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Near-Optimal Simple and Prior-Independent
AuctionsTim 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 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
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. 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)
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 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
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 distributionsprovided n ≥ poly(1/ε).
key point: sample complexity bound is distribution-independent (requires regularity)
22
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
23
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).
24
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]
25
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
26
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
27
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
28
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.
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 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*
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 ifF is regular
0 1
34
Proof of BK Theorem
opt revenue = R(q*)
selling probability q
expected revenue
R(q)
0 1
q*
35
Proof of BK Theorem
opt revenue = R(q*)
selling probability q
expected revenue
R(q)
0 1
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 ifF is regular
0 1
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 ifF is regular
0 1
q*