Pure and Bayes-Nash Price of Anarchy for GSP Renato Paes Leme Éva Tardos CornellCornell & MSR

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Selling one Ad Slot $2 $5 $7 $3 Prospective advertisers

Selling one Ad Slot $2 $5 $7 $3 Pays $5 per click Vickrey Auction -Truthful - Efficient - Simple - …

Auction Model b1b1 b1b1 b2b2 b2b2 b3b3 b3b3 b4b4 b4b4 b5b5 b5b5 b6b6 b6b6 $$$ $$ $ $

Auction Model b1b1 b1b1 b2b2 b2b2 b3b3 b3b3 b4b4 b4b4 b5b5 b5b5 b6b6 b6b6 $$$ $$ $ $ Vickrey Auction VCG Auction Generalized Second Price Auction -Truthful - Efficient - Simple (?) - … - Not Truthful - Not Efficient - Even Simpler - …

Our Results Generalized Second Price Auction (GSP), although not optimal, has good social welfare guarantees: for Pure Price of Anarchy 8 for Bayes-Nash Price of Anarchy GSP with uncertainty

(Simplified) Model α j : click-rate of slot j v i : value of player i b i : bid (declared value) Assumption: b i ≤ v i Since playing b i > v i is dominated strategy. α1α1 α2α2 α3α3 b1b1 b1b1 b2b2 b2b2 b3b3 b3b3 v1v1 v2v2 v3v3

(Simplified) Model v1v1 v2v2 v3v3 α1α1 α2α2 α3α3 b1b1 b1b1 b2b2 b2b2 b3b3 b3b3 pays b 1 per click

(Simplified) Model vivi αjαj bibi bibi j = σ(i)i = π(j) u i (b) = α σ(i) ( v i - b π(σ(i) + 1) ) Utility of player i : σ = π -1

Model vivi αjαj bibi bibi j = σ(i)i = π(j) σ = π -1 next highest bid u i (b) = α σ(i) ( v i - b π(σ(i) + 1) ) Utility of player i :

Model vivi αjαj bibi bibi j = σ(i)i = π(j) Nash equilibrium: σ = π -1 u i (b i,b -i ) ≥ u i (b’ i,b -i ) Is truth-telling always Nash ?

Example Non-truthful α 1 = 1 α 2 = 0.9 b 1 = 2 v 1 = 2 v 2 = 1 b 1 = 1 b 1 = 0.9 u 1 = 1 (2-1)u 1 = 0.9(2-0)

Measuring inefficiency vivi αjαj bibi bibi j = σ(i)i = π(j) σ = π -1 Social welfare = ∑ i v i α i ∑ i v i α σ(i) Optimal allocation =

Measuring inefficiency Price of Anarchy = max = Opt SW(Nash) Nash

Main Theorem 1 Thm: Pure Price of Anarchy ≤ If b i ≤ v i and (b 1 …b n ) are bid in equilibrium, then for the allocation σ : ∑ i v i α σ(i) ≥ ∑ i v i α i Previously known [EOS, Varian]: Price of Stability = 1

GSP as a Bayesian Game Modeling uncertainty:

GSP as a Bayesian Game b b b ?

GSP as a Bayesian Game b b b b b b b b b b b b b b b b b b b b b b b b b b Idea: Optimize against a distribution.

Bayes-Nash solution concept Thm: Bayes-Nash PoA ≤ 8 Bayes-Nash models the uncertainty of other players about valuations Values v i are independent random vars Optimize against a distribution

Bayesian Model V1V1 V2V2 V3V3 v 1 ~ v 2 ~ v 3 ~ α1α1 α2α2 α3α3 b 1 (v 1 ) b 2 (v 2 ) b 3 (v 3 )

Model ViVi v i ~ αjαj b i (v i ) j = σ(i)i = π(j) E[u i (b i,b -i )|v i ] ≥ E[u i (b’ i,b -i )|v i ] Bayes-Nash equilibrium: Expectation over v -i

v i are random variables μ(i) = slot that player i occupies in Opt (also a random variable) Bayes-Nash PoA = Bayes-Nash Equilibrium E[ ∑ i v i α μ(i) ] E[ ∑ i v i α σ(i) ] Previously known [G-S]: Price of Stability ≠ 1

Sketch of the proof α2α2 α3α3 Opt α1α1 v1v1 v2v2 v3v3 αiαi α σ(j) vjvj v π(i) therefore: α σ(j) αiαi vjvj v π(i) ≥ 1 2 ≥ 1 2 or Simple and intuitive condition on matchings in equilibrium. α σ(j) αiαi vjvj v π(i) + ≥ 1

αiαi α σ(j) vjvj v π(i) α σ(j) αiαi vjvj v π(i) + ≥ 1 Need to show only for i π(j). It is a combination of 3 relations: α σ(j) ( v j – b π(σ(j)+1) ) ≥ α i ( v j – b π(i) ) [ Nash ] b π(σ(j)+1) ≥ 0 b π(i) ≤ v π(i) [conservative] Sketch of the proof

α σ(j) αiαi vjvj v π(i) + ≥ 1 2 SW = ∑ i α σ(i) v i + α i v π(i) = v π(i) = ∑ i α i v i ≥ α σ(i) αiαi vivi + ≥ ∑ i α i v i = Opt

Proof idea: new structural condition v i α σ(i) + α i v π(i) ≥ α i v i v i E[ α σ(i) |v i ] + E[ α μ( i ) v πμ (i) |v i ] ≥ ¼ v i E[ α μ( i ) |v i ] Bayes Nash version: Pure Nash version: αiαi α σ(j) vjvj v π(i)

Upcoming results [Lucier-Paes Leme-Tardos] Improved Bayes-Nash PoA to Valid also for correlated distributions Future directions: Tight Pure PoA (we think it is 1.259)