Sponsored Search Markets (from Networks, Crowds, and Markets: Reasoning About a Highly Connected World)

Reading seminar

Chapter 15 Sponsored Search Markets
from Networks, Crowds, and Markets: Reasoning About a
Highly Connected World

Junpei Kawamoto

Advertising Tied to Search Behavior
 Search engine meets search advertising
 Early advertising was sold on the basis of “impression” like
magazine or newspapers ad.
 It was not efficient way
 Showing ad of calligraphy pens to all people vs. to people searching
“calligraphy pen”.

 Search engine queries are potent way to get users to express
their intent.

 Keywords-bases advertisement.
 How to sell keywords-based ad?

2 Chapter 15 - Sponsored Search Markets

Advertising Tied to Search Behavior
 Paying per click
 Cost-per-click (CPC) model
 You only pay when a user actually clicks on the ad.
 Clicking on an ad represents an even stronger indication of
intent than simply issuing a query.
 Setting prices through an auction
 How should a search engine set the prices pre click for
different queries?
 One easy way is the search engine posts prices.
 However, there are lots of queries and impossible to decide prices for
those all queries.
 Auction protocols is a good way to decide prices.


Overview
1. Advertising as a Matching Market
2. Encouraging Truthful Bidding
in Matching markets: The VCG Principle
3. Analyzing the VCG Procedure:
Truth-Telling as a Dominant Strategy
4. The Generalized Second Price Auction
5. Equilibria of the Generalized Second Price Auction
6. Ad Quality
7. Complex Queries and Interactions Among Keywords


Advertising as a Matching Market
 Click through Rates and Revenues Per Click.

click through revenues
slots advertisers
rates per click
10 a x 3

5 b y 2

2 c z 1



slots advertisers
rates per click
10 a x 3

Click through rates: e.g. # of clicks per hour.
5
Assuming: b y 2
1. Advertisers know the click through rates;
2. Click through rates depend only on slot not contents of the ads;
3. Click through rates don’t depend on ads on other slots.
2 c z 1



slots advertisers
rates per click
10 a x 3

The expected amount of revenues advertisers receive per user clicking on the ad.
Assuming: 5 b y 2
1. These values are intrinsic to the advertisers,
2. doesn’t depend on what was being shown on the page.

2 c z 1


 Constructing a Matching Market

The basic ingredients of matching market from Chapter 10.
 The participants in a matching market consists of a set of
buyers and a set of sellers.
 Each buyer j has a valuation for the item offered by each seller
i. This valuation can depend on the identities of both the buyer
and the seller, and we denote vij.
 The goal is to match up buyers with sellers, in such a way that
no buyers purchases two different items, and the same item
isn’t sold two different buyers.


 ri: the click through rate of slot i
 vj: the revenue per click of advertiser j
 vij = rivj: the benefit advertiser j receives from an ad in slot i

Advertiser j’s valuation for slot i in the language of matching market,

 Slots = Sellers
 Advertisers = Buyer

The problem is to assign sellers to buyers in a matching market.


Advertisers’ valuations for the slots
slots advertisers valuations
rates per click
10 a x 3 30, 15, 6

5 b y 2 20, 10, 4

2 c z 1 10, 5, 2


rates per click
10 a x 3 30, 15, 6

5 b y 2 20, 10, 4

2 c z 1 10, 5, 2

the benefit advertiser j receives from an ad in slot i, vij = rivj

rates per click
10 a x 3 30, 15, 6

5 b y 2 20, 10, 4

2 c z 1 10, 5, 2


 That is a matching market with a special structure
 The valuations of one buyer simply form a multiple of the valuations
of any other buyers.

 Assumption:
 # of slots = # of advertisers
 If not, we can add fictitious slots or advertises.
 The click through rates of fictitious slots = 0
 The revenue pre clicks of fictitious advertises = 0


 Obtaining Market-Clearing Prices

Using the framework from Chapter 10
 Each seller i announces a price pi for his item.
 Each buyer j evaluates her payoff for choosing a particular seller i
 it’s equal to the valuation minus the price for this seller’s item, vij – pi
 We then build a perfect-seller graph by linking each buyers to the
seller or sellers from which she gets the highest payoff.
 The prices are market-clearing if this graph has a perfect matching
 in this case, we can assign distinct items to all the buyers in such a way
that each buyers gets an item that maximizes her payoff.


 Market-clearing prices exists for every matching market.
 We have a procedure to construct market-clearing prices.
 Market-clearing prices always maximizes the buyer’s total
valuations for the items they get.

From Chapter 10


Finally obtained the perfect-seller graph.
rates per click
10 a x 3 30, 15, 6

5 b y 2 20, 10, 4

2 c z 1 10, 5, 2


 This construction of prices can only be carried out by
search engine if it actually knows the valuations of the
advertisers.

 Next, we consider how to set prices in a setting where
the search engine doesn’t know these valuations; it must
rely on advertisers to report them without being able to
know whether this reporting is truthful.


Encouraging Truthful Bidding
 What would be a good price-setting procedure when the search
engine doesn’t know advertises’ valuations?
 In the early days, variants of the 1st-price auction used.
 Recall from Chapter 9, in 1st-price auction, bidders under-report.

 In the case of a single-item auction, 2nd-price auction is a solution.
 Truthful bidding is a dominant strategy.
 Avoid many of the pathologies associated with more complex auctions.

 What is the analogue of the 2nd-price auction for advertising
markets with multiple slots?
 How can we define a price-setting procedure for matching markets so
that truthful reporting of valuations is a dominant strategy for buyers?


 The VCG Principle
 How to generalize 2nd-price single-item auction to multi-item one.
 A less obvious view of 2nd-price auction
 The 2nd-price auction produces an allocation that maximizes social welfare.
 The winner is charged an amount equal to the “harm” he causes the other
bidders by receiving the item.
1 Gets the apple
v1
v2 2

vn

n
v1 > v 2 > … > vn

1 If bidder 1 weren’t present
v1
v2 2 Bidder 2 got the apple, which
was worth of v2

vn Nothing was changed for
bidder 3-n. (we ignore them)
n
v1 > v 2 > … > vn

1 The presence of bidder 1
v1 causes the “harm” v2 to
v2 2 bidder 2. (Bidder 2 lost an
item worthy of v2)

Therefore, bidder 1 must
vn pay v2 as the price of the
apple.
n
v1 > v 2 > … > vn


Vickrey-Clarke-Groves (VCG) principles


 Applying the VCG Principles to Matching Markets
 We have a set of buyers and sellers
 # of buyers = # of sellers. (if not, we can add fictitious buyers and sellers.)
 A buyer j has a valuation vij for the item i.
 Each buyer knows only their own valuations.
independent, private values
 Each buyer doesn’t care anyone else.

 Procedure
 Assign items to buyers so as to maximize total valuations.
 Compute price with the VCG principles


For example
price slots advertisers valuations

a x 30, 15, 6

30 for a, 15 for b, 6 for c

b y 20, 10, 4

c z 10, 5, 2


For example
If x weren’t present
a x 30, 15, 6

y got slot a,
which was worthy of 20
b y 20, 10, 4

z got slot b,
which was worthy of 5
c z 10, 5, 2


For example
In fact, x is present.
a x 30, 15, 6
y gets b, which is worthy of
10, instead of a, which is
worth of 20. Harm = 20-10
b y 20, 10, 4
z gets c, which is worthy of
2, instead of b, which is
c z 10, 5, 2


For example
10+3 = 13 In face, x is present.
a x 30, 15, 6
y gets b, which is worthy of
10, instead of a, which is
b y 20, 10, 4
z gets c, which is worthy of
2, instead of b, which is
c z 10, 5, 2


For example

13 a x 30, 15, 6

3 b y 20, 10, 4

0 c z 10, 5, 2


 To generalize this pricing,
 S: the set of sellers
 B: the set of buyers
 VSB: the maximum total valuations over all possible matching for S & B
 S – i: the set of sellers without i
 B – j: the set of buyers without j
 VS-iB-j: the maximum total valuations of S-i & B-j

 VCG price pij that we charge buyer j for item i
 pij = VSB-j – VS-iB-j
Total valuations if j Total valuations in
weren’t present such a case j gets i.

 The VCG Price-Setting Procedure.
 We assume there’s a single price-setting authority.
 Collection information, assign items, and charge prices.

 The Procedure
1. Ask buyers to announce valuations for items. (need truthful report)
2. Choose a socially optimal assignment of items to buyers. (can do it)
3. Charge each buyer the appropriate VCG price. (pij = VSB-j – VS-iB-j)


 Market-clearing price vs.VCG price.
 Market-clearing price is a posted price, based on English
auction.
 VCG price is a personalized price and based on 2nd-price
auction.

 2nd-price auction vs.VCG prices
 The basic idea is “harm-done-to-others” principle.
 2nd-price auction for single item,VCG prices for multi items.
 2nd-price auction is a VCG price auction.
 Suppose 1 real item & n – 1 fictitious items.


Analyzing the VCG Procedure:
 We’ll show the following claim.
If items are assigned and prices computed according to
the VCG procedure, then truthfully announcing valuations
is a dominant strategy for each buyer, and the resulting
assignment maximizes the total valuations of any perfect
matching of slots and advertisers.



obvious if buyers report their valuations truthfully, then the
assignment of items is designed to maximize the total
valuations by definition.



We’ll focus the first part.


 Suppose when the buyer j announce her valuations
truthfully, then she is assigned to item i.
 Buyer j’s payoff = vij – pij

 If j lies, there’re two cases,
 j lies but gets item i
 her payoff won’t change because pij is independent with her bit.
 j lies and gets item h = VSB, because we assume
assignment j to i is optimal.
 her payoff is vhj – phj
 vij – pij = vij – (VSB-j – VS-iB-j) = vij + VS-iB-j – VSB-j = VSB – VSB-j
 vhj – phj = vhj – (VSB-j – VS-hB-j) = vhj + VS-hB-j – VSB-j ≤ VSB – VSB-j
≤ VSB


 Suppose when the buyer j announce her valuations
truthfully, then she is assigned to item i.
 Buyer j’s payoff = vij – pij

 If j lies, there’re two cases,
 j lies but gets item i
vij – pij ≥ vhj – phj
 her payoff won’t change because ij is independent with her bit.
That is, when the buyer j lies, her payoffpbecome smaller or doesn’t change.
 j lies and gets item h incentiveSB, because we assume
(no = V to tell lie)
assignment j to i is optimal.
 her payoff is vhj – phj
 vij – pij = vij – (VSB-j – VS-iB-j) = vij + VS-iB-j – VSB-j = VSB – VSB-j
 vhj – phj = vhj – (VSB-j – VS-hB-j) = vhj + VS-hB-j – VSB-j ≤ VSB – VSB-j

≤ VSB


 The VCG Procedure would make buyers happy.
 But it’s not clear the VCG procedure is the best way to
generate revenue for the search engine.
 Determining which procedure will maximize seller revenue is a
current research topic.


The Generalized Second Price Auction
 the Generalized Second Price (GSP) auction (by Google)
 a generalization of 2nd-price auction
 it’s up to the advertiser whether they bit true valuations
For example
price slots advertisers bids

b2 a x b1

b3 b y b2

bn c z bn-1

b1 > b2 > … > bn

 GSP has a number of pathologies VCG avoids:
 Truth-telling might not constitute a Nash equilibrium;
 There can be multiple Nash equilibrium;
 Some assignments don’t maximize the total valuations.

 Good news
 GSP has at least one Nash equilibrium,
 Among them, there is one which maximize the total valuation.

 However, how to maximize search engine’s revenue is a
still current research topic.


 Truth-telling may not be an equilibrium.

slots advertisers
rates per click
10 a x 7

4 b y 6

0 c z 1



slots advertisers price
rates per click
10 a x 7 10×6

4 b y 6 4×1

0 c z 1 0



slots advertisers price payoff
rates per click
10 a x 7 60 70 - 60

10×7

4 b y 6 4 24 - 4

0 c z 1 0



slots advertisers bid x lies
rates per click
10 a x 7 5

4 b y 6 6

0 c z 1 1



slots advertisers bid
rates per click
10 a x 7 5

4 b y 6 6

Changed

0 c z 1 1



slots advertisers bid price
rates per click
10 a x 7 5 4×1

4 b y 6 6 50

0 c z 1 1 0
x is the 2nd highest bidder so
use 3rd bid for the price


slots advertisers bid price payoff
rates per click
10 a x 7 5 4 28 - 4

4 b y 6 6 50 24 - 4

0 c z 1 1 0



rates per click
10 a x 7 5 4 28 - 4
When x tell truth, his payoff = 70 – 60 = 10,
When x tell lie, his payoff = 28 – 4 = 24.

4 b y 6 6 50 24 - 4

In this case, truth-telling is not an equilibrium.

0 c z 1 1 0


 Multiple and non-optimal equilibria.
click through
rates
10 a x 5

4 b y 4

0 c z 2
Both are Nash equilibrium.
click through
rates
10 a x 3
4 b y 5

0 c z 1


rates per clicks
10 a x 7 5 40 70-40

4 b y 6 4 8 24-8

0 c z 1 2 0 0

click through
rates
10 a x 3
4 b y 5

0 c z 1


rates per clicks
10 a x 7 5→3 8 28-8

4 b y 6 4 30 60-30

0 c z 1 2 0 0

click through Less than when bid 5.
slots advertisers So not bid
motivated bidding lower.
rates
10 a x 3
4 b y 5

0 c z 1


rates per clicks
10 a x 7 5 8 28-8

4 b y 6 4→6 50 60-50

0 c z 1 2 0 0

click through Less than when bid 4.
slots advertisers So not bid
motivated bidding lower.
rates
10 a x 3
4 b y 5

0 c z 1


rates per clicks
10 a x 7 5 40 70-40

4 b y 6 4 8 24-8

0 c z 1 2 0 0

click through
slots advertisers revenues bid price payoff
rates per clicks
10 a x 7 3 4 28-4

4 b y 6 5 30 60-30

0 c z 1 1 0 0


rates per clicks
10 a x 7 5 40 70-40
Total payoff
4 b y 6 4 8 24-8
46
0 c z 1 2 0 0

click through
slots advertisers revenues bid price payoff
rates per clicks
10 a x 7 3 4 28-4
Total payoff
4 b y 30 60-30
6 545
0 c z 1 1 0 0


 The Revenue of GSP and VCG
 The search engine’s revenue is depend on which equilibrium is
selected.
slots advertisers bid price revenue
rates per clicks
10 a x 7 5 40

4 b y 6 4 8 48

0 c z 1 2 0

10 a x 7 3 4

4 b y 6 5 30 34

0 c z 1 1 0

 VCG for the same example
rates per clicks
10 a x 7 70, 28, 0

4 b y 6 60, 24, 0

0 c z 1 10, 4, 0

price revenue
10 a x 7 40

4 b y 6 4 44

0 c z 1 0


 Does GSP or VCG provide more revenue to the search
engine?

Ans.
It’s depend on which equilibrium of the GSP the advertises use.


Equilibria of the Generalized Second Price
Auction
 GSP always has a Nash equilibrium
 Assuming # of slots = # of advertisers
click through
slots advertisers
rate
r1 1 1

r2 2 2

rn n n

Decreasing order of click Decreasing order of
through rates valuations per clicks


Auction
 Assuming # of slots = # of advertisers
click through
slots advertisers
rate
r1 1 1

r2 2 2

rn n n

Obtaining Market-Clearing
Prices (from Chapter 10)


Auction
 Think about a set of matching market prices {p1, …, pn}
 We can make them (from Chapter 10)
 And think about a set of price per click
 p*i = pi / ri (ri: click through rate)


Auction
 p*1 ≥ p*2 ≥ … ≥ p*n
 To see why that is true, we think p*j & p*k (j < k) (Goal: p*j ≤ p*k)
 In slot k, total payoff is (vk – p*k)rk
 In slot j, total payoff is (vk – p*j)rj
 If rj > rk but slot k is preferred, then payoff of k ≥ payoff of j
 (vk – p*k)rk ≥ (vk – p*j)rj → vk – p*k ≥ vk – p*j → p*j ≥ p*k
 If rk ≥ rj
 pj ≥ pk → p *j ≥ p*k


Auction
 We simply have advertiser j place a bid of p*j-1
 And advertiser 1 place any bid larger than p*1


1 1 x (>p*1)

2 2 p*1

n n p*n-1


Auction
 We simply have advertiser j place a bid of p*j-1
 And advertiser 1 place any bid larger than p*1
price

p*1 1 1 x (>p*1)

p*2 2 2 p*1

p*n = 0 n n p*n-1


Auction
 Why do the bids form a Nash equilibrium?
 Considering an advertise j in slot j,
 If it were to lower its bid and move to slot k, then its price would be p*k
 But since the prices are market-clearing, j is at least as happy with its
current slot at its current prices as it would be with k’s current slot at k’s
current price.

 If the advertiser j bids higher and move to slot i, the bid = p*i
price bid
p*i+1 i i p*i Advertiser j must pay higher that the
current price of slot i
price bid
p*i i j x > p*i
p*i+1 i+1 i p*i

Auction
 Why do the bids form a Nash equilibrium?
 Therefore the below bids is a Nash equilibrium

price

p*1 1 1 x (>p*1)

p*2 2 2 p*1

p*n= 0 n n p*n-1


Ad Quality
 The assumption of a fixed click through rate.
 We assumed fixed click through rates rj
 In practice, it’s also depend on the contents of ad.

 Search engine worried about a low-quality advertiser bids
very highly.
 The click through rate will be small.
 The revenue of the search engine also will be small!


Ad Quality
 The role of ad quality.
 Uses an estimated ad quality factor qj for advertiser j.
 The click through rate is qjri (instead of ri)
 (if qi = 1 for all i, it is previous model we discussed. )
 The valuations of advertiser j for slot i, vij = qjrivj
 The bid bj of advertiser j is changed to pjbj
 The price is the minimum bid the advertiser would need in
order to hold his current position.

 The analysis of this version’s GSP is equal to normal GSP.


Ad Quality
 The mysterious nature of ad quality.
 How is ad quality computed?
 Can estimate by actually observing the click through rate of the ad.
 But search engines don’t tell the actual way.

 How does the behavior of a matching market such as this one
change when the precise rules of the allocation procedure are
being kept secret?
 A still topic for potential research.


Complex Queries and Interactions Among
Keywords
 Example 1.
 A company selling ski vacation package to Switzerland.
 “Switzerland”, “Swiss vacations”, “Swiss hotels”, …

 With a fixed budget, how should the company go about
dividing its budget across different keywords?

 Still challenging problem
 † is a current research working on this problem.

†Paat Rusmevichientong and David P. Williamson. An adaptive algorithm for selecting
profitable keywords for search-based advertising services. In Proc. 7th ACM
Conference on Electronic Commerce, pages 260–269, 2006.

Complex Queries and Interactions Among
Keywords
 Example 2.
 Some users query “Zurich ski vacation trip December”
 No advertiser bids this key words.
 Which ads should the search engine show?
 Even if relevant advertisers can be identified, how much should
they be charged for a click?
 Existing search engines get some agreements.
 This is very interesting potential further research.


Sponsored Search Markets (from Networks, Crowds, and Markets: Reasoning About a Highly Connected World)

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