Pure and Bayes -Nash Price of Anarchy for GSP

1. Pure and Bayes-Nash Price of Anarchy for GSP RenatoPaesLemeÉvaTardos Cornell Cornell & MSR

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3. Keyword Auctions

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

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

7. Auction Model b1 $$$ $$$ b3 b2 $$ b4 b5 $$ $ b6 $

8. Auction Model b1 $$$ Vickrey Auction VCG Auction $$$ b3 b2 • Truthful • Efficient • Simple (?) • … $$ Generalized Second Price Auction • Not Truthful • Not Efficient • Even Simpler • … b4 b5 $$ $ b6 $

9. Our Results • Generalized Second Price Auction (GSP), although not optimal, has good social welfare guarantees: • 1.618 for Pure Price of Anarchy • 8 for Bayes-Nash Price of Anarchy • GSP with uncertainty

10. (Simplified) Model • αj : click-rate of slot j • vi : value of player i • bi : bid (declared value) • Assumption: bi≤ vi Since playing bi> vi is dominated strategy. b1 α1 v1 α2 b2 v2 α3 v3 b3

11. (Simplified) Model pays b1 per click v1 b1 α1 b2 v2 α2 α3 v3 b3

12. (Simplified) Model σ = π-1 bi αj vi i = π(j) j = σ(i) Utility of player i : ui(b) = ασ(i) ( vi - bπ(σ(i) + 1))

13. Model σ = π-1 bi αj vi i = π(j) j = σ(i) Utility of player i : ui(b) = ασ(i) ( vi - bπ(σ(i) + 1)) next highest bid

14. Model σ = π-1 bi αj vi i = π(j) j = σ(i) Nash equilibrium: ui(bi,b-i) ≥ ui(b’i,b-i) Is truth-telling always Nash ?

15. Example Non-truthful u1 = 1 (2-1) u1 = 0.9(2-0) v1 = 2 b1 = 2 b1 = 0.9 α1 = 1 v2 = 1 b1 = 1 α2 = 0.9

16. Measuring inefficiency σ = π-1 bi αj vi i = π(j) j = σ(i) Social welfare = ∑iviασ(i) Optimal allocation = ∑iviαi

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

18. Main Theorem 1 • Thm: Pure Price of Anarchy ≤ 1.618 If bi≤ vi and (b1…bn) are bid in equilibrium, then for the allocation σ : ∑iviασ(i) ≥ 1.618-1∑iviαi • Previously known [EOS, Varian]: Price of Stability = 1

19. GSP as a Bayesian Game Modeling uncertainty:

20. GSP as a Bayesian Game b b ?

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

22. Bayes-Nash solution concept • Bayes-Nash models the uncertainty of other players about valuations • Values vi are independent random vars • Optimize against a distribution • Thm:Bayes-NashPoA≤ 8

23. Bayesian Model v1 ~ V1 b1(v1) α1 b2(v2) v2 ~ V2 α2 α3 v3 ~ b3(v3) V3

24. Model bi(vi) αj vi ~ Vi i = π(j) j = σ(i) Bayes-Nash equilibrium: E[ui(bi,b-i)|vi] ≥ E[ui(b’i,b-i)|vi] Expectation over v-i

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

26. Sketch of the proof v1 α1 αi α2 vπ(i) ασ(j) v2 vj + ≥ 1 vj αi α3 ασ(j) therefore: vπ(i) v3 vπ(i) ασ(j) 1 1 Simple and intuitive condition on matchings in equilibrium. ≥ ≥ or vj αi 2 2 Opt

27. Sketch of the proof αi vj ασ(j) vπ(i) vπ(i) ασ(j) + ≥ 1 vj αi Need to show only for i < j and π(i) > π(j). It is a combination of 3 relations: ασ(j) ( vj – bπ(σ(j)+1) ) ≥αi ( vj – bπ(i) ) [ Nash ] bπ(i) ≤ vπ(i) [conservative] bπ(σ(j)+1) ≥0

28. Sketch of the proof vπ(i) ασ(j) + 2 SW = ∑iασ(i) vi + αivπ(i) = ≥ 1 vj αi vπ(i) ασ(i) = ∑iαivi ≥ + vi αi ≥ ∑iαivi = Opt

29. Proof idea:new structural condition Pure Nash version: αi vj vi ασ(i) +αivπ(i) ≥αivi ασ(j) vπ(i) Bayes Nash version: viE[ασ(i)|vi] + E[αμ(i) vπμ (i)|vi] ≥ ¼ viE[αμ(i)|vi]

30. Upcoming results[Lucier-PaesLeme-Tardos] • Improved Bayes-Nash PoA to 3.164 • Valid also for correlated distributions • Future directions: • Tight Pure PoA (we think it is 1.259)