Econ 145: Practice Questions I
Ichiro Obara
February 5, 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.
(b) Find all e
ﬃ
cient allocations. Explain how you found all the e
ﬃ

cient allocations.
(c) In general, the set of e
ﬃ
cient allocations is larger than the core
allocation (given a certain initial assignment). Can you
f
nd an
example of room assignment problem (with four students and
four rooms) such that these two sets coincide? If so,
f
nd one. If
not, explain why.
2. Consider a room assignment problem with 3 students and 3 rooms
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