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Econ 145: Practice Questions I
Suggested Answer
February 8, 2010
1. Consider a room assignment problem with 4 students and 4 rooms
{
h
1
,...,h
4
}
.
Room
h
i
is initially
assigned to student
i.
Suppose that the preferences of the students are as follows
h
2
Â
1
h
4
Â
1
h
1
Â
1
h
3
h
2
Â
2
h
1
Â
2
h
3
Â
2
h
4
h
1
Â
3
h
3
Â
3
h
4
Â
3
h
2
h
3
Â
4
h
4
Â
4
h
1
Â
4
h
2
where
Â
i
is student
i
’s (strict) preference over the rooms. Answer following questions.
(a) Is the initial allocation e
ﬃ
cient? Explain why.
Answer:
We know that the TTC algorithm with an e
ﬃ
cient initial allocation leads the same allocation (See
Problem Set I Problem 1 (a)). So we here simply apply the TTC algorithm, then compare the
resulting allocation with the initial allocation (
h
1
,h
2
3
4
).
Step 1:
We have initial assignment (
h
1
2
3
4
)/ Each student wants to obtain
h
1
→
by 1
h
2
h
2
→
by 2
h
2
h
3
→
by 3
h
1
h
4
→
by 4
h
3
Then, we can
f
nd a trading cycle
h
2
→
by 2
h
2
(self trading cycle)
So the TTC algorithm assigns
³
h
1
assigned
2
3
4
´
.
Step 2:
We have unassigned rooms
{
h
1
3
4
}
.
Each remaining student (student 1, 3, and 4) wants to obtain
h
1
→
by 1
h
4
h
3
→
by 3
h
1
h
4
→
by 4
h
3
1
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View Full DocumentThen, we can
f
nd a trading cycle
h
1
→
by 1
h
4
→
by 4
h
3
→
by 3
h
1
·
So, the TTC algorithm assigns
³
h
assigned
4
,h
assigned
2
assigned
1
assigned
3
´
.
Clearly, the above allocation is di
f
erent from initial allocation (
h
1
2
3
3
). Thus, initial allocation
is not e
ﬃ
cient.
(b) Find all e
ﬃ
cient allocations. Explain how you found all the e
ﬃ
cient allocations.
Answer:
As we discussed Problem Set I Problem 2 (a), there are two methods to
f
nd all e
ﬃ
cient allocations
Method I:
TTC Algorithm with All Possible Initial Assignments
Method II:
Serial Dictatorship with All Possible Orders
We here apply Method II.
Rewriting preferences
h
2
Â
1
h
4
Â
1
h
1
Â
1
h
3
h
2
Â
2
h
1
Â
2
h
3
Â
2
h
4
h
1
Â
3
h
3
Â
3
h
4
Â
3
h
2
h
3
Â
4
h
4
Â
4
h
1
Â
4
h
2
We generate all possible order of serial dictatorships. There are 4 students and 4
×
3
×
2
×
1=24
ways of orders.
Order of Dictatorships
Allocation
Type
Order of Dictatorships
Allocation
Type
1
→
2
→
3
→
4(
h
2
1
3
4
)I
2
→
1
→
3
→
h
4
2
1
3
)V
1
→
2
→
4
→
3(
h
2
1
4
3
I
2
→
1
→
4
→
h
4
2
1
3
1
→
3
→
2
→
h
2
3
1
4
I
I
2
→
3
→
1
→
h
4
2
1
3
1
→
3
→
4
→
2(
h
2
4
1
3
V
2
→
3
→
4
→
1(
h
4
2
1
3
1
→
4
→
2
→
h
2
1
4
3
I
2
→
4
→
1
→
h
4
2
1
3
1
→
4
→
3
→
h
2
4
1
3
V
2
→
4
→
3
→
h
4
2
1
3
Order of Dictatorships
Allocation
Type
Order of Dictatorships
Allocation
Type
3
→
1
→
2
→
h
2
3
1
4
I
I
4
→
1
→
2
→
h
2
1
4
3
I
3
→
1
→
4
→
h
2
4
1
3
V
4
→
1
→
3
→
h
2
4
1
3
V
3
→
2
→
1
→
h
4
2
1
3
4
→
2
→
1
→
h
4
2
1
3
3
→
2
→
4
→
h
4
2
1
3
4
→
2
→
3
→
h
4
2
1
3
3
→
4
→
1
→
h
2
4
1
3
V
4
→
3
→
1
→
h
2
4
1
3
V
3
→
4
→
2
→
h
4
2
1
3
4
→
3
→
2
→
h
4
2
1
3
Thus, there are
f
ve e
ﬃ
cient allocations
⎧
⎪
⎨
⎪
⎩
(
h
2
1
3
4
)

{z
}
type I
,
(
h
2
1
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 Winter '10
 Obara

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