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Near - Optimal Simple and Prior-Independent Auctions

Near - Optimal Simple and Prior-Independent Auctions. Tim Roughgarden (Stanford). Motivation. Optimal auction design: what's the point? One primary reason: suggests auction formats likely to perform well in practice. Exhibit A: single-item Vickrey auction.

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Near - Optimal Simple and Prior-Independent Auctions

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  1. Near-Optimal Simple and Prior-Independent Auctions Tim Roughgarden(Stanford)

  2. Motivation Optimal 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. The Dark Side Issue: 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. Alternative Approach Standard Approach: solve for optimal auction over huge set, hope optimal solution is reasonable Alternative: 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. Talk Outline • Reserve-price-based auctions have near-optimal revenue [Hartline/Roughgarden EC 09] • i.e., auctions can be approximately optimal without being complex • Prior-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. Optimal Auctions • Theorem [Myerson 81]: solves for optimal auction in “single-parameter” contexts. • independent but non-identical bidders • known 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. Example Settings Example #1:flexible (OR) bidders. • bidder i has private value vi for receiving anygood in a known set Si Example #2: single-minded (AND) bidders. • bidder i has private value vi for receiving every good in a known set Si

  9. Reserve-Based Auctions • Protagonists: “simple reserve-based auctions”: • remove bidders who don’t clear their reserve • maximize 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. Reserve-Based Auctions • Recall: “simple reserve-based” auction: • remove bidders who don’t clear their reserve • maximize 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 assumptions • Moral: simple auction formats usually good enough.

  11. A Simple Lemma Lemma: 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. An Open Question Setup: single-item auction. • n bidders, independent non-identical known distributions • assume 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. More On Simple vs. Optimal Sequential 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. Prior-Independent Auctions • Goal:prior-independent auction = almost as good as if underlying distribution known up front • no matter what the distribution is • should be simultaneously near-optimal for Gaussian, exponential, power-law, etc. • distribution used only in analysis of the auction, not in its design • Related: “detail-free auctions”/”Wilson’s critique”

  16. Bulow-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. Bulow-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. Prior-Independent Auctions • Goal:prior-independent auction = almost as good as if underlying distribution known up front • Theorem: [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. The Single Sample Mechanism • pick a reserve bidderiruniformly at random • run the VCG mechanism on the non-reserve bidders, let T = winners • final winnersare bidders i such that: • i belongs to T; AND • i's valuation ≥ ir's valuation

  20. Main Result Theorem 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 distributions • MHR: f(x)/(1-F(x)) is nondecreasing • a ≈ 50% fraction of optimal for matroid settings + regular distributions • matroids = generalization of flexible (OR) bidders

  21. Beyond a Single Sample Theorem 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 distributions • a (1/e)-ε fraction of optimalwelfare for arbitrary downward-closed settings + MHR distributions provided n ≥ poly(1/ε). • key point: sample complexity bound is distribution-independent (requires regularity)

  22. Supply-Limiting Mechanisms Idea: 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. Supply-Limiting Mechanisms Idea: 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 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).

  24. Supply-Limiting Mechanisms Idea: 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. Example: Unlimited Supply Simple 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. Example: Unlimited Supply Simple 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 Proof: VCG (n bidders, n/2 goods) ≥ OPT(n/2 bidders, n/2 goods) by Bulow- Klemperer

  27. Example: Unlimited Supply Simple 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 Proof: VCG (n bidders, n/2 goods) ≥ OPT(n/2 bidders, n/2 goods) ≥ ½ OPT(n bidders, n goods) by Bulow- Klemperer obvious here, true more generally

  28. Example: Multi-Item Auctions Harder Special Case: VCG with supply limit n/2 is 4-approximation with n heterogeneous goods. • n bidders, valuations from regular distribution • independent across bidders and goods • identical 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. Open Questions • better approximations, more problems, risk averse bidders, etc. • lower bounds for prior-independent auctions • even restricting to the single-sample paradigm • what’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. Bulow-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. Reformulation of BK Theorem Intuition: 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 n Reformulation 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. Proof of BK Theorem expected revenue R(q) 0 1 selling probability q

  33. Proof of BK Theorem concave if and only if F is regular expected revenue R(q) 0 1 selling probability q

  34. Proof of BK Theorem expected revenue R(q) • opt revenue = R(q*) q* 0 1 selling probability q

  35. Proof of BK Theorem expected revenue R(q) • opt revenue = R(q*) q* 0 1 selling probability q

  36. Proof of BK Theorem expected revenue R(q) • opt 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 0 1 selling probability q

  37. Proof of BK Theorem expected revenue R(q) • opt 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 0 1 selling probability q

  38. Proof of BK Theorem concave if and only if F is regular expected revenue R(q) • opt 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 q* 0 1 selling probability q

  39. Proof of BK Theorem concave if and only if F is regular expected revenue R(q) • opt 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*) q* 0 1 selling probability q

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