ECON 383
Practice Problems from Chapter 10
4, 6, 7, 8, 9, 10
H. K. Chen (SFU)
ECON 383
1/ 7
Chapter 10 — Ex.4
4. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{x, y, z}, with the following valuations:
Buyer j
x
y
z
vb,j
9
3
6
va,j
12
10
8
vc,j
8
6
5
Suppose pa = 3, pb = 1, and pc = 0. Is this set of prices market clearing?
If so, what is the resulting matching? If not, which seller(s) should raise
their price(s)?
Construct the preferred-seller graph
The set {x, y} is constricted, and so
N ({x, y}) = {a} should raise price
according to the auction procedure
H. K. Chen (SFU)
ECON 383
pi
Seller
Buyer
va,j , vb,j , vc,j
3
a
x
12, 9, 8
1
b
y
10, 3, 6
0
c
z
8, 6, 5
2/ 7
Chapter 10 — Ex.6
6. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{x, y, z}, with the following valuations:
Buyer j
x
y
z
vb,j
3
5
8
va,j
6
10
7
vc,j
2
4
6
Suppose pa = 4, pb = 1, and pc = 0. Is this set of prices market clearing?
If so, what is the resulting matching? If not, which seller(s) should raise
their price(s)?
Construct the preferred-seller graph
A perfect matching is possible given the
current set of prices, hence they are
market-clearing.
H. K. Chen (SFU)
ECON 383
pi
Seller
Buyer
va,j , vb,j , vc,j
4
a
x
6, 3, 2
1
b
y
10, 5, 4
0
c
z
7, 8, 6
3/ 7
Chapter 10 — Ex.7
7. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{x, y, z}, with the following valuations:
Buyer j
x
y
z
vb,j
8
6
6
va,j
6
5
3
vc,j
7
6
5
Suppose pa = 2, pb = 5, and pc = 4. Is this set of prices market clearing?
If so, what is the resulting matching? If not, which seller(s) should raise
their price(s)?
Construct the preferred-seller graph
The set {x, y} is constricted, and so
N ({x, y}) = {a} should raise price
according to the auction procedure
H. K. Chen (SFU)
ECON 383
pi
Seller
Buyer
va,j , vb,j , vc,j
2
a
x
6, 8, 7
5
b
y
5, 6, 6
4
c
z
3, 6, 5
4/ 7
Chapter 10 — Ex.8
8. Two sellers, {a, b}, each with one house to sell. Two buyers, {x, y},
with the following valuations:
Buyer j
x
y
va,j
7
4
vb,j
5
1
Determine the market-clearing prices using the bipartite graph auction.
Round 1
pi Seller
Buyer va,j , vb,j
Step 1. Set pi = 0 for all j
Step 2. Construct preferred seller graph
Step 3. Note that {x, y} is constricted,
and that {a} = N ({x, y})
H. K. Chen (SFU)
ECON 383
0
a
x
7, 5
0
b
y
4, 1
5(a)/ 7
Chapter 10 — Ex.8
8. Two sellers, {a, b}, each with one house to sell. Two buyers, {x, y},
with the following valuations:
va,j
7
4
Buyer j
x
y
vb,j
5
1
Determine the market-clearing prices using the bipartite graph auction.
Step 4. Raise pa by 1
Step 5. Normalize lowest price to 0
pi
Seller
Buyer
va,j , vb,j
1
a
x
7, 5
0
b
y
4, 1
This step is redundant here.
Round 2
Repeat Steps 2–5 once
In this iteration, the prices
(pa , pb ) = (1, 0) are not
market-clearing
H. K. Chen (SFU)
ECON 383
5(b)/ 7
Chapter 10 — Ex.8
8. Two sellers, {a, b}, each with one house to sell. Two buyers, {x, y},
with the following valuations:
Buyer j
x
y
va,j
7
4
vb,j
5
1
Determine the market-clearing prices using the bipartite graph auction.
Round 3
pi Seller
Buyer va,j , vb,j
Now pa = 2 and pb = 0
Step 2. Construct preferred seller graph
Step 3. Note that a perfect matching is
possible
2
a
x
7, 5
0
b
y
4, 1
So the market-clearing prices are
(pa , pb ) = (2, 0)
H. K. Chen (SFU)
ECON 383
5(c)/ 7
Chapter 10 — Ex.9
9. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{a, b, c}, with the following valuations:
Buyer j
x
y
z
vb,j
6
8
2
va,j
3
2
1
vc,j
4
1
3
Use the bipartite graph auction to find the market-clearing prices.
Round 1
pi Seller
Buyer va,j , vb,j , vc,j
Step 1. Set pi = 0 for all j
Step 2. Construct preferred seller graph
Step 3. Note that {x, y} is constricted,
and that {b} = N ({x, y})
H. K. Chen (SFU)
ECON 383
0
a
x
3, 6, 4
0
b
y
2, 8, 1
0
c
z
1, 2, 3
6(a)/ 7
Chapter 10 — Ex.9
9. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{a, b, c}, with the following valuations:
Buyer j
x
y
z
vb,j
6
8
2
va,j
3
2
1
vc,j
4
1
3
Use the bipartite graph auction to find the market-clearing prices.
Step 4. Raise pb by 1
Step 5. Normalize lowest price to 0
pi
Seller
Buyer
va,j , vb,j , vc,j
0
a
x
3, 6, 4
1
b
y
2, 8, 1
0
c
z
1, 2, 3
This step is redundant here.
Round 2
Repeat Steps 2–5 once
In this iteration, the prices
(pa , pb , pc ) = (0, 1, 0) is not
market-clearing
H. K. Chen (SFU)
ECON 383
6(b)/ 7
Chapter 10 — Ex.9
9. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{a, b, c}, with the following valuations:
Buyer j
x
y
z
vb,j
6
8
2
va,j
3
2
1
vc,j
4
1
3
Use the bipartite graph auction to find the market-clearing prices.
Round 3
pi Seller
Buyer va,j , vb,j , vc,j
Now (pa , pb , pc ) = (0, 2, 0)
Step 2. Construct preferred seller graph
Step 3. Note that {x, y, z} is
constricted, and that
{b, c} = N ({x, y, z})
H. K. Chen (SFU)
ECON 383
0
a
x
3, 6, 4
2
b
y
2, 8, 1
0
c
z
1, 2, 3
6(c)/ 7
Chapter 10 — Ex.9
9. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{a, b, c}, with the following valuations:
Buyer j
x
y
z
vb,j
6
8
2
va,j
3
2
1
vc,j
4
1
3
Use the bipartite graph auction to find the market-clearing prices.
Step 4. Raise pb and pc by 1
Step 5. Normalize lowest price to 0
pi
Seller
Buyer
va,j , vb,j , vc,j
0
a
x
3, 6, 4
3
b
y
2, 8, 1
1
c
z
1, 2, 3
This step is redundant here.
Round 4
Step 2. Construct preferred seller graph
A perfect matching is found. So
(0, 3, 1) clears the market
H. K. Chen (SFU)
ECON 383
6(d)/ 7
Chapter 10 — Ex.10
10. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{a, b, c}, with the following valuations:
Buyer j
x
y
z
vb,j
7
9
10
va,j
9
5
11
vc,j
4
7
8
Use the bipartite graph auction to find the market-clearing prices.
Round 1
pi Seller
Buyer va,j , vb,j , vc,j
Step 1. Set pi = 0 for all j
Step 2. Construct preferred seller graph
Step 3. Note that {x, z} is constricted,
and that {a} = N ({x, z})
Note that {x, y, z} is also a
0
a
x
9, 7, 4
0
b
y
5, 9, 7
0
c
z
11, 10, 8
constriction, with
{a, b} = N ({x, y, z})
H. K. Chen (SFU)
ECON 383
7(a)/ 7
Chapter 10 — Ex.10
10. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{a, b, c}, with the following valuations:
Buyer j
x
y
z
vb,j
7
9
10
va,j
9
5
11
vc,j
4
7
8
Use the bipartite graph auction to find the market-clearing prices.
Step 4. Raise pa by 1
Step 5. Normalize lowest price to 0
pi
Seller
Buyer
va,j , vb,j , vc,j
1
a
x
9, 7, 4
0
b
y
5, 9, 7
0
c
z
11, 10, 8
This step is redundant here.
Round 2
Step 2. Construct preferred seller graph
Step 3. {x, y, z} is constricted, with
{a, b} = N ({x, y, z})
H. K. Chen (SFU)
ECON 383
7(b)/ 7
Chapter 10 — Ex.10
10. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{a, b, c}, with the following valuations:
Buyer j
x
y
z
vb,j
7
9
10
va,j
9
5
11
vc,j
4
7
8
Use the bipartite graph auction to find the market-clearing prices.
Step 4. Raise pa and pb by 1
Step 5. Normalize lowest price to 0
pi
Seller
Buyer
va,j , vb,j , vc,j
2
a
x
9, 7, 4
1
b
y
5, 9, 7
0
c
z
11, 10, 8
This step is redundant here.
Round 3
Step 2. Construct preferred seller graph
Step 3. {x, y, z} is constricted, with
{a, b} = N ({x, y, z})
H. K. Chen (SFU)
ECON 383
7(c)/ 7
Chapter 10 — Ex.10
10. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{a, b, c}, with the following valuations:
Buyer j
x
y
z
vb,j
7
9
10
va,j
9
5
11
vc,j
4
7
8
Use the bipartite graph auction to find the market-clearing prices.
Step 4. Raise pa and pb by 1
Step 5. Normalize lowest price to 0
pi
Seller
Buyer
va,j , vb,j , vc,j
3
a
x
9, 7, 4
2
b
y
5, 9, 7
0
c
z
11, 10, 8
This step is redundant here.
Round 4
Step 2. Construct preferred seller graph
A perfect matching is found. So
(3, 2, 0) clears the market
H. K. Chen (SFU)
ECON 383
7(d)/ 7