Tim Roughgarden Stanford 2 Motivation Optimal auction design whats the point One primary reason suggests auction formats likely to perform well in practice Exhibit A singleitem ID: 1002264
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1. Near-Optimal Simple and Prior-Independent AuctionsTim Roughgarden (Stanford)
2. 2MotivationOptimal auction design: what's the point?One primary reason: suggests auction formats likely to perform well in practice.Exhibit A: single-item Vickrey auction.maximizes welfare (ex post) [Vickrey 61]with suitable reserve price, maximizes expected revenue with i.i.d. bidder valuations [Myerson 81]
3. 3The Dark SideIssue: in more complex settings, optimal auction can say little about how to really solve problem.Example: single-item auction, independent but non-identical bidders. To maximize revenue:winner = use highest "virtual bid"charge winner its "threshold bid”“complex”: may award good to non-highest bidder (even if multiple bidders clear their reserves)
4. 4Alternative ApproachStandard Approach: solve for optimal auction over huge set, hope optimal solution is reasonableAlternative: optimize only over "plausibly implementable" auctions.Sanity Check: want performance of optimal restricted auction close to that of optimal (unrestricted) auction.if so, have theoretically justified and potentially practically useful solution
5. 5Talk OutlineReserve-price-based auctions have near-optimal revenue [Hartline/Roughgarden EC 09]i.e., auctions can be approximately optimal without being complexPrior-independent auctions [Dhangwotnatai/Roughgarden/Yan EC 10], [Roughgarden/Talgam-Cohen/Yan EC 12]i.e., auctions can be approximately optimal without a priori knowledge of valuation distribution
6. Simple versus Optimal Auctions(Hartline/Roughgarden EC 2009)
7. 7Optimal AuctionsTheorem [Myerson 81]: solves for optimal auction in “single-parameter” contexts.independent but non-identical biddersknown distributions (will relax this later)But: optimal auctions are complex, and very sensitive to bidders’ distributions.Research agenda: approximately optimal auctions that are simple, and have little or no dependence on distributions.
8. 8Example SettingsExample #1: flexible (OR) bidders.bidder i has private value vi for receiving any good in a known set SiExample #2: single-minded (AND) bidders.bidder i has private value vi for receiving every good in a known set Si
9. 9Reserve-Based AuctionsProtagonists: “simple reserve-based auctions”:remove bidders who don’t clear their reservemaximize welfare amongst those left charge suitable prices (max of reserve and the price arising from competition) Question: is there a simple auction that's almost as good as Myerson's optimal auction?
10. 10Reserve-Based AuctionsRecall: “simple reserve-based” auction:remove bidders who don’t clear their reservemaximize welfare amongst those left charge suitable prices (max of reserve and the price arising from competition)Theorem(s): [Hartline/Roughgarden EC 09]: simple reserve-based auctions achieve a 2-approximation of expected revenue of Myerson’s optimal auction.under mild assumptions on distributions; better bounds hold under stronger assumptionsMoral: simple auction formats usually good enough.
11. 11A Simple LemmaLemma: Let F be an MHR distribution with monopoly price r (so ϕ(r) = 0). For every v ≥ r: r + ϕ(v) ≥ v.Proof: We have r + ϕ(v) = r + v - 1/h(v) [defn of ϕ] ≥ r + v - 1/h(r) [MHR, v ≥ r] = v. [ϕ(r) = 0]
12. 12An Open QuestionSetup: single-item auction.n bidders, independent non-identical known distributionsassume distributions are “regular”protagonists: Vickrey auction with some anonymous reserve (i.e., an eBay auction)Question: what fraction of optimal (Myerson) expected revenue can you get?correct answer somewhere between 25% and 50%
13. 13More On Simple vs. OptimalSequential Posted Pricing: [Chawla/Hartline/Malec/Sivan STOC 10], [Bhattacharya/Goel/Gollapudi/Munagala STOC 10], [Chakraborty/Even-Dar/Guha/Mansour/Muthukrishnan WINE 10], [Yan SODA 11], …Item Pricing: [Chawla//Malec/Sivan EC 10], …Marginal Revenue Maximization: [Alaei/Fu/Haghpanah/Hartline/Malekian 12]Approximate Virtual Welfare Maximization: [Cai/Daskalakis/Weinberg SODA 13]
14. Prior-Independent Auctions(Dhangwotnatai/Roughgarden/Yan EC 10; Roughgarden/Talgam-Cohen/Yan EC 12)
15. 15Prior-Independent AuctionsGoal: prior-independent auction = almost as good as if underlying distribution known up frontno matter what the distribution isshould be simultaneously near-optimal for Gaussian, exponential, power-law, etc.distribution used only in analysis of the auction, not in its designRelated: “detail-free auctions”/”Wilson’s critique”
16. 16Bulow-Klemperer ('96)Setup: single-item auction. Let F be a known valuation distribution. [Needs to be "regular".]Theorem: [Bulow-Klemperer 96]: for every n: Vickrey's revenue ≥ OPT's revenue [with (n+1) i.i.d. bidders] [with n i.i.d. bidders]Interpretation: small increase in competition more important than running optimal auction.
17. 17Bulow-Klemperer ('96)Theorem: [Bulow-Klemperer 96]: for every n: Vickrey's revenue ≥ OPT's revenue [with (n+1) i.i.d. bidders] [with n i.i.d. bidders]Consequence: [taking n = 1] For a single bidder, a random reserve price is at least half as good as an optimal (monopoly) reserve price.
18. 18Prior-Independent AuctionsGoal: prior-independent auction = almost as good as if underlying distribution known up frontTheorem: [Dhangwatnotai/Roughgarden/Yan EC 10] there are simple such auctions with good approximation factors for many problems.ingredient #1: near-optimal auctions only need to know suitable reserve prices [Hartline/Roughgarden 09]ingredient #2: bid from a random player good enough proxy for an optimal reserve price [Bulow/Klemperer 96]Moral: good revenue possible even in “thin” markets.
19. 19The Single Sample Mechanismpick a reserve bidder ir uniformly at randomrun the VCG mechanism on the non-reserve bidders, let T = winnersfinal winners are bidders i such that:i belongs to T; ANDi's valuation ≥ ir's valuation
20. 20Main ResultTheorem 1: [Dhangwotnotai/Roughgarden/Yan EC 10] the expected revenue of the Single Sample mechanism is at least:a ≈ 25% fraction of optimal for arbitrary downward-closed settings + MHR distributionsMHR: f(x)/(1-F(x)) is nondecreasinga ≈ 50% fraction of optimal for matroid settings + regular distributionsmatroids = generalization of flexible (OR) bidders
21. 21Beyond a Single SampleTheorem 2: [Dhangwotnotai/Roughgarden/Yan EC 10] the expected revenue of Many Samples is at least:a 1-ε fraction of optimal for matroid settings + regular distributionsa (1/e)-ε fraction of optimal welfare for arbitrary downward-closed settings + MHR distributions provided n ≥ poly(1/ε).key point: sample complexity bound is distribution-independent (requires regularity)
22. 22Supply-Limiting MechanismsIdea: let the “right” prices emerge endogenously from competition (e.g., using the VCG mechanism).Problem: what goods are not scarce?e.g., unlimited supply --- VCG nets zero revenue
23. 23Supply-Limiting MechanismsIdea: let the “right” prices emerge endogenously from competition (e.g., using the VCG mechanism).Problem: what goods are not scarce?e.g., unlimited supply --- VCG nets zero revenueSolution: artificially limit supply.Main Result: [Roughgarden/Talgam-Cohen/Yan EC 12] VCG with suitable supply limit O(1)-approximates optimal revenue for many problems (even multi-parameter).
24. 24Supply-Limiting MechanismsIdea: let the “right” prices emerge endogenously from competition (e.g., using the VCG mechanism).Solution: artificially limit supply.Main Result: [Roughgarden/Talgam-Cohen/Yan EC 12] VCG with suitable supply limit O(1)-approximates optimal revenue for many problems (even multi-parameter).Related: [Devanur/Hartline/Karlin/Nguyen WINE 11]
25. 25Example: Unlimited SupplySimple Special Case: VCG with supply limit n/2 is 2-approximation for unlimited supply settings (unit-demand bidders).n bidders, valuations i.i.d. from regular distribution
26. 26Example: Unlimited SupplySimple Special Case: VCG with supply limit n/2 is 2-approximation for unlimited supply settings (unit-demand bidders).n bidders, valuations i.i.d. from regular distributionProof: VCG (n bidders, n/2 goods) ≥ OPT(n/2 bidders, n/2 goods) by Bulow- Klemperer
27. 27Example: Unlimited SupplySimple Special Case: VCG with supply limit n/2 is 2-approximation for unlimited supply settings (unit-demand bidders).n bidders, valuations i.i.d. from regular distributionProof: VCG (n bidders, n/2 goods) ≥ OPT(n/2 bidders, n/2 goods) ≥ ½ OPT(n bidders, n goods)by Bulow- Klempererobvious here,true more generally
28. 28Example: Multi-Item AuctionsHarder Special Case: VCG with supply limit n/2 is 4-approximation with n heterogeneous goods.n bidders, valuations from regular distributionindependent across bidders and goodsidentical across bidders (but not over goods)Proof: boils down to a new BK theorem: expected revenue of VCG with supply limit n/2 at least 50% of OPT with n/2 bidders.
29. 29Open Questionsbetter approximations, more problems, risk averse bidders, etc.lower bounds for prior-independent auctionseven restricting to the single-sample paradigmwhat’s the optimal way to use a single sample?do prior-independent auctions imply Bulow-Klemperer-type-results?other interpolations between average-case and worst-case (e.g., [Azar/Daskalakis/Micali SODA 13])
30. 30Bulow-Klemperer ('96)Setup: single-item auction. Let F be a known valuation distribution. [Needs to be "regular".]Theorem: [Bulow-Klemperer 96]: for every n: Vickrey's revenue ≥ OPT's revenue [with (n+1) i.i.d. bidders] [with n i.i.d. bidders]Interpretation: small increase in competition more important than running optimal auction.a "bicriteria bound"!
31. 31Reformulation of BK TheoremIntuition: if true for n=1, then true for all n. recall OPT = Vickrey with monopoly reserve r*follows from [Myerson 81]relevance of reserve price decreases with nReformulation for n=1 case: 2 x Vickrey's revenue Vickrey's revenue with n=1 and random ≥ with n=1 and opt reserve [drawn from F] reserve r*
32. 32Proof of BK Theoremselling probability qexpected revenue R(q)01
33. 33Proof of BK Theoremselling probability qexpected revenue R(q)concave if and only ifF is regular01
34. 34Proof of BK Theoremopt revenue = R(q*)selling probability qexpected revenue R(q)01q*
35. 35Proof of BK Theoremopt revenue = R(q*)selling probability qexpected revenue R(q)01q*
36. 36Proof of BK Theoremopt revenue = R(q*)revenue of random reserve r (from F) = expected value of R(q) for q uniform in [0,1] = area under revenue curveselling probability qexpected revenue R(q)01
37. 37Proof of BK Theoremopt revenue = R(q*)revenue of random reserve r (from F) = expected value of R(q) for q uniform in [0,1] = area under revenue curveselling probability qexpected revenue R(q)01
38. 38Proof of BK Theoremopt revenue = R(q*)revenue of random reserve r (from F) = expected value of R(q) for q uniform in [0,1] = area under revenue curveselling probability qexpected revenue R(q)concave if and only ifF is regular01q*
39. 39Proof of BK Theoremopt revenue = R(q*)revenue of random reserve r (from F) = expected value of R(q) for q uniform in [0,1] = area under revenue curve ≥ ½ ◦ R(q*)selling probability qexpected revenue R(q)concave if and only ifF is regular01q*