Weighted Congestion Games: Price of Anarchy, Universal Worst-Case Examples, and Tightness

23 Pages·2014·0.58 MB·English

X
Weighted Congestion Games: The Price of Anarchy, Universal
Worst-Case Examples, and Tightness
KSHIPRABHAWALKAR,StanfordUniversity,Stanford,CA,USA
MARTINGAIRING,UniversityofLiverpool,Liverpool,U.K.
TIMROUGHGARDEN,StanfordUniversity,Stanford,CA,USA.
Wecharacterizethepriceofanarchy(POA)inweightedcongestiongames,asafunctionoftheallowablere-
sourcecostfunctions.Ourresultsprovideasthoroughanunderstandingofthisquantityasisalreadyknown
fornonatomicandunweightedcongestiongames,andtaketheformofuniversal(costfunction-independent)
worst-caseexamples.Onenoteworthybyproductofourproofsisthefactthatweightedcongestiongamesare
“tight,”whichimpliesthattheworst-casepriceofanarchywithrespecttopureNashequilibria,mixedNash
equilibria,correlatedequilibria,andcoarsecorrelatedequilibriaarealwaysequal(undermildconditions
ontheallowablecostfunctions).Anotheristhefactthat,likenonatomicbutunlikeatomic(unweighted)
congestiongames,weightedcongestiongameswithtrivialstructurealreadyrealizetheworst-casePOA,at
leastforpolynomialcostfunctions.
Wealsoproveanewresultaboutunweightedcongestiongames:theworst-casepriceofanarchyinsym-
metricgamesisaslargeasintheirmoregeneralasymmetriccounterparts.
CategoriesandSubjectDescriptors:F.2.2[AnalysisofAlgorithmsandProblemComplexity]:Nonnu-
mericalAlgorithmsandProblems
GeneralTerms:Algorithms,Economics,Theory
AdditionalKeyWordsandPhrases:PriceofAnarchy,weightedcongestiongames,Nashequilibria
ACMReferenceFormat:
KshipraBhawalkar,MartinGairing,andTimRoughgarden,2014.WeightedCongestionGames:ThePrice
ofAnarchy,UniversalWorst-CaseExamplesandTightness.ACMTrans.Econ.Comp.X,X,ArticleX(June
2014),23pages.
DOI:http://dx.doi.org/10.1145/0000000.0000000
1. INTRODUCTION
The class of congestion games is expressive enough to capture a number of other-
wise unrelated applications — including routing, network design, and the migration
ofspecies(seereferencesinRoughgarden[2006])—yetstructuredenoughtopermita
usefultheory.Suchagamehasagroundsetofresources,andeachstrategyofaplayer
istoselectasubsetofthem(e.g.,apathinanetwork).Eachresourcehasaunivariate
cost function that depends on the load induced by the players that use it, and each
A previous version of this work appeared in the proceedings of the European Symposium on Algo-
rithms(ESA),2010.
ThefirstauthorwassupportedbyaStanfordGraduateFellowshipandNSFgrantCCF-1016885.Thesecond
authorwassupportedbyafellowshipwithinthePostdoc-ProgrammeoftheGermanAcademicExchange
Service(DAAD).ThethirdauthorwassupportedinpartbyNSFgrantsCCF-0448664andCCF-1016885,
an ONR Young Investigator Award, an ONR PECASE Award, an AFOSR MURI grant, and an Alfred P.
SloanFellowship.
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DOI:http://dx.doi.org/10.1145/0000000.0000000
ACMTransactionsonEconomicsandComputation,Vol.X,No.X,ArticleX,Publicationdate:June2014.
X:2 K.Bhawalkaretal.
playeraspirestominimizethesumoftheresources’costsinitschosenstrategy(given
thestrategieschosenbytheotherplayers).
Congestiongameshaveplayedastarringroleinrecentresearchonquantifyingthe
inefficiencyofgame-theoreticequilibria.TheyarerichenoughtoencodethePrisoner’s
Dilemma,andmoregenerallycanhaveNashequilibriainwhichthesumoftheplay-
ers’costsisarbitrarilylargerthanthatinaminimum-costoutcome.Thustheresearch
goalistounderstandhowtheparametersofacongestiongamegoverntheinefficiency
ofitsequilibria,andinparticulartoestablishusefulsufficientconditionsthatguaran-
teenear-optimalequilibria.
A simple observation is that the inefficiency of equilibria in a congestion game de-
pends fundamentally on the “degree of nonlinearity” of the cost functions. Because of
this,weidentifya“thoroughunderstanding”oftheinefficiencyofequilibriainconges-
tiongameswithasimultaneoussolutiontoeverypossiblespecialcaseofcostfunctions.
Inmoredetail,anidealtheorywouldincludethefollowingingredients.
(1) For every set C of allowable resource cost functions, a relatively simple recipe for
computingthelargest-possiblepriceofanarchy(POA)—theratiobetweenthesum
ofplayers’costsinanequilibriumandinaminimum-costoutcome—incongestion
gameswithcostfunctionsinC.
(2) For analytically simple classes C like bounded-degree polynomials, an exact for-
mulafortheworst-casePOAincongestiongameswithcostfunctionsinC.
(3) An understanding of the “game complexity” required for the worst-case POA to
be realized. Ideally, such a result should refer only to the strategy sets and be
independentoftheallowablecostfunctionsC.
(4) An understanding of the equilibrium concepts — roughly equivalently, the ratio-
nalityassumptionsneeded—towhichthePOAguaranteesapply.
The earliest example of such a theory is for nonatomic congestion games, where
there is a continuum of players, each of which is infinitesimally small. (The load on
a resource is defined as the mass or measure of players that choose a strategy that
includes it.) For every class C that satisfies mild conditions, the worst-case POA of
nonatomic congestion games with cost functions in C is already realized in the sim-
plestofexamples—symmetrictwo-resourcegames(i.e.,two-nodetwo-linknetworks)
— where one resource has a constant cost function and the other resource has, intu-
itively,the“steepest”costfunctioninC [Roughgarden2003;RoughgardenandTardos
2004]. This result can be interpreted as the third ingredient above — a cost function-
independent characterization of the (essentially trivial) necessary and sufficient con-
ditions on the strategy sets needed to realize the worst-case POA over all congestion
gameswithcostfunctionsfromthepermittedclass.Thischaracterizationimmediately
implies,inprinciple,arecipeforcomputingtheworst-casePOAwithrespecttoasetC
ofallowablecostfunctions—justtryeachcostfunctionfromCinatrivialtwo-resource
example,andremembertheworstexamplefound.Italsoeasilyleadstoprecisenumer-
icalboundsontheworst-casePOAforsimpleclassesofcostfunctions,suchasabound
of≈d/lndwhenC isthesetofpolynomialswithdegreeatmostdandnonnegativeco-
efficients[Correaetal.2004;Roughgarden2003].Blumetal.[2010]latersuppliedthe
final component of this research agenda by showing that all of these POA bounds ap-
plyunderrelativelyweakbehavioralassumptions—specifically,thatallusersachieve
vanishing average regret over repeated plays of the game. More recently, Aland et al.
[2011], Awerbuch et al. [2005], Christodoulou and Koutsoupias [2005b], and Rough-
garden[2009]providedthefouringredientsaboveforatomiccongestiongames,where
thereisafinitenumberofplayersandtheloadonaresourceisdefinedasthenumber
ofplayersthatuseit.
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WeightedCongestionGames:ThePOA,UniversalWorst-CaseExamples,andTightness X:3
This paper is about the fundamental model of weighted congestion games [Milch-
taich 1996; Rosenthal 1973], where each player i has a weight w and the load on a
i
resource is defined as the sum of the weights of the players that use it. Such weights
modelnon-uniformresourceconsumptionamongtheplayersandcan,ofcourse,berel-
evantformanyreasons:formodelingdifferentamountsoftraffic(e.g.,byInternetSer-
viceProvidersfromdifferent“tiers”);fordifferentdurationsofresourceusage[Shapley
1953]; and even for collusion among several identical users, who can be thought of as
asingle“virtual”playerwithweightequaltothenumberofcolludingplayers[Fotakis
etal.2008;Hayrapetyanetal.2006].
Our main contribution is a thorough understanding of the worst-case POA in
weighted congestion games, in the form of the four ingredients listed above. Inter-
esting byproducts of our proofs include the fact that weighted congestion games are
tight in the sense of Roughgarden [2009], and thus the worst-case POA with respect
to pure Nash equilibria, mixed Nash equilibria, correlated equilibria, and coarse cor-
related equilibria are always equal (under mild conditions on C); and the fact that,
likenonatomicbutunlikeatomic(unweighted)congestiongames,weightedcongestion
games with trivial structure already realize the worst-case POA, at least for polyno-
mialcostfunctions.
1.1. OverviewofResults
Result 1: Exact POA of general weighted congestion games with general
cost functions. We provide the first characterization of the exact POA of general
weighted congestion games with general cost functions. For a given set of cost func-
tionsC,thepropertiesofthefunctionsinthisclassdeterminecertain“feasiblevalues”
of two parameters λ,µ, which lead to upper bounds of the form λ/(1−µ). The best
upper bound that can be obtained using this two-parameter approach is denoted by
ζ(C). The hard work then lies in proving that there always exists a weighted conges-
tiongamethatrealizesthisupperbound.Theabstractapproachistomakeuseofthe
inequalitiesusedintheupperboundproof—inthespiritofcomplementaryslackness
argumentsinlinearprogramming—withthecostfunctionsandloadsontheresources
thatmaketheseinequalitiestightemployedintheworst-caseexample.Ultimately,we
canexhibitexampleswithPOAarbitrarilyclosetoourupperboundofζ(C).
This approach is similar to the one taken by Roughgarden [2009] for unweighted
congestion games, although non-uniform player weights create additional technical
issuesthatnecessitatecompletelydifferentconstructions.
A side effect of our upper bound proof is that ζ(C) is actually the “Robust POA” de-
finedbyRoughgarden[2009].EveryboundontherobustPOAofagameautomatically
has numerous consequences [Roughgarden 2009], and in particular upper bounds on
the POA of mixed Nash equilibria, correlated equilibria, and coarse correlated equi-
libria. (See Section 2 for definitions.) The bound of ζ(C) is valid under much weaker
rationality assumptions on players than are needed to justify convergence to pure or
mixed Nash equilibria. Since we establish a matching lower bound for the pure POA
—whichobviouslyappliesalsotothethreemoregeneralequilibriumconcepts—our
boundofζ(C)istheexactPOAwithrespecttoalloftheseequilibriumconcepts.
Exact robust POA bounds were previously established for nonatomic and atomic
unweighted congestion games [Roughgarden 2009], which are potential games and
hencepossesspureNashequilibria.Weightedcongestiongamesdonothavethesenice
properties.TheyarenotpotentialgamesandapureNashequilibriumdoesnotalways
exist(see Harksetal.[2011]andHarksandKlimm[2012]forcharacterizations),yet
ourresultsshowthattherobustPOAisexact.
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X:4 K.Bhawalkaretal.
TableI.ValuesofexactPOAboundsforpolynomialcostfunctions.α(d)isthenon-atomicPOA[Roughgar-
denandTardos2004],γ(d)istheunweightedatomicPOA[ChristodoulouandKoutsoupias2005b;Aland
etal.2011],andζ(d)istheweightedatomicPOA[Awerbuchetal.2005;Alandetal.2011].
d α(d) γ(d) ζ(d)
1 4/3 2.5 2.618
2 1.626 9.583 9.909
3 1.896 41.54 47.82
4 2.151 267.6 277.0
d O(d/logd) O((d/logd)d) O((d/logd)d)
Result 2: Exact POA of weighted congestion games on parallel links with
polynomial cost functions. We prove that, for polynomial cost functions with non-
negativecoefficientsandmaximumdegreed,theworst-casePOAisrealizedonanet-
work of parallel links (for each d). We note that even for affine cost functions, only
partial results were previously known about the worst-case POA of weighted conges-
tion games in networks of parallel links. Thus our work implies, for the first time for
weightedcongestiongames,thattheworst-casePOAisessentiallyindependentofthe
allowable network topologies in the sense of Roughgarden [2003], at least for polyno-
mial cost functions. This result stands in contrast to unweighted congestion games,
where the worst-case POA in networks of parallel links (as well as some slight gen-
eralizations) is the same as that in nonatomic congestion games, denoted α(C), which
in turn is provably smaller than the worst-case POA in general (atomic) congestion
games [Anshelevich et al. 2008; Fotakis 2010; Holzman and Law-Yone 2003] 1. See
TableIforsomeconcreteboundsforpolynomialcostfunctions.
Result 3: POA of symmetric unweighted congestion games is as large as
asymmetric ones. Ourfinalresultcontributestounderstandinghowtheworst-case
POA of unweighted congestion games depends on the game complexity. We show that
the POA of symmetric unweighted congestion games with general cost functions is
the same value γ(C) as that for asymmetric unweighted congestion games. A result
of this form was previously known only for affine cost functions [Christodoulou and
Koutsoupias 2005b]. With this result, we can conclude that the known gap between
theworst-casePOAα(C)ofnetworksofparallellinksandtheworst-casePOAγ(C)of
general unweighted atomic congestion games is located inside the class of symmetric
games,ratherthanbetweensymmetricandasymmetricgames.
1.2. FurtherRelatedWork
Koutsoupias and Papadimitriou [1999] initiated the study of the POA of mixed Nash
equilibria in weighted congestion games on parallel links, but with a different objec-
tive function: the expected maximum of the players’ costs. Lu¨cking et al. [2008] then
developed results for the pure POA of unweighted congestion games on parallel links
(withthestandardobjectivefunction)TheyshowedthattheexactPOAofunweighted
congestion games on parallel links with linear cost functions is 4/3, and when all the
links have the same linear cost function it drops to 9/8. The first results for general
networkswereobtainedindependentlybyChristodoulouandKoutsoupias[2005b]and
Awerbuchetal.[2005].ChristodoulouandKoutsoupias[2005b,a]establishedanexact
POA bound of 5/2 for unweighted congestion games with affine cost functions. They
obtainedthesameboundforthePOAofpureNashequilibria,mixedNashequilibria,
1ThepriceofanarchyboundforunweightedcongestiongamesonparallellinksfollowsfromresultsinAn-
shelevich et al. [2008] and Holzman and Law-Yone [2003]. Anshelevich et al. [2008] establish a Price of
StabilityboundforunweightedcongestiongameswhichisthesameasthenonatomicPOAbound.Holz-
man and Law-Yone [2003] show that for a slight generalization of parallel link networks the pure Nash
equilibriumisunique.TheboundwasalsoindependentlyestablishedbyFotakis[2010].
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and correlated equilibria [Christodoulou and Koutsoupias 2005a]. They also provided
an asymptotic POA upper bound of dΘ(d) for cost functions that are polynomials with
nonnegativecoefficientsanddegreeatmostd.
Intheirconcurrentpaper,Awerbuchetal.[2005]providedtheexactPOAofweighted
congestiongameswithaffinecostfunction(namely,1+φ≈2.618whereφisgoldenra-
tio). They also provided an upper bound of dΘ(d) for the POA of weighted congestion
games with cost functions that are polynomials with nonnegative coefficients and de-
greeatmostd.LaterAlandetal.[2011]obtainedexactPOAboundsforbothweighted
andunweightedcongestiongameswithcostfunctionsthatarepolynomialswithnon-
negativecoefficients.
Caragiannis et al. [2011] analyzed asymmetric singleton congestion games with
affine cost functions. They gave lower bounds which established a worst-case POA of
5/2forunweightedcongestiongamesandaPOAof1+φfortheweightedcase.Gairing
and Schoppmann [2007] provided a detailed analysis of the POA of singleton conges-
tion games. They generalized the results in Caragiannis et al. [2011] to polynomial
cost functions, showing that the worst-case POA in asymmetric singleton games is as
largeasingeneralgames.Forsymmetricsingletoncongestiongames(i.e.,networksof
parallellinks)andpolynomialcostfunctionsofmaximumdegreed,theyshowalower
bound of the (d+1)th Bell number B . This last result is subsumed by our second
d+1
contribution.
Finallyusingtheresultspresentedintheconferenceversionofthispaperasabuild-
ingblock,Roughgarden[2012]establishedanupperboundontheBayes-NashPOAin
anincompleteinformationcongestiongamewheretheplayers’weightsortheirsetsof
permittedstrategiesaredrawnfromaproductdistribution.Thelowerboundconstruc-
tionpresentedinthispaperprovesthattheBayes-NashPOAupperboundestablished
thereistight.
2. PRELIMINARIES
Congestion Games. A general weighted congestion game Γ is composed of a set of
N players N, a set of resources E and a set C of non-negative, non-decreasing cost
functionsfromR+ →R+.ForeachplayeriinN aweightw andastrategysetS ⊆2E
i i
arespecified.Thecongestiononaresourceisthetotalweightofallplayersusingthat
resource and the associated congestion cost is specified by a function c ∈ C of the
e
congestiononthatedge.
Anoutcomeisachoiceofstrategiess=(s ,s ,...,s )byplayerswiths ∈S .Then
1 2 N i i
(cid:80)
congestiononaresourceeintheoutcomesisgivenbyx = w .Thecostofa
(cid:80) e i∈N:e∈si i
playeriisC (s) = w c (x ).Thesocialcostofanoutcomeisthesumofplayer’s
i i e∈si e e
costsi.e.C(s)=(cid:80)N C (s).ThetotalcostcanalsobewrittenasC(s)=(cid:80) x c (x ).
i=1 i e∈E e e e
Aspecialtypeofcongestiongamesisunweightedcongestiongameswhereallplayers
haveunitweight.Acongestiongameissymmetricwhenallplayershavethesameset
of strategies, so S = S ⊆ 2E for all i. A game is called singleton if every strategy for
i
everyplayerhasasingleresourceinit.Asymmetricsingletoncongestiongameiswhere
allplayershaveaccesstothesamesetofsingletonstrategies.Thisiscommonlyknown
asacongestiongameonparallellinks.Anetworkcongestiongameisacongestiongame
inwhichtheresourcescorrespondtoedgesinanunderlyingnetworkandstrategiesfor
aplayeriaregivenbypathsfromavertexs toanothervertext .Figure1illustrates
i i
therelationshipsbetweenthesedifferentclasses.
Pure Nash Equilibrium. A pure Nash equilibrium is an outcome where each
player chooses a single strategy to play and no player has an incentive to deviate
from its current strategy. In the context of weighted congestion games an outcome s
isapureNashequilibriumifforeachplayeriands∗ ∈ S ,analternativestrategyfor
i i
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X:6 K.Bhawalkaretal.
Weighted
Unweighted
Asymmetric
Symmetric
Parallel
Links
Singleton
Network
Fig.1. Classesofcongestiongamesbasedonstructure
playeri,thefollowingholds:
C (s)≤C (s ,s∗). (1)
i i −i i
Here(s ,s∗)denotestheoutcomethatresultswhenplayerichangesitsstrategyins
−i i
froms tos∗
i i
ApureNashequilibriumneednothavetheminimum-possiblesocialcost.ThePOA
[KoutsoupiasandPapadimitriou1999]captureshowmuchworseNashequilibriaare
compared to the cost of the best social outcome. For a congestion game Γ, if s denotes
a Nash equilibrium with the worst social cost and s∗ an outcome with the best social
cost,thenthePOAisdefinedasC(s)/C(s∗).ThePOAofaclassofgamesistheworst
POAamongallgamesintheclass.
Other Equilibrium Concepts. As pure Nash equilibria do not always exist in
weighted congestion games, more general equilibrium concepts have been introduced
in the literature to remedy such situations, and also to weaken the rationality as-
sumptionsrequiredtojustifyconvergencetoequilibrium.Wenextreviewtheconcepts
ofmixedNash,correlated,andcoarsecorrelatedequilibria.
Aset(σ ,··· ,σ )ofindependentprobabilitydistributionsoverplayers’strategysets
1 N
isamixedNashequilibriumif
E [C (s)]≤E [C (s ,s(cid:48))]
s∼σ i s−i∼σ−i i −i i
holdsforeveryiands(cid:48) ∈S .Hereσ isaproductdistributionofallσ ’sotherthanσ
i i −i j i
ands denotesastrategydrawnfromthisdistribution.Acorrelatedequilibriumisa
−i
jointprobabilitydistributionσ overtheoutcomesofthegamesatisfying
E [C (s)|s ]≤E [C (s ,s(cid:48))|s ]
s∼σ i i s∼σ i −i i i
foreveryiands ,s(cid:48) ∈S .Acoarsecorrelatedequilibriumisgivenbyajointdistribution
i i i
σ overtheoutcomesofthegamesatisfying
E [C (s)]≤E [C (s ,s(cid:48))]
s∼σ i s∼σ i −i i
foralliands(cid:48) ∈S .
i i
For each of these equilibrium concepts the corresponding POAs (mixed POA, cor-
related POA, coarse correlated POA) are defined as the ratio of the cost of the worst
equilibriumoutcometotheoptimalsocialoutcome.AllpureNashequilibriacanberep-
resentedasmixedNashequilibria,allmixedNashequilibriaarecorrelatedequilibria,
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and all correlated equilibria are coarse correlated equilibria. Thus the corresponding
POAscanonlybenondecreasing.
Roadmap. The rest of the paper is organized as follows. Section 3 describes the
upperboundforweightedcongestiongames.Thelowerboundconstructionsforasym-
metric weighted congestion games and weighted congestion games on parallel links
are included in Section 4. Section 5 describes the lower bound construction for sym-
metricunweightedcongestiongames.WeconcludeinSection6.
3. POAOFWEIGHTEDCONGESTIONGAMES
In this section we provide an upper bound on the POA of weighted congestion games
withgeneralcostfunctions.
ForaclassoffunctionsCtheupperboundisparameterizedbytwoparametersλand
µ.Consider
A(C)={(λ,µ):µ<1;x∗c(x+x∗)≤λx∗c(x∗)+µxc(x)}. (2)
Here the constraints range over all functions c ∈ C and reals x ≥ 0 and x∗ > 0. Each
pair (λ,µ) in A(C) yields an upper bound on the pure POA of weighted congestion
games.Weestablishthefollowing,
PROPOSITION 3.1 (POA UPPER BOUND). ForaclassoffunctionsC,if(λ,µ)∈A(C)
then every weighted congestion game with cost functions in C has pure POA at most
λ/(1−µ).
PROOF. For a congestion game, let s denote a Nash equilibrium outcome and s∗
denote an outcome that minimizes the social cost. Let x and x∗ denote the loads on
e e
edgeeinoutcomessands∗ respectively.TheNashcondition(1)impliesthat
∀i, C (s)≤C (s ,s∗).
i i −i i
Summingoverallplayersweget
N N
(cid:88) (cid:88)
C(s)= C (s)≤ C (s ,s∗). (3)
i i −i i
i=1 i=1
Sinceλ,µsatisfyequation(2),
k
(cid:88) (cid:88) (cid:88)
C (s ,s∗)≤ w c (x +w )
i −i i i e e i
i=1 e∈Ei:e∈s∗
i
(cid:88)
≤ x∗c (x +x∗)
e e e e
e∈E
(cid:88)
≤ λx∗c (x∗)+µx c (x )
e e e e e e
e∈E
=λC(s∗)+µC(s). (4)
Here the second inequality follows from the fact that for every edge (cid:80) w = x∗
i:e∈s∗ i e
i
andhencew ≤x∗.Combiningequations(3)and(4)weget
i e
C(s)≤λC(s∗)+µC(s). (5)
Rearranging,
C(s) λ
POA= ≤ .
C(s∗) 1−µ
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EveryupperboundprovedusingProposition3.1isa“smoothnessargument”inthe
sense of Roughgarden [2009], and thus automatically applies to (among other things)
alloftheequilibriumconceptsdefinedinSection2.
WedenotethebestupperboundimpliedbyProposition3.1byζ(C).
Definition 3.2. ForaclassoffunctionsC,define
(cid:26) (cid:27)
λ
ζ(C):=inf :(λ,µ)∈A(C) . (6)
1−µ
Defineζ(C):=+∞ifA(C)isempty.
4. LOWERBOUNDSFORWEIGHTEDCONGESTIONGAMES
Inthissectionwedescribetwodifferentlowerboundsthatmatchtheupperboundζ(C)
giveninDefinition3.2.ThefirstlowerboundappliestoeveryclassC ofallowablecost
functions satisfying a mild technical condition, and makes use of asymmetric conges-
tion games. The second lower bound applies only to polynomial cost functions with
nonnegativecoefficients,butusesonlynetworksofparallellinks.
Foreachlowerbound,wearegivenaclassofcostfunctionsC,andwewilldescribea
seriesofgameswithPOAapproachingζ(C).Foreachgamewespecifyplayerweights,
player strategies, and congestion cost functions on the resources. Additionally, we de-
scribetwooutcomessands∗.Justifyingtheexampleasalowerboundrequirescheck-
ingthatsisa(pure)Nashequilibriumandthattheratioofcostsoftheoutcomessand
s∗ isclosetoζ(C).
Our lower bound constructions are guided by the aspiration to simultaneously sat-
isfy all of the inequalities in the proof of Proposition 3.1 exactly. This goal translates
tothefollowingconditions.
(a)Intheoutcomes,eachplayerisindifferentbetweenitsstrategys andthedeviation
i
s∗.
i
(b)Foreachplayeri,thestrategiess ands∗ aredisjoint.
i i
(c)Cost functions and congestion on resources in the outcomes s and s∗ correspond to
tuples(c,x,x∗)thatcorrespondtobindingconstraintsintheinfimumin(6).
(d)Intheoutcomes∗,eachresourceisusedbyasingleplayer.
Webelievethatsatisfyingalloftheseconditionssimultaneouslyisimpossible(andcan
proveitforcongestiongamesonparallellinks).Nevertheless,weareableto“mostly”
satisfytheseconditions,whichpermitsanasymptoticlowerboundofζ(C)asthenum-
berofplayersandresourcestendtoinfinity.
4.1. WeightedCongestionGameswithGeneralCostFunctions
Now we present the lower bound examples that obtain POA arbitrarily close to ζ(C)
formostclassesofcostfunctionsC.WeassumethattheclassC isclosedunderscaling
and dilation, meaning that if c(x) ∈ C and r ∈ R+, then rc(x) and c(rx) are also in C.
Standard scaling and replication tricks (see [Roughgarden 2003]) imply that the first
assumption is without loss of generality. The second assumption is not without loss
butissatisfiedbymostnaturalclassesofcostfunctions.
Because our assumptions on C are so weak, we have to proceed at a quite abstract
level. The first step is to study deeply the optimization problem implicit in Defini-
tion3.2,whereconstraintsonthefeasiblevaluesofλ,µprohibitthevalueofζ(C)from
gettingarbitrarilylow.Togenericallyconstructlowerboundexamples,wethenexam-
inethesetofconstraintstofindonesthatarebinding,andusethemtogetclosetothe
valuesof(λ,µ)thatyieldζ(C).
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First,weobservethatscalinganddilationdoesnotchangethesetofconstraintsin
thedefinitionofthesetA(C).ThesetofcostfunctionsC canthenbeseenascomposed
of a number of disjoint equivalence classes (where the relation is differing by scaling
and dilation). Henceforth whenever we speak of a cost function, we think of it as a
representativeofitsequivalenceclass.
The next lemma identifies one or two of the defining constraints of A(C) that cap-
turesthequantityζ(C)toarbitraryprecision.
LEMMA 4.1 (CHARACTERIZATION OF BINDING HALF-PLANES). If ζ(C) > 1, then
forevery(cid:15)suchthat0<(cid:15)<ζ(C)−1,oneofthefollowingtwoholdstrue.
(1) Thereexistsacostfunctioncandvaluesx≥0,x∗ >0suchthatx∗c(x+x∗)≥xc(x)
and
xc(x)
≥ζ(C)−(cid:15).
x∗c(x∗)
(2) There exist cost functions c ,c and values x ,x ≥ 0 and x∗,x∗ > 0 such that, if
1 2 1 2 1 2
(λ,µ)satisfy
x∗c (x +x∗)=λx∗c (x∗)+µx c (x )
1 1 1 1 1 1 1 1 1 1
x∗c (x +x∗)=λx∗c (x∗)+µx c (x ),
2 2 2 2 2 2 2 2 2 2
then λ/(1−µ) > ζ(C)−(cid:15). Moreover, x∗c (x +x∗) ≤ x c (x ) and x∗c (x +x∗) ≥
1 1 1 1 1 1 1 2 2 2 2
x c (x ).
2 2 2
PROOF. Foracostfunctionc∈C,x≥0,andx∗ >0,letHc,x,x∗ denotethehalf-plane
x∗·c(x+x∗)≤λ·x∗·c(x∗)+µ·x·c(x)
and ∂H the boundary of this half-plane. Recall from (2) that these are the half-
c,x,x∗
planes that define the set A(C) of feasible pairs (λ,µ) for the set C of cost functions.
Also,define
x·c(x) x·c(x)
β = and ζ = .
c,x,x∗ x∗·c(x+x∗) c,x,x∗ x∗·c(x∗)
Fix a positive (cid:15) < ζ(C)−1 and let ζ(cid:48) = ζ(C)−(cid:15)/2. If ζ(C) is not finite, set ζ(cid:48) = 1/(cid:15). We
writeL forthelineλ+ζ(cid:48)·µ=ζ(cid:48) intheλ,µplane.
ζ(cid:48)
If we think of a boundary line ∂H as specifying µ as a function of λ, then this
c,x,x∗
line has slope −1/ζ and µ-intercept 1/β . The half-space H consists of
c,x,x∗ c,x,x∗ c,x,x∗
everything“northeast”ofitsboundary.
Consider the half-planes with β ≤ 1. In the lucky event that there is such a
c,x,x∗
half-plane with ζ ≥ ζ(cid:48), we are done: this choice of c,x,x∗ satisfies the conditions
c,x,x∗
ofthefirstcaseofthelemma.Fortherestoftheproof,weassumethatζ < ζ(cid:48) for
c,x,x∗
everyhalf-planewithβ ≤1.
c,x,x∗
Weconsidertwocases.Todefinethem,pickanarbitrarycostfunctionc withc (1)>
1 1
0 — since C is closed under dilation, such a function exists — and a sufficiently large
value of x so that ζ > ζ(cid:48). Our standing assumption implies that β > 1.
1 c1,x1,1 c1,x1,1
Define (λˆ,µˆ) as the unique point of intersection of ∂H and L . Since the former
c1,x1,1 ζ(cid:48)
linehasalargerslope(−1/ζ vs.−1/ζ(cid:48))andasmallerµ-intercept(1/β vs.1)
c1,x1,1 c1,x1,1
thanthelatter,λˆ >0andhenceµˆ <1.
Forthefirstcase,weassumethatthereexistsahalf-planeH withβ <1
c2,x2,x∗2 c2,x2,x∗2
whoseboundaryintersectsthelineL atapoint(λ ,µ )withµ <µˆ.Equivalently,the
ζ(cid:48) 2 2 2
line∂H intersectsL totherightofwhere∂H intersectsL .Sincetheµ-
c2,x2,x∗2 ζ(cid:48) c1,x1,1 ζ(cid:48)
interceptsof∂H and∂H (namely,1/β > 1and1/β < 1)areon
c2,x2,x∗2 c1,x1,1 c2,x2,x∗2 c1,x1,1
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X:10 K.Bhawalkaretal.
eithersideofthatofL (namely,1)andλˆ >0,thisimpliesthattheintersection(λ,µ)
ζ(cid:48)
of∂H and∂H liesonthe“northeastside”ofL .Itfollowsthatλ+ζ(cid:48)µ≥ζ(cid:48).
c1,x1,1 c2,x2,x∗2 ζ(cid:48)
Thus,c ,c ,x ,x ,1,x∗,λ,µsatisfytheconditionsinthesecondcaseofthelemma.
1 2 1 2 2
Finally, assume that all half-planes H with β < 1 have boundaries that
c,x,x∗ c,x,x∗
intersect the line L at points (λ,µ) with µ ≥ µˆ. Let µ∗ denote the infimum of all µ-
ζ(cid:48)
coordinates of such intersections. Under our standing assumption, every such bound-
ary∂H hasasmallerslope(−1/ζ vs.−1/ζ(cid:48))andalargerµ-intercept(1/β
c,x,x∗ c,x,x∗ c1,x1,1
vs. 1) than L , and hence intersects L at a point (λ,µ) with 1 > µ ≥ µˆ. Thus,
ζ(cid:48) ζ(cid:48)
1>µ∗ ≥µˆ.
We now find appropriate (c ,x ,x∗) and (c ,x ,x∗) with β ≥ 1 and β <
1 1 1 2 2 2 c1,x1,x∗1 c2,x2,x∗2
1, such that the corresponding half-plane boundaries intersect L at points (λ ,µ )
ζ(cid:48) 1 1
and(λ ,µ )withµ ,µ veryclosetoµ∗.Letδ = (cid:15)·(1−µ∗) >0.Considerthepoint(ζ(cid:48)·(1−
2 2 1 2 4·ζ(cid:48)−(cid:15)
µ∗+δ),µ∗−δ)ofL .Thispointisfeasibleforallconstraintscorrespondingto(c,x,x∗)
ζ(cid:48)
with β < 1. Since ζ(cid:48) < ζ(C), this point cannot belong to the feasible set A(C) and
c,x,x∗
hencethereexists(c ,x ,x∗)withβ ≥1suchthatthepoint(ζ(cid:48)·(1−µ∗+δ),µ∗−δ)
1 1 1 c1,x1,x∗1
violatesthecorrespondingconstraint.Notethatthepoint(0,1)ofL liesinH .
ζ(cid:48) c1,x1,x∗1
Thisimpliesthat∂H intersectsL atapoint(λ ,µ )withµ ≥µ∗−δ.Moreover,
c1,x1,x∗1 ζ(cid:48) 1 1 1
λ +ζ(cid:48)·µ =ζ(cid:48).
1 1
Ifµ >µ∗,thenwecanfind(c ,x ,x∗)withβ <1thatintersectsL at(λ ,µ )
1 2 2 2 c2,x2,x∗2 ζ(cid:48) 2 2
with µ∗ ≤ µ ≤ µ . Then, similarly to the previous case, ∂H and ∂H
2 1 c1,x1,x∗1 c2,x2,x∗2
intersectatapoint(λ,µ)suchthatλ/(1−µ)≥ζ(cid:48),completingtheproof.
We can now assume that µ∗ − δ ≤ µ ≤ µ∗. By the definition of µ∗, there ex-
1
ist (c ,x ,x∗) such that ∂H intersects L at (λ ,µ ), with µ∗ ≤ µ ≤ µ∗ + δ.
2 2 2 c2,x2,x∗2 ζ(cid:48) 2 2 2
Notethatµ ≥µ andλ +ζ(cid:48)·µ =ζ(cid:48).
2 1 2 2
Let (λ,µ) be the point where ∂H and ∂H intersect. Both these bound-
c1,x1,x∗1 c2,x2,x∗2
aries have negative slopes, which means (λ,µ) lies in the triangle formed by the
points (λ ,µ ), (λ ,µ ), and (λ ,µ ). Then λ/(1−µ) ≥ λ /(1−µ ). Since λ −λ =
1 1 2 2 2 1 2 1 1 2
ζ(cid:48)·(µ −µ )≤2·ζ(cid:48)·δ,wehave
2 1
λ λ λ −λ
2 = 1 − 1 2
1−µ 1−µ 1−µ
1 1 1
2·ζ(cid:48)·δ
≥ ζ(cid:48)−
1−µ∗+δ
(cid:15)
≥ ζ(cid:48)− .
2
This proves that the conditions of the second case in the statement of the lemma
hold.
Aslongasζ(C)>1,theaboveLemmaidentifiesoneortwobindingconstraintsthat
closelyapproximatetheupperbound.Weusetheseinourlowerboundconstruction.
Fix an (cid:15) > 0. First consider the case when there are two binding constraints.
Thelemmaguaranteestriples(c¯ ,x ,x∗),(c¯ ,x ,x∗)suchthatthecorrespondinghalf-
1 1 1 2 2 2
planesintersectat(λ,µ)withλ/(1−µ)=ζ >ζ(C)−(cid:15).
(cid:15)
Let z = x /x∗ and z = x /x∗. For any w > 0, we can identify functions c ,c that
1 1 1 2 2 2 1 2
aredilatedversionsofc¯ ,c¯ suchthat(λ,µ)satisfy:
1 2
c (w·(z +1))=λ·c (w)+µ·z ·c (w·z )
1 1 1 1 1 1
c (w·(z +1))=λ·c (w)+µ·z ·c (w·z ). (7)
2 2 2 2 2 2
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