An Optimal Mechanism for Sponsored Search Auction Dinesh Garg Computer Science & Automation, Indian Institute of Science, Bangalore, INDIA 1 e - Enterprises Lab, CSA, IISc
Introduction Problem Definition 2 e - Enterprises Lab, CSA, IISc
Introduction Problem Definition: Sequence of Queries User 1 Google User 2 User N Q1 Q2 Q1 Q3 Q2 Q1 Q2 Q3 3 e - Enterprises Lab, CSA, IISc
Introduction Problem Definition: Bids, Valuations, and Click Probabilities b 1 Q 1 Search Results Sponsored Links b 1 i 1 2 2 2 i 2 3 b m n im n Advertisers CPC 4 e - Enterprises Lab, CSA, IISc
Introduction Problem Definition: Bids, Valuations, and Click Probabilities b b , , b Bid vector of advertiser s 1 n (1) ( n ) b , , b Decreasing ordering of the bids Value derived out of a click by advertiser i i Type of advertiser i Set of types of advertiser i i ( , , ) Type vector of advertiser s n 1 th th Click probabilit y of Ad in position i j ij (AAE Assumption) 1 i N 0 i i im 1 2 5 e - Enterprises Lab, CSA, IISc
Introduction Problem Definition: Search Engine’s Problem Allocation Rule 1 Who should be allocated what ? 2 if advertiser i is allocated slot j 1 ( ) y ij b o/w 0 m Payment Rule ( m ) b Which advertiser should be charged what price ? ( 1 ) b p i ( b ) Price that is charged from advertiser i Google ( 2 ) b for per click 6 e - Enterprises Lab, CSA, IISc
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 J. Feng , “Optimal Mechanism for selling a set of Commonly Ranked Objects” , Mimeo , February 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 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 H. R. Varaian , “ Position Auctions ” , Mimeo , February 2006 7 e - Enterprises Lab, CSA, IISc
Outline 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 8 e - Enterprises Lab, CSA, IISc
Generalized First Price (GFP) y ( b ) 1 11 b 2 1 y ( b ) 0 12 Q p ( b ) 2 1 Search Results Sponsored Links ( ) y b 0 b . 1 5 21 1 2 ( ) y b 1 22 p ( b ) . 1 5 2 2 y ( b ) 0 b 1 31 3 y ( b ) 0 32 p ( b ) 0 3 ( , . , ) b 2 1 5 1 9 e - Enterprises Lab, CSA, IISc
Generalized First Price (GFP) Allocation Rule Allocate the slots in decreasing order of bids ( ) j if b b and j min( m , n ) 1 i y ( b ) ij o/w 0 Payment Rule For every user click, charge the advertiser his bid b if advertiser i ' s Ad is displayed i ( ) p b i o/w 0 Introduced by Overture in 1997 10 e - Enterprises Lab, CSA, IISc
Generalized Second Price (GSP) Allocation Rule Yahoo Rule ( m ) b 1 Allocate the slots in decreasing order of bids Greedy Rule 2 ( 1 ) b Allocate 1 st slot to advertiser i arg max b 1 i 1 i i N ( 2 ) b Allocate 2 nd slot to advertiser arg max m i b i i 2 2 i N \ i 1 Google Rule Allocate the slots in decreasing order of Ranking Score b CTR Ranking Score = i i Introduced by Google in 2002 (Above facts are based on literatur e) 11 e - Enterprises Lab, CSA, IISc
Generalized Second Price (GSP) Payment Rule 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 ( m ) b ( 1 ) b Google ( 2 ) b 12 e - Enterprises Lab, CSA, IISc
Generalized Second Price (GSP) y ( b ) 1 11 b 2 y ( b ) 0 1 12 Q p ( b ) . 1 5 1 Search Results Sponsored Links y ( b ) 0 . b 1 5 21 1 2 y ( b ) 1 22 p ( b ) 1 2 2 y ( b ) 0 b 1 31 3 ( ) y b 0 32 ( ) p b 0 3 ( , . , ) b 2 1 5 1 13 e - Enterprises Lab, CSA, IISc
Generalized Second Price (GSP) Allocation Rule Greedy b CTR 11 m 1 1 1 b CTR n nm n n 1 Yahoo Google m m CTR CTR y i ij i ij ij j 1 j 1 14 e - Enterprises Lab, CSA, IISc
Generalized Second Price (GSP) Learning CTR and Click Probabilities Average over Fixed Time Window C C ij i CTR ; i ij I I X X X X i ij T Average over Fixed Impression Window C C ij i ; CTR X X X i ij X 1000 1000 I 1000 i Average over Fixed Click Window 100 100 ; CTR i ij X X X I I X i ij C 100 i 15 e - Enterprises Lab, CSA, IISc
Generalized Second Price (GSP) Relationship Among Allocation Rules (AE) Problem Proposition Max Let click probabilities satisfy AAE assumption n m ( ) b y b Greedy allocation rule is an optimal solution i ij ij i j 1 1 of the (AE) Problem n b v y ( b ) If click probabilities depend only on identity of i i the advertiser and are independent of the i 1 CTR position of the Ad, i.e. s.t. i 1 i 2 im i then greedy rule and Google rule result in the n same allocation y ( b ) j M 1 ij i 1 If click probabilities depend only on position m of the Ad and are independent of the identity y ( b ) i N 1 ij of the advertiser, i.e. 1 j 2 j nj j j 1 then greedy rule and Yahoo! rule result in the , y i N j M 0 same allocation ij 16 e - Enterprises Lab, CSA, IISc
Vickrey-Clarke-Groves (VCG) ( m ) b Allocation Rule 1 Solution of (AE) Problem 2 ( 1 ) Same as Yahoo! allocation under the assumption b that click probability depends only on position ( 2 ) b m Payment Rule * * ( ) ( ( )) ( ( )) t b b v y b b v y b i j j i j j ( m ) b j i j i ( j ) ( ) t b ( 1 ) b ( j ) ( ) p b Google j ( 2 ) b 17 e - Enterprises Lab, CSA, IISc
Vickrey-Clarke-Groves (VCG) Payment Rule ( ) m n Case 1 m 1 1 ( k ) ( m ) 1 m 1 if ( ) b b j m 1 1 k k j j j ( j ) ( m ) 1 p ( b ) b if j m if m j n 0 ( n m ) Case 2 n 1 1 ( k ) 1 if ( ) b j n 1 1 ( j ) ( ) k p b k j j if j n 0 ( ) where k k k 1 18 e - Enterprises Lab, CSA, IISc
Vickrey-Clarke-Groves (VCG) ( ) y b 1 11 b . 2 0 ( ) y b 1 0 12 Q ( ) . 2 p b 1 5 1 1 3 Search Results Sponsored Links 1 ( ) y b 0 21 b . 1 5 1 2 ( ) y b 1 22 p ( b ) 1 2 2 y ( b ) 0 . 31 b 1 0 3 y ( b ) 0 32 p ( b ) 0 3 19 e - Enterprises Lab, CSA, IISc
Outline 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 20 e - Enterprises Lab, CSA, IISc
Optimal (OPT) ( m ) J Allocation Rule 1 : if 0 j n J b 0 1 ( 1 ) i i 2 J ( ) j : if y b j m J b J 1 1 ij i i ( j ) m j n : if J b J 0 ( 2 ) J i i m ( b ) 1 th where is the highest value among i i ( j ) j J ( b ) b J i i i ( b ) i i (Assumption: ( ) is non decreasing: True for Uniform, Exponential) J i b i Proposition Advertisers are symmetric , i.e. For a given bid vector b, the OPT results in the same allocation as n 1 2 (.) (.) (.) the GSP and the VCG, i.e. allocate 1 2 n in decreasing order of bids (.) , , J i i n 0 1 21 e - Enterprises Lab, CSA, IISc
Optimal (OPT) where Payment Rule r is the position at ( m n ) Case 1 which advertiser i is allocated m 1 1 m z ( b ) z ( b ) if r ( m ) 1 1 k ik i im i k r r r ( , ) ( ) if r p b b z b m ( ) i i i im i k k k 1 o/w 0 z ij b ( ) is the i ( ) n m Case 2 minimum bid for the advertiser i which can make n 1 1 n z ( b ) z ( b ) if r ( n ) 1 1 th j him win slot k ik i in i k r r r against the bid ( , ) ( ) if r p b b z b n i i i in i vector from b o/w i 0 other advertisers 22 e - Enterprises Lab, CSA, IISc
Optimal (OPT) Payment Rule when Advertisers are Symmetric [ , ] L U n 1 2 (.) (.) (.) 1 2 n ( ) m n Case 1 m 1 1 ( ) ( ) k 1 m 1 m b b if j ( m ) 1 1 k k r r r ( m ) 1 ( , ) if p b b b j m i i i if m j n 0 ( ) n m Case 2 n 1 1 ( ) k 1 n if ( ) b L j n 1 1 p ( b , b ) k i i i k r r r L if j n 23 e - Enterprises Lab, CSA, IISc
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