Near-Optimal Simple and Prior-Independent

AuctionsTim Roughgarden (Stanford)

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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]

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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)

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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

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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)

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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.

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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

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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?

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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.

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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]

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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%

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More 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]

Prior-Independent Auctions

(Dhangwotnatai/Roughgarden/Yan EC 10; Roughgarden/Talgam-Cohen/Yan EC 12)

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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”

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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.

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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.

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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.

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The Single Sample Mechanism1. 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. i belongs to T; AND2. i's valuation ≥ ir's valuation

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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

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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 distributionsprovided n ≥ poly(1/ε).

key point: sample complexity bound is distribution-independent (requires regularity)

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Supply-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

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Supply-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

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).

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Supply-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]

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Example: 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

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Example: 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

Proof: VCG (n bidders, n/2 goods) ≥ OPT(n/2 bidders, n/2 goods)

by Bulow- Klemperer

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Example: 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

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

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Example: Multi-Item AuctionsHarder 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.

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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])

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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"!

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Reformulation 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

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*

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Proof of BK Theorem

selling probability q

expected revenue

R(q)

0 1

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Proof of BK Theorem

selling probability q

expected revenue

R(q)

concave if and only ifF is regular

0 1

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Proof of BK Theorem

opt revenue = R(q*)

selling probability q

expected revenue

R(q)

0 1

q*

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Proof of BK Theorem

opt revenue = R(q*)

selling probability q

expected revenue

R(q)

0 1

q*

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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

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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

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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 ifF is regular

0 1

q*

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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 ifF is regular

0 1

q*