Computation and Incentives in Combinatorial

Public Projects

Michael SchapiraYale 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

Agentscomputatio

nal efficiency

incentive-compatibilit

y

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 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.

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 i vi(S)

S [m], |S|=k

Complement-Free Hierarchy

[Lehmann-Lehmann-Nisan]

Fractionally-

Subadditive

(“XOS”)

Complement-Free

(Subadditive)

Submodular

Gross Substitu

te

Unit-Demand

(“XS”)

Multi-Unit-

Demand (“OXS”)

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

Fractionally-

Subadditive

(“XOS”)

Complement-Free

(Subadditive)

Submodular

Gross Substitu

te

Unit-Demand

(“XS”)

Multi-Unit-

Demand (“OXS”)

Capped Additive(“Budget-Additive”)

Coverage

CPPP

combinatorialauctions

even for n=1

Complement-Free Hierarchy: Approximability

Fractionally-

Subadditive

(“XOS”)

Complement-Free

(Subadditive)

Submodular

Gross Substitu

te

Unit-Demand

(“XS”)

Multi-Unit-

Demand (“OXS”)

Capped Additive(“Budget-Additive”)

Coverage

CPPP

combinatorialauctions

Complement-Free Hierarchy: Area of Interest

Submodular

Gross Substitu

te

Unit-Demand

(“XS”)

Multi-Unit-

Demand (“OXS”)

Capped Additive(“Budget-Additive”)

Coverage

Unit-Demand

(“XS”)

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

resources0 0 0 11

Each user only wants (value 1) at most two resources and does not want (value 0) all

others.

user

• 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?

2-{0,1}-Unit-Demand

all sets of resourcesof size k

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 S in R v1(S)+…+vn(S)

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

A

•Thm [S-Singer]: There exists a computationally-efficient MIR mechanism for CPPP with complement-free 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 computationally-efficient MIR mechanism for CPPP with 2-{0,1}-unit-demand valuations that has appx ratio ½.– Simply choose the k most demanded resources.

2-{0,1}-Unit-Demand

•What about truthful mechanisms for CPPP with unit-demand valuations?– Characterization?

•No techniques?– VCG– random sampling– LP

Open Question

Truthfulness With One Player?

The interests of the player and mechanism are aligned (value =

social welfare)

playermechanism

What do you

want?

I want this!

Take it!

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 >0

such that for every set of resources S:

v(S) = |Uj in S Tj|

• 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?

CPPP With a Coverage Valuation

• 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.

Hardness of Truthfulness With One Player?

3 Challenges

Complexity theory

mechanism design

combinatorics

(hardness of truthful

mechanisms)

(characterization of truthful

mechanisms)

(structure of truthful

mechanisms)

•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.

The Proof: Overview

• If a computationally-efficient MIR mechanism A has appx ratio better than 1/√m then |RA| ≥ 2m (for some constant >0).– probabilistic construction.

• So, a MIR mechanism A that has appx ratio better than 1/√m optimizes over exponentially many outcomes.

Proof of Inapproximability Lemma (sketch)

All sets of resources of size k

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! RA

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|) ).

• RA is a collection of subsets of the universe of resources.

The Reduction

• We know that |RA| ≥ 2m (for some constant .

• Hence, there is a set of resources of size m (for some constant >0) that is shattered by RA.

• We can now show that the MIR mechanism A solves exactly a smaller (but not too small!) CPPP with one coverage valuation!

RA

Truthfulness With One Player?

Somewhat Strange…Do we need to rethink the

framework?

playermechanism

What do you

want?

I don’t know!

Positive Results for CPPP

Submodular

Gross Substitu

te

Unit-Demand

(“XS”)

Multi-Unit-

Demand (“OXS”)

Capped Additive(“Budget-Additive”)

Coverage

FPTAS for constant n’s

optimal algorithm for n=2

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 i vi(Si) is maximized

What About Combinatorial Auctions?

Complexity theory

mechanism design

combinatorics

embedding hard problems in

partial rangesTruthful=MIR

(VC dimension)

consider only MIR

generalize the VC dimension to

handle 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

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

A 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.

• 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 YouThank You