Sponsored SearchSponsored Search

Presenter:

Lory Al Moakar

Outline

• Motivation• Problem Definition • VCG solution • GSP(Generalized Second Price)

• GSP vs. VCG• Is GSP incentive compatible? • GSP has a Nash Equilibrium formed of market clearing prices. • Some GSP Nash Equilibria may not be socially optimal.

• Conclusion• A note on further issues

Motivation

• Search is by far the most lucrative area, accounting for 40 percent of the total online ad spending in the U.S., according to JupiterResearch.

• Search advertising is expected to grow from $4.2 billion in 2005 to $7.5 billion in 2010, … JupiterResearch has forecast.

Snap Shots of Adwords

Problem Definition• N advertisers • M slots such that M ≤ N • rj the clickthrough rate for slot j i.e. the number of clicks

the ad will receive if it is listed in slot j.• Vi the declared value an advertiser is willing to pay per

click • Vi* is the true value an advertiser is willing to pay per

click • The value an advertiser i places on his/her ad in the jth slot

Vij = Vi × rj

• Xij = 1 if advertiser i is assigned slot j• Social Objective: find a perfect matching that maximizes the

total valuation of the advertisers over all perfect matchings i j

ijij VXmax

Example

• 3 advertisers:

• 3 slotsVij* a1 a2 a3

s1 6 4 1

s2 3 2 1

s3 0 0 0

a1 a2 a3

Vi* 3 2 1

s1 s2 s3

rj 2 1 0

a1•

a2•

a3•

•s1

•s2

•s3

6

2

0

Review on VCG mechanism with Clark-Pivot Rule

• Player i pays the mechanism :

• Pi = the best the other players can do if player i was not there – the best the other players can do when player i is there

ij ij

jb

i aVbVP )()(max

Another objective: Incentive Compatible using VCG

• Assume advertiser i is assigned slot j if i tells the truth.• VN-i

M is the maximum valuation over all perfect matchings of all slots and all

advertisers except i • VN-i

M-j is the maximum valuation over all perfect matchings of all slots except j and all

advertisers except i • The harm that i does by showing up is:

Pj = VN-iM – VN-i

M-j

• Pj is the VCG price for advertiser i assigned to slot j.

Is this incentive compatible? Assume i lies and is assigned slot k ≠ j which is the slot i gets if it does not lie.

• Gain from lying = Vik* – Pk • Gain from telling the truth = Vij* – Pj • To be incentive compatible, we need to prove that the gain from lying ≤ gain

from telling the truth

• prove that [Vij* – Pj ] – [Vik* – Pk ] ≥ 0

• Proof:

• Vik* – Pk = Vik* – [ VN-iM – VN-i

M-k]

• Vij* – Pj = Vij* – [ VN-iM – VN-i

M-j]

• [Vij* – Pj ] – [Vik* – Pk ] = Vij* + VN-iM-j – [Vik* + VN-i

M-k]

= optimal valuation – (total valuation when i gets k) ≥ 0 This mechanism is incentive compatible.

GSP ( Generalized Second Price) • GSP assigns slots to highest M bids • GSP charges advertiser who gets slot j the bid of the advertiser

in slot j-1. • used by Google and a variation of it used by yahoo• Excerpt from Google’s AdWords Ad:

Is GSP incentive compatible?

Counterexample:

• 2 slots with r1 = 10 and r2 = 4

• 3 advertisers with V1* = 7, V2* = 6 and V3* = 1

• if a1 bids truthfully, he/she will be assigned to s1 pays 60 and gains 70 with net profit of 10

• if a1 lies and bids 5, he/she will be assigned to s2 pays 4 and gains 28 with net profit of 24.

• a1 is better off with lying!

GSP vs. VCG

• Consider this example.

• 2 slots (r1 = 10 and r2 = 4) add r3 = 0

• 3 advertisers (V1* = 7, V2* = 6 and V3* = 1)

• VCG prices are• P1 = 40 P2 = 4 Total 44

• GSP prices are• P1 = 60 P2 = 4 Total 64

Why do search engines choose not to use VCG?

• PVCGi = VN-iM – VN-i

M-i

• VN-iM =

• VN-iM-i =

• PVCGi =

1

1

1 1

k

i

k

N

ik

kkk rVrV

1

1 1

i

k

N

ik

kkkk rVrV

M

ikkrrVk 1k

1( )( )

Why do search engines choose not to use VCG?-cont’d

• PVCGi+1 =

• ΔVCG = PVCGi – PVCGi+1 = Vi+1(ri – ri+1)

• ΔGSP = PGSPi – PGSPi+1= Vi+1×ri – Vi+2×ri+1

• ΔGSP – ΔVCG = ri (Vi+1 – Vi+2 ) ≥ 0 an advertiser pays less using VCG a search engine gains less using VCG

M

ikkk rrV 1k

2( )( )

Is there a social optimal equilibrium in GSP?

• Consider the previous example:• 2 slots with r1 = 10 and r2 = 4 r3 = 0 • 3 advertisers with V1* = 7, V2* = 6 and V3* = 1• The market clearing prices per click for s1 , s2 and s3

respectively are (4, 1, 0) • Are there bids that would result in these prices for the

slots? • Yes, a1 bids higher than 4, a2 bids 4, and a3 bids 1 a1 gets s1 and pays 4 per click payoff = 70 – 40 = 30 a2 gets s2 and pays 1 per click payoff = 24 – 4 = 20 a3 gets s3 and pays 0 per click payoff = 0

Is there a social optimal equilibrium in GSP?-cont’d

• a1 lowers its bid to 4 - ε gets s2 payoff = 28 – 4 = 24 < 30 a1 does not have the

incentive to lower its bid • a2 raises its bid to 5 gets s1

payoff = 60 – 40 = 20 a2 does not have the incentive to lower its bid

• This is a Nash equilibrium

Vi* Vi ri Slot payoff

1 7 4+ε 10 1 30

2 6 4 4 2 20

3 1 1 0 3 0

General Argument: Is there a social optimal equilibrium in GSP?

• N advertisers sorted in decreasing order of valuation per click

• M slots + (N-M) slots with 0 click-through rate• sort slots in decreasing order of click-though rates • Consider any set of market clearing prices:• Since for any set of market-clearing prices, a perfect

matching in the resulting preferred seller graph maximizes the total valuation of each valuation for the slot it gets

the advertiser with the highest valuation per click gets the top slot

advertiser i gets slot i

Proof that there is an optimal equilibrium in GSP

• Plan of the proof:1.Construct a set of bids that produces a set of market

clearing prices2.These bids form a Nash equilibrium

1. Construct a set of bids that produces a set of market clearing prices

Consider a set of market clearing prices pj

price per click for slot j : pj* = pj / rj

show that p1* p2* p3* …… pM*

Proof that there is an optimal equilibrium in GSP

Proof:• Consider any arbitrary slots j and k such that j < k

show that pj* pk*• advertiser k prefers slot k to j (by property of market clearing prices)• advertiser k ‘s payoff in slot k is (Vk* – pk*) rk

• advertiser k ’s payoff in slot j is (Vk* – pj*) rj • k’s payoff per click in slot k = Vk* – pk*• k’s payoff per click in slot j = Vk* – pj*• since rj rk but advertiser k prefers slot k to slot j

Vk* – pk* Vk* – pj* pj* pk*• Therefore, if advertiser i bids Vi such that Vi = pi-1* then the prices

for the slots are market clearing prices.

Proof that there is an optimal equilibrium in GSP

2. These bids form a Nash equilibriumShow that with the above bids, no advertiser wants to lower his/her

bid and no advertiser wants to raise his/her price.

Proof:• advertiser j in slot j lowers its bid and gets slot k

where j < k • but since prices are market clearing advertiser j prefers slot j

over any other slot for its current price. • advertiser j in slot j raises its bid to get slot i but p i > pj • in order to get slot i, j has to bid higher than what advertiser i is

paying new pi > current pi

Proof that there is an optimal equilibrium in GSP

• since the current prices are market clearing prices advertiser j doesn’t want slot i at the current price

so j does not want this slot at a higher price this set of bids forms a Nash Equilibrium

• Therefore: GSP has one good Nash Equilibrium in which advertisers get matched to slots in a way that maximizes social welfare i.e. the total valuation of all advertisers for their slots.

Are all GSP Nash Equilibria socially optimal?

Counterexample:• 3 slots with • r1 = 10, r2 = 4, r3 = 0

• if a3 moves, he/she pays 3 which results in a negative payoff

• if a2 moves, he/she pays 3 and gets a payoff 24-3 = 21 less than 30.

• if a1 moves, he/she pays 5 and gets a payoff 70-50= 20 less than 24

a Nash equilibrium

Vi* Vi Slot payoff

1 7 3 s2 24-3=21

2 6 5 s1 60-30=30

3 1 1 s3 0

Are all GSP Nash Equilibria socially optimal?

• Is this Nash market clearing?• No, since a1 would prefer s1 for its current price 30

and get a payoff of 70 – 30 = 40 instead of s2 but if a1 increases its bid then it has to pay 50 and get a payoff of 20 < 24 (its current payment).

Vi* Vi Slot payoff

1 7 3 s2 24-3=21

2 6 5 s1 60-30=30

3 1 1 s3 0

Conclusions

VCG • the worst equilibrium

for the search engines • the best equilibrium

for the advertisers• Incentive compatible

GSP • Not clear if it

maximizes revenue• Has one optimal

equilibrium• truth-telling is

generally not an equilibrium strategy

Further Issues

• A more general model : vij = vi ×αij • αij is the probability that a user will click on this ad if

advertiser i is in the jth slot• What happens when a user specifies a budget and a set of

keywords to bid on?• Which ads to show given a set of keywords and a search query

asking for similar keywords but not the exact ones? If a click occurs, then how much to charge the chosen advertisers who did not bid on these keywords?

• How many ads to show? Does more ads mean more revenue?• Is pay-per-click the best model? How about pay-per-sale or pay-

per-action to stop robot clicks?• Does the identity of the other advertisers in the other slots affect

an advertiser’s click-through-ratio?

References

• David Easley and Jon Kleinberg, Lecture Notes on• Keyword Based Advertising

• Matching Buyers and Sellers

• Chapter 28: Sponsored Search Auctions from Algorithmic Game Theory by Nisan, Roughgarden, Tardos and Vazirani.

• www.google.com/adsense/afs.pdf