +
p
2
u
(
y
0
) +
p
3
u
(
z
)
(2)
U
(
C
)
=
p
1
u
(
x