Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

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by  George Christodoulou

Improving the Price of Anarchy for Selfish Routing via
Coordination Mechanisms
GeorgeChristodoulou KurtMehlhorn† EvangeliaPyrga‡
∗
Abstract
Wereconsiderthewell-studiedSelfishRoutinggamewithaffinelatencyfunctions.ThePriceof
2
Anarchyforthisclassofgamestakesmaximumvalue4/3; thismaximumisattainedalreadyfora
1
0 simplenetworkoftwoparallellinks,knownasPigou’snetwork.Weimproveuponthevalue4/3by
2 meansofCoordinationMechanisms.
Weincreasethelatencyfunctionsoftheedgesinthenetwork,i.e.,ifℓ (x)isthelatencyfunction
b e
e ofanedgee,wereplaceitbyℓˆe(x)withℓe(x) ℓˆe(x)forallx. Thenanadversaryfixesademand
≤
F rateasinput. TheengineeredPriceofAnarchyofthemechanismisdefinedastheworst-caseratio
oftheNashsocialcostinthemodifiednetworkovertheoptimalsocialcostintheoriginalnetwork.
3
1 Formally, ifCˆN(r) denotesthe cost of the worstNash flow in the modifiednetworkfor rate r and
C (r)denotesthecostoftheoptimalflowintheoriginalnetworkforthesameratethen
opt
]
T
Cˆ (r)
G ePoA=max N .
. r≥0 Copt(r)
s
c Wefirstexhibitasimplecoordinationmechanismthatachievesforanynetworkofparallellinks
[
an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our ba-

1 sic mechanism gives 5/4 = 1.25. Then, for the case of two parallel links, we describe an optimal
v mechanism;itsengineeredPriceofAnarchyliesbetween1.191and1.192.
7
7
8
1 Introduction
2
.
2
Wereconsiderthewell-studiedSelfishRoutinggamewithaffinecostfunctionsandaskwhetherincreas-
0
2 ing the cost functions can reduce the cost ofaNash flow1. In other words, the increased cost functions
1
should induce a user behavior that reduces cost despite the fact that the cost is now determined by in-
:
v creasedcostfunctions. Weanswerthequestion positivelyinthefollowingsense. ThePriceofAnarchy,
i
X defined as the maximum ratio of Nash cost to optimal cost, is 4/3 for this class of games. We show
r thatincreasing costscanreducethepriceofanarchy toavaluestrictly below4/3atleastforthecaseof
a
networksofparallellinks. Foranetworkoftwoparallellinks,wereducethepriceofanarchytoavalue
between 1.191 and 1.192 and prove that this is optimal. In order to state our results precisely, weneed
somedefinitions.
We consider single-commodity congestion games on networks, defined by a directed graph G =
(V,E), designated nodes s,t V, and aset ℓ=(ℓ ) of non-decreasing, non-negative functions; ℓ is
e e E e
∈ ∈
the latency function of edge e E. Let P be the set of all paths from s to t, and let f(r) be a feasible
∈
s,t-flow routing r units of flow. Forany p P, let f (r) denote theamount offlow that f(r) routes via
p
∈
path p. For ease of notation, when r is fixed and clear from context, we will write simply f f instead
, p
UniversityofLiverpool,[email protected]
∗
†Max-Planck-Institutfu¨rInformatik,Saarbru¨cken,[email protected]
‡TechnischeUniversita¨tMu¨nchen,[email protected]
1Apreliminaryversionofthisworkappearedin[8]
ℓ2(x)=1 f2∗(r)=max(0,r−1/2) f2N(r)=max(0,r−1)
ℓ1(x)=x f1∗(r)=min(1/2,r) f1N(r)=min(1,r)
PoA(r)
ℓˆ (x)=1
2
4/3
1
1 1 r x forx 1/2
2 ℓˆ (x)= ≤
1 (¥ forx>1/2
Figure 1: Pigou’s network: We show the original network, the optimal flow and the Nash flow as a
function of the rate r, respectively, the Price of Anarchy as a function of the rate (PoA(r) is 1 for
r 1/2, then starts togrow until itreaches itsmaximum of4/3 atr=1, andthen decreases again and
≤
approaches 1 as r goes to infinity), and finally the modified latency functions. We obtain ePoA(r)=1
forallr inthecaseofPigou’snetwork.
of f(r),f (r). By definition, (cid:229) f =r. Similarly, for any edge e E, let f be the amount of flow
p p P p e
going through e. We define the l∈atency of p under flow f as ℓ (f)=∈(cid:229) ℓ (f ) and the cost of flow
p e p e e
f asC(f)=(cid:229) f ℓ (f ) and useC (r) to denote the minimum cost o∈f any flowof rate r. Wewill
e E e e e opt
∈ ·
refertosuchaminimumcostflowasanoptimalflow(Opt). Afeasibleflow f thatroutesrunitsofflow
from s to t is at Nash (or Wardrop [29]) Equilibrium2 if for p ,p P with f >0, ℓ (f) ℓ (f).
1 2 ∈ p1 p1 ≤ p2
We useC (r) to denote the maximum cost of a Nash flow for rate r. The Price of Anarchy (PoA)[23]
N
(fordemandr)isdefinedas
C (r)
N
PoA(r)= and PoA=maxPoA(r).
Copt(r) r>0
PoAis bounded by 4/3 inthe case ofaffine latency functions ℓ (x)=a x+b witha 0and b 0;
e e e e e
≥ ≥
see [28, 13]. The worst-case is already assumed for a simple network of two parallel links, known as
Pigou’snetwork;seeFigure1.
ACoordination Mechanism3 replacesthecostfunctions (ℓ ) byfunctions4 ℓˆ=(ℓˆ ) suchthat
e e E e e E
ℓˆ (x) ℓ (x) for all x 0. LetCˆ(f)be the cost offlow f when f∈or each edge e E,ℓˆ is u∈sed instead
e e e
≥ ≥ ∈
ofℓ andletCˆ (r)bethemaximumcostofaNashflowofraterforthemodifiedlatencyfunctions. We
e N
definetheengineered PriceofAnarchy(fordemandr)as
Cˆ (r)
N
ePoA(r)= and ePoA=maxePoA(r).
Copt(r) r>0
Westressthattheoptimalcostreferstotheoriginallatencyfunctions ℓ.
2Thisassumescontinuityandmonotonicityofthelatencyfunctions.Fornon-continuousfunctions,seethediscussionlater
inthissection.
3Technically,weconsidersymmetriccoordinationmechanismsinthiswork,asdefinedin[9]i.e.,thelatencymodifications
affecttheusersinasymmetricfashion.
4Onecaninterpretthedifferenceℓˆe ℓeasaflow-dependenttollimposedontheedgee.
−
2
Non-continuous Latency Functions: In the previous definition, as it will become clear in Section 2,
it is important to allow non-continuous modified latencies. However, when we move from continuous
tonon-continuous latencyfunctions, Wardropequilibria donotalwaysexist. Non-continuous functions
have been studied by transport economists to model the effects of step-function congestion tolls and
trafficlights. Severalnotions ofequilibrium thathandle discontinuities have beenproposed intheliter-
ature5. The ones that are closer in spirit to Nash equilibria, are those proposed by Dafermos6 [15] and
Berstein and Smith [4]. According to the Dafermos’ [15] definition of user optimization, a flow is in
equilibriumifnosufficientlysmallfractionoftheusersonanypath,candecreasethelatencytheyexperi-
encebyswitchingtoanotherpath7. BersteinandSmith[4]introducedtheconceptofUserEquilibrium,
weakeningfurthertheDafermosequilibrium, takingthefractionoftheuserstothelimitapproaching 0.
The main idea of their definition is to capture the notion of the individual commuter, that is implicit in
Wardrop’sdefinitionforcontinuousfunctions. TheDafermosequilibriumontheotherhandisastronger
conceptthatcapturesthenotionofcoordinated deviations bygroupsofcommuters.
WeadopttheconceptofUserEquilibrium. Formally,wesaythatafeasibleflow f thatroutesrunits
offlowfromstot isaUserEquilibrium, iffforall p ,p Pwith f >0,
1 2 ∈ p1
ℓ (f) liminfℓ (f +e 1 e 1 ), (1)
p1 ≤ e 0 p2 p2− p1
↓
where1 denotes theflowwhereonlyoneunitpassesalongapath p.
p
NotethatforcontinuousfunctionstheabovedefinitionisidenticaltotheWardropEquilibrium. One
hastobecareful whendesigning aCoordination Mechanism withdiscontinuous functions, because the
existence of equilibria isnot always guaranteed8. Itisimportant to emphasize, that all themechanisms
that we suggest in this paper use both lower semicontinuous and regular9 latencies, and therefore User
Equilibrium existence is guaranteed due to the theorem of [4]. Moreover, since our modified latencies
arenon-decreasing, allUserEquilibriaarealsoDafermos-Sparrowequilibria. Fromnowon,wereferto
theUserEquilibria asNashEquilibria, orsimplyNashflows.
OurContribution: WedemonstratethepossibilityofreducingthePriceofAnarchyforSelfishRouting
viaCoordination Mechanisms. Weobtainthefollowingresultsfornetworksofkparallel links.
iforiginalandmodifiedlatencyfunctionsarecontinuous,noimprovementispossible,i.e.,ePoA
• ≥
PoA;seeSection2.
for the case of affine cost functions, wedescribe a simple coordination mechanism that achieves
• an engineered Price of Anarchy strictly less than 4/3; see Section 3. The functions ℓˆ are of the
e
form
ℓ (x) forx r
ℓˆ (x)= e ≤ e (2)
e (¥ forx>re.
Forthecaseoftwoparallel links,themechanism gives5/4(seeSection3.1),forPigou’snetwork
itgives1,seeFigure1.
Forthecaseoftwoparallellinkswithaffinecostfunctions,wedescribeanoptimal10 mechanism;
•
its engineered Price of Anarchy lies between 1.191 and 1.192 (see Sections 4 and 5). It uses
5See[26,24]foranexcellent exposureoftherelevantconcepts, therelationamongthem, aswellasforconditions that
guaranteetheirexistence.
6In[15],Dafermosweakenedtheorginaldefinitionby[14]tomakeitclosertotheconceptofNashEquilibrium.
7SeeSection5foraformaldefinition.
8Seefor example[16,4]forexampleswhereequilibriadonot existevenforthesimplest caseof twoparallellinksand
non-decreasingfunctions.
9See[4]foradefinitionofregularfunctions.
10ThelowerboundthatweprovideinSection5holdsforalldeterministiccoordinationmechanismsthatusenon-decreasing
modifiedlatencies,withrespecttobothnotionsofequilibriumdescribedinthepreviousparagraph.
3
modifiedcostfunctions oftheform
ℓ (x) forx r andx u
ℓˆ (x)= e ≤ e ≥ e (3)
e
(ℓe(ue) forre<x<ue.
The Price of Anarchy is a standard measure to quantify the effect of selfish behavior. There is a vast
literature studying the Price of Anarchy for various models of selfish routing and scheduling problems
(see [25]). We show that simple coordination mechanisms can reduce the Price of Anarchy for selfish
routinggamesbelowthe4/3worstcasefornetworksofparallellinksandaffinecostfunctions.
Webelievethatourargumentsextendtomoregeneralcostfunctions,e.g.,polynomialcostfunctions.
However,therestriction toparallellinksiscrucialforourproof. Weleaveitasamajoropenproblemto
proveresultsforgeneralnetworksoratleastmoregeneralnetworks, e.g.,series-parallel networks.
Implementation: We discuss the realization of the modified cost function in a simple traffic scenario
where the driving speed on a link is a decreasing function of the flow on the link and hence the transit
time is an increasing function. The step function in (3) can be realized by setting a speed limit corre-
sponding to transit time ℓ (u ) once the flow is above r . The functions in (2) can be approximately
e e e
realized byaccess control. Inanytimeunitonly r itemsareallowed toenter thelink. Iftheusage rate
e
ofthelinkisabover ,thequeueinfrontofthelinkwillgrowindefinitelyandhencetransittimewillgo
e
toinfinity.
Related Work: The concept of Coordination Mechanisms was introduced in (the conference version
of)[9]. Coordination Mechanisms havebeenusedtoimprovethePriceofAnarchyinscheduling prob-
lemsforparallelandrelatedmachines[9,19,22]aswellasforunrelatedmachines[3,6];theobjectiveis
makespanminimization. Veryrecently, [10]considered asanobjectivetheweightedsumofcompletion
times. Truthfulcoordination mechanismshavebeenstudied in[1,7,2].
Another very well-studied attempt to cope with selfish behavior is the introduction of taxes (tolls)
ontheedges ofthenetwork inselfishrouting games[11,18,20,21,17,5]. Thedisutility ofaplayer is
modified and equals her latency plus some toll for every edge that is used in her path. Itis wellknown
(seeforexample[11,18,20,21])thatso-called marginal costtolls,i.e.,ℓˆ (x)=ℓ (x)+xℓ (x),resultin
e e ′e
aNashflowthatisequaltotheoptimum flowfortheoriginalcostfunctions.11 Roughgarden [27]seeks
a subnetwork of a given network that has optimal Price of Anarchy for a given demand. [12] studies
thequestion whethertollscanreducethecostofaNashequilibrium. Theyshowthatfornetworks with
affine latencies, marginal cost pricing does not improve the cost of a flow at Nash equilibrium, as well
asthatthemaximumpossible benefitthatonecangetisnomorethanthatofedgeremoval.
Discussion: The results of this paper are similar in spirit to the results discussed in the previous para-
graph, but also very different. The above papers assume that taxes or tolls are determined with full
knowledge of the demand rate r. Our coordination mechanisms must a priori decide on the modified
latencyfunctionswithoutknowledgeofthedemand;itmustdeterminethemodifiedfunctionsℓˆandthen
an adversary selects the input rate r. More importantly, our target objectives are different; we want to
minimizetheratio ofthemodifiedcost (taking intoaccount theincrease ofthelatencies) overtheorig-
inal optimal cost. Our simple strategy presented in Section 3 can be viewed as ageneralization of link
removal. Removal of a link reduces the capacity of the edge to zero, our simple strategy reduces the
capacity toathreshold r .
e
11ItisimportanttoobservethatalthoughtheNashflowisequaltotheoptimumflow,itscostwithrespecttothemarginal
costfunctioncanbetwiceaslargeasitscostwithrespecttotheoriginalcostfunction.ForPigou’snetwork,themarginalcosts
areℓˆ1(x)=2xandℓˆ2(x)=1. ThecostofaNashflowofraterwithr 1/2is2r2withrespecttomarginalcosts;thecostof
thesameflowwithrespecttotheoriginalcostfunctionsisr2. ≤
4
2 Continuous Latency Functions Yield No Improvement
The network in this section consists of k parallel links connecting s to t and the original latency func-
tions are assumed tobe continuous and non-decreasing. Weshow that substituting them by continuous
functions bringsnoimprovement.
Lemma 1. Assume that the original functions ℓ are continuous and non-decreasing. Consider some
e
modified latency functions ℓˆandsomerater forwhichthereisaNashEquilibrium flow fˆsuchthat the
latencyfunction ℓˆ iscontinuous at fˆ(r)forall1 i k. ThenePoA(r) PoA(r).
i i
≤ ≤ ≥
Proof. It is enough to show thatCˆ (r) C (r). Let f be a Nash flow for rate r and the original cost
N N
≥
functions. If f = fˆ, the claim is obvious. If fˆ= f, there must be a j with fˆ(r) > f (r). The local
j j
continuity of ℓˆ at fˆ(r), implies that ℓˆ(fˆ(r)) =6 ℓˆ (fˆ (r)), for all i,i k such that fˆ(r),fˆ (r) > 0.
i i i i i i ′ i i
′ ′ ≤ ′
Therefore,
k
Cˆ (r)=Cˆ(fˆ(r))= (cid:229) fˆ(r)ℓˆ(fˆ(r))=r ℓˆ (fˆ(r)) r ℓ (fˆ(r)) r ℓ (f (r))
N i i i j j j j j j
· ≥ · ≥ ·
i=1
sinceℓˆ (x) ℓ (x)forallxandℓ isnon-decreasing. Since f isaNashflowwehaveℓ(f(r)) ℓ (f (r))
j j j i i j j
≥ ≤
foranyiwith f(r)>0. Thus
i
k
(cid:229)
C (r)= f(r)ℓ(f(r)) r ℓ (f (r)).
N i i i j j
≤ ·
i=1
3 A Simple Coordination Mechanism
Let ℓ(x) = ax+b = (x+g )/l be the latency function of the i-th link, 1 i k. We call l the
i i i i i i
≤ ≤
efficiencyofthelink. Weorderthelinksinorderofincreasingb-valueandassumeb <b <...<b as
1 2 k
two links with the same b-value may be combined (by adding their efficiencies). We may also assume
a >0fori<k;ifa =0,linksi+1andhigherwillneverbeused. Wesaythatalinkisusedifitcarries
i i
positiveflow. ThefollowingtheoremsummarizessomebasicfactsaboutoptimalflowsandNashflows;
itisprovedby straightforward calculations.12 Westatethe theorem forthecase thata ispositive. The
k
theorem isreadily extended tothecasea =0byletting a gotozeroanddetermining thelimitvalues.
k k
Wewillonlyusethetheorem insituations, wherea >0.
k
Theorem1. Letb <b <...<b andl ´ foralli. LetL =(cid:229) l andG =(cid:229) g . Considera
1 2 k i j i j i j i j j
fixed rate r and let f and fN, 1 i k, b∈e the optimal flow and the≤Nash flow for ra≤te r respectively.
i∗ i ≤ ≤
Let
r = (cid:229) (b b)L .
j i+1 i i
−
1 i<j
≤
Then Nash uses link j for r>r and Opt uses link j for r>r /2. If Opt uses exactly j links at rate r
j j
then
rl G l
f = i +d /2, whered = j i g
i∗ L i i L i
−
j j
12InaNashflowallusedlinkshavethesamelatency. Thus,if jlinksareusedatraterand fN istheflowonthei-thlink,
i
thena1f1N+b1=...=ajfNj +bj≤bj+1andr= f1N+...+fNj . Thevaluesforrj and fiN followfromthis. Similarly,inan
optimalflowallusedlinkshavethesamemarginalcosts.
5
and
C (r)= 1 r2+G r (cid:229) d i2 = 1 r2+G r C ,whereC = (cid:229) (b b)2l l /(4L ).
opt L j − 4l L j − j j h− i h i j
j i j i j i h j !
(cid:0) (cid:1) ≤ (cid:0) (cid:1) ≤ ≤
IfNashusesexactly jlinksatraterthen
rl 1
fN = i +d and C (r)= r2+G r .
i L i N L j
j j
(cid:0) (cid:1)
Ifs<randOptusesexactly jlinksatsandrthen
1
C (r)=C (s)+ (r s)2+(G +2s)(r s) .
opt opt L − j −
j
(cid:0) (cid:1)
Ifs<randNashusesexactly jlinksatsandrthen
1
C (r)=C (s)+ (r s)2+(G +2s)(r s) .
N N L j
− −
j
(cid:0) (cid:1)
Finally,G +r =b L andG +r =b L .
j j j j j 1 j j j 1
− −
Wenext define our simple coordination mechanism. It isgoverned by parameters R , R ,...,R ;
1 2 k 1
R 2 for all i. We call the j-th link super-efficient if l >R L . In Pigou’s network, the seco−nd
i j j 1 j 1
link≥is super-efficient for any choice of R since l =¥ and l −=1.−Super-efficient links are the cause
1 2 1
of high Price of Anarchy. Observe that Opt starts using the j-th link at rate r /2 and Nash starts using
j
itatrater . Ifthe j-thlinkissuper-efficient, Optwillsendasignificant fraction ofthetotalflowacross
j
the j-th link and this will result in a high Price of Anarchy. Our coordination mechanism induces the
Nash flow to use super-efficient links earlier. The latency functions ℓˆ are defined as follows: ℓˆ =ℓ if
i i i
there is no super-efficient link j>i; in particular the latency function of the highest link (= link k) is
unchanged. Otherwise, we choose a threshold value T (see below) and set ℓˆ(x)=ℓ(x) for x T and
i i i i
ℓˆ(x)=¥ forx>T. Thethreshold valuesarechosensothatthefollowingbehavior results. We≤callthis
i
behavior modifiedNash(MN).
AssumethatOptuseshlinks,i.e.,r /2 r r /2. Ifl RL foralli,1 i<h,MNbehaves
h h+1 i+1 i i
≤ ≤ ≤ ≤
like Nash. Otherwise, let j be minimal such that link j+1is super-efficient; MN changes its behavior
at rate r /2. More precisely, it freezes the flow across the first j links at their current values when
j+1
the total flow is equal to r /2 and routes any additional flow across links j+1 to k. The thresholds
j+1
for the lower links are chosen in such a way that this freezing effect takes place. The additional flow
is routed by using the strategy recursively. In other words, let j +1, ..., j +1 be the indices of the
1 t
super-efficient links. ThenMNchanges behavior atrates r /2. Atthis ratethe flowacross links 1to
ji+1
j isfrozenandadditional flowisroutedacrossthehigherlinks.
i
WeuseC (r)=CˆR1,...,Rk 1(r)todenotethecostofMNatraterwhenoperatedwithparametersR
MN N − 1
to R . Then ePoA(r)=C (r)/C (r). For the analysis of MN we use the following strategy. We
k 1 MN opt
−
first investigate the benign case when there is no super-efficient link. In the benign case, MN behaves
like Nash and the worst case bound of 4/3 on the PoA can never be attained. More precisely, we will
exhibit a function B(R ,...,R ) which is smaller than 4/3 for all choices of the R’s and will prove
1 k 1 i
−
C (r) B(R ,...,R )C (r). Wetheninvestigate thenon-benign case. Wewillderivearecurrence
MN 1 k 1 opt
≤ −
relationfor
CˆR1,...,Rk 1(r)
ePoA(R ,...,R )=max N −
1 k 1
− r Copt(r)
In the case of a single link, i.e., k = 1, MN behaves like Nash which in turn is equal to Opt. Thus
ePoA()=1. The coming subsections are devoted to the analysis of two links and more than two links,
respectively.
6
3.1 Two Links
Themodifiedalgorithm isdetermined byaparameter R>1. Ifl Rl ,modifiedNashisidentical to
2 1
≤
Nash. If l >Rl , the modified algorithm freezes the flow across the first link at r /2 once it reaches
2 1 2
this level. In Pigou’s network we have ℓ (x) = x and ℓ (x) = 1. Thus l = ¥ . The modified cost
1 2 2
functions areℓˆ (x)=ℓ (x)andℓˆ (x)=xforx r /2=1/2andℓˆ (x)=¥ forx>1/2. TheNashflow
2 2 1 2 1
≤
with respect to the modified cost function is identical to the optimum flow in the original network and
Cˆ (f )=C(f ). ThusePoA=1forPigou’snetwork.
N ∗ ∗
Theorem 2. Forthe case oftwo links, ePoA max(1+1/R,(4+4R)/(4+3R). Inparticular ePoA=
≤
5/4forR=4.
Proof. Consider first the benign case l RL . There are three regimes: for r r /2, Opt and Nash
2 1 2
≤ ≤
behave identically. For r /2 r r , Opt uses both links and Nash uses only the first link, and for
2 2
≤ ≤
r r ,OptandNashusebothlinks. PoA(r)isincreasingforr r anddecreasingforr r . Theworst
2 2 2
≥ ≤ ≥
case isatr=r . ThenPoA(r )=C (r )/C (r )=C (r )/(C (r ) C )=1/(1 C /C (r )). We
2 2 N 2 opt 2 N 2 N 2 2 2 N 2
− −
upper-bound C /C (r ). Recall that r =(b b )l , r +G =b l andC (r )=(1/l (r2+G r ).
2 N 2 2 2− 1 1 2 1 2 1 N 2 1 2 1 2
Weobtain
C (b b )2l l (b b )2l l l 1
2 2 1 1 2 2 1 1 2 2
= − = − .
C (r ) 4L (1/l )(r2+g r ) 4L (1/l )(b b )l b l ≤ 4L ≤ 4(1+1/R)
N 2 2 1 2 1 2 2 1 2− 1 1 2 1 2
ThusPoA(r) B(R):=1/(1 R/(4(R+1)))=(4+4R)/(4+3R).
≤ −
Wecometothecasel >RL : Therearetworegimes: forr r /2,OptandMNbehaveidentically.
2 1 2
≤
For r >r /2, Opt uses both links and MN routes r /2 over the first link and r r /2 over the second
2 2 2
−
link. Thusforr r /2:
2
≥
ePoA(r)= CN(r) = Copt(r2/2)+(r−rl22/2)2 +b2(r−r2/2) L 2 1+1/R.
Copt(r) Copt(r2/2)+(r−Lr2/2)2 +b2(r r2/2) ≤ l 2 ≤
2 −
3.2 Many Links
Asalreadymentioned,wedistinguishcases. Wefirststudythebenigncasel RL foralli,1 i<k,
i+1 i i
≤ ≤
andthendealwiththenon-benign case.
TheBenignCase: Weassume l RL foralli, 1 i<k. ThenMNbehaves like Nash. Wewill
i+1 i i
≤ ≤
show ePoA B(R ,...,R )<4/3; here B stands for benign case or base case. Our proof strategy is
1 k 1
≤ −
as follows; wewill first show (Lemma2)that for the i-th link the ratio of Nash flowto optimal flow is
bounded by 2L /(L +L ). This ratio is never more than two; in the benign case, it is bounded away
k i k
fromtwo. WewillthenusethisfacttoderiveaboundonthePriceofAnarchy(Lemma4).
Lemma2. LethbethenumberoflinksthatOptisusing. Then
fN 2L
i h
f ≤ L +L
i∗ i h
fori h. Ifl R L forall j,then
j+1 j j
≤ ≤
2L 2P
h
,
L +L ≤ P+1
i h
whereP:=R (cid:213) (1+R).
1 1<i<k i
·
7
Proof. Fori> j,theNashflowonthei-thlinkiszeroandtheclaimisobvious. Fori j,wecanwrite
≤
theNashandtheoptimalflowthroughlinkias
fN =rl /L +(G l /L g ) and f =rl /L +(G l /L g )/2
i i j j i j i i∗ i h h i h i
− −
Thereforetheirratioasafunction ofris
fN L 2r+2G 2bL
F(r)= i = h j− i j
f L · 2r+G bL
i∗ j h− i h
ThesignofthederivativeF (r)isequaltothesignofG bL 2G +2bL andhenceconstant. Thus
′ h i h j i j
− −
F(r)attainsitsmaximumeitherforr orforr . Wehave
j j+1
L 2r +2G 2bL L 2(b b)L
h j+1 j i j h j+1 i j
F(r ) − = −
j+1 ≤ L · 2r +G bL L ·2b L 2G +G bL
j j+1 h i h j j+1 j+1 j+1 h i h
− − −
2(b b)L 2(b b)L
j+1 i h j+1 i h
= − = −
(cid:229) (2b 2b )l +(cid:229) (b b)l (cid:229) (2b b b)l +(cid:229) (b b)l
g j+1 j+1 g g g h g i g g j j+1 g i g j<g h g i g
≤ − ≤ − ≤ − − ≤ −
2(b b)L
j+1 i h
= −
(cid:229) (2b b b)l +(cid:229) (2b b b)l +(cid:229) (b b)l
g i j+1 g i g i<g j j+1 g i g j<g h g i g
≤ − − ≤ − − ≤ −
2(b b)L
j+1 i h
−
≤ (cid:229) 2(b b)l +(cid:229) (b b)l
g i j+1 i g i<g h j+1 i g
≤ 2L − 2≤L −
h h
= =
(cid:229) 2l +(cid:229) l L +L
g i g i<g h g i h
≤ ≤
and
L 2r +2G 2bL L 2(b b)L
h j j i j h j i j
F(r ) − = −
j ≤ L · 2r +G bL L ·2b L 2G +G bL
j j h i h j j j j h i h
− − −
2(b b)L 2(b b)L
j i h j i h
= − = −
(cid:229) (2b 2b )l +(cid:229) (b b)l (cid:229) (2b b b)l +(cid:229) (b b)l
g j j g g g h g i g g j j g i g j<g h g i g
≤ − ≤ − ≤ − − ≤ −
2(b b)L
j i h
= −
(cid:229) (2b b b)l +(cid:229) (2b b b)l +(cid:229) (b b)l
g i j g i g i<g j j g i g j<g h g i g
≤ − − ≤ − − ≤ −
2(b b)L 2L 2L
j i h h h
− = =
≤ (cid:229) 2(b b)l +(cid:229) (b b)l (cid:229) 2l +(cid:229) l L +L
g i j i g i<g h j i g g i g i<g h g i h
≤ − ≤ − ≤ ≤
Ifl R L forall j,thenL =l +L (1+R )L forall jandhenceL L PL .
j+1 j j j+1 j j j j h k 1
≤ ≤ ≤ ≤
Lemma3. Foranyreals m ,a ,andb with1≤m ≤2anda /b ≤m ,ba ≤ mm−21a 2+b 2.
Proof. Wemayassume b 0. Ifb =0, thereisnothing toshow. Soassume b >0andleta /b =dm
≥
forsomed 1. Weneedtoshow(dividethetargetinequalitybyb 2)dm (m 1)d 2+1orequivalently
≤ ≤ −
md (1 d ) (1 d )(1+d ). Thisinequality holdsford 1and m 2.
− ≤ − ≤ ≤
Lemma4. If fN/f m 2foralli,thenPoA m 2/(m 2 m +1). Ifl R L forall j,then
i i∗≤ ≤ ≤ − j ≤ j j
4P2
PoA B(R ,...,R ):= ,
≤ 1 k−1 3P2+1
whereP=R (cid:213) (1+R ).
1 1<j<k j
·
8
Proof. AssumethatNashuses jlinksandletLbethecommonlatencyofthelinksusedbyNash. Then
L=a fN+b fori jandL b =a fN+b fori> j. Thus
i i i ≤ ≤ i i i i
CN(r)=Lr=(cid:229) Lfi∗≤(cid:229) (aifiN+bi)fi∗≤ m m−21(cid:229) ai(fiN)2+(cid:229) (ai(fi∗)2+bifi)
i i i i
m 1
− C (r)+C (r)
≤ m 2 N opt
and hence PoA m 2/(m 2 m +1). If l R L for all j, we may use m = 2P/(P+1) and obtain
j j j
≤ − ≤
PoA 4P2/(3P2+1).
≤
TheGeneral Case: Wecometo the case where l RL for some i. Let j be the smallest such i.
i+1 i i
≥
Forr r /2,MNandOptuseonlylinks1to jandweareinthebenigncase. HenceePoAisbounded
j+1
≤
byB(R ,...,R )<4/3. MNroutestheflowexceeding r /2exclusively onhigherlinks.
1 j 1 j+1
−
Lemma5. MNdoesnotuselinksbeforeOpt.
Proof. Consideranyh> j+1. MNstartstouselinkhats =r /2+(cid:229) (b b)(L L )and
h j+1 j+1 i<h i+1 i i j
Optstartstouseitatr /2=r /2+(cid:229) (b b)L /2. Wehaves ≤ r /2sin−ceL L− L /2
h j+1 j+1 i<h i+1 i i h h i j i
≤ − ≥ − ≥
fori> j.
We need to bound the cost of MN in terms of the cost of Opt. In order to do so, we introduce an
intermediate flow Mopt (modified optimum) that we can readily relate to MN and to Opt. Mopt uses
links 1 to j to route r /2 and routes f =r r /2 optimally across links j+1 to k. Let f and fm
j+1 − j+1 i∗ i
betheoptimalflowsandtheflowsofMopt,respectively, atrater. Letr =(cid:229) f r /2bethetotal
s i≤j i∗≥ j+1
flowroutedacrossthefirst jlinksintheoptimalflow(thesubscript sstandsforsmall)andlet
r r /2
j+1
t = −
r r
s
−
Wewillshowt 1+1/R below. WenextrelatethecostofMoptonlinks j+1toktothecostofOpt
j
≤
ontheselinks. Tothisendwescaletheoptimalflowontheselinksbyafactoroft,i.e.,weconsider the
following flowacross links j+1tok: onlinki, j+1 i k,itroutest f . Thetotalflowonthehigh
≤ ≤ · i∗
links (= links j+1 to k) is r r /2 and hence Mopt incurs at most the cost of this flow on its high
j+1
−
links. Thus
(cid:229) ℓ(fm)fm (cid:229) ℓ(tf )tf t2 (cid:229) ℓ(f )f .
i i i i i∗ i∗ i i∗ i∗
≤ ≤
i>j i>j i>j !
The cost of MN on the high links is at most ePoA(R ,...,R ) times this cost by the induction
j+1 k 1
−
hypothesis. We can now bound the cost of MN as follows: (Kurt changes this on September 1st; I left
theoldversioninthedocument; theoldversioncomesfirst,thenewversioncomesthen).
C (r)=C (r /2)+C (flow f acrosslinks j+1tok)
MN N j+1 MN
B(R ,...,R )C (r /2)+t2ePoA(R ,...,R ) (cid:229) ℓ(f )f
1 j 1 opt j+1 j+1 k 1 i i∗ i∗
≤ − − i>j !
B(R ,...,R )C (r )+t2ePoA(R ,...,R ) (cid:229) ℓ(f )f
1 j 1 opt s j+1 k 1 i i∗ i∗
≤ − − i>j !
max(B(R ,...,R ),t2ePoA(R ,...,R ))C (r)
1 j 1 j+1 k 1 opt
≤ − −
9
C (r)=C (r /2)+C (flow f acrosslinks j+1tok)
MN N j+1 MN
B(R ,...,R )C (r /2)+t2ePoA(R ,...,R ) (cid:229) ℓ(f )f
1 j 1 opt j+1 j+1 k 1 i i∗ i∗
≤ − − i>j !
B(R ,...,R ) (cid:229) ℓ(f )f +t2ePoA(R ,...,R ) (cid:229) ℓ(f )f
1 j 1 i i∗ i∗ j+1 k 1 i i∗ i∗
≤ − i j ! − i>j !
≤
max(B(R ,...,R ),t2ePoA(R ,...,R ))C (r)
1 j 1 j+1 k 1 opt
≤ − −
Lemma6. t 1+1/R .
j
≤
Proof. Assume that Opt uses h links where j+1 h k. Then r /2 r r /2. Let r=r /2+d .
h h+1 h
AccordingtoTheorem1, f =rl /L +(G l /L ≤ g≤)/2andhence ≤ ≤
i∗ i h h j h− i
r L 1 G L
r = h +d j + h j G .
s 2 L 2 L − j
h h
(cid:18) (cid:19)
(cid:16) (cid:17)
SinceG +r =b L andG +r =b L (seeTheorem1),thissimplifiesto
h h h h j j j j
L d b L G L 1 G L L d 1 L d
rs= L j + h h2− hL j +2 Lh j −bjL j+rj = L j +2((bh−bj)L j+rj)= L j +rs∗,
h h h h h
(cid:18) (cid:19)
wherer = 1((b b )L +r ). Wecannowboundt.
s∗ 2 h− j j j
r r /2 r /2+d r /2 r /2 r /2 1
j+1 h j+1 h j+1
t = − = − max − , .
r−rs rh/2+d −rs∗−(L jd )/L h ≤ rh/2−rs∗ (1−LL hj)!
Nextobservethat
rh/2 rj+1/2 (cid:229) j+1 i<h(bi+1 bi)L i (cid:229) j+1 i<h(bi+1 bi)L i
r /−2 r = (cid:229) (b ≤ b)L − (b b )L = (cid:229) ≤(b b−)(L L )
h − s∗ j≤i<h i+1− i i − h− j j j+1≤i<h i+1− i i− j
L L L +l 1
(cid:0)max i = (cid:1) j+1 = j j+1 1+ .
≤ j+1 i<hL i L j L j+1 L j l j+1 ≤ Rj
≤ − −
Thesecondtermintheupperboundfort isalsobounded bythisquantity.
Wesummarizethediscussion.
Lemma7. Foreverykandevery jwith1 j<k. Ifl >R L andl RL fori< jthen
j+1 j j i i i+1
≤ ≤
2
1
ePoA(R ,...,R ) max B(R ,...,R ), 1+ ePoA(R ,...,R ) .
1 k 1 1 j 1 j+1 k 1
− ≤ − Rj − !
(cid:18) (cid:19)
Wearenowreadyforourmaintheorem.
Theorem3. Foranyk,thereisachoiceoftheparameters R toR suchthattheengineered Priceof
1 k 1
−
Anarchywiththeseparametersisstrictlylessthan4/3.
10