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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. maximizes welfare (ex post) [Vickrey 61] with suitable reserve price, maximizes expected revenue with i.i.d. bidder valuations [Myerson 81] 2 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) 3 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 4 Talk Outline Reserve-price-based auctions have nearoptimal revenue [Hartline/Roughgarden EC 09] 1. i.e., auctions can be approximately optimal without being complex Prior-independent auctions [Dhangwotnatai/Roughgarden/Yan EC 10], [Roughgarden/Talgam-Cohen/Yan EC 12] 2. i.e., auctions can be approximately optimal without a priori knowledge of valuation distribution 5 Simple versus Optimal Auctions (Hartline/Roughgarden EC 2009) 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. 7 Example Settings Example #1: flexible (OR) bidders. bidder i has private value vi for receiving any good 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 8 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? 9 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. 10 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) ≥ r + v - 1/h(r) = v. [defn of ϕ] [MHR, v ≥ r] [ϕ(r) = 0] 11 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% 12 More On Simple vs. Optimal Sequential Posted Pricing: [Chawla/Hartline/Malec/Sivan STOC 10], [Bhattacharya/Goel/Gollapudi/Munagala STOC 10], [Chakraborty/EvenDar/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] 13 Prior-Independent Auctions (Dhangwotnatai/Roughgarden/Yan EC 10; Roughgarden/Talgam-Cohen/Yan EC 12) 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” 15 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 [with (n+1) i.i.d. bidders] ≥ OPT's revenue [with n i.i.d. bidders] Interpretation: small increase in competition more important than running optimal auction. 16 Bulow-Klemperer ('96) Theorem: [Bulow-Klemperer 96]: for every n: Vickrey's revenue [with (n+1) i.i.d. bidders] ≥ OPT's revenue [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. 17 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. 18 The Single Sample Mechanism 1. pick a reserve bidder ir uniformly at random 2. run the VCG mechanism on the non-reserve bidders, let T = winners 3. final winners are bidders i such that: 1. 2. i belongs to T; AND i's valuation ≥ ir's valuation 19 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 20 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 optimal welfare for arbitrary downward-closed settings + MHR distributions provided n ≥ poly(1/ε). key point: sample complexity bound is distribution-independent (requires regularity) 21 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 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 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). 23 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] 24 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 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 Proof: VCG (n bidders, n/2 goods) ≥ OPT(n/2 bidders, n/2 goods) by BulowKlemperer 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 BulowKlemperer ≥ ½ OPT(n bidders, n goods) obvious here, true more generally 27 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. 28 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 BulowKlemperer-type-results? other interpolations between average-case and worst-case (e.g., [Azar/Daskalakis/Micali SODA 13]) 29 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 [with (n+1) i.i.d. bidders] ≥ OPT's revenue [with n i.i.d. bidders] Interpretation: small increase in competition more important than running optimal auction. a "bicriteria bound"! 30 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 with n=1 and random ≥ reserve [drawn from F] Vickrey's revenue with n=1 and opt reserve r* 31 Proof of BK Theorem expected revenue R(q) 0 selling probability q 1 32 Proof of BK Theorem concave if and only if F is regular expected revenue R(q) 0 selling probability q 1 33 Proof of BK Theorem expected revenue R(q) q* 0 selling probability q 1 opt revenue = R(q*) 34 Proof of BK Theorem expected revenue R(q) q* 0 selling probability q 1 opt revenue = R(q*) 35 Proof of BK Theorem expected revenue R(q) 0 selling probability q 1 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 36 Proof of BK Theorem expected revenue R(q) 0 selling probability q 1 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 37 Proof of BK Theorem concave if and only if F is regular expected revenue R(q) q* 0 selling probability q 1 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 38 Proof of BK Theorem concave if and only if F is regular expected revenue R(q) q* 0 selling probability q 1 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*) 39