On the Hardness of Being
Truthful
Michael Schapira
Yale University and UC Berkeley
Joint work with Christos Papadimitriou
and Yaron Singer (2008),
and with Elchanan Mossel, Christos
Papadimitriou and Yaron Singer (2009)
On the Agenda
• Algorithmic mechanism design
• An impossibility result: Truthfulness and
computation can clash!
• Extending our results to combinatorial
auctions
• Open questions and directions for future
research
Designing Algorithms for
Environments With Selfish Agents
• Computational concerns:
bounded computational resources
optimization
…
• Economic concerns:
truthful behaviour
fairness
…
computational
efficiency
incentivecompatibility
Algorithmic Mechanism Design
• Can these different desiderata coexist?
• The central problem in Algorithmic
Mechanism Design [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.
• Goal: find a partition of the items
between the bidders S1,…,Sn such that
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.
Can Truth and Computation
Coexist?
Computation
Hard
(Clique)
Easy
(in APX, e.g.,
matching)
Incentives
Easy
(socialwelfare max.
in auctions)
Hard
(max-min
fairness in
auctions)
easy + easy = easy?
NO!
[“canonically hard” problems – Feigenbaum-Shenker]
[Papadimitriou-S-Singer]
Combinatorial Public Project
Problem (CPPP)
Motivation: Find the best overlay network.
source nodes
potential overlay nodes
destination nodes
Combinatorial Public Projects
• Set of n agents; Set of m resources;
• Each agent i has a valuation function:
vi : 2[m] → R≥0
• Objective: 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
Assumptions Regarding Each
Valuation Function
– Normalized:
v(∅) = 0
– Non-decreasing:
v(S) ≤ v(T)
S  T
– Submodular:
v(Sυ{j}) − v(S) ≥ v(Tυ{j}) − v(T)
S  T
Special Cases
• Elections for a committee: The
agents are voters, resources are
potential candidates.
• Overlay networks: The agents are
source nodes, resources are potential
overlay nodes.
Are Combinatorial Public Projects
Easy?
• Computational Perspective:
A 1-1/e approximation ratio is achievable (not
truthful!)
 A tight lower bound exists [Feige].
• Strategic Perspective:
A truthful solution is achievable via VCG
payments (but NP-hard to obtain)
• What about achieving both simultaneously?
Truth and Computation Don’t Mix
– Theorem (Informal): [Papadimitriou-S-Singer]
Any algorithm for CPPP that is both
truthful and computationally-efficient
does not have an approximation ratio
better than 1/√m
• Even for n=2.
• Tight! [S-Singer].
– Two models:
• Communication complexity.
• Computational complexity.
Combinatorial Public Projects:
The Proof
mechanism
design
(what do truthful
algorithms look
like?)
Complexity
theory
(the hardness of
truthful algorithms)
combinatorics
(the combinatorial
properties of
truthful algorithms)
Communication Complexity
–Theorem:
Any truthful algorithm for
CPPP that approximates better
than 1/√m requires exponential
communication.
Proving the Lower Bound
– Lemma 1: Any maximal-in-range (MIR)
algorithm for CPPP that approximates
better than 1/√m requires exponential
communication in m.
– Lemma 2 (!): An algorithm for the
combinatorial public project problem is
truthful iff it is MIR;
Maximal-In-Range Algorithms
(= VCG-Based)
Definition:
A is MIR if there is some
RA  {|S | = k| S  [m]}
s.t. A(v1,v2) = argmax S in R v1(S) + v2(S)
A
*
we shall refer to RA as A’s range.
RA
all sets of resources
of size k
Lower Bound For MIR
– Inapproximability Lemma: Any MIR algorithm for CPPP
that approximates better than 1/√m requires
exponential communication in m.
– Proof in two steps: [Dobzinski-Nisan]
• Proposition 1: In order to get an approximation
better than 1/√m, the range must be exponentially
large (in m).
 Even for n=1. Simple (succinctly described) valuations.
• Proposition 2: Maximizing over a range RA requires
communicating |RA| bits.
 Even for n=2. We use the fact that valuations can be exponentially long.
Characterization Lemma
– Characterization Lemma: An algorithm for
CPPP is truthful iff it is MIR
– Theorem (Roberts 79): For unrestricted
valuation functions any truthful algorithm is
MIR.
• Actually, affine maximizer…
– We use machinery from simplified proofs of
Roberts’ Theorem [Lavi-Mu’alem-Nisan].
• But… our domain is severely restricted!
• But… our domain isn’t open!
Characterizing Truthfulness (cntd)
single-parameter
domains
combinatorial auctions,
combinatorial pubic projects,
…
?
Many
non-MIR algorithms
unrestricted valuations
Only MIR
(Roberts 1979)
(truthfulness
is well-understood)
Not always MIR
[auction environments: Lavi-Mu’alem-Nisan, Bartal-Gonen-Nisan]
Truthful = MIR
for CPPP!
Computational Hardness of
Truthfulness
– To prove our results we had to assume
that the ``input’’ can be exponential in
m.
Realistic?
– If users have succinctly described
valuations then computationalcomplexity techniques are required.
No such impossibility results are known.
Our Proof Revisited
– Characterization Lemma: an algorithm is truthful
iff it is an affine-maximizer.
Observation: The proof only requires succinctly-described
valuations.
– Inapproximability Lemma: Any affine maximizer
which approximates better than √m requires
exponential communication.
Proposition 1: In order to get an approximation
better than √m, the range must be exponential.
Proposition 2: Maximizing over a range RA
requires communicating |RA| bits.
New Proof
– Characterization Lemma: an algorithm is truthful
iff it is an affine-maximizer.
– Inapproximability Lemma: No affine maximizer
can approximate better than √m unless [computational
assumption] is false.
Proposition 1: In order to get an approximation
better than √m, the range must be exponential.
New Challenge: Maximizing over an exponentialsize range in polynomial time implies that
[computational assumption] is false.
New technique.
Computational Complexity Hardness
– For many families of succinctly described
valuations CPPP is NP-hard.
Special case: MAX-K-COVER
[Feige]
– So, optimizing over the set of all possible
solutions is hard.
– What about optimizing over a set of solutions of
exponential size?
Intuition - also hard!
All sets of resources
of size k
RA
Analogous Problem: SATL
– You are given a language L  {0,1}n s.t.
a
n
L is exponentially dense, i.e., |L| ≥ 2
(for some constant 0<a≤1)
– SATL: Given a CNF boolean formula
determine whether or not it has a
satisfying assignment in L.
– Conjecture: SATL is NP-hard for
every exponentially dense L.
Intuition
– Let L = {s| s is of the form 00…0xx…x}
n/2
– For this L, SATL is obviously NP-hard.
– General approach: Find a smaller SAT
hiding in SATL.
Not too small!
n/2
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|) ).
• Think of L as a collection of subsets of the
universe of variables.
Sauer-Shelah Lemma (for SATL)
– Let L be some exponentially dense
language.
– Then, there exists a set N of nb variables
(for some constant 0<b≤1) s.t. all
assignments for these variables are in L.
– Are we done? Did we prove that SATL is
NP-hard?
No!
– We do not know how to find (approximate) N in
polynomial time.
Hard!
[Papadimitriou-Yannakakis, Schaefer, Mossel-Umans]
– Theorem: If SATL is in P then SAT has
polynomial-sized circuits.
– What about a probabilistic reduction from
SAT?
A naïve approach fails.
Ajtai’s probabilistic version of the SauerShelah Lemma helps in our case!
What about SATL? (CIRCUIT-SATL is different…)
So…
– Truthulness and computation can clash!
• In two complexity models.
– APX is not preserved under truthfulness
(unlike P and NP).
Back to 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.
• Goal: find a partition of the items
between the bidders S1,…,Sn such that
social welfare Si vi(Si) is maximized
Sounds Familiar?
• Easy from an economic perspective.
– VCG!
• Easy to solve computationally in many
intersting cases.
Huge Gaps!
non-truthful:
constant
approximation
ratios
(subadditive,
submodular)
truthful:
?
non-constant
approximations
are known
(subadditive,
submodular)
What About Combinatorial
Auctions?
mechanism
design
(Characterization
of
consider a specific
truthful
class of algorithms,
algorithms
based
Roberts’
(MIR
= on
VCG
based).
Theorem)
Complexity
theory
(the embedding of
NP-hard problems)
combinatorics
generalize the VC
dimension to handle
(VC
dimension)
partitions
of a
universe.
The Case of 2 Bidders
• Not trivial even for n=2!
• Not trivial even for if bidders’ valuation
functions are of a very restricted form.
“Simple” 2-bidder
Combinatorial Auctions
• A set of m items for sale {1,…m}.
• 2 bidders. Each bidder i has an additive valuation
with a spending constraint vi.
– per-item values ai1,…,aim
– “maximum spending” value bi
– For every bundle S, vi(S)=min {S j in S aij , bi},
• Goal: find a partition of the items between the 2
bidders (S1,S2) such that social welfare
v1(S1)+v2(S2) is maximized
Truthful vs. Unrestricted
Algortihms
• A non-truthful FPTAS exists [AndelmanMansour]
• A simple MIR algorithm obtains a ½approximation ratio.
– Simply bundle all items together.
– The best truthful appx. for this problem to
date.
• Is this the best we can do?
Yes! (Sort Of…)
• Theorem [Mossel-Papadimitriou-S-Singer]:
No computationally-efficient MIR mechanism M
obtains an appx-ratio of ½+e in the allocate-allitems case.
– unless NP has polynomial size circuits.
• Techniques similar to those used in the proof for
CPPP.
• Removing the allocate-all-items assumption is not
trivial!
– If we just allocate unallocated items arbitrarily we might
lose the MIR property!
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
M 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.
An Analogue of the Sauer-Shelah
Lemma [Mossel-Papadimitriou-S-Singer]
• Definition: A partition of a universe is a
pair of disjoint subsets of the universe.
– Does not necessarily exhaust the universe!
– We refer to a partition that does exhaust
the universe as a “covering partition”.
• Definition: Two partitions, (T1,T2) and
(T’1,T’2) , are said to be b-far (or at
distance b) if |T1 U T’2| + |T’1 U T2| ≥ b.
An Analogue of the Sauer-Shelah
Lemma [Mossel-Papadimitriou-S-Singer]
• Lemma: Let d > 0 be some constant. Let R be a
collection of partitions of a universe U, such
that every two partitions in R are d|U|-far.
Then, there exists a subset E of U such that:
– R’s projection on E contains all covering
partitions of E.
– |E| ≥ W( log(|R|)/log(|U|) ).
Directions for Future Research
• Relaxing the computational assumptions.
• Characterizing truthful algorithms for
combinatorial auctions.
• Lower bounds for MIR algorithms for
combinatorial auctions.
– Recent results by Buchfuhrer and Umans obtained
using our techniques and new ones.
• Many intriguing questions regarding the VC
dimension of partitions.
Thank You