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 1. Reserve-price-based auctions have near- optimal revenue [Hartline/Roughgarden EC 09]  i.e., auctions can be approximately optimal without being complex 2. 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

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 v i for receiving any good in a known set S i Example #2: single-minded (AND) bidders. bidder i has private value v i for receiving every good in a known set S i

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]

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 1. pick a reserve bidder i r 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. i belongs to T; AND 2. i's valuation ≥ i r '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 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 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 selling probability q expected revenue R(q) 0 1

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

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

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

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

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

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

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