Sponsored Search Equilibria for Conservative Bidders Renato Paes - PowerPoint PPT Presentation

Sponsored Search Equilibria for Conservative Bidders Renato Paes Leme Éva Tardos Cornell University

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Auction Model $$$ b 1 $$$ b 3 b 2 $$ b 4 b 5 $$ $ b 6 $

Auction Mechanisms VCG Myerson’s Reserve Mechanism prices - Not truthful - Doesn’t maximize Social GSP Welfare - Simple and Natural - Good balance between revenue and social welfare

Main Result Under some assumptions, any Nash equilibrium in GSP is within a factor of 1.618 to the optimal social welfare. 1 5 Golden ratio: 1 . 618 2 Today we prove a factor of 2.

Model • n advertisers and v 1 n slots α 1 • Each advertiser v 2 α 2 has a value v i v 3 α 3 • Each advertiser v 4 submits a bid b i α 4 • Each slot has a v 5 α 5 click-through- rate α i … … v 1 ≥ v 2 ≥ … ≥ v n α 1 ≥ α 2 ≥ … ≥ α n

Model • Advertisers are b 1 v 1 α 1 ordered by bids and assigned to b 2 v 2 α 2 slots b 3 v 3 α 3 • They are charged b 4 v 4 α 4 the next highest b 5 v 5 bid α 5 … … v 1 ≥ v 2 ≥ … ≥ v n α 1 ≥ α 2 ≥ … ≥ α n

Model • Utility of player i v 1 α 1 when assigned to slot j: v 2 α 2 u i = α j (v i – b π (j+1) ) v 3 α 3 • Allocation π v 4 α 4 π (j) is the bidder v 5 allocated in slot j α 5 … …

Separable Click Through Rates • More general v 1 b 1 γ 1 α 1 model v 2 b 2 γ 2 α 2 • Quality score γ v 3 b 3 γ 3 • Same bounds α 3 • Today: stick with v 4 b 4 γ 4 α 4 simplest model v 5 b 5 γ 5 α 5 … …

Nash Equilibrium • A set of bids (b 1 , …, b n ) and its corresponding v π (i) assignment π is a α i Nash equilibrium if: α i ( v π (i) – b π (i+1) ) ≥ α j ( v π (i) – b π (j) ) j < i α i ( v π (i) – b π (i+1) ) ≥ α j ( v π (i) – b π (j+1) ) j > i

Nash Equilibrium • A set of bids (b 1 , …, b n ) and its corresponding assignment π is a Nash equilibrium if: α i ( v π (i) – b π (i+1) ) ≥ α j ( v π (i) – b π (j) ) j < i α i ( v π (i) – b π (i+1) ) ≥ α j ( v π (i) – b π (j+1) ) j > i • Social Welfare of an assignment: SW = ∑ j α j v π (j)

There are good equilibria … • Theorem [Edelman & Ostrovsky & Schwarz, Varian]: There is always a Nash equilibrium for GSP maximizing social welfare. v 1 α 1 v 2 α 2 v 3 α 3

… and bad equilibria α i b i v i 0 1 1 r 1-r 0 This is a Nash equilibrium with Social Welfare = r. Optimum Social Welfare = 1. Arbitrarily large gap: 1/r  ∞ But this configuration is very unnatural, since the second player is taking a lot of risk.

Conservative Assumption • Assuming bidders are conservative, i.e., no one bids above its valuation: b i ≤ v i we can prove that each Nash is within a factor of 1.618 to the optimal. SW (OPT) Price of anarchy: SW(Nash)

Conservative Assumption • Assuming bidders are conservative, i.e., no one bids above its valuation: b i ≤ v i we can prove that each Nash is within a factor of 1.618 to the optimal. • Related result: [Lahaie] proves a bound on the price of anarchy supposing a good separation of the click-through-rates.

Weakly feasible assignment Lemma: If π is an allocation in a Nash equilibrium under the conservative assumption, then: v π (i) α j + ≥ 1 v π (j) α i v π (j) α i therefore: Weakly feasible assignments v π (i) ≥ 1 α j ≥ 1 v π (i) or α j v π (j) α i 2 2

Weakly feasible assignment Lemma: If π is an allocation in a Nash equilibrium under the conservative assumption, then: v π (i) α j + ≥ 1 v π (j) α i Proof: Need to prove only if i < j and π (i) > π (j). It is a combination of 3 relations: α j ( v π (j) – b π (j+1) ) ≥ α i ( v π (j) – b π (i) ) [ Nash ] b π (i) ≤ v π (i) [conservative] b π (j+1) ≥ 0

Some intuition… v π (i) α j + ≥ 1 v π (j) α i • If values v i are very close then their order doesn’t influence social welfare much • If values v i are well separated, then permutations producing bad social welfare are not weakly feasible More symmetric and easy to use.

Factor of 2 Theorem: Any conservative Nash equilibrium is within a factor of 2 to the optimum. Theorem: Any weakly feasible assignment is within a factor of 2 to the optimum.

Factor of 2 Proof: Induction on the number of slots. 1 By the lemma: 1 α i ≥ 1 v j ≥ 1 or α 1 v 1 2 2 i In the first case, remove bidder 1 and j slot i and apply inductive hypothesis … …

Factor of 2 Proof: Applying the induction hypothesis:. ∑ k≠i α k v π (k) ≥ ½ ( α 2 v 1 + … + α i v i-1 + α i+1 v i+1 + … + α n v n ) ≥ ½ ( α 2 v 2 + … + α i v i + α i +1 v i+1 + … + α n v n ) ∑ k α k v π (k) = α i v 1 + ∑ k≠j α k v π (k) ≥ ½ α 1 v 1 + ½ ∑ k>1 α k v k Using the Lemma in its full potential gives us the 1.618 bound.

What else can we do: • Bound of 1.618 • Same bounds for separable click- through-rates: quality score • Similar bounds for γ -conservative bidders: γ b i ≤ v i