Multi-item Auctions / Matching
Economics 383 - Auction Theory
Instructor: Songzi Du
Simon Fraser University
October 2, 2015
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
1 / 20
Terminologies
Bipartite Graph
In mathematics, a graph is a set of
objects (nodes) where some pairs of
them are connected by links (edges)
c
e
b
d
a
g
ECON 383 (SFU)
Multi-item Auctions / Matching
f
October 2, 2015
2 / 20
Terminologies
Bipartite Graph
In mathematics, a graph is a set of
objects (nodes) where some pairs of
them are connected by links (edges)
c
e
b
d
a
g
A bipartite graph is a graph whose
nodes can be divided into two disjoint
subsets, O and B, such that every edge
connects a node in O to one in B.
ECON 383 (SFU)
Multi-item Auctions / Matching
f
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
2 / 20
Terminologies
Perfect Matching
A perfect matching occurs when
there are equal number of nodes in
the sets O and B
each node in O is connected by an
edge to a node in B
no two nodes in O are connected to
the same node in B
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
3 / 20
Terminologies
Perfect Matching
A perfect matching occurs when
there are equal number of nodes in
the sets O and B
each node in O is connected by an
edge to a node in B
no two nodes in O are connected to
the same node in B
A perfect matching can also be viewed
as a choice of edges in a bipartite graph
so that each node is the endpoint of
exactly one of the chosen edges.
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
3 / 20
Terminologies
Constricted Set
Given a subset of nodes S ⊆ B in a
bipartite graph, define its neighbor set
N (S) to be the set of nodes in O that
are connected by edges to nodes in S.
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
4 / 20
Terminologies
Constricted Set
Given a subset of nodes S ⊆ B in a
bipartite graph, define its neighbor set
N (S) to be the set of nodes in O that
are connected by edges to nodes in S.
If S = {5, 3}, then N (S) = {a, b, c, d}
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
4 / 20
Terminologies
Constricted Set
Given a subset of nodes S ⊆ B in a
bipartite graph, define its neighbor set
N (S) to be the set of nodes in O that
are connected by edges to nodes in S.
If S = {5, 3}, then N (S) = {a, b, c, d}
If S = {1, 4}, then N (S) =
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
4 / 20
Terminologies
Constricted Set
Given a subset of nodes S ⊆ B in a
bipartite graph, define its neighbor set
N (S) to be the set of nodes in O that
are connected by edges to nodes in S.
If S = {5, 3}, then N (S) = {a, b, c, d}
If S = {1, 4}, then N (S) = {b, e}
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
4 / 20
Terminologies
Constricted Set
Given a subset of nodes S ⊆ B in a
bipartite graph, define its neighbor set
N (S) to be the set of nodes in O that
are connected by edges to nodes in S.
If S = {5, 3}, then N (S) = {a, b, c, d}
If S = {1, 4}, then N (S) = {b, e}
A set S ⊆ B is constricted if its size is
strictly greater than the size of its
neighbor set; that is, if |S| > |N (S)|
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
4 / 20
Terminologies
Constricted Set
Given a subset of nodes S ⊆ B in a
bipartite graph, define its neighbor set
N (S) to be the set of nodes in O that
are connected by edges to nodes in S.
If S = {5, 3}, then N (S) = {a, b, c, d}
If S = {1, 4}, then N (S) = {b, e}
A set S ⊆ B is constricted if its size is
strictly greater than the size of its
neighbor set; that is, if |S| > |N (S)|
For example, take S = {1, 2, 3}, so
that N (S) =
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
4 / 20
Terminologies
Constricted Set
Given a subset of nodes S ⊆ B in a
bipartite graph, define its neighbor set
N (S) to be the set of nodes in O that
are connected by edges to nodes in S.
If S = {5, 3}, then N (S) = {a, b, c, d}
If S = {1, 4}, then N (S) = {b, e}
A set S ⊆ B is constricted if its size is
strictly greater than the size of its
neighbor set; that is, if |S| > |N (S)|
For example, take S = {1, 2, 3}, so
that N (S) = {a, b}
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
4 / 20
Terminologies
Constricted Set
Given a subset of nodes S ⊆ B in a
bipartite graph, define its neighbor set
N (S) to be the set of nodes in O that
are connected by edges to nodes in S.
If S = {5, 3}, then N (S) = {a, b, c, d}
If S = {1, 4}, then N (S) = {b, e}
A set S ⊆ B is constricted if its size is
strictly greater than the size of its
neighbor set; that is, if |S| > |N (S)|
For example, take S = {1, 2, 3}, so
that N (S) = {a, b}
Clearly, |S| = 3 > 2 = |N (S)|
ECON 383 (SFU)
Multi-item Auctions / Matching
a
5
b
4
c
3
d
2
e
1
O
B
October 2, 2015
4 / 20
Terminologies
Constricted Set
Given a subset of nodes S ⊆ B in a
bipartite graph, define its neighbor set
N (S) to be the set of nodes in O that
are connected by edges to nodes in S.
If S = {5, 3}, then N (S) = {a, b, c, d}
If S = {1, 4}, then N (S) = {b, e}
A set S ⊆ B is constricted if its size is
strictly greater than the size of its
neighbor set; that is, if |S| > |N (S)|
For example, take S = {1, 2, 3}, so
that N (S) = {a, b}
Clearly, |S| = 3 > 2 = |N (S)|
a
5
b
4
c
3
d
2
e
1
O
B
If we interpret B and O as sets of buyers and sellers respectively, then
constriction represents over-demand
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
4 / 20
Matching Theorem
Matching Theorem
A bipartite graph (with equal number of nodes in O and B) has a perfect
matching if and only if it has no constricted set.
Interpretation:
If there is over-demand for a subset of sellers, then it is not possible
to match buyers and sellers in a 1-to-1 fashion.
The failure to match buyers and sellers in a 1-to-1 fashion can only be
caused by over-demand.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
5 / 20
Market-Clearing Prices
Preferred Seller Graph
A set of buyers B and a set of objects (sellers) O in a bipartite graph
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
6 / 20
Market-Clearing Prices
Preferred Seller Graph
A set of buyers B and a set of objects (sellers) O in a bipartite graph
Let the price of each object i ∈ O be pi
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
6 / 20
Market-Clearing Prices
Preferred Seller Graph
A set of buyers B and a set of objects (sellers) O in a bipartite graph
Let the price of each object i ∈ O be pi
Let buyer j ∈ B value object i ∈ O at vi,j ; each buyer wants only one
object
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
6 / 20
Market-Clearing Prices
Preferred Seller Graph
A set of buyers B and a set of objects (sellers) O in a bipartite graph
Let the price of each object i ∈ O be pi
Let buyer j ∈ B value object i ∈ O at vi,j ; each buyer wants only one
object
For each j, let Dj (p) be the set of solutions to
maxi vi,j − pi
s.t. vi,j − pi ≥ 0
Dj (p) could contain multiple solutions; Dj (p) could also contain no
solution.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
6 / 20
Market-Clearing Prices
Preferred Seller Graph
A set of buyers B and a set of objects (sellers) O in a bipartite graph
Let the price of each object i ∈ O be pi
Let buyer j ∈ B value object i ∈ O at vi,j ; each buyer wants only one
object
For each j, let Dj (p) be the set of solutions to
maxi vi,j − pi
s.t. vi,j − pi ≥ 0
Dj (p) could contain multiple solutions; Dj (p) could also contain no
solution.
Connect each j ∈ B to his demands, that is, objects in Dj (p).
If Dj (p) is an empty set, then no edge is drawn.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
6 / 20
Market-Clearing Prices
Preferred Seller Graph
A set of buyers B and a set of objects (sellers) O in a bipartite graph
Let the price of each object i ∈ O be pi
Let buyer j ∈ B value object i ∈ O at vi,j ; each buyer wants only one
object
For each j, let Dj (p) be the set of solutions to
maxi vi,j − pi
s.t. vi,j − pi ≥ 0
Dj (p) could contain multiple solutions; Dj (p) could also contain no
solution.
Connect each j ∈ B to his demands, that is, objects in Dj (p).
If Dj (p) is an empty set, then no edge is drawn.
The resulting graph is the preferred seller/object graph
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
6 / 20
Market-Clearing Prices
Preferred Seller Graph
A set of buyers B and a set of objects (sellers) O in a bipartite graph
Let the price of each object i ∈ O be pi
Let buyer j ∈ B value object i ∈ O at vi,j ; each buyer wants only one
object
For each j, let Dj (p) be the set of solutions to
maxi vi,j − pi
s.t. vi,j − pi ≥ 0
Dj (p) could contain multiple solutions; Dj (p) could also contain no
solution.
Connect each j ∈ B to his demands, that is, objects in Dj (p).
If Dj (p) is an empty set, then no edge is drawn.
The resulting graph is the preferred seller/object graph
Market-Clearing Prices
A set of prices {pi }i∈O is market-clearing if the resulting preferred
seller graph has a perfect matching.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
6 / 20
Example of Market-clearing Prices
Four houses in different neighborhoods. Four buyers: Alice, Bob,
Clair, Doug, with values:
Alice
Bob
Clair
Doug
Yaletown
4
5
4
2
Downtown Eastside
1
0
0
0
Kitsilano
5
5
8
5
Metrotown
3
2
5
1
Do prices pY = 2, pE = 1, pK = 2, pM = 1 clear the market?
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
7 / 20
Market-clearing prices maximize social welfare
If m is a matching (j is matched to m(j), for every buyer j), the social
welfare generated by m is:
∑ vm(j),j
j∈B
Social planner wants to solve:
max ∑ vm(j),j
m
j∈B
among all feasible matching m.
Theorem
If {pi } are market-clearing prices, then any perfect matching associated
with these prices maximize the social welfare.
Market-clearing prices automatically maximize the social welfare, work
as well as a social planner.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
8 / 20
Ascending-price Auction to find market-clearing prices
Input: vi,j , for every buyer j ∈ B and object i ∈ O.
1. Set initially pi = 0 for every i ∈ O.
2. Construct the preferred-object graph, and check if there is a perfect
matching.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
9 / 20
Ascending-price Auction to find market-clearing prices
Input: vi,j , for every buyer j ∈ B and object i ∈ O.
1. Set initially pi = 0 for every i ∈ O.
2. Construct the preferred-object graph, and check if there is a perfect
matching.
3. If yes, done.
4. If not, find a constricted set of buyers, S ⊂ B, and the over-demanded
objects N (S).
5. For each object in N (S), increase its price by one.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
9 / 20
Ascending-price Auction to find market-clearing prices
Input: vi,j , for every buyer j ∈ B and object i ∈ O.
1. Set initially pi = 0 for every i ∈ O.
2. Construct the preferred-object graph, and check if there is a perfect
matching.
3. If yes, done.
4. If not, find a constricted set of buyers, S ⊂ B, and the over-demanded
objects N (S).
5. For each object in N (S), increase its price by one.
6. If the lowest price among all objects is no longer zero, decrease every
price by one.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
9 / 20
Ascending-price Auction to find market-clearing prices
Input: vi,j , for every buyer j ∈ B and object i ∈ O.
1. Set initially pi = 0 for every i ∈ O.
2. Construct the preferred-object graph, and check if there is a perfect
matching.
3. If yes, done.
4. If not, find a constricted set of buyers, S ⊂ B, and the over-demanded
objects N (S).
5. For each object in N (S), increase its price by one.
6. If the lowest price among all objects is no longer zero, decrease every
price by one.
7. Go to Step 2.
Output: pi that are market-clearing prices, and an associated perfect
matching.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
9 / 20
Example of Ascending Price Auction
Six students: Alice, Bob, Clair, Doug, Elle, Frank. Four letter grades:
A, B, C, D.
The professor has to give 1 A, 2 B’s, 2 C’s, and 1 D (department
guideline).
The students’ willingness-to-pay:
Alice
Bob
Clair
Doug
Elle
Frank
ECON 383 (SFU)
A
10
12
8
7
9
10
B
9
11
7
5
2
8
C
8
10
3
2
1
3
Multi-item Auctions / Matching
D
0
0
0
0
0
0
October 2, 2015
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Example of Ascending Price Auction
Grade
Student
0
A
a
10, 9, 9, 8, 8, 0
B1
b
12, 11, 11, 10, 10, 0
0
0
B2
c
8, 7, 7, 3, 3, 0
d
7, 5, 5, 2, 2, 0
0
C1
C2
e
9, 2, 2, 1, 1, 0
0
f
10, 8, 8, 3, 3, 0
0
D
Round 1: constricted set S = {a, b, c, d, e, f }, N (S) = {A}.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
11 / 20
Example of Ascending Price Auction
Grade
Student
1
A
a
10, 9, 9, 8, 8, 0
B1
b
12, 11, 11, 10, 10, 0
0
0
B2
c
8, 7, 7, 3, 3, 0
d
7, 5, 5, 2, 2, 0
0
C1
C2
e
9, 2, 2, 1, 1, 0
0
f
10, 8, 8, 3, 3, 0
0
D
Round 2: constricted set S = {d, e, f }, N (S) = {A}.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
12 / 20
Example of Ascending Price Auction
Grade
Student
2
A
a
10, 9, 9, 8, 8, 0
B1
b
12, 11, 11, 10, 10, 0
0
0
B2
c
8, 7, 7, 3, 3, 0
d
7, 5, 5, 2, 2, 0
0
C1
C2
e
9, 2, 2, 1, 1, 0
0
f
10, 8, 8, 3, 3, 0
0
D
Round 3: constricted set S = {a, b, c}, N (S) = {B1 , B2 }.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
13 / 20
Example of Ascending Price Auction
Grade
Student
2
A
a
10, 9, 9, 8, 8, 0
B1
b
12, 11, 11, 10, 10, 0
1
1
B2
c
8, 7, 7, 3, 3, 0
d
7, 5, 5, 2, 2, 0
0
C1
C2
e
9, 2, 2, 1, 1, 0
0
f
10, 8, 8, 3, 3, 0
0
D
Round 4: constricted set S = {d, e, f }, N (S) = {A}.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
14 / 20
Example of Ascending Price Auction
Grade
Student
3
A
a
10, 9, 9, 8, 8, 0
B1
b
12, 11, 11, 10, 10, 0
1
1
B2
c
8, 7, 7, 3, 3, 0
d
7, 5, 5, 2, 2, 0
0
C1
C2
e
9, 2, 2, 1, 1, 0
0
f
10, 8, 8, 3, 3, 0
0
D
Round 5: constricted set S = {a, b, c, d, e, f }, N (S) = {A, B1 , B2 , C1 , C2 }.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
15 / 20
Example of Ascending Price Auction
Grade
Student
4
A
a
10, 9, 9, 8, 8, 0
B1
b
12, 11, 11, 10, 10, 0
2
2
B2
c
8, 7, 7, 3, 3, 0
d
7, 5, 5, 2, 2, 0
1
C1
C2
e
9, 2, 2, 1, 1, 0
1
f
10, 8, 8, 3, 3, 0
0
D
Round 6: constricted set S = {a, b, c, d, e, f }, N (S) = {A, B1 , B2 , C1 , C2 }.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
16 / 20
Example of Ascending Price Auction
Grade
Student
5
A
a
10, 9, 9, 8, 8, 0
B1
b
12, 11, 11, 10, 10, 0
3
3
B2
c
8, 7, 7, 3, 3, 0
d
7, 5, 5, 2, 2, 0
2
C1
C2
e
9, 2, 2, 1, 1, 0
2
f
10, 8, 8, 3, 3, 0
0
D
Round 7: constricted set S = {a, b, c, d, e, f }, N (S) = {A, B1 , B2 , C1 , C2 }.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
17 / 20
Example of Ascending Price Auction
Grade
Student
6
A
a
10, 9, 9, 8, 8, 0
B1
b
12, 11, 11, 10, 10, 0
4
4
B2
c
8, 7, 7, 3, 3, 0
d
7, 5, 5, 2, 2, 0
3
C1
C2
e
9, 2, 2, 1, 1, 0
3
f
10, 8, 8, 3, 3, 0
0
D
Round 8: constricted set S = {a, b, c, d, e, f }, N (S) = {A, B1 , B2 , C1 , C2 }.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
18 / 20
Example of Ascending Price Auction
Grade
Student
7
A
a
10, 9, 9, 8, 8, 0
B1
b
12, 11, 11, 10, 10, 0
5
5
B2
c
8, 7, 7, 3, 3, 0
d
7, 5, 5, 2, 2, 0
4
C1
C2
e
9, 2, 2, 1, 1, 0
4
f
10, 8, 8, 3, 3, 0
0
D
Round 9: a perfect matching exists.
Student’s welfare: 8 + 10 + 7 + 0 + 9 + 8 = 42; Professor’s revenue:
ECON 383 (SFU)
7 + 5 + 5 + 4 + 4 + 0 = 25.
Multi-item Auctions / Matching
October 2, 2015
19 / 20
Relating to Second Price Auction
A single object, four buyers: v1 = 5, v2 = 3, v3 = 2 and v4 = 0.
Two identical objects, four buyers each wanting only one unit:
v1 = 5, v2 = 3, v3 = 2 and v4 = 0.
ECON 383 (SFU)
Multi-item Auctions / Matching
October 2, 2015
20 / 20