Position Auctions with Bidder- Specific Minimum Prices Eyal Even-DarGoogle Jon Feldman Google Yishay Mansour Tel-Aviv Univ., Google S. Muthukrishnan Google

Sponsored Search Monetization of search results Search engine needs to balance – advertiser efficiency – user experience – revenue

First Model Advertisers submit per-click bids b(i) Position effect q(1) > q(2) > … > q(k). Separability: Pr[click(i,j)] = p(i) q(j) Rank ads by b(i) p(i) – Under separability, maximizes efficiency GSP: charge “next price” b(i+1) p(i+1)/p(i) VCG: charge effect on others’ efficiency

Envy-free equilibria GSP not truthful, but admits an efficient “envy-free” equilibrium with same outcome as VCG. [EOS, V, AGM] “Envy-free” equilibrium = Every bidder would rather have her own (position, price) than any other available. – Stronger than Nash Equilibria in GSP

Reserve Prices Pays for SE’s loss in quality Boosts revenue (in undersold auctions) In many cases, reserve prices should be bidder-specific: – Both Google and Yahoo use them – AdWords FAQ: “Minimum prices are based on the quality and relevance of the keyword, its ad, and associated landing page.”

Our work How do bidder-specific reserve prices affect GSP? - GSP equilibria no longer efficient - “Envy locality” no longer holds - Despite this, GSP with bidder-specific reserve prices still has an envy-free equilibrium. [Main result]

Warm-up: VCG How do bidder-specific reserve prices affect the equilibria of VCG? – Naïve application of reserve prices breaks truthfulness: BidderValueReserve price A$1.500 B$1.25$1.00 C$0.500 D$0.250 VCG price/click for bidder B: Position 1: 1.5/2 +.5/4 +.25/4 = 15/16 Position 2: (.5/4 +.25/4) / (1/2) = 3/8 Position effects: q(1)=1, q(2)=1/2, q(3)=1/4

Fixes to VCG with reserve prices “Virtual values”: – For price(i), use max{b(j),R(i)} instead of b(j). – Efficient, truthful, not envy-free. “Minimum price offset” – Subtract R(i) from bids: b’(i) = b(i) – R(i), then run VCG. – Truthful (easy), efficient in v’(i) = v(i) – R(i)

GSP with bidder-spec. reserves Bad news: Not necessarily efficient BidderValueReserve price A$1.000 B$0.68$0.67 Position effects: q(1) = 1, q(2) = 1/2 Bidder A: Profit at 1 st pos = 1 ($ b(B)) < $0.33 Profit at 2 nd pos = ½ ($1.00 – 0) = $0.50 Bidder A will underbid bidder B in any equilibrium.

GSP with bidder-spec. reserves Bad news: no envy locality – simple example in paper: locally high reserve prices, bargain at the bottom. …thus, global argument is needed to show envy-freeness

GSP with bidder-spec. reserves Good news: Theorem: The GSP auction with arbitrary bidder-specific minimum prices admits an envy- free equilibrium

GSP with bidder-spec. reserves Proof setup: – Slot prices define bipartite “best response” graph modeling envy – Matching in graph that hits all slots implies equilibrium assignment Tâtonnement process to raise prices: – Maintain matching on slot prefix (Hall’s thm) – Grow prefix by increasing prices – Prove if not all slots in matching, can proceed

Revenue Theorem: Let P vcg (j) = price at pos. j under VCG without reserve prices; Let P res (j) = envy-free price at pos. j under GSP with reserve prices; Then, assuming all bidders have v(i) > R(i), we have P res (j) ≥ P vcg (j)

Conclusions Bidder-specific reserve prices are important tools used by search engines. In VCG, naïve application can break truthfulness, but there are fixes In GSP, reserve prices can hurt efficiency, only help revenue, complicate bidder dynamics, but equilibrium still exists.

Future work Relationship of VCG variants, GSP equilibrium? Equilibrium discovery? Position-specific reserve prices? – [Gonen, Vassilvitskii, tomorrow] Minimum quality score (ctr)?