Homework 8
(Total 20 points), Due on
April 29, Tuesday
.
Q1
: There are 6 individuals. Each individual
i
= 1
,
2
,
3
,
4
,
5
,
6 is endowed with an
indivisible object
ω
i
. Their preferences over objects are listed as
1
2
3
4
5
6
ω
2
ω
6
ω
4
ω
6
ω
2
ω
5
ω
5
ω
3
ω
3
ω
5
ω
5
ω
3
ω
4
ω
1
ω
2
ω
1
ω
1
ω
6
ω
1
ω
5
ω
5
ω
4
ω
4
ω
1
ω
3
ω
2
ω
6
ω
3
ω
6
ω
4
ω
6
ω
4
ω
1
ω
2
ω
3
ω
2
Perform the (Gale’s) toptradingcycle algorithm to find the core allocation.
Answer
:
Round 1:
1
→
ω
2
, 2
→
ω
6
, 3
→
ω
4
, 4
→
ω
6
, 5
→
ω
2
, 6
→
ω
5
.
There is one cycle 2
→
6
→
5
→
2. 2 gets
ω
6
, 6 gets
ω
5
, 5 gets
ω
2
and leave.
Round 2: 1
→
ω
4
, 3
→
ω
4
, 4
→
ω
1
. There is one cycle 1
→
4
→
1. 1 gets
ω
4
, 4
gets
ω
1
and leave.
Round 3: 3
→
ω
3
. 3 gets
ω
3
.
Summing up, the core allocation is (
ω
4
, ω
6
, ω
3
, ω
1
, ω
2
, ω
5
).
Q2
: There are four workers and four firms. Workers are denoted by
w
1
, w
2
, w
3
, w
4
respectively, firms are denoted by
f
1
, f
2
, f
3
, f
4
respectively. Each worker can work
for at most one firm, and each firm can hire at most one worker. Their preferences
over the opposite side are as follows:
w
1
w
2
w
3
w
4
f
1
f
2
f
3
f
4
f
2
f
1
f
1
f
2
w
1
w
3
w
1
w
3
f
1
f
3
f
2
f
3
w
2
w
4
w
4
w
1
f
3
f
2
f
4
f
4
w
3
w
1
w
2
w
4
f
4
f
4
f
3
f
1
w
4
w
2
w
3
w
2
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 Spring '10
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