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