The Price of Anarchy of Relax-and-Round

15 Pages·2015·0.27 MB·English

Algorithms as Mechanisms: The Price of Anarchy of Relax-and-Round
PAULDU¨TTING,LondonSchoolofEconomics
THOMASKESSELHEIM,Max-Planck-Institutfu¨rInformatik
E´VATARDOS,CornellUniversity
Many algorithms, that are originally designed without explicitly considering incentive properties, are
later combined with simple pricing rules and used as mechanisms. The resulting mechanisms are often
naturalandsimpletounderstand.Buthowgoodarethesealgorithmsasmechanisms?Truthfulreporting
ofvaluationsistypicallynotadominantstrategy(certainlynotwithapay-your-bid,first-pricerule,butit
islikelynotagoodstrategyevenwithacriticalvalue,orsecond-pricestyleruleeither).Ourgoalistoshow
thatawideclassofapproximationalgorithmsyieldsthiswaymechanismswithlowPriceofAnarchy.
The seminal result of Lucier and Borodin [2010] shows that combining a greedy algorithm that is an
α-approximationalgorithmwithapay-your-bidpaymentruleyieldsamechanismwhosePriceofAnarchy
isO(α).Inthispaperwesignificantlyextendtheclassofalgorithmsforwhichsucharesultisavailable
byshowingthatthiscloseconnectionbetweenapproximationratioontheonehandandPriceofAnarchy
ontheotheralsoholdsforthedesignprincipleofrelaxationandroundingprovidedthattherelaxationis
smoothandtheroundingisoblivious.
Wedemonstratethefar-reachingconsequencesofourresultbyshowingitsimplicationsforsparsepack-
ing integer programs, such as multi-unit auctions and generalized matching, for the maximum traveling
salesmanproblem,forcombinatorialauctions,andforsinglesourceunsplittableflowproblems.Inallthese
problems our approach leads to novel simple, near-optimal mechanisms whose Price of Anarchy either
matchesorbeatstheperformanceguaranteesofknownmechanisms.
CategoriesandSubjectDescriptors:F.2[TheoryofComputation]:AnalysisofAlgorithmsandProblem
Complexity;J.4[ComputerApplications]:SocialandBehavioralSciences—Economics
AdditionalKeyWordsandPhrases:AlgorithmicGameTheory,PriceofAnarchy,Smoothness
1. INTRODUCTION
Mechanismdesign—or“reverse”gametheory—isconcernedwithprotocols,ormech-
anisms, through which potentially selfish agents interact with one another. The basic
assumption is that the data is held by the agents, who may behave strategically. The
goalisthentoachieveoutcomesthatapproximatethesocialoptimuminawiderange
ofstrategicequilibria.
Themostsweepingpositiveresultthatonecouldpossiblyhopeforinthiscontext—
withsomeprofessionalbiasofcourse—isageneralreductionfrommechanismdesign
to algorithm design, showing that mechanism design is just as “easy” as algorithm
E´.TardosissupportedinpartbyNSFgrantsCCF-0910940andCCF-1215994,ONRgrantN00014-08-1-
0031,aYahoo!ResearchAllianceGrant,andaGoogleResearchGrant.
Author addresses: P. Du¨tting, Department of Mathematics, London School of Economics, Houghton
Street,LondonWC2A2AE,UK;T.Kesselheim,Max-Planck-Institutfu¨rInformatik,CampusE14,66123
Saarbru¨cken,Germany;E´.Tardos,DepartmentofComputerScience,CornellUniversity,GatesHall,Ithaca,
NY14853,USA.
Emails:[email protected],[email protected],[email protected].
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http://dx.doi.org/10.1145/2764468.2764486
2 P.Du¨ttingetal.
design. Specifically, one could hope that using algorithms as they are and charging
bidderstheirrespectivebidsyieldsmechanismswhoseequilibriaareclosetooptimal.
Why would this be appealing? Such a result would make the entire toolbox of algo-
rithm design available to mechanism design, significantly broadening the tools cur-
rently available. It would also “hide” the incentives aspect from the designer, who
wouldthennolongerneedtoworryaboutpossiblemanipulationsthroughtheagents.
He could simply focus on the problem of computing near optimal solutions for the
claimedinput.Finally,theresultingmechanismswouldenjoyasimplicitywellbeyond
thatfoundinmoststate-of-theartmechanisms.
Our goal in this paper is to identify general algorithm design principles that work
wellwhenusedasmechanisms.Wecannotexpectthistobethecaseforallalgorithms.
Identifying algorithm design principles that automatically work well as mechanisms
would, in some sense, give us the vocabulary to which we — as algorithm designers
— should confine ourselves if we expect that our algorithm will be used in strategic
environments.Specifically,itwouldequipuswiththetoolstodesignsimpleandrobust
mechanismsforsuchsettings.
Lucier and Borodin [2010] showed that greedy algorithms have this property: Any
equilibriumofagreedyalgorithmthatisanα-approximationalgorithmiswithinO(α)
of the optimal solution. Our main result is to show that the common design principle
ofrelaxationandroundingasin—relaxintegerlinearprogramtofractionaldomain,
solve relaxation to optimality, and then convert into integer solution via randomized
rounding — also preserves the approximation guarantee as Price of Anarchy guaran-
teeprovidedthattherelaxationissmoothandtheroundingprocessisoblivious(more
onthisbelow).
This result has — as we show — far-reaching consequences in mechanism design:
Itleadstonovelsimple,yetnear-optimalmechanismsforsparsepackingintegerpro-
grams,suchasmulti-unitauctionsandgeneralizedmatching,forthemaximumtrav-
elingsalesmanproblem,forcombinatorialauctionsandforsinglesourceroutingprob-
lems. In all cases we obtain Price of Anarchy bounds that match or beat known Price
of Anarchy guarantees, or they are the first non-trivial guarantees for the respective
problem.
1.1. OurContributions
Our results concern the algorithmic blueprint of relaxation and rounding (see, e.g.,
[Vazirani 2001]). In this approach a problem Π is relaxed to a problem Π(cid:48), with the
purpose of rendering exact optimization computationally tractable. Having found the
optimalrelaxedsolutionx(cid:48),anotheralgorithmderivesasolutionxtotheoriginalprob-
lem. This process is typically called rounding. The best-known example are integer
linearprogramswhicharerelaxedtofractionaldomains.
Manyroundingschemesintextbooksaswellashighlysophisticatedonesareobliv-
ious.Thatis,theydonotrequireknowledgeoftheobjectivefunction.Uptothispoint,
to the best of our knowledge, this property—though wide-spread—has never proven
useful. In this paper, we show that oblivious rounding schemes preserve bounds on
the Price of Anarchy. That is, applying an α-approximate oblivious rounding scheme
on a problem with a Price of Anarchy bound β, the combined mechanism has Price of
AnarchyatmostO(αβ).
Wethustranslatetherelax-and-roundapproachfromalgorithmdesignintomecha-
nism design: If we relax a problem into a problem with Price of Anarchy β and round
thesolutiontotherelaxedproblemwithanα-approximateobliviousroundingscheme,
theresultingmechanismhasaPriceofAnarchyofO(αβ).Thebounddoesnotonlyap-
ply to Nash equilibria, but also extends to the Bayesian setting as well as to learning
AlgorithmsasMechanisms:ThePriceofAnarchyofRelax-and-Round 3
outcomes (coarse correlated equilibria). See Section 9 for discussion on the existence
andcomputationalcomplexityoffindingsuchoutcomes.
Main Result. Our main result leverages the power of the smoothness framework of
Roughgarden[2009,2012]andSyrgkanisandTardos[2013].
At the heart of this framework is the notion of a (λ,µ)-smooth mechanism, where
λ,µ≥0.Themainresultisthatamechanismthatis(λ,µ)-smoothachievesaPriceof
Anarchyofβ(λ,µ)=max(1,µ)/λwithrespecttoabroadrangeofequilibriumconcepts
includinglearningoutcomes.Furthermore,thesimultaneousandsequentialcomposi-
tion of (λ,µ)-smooth mechanisms is again (λ,µ)-smooth. Ideally, λ = 1 and µ ≤ 1 in
whichcasethisresulttellsusthatallequilibriaofthemechanismaresociallyoptimal;
otherwise,ifλ<1orµ>1,thenthisresulttellsuswhichfractionoftheoptimalsocial
welfarethemechanismisguaranteedtogetatanyequilibrium.
The other crucial ingredient to our main result is the notion of an α-approximate
oblivious rounding scheme, where α ≥ 1. This is a (possibly randomized) rounding
scheme for translating a solution x(cid:48) to the relaxed problem Π(cid:48) into a solution x to the
originalproblemΠsothatforallpossiblevaluationprofileseachagentisguaranteed
toget,inexpectation,a1/α-fractionofthevaluethatitwouldhavehadforthesolution
totherelaxedproblem.
Clearly an α-approximate oblivious rounding scheme, when combined with opti-
mallysolvingtherelaxedproblem,leadstoanapproximationratioofα.Weshowthat
italsoapproximatelypreservesthePriceofAnarchyoftherelaxation.Wefocusonpay-
your-bidmechanismsforconcreteness.Ourresultactuallyappliestoabroadrangeof
mechanisms and can also be extended to include settings where the relaxation is not
solvedoptimally;wediscusstheseextensionsinSection8.
THEOREM 1.1 (MAIN THEOREM, INFORMAL). Consider problem Π and a relax-
ationΠ(cid:48).Supposethepay-your-bidmechanismM forΠisderivedfromthepay-your-bid
mechanismM(cid:48) forΠ(cid:48).IfM(cid:48) is(λ,µ)-smooth,thenM is(λ/(2α),µ)-smooth.
COROLLARY 1.2. The Price of Anarchy established via smoothness of mechanism
M(cid:48) of β translates into a smooth Price of Anarchy bound for mechanism M of 2αβ
extendingtobothBayesianNashequilibriaandlearningoutcomes.
Ourmaintheoremcanbestrengthenediftherelaxationsatisfiesaslightlystronger
smoothness condition, also parametrized by λ and µ, which all our application do. In
thiscasewecanshowthatthederivedmechanismis(λ/α,µ)-smooth;andthecorollary
wouldread“aPriceofAnarchyofβ translatesintoaPriceofAnarchyofαβ.”
Applications. We demonstrate the far-reaching consequences of our result by applying
it to a broad range of optimization problems. For each of these problems we show the
existence of a smooth relaxation and the existence of an oblivious rounding scheme.
We note that in all of our applications, it is important to use the relaxation to show
smoothnessoftheproblem.Forexample,optimallysolvingtheoriginal(integer)prob-
lemwould giveaveryhighPriceofAnarchy.
Sparse Packing Integer Programs. The first problem we consider are multi-unit auc-
tions with n bidders and m items, where bidders have unconstrained valuations.
The underlying optimization problem has a natural LP relaxation, which we show is
(1/2,1)-smooth. Using the 8-approximate oblivious rounding scheme of [Bansal et al.
2010], our framework yields a constant PoA. This is quite remarkable as solving the
integraloptimizationproblemleadstoaPoAthatgrowslinearlyinnandm.
Wethenconsiderthegeneralizedassignmentprobleminwhichnbiddershaveunit-
demandvaluationsforacertainamountofoneofkservicesandallocationsofservices
to bidders must respect the limited availability of each service. For this problem we
4 P.Du¨ttingetal.
also show (1/2,1)-smoothness, and use the 8-approximate oblivious rounding scheme
of[Bansaletal.2010]toobtainaconstantPoA.
Boththeseresultsareinfactspecialcasesofageneralresultregardingsparsepack-
ing integer programs (PIP) that we show. Namely, the pay-your-bid mechanism that
solves the canonical relaxation of a PIP with column sparsity d is (1/2,d+1)-smooth.
Multi-unit auctions and the generalized assignment problem have d = 1; combina-
torial auctions in which each bidder is interested in at most d items simultaneously
have d ≥ 1. For general PIPs the rounding scheme of [Bansal et al. 2010] is O(d)-
approximate.WegetaPoAofO(d2).
MaximumTravelingSalesman.Oursecondapplicationisthemaximizationvariantof
theclassictravelingsalesmanproblem(max-TSP).Wethinkoftheproblemasagame
where each edge e has a value for being included, and the goal of the mechanism is
to select a TSP of maximum total value. The classic algorithm for this problem is a
2-approximation[Fisheretal.1979].Itproceedsbycomputingacyclecover,dropping
anedgefromeachcycle,andconnectingtheresultingpathsinanarbitrarymannerto
obtainasolution.Weprovethiscanbethoughtoffasa2-approximateobliviousround-
ing schemeand show,through anovel combinatorialargument, thatthe relaxationis
(1/2,3)-smooth.WethusobtainaPriceofAnarchyof12.
The best approximation guarantee for max-TSP is a 3/2-approximation due to Ka-
planetal.[2003].Thesameapproximationratioisachievedbya(muchsimpler)algo-
rithmofPaluchetal.[2012].Weshowthatthisalgorithm—justasthebasicalgorithm
—canbeinterpretedasarelax-and-roundalgorithm.Generalizingtheargumentsfor
the basic algorithm to the (different) relaxation used in this interpretation, we show
thatthisalgorithmachievesaPriceofAnarchythatisbyafactor3/4betterthanthe
PriceofAnarchyofthebasicalgorithm.
These examples are especially interesting as they show how a seemingly combina-
torial algorithm can be re-stated within our framework. They also represent the first
non-trivialPoAboundsforthisproblem.
Combinatorial Auctions. We also consider the “canonical” mechanism design problem
of combinatorial auctions in which valuations are restricted to come from a certain
class.Ourfirstresultconcernsfractionallysubadditive,orXOS,valuations[Lehmann
etal.2006].Weshowthatthepay-your-bidmechanismforthecanonicalLPrelaxation
is (1/2,1)-smooth. Using Feige’s ingenious e/(e−1)-approximate oblivious rounding
scheme[Feige2009],ourmainresultimpliesanupperboundonthePriceofAnarchy
of4e/(e−1).
WethenshowhowtoextendthisresulttotherecentlyproposedhierarchyofMPH-
k valuations [Feige et al. 2014]. Levels of the hierarchy correspond to the degree of
complementarity in a given function. The lowest level k = 1 coincides with the class
of XOS/fractionally subadditive valuations; the highest level k = m can be shown to
comprise all monotone valuation functions. We show that for MPH-k valuations the
LP relaxation is (1/2,k + 1) smooth. Together with the O(k)-approximate oblivious
roundingschemeof[Feigeetal.2014]weobtainaPriceofAnarchyofΘ(k2).
These results nicely complement the recent work on “simple auctions” such as
[Christodoulou et al. 2008; Bhawalkar and Roughgarden 2011; Feldman et al. 2013;
Du¨ttingetal.2013;Roughgarden2014],answeringanopenquestionofBabaioffetal.
[2014]regardingthePriceofAnarchyofdirectmechanismbasedonapproximational-
gorithmsinthesesettings.Theadvantageofhavingadirectmechanismforthisprob-
lemisthatonecanconsidersimplebiddingstrategies(suchasbiddinghalfthevalue)
toestablishtheperformanceguarantees,whereasinindirectmechanismssuchascom-
binatorialauctionswithitembiddingthecomputationaleffortiseffectivelyshiftedto
thebidders.
AlgorithmsasMechanisms:ThePriceofAnarchyofRelax-and-Round 5
Single Source Unsplittable Flow. The final problem that we consider are multi-
commodity flow (MCF) problems with a single source (or target). In these problems
wearegivenacapacitated,directednetworkandasetofrequestsconsistingofatar-
get (a source) and a demand, corresponding to requests of, say different information,
heldatthesource.Thegoalistomaximizethetotaldemandrouted(orthetotalvalue
ofthedemandrouted),subjecttofeasibility.Weassumeeachplayerhasademandfor
some flow to be routed from a shared source to a terminal specific to the player, and
theplayerhasaprivatevalueforroutingthisflow.
ForthisproblemweshowthatthenaturalLPrelaxationis(1/2,1)-smooth.A(1+(cid:15))-
approximateobliviousroundingschemeforhighenoughcapacitiesisobtainedthrough
an adaptation of the “original” randomized rounding algorithm of [Raghavan 1988;
RaghavanandThompson1987].ThisyieldsaPoAof2(1+(cid:15)).
AninterestingfeatureofthisresultisthattheLPcanbesolvedgreedilythrougha
variantofFord-Fulkersonwhichallowsustoexploittheknownconnectiontosmooth-
ness [Lucier and Borodin 2010; Syrgkanis and Tardos 2013]. Crucially, the reference
totheseresultshastobeonthefractionallevel,asagreedyprocedureontheintegral
levelachievesasignificantlyworseapproximationguarantee.
1.2. RelatedWork
Our work is closely related to the literature on so-called “back-box reductions”, which
hasledtosomeofthemostimpressiveresultsinalgorithmicmechanismdesign(such
as[LaviandSwamy2005;Briestetal.2005;DughmiandRoughgarden2014;Dughmi
et al. 2011; Babaioff et al. 2010, 2013]). This approach takes an algorithm, and aims
toimplementthealgorithm’soutcomeviaagame.Tothisendittypicallymodifiesthe
algorithm andaddsasophisticatedpaymentscheme.Ourapproachisdifferentinthat
weconsideranalgorithmwithoutanymodification,introduceasimplepaymentrule,
suchasthe“payyourbid”rule,andunderstandtheexpectedoutcomesoftheresulting
game.
Lavi and Swamy [2005] use randomized meta rounding [Carr and Vempala 2002]
to turn LP-based approximation algorithms for packing domains into truthful-in-
expectationmechanisms.Ourresultissimilarinspiritasitdemonstratestheimplica-
tions of obliviousness for non-truthful mechanism design. The property that we need,
however,islessstringentandsharedbymostroundingalgorithms.Anotherimportant
differenceisthatourapproachisnotlimitedtopackingdomains.
Briest et al. [2005] show how pseudo-polynomial approximation algorithms for
single-parameterproblemscanbeturnedintoatruthfulfullypolynomial-timeapprox-
imationschemes(FPTAS).DughmiandRoughgarden[2014]provethateverywelfare-
maximizationproblemthatadmitsaFPTASandcanbeencodedasapackingproblem
also admits a truthful-in-expectation randomized mechanism that is an FPTAS. Un-
like our approach these approaches are limited to single-parameter problems, or to
multi-parameterproblemswithpackingstructure.
Dughmi et al. [2011] present a general framework that also looks at the fractional
relaxationoftheproblem.Theyshowthatiftheroundingprocedurehasacertainprop-
erty, which they refer to as convex rounding, then the resulting algorithm is truthful.
They instantiate this framework to design a truthful-in-expectation mechanism for
CAs with matroid-rank-sum valuations (which are strictly less general than submod-
ular).Themaindifferencetoourworkisthatstandardroundingproceduresareoften
obliviousbuttypicallynotconvex.
Babaioff et al. [2010, 2013] show how to transform a monotone or cycle-monotone
algorithm into a truthful-in-expectation mechanism using a single call to the algo-
rithm. The resulting mechanism coincides with the algorithm with high probability.
6 P.Du¨ttingetal.
Thisworkdiffersfromoursinthatitonlyappliestomonotoneorcycle-monotonealgo-
rithms.
By insisting on truthfulness, or truthfulness-in-expectation, as a solution concept,
all these approaches face certain natural barriers to how good they can get (see, e.g.,
[Papadimitriouetal.2008;Chawlaetal.2012]).Inaddition,theytypicallydonotlead
tosimple,practicalmechanisms.Forexample,despiterunningtimestechnicallybeing
polynomial, these mechanisms require far more computational effort than standard
approximationalgorithmsfortheunderlyingoptimizationproblem.Insomecases,the
reduction yields mechanisms in which the approximation guarantee is tight on every
singleinstance(notonlyintheworstcase).Thatis,evenwhentheoptimizationprob-
lemistrivial,themechanismsacrificesthesolutionqualityforincentives.
1.3. Organization
WeformallydefineourmodelinSection2.Section3presentsthemeta-theoremswith
proofs.Sections4through7discussapplications.Weonlygiveproofsketchesforthese
results here. Details can be found in the full version. Section 8 presents possible ex-
tensionsofourframework.Weconcludewithadiscussionorourresultsanditsimpli-
cationsinSection9.
2. PRELIMINARIES
AlgorithmDesignBasics. WeconsidermaximizationproblemsΠinwhichthegoalisto
determineafeasibleoutcomex∈Ωthatmaximizestotalweightgivenbyw(x)fornon-
negativeaweightfunctionw: Ω→R .ApotentiallyrandomizedalgorithmAreceives
≥0
the functions w as input and computes an output A(w) ∈ Ω. The algorithm is an α-
approximationalgorithm,forα≥1,ifforallweightsw,E[w(A(w))]≥ 1 ·max w(x).
α x∈Ω
We are interested in relax-and-round algorithms. These algorithms first relax the
problemΠtoΠ(cid:48)byextendingthespaceoffeasibleoutcomestoΩ(cid:48) ⊇Ωandgeneralizing
weight functions w to all x ∈ Ω(cid:48). They compute an optimal solution x(cid:48) ∈ Ω(cid:48) to the
relaxed problem. Then a solution x ∈ Ω of the original problem is derived based on
x(cid:48) ∈Ω(cid:48),typicallyviarandomizedrounding.
A rounding algorithm is oblivious if it does not require knowledge of the actual ob-
jectivefunctionw,beyondthefactthatx(cid:48) wasoptimizedwithrespecttow.Formally,a
roundingschemeisanα-approximateobliviousroundingschemeif,givensomerelaxed
solution x(cid:48), it computes a solution x such that for all w, E[w(x)] ≥ 1w(x(cid:48)). Clearly, a
α
relax-and-round algorithm based on an α-approximate oblivious rounding scheme is
anα-approximationalgorithm.
Mechanism Design Basics. Our results apply to general multi-parameter mechanism
design problems Π in which agents N = {1,...,n} interact to select an element from
a set Ω of outcomes. Each agent has a valuation function v : Ω → R . We use v for
i ≥0
the valuation profile that specifies a valuation for each agent, and v to denote the
−i
valuationsoftheagentsotherthani.Thequalityofanoutcomex ∈ Ωismeasuredin
(cid:80)
termsofitssocialwelfare v (x).
i∈N i
We consider direct mechanisms M that ask the agents to report their valuations.
We refer to the reported valuations as bids and denote them by b. The mechanism
uses outcome rule f to compute an outcome f(b) ∈ Ω and payment rule p to compute
payments p(b) ∈ R . Both the computation of the outcome and the payments can
≥0
be randomized. We are specifically interested in pay-your-bid mechanisms, in which
agentsareaskedtopaywhattheyhavebidontheoutcometheyget.Inotherwords,in
apay-your-bidmechanismM =(f,p),p (b)=b (f (b)).Weassumethattheagentshave
i i i
quasi-linearutilitiesandthattheyareriskneutral.Thatis,weassumethatagenti’s
expectedutilityinmechanismM =(f,p)isgivenbyu (b,v )=E[v (f(b))]−E[p (b)].
i i i i
AlgorithmsasMechanisms:ThePriceofAnarchyofRelax-and-Round 7
For the game-theoretic analysis we distinguish two settings. In the complete infor-
mationsettingagentsknoweachothers’valuations,andapotentiallyrandomizedbid
profilebthatmaydependonv isamixedNashequilibriumifforallagentsi∈N and
possible deviations b(cid:48) that may depend on v, E [u (b,v )] ≥ E [u ((b(cid:48),b ),v )]. In
i b i i b(cid:48)i,b−i i i −i i
the incomplete information setting valuations are drawn from independent distribu-
tions D , and each agent i ∈ N knows its own valuation v and the distributions D
i i −i
from which the other agents valuations are drawn. A mixed Bayes-Nash equilibrium
isapotentiallyrandomizedbidprofileb thatmaydependonthisagent’svaluationv
i i
andthedistributionsD fromwhichtheotheragents’valuationsaredrawnsuchthat
−i
forallagentsi∈N andpotentialdeviationsb(cid:48) whicharealsoallowedtodependonv
i i
andD ,E [u (b,v )]≥E [u ((b(cid:48),b ),v )].
−i b,v−i i i b(cid:48)i,b−i,v−i i i −i i
PriceofAnarchy. WeevaluatethequalityofmechanismsbytheirPriceofAnarchy.The
Price of Anarchy with respect to Nash equilibria (PoA) is the worst ratio between the
optimal social welfare and the expected welfare in a mixed Nash equilibrium. Simi-
larly, the Price of Anarchy with respect to Bayes-Nash equilibria (BPoA) is the worst
ratiobetweentheoptimalexpectedsocialwelfareandtheexpectedwelfareinamixed
Bayes-Nash equilibrium. Formally, define NASH(v) and BNASH(D) as the set of all
mixedNashandmixedBayesNashequilibriarespectively.Then,
(cid:80) (cid:2)(cid:80) (cid:3)
max v (x) maxE v (x)
i∈N i i∈N i
PoA=max max x∈Ω andBPoA=max max x∈Ω .
(cid:2)(cid:80) (cid:3) (cid:2)(cid:80) (cid:3)
v b∈NASH(v)E i∈Nvi(f(b)) D b∈BNASH(D) E i∈Nvi(f(b))
TheSmoothnessFramework. An important ingredient in our result is the following no-
tion of a smooth mechanism of Syrgkanis and Tardos [2013]. A mechanism is (λ,µ)-
smooth for λ,µ ≥ 0 if for all valuation profiles v and all bid profiles b there exists a
possibly randomized strategy b(cid:48) for every agent i that may depend on the valuation
i
profilev ofallagentsandthebidb ofthatagentsuchthat
i
(cid:88) (cid:88) (cid:88)
E[u ((b(cid:48),b ),v )]≥λ·max v (x)−µ· E[p (b)] .
i i −i i i i
x∈Ω
i∈N i∈N i∈N
THEOREM 2.1 (SYRGKANIS AND TARDOS [2013]). Ifamechanismis(λ,µ)-smooth
andagentshavethepossibilitytowithdrawfromthemechanism,thentheexpectedso-
cialwelfareatanymixedNashormixedBayes-Nashequilibriumisatleastλ/max(µ,1)
oftheoptimalsocialwelfare.
As shown in [Syrgkanis and Tardos 2013], (λ,µ)-smoothness also implies a bound
of max(µ,1)/λ on the Price of Anarchy for correlated equilibria, also known as learn-
ing outcomes. Furthermore, the simultaneous and sequential composition of multiple
(λ,µ)-smoothmechanismsisagain(λ,µ)-smooth.Fordetails,ontheprecisedefinitions
andstatementsbeyondNashequilibria,see[SyrgkanisandTardos2013].
In fact, our smoothness proofs show an even slightly stronger property, semi-
smoothness as defined by [Caragiannis et al. 2015]: the deviation strategy b(cid:48) only
i
depends on the respective agent’s valuation v , but not on the agent’s bid b or the
i i
other agents’ valuations v . Therefore, the same Price of Anarchy bounds also apply
−i
tocoarsecorrelatedequilibriaandBayes-Nashequilibriawithcorrelatedtypes.
3. OBLIVIOUSROUNDINGANDSMOOTHRELAXATIONS
In this section, we show our main theorem. We consider mechanisms for a problem Π
thatareconstructedasfollows.First,onecomputesanoptimalsolutionx(cid:48) toarelaxed
problem Π(cid:48) that maximizes the declared welfare. That is, it maximizes (cid:80) b (x(cid:48)).
i∈N i
Afterwards,anα-approximateobliviousroundingschemeisappliedtoderiveafeasible
8 P.Du¨ttingetal.
solution x to the original problem Π. Each bidder is charged b (x), i.e., his declared
i
valueofthisoutcome.
THEOREM 3.1 (MAIN RESULT). Consider problem Π and a relaxation Π(cid:48). Given a
pay-your-bidmechanismM(cid:48) =(f(cid:48),p(cid:48))thatis(λ,µ)-smoothwheref(cid:48)isanexactdeclared
welfaremaximizerfortherelaxationΠ(cid:48).Thenapay-your-bidmechanismM =(f,p)for
theoriginalproblemΠthatisobtainedfromtherelaxationthroughanα-approximate
obliviousroundingschemeis(λ/(2α),µ)-smooth.
In many applications, smoothness is shown by the deviation strategy of reporting
halfone’struevalue.Firstweshowthat,whilegenerallythedeviationstrategyb(cid:48) can
i
bearbitrary,itissufficienttoconsideronlythisdeviationb(cid:48) = 1v .Weexploitthefact
i 2 i
thatf(cid:48) performsexactoptimization.
LEMMA 3.2. Given a pay-your-bid mechanism M = (f,p) that is (λ,µ)-smooth
where f is an exact declared welfare maximizer. Then M is (λ/2,µ)-smooth for devi-
ations to half the value. That is, for all bid vectors b and bids b(cid:48) = 1v for all i ∈ N,
i 2 i
(cid:80) u ((b(cid:48),b ),v )≥ λOPT(v)−µ(cid:80) p (b).
i∈N i i −i i 2 i∈N i
PROOF. We first use (λ,µ)-smoothness of M. For any valuations, there have to be
deviationbidsfulfillingtherespectiveconditions.So,inparticular,letuspretendthat
eachbidderihasvaluation 1v .Bysmoothness,therearebidsb(cid:48)(cid:48) againstbsuchthat
2 i i
(cid:88) (cid:18) 1 (cid:19) (cid:16)v(cid:17) (cid:88)
u (b(cid:48)(cid:48),b ), v ≥λOPT −µ p (b) . (1)
i i −i 2 i 2 i
i∈N i∈N
The next step is to relate the sum of utilities (cid:80) u ((b(cid:48),b ),v ) =
i∈N i i −i i
(cid:80) 1v (f (b(cid:48),b ))=(cid:80) b(cid:48)(f (b(cid:48),b ))thatagentswithvaluationsvgetinM when
i∈N 2 i i i −i i∈N i i i −i
theyunilaterallydeviatefrombtob(cid:48) tothesumofutilities(cid:80) u ((b(cid:48)(cid:48),b ),1v )that
i i∈N i i −i 2 i
theygetinM withvaluations 1v andunilateraldeviationsfrombtob(cid:48)(cid:48).
2 i
Theallocationfunctionf optimizesexactlyoveritsoutcomespace.Therefore,itcan
be used to implement a truthful mechanism MVCG = (f,pVCG) by applying VCG pay-
ments.AsVCGpaymentsarenon-negative,weget
1
u ((b(cid:48),b ),v )= v (f(b(cid:48),b ))=b(cid:48)(f(b(cid:48),b ))≥b(cid:48)(f(b(cid:48),b ))−pVCG(b(cid:48),b ) .
i i −i i 2 i i −i i i −i i i −i i −i
Observe that the latter term is exactly the utility bidder i receives in MVCG if his
valuation and bid is b(cid:48). As MVCG is truthful, this term is maximized by reporting the
i
true valuation. In other words, it can only decrease, if bidder i changes his bid to b(cid:48)(cid:48)
i
(keepingthevaluationb(cid:48)).Thatis,
i
u ((b(cid:48),b ),v )≥b(cid:48)(f(b(cid:48),b ))−pVCG(b(cid:48),b )≥b(cid:48)(f(b(cid:48)(cid:48),b ))−pVCG(b(cid:48)(cid:48),b ) .
i i −i i i i −i i −i i i −i i i −i
Finally,weusethatpVCGisnolargerthanpbecauseVCGpaymentsneverexceedbids,
i.e.,pVCG(b(cid:48)(cid:48),b ) ≤ b(cid:48)(cid:48)(f(b(cid:48)(cid:48),b )) = p (b(cid:48)(cid:48),b ).Byfurthermorechangingb(cid:48) backto 1v ,
i i −i i i −i i i −i i 2 i
weget
(cid:18) (cid:19)
1 1
u ((b(cid:48),b ),v )≥ v (f(b(cid:48)(cid:48),b ))−p (b(cid:48)(cid:48),b )=u (b(cid:48)(cid:48),b ), v .
i i −i i 2 i i −i i i −i i i −i 2 i
Summingthisinequalityoveralli∈N andcombiningitwithinequality(1),weget
(cid:88) (cid:16)v(cid:17) (cid:88) λ (cid:88)
u ((b(cid:48),b ),v )≥λOPT −µ p (b)= OPT(v)−µ p (b) .
i i −i i 2 i 2 i
i∈N i∈N i∈N
AlgorithmsasMechanisms:ThePriceofAnarchyofRelax-and-Round 9
Itremainstoshowthatsmoothnessoftherelaxationfordeviationstohalfthevalue,
implies smoothness of the derived mechanism for the original problem. As it is often
possible to directly show smoothness for deviations to half the value, we state the
following stronger version of Theorem 3.1 for relaxations that are (λ,µ)-smooth for
deviationstohalfthevalue.
Theorem 3.1 follows by first using Lemma 3.2 to argue that unconstrained (λ,µ)-
smoothness of the relaxation implies (λ/2,µ)-smoothness for deviations to half the
valueandthenusingTheorem3.1(cid:48) toshowthatthederivedmechanismis(λ/(2α),µ)-
smooth.
THEOREM 3.1(cid:48) (STRONGER VERSION OF MAIN THEOREM). If the pay-your-bid
mechanism M(cid:48) = (f(cid:48),p(cid:48)) that solves the relaxation Π(cid:48) optimally is (λ,µ)-smooth for
deviations to b(cid:48) = 1v , then the pay-your-bid mechanism M = (f,p) for Π that is ob-
i 2 i
tained from the relaxation through an α-approximate oblivious rounding scheme is
(λ/α,µ)-smooth.
PROOF. For any bid vector b, denote the utility of agent i ∈ N under mechanism
M =(f,p)byu (b,v)=v (f (b))−p (b)andundermechanismM(cid:48) =(f(cid:48),p(cid:48))byu(cid:48)(b,v)=
i i i i i
v (f(cid:48)(b))−p(cid:48)(b).
i i i
For each bidder i, we consider the unilateral deviation by b(cid:48) = 1v . As M is a pay-
i 2 i
your-bid mechanism, bidder i’s utility when bidding b(cid:48) against b can be expressed
i −i
by
1
E[u ((b(cid:48),b ),v )]=E[v (f(b(cid:48),b ))−p (b(cid:48),b )]= E[v (f(b(cid:48),b ))] .
i i −i i i i i i i −i 2 i i −i
Next we use that the outcome f(b(cid:48),b ) is derived from f(cid:48)(b(cid:48),b ) by applying an α-
i −i i −i
approximate oblivious rounding scheme by considering the weight function in which
w = v for all i and concluding that E[v (f(b(cid:48),b ))] ≥ 1v (f(cid:48)(b(cid:48),b )). That is, for
i i i i −i α i i −i
bidderi’sutility,weget
1 1
E[u ((b(cid:48),b ),v )]≥ v (f(cid:48)(b(cid:48),b ))= u(cid:48)((b(cid:48),b ),v ) ,
i i −i i 2α i i −i α i i −i i
wherethelaststepusesthatM(cid:48) isapay-your-bidmechanismaswell.
Next,weapplythefactthatM(cid:48) is(λ,µ)-smoothfordeviationstob(cid:48) = 1v .Wegetfor
i 2 i
thesumofutilitiesinM that
(cid:32) (cid:33)
(cid:88) 1 (cid:88) 1 (cid:88)
E[u ((b(cid:48),b ),v )]≥ u(cid:48)((b(cid:48),b ),v )≥ λOPT(v)−µ p(cid:48)(b) .
i i −i i α i i −i i α i
i∈N i∈N i∈N
To bound the terms p(cid:48)(b), we use once more the fact that we are applying an α-
i
approximateobliviousroundingscheme,thistimetoderivef(b)fromf(cid:48)(b)andconsid-
eringtheweightfunctioninwhichw =b foralli.Thisimplies
i i
p(cid:48)(b)=b (f(cid:48)(b))≤αE[b (f (b))]=αE[p (b)] .
i i i i i i
Overall,weget
(cid:88) 1 (cid:88)
E[u ((b(cid:48),b ),v )]≥ λOPT(v)−µ p (b) ,
i i −i i α i
i∈N i∈N
asclaimed.
4. SPARSEPACKINGINTEGERPROGRAMS
In a sparse packing integer program (PIP) each bidder i can be served in K possi-
ble ways. The fact whether bidder i gets option k is represented by a binary variable
10 P.Du¨ttingetal.
(cid:80)
x ∈ {0,1}. Each bidder i can only get one option, that is x ≤ 1 for each i.
i,k k∈[K] i,k
Furthermore, matrix A and vector c represent packing constraints between the bid-
ders, requiring that Ax ≤ c. Each bidder’s valuation depends on the option that he
(cid:80)
is served by. That is v can be expressed as v (x) = v x . The goal is to find
i i k∈[K] i,k i,k
(cid:80)
max v (x)subjecttofeasibility.
i∈N i
We consider the relaxation of this integer program in which the binary variables
x ∈ {0,1} are replaced with non-negative variables x ≥ 0. The interpretation
i,k i,k
is that x is a fractional allocation of option k to bidder i, and no bidder i can be
i,k
assignedmorethanthefractionalequivalentofoneoption.ThisrelaxationisaLPand
canthereforebesolvedinpolynomialtime.
Thecolumnsparsitydisthemaximumnumberofnon-zeroentriesinasinglecolumn
ofA.Formally,foreachvariablex ,letS bethesetofconstraintsinAwithanon-
j,k j,k
zero coefficient, that is, S = {(cid:96) | A (cid:54)= 0}. Now d = max |S |. Examples with
j,k (cid:96),j,k j,k j,k
d=1aremultiunit-auctionswithunconstrainedvaluationsorunitdemandauctions,
where each player wants at most one item, possibly with player dependent capacity
constraints, like makespan constraints in a generalized assignment problem; or more
generally, combinatorial auctions in which each bidder is interested in bundles of at
mostditemsareanexamplewithd≥1.
THEOREM 4.1. There is an oblivious rounding based, pay-your-bid mechanism for
d-sparsepackingintegerprogramsthatachievesaPriceofAnarchyof32ford=1and
of16d(d+1)forgenerald.
PROOF SKETCH. An 8d-approximate oblivious rounding scheme is available from
[Bansal et al. 2010]. We show that the LP relaxation of a d-sparse PIP is (1/2,d+1)-
smooth for deviations to b(cid:48) = 1/2v . Theorem 3.1(cid:48) then implies the Price of Anarchy
i i
guarantee.
Toestablishsmoothnesswefirstshowthatthemechanismis(1,µ)-smoothfordevia-
2
tionstob(cid:48) = 1v withµdefinedasfollows.Denotingtheoptimaldeclaredwelfareforca-
i 2 i
pacityvectorcandbidvectorbbyWb(c),wedefineµ>0tobethesmallestvaluesuch
thatforallfeasibleallocationsx¯,(cid:80)i∈N(Wb−i(c)−Wb−i(c−A(x¯i,0)))≤µ(cid:80)i∈NWb(c).
We then show that µ = (d+1) is a valid solution to this problem through the fol-
lowingscalingargument:ConstructfromWb(c)afeasibleallocationxˆ−i forcapacities
c−A(x¯ ,0) by setting xˆ−i = (1−δi )xˆ , where δi = max (A(x¯i,0))(cid:96). Then for
i j,k j,k j,k j,k (cid:96)∈Sj,k c(cid:96)
all j and k, (cid:80) δi ≤ (cid:80) (cid:80) (A(x¯i,0))(cid:96) = (cid:80) (cid:80) (A(x¯i,0))(cid:96) ≤ |S | ≤ d and
therefore (cid:80)i(cid:54)=j,ii∈Nj,k(1 − δji,ik) ≥(cid:96)∈Snj,k− d −c(cid:96) 1. It follo(cid:96)w∈Ssj,tkhati (cid:80)i∈c(cid:96)NWb−i(c −j,kA(x¯i,0)) ≥
(cid:16) (cid:17)
(cid:80)j∈N(cid:80)i(cid:54)=j,i∈N(cid:80)kbj,k 1−δji,k xˆj,k ≥(n−d−1)Wb(c),andtherefore(cid:80)i∈N(Wb−i(c)−
Wb−i(c−A(x¯i,0)))≤(d+1)Wb(c).
In stark contrast, as we show in the full version, the mechanism that solves the
integralproblemoptimallyhasanunboundedPriceofAnarchyevenwhend=1.
5. SINGLESOURCEUNSPLITTABLEFLOW
Weconsiderthesinglesourceweightedunsplittablemulti-commodityflowproblemin
which we are given a graph G = (V,E) with edge capacities c for each edge e ∈ E.
e
All bidders share a source node s and each bidder i has a sink node t . He asks for a
i
path connecting s and t fulfilling his demand d . His value for this is v , and he has
i i i
novalueforlessflowthanhisdemand.Weassumethatthesinkt anddemandd for
i i
eachplayeriscommonknowledge,sotheplayer’sbidisaclaimedvalue,whichwillbe
denotedbyb .
i