Exact Price of Anarchy for Polynomial Congestion Games

12 Pages·2005·0.17 MB·English

Exact Price of Anarchy for Polynomial Congestion
Games ?
SebastianAland,DominicDumrauf,MartinGairing,BurkhardMonien,andFlorian
Schoppmann
DepartmentofComputerScience,ElectricalEngineeringandMathematics
UniversityofPaderborn,Fu¨rstenallee11,33102Paderborn,Germany
{sebaland,masa,gairing,bm,fschopp}@uni-paderborn.de
Abstract. We show exact values for the price of anarchy of weighted and un-
weightedcongestiongameswithpolynomiallatencyfunctions.Thegivenvalues
alsoholdforweightedandunweightednetworkcongestiongames.
1 Introduction
Motivation and Framework. Large scale communication networks, like e.g. the in-
ternet,oftenlackacentralregulationforseveralreasons:Thesizeofthenetworkmay
be too large, or the users may be free to act according to their private interests. Even
cooperation among the users may be impossible due to the fact that users may not
even know each other. Such an environment—where users neither obey some central
controlinstancenorcooperatewitheachother—canbemodeledasanon-cooperative
game[18].
One of the most widely used solution concepts for non-cooperative games is the
conceptofNashequilibrium.ANashequilibriumisastateinwhichnoplayercanim-
provehisobjectivebyunilaterallychanginghisstrategy.ANashequilibriumiscalled
pureifallplayerschooseapurestrategy,andmixed ifplayerschooseprobabilitydis-
tributionsoverstrategies.
Rosenthal [25] introduced a special class of non-cooperative games, now widely
knownascongestiongames.Here,thestrategysetofeachplayerisasubsetofthepower
setofgivenresources.Theplayersshareaprivatecostfunction,definedasthesum(over
theirchosenresources)offunctionsinthenumberofplayerssharingthisresource.Later
Milchtaich [20] considered weighted congestion games as an extension to congestion
gamesinwhichtheplayershaveweightsandthusdifferentinfluenceonthecongestion
oftheresources.Weightedcongestiongamesprovideuswithageneralframeworkfor
modeling any kind of non-cooperative resource sharing problem. A typical resource
sharing problem is that of routing. In a routing game the strategy sets of the players
correspond to paths in a network. Routing games where the demand of the players
cannot be split among multiple paths are also called (weighted) network congestion
games.Anothermodelforselfishrouting—thesocalledWardropmodel—wasalready
?This work has been partially supported by the DFG-SFB 376 and by the European Union
withinthe6thFrameworkProgrammeundercontract001907(DELIS).
studiedinthe1950’s(seee.g.[3,29])inthecontextofroadtrafficsystems,wheretraffic
flows can be split arbitrarily. The Wardrop model can be seen as a special network
congestiongamewithinfinitelymanyplayerseachcarryinganegligibledemand.
Inordertomeasurethedegradationofsocialwelfareduetotheselfishbehaviorof
theplayers,KoutsoupiasandPapadimitriou[16]introducedaglobalobjectivefunction,
usuallycoinedassocialcost.Theydefinedthepriceofanarchy,alsocalledcoordination
ratio,astheworst-caseratiobetweenthevalueofsocialcostinaNashequilibriumand
thatofsomesocialoptimum.Thus,thepriceofanarchymeasurestheextenttowhich
non-cooperation approximates cooperation. The price of anarchy directly depends on
the definition of social cost. Koutsoupias and Papadimitriou [16] considered a very
simpleweightednetworkcongestiongameonparallellinks,nowknownasKP-model.
For this model they defined the social cost as the expected maximum latency. For the
Wardrop model, Roughgarden and Tardos [28] considered social cost as the total la-
tency,whichisameasureforthe(weighted)totaltraveltime.Awerbuchetal.[1]and
Christodoulou and Koutsoupias [5] considered the total latency for congestion games
withafinitenumberofplayerswithnon-negligibledemands.Inthissetting,theyshow
asymptoticboundsonthepriceofanarchyforweighted(andunweighted)congestion
gameswithpolynomiallatency(cost)functions.Here,allpolynomialsareofmaximum
degree d and have non-negative coefficients. For the case of linear latency functions
theygiveexactboundsonthepriceofanarchy.
Contribution and Comparison. In this work we prove exact bounds on the price
of anarchy for unweighted and weighted congestion games with polynomial latency
functions.Weusethetotallatencyassocialcostmeasure.Thisimprovesonresultsby
Awerbuchetal.[1]andChristodoulouandKoutsoupias[5],wherenon-matchingupper
andlowerboundsaregiven.
Wenowdescribeourfindingsinmoredetail.
– For unweighted congestion games we show that the price of anarchy (PoA) is ex-
actly
(k+1)2d+1−kd+1(k+2)d
PoA= ,
(k+1)d+1−(k+2)d+(k+1)d−kd+1
where k = bΦ c and Φ is a natural generalization of the golden ratio to larger
d d
dimensionssuchthatΦ isthesolutionto(Φ +1)d =Φd+1.Priortothispaperthe
d d d
best known upper and lower bounds were shown to be of the form dd(1−o(1)) [5].
However,thetermo(1)stillhidesagapbetweentheupperandthelowerbound.
– Forweightedcongestiongamesweshowthatthepriceofanarchy(PoA)isexactly
PoA=Φd+1.
d
This result closes the gap between the so far best upper and lower bounds of
O(2ddd+1)andΩ(dd/2)from[1].
Weshowthattheabovevaluesonthepriceofanarchyalsoholdforthesubclassesof
unweightedandweightednetworkcongestiongames.
Forourupperboundsweuseasimilaranalysisasin[5].Thecoreofouranalysisis
todetermineparametersc andc suchthat
1 2
y·f(x+1)≤c ·x·f(x)+c ·y·f(y) (1)
1 2
2
forallpolynomiallatencyfunctionsofmaximumdegreedandforallrealsx,y≥0.For
thecaseofunweighteddemandsitsufficestoshow(1)forallintegersx,y.Inorderto
provetheirupperboundChristodoulouandKoutsoupias[5]lookedat(1)withc = 1
1 2
and gave an asymptotic estimate for c . In our analysis we optimize both parameters
2
c ,c .Thisoptimizationprocessrequiresnewideasandisnon-trivial.
1 2
Table 1 shows a numerical comparison of our bounds with the previous results of
Awerbuchetal.[1]andChristodoulouandKoutsoupias[5].
For d ≥ 2, the table only gives the respective lower bounds that are given in the
cited works (before any estimates are applied). Values in parentheses denote cases in
whichtheboundforlinearfunctionsisbetterthanthegeneralcase.
In [1, Theorem 4.3], a construction scheme for networks is described with price
of anarchy approximating 1P∞ kd which yields the d-th Bell number. In [5, Theo-
e k=1 k!
rem 10], a network with price of anarchy (N−1)d+2 is given, with N being the largest
N
integerforwhich(N −1)d+2 ≤Ndholds.
Thecolumnwiththeupperboundfrom[5] is computed byusing(1)withc = 1
1 2
andoptimizingc withhelpofouranalysis.Thus,thecolumnshowsthebestpossible
2
boundsthatcanbeshownwithc = 1.
1 2
unweightedPoA weightedPoA
d Φd Ourexactresult UpperBound[5] Lowerbound[5] Ourexactresult Lowerbound[1]
1 1.618 2.5 2.5 2.5 2.618 2.618
2 2.148 9.583 10 (2.5) 9.909 (2.618)
3 2.630 41.54 47 (2.5) 47.82 5
4 3.080 267.6 269 21.33 277.0 15
5 3.506 1,514 2,154 42.67 1,858 52
6 3.915 12,345 15,187 85.33 14,099 203
7 4.309 98,734 169,247 170.7 118,926 877
8 4.692 802,603 1,451,906 14,762 1,101,126 4,140
9 5.064 10,540,286 20,241,038 44,287 11,079,429 21,147
10 5.427 88,562,706 202,153,442 132,860 120,180,803 115,975
Table1.Comparisonofourresultsto[5]and[1]
RelatedWork. ThepapersmostcloselyrelatedtoourworkarethoseofAwerbuchet
al. [1] and Christodoulou and Koutsoupias [5,4]. For (unweighted) congestion games
and social cost defined as average private cost (which in this case is the same as total
latency) it was shown that the price of anarchy of pure Nash equilibria is 5 for linear
2
latencyfunctionsanddΘ(d) forpolynomiallatencyfunctionsofmaximumdegreed[1,
5].Theboundof 5 forlinearlatencyfunctionalsoholdsforthecorrelatedandthusalso
2
forthemixedpriceofanarchy[4].Forweighted congestiongamesthemixedpriceof
√
anarchyfortotallatencyis 3+ 5 forlinearlatencyfunctionsanddΘ(d) forpolynomial
2
latencyfunctions[1].
The price of anarchy [24], also known as coordination ratio, was first introduced
andstudiedbyKoutsoupiasandPapadimitriou[16].Asastartingpointoftheirinvesti-
gationtheyconsideredasimpleweightedcongestiongameonparallellinks,nowknown
asKP-model.IntheKP-modellatencyfunctionsarelinearandsocialcostisdefinedas
themaximumexpectedcongestiononalink.Inthissetting,thereexisttightboundson
thepriceofanarchyofΘ( logm )foridenticallinks[7,15]andΘ( logm )[7]for
loglogm logloglogm
3
relatedlinks.ThepriceofanarchyhasalsobeenstudiedforvariationsoftheKP-model,
namelyfornon-linearlatencyfunctions[6,12],forthecaseofrestrictedstrategysets[2,
10],forthecaseofincompleteinformation[14]andfordifferentsocialcostmeasures
[11,17].Inparticular Lu¨ckinget al.[17] studythetotal latency(they callitquadratic
social cost) for routing games on parallel links with linear latency functions. For this
model they show that the price of anarchy is exactly 4 for case of identical player
3
weightsand 9 forthecaseofidenticallinksandarbitraryplayerweights.
8
The class of congestion games was introduced by Rosenthal [25] and extensively
studiedafterwards(seee.g.[8,20,21]).InRosenthal’smodelthestrategyofeachplayer
isasubsetofresources.Resourceutilityfunctionscanbearbitrarybuttheyonlydepend
onthenumberofplayerssharingthesameresource.Rosenthalshowedthatsuchgames
alwaysadmitapureNashequilibriumusingapotentialfunction.MondererandShap-
ley [21] characterize games that possess a potential function as potential games and
show their relation to congestion games. Milchtaich [20] considers weighted conges-
tion games with player specific payoff functions and shows that these games do not
admitapureNashequilibriumingeneral.Fotakisetal.[8,9]considerthepriceofan-
archyforsymmetricweightednetworkcongestiongamesinlayerednetworks[8]and
forsymmetric(unweighted)networkcongestiongamesingeneralnetworks[9].Inboth
casestheydefinesocialcostasexpectedmaximumlatency.Forasurveyonweighted
congestiongameswereferto[13].
InspiredbytheariseninterestinthepriceofanarchyRoughgardenandTardos[28]
re-investigatedtheWardropmodelandusedthetotallatencyasasocialcostmeasure.
Inthiscontextthepriceofanarchywasshowntobe 4 forlinearlatencyfunctions[28]
3
andΘ( d )[26]forpolynomiallatencyfunctionsofmaximumdegreed.Anoverview
logd
onresultsforthismodelcanbefoundintherecentbookofRoughgarden[27].
Roadmap. The rest of this paper is organized as follows. In Section 2 we give an
exact definition of weighted congestion games. We present exact bounds on the price
ofanarchyforunweightedcongestiongamesinSection3andforweightedcongestion
gamesinSection4.Duetolackofspaceweomitsomeoftheproofs.
2 Notations
General. Forallintegersk ≥ 0,wedenote[k] = {1,...,k},[k] = {0,...,k}.For
0
all integers d > 0, let Φ ∈ R+ denote the number for which (Φ +1)d = Φd+1.
d d d
Clearly,Φ coincideswiththegoldenratio.Thus,Φ isanaturalgeneralizationofthe
1 d
goldenratiotolargerdimensions.
Weighted Congestion Games. A weighted congestion game Γ is a tuple
Γ = (cid:0)n,E,(w ) ,(S ) ,(f ) (cid:1). Here, n is the number of players (or users)
i i∈[n] i i∈[n] e e∈E
and E is the finite set of resources. For every player i ∈ [n], w ∈ R+ is the weight
i
and S ⊆ 2E is the strategy set of player i. Denote S = S ×...×S and S =
i 1 n −i
S × ... × S × S ... × S . For every resource e ∈ E, the latency function
1 i−1 i+1 n
f : R+ → R+ describesthelatencyonresourcee.Weconsideronlypolynomialla-
e
tency functions with maximum degree d and non-negative coefficients, that is for all
e ∈ E thelatencyfunctionisoftheformf (x) = Pd a ·xj witha ≥ 0forall
e j=0 e,j e,j
j ∈[d] .
0
4
Ina(unweighted)congestiongame,theweightsofallplayersareequal.Thus,the
private cost of a player only depends on the number of players choosing the same re-
sources.
StrategiesandStrategyProfiles. Apurestrategyforplayeri ∈ [n]issomespecific
s ∈S whereasamixedstrategyP =(p(i,s )) isaprobabilitydistributionover
i i i i si∈Si
S ,wherep(i,s )denotestheprobabilitythatplayerichoosesthepurestrategys .
i i i
Apurestrategyprofileisann-tuples = (s ,...,s ) ∈ S whereasamixedstrat-
1 n
egy profile P = (P ,...,P ) is represented by an n-tuple of mixed strategies. For a
1 n
Q
mixedstrategyprofilePdenotebyp(s)= p(i,s )theprobabilitythattheplay-
i∈[n] i
erschoosethepurestrategyprofiles = (s ,...,s ).Followingstandardgametheory
1 n
notation,wedenoteP = (P ,...,P ,P ,...,P )asthe(mixed)strategypro-
−i 1 i−1 i+1 n
fileofallplayersexceptplayeriand(P ,Q )asthestrategyprofilethatresultsfrom
−i i
PifplayerideviatestostrategyQ .
i
P
PrivateCost. Fixanypurestrategyprofiles,anddenotebyl (s)= w the
e i∈[n],si3e i
loadonresourcee ∈ E.Theprivatecostofplayeri ∈ [n]inapurestrategyprofiles
isdefinedbyPC (s)=P f (l (s)).ForamixedstrategyprofileP,theprivatecost
i e∈si e e
ofplayeri∈[n]is
X
PC (P)= p(s)·PC (s).
i i
s∈S
Social Cost. Associated with a weighted congestion game Γ and a mixed strategy
profilePisthesocialcostSC(P)asameasureofsocialwelfare.Inparticularweuse
theexpectedtotallatency,thatis,
X X
SC(P)= p(s) l (s)·f (l (s))
e e e
s∈S e∈E
X X X
= p(s) w ·f (l (s))
i e e
s∈S i∈[n]e∈si
X
= w ·PC (P).
i i
i∈[n]
The optimum associated with a weighted congestion game is defined by OPT =
min SC(P).
P
NashEquilibriaandPriceofAnarchy. Weareinterestedinaspecialclassof(mixed)
strategyprofilescalledNashequilibria[22,23]thatwedescribehere.Givenaweighted
congestiongameandanassociatedmixedstrategyprofileP,aplayeri∈[n]issatisfied
ifhecannotimprovehisprivatecostbyunilaterallychanginghisstrategy.Otherwise,
playeriisunsatisfied.ThemixedstrategyprofilePisaNashequilibriumifandonly
ifallplayersi ∈ [n]aresatisfied,thatis,PC (P) ≤ PC (P ,s )foralli ∈ [n]and
i i −i i
s ∈S .
i i
Note,thatifthisinequalityholdsforallpurestrategiess ∈ S ofplayeri,thenit
i i
also holds for all mixed strategies over S . Depending on the type of strategy profile,
i
wedifferbetweenpureandmixedNashequilibria.
Thepriceofanarchy,alsocalledcoordinationratioanddenotedPoA,isthemaxi-
mumvalue,overallinstancesΓ andNashequilibriaP,oftheratio SC(P).
OPT
5
3 PriceofAnarchyforUnweightedCongestionGames
In this section, we prove the exact value for the price of anarchy of unweighted con-
gestiongameswithpolynomiallatencyfunctions.Westartwithtwotechnicallemmas
whicharecrucialfordeterminingc andc in(1)andthusforprovingtheupperbound
1 2
Theorem 1. In Theorem 2 we give a matching lower bound which also holds for un-
weightednetworkcongestiongames(Corollary1).
Lemma1. Let0≤c<1andd∈N then
0
((cid:18)x+1(cid:19)d (cid:18)x(cid:19)d+1) n o
max −c· =max (x+1)d−c·xd+1 .
x∈N0,y∈N y y x∈N0
Lemma2. Letd∈Nand
F ={g(d) :R→R|g(d)(x)=(r+1)d−x·rd+1,r∈R }
d r r ≥0
beaninfinitesetoflinearfunctions.Furthermore,letγ(s,t)fors,t ∈ R ands 6= t
≥0
denote the intersection abscissa of g(d) and g(d). Then it holds for any s,t,u ∈ R
s t ≥0
withs<t<uthatγ(s,t)>γ(s,u)andγ(u,s)>γ(u,t).
Theorem1. For unweighted congestion games with polynomial latency functions of
maximumdegreedandnon-negativecoefficients,wehave
(k+1)2d+1−kd+1(k+2)d
PoA≤ , wherek=bΦ c.
(k+1)d+1−(k+2)d+(k+1)d−kd+1 d
Proof. LetP = (P ,...,P )bea(mixed)NashequilibriumandletQ = (Q ,...,Q )
1 n 1 n
be a pure strategy profile with optimum social cost. Since P is a Nash equilibrium,
playeri∈[n]cannotimprovebyswitchingfromstrategyP tostrategyQ .Thus,
i i
X X
PC (P)= p(s) f (l (s))≤PC (P ,Q )
i e e i −i i
s∈S e∈si
 
X X X
= p(s) fe(le(s))+ fe(le(s)+1)
s∈S e∈Qi∩si e∈Qi\si
X X
≤ p(s) f (l (s)+1).
e e
s∈S e∈Qi
Summingupoverallplayersi∈[n]yields
X X X X X X
SC(P)= p(s) f (l (s))≤ p(s) f (l (s)+1)
e e e e
i∈[n]s∈S e∈si i∈[n]s∈S e∈Qi
X X
= p(s) l (Q)·f (l (s)+1).
e e e
s∈S e∈E
Now, l (Q) and l (s) are both integer, since Q and s are both pure strategy profiles.
e e
Thus,bychoosingc ,c suchthat
1 2
y·f(x+1)≤c ·x·f(x)+c ·y·f(y) (2)
1 2
6
forallpolynomialsf withmaximumdegreedandnon-negativecoefficientsandforall
x,y ∈N ,weget
0
X X
SC(P)≤ p(s) [c l (s)f (l (s))+c l (Q)f (l (Q))]=c ·SC(P)+c ·SC(Q).
1 e e e 2 e e e 1 2
s∈S e∈E
With c1 < 1 it follows that SSCC((QP)) ≤ 1−c2c1. Since P is an arbitrary (mixed) Nash
equilibriumweget
c
PoA≤ 2 . (3)
1−c
1
Infact,c andc dependonthemaximumdegreed,however,forthesakeofreadability
1 2
weomitthisdependenceinournotation.
We will now show how to determine constants c and c such that Inequality (2)
1 2
holdsandsuchthattheresultingupperboundof c2 isminimal.Todoso,wewillfirst
1−c1
showthatitsufficestoconsiderInequality(2)withy =1andf(x)=xd.
Sincef isapolynomialofmaximumdegreedwithnon-negativecoefficients,itis
sufficienttodeterminec andc thatfulfill(2)forf(x)=xrforallintegers0≤r ≤d.
1 2
Soletf(x)=xr forsome0≤r ≤d.Inthiscase(2)reducesto
y·(x+1)r ≤c ·xr+1+c ·yr+1. (4)
1 2
Foranygivenconstant0 ≤ c < 1letc (r,c )betheminimumvalueforc suchthat
1 2 1 2
(4)holds,thatis
(cid:26)y(x+1)r−c ·xr+1(cid:27) ((cid:18)x+1(cid:19)r (cid:18)x(cid:19)r+1)
c (r,c )= max 1 = max −c · .
2 1 x∈N0,y∈N yr+1 x∈N0,y∈N y 1 y
Notethat(4)holdsforanyc wheny =0.ByLemma1wehave
2
c (r,c )=max(cid:8)(x+1)r−c ·xr+1(cid:9). (5)
2 1 1
x∈N0
Now, c (r,c ) is the maximum of infinitely many linear functions in c ; one for
2 1 1
eachx∈N .DenoteF asthe(infinite)setoflinearfunctionsdefiningc (r,c ):
0 r 2 1
F :={g(r) :(0,1)→R|g(r)(c )=(x+1)r−c ·xr+1,x∈N }
r x x 1 1 0
Forthepartialderivativeofanyfunction(x,r,c )7→g(r)(c )weget
1 x 1
∂((x+1)r−c ·xr+1)
1 =(x+1)r·ln(x+1)−c ·xr+1·ln(x)
∂r 1
>ln(x+1)(cid:2)(x+1)r−c ·xr+1(cid:3)≥0,
1
for(x+1)r−c ·xr+1 ≥0,thatis,forthepositiverangeofthechosenfunctionfrom
1
F . Thus, the positive range of (x+1)d −c ·xd+1 dominates the positive range of
r 1
(x+1)r−c ·xr+1forall0≤r ≤d.Sincec (r,c )>0forall0≤r ≤d,itfollows
1 2 1
thatc (d,c ) ≥ c (r,c ),forall0 ≤ r ≤ d.Thus,withoutlossofgenerality,wemay
2 1 2 1
assumethatf(x)=xd.
Fors,t∈R ands6=tdefineγ(s,t)astheintersectionabscissaofg(d) andg(d)
≥0 s t
(asinLemma2).Nowconsidertheintersectionofthetwofunctionsg(d)andg(d) from
v v+1
F forsomev ∈ N.Weshowthatthisintersectionliesaboveallotherfunctionsfrom
d
F .
d
7
– First consider any function g(d) with z > v +1. We have g(d)(0) > g(d) (0) >
z z v+1
g(d)(0). Furthermore, by Lemma 2 we get γ(v,z) < γ(v,v +1). It follows that
v
g(d)(γ(v,v+1))>g(d)(γ(v,v+1)).
v z
– Nowconsideranyfunctiong(d) withz < v.Wehaveg(d) (0) > g(d)(0) > g(d)(0).
z v+1 v z
Furthermore, by Lemma 2 we get γ(v,z) > γ(v,v + 1). Again, it follows that
g(d)(γ(v,v+1))>g(d)(γ(v,v+1)).
v z
Thus,allintersectionsoftwoconsecutivelinearfunctionsfromF lieonc (d,c ).
d 2 1
By (3), any point that lies on c (d,c ) gives an upper bound on PoA. Let k be the
2 1
largestintegersuchthat(k+1)d ≥kd+1,thatisk=bΦ c.Then(k+2)d <(k+1)d+1.
d
Choosec andc attheintersectionofthetwolinesfromF withx=kandx=k+1,
1 2 d
thatisc =(k+1)d−c ·kd+1andc =(k+2)d−c ·(k+1)d+1.Thus,
2 1 2 1
(k+2)d−(k+1)d (k+1)2d+1−(k+2)d·kd+1
c = and c = .
1 (k+1)d+1−kd+1 2 (k+1)d+1−kd+1
Notethatbythechoiceofkwehave0<c <1.
1
Itfollowsthat
(k+1)2d+1−kd+1(k+2)d
PoA≤ .
(k+1)d+1−(k+2)d+(k+1)d−kd+1
Thiscompletestheproofofthetheorem. ut
Theorem2. For unweighted congestion games with polynomial latency functions of
maximumdegreedandnon-negativecoefficients,wehave
(k+1)2d+1−kd+1(k+2)d
PoA≥ , wherek=bΦ c.
(k+1)d+1−(k+2)d+(k+1)d−kd+1 d
Proof. Given the maximum degree d ∈ N for the polynomial latency functions, we
constructacongestiongameforn≥k+2playersand|E|=2nfacilities.
We divide the set E into two subsets E := {g ,...,g } and E := {h ,...,h }.
1 1 n 2 1 n
Eachplayerihastwopurestrategies,P :={g ,...,g ,h ,...,h }andQ :=
i i+1 i+k i+1 i+k+1 i
{g ,h }whereg :=g andh :=h forj >n.I.e.S ={Q ,P }.
i i j j−n j j−n i i i
EachofthefacilitiesinE sharethelatencyfunctionx7→axdforana∈R (yet
1 >0
tobedetermined)whereasthefacilitiesinE havelatencyx7→xd.
2
Obviously, the optimal allocation Q is for every player i to choose Q . Now we
i
determine a value for a such that the allocation P := (P ,...,P ) becomes a Nash
1 n
Equilibrium,i.e.,eachplayeriissatisfiedwithP,thatisPC (P)≤PC (P ,Q )forall
i i −i i
i∈[n],orequivalentlyk·a·kd+(k+1)·(k+1)d ≤a·(k+1)d+(k+2)d.Resolving
tothecoefficientagives
(k+1)d+1−(k+2)d
a≥ >0. (6)
(k+1)d−kd+1
Because (k + 1)d 6= kd+1, due to either k + 1 or k being odd and the other being
even, a is well defined and positive. Now since for any player i the private costs are
PC (Q)=a+1andPC (P)=a·kd+1+(k+1)d+1,itfollowsthat
i i
P
SC(P) = Pi∈[n]PCi(P) = a·kd+1+(k+1)d+1. (7)
SC(Q) PC (Q) a+1
i∈[n] i
8
Providedthat(k+1)d ≥kd+1,itisnothardtoseethat(7)ismonotonicallydecreasing
ina.Thus,weassumeequalityin(6),whichthengives
SC(P) (k+1)2d+1−kd+1(k+2)d
PoA≥ = .
SC(Q) (k+1)d+1−(k+2)d+(k+1)d−kd+1
Thiscompletestheproofofthetheorem. ut
Corollary1. ThelowerboundinTheorem2onPoAalsoholdsforunweightednetwork
congestiongames.
4 PriceofAnarchyforWeightedCongestionGames
Inthissection,weprovetheexactvalueforthepriceofanarchyofweightedcongestion
gameswithpolynomiallatencyfunctions.TheproofoftheupperboundinTheorem3
hasasimilarstructureastheonefortheunweightedcase(cf.Theorem1).InTheorem4
we give a matching lower bound which also holds for weighted network congestion
games (Corollary 2). Corollary 3 shows the impact of player weights to the price of
anarchy.
Theorem3. Forweightedcongestiongameswithpolynomiallatencyfunctionsofmax-
imumdegreedandnon-negativecoefficientswehavePoA≤Φd+1.
d
Theorem4. Forweightedcongestiongameswithpolynomiallatencyfunctionsofmax-
imumdegreedandnon-negativecoefficients,wehavePoA≥Φd+1.
d
Proof. Given the maximum degree d ∈ N for the polynomial latency functions, set
k ≥ max{(cid:0) d (cid:1),2}. Note, that (cid:0) d (cid:1) = max (cid:0)d(cid:1). We construct a congestion
bd/2c bd/2c j∈[d]0 j
gameforn=(d+1)·kplayersand|E|=nfacilities.
WedividethesetE intod+1partitions:Fori ∈ [d] ,letE := {g ,...,g },
0 i i,1 i,k
witheachg sharingthelatencyfunctionx 7→ a ·xd.Thevaluesofthecoefficients
i,j i
a willbedeterminedlater.Forsimplicityofnotation,setg := g forj > k in
i i,j i,j−k
thefollowing.
Similarly,wepartitionthesetofplayers[n]:Fori∈[d] ,letN :={u ,...,u }.
0 i i,1 i,k
TheweightofeachplayerinsetN isΦi,sow =Φi foralli∈[d] ,j ∈[k].
i d ui,j d 0
Now,foreverysetN ,eachplayeru ∈N hasexactlytwostrategies:
i i,j i
(
{g ,...,g ,g } fori=1tod
Qui,j :={gi,j} and Pui,j := {gd,j+1} d,j+(di) i−1,j fori=0
d,j+1
NowletQ:=(Q ,...,Q )andP:=(P ,...,P )bestrategyprofiles.Thefacilities
1 n 1 n
ineachsetE thenhavethefollowingloadsforQandP:
i
loadoneveryfacilitye∈E
i
i l (Q) l (P)
e e
d Φd Pd (cid:0)d(cid:1)Φl =(Φ +1)d =Φd+1
d l=0 l d d d
0tod−1 Φi Φi+1
d d
9
ForPtobecomeaNashEquilibrium,weneedtofulfillthefollowingNashinequalities
foreachsetN ofplayers:
i
i Nashinequalitytofulfill
PC (P)=(cid:0)d(cid:1)·a ·(Φd+1)d+a ·(Φi)d
1tod ui,j i d d i−1 d
≤a ·(Φi+1+Φi)d =PC (P ,Q )
i d d ui,j −ui,j ui,j
0 PC (P)=a ·(Φd+1)d ≤a ·(Φ +1)d =PC (P ,Q )
u0,j d d 0 d u0,j −u0,j u0,j
Replacing“≤”by“=”yieldsahomogeneoussystemoflinearequations,i.e.,thesystem
B ·a=0whereB isthefollowing(d+1)×(d+1)matrix:
d d
−Φd2+d+1+Φd2+d Φd2 0 ··· ··· 0 
d d d
 (cid:0) d (cid:1)Φd2+d −Φd2+1 ... ... 
 d−1 d d 
 ... 0 ... 
 
 ... ... ... ... ... 
 
Bd = (cid:0)di(cid:1)Φ...dd2+d 0... ··· 0 −Φidd0+d+1 Φ..id.d .0.. ··· 0...  (8)
 
 ... ... 0 
 
 ... ... ... ... Φdd 
Φd2+d 0 ··· 0 ··· 0 −Φd+1
d d
and a := (a ...a )t. Obviously, a solution to this system fulfills the initial Nash in-
d 0
equalities.Notethat
(Φi+1+Φi)d =(Φi)d·(Φ +1)d =Φid+d+1.
d d d d d
Claim. The(d+1)×(d+1)matrixB from(8)hasrankd.
d
Proof. Weusethewell-knownfactfromlinearalgebrathatifamatrixC resultsfrom
another matrix D by adding a multiple of one row (or column) to another row (or
column,respectively)thenrank(C)=rank(D).
NowconsiderthematrixC thatresultsfromaddingrowj multipliedbythefactor
d
Φ−1 torowj−1,sequentiallydoneforj = d+1,d,...,2.Obviously,C isalower
d d
triangular matrix with nonzero elements only in the first column and on the principal
diagonal.
ForthetopleftelementofC weget
d
d ! d ! !
−Φd2+d+1+X d Φd2+j =Φd2 · −Φd+1+X d Φj =0.
d j d d d j d
j=0 j=0
| {z }
(Φd+1)d
SinceallelementsontheprincipaldiagonalofC —withthejustshownexception
d
ofthefirstone—arenonzero,itiseasytoseethatC (andthusalsoB )hasrankd. ut
d d
10