Tight Inefficiency Bounds for Perception-Parameterized Affine Congestion Games

0.33 MB·

by  Pieter Kleer

Tight Inefficiency Bounds for Perception-Parameterized
Affine Congestion Games
PieterKleer1andGuidoScha¨fer1,2
1 CentrumWiskunde&Informatica(CWI),NetworksandOptimizationGroup
2 VrijeUniversiteitAmsterdam,DepartmentofEconometricsandOperationsResearch
Amsterdam,TheNetherlands
[email protected], [email protected]
Abstract. Congestion games constitute an important class of non-cooperative games which
wasintroducedbyRosenthalin1973.Inrecentyears,severalextensionsofthesegameswere
proposedtoincorporateaspectsthatarenotcapturedbythestandardmodel.Examplesofsuch
7 extensionsincludetheincorporationofrisksensitiveplayers,themodelingofaltruisticplayer
1
behaviorandtheimpositionoftaxesontheresources.Theseextensionswerestudiedintensively
0
withthegoaltoobtainapreciseunderstandingoftheinefficiencyofequilibriaofthesegames.
2
In this paper, we introduce a new model of congestion games that captures these extensions

n (andadditionalones)inaunifyingway.Thekeyideahereistoparameterizeboththeperceived
a costofeachplayerandthesocialcostfunctionofthesystemdesigner.Intuitively,eachplayer
J
perceivestheloadinducedbytheotherplayersbyanextentofρ≥0,whilethesystemdesigner

6 estimates that each player perceives the load of all others by an extent of σ ≥0. The above
2
mentionedextensionsreducetospecialcasesofourmodelbychoosingtheparametersρandσ

accordingly.Asinmostrelatedworks,weconcentrateoncongestiongameswithaffinelatency
]
T functionshere.
G Despite the fact that we deal with a more general class of congestion games, we manage to
derivetightboundsonthepriceofanarchyandthepriceofstabilityforalargerangeofpa-
.
s rameters. Our bounds provide a complete picture of the inefficiency of equilibria for these
c
perception-parameterizedcongestiongames.Asaresult,weobtaintightboundsontheprice
[
ofanarchyandthepriceofstabilityfortheabovementionedextensions.Ourresultsalsoreveal

1 howoneshould“design”thecostfunctionsoftheplayersinordertoreducethepriceofanar-
v chy.Somewhatcounterintuitively,ifeachplayercaresaboutallotherplayerstotheextentof
4 0.625only(insteadof1inthestandardsetting)thepriceofanarchyreducesfrom2.5to2.155
1 andthisisbestpossible.
6
7
0
.
1
0
7
1
:
v
i
X
r
a
1 Introduction
Congestion games constitute an important class of non-cooperative games which was introduced
by Rosenthal in 1973 [13]. In a congestion game, we are given a set of resource from which a set
ofplayerscanchoose.Eachresourceisassociatedwithacostfunctionwhichspecifiesthecostof
this resource depending on the total number of players using it. Every player chooses a subset of
resources(fromasetofresourcesubsetsavailabletoher)andexperiencesacostequaltothesumof
thecostsofthechosenresources.Congestiongamesareboththeoreticallyappealingandpractically
relevant.Forexample,theyhaveapplicationsinnetworkrouting,resourceallocationandscheduling
problems.
Rosenthal[13]provedthateverycongestiongamehasapureNashequilibrium,i.e.,astrategy
profile such that no player can decrease her cost by unilaterally deviating to another feasible set
of resources. This result was established through the use of an exact potential function (known as
Rosenthal potential) satisfying that the cost difference induced by a unilateral player deviation is
equal to the potential difference of the respective strategy profiles. In fact, Monderer and Shapley
[11]showedthattheclassofgamesadmittinganexactpotentialfunctionisisomorphictotheclass
ofcongestiongames.
One of the main research directions in algorithmic game theory focusses on quantifying the
inefficiencycausedbyselfishbehavior.TheideaistoassessthequalityofaNashequilibriumrelative
toanoptimaloutcome.Herethequalityofanoutcomeismeasuredintermsofagivensocialcost
objective(e.g.,thesumofthecostsofallplayers).KoutsoupiasandPapadimitriou[10]introduced
thepriceofanarchyastheratiobetweentheworstsocialcostofaNashequilibriumandthesocial
costofanoptimum.Anshelevichetal.[1]definedthepriceofstabilityastheratiobetweenthebest
socialcostofaNashequilibriumandthesocialcostofanoptimum.
Inrecentyears,severalextensionsofRosenthal’scongestiongameswereproposedtoincorporate
aspects which are not captured by the standard model. For example, these extensions include risk
sensitivityofplayersinuncertainsettings[12],altruisticplayerbehavior[4,5]andcongestiongames
withtaxes[3,14].WeelaborateinmoredetailontheseextensionsinSection2.Thesegameswere
studiedintensivelywiththegoaltoobtainapreciseunderstandingofthepriceofanarchy.
In this paper, we introduce a new model of congestion games, which we term perception-
parameterized congestion games, that captures all these extensions (and more) in a unifying way.
Thekeyideahereistoparameterizeboththeperceivedcostofeachplayerandthesocialcostfunc-
tion.Intuitively,eachplayerperceivestheloadinducedbytheotherplayersbyanextentofρ ≥0,
while the system designer estimates that each player perceives the load of all others by an extent
of σ ≥0. The above mentioned extensions reduce to special cases of our model by choosing the
parametersρ andσ accordingly.
Despite the fact that we deal with a more general class of congestion games, we manage to
derive tight bounds on the price of anarchy and the price of stability for a large range of parame-
ters. Our bounds provide a complete picture of the inefficiency of equilibria for these perception-
parameterizedcongestiongames.Asaconsequence,weobtaintightboundsonthepriceofanarchy
andthepriceofstabilityfortheabovementionedextensions.Whilethepriceofanarchyboundsare
(mostly)knownfrompreviousresults,thepriceofstabilityresultsarenew.Asin[3,4,5,12,14],we
focushereoncongestiongameswithaffinecostfunctions.3
We illustrate our model by means of a simple example; formal definitions of our perception-
parameterizedcongestiongamesaregiveninSection2.Supposewearegivenasetofmresources
andthateveryplayerhastochoosepreciselyoneoftheseresources.Thecostofaresourcee∈[m]4
is given by a cost function c that maps the load on e to a real value. In the classical setting, the
e
loadofaresourceeisdefinedasthetotalnumberofplayersx usinge.Thatis,thecostthatplayer
e
i experiences when choosing resource e is c (x ). In contrast, in our setting players have different
e e
perceptionsoftheloadinducedbytheotherplayers.Moreprecisely,theperceivedload ofplayeri
choosingresourceeis1+ρ(x −1),whereρ ≥0isaparameter.Consequently,theperceivedcost
e
ofplayeriforchoosingeisc (1+ρ(x −1)).Notethatasρ increasesplayerscaremoreaboutthe
e e
3Conceptually,ourmodelcaneasilybeadaptedtomoregenerallatencyfunctions;onlythederivationofthe
respectiveboundsisanalyticallymuchmoreinvolved.
4Givenapositiveintegerm,weuse[m]torefertotheset{1,...,m}.
1
Model Parameters PoA Reference
Classical ρ=σ=1 5/2 [6]
Altruism(1) σ=1,1≤ρ≤2 (4ρ+1)/(1+ρ) [4,5]
Altruism(2) σ=1,2≤ρ≤∞ ρ+1 [5]
Uncertainty:riskneutral-players σ=ρ=1/2 5/3 [12]
Uncertainty:Wald’sminimaxprinciple σ=1/2,ρ=1 2 [2,12]
Constantuniversaltaxes σ=1,ρ=h(1). 2.155 [3]
Generalizedaffinecongestiongames − ∞ [∗]
Model Parameters PoS Reference
Classical ρ=σ=1 1.577 [3]
√ √
Altruism(1) σ=1,1≤ρ≤2 ( 3+1)/( 3+ρ−1) [∗]
Uncertainty:riskneutral-players σ=ρ=1/2 1.447 [∗]
Uncertainty:Wald’sminimaxprinciple σ=1/2,ρ=1 1 [∗]
Constantuniversaltaxes σ=1,ρ=h(1) 2.013 [∗]
Generalizedaffinecongestiongames − 2 [∗]
Table1.Anoverviewof(tight)priceofanarchyandpriceofstabilityresultsforcertainvaluesofρandσ.Here
h(1)≈0.625(seeTheorem1foraformaldefinition).Anasteriskindicatesthatthisresultisnew.
presence of other players.5 In addition, we introduce a similar parameter σ ≥0 for the social cost
objective.Intuitively,thiscanbeseenasthesystemdesigner’sestimateofhoweachplayerperceives
theloadoftheotherplayers.Inourexample,thesocialcostisdefinedas∑e∈[m]ce(1+σ(xe−1))xe.
OurResults. InSection3,weproveaboundof
(cid:26) (cid:27)
2ρ(1+σ)+1
max ρ+1, (1)
ρ+1
onthepriceofanarchyofaffinecongestiongamesforalargerangeorparameters(ρ,σ)(seealso
Figure1foranillustration).Forgeneralaffinecongestiongamesweprovethatthisboundistight.
Further,evenforthespecialcaseofsymmetricnetworkcongestiongamesweshowthatthebound
(2ρ(1+σ)+1)/(ρ+1)isasymptoticallytight(ontherangeforwhichitisattained).
InSection4,wegiveaboundof
(cid:112)
σ(σ+2)+σ
(2)
(cid:112)
σ(σ+2)+ρ−σ
on the price of stability of affine congestion games for a large range or pairs (ρ,σ) (see below
for details). For general affine congestion games we show that this bound is tight. For symmetric
networkcongestiongameswegiveabetter(tight)boundonthepriceofstabilityforthecaseσ =1
andρ ≥0arbitrary(detailsaregiveninSection5).
An overview of the price of anarchy and the price of stability results that we obtain from (1)
and(2)forseveralapplicationsknownintheliteratureisgiveninTable1.Therespectivereferences
wheretheseboundswereestablishedfirstaregivenintherightmostcolumn.Theconnectionbetween
theseapplicationsandourmodelisdiscussedindetailattheendofSection2.
Inlightoftheabovebounds,weobtainan(almost)completepictureoftheinefficiencyofequi-
libria(parameterizedbyρandσ);forexample,seeFigure3foranillustrationofthepriceofanarchy
ifσ =1.Notethatthepriceofanarchydecreasesfrom 5 forρ=1to2.155forρ=h(1)≈0.625.6
2
WereferthereadertoSection5forfurtherremarksanddiscussionsoftheresults.
5In general, the parameter ρ can be player- or resource-specific, but in this work we concentrate on the
homogenouscase(i.e.,allplayershavethesameperceptionparameter).
6Thepriceofanarchyforρ=h(1)wasfirstestablishedbyCaragiannisetal.[3].However,ourboundsreveal
thatthepriceofanarchyisinfactminimizedatρ=h(1)(seealsoFigure3).
2
2 OurModel,ApplicationsandRelatedWork
Wefirstformallyintroduceourmodelofcongestiongameswithparameterizedperceptions.Wethen
show that our model subsumes several other models that were studied in the literature as special
cases.
CongestionGames. AcongestiongameΓ isgivenbyatuple(N,E,(S) ,(c ) )whereN=[n]
i i∈N e e∈E
isthesetofplayers,E thesetofresources(orfacilities),S ⊆2E thesetofstrategiesofplayeri,and
i
c :R →R the cost function of facility e. Given a strategy profile s=(s ,...,s )∈×S, we
e ≥0 ≥0 1 n i i
definex asthenumberofplayersusingresourcee,i.e.,x =x (s)=|{i∈N:e∈s}|.IfS =S for
e e e i i j
alli,j∈N,thegameiscalledsymmetric.ForagivengraphG=(V,E),wecallΓ a(directed)network
congestiongameifforeveryplayerithereexists,t ∈V suchthatS isthesetofall(directed)(s,t)-
i i i i i
pathsinG.Anaffinecongestiongamehascostfunctionsoftheformc (x)=a x+b witha ,b ≥0.
e e e e e
Ifb =0foralle∈E,thegameiscalledlinear.
e
OurModel. Weintroduceourunifyingmodelofperception-parameterizedcongestiongameswith
affinelatencyfunctions.Forafixedparameterρ ≥0,wedefinethecostofplayeri∈N by
Cρ(s)= ∑c (1+ρ(x −1))=a [1+ρ(x −1)]+b (3)
i e e e e e
e∈si
foragivenstrategyprofiles=(s ,...,s ).Forafixedparameterσ ≥0,thesocialcostofastrategy
1 n
profilesisgivenby
Cσ(s)=∑Cσ(s)= ∑x (a [1+σ(x −1)]+b ). (4)
i e e e e
i e∈E
Werefertothecaseρ=σ =1astheclassicalcongestiongamewithcostfunctionsc (x)=a x+b
e e e
foralle∈E.
InefficiencyofEquilibria. AstrategyprofilesisaNashequilibriumifforallplayersi∈N itholds
thatCρ(s)≤Cρ(s(cid:48),s )foralls(cid:48)∈S,where(s(cid:48),s )denotesthestrategyprofileinwhichplayeri
i i i −i i i i −i
playss(cid:48)andalltheotherplayerstheirstrategyins.Thepriceofanarchy(PoA)andpriceofstability
i
(PoS)ofagameΓ aredefinedas
max Cσ(s) min Cσ(s)
s∈NE s∈NE
PoA(Γ,ρ,σ)= and PoS(Γ,ρ,σ)= ,
mins∗∈×iSiCσ(s∗) mins∗∈×iSiCσ(s∗)
where NE = NE(ρ) denotes the set of Nash equilibria with respect to the player costs as de-
fined in (3). For a collection of games H we define PoA(H,ρ,σ) = sup PoA(Γ,ρ,σ) and
Γ∈H
PoS(H,ρ,σ)=sup PoS(Γ,ρ,σ). Rosenthal [13] shows that classical congestion games (i.e.,
Γ∈H
ρ =σ =1)haveanexactpotentialfunction:Φ :×S →Risanexactpotentialfunctionforacon-
i i
gestiongameΓ ifforeverystrategyprofiles,foreveryi∈Nandeverys(cid:48)∈S: Φ(s)−Φ(s ,s(cid:48))=
i i −i i
Ci(s)−Ci(s−i,s(cid:48)i).TheRosenthalpotentialΦ(s)=∑e∈E∑xke=1ce(k)isanexactpotentialfunctionfor
classicalcongestiongames.
Applications. Wereviewvariousmodelsthatfallwithin,orarerelatedto,theframeworkproposed
above(forcertainvaluesofρ andσ).Thesemodelssometimesinterprettheparametersdifferently
thanexplainedabove.
Altruism[4,5].Wecanrewritethecostofplayerias
Cρ(s)= ∑(a x +b )+(ρ−1)a (x −1).
i e e e e e
e∈si
The term (ρ−1)a (x −1) can be interpreted as a “dynamic” (meaning load-dependent) tax that
e e
players using resource e have to pay.7 For 1≤ρ ≤∞ and σ =1, this model is equivalent to the
altruisticplayersettingproposedbyCaragiannisetal.[4].Chenetal.[5]alsostudythismodelof
7Infact,forρ=2thiscorrespondstothedynamictaxesasproposedinatechnicalreportbySingh[14].
3
altruismfor1≤ρ ≤2andσ =1.(Theequivalencebetweenthealtruisticmodelandourmodelis
provedinLemma3intheappendix.)
Constanttaxes[3].Wecanrewritethecostofplayerias
Cρ(s)= ∑ρa x +(1−ρ)a +b .
i e e e e
e∈si
NowastrategyprofilesisaNashequilibriumifforeveryplayeriandeverys(cid:48)∈S
i i
∑ρa x +(1−ρ)a +b ≤ ∑ ρa x +(1−ρ)a +b + ∑ ρa (x +1)+(1−ρ)a +b .
e e e e e e e e e e e e
e∈si e∈si∩s(cid:48)i e∈s(cid:48)i\si
Dividingbyρ givesthatsisalsoaNashequilibriumforthecostfunctions
b 1−ρ
Tρ(s)= ∑a x + e +∑ a .
i e e ρ ρ e
e∈si e∈si
That is, s is a Nash equilibrium in a classical congestion game in which players take into account
constant resource taxes of the form (1−ρ)/ρ·a . Caragiannis, Kaklamanis and Kanellopoulos
e
[3] study this type of taxes, which they call universal tax functions, for ρ satisfying (1−ρ)/ρ =
√
3/2 3−2. They consider these taxes to be refundable, i.e., they are not taken into account in the
socialcost,whichisequivalenttothecaseσ =1.Notethatthefunctionτ :(0,1]→[0,∞)defined
byτ(ρ)=(1−ρ)/ρ isbijective.8
Risksensitivityunderuncertainty[12].Nikolova,PiliourasandShamma[12]considerconges-
tiongamesinwhichthereisa(non-deterministic)orderoftheplayersoneveryresource.Aplayer
isonlyaffectedbyplayersinfrontofher.Thatis,theloadonresourceeforplayeriinastrictorder-
ingr,wherer (i)denotesthepositionofplayeri,isgivenbyx (i)=|{j∈N:r (j)≤r (i)}|.The
e e e e
costofplayeriisthenCi(s)=∑e∈sice(xe(i)).Notethatxe(i)isarandomvariableiftheorderingis
non-deterministic.Thesocialcostofthemodelisdefinedbythesumofallplayercosts,
C12(s)= ∑aexe(xe+1)+be
2
e∈E
which is independent of the ordering r. (This holds because in every ordering there is always one
playerfirst,oneplayersecond,andsoon.)Notethatthesocialcostcorrespondstothecaseσ = 1
2
in our framework. Nikolova et al. [12] study various risk attitudes towards the ordering r that is
assumed to have a uniform distribution over all possible orderings. The two relevant attitudes are
that of risk-neutral players and players applying Wald’s minimax principle. Risk-neutral players
define their cost as the expected cost under the ordering r, which correspond to the case ρ = 1 in
2
(3).Thiscanroughlybeinterpretedasthatplayersexpecttobescheduledinthemiddleonaverage.
Wald’s minimax principle implies that players assume a worst-case scenario, i.e., being scheduled
lastonalltheresources.Thiscorrespondstothecaseρ =1.
ApproximateNashequilibria[7].SupposethatsisaNashequilibriumunderthecostfunctions
definedin(3).Then,inparticular,wehave
C1(s)≤Cρ(s)≤Cρ(s(cid:48),s )≤ρC1(s(cid:48),s )
i i i i −i i i −i
foranyplayeriands(cid:48)∈S andρ ≥1.Thatis,wehaveC1(s)≤ρ·C1(s(cid:48),s )whichmeansthatthe
i i i i i −i
strategyprofilesisaρ-approximateequilibrium,asstudiedbyChristodoulou,KoutsoupiasandSpi-
rakis[7].Inparticular,thisimpliesthatanyupperboundonthepriceofanarchy,orpriceofstability,
inourframeworkyieldsanupperboundonthepriceofstabilityforρ-approximateequilibriaforthe
sameclassofgames.
Generalized affine congestion games. Let A(cid:48) denote the class of all congestion games Γ for
whichallresourceshavethesamecostfunctionc(x)=ax+b,wherea=a(Γ)andb=b(Γ)satisfy
a≥0anda+b>0.Theclassofaffinecongestiongameswithnon-negativecoefficientsiscontained
8Thisrelationbetweenaltruism(orspite)andconstanttaxesisalsomentionedbyCaragiannisetal.[4].
4
ρ ρ=2σ
ρ+1
2ρ(1+σ)+1
ρ=σ
ρ+1
ρ=h(σ)
?
0 1 1 σ
2
Fig.1.Theboundρ+1holdsforρ ≥2σ ≥1.Thebound(2ρ(1+σ)+1)/(1+ρ)holdsforσ ≤ρ ≤2σ.
Roughly speaking, this bound also holds for h(σ)≤ρ ≤σ, but our proof of Theorem 1 only works for a
discretizedrangeofσ (hencetheverticaldottedlinesinthisarea).ThefunctionhisgiveninTheorem1.
inA(cid:48) sinceeverysuchgamecanalwaysbetransformed9 intoagameΓ(cid:48) witha =1andb =0for
e e
all resources e∈E(cid:48), where E(cid:48) is the resource set ofΓ(cid:48). Without loss of generality we can assume
that a+b=1, since the cost functions can be scaled by 1/(a+b). The cost functions ofΓ ∈A(cid:48)
canthenequivalentlybewrittenasc(x)=ρx+(1−ρ)forρ ≥0.Thisispreciselythedefinitionof
Cρ(s)(witha =1andb =0takenthere).Inparticular,ifwetakeσ =ρ,meaningthatCρ(s)=
i e e
∑ Cρ(s),wehave
i∈N i
PoA(A(cid:48))=sup PoA(A,ρ,ρ) and PoS(A(cid:48))=sup PoS(A,ρ,ρ),
ρ≥0 ρ≥0
whereAdenotestheclassofaffinecongestiongameswithnon-negativecoefficients.
Duetopagelimitationssomematerialisomittedfromthemaintextbelowandcanbefoundinthe
appendix.
3 PriceofAnarchy
Inthissectionwederiveanupperboundonthepriceofanarchyforaffinecongestiongamesof
(cid:26) (cid:27)
2ρ(1+σ)+1
max ρ+1,
ρ+1
forawiderangeofpairs(ρ,σ)thatcapturesallknown(tous)priceofanarchyresultsintheliter-
aturethatfallwithinourmodel.Weshowthatthisboundistightforgeneralcongestiongames.We
alsoshowthatthebound(2ρ(1+σ)+1)/(ρ+1)is(asymptotically)tightforsymmetricnetwork
congestiongames,fortherangeof(ρ,σ)onwhichitisattained.
WeneedthefollowingtechnicalresultfortheproofofthemainresultinTheorem1:
Lemma1. LetsbeaNashequilibriumunderthecostfunctionsCρ(s)andlets∗ beaminimizerof
i
Cσ(·).Forρ,σ ≥0fixed,ifthereexistα(ρ,σ),β(ρ,σ)≥0suchthat
(1+ρ·x)y−ρ(x−1)x−x≤−β(ρ,σ)(1+σ(x−1))x+α(ρ,σ)(1+σ(y−1))y
forallnon-negativeintegersxandy,thenβ(ρ,σ)Cσ(s)≤α(ρ,σ)Cσ(s∗).
Theorem1. LetsbeaNashequilibriumunderthecostfunctionsCρ(s)andlets∗beaminimizerof
i
Cσ(·).Then
Cσ(s) 2ρ(1+σ)+1
≤ (5)
Cσ(s∗) ρ+1
9ThistransformationcanbedoneinsuchawaythatbothPoAandPoSofthegamedonotchange.Foraproof
thereaderisreferredto,e.g.,[5,Lemma4.3].
5
i) if 1 ≤σ ≤ρ ≤2σ,
2
ii) ifσ =1andh(σ)≤ρ ≤2σ,
(cid:112)
whereh(σ)=g(1+σ+ σ(σ+2),σ)istheoptimumofthefunction
σ(a2−1)
g(a,σ)= .
(1+σ)a2−(2σ+1)a+2σ(σ+1)
Furthermore,thereexistsafunction∆ =∆(σ)(specifiedintheappendix)satisfyingforeveryfixed
σ ≥1/2:if∆(σ )≥0,then(5)istrueforallh(σ )≤ρ ≤2σ .
0 0 0 0
Proof (Sketch).Forthefunctionsα(ρ,σ)=(2ρ(1+σ)+1)/(1+2σ)andβ(ρ,σ)=(1+ρ)/(1+
2σ),weprovetheinequalityinLemma1.Weshowthat,forcertainfunctions f and f ,thesmallest
1 2
ρ satisfyingtheinequalityofLemma1isgivenbythequantity
f (x,y,σ)
h(σ)= sup − 2 .
x,y∈N:f1(x,y,σ)>0 f1(x,y,σ)
Wedividetheset(x,y)∈N×Ninlinesoftheformx=ayanddeterminethesupremumoverevery
line. After that we take the supremum over all lines, which then gives the desired result. We first
showthatthecasex≤ywhichistrivial.Wethenfocusony>x.Inthiscasewewanttodetermine
f (ay,y,σ)
h(σ)= sup sup− 2 .
a∈R>1y>1 f1(ay,y,σ)
We show that h(σ)=max{γ (σ),γ (σ)} for certain functions γ and γ . Numerical experiments
1 2 1 2
suggestthat∆(σ):=γ (σ)−γ (σ)≥0,thatis,themaximumisalwaysattainedforγ (whichisthe
1 2 1
definitionofhgiveninthestatementofthetheorem).Inparticular,thismeansthatif,forafixedσ,
thenon-negativityof∆(σ)ischecked,thentheproofindeedyieldstheinequalityofLemma1for
h(σ)≤ρ ≤2σ. (cid:116)(cid:117)
Numerical experiments suggest that ∆(σ) is non-negative for all σ ≥1/2. We emphasize that
forafixedσ,with∆(σ)≥0,theproofthattheinequalityholdsforallh(σ)≤ρ ≤2σ isexactin
theparameterρ.ThefirsttwocasesofTheorem1captureallthepriceofanarchyresultsfromthe
literature.
WenextshowthattheboundofTheorem1isalsoan(asymptotic)lowerboundforlinearsym-
metricnetworkcongestiongames.10Thisimprovesaresultintherisk-uncertaintymodelofPiliouras
etal.[12],whoonlyproveasymptotictightnessforsymmetriclinearcongestiongames(fortheirre-
spectivevaluesofρ andσ).ItalsoimprovesaresultinthealtruismmodelbyChenetal.[5],who
showtightnessonlyforgeneralcongestiongames.Theproofisageneralizationoftheconstruction
of Correa et al. [8], who proved that for the classical case ρ =σ =1, the Price of Anarchy upper
boundof5/2,shownin[6],isasymptoticallytightforsymmetricnetworkcongestiongames.
Theorem2. Forρ,σ>0fixed,andplayerswithcostfunctionsCρ(s),thereexistsymmetricnetwork
i
linearcongestiongamessuchthat
Cσ(s) 2ρ(1+σ)+1
≥ −ε
Cσ(s∗) ρ+1
foranyε >0,wheresisaNashequilibrium,ands∗asociallyoptimalstrategyprofile.
Proof. Weconstructasymmetricnetworklinearcongestiongamewithnplayers.Wefirstdescribe
thegraphtopologyusedintheproofofTheorem5in[8](usingsimilarnotationandterminology).
ThegraphGconsistsofnprincipaldisjointpaths,calledP,...,P fromstot (horizontalpathsin
1 n
Figure2),eachconsisting2n−1arcs(andhence2nnodes).Withe the j-tharconpathiisdenoted
i,j
fori=1,...,nand j=1,...,2n−1.Also,v denotesthe j-thnodeonpathifori=1,...,nand
i,j
j=1,...,2n.Therearealson(n−1)connectingarcs:foreverypathithereisanarc(v ,v )
i,2k+1 i−1,2k
6
ρx x ρx x ρx x ρx
(1+ρ)x (1+ρ)x
v w
1 1
s t
v
2
w
2
0 1 2 3 4 5 6 7 8
Fig.2.Illustrationoftheinstanceforn=5.Thedashed(blue)pathindicatesthestrategyofplayer2intheNash
equilibrium.Foreveryprincipalpath,thefirstandlastarchavecost(1+ρ)x,andinbetweenthecostsalternate
betweenρxandx(startingandendingwithρx).Thediagonalconnectingarcshavecostzero.Thenumbersat
thebottomindicatethelayers.Thebold(red)subpathsindicatethetwodeviationsituationsthatareanalyzedto
provethatsisindeedaNashequilibrium.
fork=1,...,n−1,wherei−1istakenmodulon(thediagonalarcsinFigure2).For j≥1fixed,
wesaythatthearcse fori=1,...,nformthe(j−1)-thlayerofG(seeFigure2).
i,j
The cost functions are as follows. All arcs leaving s (the arcs e for i=1,...,n) and all arcs
i,1
enteringt (thearcse fori=1,...,n)havelatencyc (x)=(1+ρ)x.Foralli=1,...,n,thearcs
i,2n e
e fork=1,...,n−1havecostfunctionc (x)=ρx,whereasthearcse fork=1,...,n−2
i,2k−1 e i,2k
havecostfunctionc (x)=x.Allotherarcs(thediagonalconnectingarcs)havecostzero.
e
ThefeasiblestrategyprofiletinwhichplayeriusesprincipalpathP,foralli=1,...,nhassocial
i
costCσ(t)=n(2(1+ρ)+(n−1)ρ+(n−2))=n((1+ρ)n+ρ).ANashequilibriumisgivenby
the strategy profile in which every player k uses the following path: she starts with arcs e and
k,1
e ,thenusesallarcsoftheforme ,e ,e for j=1,...,n−1,andendswitharcs
k,2 k+j,2j k+j,2j+1 k+j,2j+2
e ,e (andusesallconnectingarcsinbetween).11Notethatallthe(principal)arcs
k+n−1,2n−2 k+n−1,2n−1
oflayer j haveload1is j iseven,andload2if j isodd.Thesocialcostofthisprofileisgivenby
Cσ(s)=n(2(1+ρ)+(n−1)·2·ρ(1+σ(2−1))+n−2)=n((1+2ρ(1+σ))n−2ρσ).Itfollows
that Cσ(s)/Cσ(t)↑(1+2ρ(1+σ))/(1+ρ) as n→∞. We now show that the above mentioned
strategyprofilesisindeedaNashequilibrium.
Fixsomeplayer,sayplayer2,asinFigure2,andsupposethatthisplayerdeviatestosomepath
Q. Let j be the first layer in which P and Q overlap. Note that j must be odd. The costCρ(s) of
2 2
player2,onthesubpathofP leadingtothefirstoverlappingarcwithQ,isatmost
2
j−1 j−1
(1+ρ)+ ·[2·[ρ(1+ρ(2−1))]+1]+ρ(1+ρ(2−1))=(1+ρ)2+ (1+2ρ(1+ρ))
2 2
ThesubpathofQleadingtothefirstoverlappingarcwithP hasCρ(Q,s )asfollows.Sheusesat
2 i −i
leastonearcineveryoddlayer(beforetheoverlappinglayer)withaloadof3andonearcofevery
evenlayer(beforetheoverlappingarc)withload2,meaningthatthecostofplayerionthesubpath
ofQisatleast
j−1 j−1
(1+ρ)(1+ρ(2−1))+ ·[(ρ(1+ρ(3−1)))+(1+ρ(2−1))]=(1+ρ)2+ ·(2ρ+1)(1+ρ)
2 2
Since1+2ρ(1+ρ)<(2ρ+1)(1+ρ)forallρ≥0,iffollowsthatthecostofplayerionthesubpath
ofP isnoworsethanthatofthesubpathofQ,whenplayer2deviatesfromP toQ.Iffollowsthat
2 2
itsufficestoshowthatP isanequilibriumstrategyinswithrespecttodeviationsQthatoverlapon
2
the first arc e with P. A similar argument shows that it also suffices to look at deviations Q for
2,1 2
whichQandP overlaponthelastarce ofP.
2 2,2n−1 2
10Intheappendix(Theorem8)weshow(non-asymptotic)tightnessforgeneralcongestiongames.
11ThisissimilartotheconstructioninTheorem5[8].
7
NowsupposethatP andQdonotoverlaponsomeinternalpartofP.Notethatthefirstarcof
2 2
Q that is not contained in P, say (v ,w ) must be in an even layer, and also that the last arc, say
2 1 1
(v ,w )(whichisaconnectingarc)isinanoddlayer(notethatv (cid:54)=sandw (cid:54)=tw.l.o.g.bywhatis
2 2 1 2
saidinthepreviousparagraph).ItisnothardtoseethatthesubpathofQfromv tow containsthe
1 2
same number of even-layered arcs as the subpath of P, and the same number of odd-layered arcs
2
asthesubpathofP.However,theloadonalltheodd-layeredarcsonthesubpathofdeviationQis
2
3,whereastheloadonodd-layeredarcsinthesubpathofP betweenv andw (instrategys)is2.
2 1 2
Similarly,theloadoneveryeven-layeredarconthesubpathofdeviationQis2,whereastheloadon
evereven-layeredarcinthesubpathofP is1.HencethesubpathofdeviationQbetweenv andw
2 1 2
canneverbeprofitable. (cid:116)(cid:117)
Forρ≥2σ,wecanobtainatightboundofρ+1onthepriceofanarchy.Remarkably,thebound
itselfdoesnotdependonσ,onlytherangeofρ andσ forwhichitholds.Fortheparametersσ =1
andρ ≥2inthealtruismmodelofCaragiannisetal.[4],thisboundisknowntobetightfornon-
symmetric singleton congestion games (where all strategies consist of a single resource). We only
providetightnessforgeneralcongestiongames,buttheconstructionissignificantlysimpler.
Theorem3. LetsbeaNashequilibriumunderthecostfunctionsCρ(s)andlets∗beaminimizerof
i
Cσ(·).ThenCσ(s)/Cσ(s∗)≤ρ+1for1≤2σ ≤ρ.Furthermore,thisboundistight.
4 PriceofStability
Inthissectionwegiveatightboundforthepriceofstabilityof
(cid:112)
σ(σ+2)+σ
(cid:112)
σ(σ+2)+ρ−σ
foralargerangeofpairs(ρ,σ).Weshowthisboundtobe(asymptotically)tight.
Weneedthefollowingtechnicallemma.
Lemma2. Forallnon-negativeintegersxandy,andσ ≥0arbitrary,wehave
(cid:18) 1(cid:19)2 1 (cid:112)
x−y+ − +2σx(x−1)+( σ(σ+2)+σ)[y(y−1)−x(x−1)]≥0.
2 4
Theorem4. LetsbeabestNashequilibriumunderthecostfunctionsCρ(s)andlets∗ beamini-
i
mizerofCσ(·).Wehave
(cid:112)
Cσ(s) σ(σ+2)+σ
≤
Cσ(s∗) (cid:112)σ(σ+2)+ρ−σ
forallσ >0and
2σ
≤ρ ≤2σ.
(cid:112)
1+σ+ σ(σ+2)
AnoverviewoftheimplicationsofthisboundisgiveninTable1.Theboundof2forgeneralized
affinecongestiongamesrequiressomeadditionalarguments(seeappendix).
Proof. OurproofissimilartoatechniqueofChristodoulou,KoutsoupiasandSpirakis[7]forupper
bounding theprice of stabilityof ρ-approximate equilibria(we elaborate onthe conceptual differ-
ence in Section 5). However, for a general σ the analysis is more involved. The main technical
contributioncomesfromestablishingtheinequalityinLemma2.
WecanwriteCρ(s)=a x +b +(ρ−1)a x ,which,byRosenthal[13],impliesthat
i e e e e e
x (x +1) (x −1)x
Φρ(s):= ∑a e e +b x +(ρ−1)∑a e e
e e e e
2 2
e∈E e∈E
8
isanexactpotentialforCρ(s).
i
TheideaoftheproofistocombinetheNashinequalities,andthefactthattheglobalminimum
ofΦρ(·)isaNashequilibrium(becauseitisanexactpotential).Letsdenotetheglobalminimum
ofΦρ,ands∗ asociallyoptimalsolution.Wecanwithoutlossofgeneralityassumethata =1and
e
b =0.TheNashinequalities(asinthepriceofanarchyanalysis)yield
e
∑x (1+ρ(x −1))≤ ∑(1+ρx )x∗
e e e e
e∈E e∈E
whereasthefactthatsisaglobaloptimumofΦρ(·)yieldsΦρ(s)≤Φρ(s∗),whichreducesto
∑ρx2+(2−ρ)x ≤ ∑ρ(x∗)2+(2−ρ)x∗.
e e e e
e∈E e∈E
Ifwecanfindγ,δ ≥0,andsomeK≥1,forwhich
(cid:0)0≤(cid:1) γ(cid:2)ρ(x∗)2+(2−ρ)x∗−ρx2−(2−ρ)x (cid:3)+δ[(1+ρx )x∗−x (1+ρ(x −1)]
e e e e e e e e
≤K·x∗[1+σ(x∗−1)]−x [1+σ(x −1)], (6)
e e e e
thenthisimpliesthatCσ(s)/Cσ(s∗)≤K.Wetake
K−1 (ρ−1)K+1
δ = and γ = .
ρ 2ρ
It is not hard to see that δ ≥0 always holds, however, for γ we have to be more careful. We will
later verify for which combinations of ρ and σ the parameter γ is indeed non-negative. Rewriting
theexpressionin(6)yieldsthatwehavetofindK satisfying
f (x ,x∗,σ) (x∗)2−2x x∗+(1+2σ)x2−x∗+(1−2σ)x
K≥ 2 e e := e e e e e e .
f (x ,x∗,ρ,σ) [(1−ρ+2σ)(x∗)2−2x x∗+(1+ρ)x2+(ρ−1−2σ)x∗−(ρ−1)x ]
1 e e e e e e e e
Notethatthisreasoningisonlycorrectif f (x ,x∗,ρ,σ)≥0.Thisistruesince
1 e e
(cid:18) 1(cid:19)2 1
f (x ,x∗,ρ,σ)= x −x∗+ − +(2σ−ρ)x∗(x∗−1)+ρx (x −1)
1 e e e e 2 4 e e e e
whichisnon-negativeforallx ,x∗∈N,σ ≥0and0≤ρ ≤2σ.Furthermore,theexpressioniszero
e e
ifandonlyif(x ,x∗)∈{(0,1),(1,1)},butforthesepairsthenominatorisalsozero,andhence,the
e e
expressionin(6)isthereforesatisfiedforthosepairs.Wecanwrite
(cid:18) 1(cid:19)2 1
f (x ,x∗,σ)= x −x∗+ − +2σx (x −1)
2 e e e e 2 4 e e
andtherefore f /f = A ,where
2 1 A+(2σ−ρ)B
(cid:18) 1(cid:19)2 1
A= x −x∗+ − +2σx (x −1), B=x∗(x∗−1)−x (x −1).
e e 2 4 e e e e e e
Notethatifρ =2σ,wehave f /f =1,andhencewecantakeK=1.Otherwise,
2 1
(cid:112)
A σ(σ+2)+σ (cid:112)
≤ =:K ⇔ A+( σ(σ+2)+σ)B≥0
(cid:112)
A+(2σ−ρ)B σ(σ+2)+ρ−σ
TheinequalityontherightistruebyLemma2.
Tofinishtheproof,wedeterminethepairs(ρ,σ)forwhichtheparameterγisnon-negative.This
holdsifandonlyif
(cid:112)
σ(σ+2)+σ
(ρ−1)K+1=(ρ−1) +1≥0.
(cid:112)
σ(σ+2)+ρ−σ
Rewritingthisyieldstheboundonρ inthestatementofthetheorem. (cid:116)(cid:117)
9