An Optimal Mechanism for Sponsored Search Auction Dinesh Garg - - PowerPoint PPT Presentation

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An Optimal Mechanism for Sponsored Search Auction

Dinesh Garg

Computer Science & Automation, Indian Institute of Science, Bangalore, INDIA

e -Enterprises Lab, CSA, IISc

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Introduction

Problem Definition

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User 1 User 2 User N

Google

Q1 Q2 Q1 Q3 Q2 Q1 Q2 Q3

Introduction

Problem Definition: Sequence of Queries

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Introduction

Problem Definition: Bids, Valuations, and Click Probabilities

Advertisers CPC

Q Sponsored Links Search Results 1 2 3 m

1

b

2

b

n

b

1 2 n 1 i 2 i im

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Introduction

1 position in Ad

• f

y probabilit Click s advertiser

• f

vector Type ) , , ( advertiser

• f

types

• f

Set advertiser

• f

Type advertiser by click a

• f
• ut

derived Value bids the

• f
• rdering

Decreasing , , s advertiser

• f

vector Bid , ,

) ( (1)

N i j i i i i b b b b b

im i i th th ij n i i n n 2 1 1 1

(AAE Assumption)

Problem Definition: Bids, Valuations, and Click Probabilities

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Allocation Rule Payment Rule

Who should be allocated what ? Which advertiser should be charged what price ?

1 2 m

Google

) (1

b

) (2

b

) (m

b

Introduction

Problem Definition: Search Engine’s Problem

• /w

slot allocated is advertiser if ) ( j i b yij 1 click per for advertiser from charged is that Price ) ( i b pi

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Introduction

Recent Literature

• B. Edelman, M. Ostrovsky, and M. Schwarz, “Internet Advertising and the

Generalized Second Price Auction: Selling Billions of Dollars Worth of Keywords”, Mimeo, September, 2005

• S. Lahaie, “An Analysis of Alternative Slot Auction Designs for Sponsored

Search”, ACM Conference on Electronic Commerce (EC’06), Ann Arbor, MI, June 11-15, 2006

• H. R. Varaian, “Position Auctions”, Mimeo, February 2006
• J. Feng, “Optimal Mechanism for selling a set of Commonly Ranked Objects”,

Mimeo, February 2005

• G. Aggarwal, A. Goel, and R. Motwani, “Truthful Auction for Pricing Search

Keywords”, ACM Conference on Electronic Commerce (EC’06), Ann Arbor, MI, June 11-15, 2006

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8 Introduction Problem Definition Significance Recent Literature Three well known mechanisms Generalized First Price (GFP) Generalized Second Price (GSP) Vickrey-Clarke-Groves (VCG) A new mechanism – Optimal (OPT) Mechanism What is the best mechanism for Sponsored Search Auction? Comparison of OPT with GSP and VCG Incentive Compatibility Expected Revenue of the Search Engine Individual Rationality Computational Complexity

Outline

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Search Results Sponsored Links Q 1 2

2

1

b 5 1

2

. b 1

3

b

2 1

1 12 11

) ( ) ( ) ( b p b y b y 5 1 1

2 22 21

. ) ( ) ( ) ( b p b y b y

3 32 31

) ( ) ( ) ( b p b y b y

Generalized First Price (GFP)

) , . , ( 1 5 1 2 b

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Generalized First Price (GFP)

• /w

) , min( and if ) (

) (

1 n m j b b b y

j i ij

Allocation Rule Payment Rule

• /w

displayed is Ad s ' advertiser if ) ( i b b p

i i

Introduced by Overture in 1997 Allocate the slots in decreasing order of bids For every user click, charge the advertiser his bid

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Allocation Rule

Allocate the slots in decreasing order of bids

1 2 m

) (1

b

) (2

b

) (m

b

Yahoo Rule

Greedy Rule

Allocate 1st slot to advertiser

Google Rule

Allocate the slots in decreasing order of Ranking Score Ranking Score =

i i N i

b i

1 1

max arg

i i

CTR b

Allocate 2nd slot to advertiser

i i i N i

b i

2 2

1

\

max arg

Generalized Second Price (GSP)

Introduced by Google in 2002 (Above facts are based on literature)

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Payment Rule

Google

) (1

b

) (2

b

) (m

b

For every click, charge next highest bid + $0.01 The bottom most advertiser is charged highest disqualified bid +$0.01 charge 0 if no such bid

Generalized Second Price (GSP)

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Search Results Sponsored Links Q 1 2

2

1

b 5 1

2

. b 1

3

b

5 1 1

1 12 11

. ) ( ) ( ) ( b p b y b y 1 1

2 22 21

) ( ) ( ) ( b p b y b y

3 32 31

) ( ) ( ) ( b p b y b y

Generalized Second Price (GSP)

) , . , ( 1 5 1 2 b

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Greedy

n n nm n m

CTR CTR b b

1 1 1 1 11

Generalized Second Price (GSP)

Allocation Rule

m j ij ij i

y CTR

1 m j ij i

CTR

1

Yahoo Google

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Learning CTR and Click Probabilities

Average over Fixed Time Window

X X X X T

ij ij ij i i i

I C I C CTR ;

Average over Fixed Impression Window

1000 1000

ij ij i i

C C CTR ;

X X X X

1000

i

I

X X X X

100

i

C Average over Fixed Click Window

ij ij i i

I I CTR 100 100 ;

Generalized Second Price (GSP)

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Generalized Second Price (GSP)

Relationship Among Allocation Rules

M j N i y N i b y M j b y b y v b b y b

ij m j ij n i ij n i i i i m j ij ij i

, ) ( ) ( s.t. ) ( ) ( Max

n

1 1

1 1 1 1 1

Proposition

Greedy allocation rule is an optimal solution

• f the (AE) Problem

(AE) Problem

If click probabilities depend only on identity of the advertiser and are independent of the position of the Ad, i.e. then greedy rule and Google rule result in the same allocation

Let click probabilities satisfy AAE assumption

i im i i

CTR

2 1

If click probabilities depend only on position

• f the Ad and are independent of the identity
• f the advertiser, i.e.

then greedy rule and Yahoo! rule result in the same allocation

j nj j j 2 1

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Allocation Rule

Google

) (1

b

) (2

b

) (m

b

Payment Rule

Vickrey-Clarke-Groves (VCG)

Solution of (AE) Problem Same as Yahoo! allocation under the assumption that click probability depends only on position 1 2 m

) (1

b

) (2

b

) (m

b

i j j j i j i j j i

b y v b b y v b b t )) ( ( )) ( ( ) (

* *

j j j

b t b p ) ( ) (

) ( ) (

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n j m m j b m j b b b p

m m j m m j k k k j j

if if ) ( if ) (

) ( ) ( ) ( ) (

1 1 1

1 1 1 1

Vickrey-Clarke-Groves (VCG)

Payment Rule

where

) (

1 k k k

Case 1

) ( n m

n j n j b b p

n j k k k j j

if ) ( if ) (

) ( ) (

1 1 1

1 1

Case 2

) ( m n

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Search Results Sponsored Links Q 1 2

2

1

. b 5 1

2

. b

1

3

. b

1 2 1 12 11

3 1 5 1 1 . ) ( ) ( ) ( b p b y b y 1 1

2 22 21

) ( ) ( ) ( b p b y b y

3 32 31

) ( ) ( ) ( b p b y b y

Vickrey-Clarke-Groves (VCG)

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20 Introduction Problem Definition Significance Related Literature Three well known mechanisms Generalized First Price (GFP) Generalized Second Price (GSP) Vickrey-Clarke-Groves (VCG) A new mechanism – Optimal (OPT) What is the best mechanism for Sponsored Search Auction? Comparison of OPT with GSP and VCG Incentive Compatibility Expected Revenue of the Search Engine Individual Rationality Computational Complexity

Outline

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Allocation Rule

) ( ) ( i

if : if : if :

j i i j i i i ij

J b J n j m J b J m j b J n j b y 1 1 1

Advertisers are symmetric, i.e.

(.) (.) (.)

n n 2 1 2 1

where is the highest value among

) ( j

J

th

j

) ( ) ( ) (

i i i i i i i

b b b b J 1 (Assumption: is non decreasing: True for Uniform, Exponential)

) (

i i b

J

Optimal (OPT)

1 2 m

) (1

J

) (2

J

) (m

J

Proposition

n i Ji , , (.) 1

For a given bid vector b, the OPT results in the same allocation as the GSP and the VCG, i.e. allocate in decreasing order of bids

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is the position at which advertiser is allocated is the minimum bid for the advertiser which can make him win slot against the bid vector from

• ther advertisers
• /w

r if ) ( ) ( if ) ( ) ( ) , ( 1 1 1

1

m b z m r b z b z b b p

i im i im r m m r k i ik k r i i i

Optimal (OPT)

Payment Rule

where ) (

1 k k k

) (

i ij b

z

th

j

i

i

b

r i

Case 1

) ( n m

• /w

r if ) ( ) ( if ) ( ) ( ) , ( 1 1 1

1

n b z n r b z b z b b p

i in i in r n n r k i ik k r i i i

Case 2

) ( m n

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Optimal (OPT)

Payment Rule when Advertisers are Symmetric

Case 1

) ( n m

Case 2

) ( m n

n j m m j b m j b b b b p

m m r m m r k k k r i i i

if if ) ( if ) , (

) ( ) ( ) (

1 1 1

1 1 1 1

n j L n j L b b b p

r n n r k k k r i i i

if ) ( if ) , (

) (

1 1 1

1 1

(.) (.) (.) ] , [

n n

U L

2 1 2 1

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Optimal (OPT)

Proposition

Advertisers are symmetric, i.e. m < n

(.) (.) (.) ] , [

n n

U L

2 1 2 1

n i Ji , , (.) 1

Payment Rule OPT VCG Advertisers are symmetric, i.e. m = n

(.) (.) (.)

n n 2 1 2 1

n i Ji , , (.) 1

Payment Rule OPT VCG (up to a constant factor L)

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Example: OPT

Search Results Sponsored Links Q 1 2

2

1

. 5 1

2

.

1

3

.

1 2 1 11 1 1 1 1

3 1 5 1 1 2 1 1 1 2 2 1 1 2 1 . ) ( ; ) ( ) ( ) ( ); ( ) ( ] , [ b p b y J x x x

1 1 1 1 5 1 5 1 5 1 1 1 2 1

2 22 2 2 2 2

) ( ; ) ( . . ) . ( ) ( ); ( ) ( ; ] , [ b p b y J x x x

1 1 1 1 1 1 2 1

3 3 3 3 3 3

) ( ; ) ( ) ( ); ( ) ( ; ] , [ b p y J x x x

j

e -Enterprises Lab, CSA, IISc

26 Introduction Problem Definition Significance Related Literature Three well known mechanisms Generalized First Price (GFP) Generalized Second Price (GSP) Vickrey-Clarke-Groves (VCG) A new mechanism – Optimal (OPT) What is the best mechanism for Sponsored Search Auction? Comparison of OPT with GSP and VCG Incentive Compatibility Expected Revenue of the Search Engine Individual Rationality Computational Complexity

Outline

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What is the best Mechanism for Sponsored Search Auction?

Search Engine’s View Points

Economic and Computational Performance measures The advertisers’ equilibrium bidding strategy profile Effect of on performance measures

(.) , (.),

* * n

s s1 (.) , (.),

* * n

s s1

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28 Revenue Maximization Individual Rationality (IR) Incentive Compatibility (IC) Computational Complexity

Economic and Computational Performance Measures

What is the best Mechanism for Sponsored Search Auction?

e -Enterprises Lab, CSA, IISc

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1

2 n

) ( ) ( ) ( )) ( ( ), (

m 1 j

b p b y b p b y v b f u

i i ij j i i i i i 2

n

1

b

2

b

n

b

M j N i i ij

b p b y b f

,

) ( ), ( ) (

(Allocation Rule, Payment Rule)

1

i i i

B s :

, i

i i i

B

Sponsored Search Auction as a Mechanism Design Problem

What is the best Mechanism for Sponsored Search Auction?

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30 If all the advertisers are rational and intelligent and this fact is common knowledge then each advertiser’s expected bidding behavior is given by

Bayesian Nash Equilibrium (BNE)

i i i i i i i i i i i i i i

b s b f u E s s f u E

i i

i * * *

| ) )), ( , ( ( | ) )), ( ), ( ( ( Strategy profile is said to be Bayesian Nash equilibrium iff

(.) , (.),

* * n

s s1

Dominant Strategy Equilibrium (DSE)

Strategy profile is said to be dominant Strategy equilibrium iff (.) , (.),

* * n

s s1

i i i i i i i i i i i

b b b b f u b s f u

i

• i

*

, ) )), , ( ( ) )), ), ( ( (

Strategic Bidding Behavior of Advertisers

What is the best Mechanism for Sponsored Search Auction?

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31 Introduction Problem Definition Significance Related Literature Three well known mechanisms Generalized First Price (GFP) Generalized Second Price (GSP) Vickrey-Clarke-Groves (VCG) A new mechanism – Optimal (OPT) What is the best mechanism for Sponsored Search Auction? Comparison of OPT with GSP and VCG Incentive Compatibility Expected Revenue of the Search Engine Individual Rationality Computational Complexity

Outline

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Comparison of OPT with GSP and VCG

• VCG:

Follow irrespective of what the others are doing (DSE)

i i i

s ) (

*

• OPT:

Follow if all rivals are also doing so (BNE)

i i i

s ) (

*

• GSP:

Never follow strategy . Use the following BNE strategy

i i i

s ) (

*

i l i l

n m dx x s m x f m g m n dx x s m x f m g s

i i i i i i i i

if : ) ( ' ) , , ( ) , ( if : ) ( ' )) ( , , ( )) ( , ( ) (

*

1 1 1 1

k j j n i j i j n j i

C j k x f

1 2 1 1

1 )) ( ( )) ( ( ) ( ) , , (

1 1 1 1 1 1 1 1 1 k j j n i j i j n j j k n i k i k n k i

C j C k k g )) ( ( )) ( ( ) ( )) ( ( )) ( ( ) , (

Incentive Compatibility

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Comparison of OPT with GSP and VCG

Expected Revenue Earned by the Search Engine

Revenue Equivalence Theorem:

Consider a sponsored search auction setting, in which

• 1. The advertisers are risk neutral
• 2. The advertisers are symmetric
• 3. For each advertiser , we have
• 4. The advertisers draw their types independently

Consider two different mechanisms, each having symmetric and increasing Bayesian Nash equilibrium such that

• 1. For each possible the final allocation is the same
• 2. Each advertiser has same expected utility in two mechanisms for

then equilibria of two mechanisms generate the same expected revenue for the search engine

(.)

i

i

n

, ,

1

i L

i

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Comparison of OPT with GSP and VCG

Expected Revenue Earned by the Search Engine

Revenue Equivalence of GSP, VCG, and OPT Mechanisms

Consider a sponsored search auction setting, in which

• 1. The advertisers are risk neutral
• 2. The advertisers are symmetric
• 3. For each advertiser , we have
• 4. The advertisers draw their types independently
• 5. For each advertiser , we have and is non-decreasing

Consider three different auction mechanisms – GSP, VCG, and OPT. Let and be the expected revenue earned by the search engine under these three mechanisms against every query received, then

(.)

i

i

VCG GSP

R R ,

i

m n R R R n m R R R

OPT GSP VCG OPT VCG GSP

if if

OPT

R

(.)

i

J (.)

i

J

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U L m j j n j j n j n OPT

dx x x x x C j L n R ) ( )) ( ( )) ( (

1 1 1 1 U L m j j n j j n j m n m m n m OPT

dx x x x x C j x x C m n R ) ( )) ( ( )) ( ( )) ( ( )) ( (

1 1 1 1 1 1

Expected Revenue of Search Engine

Comparison of OPT with GSP and VCG

Case 1

) ( n m

Case 2

) ( m n

U L m j j n j j n j VCG

dx x x x x C j n R ) ( )) ( ( )) ( (

1 1 1 1

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X

Generalized VCG Decreasing

• rder of the

bids

OPT

Marginal Contribution Decreasing

• rder of the

bids

VCG

X X

Next Highest bid Decreasing

• rder of the

bids

GSP IR BIC DSIC Payment Allocation

Economic Performance of Auction Mechanisms

Comparison of OPT with GSP and VCG

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Allocative Efficient Individually Rational Dominant Strategy Incentive Compatible Bayesian Incentive Compatible

GFP GSP VCG OPT

Comparison of OPT with GSP and VCG

Economic Performance of Auction Mechanisms

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Comparison of OPT with GSP and VCG

Experimental Results

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Comparison of OPT with GSP and VCG

Experimental Results

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Comparison of OPT with GSP and VCG

Experimental Results

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Comparison of OPT with GSP and VCG

Experimental Results

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OPT VCG GSP Computational Complexity

n n O log

2

) , min( log n m n n O

2

) , min( log n m n n O

Comparison of OPT with GSP and VCG

Computational Performance of Auction Mechanisms

e -Enterprises Lab, CSA, IISc

43 Introduction Problem Definition Significance Related Literature Three well known mechanisms Generalized First Price (GFP) Generalized Second Price (GSP) Vickrey-Clarke-Groves (VCG) A new mechanism – Optimal (OPT) What is the best mechanism for Sponsored Search Auction? Comparison of OPT with GSP and VCG Incentive Compatibility Expected Revenue of Search Engine Individual Rationality Computational Complexity

Outline

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44 Long Term Goals versus Short Term Goals Daily Budget Learning the Valuation Distribution Assumption of Independence of Click Probability on Advertisers’ Identity Revenue Performance under Asymmetric Advertisers Click Fraud Competing Search Engines Optimal Bidding Strategy of the Advertisers

(.)

i

Future Directions

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Q ue s tio ns and Ans we rs … Thank Yo u …

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