Computation and Incentives in
Combinatorial Public Projects
Michael Schapira
Yale University and UC Berkeley
Joint work with Dave Buchfuhrer
and Yaron Singer
Take Home Messages
• Combinatorial Public Projects are cool!
• More suitable arena for exploring
truthful computation?
• Should we rethink AMD solution
concept?
Designing Algorithms for
Environments With Selfish Agents
incentivecompatibility
computational
efficiency
When can these coexist? [Nisan-Ronen]
Paradigmatic Problem:
Combinatorial Auctions
• A set of m items on sale {1,…m}.
• n bidders {1,…,n}. Each bidder i has
valuation function vi : 2[m] → R≥0.
– normalized, non-decreasing.
• Goal: find a partition of the items
between the bidders S1,…,Sn such that
the social welfare Si vi(Si) is maximized
What Do We Want?
• Quality of the solution: As close to the
optimum as possible.
• Computationally tractable: Polynomial running
time (in n and m).
• Truthful: Motivate (via payments) bidders to
report their true values.
– The utility of each agent is ui = vi(S) – pi
– Solution concepts: dominant strategies, expost Nash.
State of the Art
“It is probably fair to summarize that
most computational issues have been resolved,
while most strategic questions have remained
open… despite much work and some mild
progress…
The basic question of how well can
computationally-efficient incentive-compatible
combinatorial auctions … perform remains as open
as in the beginning of the decade, and gets my
(biased) AGT open problem of the decade
award.”
Noam Nisan
Why is This Happening?
• We do not understand truthfulness.
– Roberts’ Theorem...
• Combinatorial auctions are complex
– Too much noise… (combinatorics)
• Other approach: find “minimal”
environments where computation and
incentives clash.
– and then go back to combinatorial auctions.
Combinatorial Public Projects Problem
(CPPP) [Papadimitriou-S-Singer]
• Set of n agents; Set of m resources;
• Each agent i has a valuation function:
vi : 2[m] → R≥0
– normalized, non-decreasing.
• Goal: Given a parameter k, choose a set
of resources S* of size k which
maximizes the social welfare:
S* = argmaxi Si vi(S)
S  [m], |S|=k
Complement-Free Hierarchy
[Lehmann-Lehmann-Nisan]
Complement-Free
(Subadditive)
FractionallySubadditive
Submodular
(“XOS”)
Gross
Substitute
Multi-UnitDemand
(“OXS”)
Unit-Demand
(“XS”)
Capped Additive
(“Budget-Additive”)
Coverage
Questions:
1. Where does CPPP cease to
be tractable? (VCG!)
2. Where does CPPP cease to
be approximable?
Complement-Free Hierarchy:
Tractability
Complement-Free
(Subadditive)
FractionallySubadditive
Submodular
(“XOS”)
Gross
Substitute
Multi-UnitDemand
(“OXS”)
Unit-Demand
(“XS”)
Capped Additive
(“Budget-Additive”)
Coverage
even for n=1
combinatorial
auctions
CPPP
Complement-Free Hierarchy:
Approximability
combinatorial
auctions
Complement-Free
(Subadditive)
FractionallySubadditive
CPPP
Submodular
(“XOS”)
Gross
Substitute
Multi-UnitDemand
(“OXS”)
Unit-Demand
(“XS”)
Capped Additive
(“Budget-Additive”)
Coverage
Complement-Free Hierarchy:
Area of Interest
Submodular
Gross
Substitute
Multi-UnitDemand
(“OXS”)
Unit-Demand
(“XS”)
Capped Additive
(“Budget-Additive”)
Coverage
even for n=1
Two Simple Environments
• CPPP with unit-demand agents
– Each agent only wants one resource!
• CPPP with one coverage valuation
2-{0,1}-Unit-Demand
user
resources
0
1
0
0
1
Each user only wants (value 1) at most two
resources and does not want (value 0) all others.
2-{0,1}-Unit-Demand
• Combinatorial auctions with such
valuations are trivial.
– matching
• CPPP with such valuations is NP-hard.
– Vertex Cover
– But approximable
– (Solvable for constant n’s)
• The perfect starting point.
– What about truthful computation?
Maximal-In-Range Mechanisms
(= VCG-Based)
Definition:
A is MIR if there is some
RA  {|S | = k| S  [m]}
s.t. A(v1,…vn) = argmax
*
We shall refer to RA as A’s range.
RA
S in RA
v1(S)+…+vn(S)
all sets of resources
of size k
2-{0,1}-Unit-Demand
• Thm [S-Singer]: There exists a
computationally-efficient MIR
mechanism for CPPP with complementfree valuations with appx ratio 1/√m.
• Thm: No computationally-efficient
MIR mechanism for CPPP with 2-{0,1}unit-demand valuations has appx ratio
better than 1/√m
– unless SAT is in P/poly.
2-{0,1}-Unit-Demand
• What about general truthful
mechanisms?
• Thm: There exists a computationallyefficient MIR mechanism for CPPP
with 2-{0,1}-unit-demand valuations
that has appx ratio ½.
– Simply choose the k most demanded resources.
Open Question
• What about truthful mechanisms
for CPPP with unit-demand
valuations?
– Characterization?
• No techniques?
– VCG
– random sampling
– LP
Truthfulness With One Player?
I want
this!
What do
Take it!
you want?
mechanism
player
The interests of the player and mechanism
are aligned (value = social welfare)
CPPP With a Coverage Valuation
• Defn: A valuation v is a coverage
valuation if there is
– a universe U
– m subsets of U, T1,…,Tm
– and b>0
such that for every set of resources S:
v(S) = b|Uj in S Tj|
CPPP With a Coverage Valuation
• Computational Perspective:
A 1-1/e approximation ratio is achievable (not
truthful!)
 A tight lower bound exists [Feige].
• Strategic Perspective:
A truthful solution is trivially achievable via
VCG payments (but NP-hard to obtain)
• What about achieving both simultaneously?
Hardness of Truthfulness With
One Player?
• Thm: No computationally-efficient
and truthful mechanism for CPPP with
one coverage valuation has appx ratio
better than 1/√m
– Unless SAT is in P/poly.
– Tight
• Strengthens and simplifies a recent
result in [Papadimitriou-S-Singer]
– For n=2
– For submodular valuations.
3 Challenges
mechanism
design
(characterization of
truthful mechanisms)
Complexity
theory
(hardness of
truthful mechanisms)
combinatorics
(structure of
truthful mechanisms)
The Proof: Overview
• Characterization Lemma (informal):
Every truthful mechanism for CPPP with
one coverage valuation is MIR.
– True for all one-player mechanism design environments
• Inapproximability Lemma: No
computationally-efficient MIR
mechanism for CPPP with one coverage
valuation has appx ratio better than
1/√m
– unless SAT is in P/poly.
Proof of Inapproximability
Lemma (sketch)
• If a computationally-efficient MIR
mechanism A has appx ratio better than
1/√m then |RA| ≥ 2am (for some constant
a>0).
– probabilistic construction.
• So, a MIR mechanism A that has appx
ratio better than 1/√m optimizes over
exponentially many outcomes.
Computational Hardness
– CPPP with one coverage valuation is NP-hard.
– So, optimizing over the set of all possible
outcomes is hard.
– What about optimizing over a set of
outcomes of exponential size?
Intuition: also hard!
All sets of resources
of size k
RA
The VC Dimension
collection
of subsets
R
1
x
3
x
5
1
2
3
4
5
1
2
x
x
5
x
x
x
4
x
shattered set
universe
1
2
3
4
5
Lower Bounding the VC Dimension
• The Sauer-Shelah Lemma: Let R be a
collection of subsets of a universe U. Then,
there exists a subset E of U such that:
– E is shattered by R.
– |E| ≥ W( log(|R|)/log(|U|) ).
• RA is a collection of subsets of the
universe of resources.
The Reduction
• We know that |RA| ≥ 2am (for some
constant a>0).
• Hence, there is a set of resources of size
mb (for some constant b>0) that is
shattered by RA.
RA
• We can now show that the MIR mechanism
A solves exactly a smaller (but not too
small!) CPPP with one coverage valuation!
Truthfulness With One Player?
I don’t
know!
What do
you want?
mechanism
player
Somewhat Strange…
Do we need to rethink the framework?
Positive Results for CPPP
Submodular
Gross
Substitute
Multi-UnitDemand
(“OXS”)
Unit-Demand
(“XS”)
Capped Additive
(“Budget-Additive”)
FPTAS for constant n’s
optimal algorithm
for n=2
Coverage
Take Home Messages
• Combinatorial Public Projects are cool!
• More suitable arena for exploring
truthful computation?
• Should we rethink AMD solution
concept?
Back to Combinatorial Auctions…
[Mossel-Papadimitriou-S-Singer]
• A set of m items on sale {1,…m}.
• n bidders {1,…,n}. Each bidder i has
valuation function vi : 2[m] → R≥0.
– normalized, non-decreasing.
• Goal: find a partition of the items
between the bidders S1,…,Sn such that
social welfare Si vi(Si) is maximized
What About Combinatorial
Auctions?
mechanism
design
Truthful=MIR
consider
only MIR
Complexity
theory
embedding hard
problems in partial
ranges
combinatorics
generalize the VC
dimension
to handle
(VC dimension)
partitions of a
universe.
The Case of 2 Bidders
• Not trivial even for n=2!
• The trivial MIR mechanism: allocate the
bundle of all items to the highest bidder.
– ½ appx. ratio.
• Is this the best we can do (with MIR)?
– Yes! [Buchfuhrer et al.]
– extends to general n’s.
Intuition
MIR algorithm A
RA
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
5 items
2 bidders
A is (implicitly)
optimally solving
a 2-item auction
Intuition
• We wish to prove the existence of a subset of
items E that is “shattered” by A’s range (RA).
– “Embed” a small NP-hard auction in E.
– Not too small! (|E| ≥ ma)
• VC dimension
– We need to bound the VC dimension of collections of
partitions!
– Of independent interest.
VC Dimension of Partitions
• We want to prove an analogue of the
Sauer-Shelah Lemma for the case of
partitions of a universe.
– That do not necessarily cover the universe.
• Problem: The size of the collection of
partitions does not tell us much.
• Recent advances [Mossel-Papadimitriou-S-Singer,
Buchfuhrer-Umans, Dughmi-Fu-Kleinberg]
Directions for Future
Research
• Understanding truthful computation in the
context of CPPP with unit-demand
valuations.
• Implications for combinatorial auctions.
• Many open questions regarding the
approximability of CPPP.
• Truthfulness in single-player environments?
Thank You