ECON 383
Practice Problems from Chapter 10
11, 12, 13, 14, 15
H. K. Chen (SFU)
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Chapter 10 — Ex.11
11. Parking spaces: {a, b, c}; people: {x, y, z}. The distance between a
person j and a parking space i, di,j , is the number of blocks between i and
j. j’s valuation for i is vi,j = 8 − di,j
H. K. Chen (SFU)
ECON 383
2 / 19
Chapter 10 — Ex.11
11. Parking spaces: {a, b, c}; people: {x, y, z}. The distance between a
person j and a parking space i, di,j , is the number of blocks between i and
j. j’s valuation for i is vi,j = 8 − di,j
j
x
y
z
H. K. Chen (SFU)
va,j
ECON 383
vb,j
vc,j
2 / 19
Chapter 10 — Ex.11
11. Parking spaces: {a, b, c}; people: {x, y, z}. The distance between a
person j and a parking space i, di,j , is the number of blocks between i and
j. j’s valuation for i is vi,j = 8 − di,j
j
x
y
z
H. K. Chen (SFU)
va,j
8−2
ECON 383
vb,j
8−3
vc,j
8−6
2 / 19
Chapter 10 — Ex.11
11. Parking spaces: {a, b, c}; people: {x, y, z}. The distance between a
person j and a parking space i, di,j , is the number of blocks between i and
j. j’s valuation for i is vi,j = 8 − di,j
j
x
y
z
H. K. Chen (SFU)
va,j
6
ECON 383
vb,j
5
vc,j
2
2 / 19
Chapter 10 — Ex.11
11. Parking spaces: {a, b, c}; people: {x, y, z}. The distance between a
person j and a parking space i, di,j , is the number of blocks between i and
j. j’s valuation for i is vi,j = 8 − di,j
j
x
y
z
H. K. Chen (SFU)
va,j
6
8−1
ECON 383
vb,j
5
8−2
vc,j
2
8−5
2 / 19
Chapter 10 — Ex.11
11. Parking spaces: {a, b, c}; people: {x, y, z}. The distance between a
person j and a parking space i, di,j , is the number of blocks between i and
j. j’s valuation for i is vi,j = 8 − di,j
j
x
y
z
H. K. Chen (SFU)
va,j
6
7
ECON 383
vb,j
5
6
vc,j
2
3
2 / 19
Chapter 10 — Ex.11
11. Parking spaces: {a, b, c}; people: {x, y, z}. The distance between a
person j and a parking space i, di,j , is the number of blocks between i and
j. j’s valuation for i is vi,j = 8 − di,j
j
x
y
z
H. K. Chen (SFU)
va,j
6
7
8−2
ECON 383
vb,j
5
6
8−1
vc,j
2
3
8−2
2 / 19
Chapter 10 — Ex.11
11. Parking spaces: {a, b, c}; people: {x, y, z}. The distance between a
person j and a parking space i, di,j , is the number of blocks between i and
j. j’s valuation for i is vi,j = 8 − di,j
j
x
y
z
H. K. Chen (SFU)
va,j
6
7
6
ECON 383
vb,j
5
6
7
vc,j
2
3
6
2 / 19
Chapter 10 — Ex.11(a)
11(a) Describe how you would set up this question as a matching problem.
Who are the sellers/objects, who are the buyers?
H. K. Chen (SFU)
ECON 383
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Chapter 10 — Ex.11(a)
11(a) Describe how you would set up this question as a matching problem.
Who are the sellers/objects, who are the buyers?
The sellers/objects are the parking spaces, {a, b, c}
H. K. Chen (SFU)
ECON 383
3 / 19
Chapter 10 — Ex.11(a)
11(a) Describe how you would set up this question as a matching problem.
Who are the sellers/objects, who are the buyers?
The sellers/objects are the parking spaces, {a, b, c}
The buyers are the people living in apartments, {x, y, z}
H. K. Chen (SFU)
ECON 383
3 / 19
Chapter 10 — Ex.11(a)
11(a) Describe how you would set up this question as a matching problem.
Who are the sellers/objects, who are the buyers?
The sellers/objects are the parking spaces, {a, b, c}
The buyers are the people living in apartments, {x, y, z}
Buyers’ valuations are as calculated
va,j
j
x
6
y
7
z
6
H. K. Chen (SFU)
ECON 383
on the previous slide:
vb,j
vc,j
5
2
6
3
7
6
3 / 19
Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine the market-clearing
prices
H. K. Chen (SFU)
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Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine the market-clearing
prices
Round 1
pi Seller
Buyer va,j , vb,j , vc,j
Step 1. Set pi = 0 for all j
H. K. Chen (SFU)
ECON 383
0
a
x
6, 5, 2
0
b
y
7, 6, 3
0
c
z
6, 7, 6
4 / 19
Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine 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
H. K. Chen (SFU)
ECON 383
0
a
x
6, 5, 2
0
b
y
7, 6, 3
0
c
z
6, 7, 6
4 / 19
Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine 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 {a} = N ({x, z})
Note that {x, y, z} is also a
0
a
x
6, 5, 2
0
b
y
7, 6, 3
0
c
z
6, 7, 6
constriction, with
{a, b} = N ({x, y, z})
H. K. Chen (SFU)
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Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine the market-clearing
prices
Step 4. Raise pa by 1
H. K. Chen (SFU)
ECON 383
pi
Seller
Buyer
va,j , vb,j , vc,j
1
a
x
6, 5, 2
0
b
y
7, 6, 3
0
c
z
6, 7, 6
4 / 19
Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine 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
6, 5, 2
0
b
y
7, 6, 3
0
c
z
6, 7, 6
This step is redundant here.
H. K. Chen (SFU)
ECON 383
4 / 19
Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine 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
6, 5, 2
0
b
y
7, 6, 3
0
c
z
6, 7, 6
This step is redundant here.
Round 2
Step 2. Construct preferred seller graph
H. K. Chen (SFU)
ECON 383
4 / 19
Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine 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
6, 5, 2
0
b
y
7, 6, 3
0
c
z
6, 7, 6
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)
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4 / 19
Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine 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
6, 5, 2
1
b
y
7, 6, 3
0
c
z
6, 7, 6
This step is redundant here.
H. K. Chen (SFU)
ECON 383
4 / 19
Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine 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
6, 5, 2
1
b
y
7, 6, 3
0
c
z
6, 7, 6
This step is redundant here.
Round 3
Step 2. Construct preferred seller graph
H. K. Chen (SFU)
ECON 383
4 / 19
Chapter 10 — Ex.11(b)
11(b) Use the bipartite graph auction to determine 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
6, 5, 2
1
b
y
7, 6, 3
0
c
z
6, 7, 6
This step is redundant here.
Round 3
Step 2. Construct preferred seller graph
A perfect matching is found. So
(2, 1, 0) clears the market
H. K. Chen (SFU)
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Chapter 10 — Ex.11(c)
j
x
y
z
va,j
6
7
6
vb,j
5
6
7
vc,j
2
3
6
11(c) At a more informal level, how do the prices you determined for the
parking spaces in (b) relate to these spaces’ intuitive “attractiveness” to
the people in apartments x, y, and z? Explain.
H. K. Chen (SFU)
ECON 383
5 / 19
Chapter 10 — Ex.11(c)
j
x
y
z
va,j
6
7
6
vb,j
5
6
7
vc,j
2
3
6
11(c) At a more informal level, how do the prices you determined for the
parking spaces in (b) relate to these spaces’ intuitive “attractiveness” to
the people in apartments x, y, and z? Explain.
The market-clearing prices, (pa , pb , pc ) = (2, 1, 0), happens to be the
number of buyer demanding the objects when they are free:
a is demanded by both x and y when its price is zero
b is demanded by z when its price is zero
c is demanded by no one when its price is zero
However, this phenomenon is not general.
H. K. Chen (SFU)
ECON 383
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Chapter 10 — Ex.12
12. Two buyers’ valuations for objects a, b are as follows:
Buyer j
x
y
va,j
4
3
vb,j
1
2
Give three different sets of market-clearing prices for this market. Note
that prices must be integers.
H. K. Chen (SFU)
ECON 383
6 / 19
Chapter 10 — Ex.12
12. Two buyers’ valuations for objects a, b are as follows:
Buyer j
x
y
va,j
4
3
vb,j
1
2
Give three different sets of market-clearing prices for this market. Note
that prices must be integers.
To clear the market, need to have a perfect matching, i.e. assigning
different objects to different buyers
H. K. Chen (SFU)
ECON 383
6 / 19
Chapter 10 — Ex.12
12. Two buyers’ valuations for objects a, b are as follows:
Buyer j
x
y
va,j
4
3
vb,j
1
2
Give three different sets of market-clearing prices for this market. Note
that prices must be integers.
To clear the market, need to have a perfect matching, i.e. assigning
different objects to different buyers
Notice that both x and y prefer a to b; but x prefers a more
“strongly” than y does
H. K. Chen (SFU)
ECON 383
6 / 19
Chapter 10 — Ex.12
12. Two buyers’ valuations for objects a, b are as follows:
Buyer j
x
y
va,j
4
3
vb,j
1
2
Give three different sets of market-clearing prices for this market. Note
that prices must be integers.
To clear the market, need to have a perfect matching, i.e. assigning
different objects to different buyers
Notice that both x and y prefer a to b; but x prefers a more
“strongly” than y does
x prefers a more “intensely” than y does: va,x > va,y
H. K. Chen (SFU)
ECON 383
6 / 19
Chapter 10 — Ex.12
12. Two buyers’ valuations for objects a, b are as follows:
Buyer j
x
y
va,j
4
3
vb,j
1
2
Give three different sets of market-clearing prices for this market. Note
that prices must be integers.
To clear the market, need to have a perfect matching, i.e. assigning
different objects to different buyers
Notice that both x and y prefer a to b; but x prefers a more
“strongly” than y does
x prefers a more “intensely” than y does: va,x > va,y
x’s “relative valuation” for a is higher than y’s: va,x − vb,x > va,y − vb,y
H. K. Chen (SFU)
ECON 383
6 / 19
Chapter 10 — Ex.12
12. Two buyers’ valuations for objects a, b are as follows:
Buyer j
x
y
va,j
4
3
vb,j
1
2
Give three different sets of market-clearing prices for this market. Note
that prices must be integers.
To clear the market, need to have a perfect matching, i.e. assigning
different objects to different buyers
Notice that both x and y prefer a to b; but x prefers a more
“strongly” than y does
x prefers a more “intensely” than y does: va,x > va,y
x’s “relative valuation” for a is higher than y’s: va,x − vb,x > va,y − vb,y
Naturally then, the market-clearing prices should be such that a is
assigned to x, and thus b is assigned to y
Recall that market-clearing prices always lead to optimal matching
H. K. Chen (SFU)
ECON 383
6 / 19
Chapter 10 — Ex.12
Buyer j
x
y
va,j
4
3
vb,j
1
2
To incentivize x and y to choose a and b respectively, the prices pa , pb
must be such that
H. K. Chen (SFU)
4 − pa ≥ 1 − pb
(1)
3 − pa ≤ 2 − pb
(2)
pa ≤ 4
(3)
pb ≤ 2
(4)
ECON 383
7 / 19
Chapter 10 — Ex.12
Buyer j
x
y
va,j
4
3
vb,j
1
2
To incentivize x and y to choose a and b respectively, the prices pa , pb
must be such that
4 − pa ≥ 1 − pb
(1)
3 − pa ≤ 2 − pb
(2)
pa ≤ 4
(3)
pb ≤ 2
(4)
(1) and (2) are equivalent to (by rearranging the terms):
1 + pb ≤ pa ≤ 3 + pb
H. K. Chen (SFU)
ECON 383
(5)
7 / 19
Chapter 10 — Ex.12
Therefore, any pa , pb that satisfy equations (3), (4), and (5) will clear
the market:
H. K. Chen (SFU)
pa ≤ 4
(3)
pb ≤ 2
(4)
1 + pb ≤ pa ≤ 3 + pb
(5)
ECON 383
8 / 19
Chapter 10 — Ex.12
Therefore, any pa , pb that satisfy equations (3), (4), and (5) will clear
the market:
pa ≤ 4
(3)
pb ≤ 2
(4)
1 + pb ≤ pa ≤ 3 + pb
(5)
Take pb = 0, then any pa ∈ {1, 2, 3} would satisfy (3)–(5)
So possible market-clearing prices (pa , pb ) are (1, 0), (2, 0), and (3, 0)
H. K. Chen (SFU)
ECON 383
8 / 19
Chapter 10 — Ex.12
Therefore, any pa , pb that satisfy equations (3), (4), and (5) will clear
the market:
pa ≤ 4
(3)
pb ≤ 2
(4)
1 + pb ≤ pa ≤ 3 + pb
(5)
Take pb = 0, then any pa ∈ {1, 2, 3} would satisfy (3)–(5)
So possible market-clearing prices (pa , pb ) are (1, 0), (2, 0), and (3, 0)
Take pb = 1, then any pa ∈ {2, 3, 4} would satisfy (3)–(5)
So possible market-clearing prices (pa , pb ) are (2, 1), (3, 1), and (4, 1)
H. K. Chen (SFU)
ECON 383
8 / 19
Chapter 10 — Ex.12
Therefore, any pa , pb that satisfy equations (3), (4), and (5) will clear
the market:
pa ≤ 4
(3)
pb ≤ 2
(4)
1 + pb ≤ pa ≤ 3 + pb
(5)
Take pb = 0, then any pa ∈ {1, 2, 3} would satisfy (3)–(5)
So possible market-clearing prices (pa , pb ) are (1, 0), (2, 0), and (3, 0)
Take pb = 1, then any pa ∈ {2, 3, 4} would satisfy (3)–(5)
So possible market-clearing prices (pa , pb ) are (2, 1), (3, 1), and (4, 1)
Take pb = 2, then any pa ∈ {3, 4} would satisfy (3)–(5)
So possible market-clearing prices (pa , pb ) are (3, 2) and (4, 2)
H. K. Chen (SFU)
ECON 383
8 / 19
Chapter 10 — Ex.13(a)
13. You have two units of a good, and there are four potential buyers who
value each unit at vi (each buyer wants at most one unit).
(a) Describe how you would sell these two units of goods using an
ascending-bid auction.
H. K. Chen (SFU)
ECON 383
9 / 19
Chapter 10 — Ex.13(a)
13. You have two units of a good, and there are four potential buyers who
value each unit at vi (each buyer wants at most one unit).
(a) Describe how you would sell these two units of goods using an
ascending-bid auction.
Call the two units of the good g1 , g2
H. K. Chen (SFU)
ECON 383
9 / 19
Chapter 10 — Ex.13(a)
13. You have two units of a good, and there are four potential buyers who
value each unit at vi (each buyer wants at most one unit).
(a) Describe how you would sell these two units of goods using an
ascending-bid auction.
Call the two units of the good g1 , g2
Label the four buyers as 1, 2, 3, 4, and assume, without loss of
generality, that v1 = 1, v2 = 2, v3 = 3, v4 = 4.
H. K. Chen (SFU)
ECON 383
9 / 19
Chapter 10 — Ex.13(a)
13. You have two units of a good, and there are four potential buyers who
value each unit at vi (each buyer wants at most one unit).
(a) Describe how you would sell these two units of goods using an
ascending-bid auction.
Call the two units of the good g1 , g2
Label the four buyers as 1, 2, 3, 4, and assume, without loss of
generality, that v1 = 1, v2 = 2, v3 = 3, v4 = 4.
As there are more buyers than objects, we can model this as a
matching problem by adding two dummy objects, d1 , d2 , of which
each buyer has a zero valuation
H. K. Chen (SFU)
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Chapter 10 — Ex.13(a)
The matching problem can be
represented by a graph
H. K. Chen (SFU)
ECON 383
p
Object
Buyer
valuations
g1
1
1, 1, 0, 0
g2
2
2, 2, 0, 0
d1
3
3, 3, 0, 0
d2
4
4, 4, 0, 0
10 / 19
Chapter 10 — Ex.13(a)
The matching problem can be
represented by a graph
p
Object
Buyer
valuations
Run the bipartite graph auction
0
g1
1
1, 1, 0, 0
0
g2
2
2, 2, 0, 0
0
d1
3
3, 3, 0, 0
0
d2
4
4, 4, 0, 0
H. K. Chen (SFU)
ECON 383
10 / 19
Chapter 10 — Ex.13(a)
The matching problem can be
represented by a graph
p
Object
Buyer
valuations
Run the bipartite graph auction
Round 1 fails to produce a perfect
matching
0
g1
1
1, 1, 0, 0
0
g2
2
2, 2, 0, 0
There are 5 constricted sets
0
d1
3
3, 3, 0, 0
0
d2
4
4, 4, 0, 0
H. K. Chen (SFU)
ECON 383
10 / 19
Chapter 10 — Ex.13(a)
The matching problem can be
represented by a graph
p
Object
Buyer
valuations
Run the bipartite graph auction
Round 1 fails to produce a perfect
matching
1
g1
1
1, 1, 0, 0
1
g2
2
2, 2, 0, 0
There are 5 constricted sets
0
d1
3
3, 3, 0, 0
0
d2
4
4, 4, 0, 0
Round 2 also fails to produce a perfect
matching
There is 1 constricted set
H. K. Chen (SFU)
ECON 383
10 / 19
Chapter 10 — Ex.13(a)
The matching problem can be
represented by a graph
p
Object
Buyer
valuations
Run the bipartite graph auction
Round 1 fails to produce a perfect
matching
2
g1
1
1, 1, 0, 0
2
g2
2
2, 2, 0, 0
There are 5 constricted sets
0
d1
3
3, 3, 0, 0
0
d2
4
4, 4, 0, 0
Round 2 also fails to produce a perfect
matching
There is 1 constricted set
Round 3 has a perfect matching
H. K. Chen (SFU)
ECON 383
10 / 19
Chapter 10 — Ex.13(a)
The matching problem can be
represented by a graph
p
Object
Buyer
valuations
Run the bipartite graph auction
Round 1 fails to produce a perfect
matching
2
g1
1
1, 1, 0, 0
2
g2
2
2, 2, 0, 0
There are 5 constricted sets
0
d1
3
3, 3, 0, 0
0
d2
4
4, 4, 0, 0
Round 2 also fails to produce a perfect
matching
There is 1 constricted set
Round 3 has a perfect matching
H. K. Chen (SFU)
ECON 383
10 / 19
Chapter 10 — Ex.13(a)
The matching problem can be
represented by a graph
p
Object
Buyer
valuations
Run the bipartite graph auction
Round 1 fails to produce a perfect
matching
2
g1
1
1, 1, 0, 0
2
g2
2
2, 2, 0, 0
There are 5 constricted sets
0
d1
3
3, 3, 0, 0
0
d2
4
4, 4, 0, 0
Round 2 also fails to produce a perfect
matching
There is 1 constricted set
Round 3 has a perfect matching
Therefore the market-clearing price for each unit of the object is 2
H. K. Chen (SFU)
ECON 383
10 / 19
Chapter 10 — Ex.13(b)
13(b) In the case of single-item auction, the bipartite graph procedure
yielded the simple rule from the ascending-bid (English) auction:
Rule for single-item auction
Sell to the highest bidder at the second-highest price.
In simple terms, what should be the rule for the current case of two
identical items?
H. K. Chen (SFU)
ECON 383
11 / 19
Chapter 10 — Ex.13(b)
13(b) In the case of single-item auction, the bipartite graph procedure
yielded the simple rule from the ascending-bid (English) auction:
Rule for single-item auction
Sell to the highest bidder at the second-highest price.
In simple terms, what should be the rule for the current case of two
identical items?
Rule for two-identical-item auction
Sell to the two highest bidders at the third-highest price.
H. K. Chen (SFU)
ECON 383
11 / 19
Chapter 10 — Ex.13(b)
13(b) In the case of single-item auction, the bipartite graph procedure
yielded the simple rule from the ascending-bid (English) auction:
Rule for single-item auction
Sell to the highest bidder at the second-highest price.
In simple terms, what should be the rule for the current case of two
identical items?
Rule for two-identical-item auction
Sell to the two highest bidders at the third-highest price.
Can you conjecture the rule for general n-item auction?
H. K. Chen (SFU)
ECON 383
11 / 19
Chapter 10 — Ex.13(b)
13(b) In the case of single-item auction, the bipartite graph procedure
yielded the simple rule from the ascending-bid (English) auction:
Rule for single-item auction
Sell to the highest bidder at the second-highest price.
In simple terms, what should be the rule for the current case of two
identical items?
Rule for two-identical-item auction
Sell to the two highest bidders at the third-highest price.
Can you conjecture the rule for general n-item auction?
Rule for n-identical-item auction
Sell to the n highest bidders at the (n + 1)th -highest price.
H. K. Chen (SFU)
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11 / 19
Chapter 10 — Ex.14
14. Recall that
A matching M is social welfare maximizing if it maximizes the sum of
buyers’ valuations for what they get, over all possible perfect matchings;
i.e. M is the solution to
max ∑ vm(i),i
m
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i
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Chapter 10 — Ex.14
14. Recall that
A matching M is social welfare maximizing if it maximizes the sum of
buyers’ valuations for what they get, over all possible perfect matchings;
i.e. M is the solution to
max ∑ vm(i),i
m
i
However, the sum of buyers’ valuations need not be the only criterion that
we use to judge the desirability of a matching. Sometimes we may want to
have a matching that ensures that no buyer gets a valuation that is too
small.
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Chapter 10 — Ex.14
Define the baseline of a matching m to be the minimum valuation that
any buyer has for the item they get in M; i.e.
min{vm(i),i |i = 1, . . . , n}.
A matching M is baseline maximizing if M is the solution to
n
o
max min{vm(i),i |i = 1, . . . , n}
m
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Chapter 10 — Ex.14
Define the baseline of a matching m to be the minimum valuation that
any buyer has for the item they get in M; i.e.
min{vm(i),i |i = 1, . . . , n}.
A matching M is baseline maximizing if M is the solution to
n
o
max min{vm(i),i |i = 1, . . . , n}
m
This notion of baseline is motivated by egalitarian considerations: no one
should be left too badly off.
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Chapter 10 — Ex.14
Buyer i
x
y
z
vb,j
7
9
10
va,j
9
5
11
vc,j
4
7
8
For example, consider a matching M = {a-x, b-y, c-z}
The baseline of M is 8, because 8 = min{9, 9, 8}
The social welfare of M is 9 + 9 + 8 = 26
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Chapter 10 — Ex.14
Buyer i
x
y
z
vb,j
7
9
10
va,j
9
5
11
vc,j
4
7
8
For example, consider a matching M = {a-x, b-y, c-z}
The baseline of M is 8, because 8 = min{9, 9, 8}
The social welfare of M is 9 + 9 + 8 = 26
Consider another matching M0 = {b-x, c-y, a-z}
The baseline of M0 is 7, because 7 = min{7, 7, 11}
The social welfare of M0 is 7 + 7 + 11 = 25
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Chapter 10 — Ex.14
Buyer i
x
y
z
vb,j
7
9
10
va,j
9
5
11
vc,j
4
7
8
For example, consider a matching M = {a-x, b-y, c-z}
The baseline of M is 8, because 8 = min{9, 9, 8}
The social welfare of M is 9 + 9 + 8 = 26
Consider another matching M0 = {b-x, c-y, a-z}
The baseline of M0 is 7, because 7 = min{7, 7, 11}
The social welfare of M0 is 7 + 7 + 11 = 25
M is also the baseline maximizing matching
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Chapter 10 — Ex.14(a)
14(a) Give an example of equal-sized sets of sellers and buyers, with
valuations on the buyers, so that there is no perfect matching that is both
social welfare maximizing and baseline maximizing. (In other words,
social-welfare maximization and baseline maximization should only occur
with different matchings.)
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Chapter 10 — Ex.14(a)
14(a) Give an example of equal-sized sets of sellers and buyers, with
valuations on the buyers, so that there is no perfect matching that is both
social welfare maximizing and baseline maximizing. (In other words,
social-welfare maximization and baseline maximization should only occur
with different matchings.)
Consider a two-seller (a, b) and two-buyer (x, y) example.
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Chapter 10 — Ex.14(a)
14(a) Give an example of equal-sized sets of sellers and buyers, with
valuations on the buyers, so that there is no perfect matching that is both
social welfare maximizing and baseline maximizing. (In other words,
social-welfare maximization and baseline maximization should only occur
with different matchings.)
Consider a two-seller (a, b) and two-buyer (x, y) example.
There are only two possible perfect matchings
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Chapter 10 — Ex.14(a)
14(a) Give an example of equal-sized sets of sellers and buyers, with
valuations on the buyers, so that there is no perfect matching that is both
social welfare maximizing and baseline maximizing. (In other words,
social-welfare maximization and baseline maximization should only occur
with different matchings.)
Consider a two-seller (a, b) and two-buyer (x, y) example.
There are only two possible perfect matchings
M = {a-x, b-y}
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Chapter 10 — Ex.14(a)
14(a) Give an example of equal-sized sets of sellers and buyers, with
valuations on the buyers, so that there is no perfect matching that is both
social welfare maximizing and baseline maximizing. (In other words,
social-welfare maximization and baseline maximization should only occur
with different matchings.)
Consider a two-seller (a, b) and two-buyer (x, y) example.
There are only two possible perfect matchings
M = {a-x, b-y}
M0 = {b-x, a-y}
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Chapter 10 — Ex.14(a)
14(a) Give an example of equal-sized sets of sellers and buyers, with
valuations on the buyers, so that there is no perfect matching that is both
social welfare maximizing and baseline maximizing. (In other words,
social-welfare maximization and baseline maximization should only occur
with different matchings.)
Consider a two-seller (a, b) and two-buyer (x, y) example.
There are only two possible perfect matchings
M = {a-x, b-y}
M0 = {b-x, a-y}
Our objective is to make M (only) social welfare maximizing, and
make M0 (only) baseline maximizing
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Chapter 10 — Ex.14(a)
14(a) Give an example of equal-sized sets of sellers and buyers, with
valuations on the buyers, so that there is no perfect matching that is both
social welfare maximizing and baseline maximizing. (In other words,
social-welfare maximization and baseline maximization should only occur
with different matchings.)
Consider a two-seller (a, b) and two-buyer (x, y) example.
There are only two possible perfect matchings
M = {a-x, b-y}
M0 = {b-x, a-y}
Our objective is to make M (only) social welfare maximizing, and
make M0 (only) baseline maximizing
Note that it’s the same if you make M baseline maximizing and M0
social welfare maximizing
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Chapter 10 — Ex.14(a)
M is social welfare maximizing implies
va,x + vb,y > vb,x + va,y
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(1)
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Chapter 10 — Ex.14(a)
M is social welfare maximizing implies
va,x + vb,y > vb,x + va,y
(1)
M0 is baseline maximizing implies
min{va,x , vb,y } < min{vb,x , va,y }
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(2)
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Chapter 10 — Ex.14(a)
M is social welfare maximizing implies
va,x + vb,y > vb,x + va,y
(1)
M0 is baseline maximizing implies
min{va,x , vb,y } < min{vb,x , va,y }
(2)
Suppose min{va,x , vb,y } = va,x = 0 and min{vb,x , va,y } = vb,x = 1, so
that condition (2) is satisfied.
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Chapter 10 — Ex.14(a)
M is social welfare maximizing implies
va,x + vb,y > vb,x + va,y
(1)
M0 is baseline maximizing implies
min{va,x , vb,y } < min{vb,x , va,y }
(2)
Suppose min{va,x , vb,y } = va,x = 0 and min{vb,x , va,y } = vb,x = 1, so
that condition (2) is satisfied.
Condition (1) becomes
0 + vb,y > 1 + va,y
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(1’)
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Chapter 10 — Ex.14(a)
M is social welfare maximizing implies
va,x + vb,y > vb,x + va,y
(1)
M0 is baseline maximizing implies
min{va,x , vb,y } < min{vb,x , va,y }
(2)
Suppose min{va,x , vb,y } = va,x = 0 and min{vb,x , va,y } = vb,x = 1, so
that condition (2) is satisfied.
Condition (1) becomes
0 + vb,y > 1 + va,y
(1’)
Need to choose vb,y > 0 and va,y > 1 to satisfy (1’)
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Chapter 10 — Ex.14(a)
M is social welfare maximizing implies
va,x + vb,y > vb,x + va,y
(1)
M0 is baseline maximizing implies
min{va,x , vb,y } < min{vb,x , va,y }
(2)
Suppose min{va,x , vb,y } = va,x = 0 and min{vb,x , va,y } = vb,x = 1, so
that condition (2) is satisfied.
Condition (1) becomes
0 + vb,y > 1 + va,y
(1’)
Need to choose vb,y > 0 and va,y > 1 to satisfy (1’)
Pick va,y = 2 so that min{vb,x , va,y } = vb,x = 1 is satisfied
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Chapter 10 — Ex.14(a)
M is social welfare maximizing implies
va,x + vb,y > vb,x + va,y
(1)
M0 is baseline maximizing implies
min{va,x , vb,y } < min{vb,x , va,y }
(2)
Suppose min{va,x , vb,y } = va,x = 0 and min{vb,x , va,y } = vb,x = 1, so
that condition (2) is satisfied.
Condition (1) becomes
0 + vb,y > 1 + va,y
(1’)
Need to choose vb,y > 0 and va,y > 1 to satisfy (1’)
Pick va,y = 2 so that min{vb,x , va,y } = vb,x = 1 is satisfied
Pick vb,y = 4 so that (1’) holds
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Chapter 10 — Ex.14(a)
We can verify that (va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4) lead to distinct
matchings for the two maximizations
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Chapter 10 — Ex.14(a)
We can verify that (va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4) lead to distinct
matchings for the two maximizations
Under M = {a-x, b-y},
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Chapter 10 — Ex.14(a)
We can verify that (va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4) lead to distinct
matchings for the two maximizations
Under M = {a-x, b-y},
social welfare is 0 + 4 = 4
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Chapter 10 — Ex.14(a)
We can verify that (va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4) lead to distinct
matchings for the two maximizations
Under M = {a-x, b-y},
social welfare is 0 + 4 = 4
baseline is min{0, 4} = 0
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Chapter 10 — Ex.14(a)
We can verify that (va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4) lead to distinct
matchings for the two maximizations
Under M = {a-x, b-y},
social welfare is 0 + 4 = 4
baseline is min{0, 4} = 0
Under M0 = {b-x, a-y},
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Chapter 10 — Ex.14(a)
We can verify that (va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4) lead to distinct
matchings for the two maximizations
Under M = {a-x, b-y},
social welfare is 0 + 4 = 4
baseline is min{0, 4} = 0
Under M0 = {b-x, a-y},
social welfare is 1 + 2 = 3
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Chapter 10 — Ex.14(a)
We can verify that (va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4) lead to distinct
matchings for the two maximizations
Under M = {a-x, b-y},
social welfare is 0 + 4 = 4
baseline is min{0, 4} = 0
Under M0 = {b-x, a-y},
social welfare is 1 + 2 = 3
baseline is min{1, 2} = 1
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Chapter 10 — Ex.14(a)
We can verify that (va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4) lead to distinct
matchings for the two maximizations
Under M = {a-x, b-y},
social welfare is 0 + 4 = 4
baseline is min{0, 4} = 0
Under M0 = {b-x, a-y},
social welfare is 1 + 2 = 3
baseline is min{1, 2} = 1
Thus we have achieved the objective: M is social welfare maximizing
while M0 is baseline maximizing
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Chapter 10 — Ex.14(b)
14(b) Give a yes/no answer to the following question. Explain your answer.
For any equal-sized sets of sellers and buyers, with valuations on
the buyers, is there always a set of market-clearing prices so that
the resulting preferred-seller graph contains a
baseline-maximizing perfect matching?
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Chapter 10 — Ex.14(b)
14(b) Give a yes/no answer to the following question. Explain your answer.
For any equal-sized sets of sellers and buyers, with valuations on
the buyers, is there always a set of market-clearing prices so that
the resulting preferred-seller graph contains a
baseline-maximizing perfect matching?
No. The example in part (a) is one where no market-clearing prices
can produce a perfect matching that is baseline maximizing.
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Chapter 10 — Ex.14(b)
14(b) Give a yes/no answer to the following question. Explain your answer.
For any equal-sized sets of sellers and buyers, with valuations on
the buyers, is there always a set of market-clearing prices so that
the resulting preferred-seller graph contains a
baseline-maximizing perfect matching?
No. The example in part (a) is one where no market-clearing prices
can produce a perfect matching that is baseline maximizing.
Recall that
(va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4)
and
M0 = {b-x, a-y}
M = {a-x, b-y}
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Chapter 10 — Ex.14(b)
14(b) Give a yes/no answer to the following question. Explain your answer.
For any equal-sized sets of sellers and buyers, with valuations on
the buyers, is there always a set of market-clearing prices so that
the resulting preferred-seller graph contains a
baseline-maximizing perfect matching?
No. The example in part (a) is one where no market-clearing prices
can produce a perfect matching that is baseline maximizing.
Recall that
(va,x , vb,x , va,y , vb,y ) = (0, 1, 2, 4)
and
M0 = {b-x, a-y}
M = {a-x, b-y}
Any set of market-clearing prices must lead to a social welfare
maximizing matching, which is M in this example; while M0 is the
only baseline maximizing matching.
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Chapter 10 — Ex.15
15. Consider a bipartite graph auction with equal number of sellers and
buyers. Suppose a particular seller i is every buyer’s favorite, in the sense
that vi,j ≥ vk,j for every buyer j and every other seller k. Given a set of
market-clearing prices, must it be the case that the price charged by seller
i is at least as high as the price charged by any other seller (i.e. pi ≥ pk for
any other seller k)?
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Chapter 10 — Ex.15
15. Consider a bipartite graph auction with equal number of sellers and
buyers. Suppose a particular seller i is every buyer’s favorite, in the sense
that vi,j ≥ vk,j for every buyer j and every other seller k. Given a set of
market-clearing prices, must it be the case that the price charged by seller
i is at least as high as the price charged by any other seller (i.e. pi ≥ pk for
any other seller k)?
Suppose (for contradiction) there exists a seller a whose price pa > pi
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Chapter 10 — Ex.15
15. Consider a bipartite graph auction with equal number of sellers and
buyers. Suppose a particular seller i is every buyer’s favorite, in the sense
that vi,j ≥ vk,j for every buyer j and every other seller k. Given a set of
market-clearing prices, must it be the case that the price charged by seller
i is at least as high as the price charged by any other seller (i.e. pi ≥ pk for
any other seller k)?
Suppose (for contradiction) there exists a seller a whose price pa > pi
Since prices clear the market, there is a buyer x who gets object a
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Chapter 10 — Ex.15
15. Consider a bipartite graph auction with equal number of sellers and
buyers. Suppose a particular seller i is every buyer’s favorite, in the sense
that vi,j ≥ vk,j for every buyer j and every other seller k. Given a set of
market-clearing prices, must it be the case that the price charged by seller
i is at least as high as the price charged by any other seller (i.e. pi ≥ pk for
any other seller k)?
Suppose (for contradiction) there exists a seller a whose price pa > pi
Since prices clear the market, there is a buyer x who gets object a
Market-clearing also dictates that x weakly prefers a to j:
va,x − pa ≥ vi,x − pi
H. K. Chen (SFU)
⇒
(va,x − vi,x ) + (pi − pa ) ≥ 0
ECON 383
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Chapter 10 — Ex.15
15. Consider a bipartite graph auction with equal number of sellers and
buyers. Suppose a particular seller i is every buyer’s favorite, in the sense
that vi,j ≥ vk,j for every buyer j and every other seller k. Given a set of
market-clearing prices, must it be the case that the price charged by seller
i is at least as high as the price charged by any other seller (i.e. pi ≥ pk for
any other seller k)?
Suppose (for contradiction) there exists a seller a whose price pa > pi
Since prices clear the market, there is a buyer x who gets object a
Market-clearing also dictates that x weakly prefers a to j:
va,x − pa ≥ vi,x − pi
⇒
(va,x − vi,x ) + (pi − pa ) ≥ 0
But va,x ≤ vi,x and pi < pa , so a contradiction!
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Chapter 10 — Ex.15
15. Consider a bipartite graph auction with equal number of sellers and
buyers. Suppose a particular seller i is every buyer’s favorite, in the sense
that vi,j ≥ vk,j for every buyer j and every other seller k. Given a set of
market-clearing prices, must it be the case that the price charged by seller
i is at least as high as the price charged by any other seller (i.e. pi ≥ pk for
any other seller k)?
Suppose (for contradiction) there exists a seller a whose price pa > pi
Since prices clear the market, there is a buyer x who gets object a
Market-clearing also dictates that x weakly prefers a to j:
va,x − pa ≥ vi,x − pi
⇒
(va,x − vi,x ) + (pi − pa ) ≥ 0
But va,x ≤ vi,x and pi < pa , so a contradiction!
Therefore it’s impossible to find a seller a with pa > pi . So the
statement in the question is true.
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