CSV 886 Social Economic and Information Networks
Lecture 5: Matching Markets, Sponsored Search
R Ravi
[email protected]
Simple Models of Trade
• Decentralized
– Buyers and sellers have to find each other (search
costs) and agree on price (negotiation costs)
• Central exchange with competition
– Auctions set prices
• Centralized exchange
– Intermediaries setting fixed prices
– Faster search and clearing
2
Perfect Matching Market
3
Is there a perfect assignment?
4
One reason for no perfect assignment
A constricted set : A set of students whose wish
list is smaller than the number of students
5
The ONLY reason for no perfect assignment
• If there is a constricted set there is no perfect
matching
• If there is no constricted set, there is always a
perfect matching!
6
General Matching Market
• Valuations for different items, only want one
• Buyers X, Y, Z for items A, B, C
A
B
C
X
9
7
4
Y
5
9
7
Z
11
10
8
7
Alternate view
A
B
C
X
9
7
4
Y
5
9
7
Z
11
10
8
8
Prices
• Set a price per item and let each buyer choose
the item with most payoff = (value - price).
Try (1,3,4)
A B C
X
9
7
4
Y
5
9
7
Z
11
10
8
9
Market-clearing price
• Prices such that the graph of edges from
buyers to preferred sellers has a perfect
matching. Try (4,3,1)
A B
C
X
9
7
4
Y
5
9
7
Z
11
10
8
10
Two amazing facts
• For ANY set of buyer valuations, marketclearing prices exist
• For ANY set of market-clearing prices, the
resulting assignment has maximum total
valuation (over all assignment of items to
buyers)
11
Generalized auction round
• Current prices, smallest at zero
• Check preferred seller graph for matching
• If none, find a constricted set and raise prices of
the items wished by the set by one unit
• Rescale prices by subtracting smallest from all
to make the smallest zero
12
Example – start of Round 1
13
Example – start of Round 2
14
Example – start of Round 3
15
Example – start of Round 4
16
Why does auction end?
• Energy of buyer = maximum payoff
• Energy of item/seller = price
• Energy of auction = sum of above
• All energies non-negative at all times
Δ Energy in one round?
17
Basic Models of Internet Advertising
• CPM
• CPC
• CPA
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Sponsored Search
Charging for ads
• First generation
– CPI or cost-per-impression (also called CPM or
cost-per-mille)
• Second generation
– Overture: CPC or cost-per-click
• Key question: How much to charge per click?
First Take
• First price auction
• Results (Overture, circa 2002-2003)
Second Take
• Generalized Second Price Auction
• Two generalizations
– One related to maintaining truthful bidding of
valuations (VCG)
– Other, more superficial, related to mechanics of
second price (GSP)
Setting
• Sellers: descending click-thru rates
• Buyers: descending revenue per click
Market Clearing Prices?
Problem
• Don’t know everyone’s values to announce
these clearing prices
• Need a pricing mechanism that encourages
everyone to announce their true valuations
VCG Principle
• Allocate items so as to maximize total valuation
(not payoff to buyers, which subtracts the
price)
• Winner is charged a price equal to the “harm”
(decrease) he causes to all other bidders’
valuations by getting his item
Example: Maximum value allocation
Example: Price charged to x
Reduction in y’s and z’s value due to x =
Example: Price charged to y
Reduction in x’s and z’s value due to y =
Example: Max value allocation of VCG
A
B
C
X
9
7
4
Y
5
10
7
Z
11
10
8
Price charged to X
A
B
C
X
9
7
4
Y
5
10
7
Z
11
10
8
Price charged to Y
A
B
C
X
9
7
4
Y
5
10
7
Z
11
10
8
Price charged to Z
A
B
C
X
9
7
4
Y
5
10
7
Z
11
10
8
General Setting
• S = set of sellers (items)
• B = set of buyers
• VSB = maximum valuation matching of S to B
Suppose item i is given to j in this allocation.
Charge it the VCG price
pij = VSB-j – VS-iB-j
(Value if this seller j was not there – Value if
seller j got item i)
Why does VCG induce truth telling?
• Suppose item i goes to buyer j in VCG
mechanism for a price pij
• By lying about value, suppose j can get item h
instead. For this to be of no use to j, we need
for every h other than i,
vij - pij ≥ vhj - phj
First is overall maximum, second fixes h to j
Reality is not as nice
GSP or generalized second price auction
– Rank slots in decreasing order of CTR
– Rank buyers in decreasing order of bids
– Give 1st slot to 1st buyer at price of 2nd buyer’s bid
per click
– Give 2nd slot to 2nd buyer at price of 3rd buyer’s bid
per click…
– Give ith slot to ith buyer at price of (i+1)st bid per
click (total price CTRi times bidi+1)
Example: all bid true values
Example: x underbids at 5
Truth-telling may not be equilibrium
x has incentive to underbid from 7 to 5
Payoff bidding 7 =
Payoff bidding 5 =
Redeeming fact
GSP always has a Nash equilibrium where the
prices correspond to market clearing prices
Reality
• Assumption of decreasing CTRs may not hold
if ads placed higher up has bad quality
• Fudge quality factor q in GSP:
– Sort buyers by order of qjbj rather than bj
– Each buyer pays minimum amount to keep its
current position ( = qj+1bj+1 / qj)
• Expressiveness of keywords at both sides
(concern of SEM)
Advanced Material: Sec 15.9
VCG prices treated as posted prices are the
unique market clearing with minimum total
sum
HOMEWORK
1. Give the fastest algorithm to find, for any edge
of a graph, the number of pairs of nodes
whose shortest paths go through this node.
(Extra: Use the algorithm to compute the
fraction of the total number of shortest s-t
paths that go through this edge, for a given
pair s,t).
2. Give a polynomial time algorithm for finding
the smallest affiliation network explaining a
given social network edges, or show that it is
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NP-hard.
HOMEWORK CONTD
3. Exercise 8.4.4 of the text on adding tolls in
traffic networks
4. Exercise 10 from Chapter 9 (on entry price
for auctions)
5. Exercise 11 from Chapter 10 (on matching
markets)
6. Exercise 5 from Chapter 15 (on sponsored
search markets)
DUE in class MONDAY, February 2nd. OK to
email to [email protected]
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OTHER WORK
Second Homework: one problem per day, for a
total of four, due over email by next Monday,
February 9th.
Exam: 2 hours, 4 questions.
What day?
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