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Unformatted text preview: Efficiency and Revenue in Certain Nash Equilibria of Keyword Auctions S´ ebastien Lahaie [email protected] Yahoo Research New York, NY 10018 SISHOO 2007 – p.1 Sponsored Search SISHOO 2007 – p.2 Outline • Model for keyword auctions. • Efficiency in purestrategy Nash equilibrium. • Necessary conditions for equilibrium. • Worstcase bound on efficiency. • Revenue in symmetric equilibrium. • General case. • Restricted family of weights. • Efficiency and relevance considerations. SISHOO 2007 – p.3 Outline • Model for keyword auctions. • Efficiency in purestrategy Nash equilibrium. • Necessary conditions for equilibrium. • Worstcase bound on efficiency. • Revenue in symmetric equilibrium. • General case. • Restricted family of weights. • Efficiency and relevance considerations. SISHOO 2007 – p.4 Model • K positions, N bidders. • The clickthrough rate of bidder s in positions t is e s x t , i.e. separable into 1. advertiser effect (or relevance ) e s 2. position effect x t ( x 1 > x 2 > . . . > x K ) . • Bidder s has perclick value of v s . • If bidder s obtains slot t at price of p per click, utility is e s x t ( v s p ) , i.e. quasilinear. SISHOO 2007 – p.5 Auction Rules • The auctioneer assigns a weight w s to each bidder s [Aggarwal et al. ’06]. • Bidders submit bids (reported values) b s . • Bidders are ranked in order of decreasing score w s b s . • Bidder s pays per click the lowest bid necessary to maintain its position: w s b s ≥ w s +1 b s +1 ⇒ b s ≥ w s +1 w s b s +1 • “Yahoo model”: w s = 1 . • “Google model”: w s = e s . SISHOO 2007 – p.6 Efficient Ranking • A bidder’s true score is r s = w s v s . • An allocation of slots to bidders σ : K → N maximizes the objective summationdisplay t x t w σ ( t ) v σ ( t ) if and only if bidders are ranked in decreasing order of true score. • Follows easily from the fact that x 1 > x 2 . . . > x K . • If w s = e s , the objective is the total value . SISHOO 2007 – p.7 Outline • Model for keyword auctions. • Efficiency in purestrategy Nash equilibrium. • Necessary conditions for equilibrium....
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This document was uploaded on 01/05/2012.
 Fall '09

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