On the Hardness of Being Truthful

Michael SchapiraYale 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 resourcesoptimization…

• Economic concerns: truthful behaviourfairness…

computational

efficiency

incentive-compatibilit

y

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 i 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, ex-post Nash.

Can Truth and Computation Coexist?

easy + easy = easy?“[canonically hard” problems – Feigenbaum-Shenker]

NO! ]Papadimitriou-S-Singer[

Hard(Clique)

Easy(in APX, e.g.,

matching)

Easy(social-

welfare max. in auctions) Hard

(max-min fairness in auctions)

Computation

Incentives

Combinatorial Public Project Problem (CPPP)

Motivation: Find the best overlay network.

source nodes

destination nodespotential overlay 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 i 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?

–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.

Truth and Computation Don’t Mix

Combinatorial Public Projects:The Proof

Complexity theory

mechanism design

combinatorics

(the hardness of truthful

algorithms)

(what do truthfulalgorithms look

like?)

(the combinatorialproperties of

truthful algorithms)

–Theorem: Any truthful algorithm for CPPP that approximates better than 1/√m requires exponential communication.

Communication Complexity

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;

all sets of resourcesof size k

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)

* we shall refer to RA as A’s range. RA

A

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)

unrestricted valuationssingle-parameter

domains

Only MIR(Roberts 1979)

Manynon-MIR algorithms

(truthfulnessis well-understood)

?

combinatorial auctions,combinatorial pubic projects,

…

Not always MIR]auction environments: Lavi-Mu’alem-Nisan, Bartal-Gonen-Nisan[

Truthful = MIRfor CPPP!

– To prove our results we had to assume that the ``input’’ can be exponential in m.Realistic?

– If users have succinctly described valuations then computational-complexity techniques are required.No such impossibility results are known.

Computational Hardness of Truthfulness

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 exponential-size range in polynomial time implies that [computational assumption] is false.New technique.

All sets of resources of size k

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!

RA

Analogous Problem: SATL

– You are given a language L {0,1}n s.t. L is exponentially dense, i.e., |L| ≥ 2n (for some constant 0<≤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}

– For this L, SATL is obviously NP-hard.

– General approach: Find a smaller SAT hiding in SATL.Not too small!

n/2 n/2

The VC Dimension

universe

1 x 3 x 5

1 2 3 4 5

1 2 x x 5

x x x 4 x

collectionof subsets

R

shattered set

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| ≥ ( 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 n variables (for some constant 0<≤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 Sauer-Shelah 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 i 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-constantapproximations

are known(subadditive,submodular)

What About Combinatorial Auctions?

Complexity theory

mechanism design

combinatorics

(the embedding of NP-hard problems)

(Characterization of truthful algorithms,

based on Roberts’ Theorem)

(VC dimension)

consider a specificclass of algorithms

(MIR = VCG based).

generalize the VC dimension to

handle 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 {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 [Andelman-Mansour]

• A simple MIR algorithm obtains a ½-approximation 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+ in the allocate-all-items 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

1 2 3 4 5

5 items

MIR algorithm A

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

RA

M is (implicitly)optimally solvinga 2-item auction

2 bidders

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| ≥ m)

• 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 > 0 be some constant. Let R be a collection of partitions of a universe U, such that every two partitions in R are |U|-far.

Then, there exists a subset E of U such that:

– R’s projection on E contains all covering partitions of E.

– |E| ≥ ( 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 YouThank You