Strong and Pareto Price of Anarchy in Congestion Games

20 Pages·2009·0.18 MB·English

Strong and Pareto Price of Anarchy in Congestion Games†
SteveChien‡ AlistairSinclair§
September21, 2009
Abstract
Strong Nash equilibria and Pareto-optimal Nash equilibria are natural and important strengthenings of the
Nash equilibrium concept. We study these stronger notions of equilibrium in congestion games, focusing
on the relationships between the price of anarchy for these equilibria and that for standard Nash equilibria
(which iswellunderstood). Forsymmetric congestion gameswithpolynomial orexponential latency func-
tions, we show that the price of anarchy for strong and Pareto-optimal equilibria is much smaller than the
standard price of anarchy. On the other hand, for asymmetric congestion games with polynomial latencies
the strong and Pareto prices of anarchy are essentially as large as the standard price of anarchy; while for
asymmetricgameswithexponential latenciestheParetoandstandardpricesofanarchyarethesamebutthe
strong priceofanarchy issubstantially smaller. Finally, inthespecialcaseoflinearlatencies, weshowthat
in asymmetric games the strong and Pareto prices of anarchy coincide exactly with the known value 5 for
2
standard Nash,butarestrictlysmallerforsymmetricgames.
†ApreliminaryversionofthispaperappearedinProceedingsofthe36thInternational ColloquiumonAutomata,Languages
andProgramming(ICALP),PartI,2009.
‡Microsoft Research Silicon Valley Campus, 1065 La Avenida, Mountain View CA 94043, U.S.A. Email:
[email protected].
§Computer Science Division, University of California, Berkeley CA 94720-1776, U.S.A. Email:
[email protected]. Supported in part by NSF grant 0635153. Work done in part while this author was
visitingMicrosoftResearchSiliconValleyCampus.
1 Introduction
1.1 Background
In algorithmic game theory, the price of anarchy [14] is defined as the ratio of the social cost of a worst
Nashequilibrium tothat ofasocial optimum (i.e.,anassignment ofstrategies toplayers achieving optimal
socialcost). Thishighlysuccessful andinfluential concept isfrequently thought ofasthestandard measure
ofthepotentialefficiencylossduetoindividualselfishness,whenplayersareconcernedonlywiththeirown
utility and not with the overall social welfare. However, because a Nash equilibrium guarantees only that
no single player (as opposed to no coalition) can improve his utility by moving to anew strategy, the price
ofanarchy arguably conflatestheeffectsofselfishness andlackofcoordination. Indeed, forseveral natural
classes of games, the worst-case price of anarchy is achieved at a Nash equilibrium in which a group of
selfishplayerscanallimprovetheirindividualutilitiesbymovingsimultaneouslytonewstrategies;insome
cases, theworstNashequilibrium maynot evenbePareto-optimal—i.e., itmaybepossible that agroup of
playerscanmovetonewstrategiessothateveryplayerisbetteroff(ornoworseoff)thanbefore.
In this context, two stronger equilibrium concepts perhaps better isolate the efficiency loss due only
to selfishness. A strong Nash equilibrium [5] is defined as a state in which no subset of the players may
simultaneously change their strategies so asto improveall oftheir costs. Thestrong price ofanarchy (see,
e.g., [3]) is the ratio between the cost of the worst strong equilibrium and the optimum cost. A weaker
concept that is very widely studied in the economics literature (see, e.g., [15]) is that of a Pareto-optimal
Nash equilibrium, which is defined as a Nash equilibrium for which there is no other state in which every
player is better off. (Equivalently, one may think of a Pareto-optimal equilibrium as being stable under
moves by single players or the coalition of all players, but not necessarily under arbitrary coalitions.) One
canarguethatPareto-optimality shouldbeaminimumrequirement foranyequilibrium conceptintendedto
capture the notion of selfishness, in that it should not be in every player’s self-interest to move to another
state. TheParetopriceofanarchyisthendefinedintheobvious way.†
Anaturalquestiontoaskiswhetherthestrongand/orParetopricesofanarchyaresignificantlylessthan
the standard price of anarchy. In other words, does the requirement that the equilibrium be stable against
coalitions lead to greater efficiency? We note that this question has been addressed recently for several
specific families of games in the case of the strong (though not Pareto) price of anarchy [2, 3, 10]; see the
related worksectionbelow. Inthispaper, weinvestigate thequestion forthelargeandwell-studied classof
congestion gameswithlinear, polynomial orexponential latencyfunctions.
A congestion game is an n-player game in which each player’s strategy consists of a set of resources,
and the cost of the strategy depends only on the number of players using each resource, i.e., the cost takes
the form ` (f(r)), where f(r) is the number of players using resource r, and ` is a non-negative
r r r
increasing function. A standard example is a network congestion game on a directed graph, in which each
P
player selects apath from some source tosome destination, and each edge has an associated cost function,
or “latency”, ` that increases with the number of players using it. (Throughout, we shall use the term
r
“latency” even though wewillalways be discussing general (non-network) congestion games.) Frequently
thelatencies areassumedtohaveasimpleform,suchaslinear, polynomial, orexponential.
CongestiongameswereintroducedinEconomicsbyRosenthal[19],andfurtherstudiedinaninfluential
paper by Monderer and Shapley [16]. They have since featured prominently in algorithmic game theory,
partly because they capture a large class of routing and resource allocation scenarios, and partly because
they are known to possess pure Nash equilibria [19]. The price of anarchy for congestion games is by
nowquite wellunderstood, starting withKoutsoupias and Papadimitriou [14]whoconsidered a(weighted)
congestion gameonasetofparallel edges. Theseminal workofRoughgarden andTardos [21]established
the value 4 as the price of anarchy of network congestion games with linear latencies in the nonatomic or
3
†Notethatthroughout weareassumingthatcost (orutility)isnon-transferable, i.e., playersinacoalitioncannot sharetheir
costswitheachother.Ifcostscanbeshared,thesituationisverydifferent;see[12]foradiscussionofthisalternativescenario.
1
Wardrop case [7] (where there are infinitely many players, each of whom controls an infinitesimal amount
of traffic); this was extended to polynomial latencies in [22]. The more delicate n-player case was solved
independently byAwerbuch,AzarandEpstein[6]andbyChristodoulouandKoutsoupias[8],whoobtained
thetightvalue 5 forthepriceofanarchy inthelinearcase,andtheasymptotically tightvaluekk(1−o(1)) for
2
thecaseofpolynomial latencies. Subsequently Alandetal.[1]gaveanexactvalueforthepolynomialcase.
Much less is known about strong or Pareto-optimal Nash equilibria in congestion games. Note that
such equilibria need not exist. Holzman and Law-Yone [13] give a sufficient condition for the existence
of a strong equilibrium based on the absence of a certain structural feature in the game, and also discuss
theuniqueness andPareto-optimality ofNashequilibria under thesamecondition. Forthestrong orPareto
priceofanarchy, however,thereappeartobenoresultsforgeneralcongestion games.
1.2 Results
We investigate the strong and Pareto price of anarchy for congestion games with linear, polynomial and
exponential latencies. Roughly speaking, we find that in symmetric‡ games the resulting price of anarchy
canbemuchlessthanthestandard(Nash)priceofanarchy,whileinasymmetricgamesthebehaviorismore
complicated: forlinearandpolynomial latencies allthreepricesofanarchyareessentially thesame,butfor
exponential latencies thestandard and Paretoprices ofanarchy areequal, whilethestrong priceofanarchy
is substantially smaller. (We note that this gap between symmetric and asymmetric games does not appear
forstandard Nashequilibria. Understanding the reason forthis difference maybeatopic worthy offurther
study.)
Morespecifically,weshowthatthestrongandParetopricesofanarchyforsymmetriccongestiongames
with polynomial latencies of degree k are at most 2k+1 (and that this is tight up to a constant factor); this
is in sharp contrast to the Nash price of anarchy of kk(1−o(1)) obtained in [1, 6, 8]. In the special case of
linear latency, we show that the strong and Pareto prices of anarchy are strictly less than the exact value 5
2
for standard Nash obtained in [6, 8]. For symmetric games with exponential latency αt, we show that the
strong and Pareto prices of anarchy are at most max{α,n}, while the standard Nash price of anarchy is at
leastβn,whereβ > 1isaconstant thatdependsonα.
Ontheotherhand, forasymmetric gameswithpolynomial latency ofdegreek,weshowthatthestrong
(and therefore also the Pareto) price of anarchy is kk(1−o(1)), matching the asymptotic value for standard
Nash derived in [6, 8]. Moreover, in the linear case all three prices of anarchy coincide exactly. For ex-
ponential latencies, we show that the Pareto price of anarchy is the same as for standard Nash (which is
exponentially large), andalsothatthestrong priceofanarchy ismuchsmaller; thusweexhibit aseparation
betweenstrongandParetopricesofanarchyforanatural classofgames.
Since strong and Pareto-optimal equilibria do not always exist, we should clarify the meaning of the
abovestatements. Anupperboundonthestrong(respectively, Pareto)priceofanarchyforacertainclassof
games bounds the price of anarchy whenever a strong (respectively, Pareto-optimal) equilibrium exists. A
lowerbound meansthatthere isaspecificgameintheclass thathasastrong (respectively, Pareto-optimal)
equilibrium achieving thestatedpriceofanarchy.
We now briefly highlight a few of our proof techniques. To obtain upper bounds on the Pareto (and
hence also strong) price of anarchy in symmetric games, weshow that this price of anarchy can always be
bounded abovebythemaximumratioofthecostsofindividual players atequilibrium andthesameratioat
thesocialoptimum. Thisallowsustostudytheequilibriumandtheoptimumseparately, greatlysimplifying
the analysis. We note that this fact holds for arbitrary symmetric games, not only congestion games, and
thus may beof wider interest. Ourupper bound on the Pareto price of anarchy for linear latencies requires
amuchmoreintricateanalysis, andmakesuseofamatrixM = (m ),wherem istherelativeincreasein
ij ij
playeri’scostatoptimum whenanewplayer movestoplayerj’sstrategy. Thisturnsouttobeastochastic
matrix with several useful properties. Finally, our lower bound arguments are based on constructions used
‡Agameissymmetricifallplayershavethesamesetsofallowablestrategies.
2
in [1, 6, 8], suitably modified so as to handle the stronger requirements of strong and Pareto equilibria.
(These constructions typically have the property that the social optimum is a strong Nash equilibrium, so
theyarenotimmediatelyapplicable inoursetting.)
1.3 Related work
Webriefly mention here some other related results not discussed above. A fair amount is known about the
standard price of anarchy for variations of congestion games. The previously mentioned [6] and [1] both
extend their results to congestion games with weighted players, while [8] handles the case where social
cost is defined as the maximum, rather than total player cost. This latter case is also addressed by the
papers[9,20]fortheWardropmodel.
For the strong equilibrium concept, several authors have considered the strong price of anarchy and
the existence of strong Nash equilibria in various specific classes of games, often deriving significant gaps
betweenthestrongandstandardpriceofanarchy. Forexample,Andelmanetal.[3]studyjobschedulingand
network creation games, Epstein et al. [10] cost-sharing connection games, and Albers [2] network design
games.
Other measures stronger than the standard Nash price of anarchy have also been studied recently. An-
shelevich et al. [4] consider the price of stability, or the ratio of the cost of a best Nash equilibrium to the
social optimum, for network design games. And Hayrepeyan et al. [12]define and study the “price of col-
lusion”inanalogous fashiontothestrongpriceofanarchy,withthecrucialdifferencethatcoalitions aimto
minimizenotthecostofeachoftheirmembersbutthecombinedcostofallmembers.
OntheissueofexistenceofstrongNashequilibriaincongestiongames,RozenfeldandTennenholtz[18]
follow on from the above-mentioned [13] and consider the case where the “latencies” are monotonically
decreasing.
2 Preliminaries
2.1 Equilibrium concepts and congestiongames
AgameconsistsofafinitesetofplayersP = {1,...,n},eachofwhichisassignedafinitesetofstrategies
S andacostfunctionc :S ×···×S → Nthathewishestominimize. Agameiscalledsymmetricifallof
i i 1 n
theS areidentical. Astates= (s ,...,s ) ∈ S ×···×S isanycombinationofstrategiesfortheplayers.
i 1 n 1 n
A state s is a pure Nash equilibrium if for all players i, c (s ,...,s ,...,s ) ≤ c (s ,...,s0,...,s ) for
i 1 i n i 1 i n
alls0 ∈ S ;thusataNashequilibrium, noplayercanimprovehiscostbyunilaterally changing hisstrategy.
i i
It is well known that while every (finite) game has a mixed Nash equilibrium§, not every game has a pure
Nash equilibrium. A state s = (s ,...,s ) is a Pareto-optimal Nash equilibrium if it is a pure Nash
1 n
equilibrium and there is no other state in which every player has lower¶ cost than at s; in other words, for
all s0 = (s0,...,s0 ) ∈ S ×···×S , there exists some player j ∈ P such that c (s0) ≥ c (s). A state
1 n 1 n j j
s = (s ,...,s )isastrong Nashequilibrium ifthere doesnotexistanycoalition ofplayers thatcanmove
1 n
insuchawaythateverymemberofthecoalition payslowercostthanatequilibrium; i.e. foranyotherstate
s0 6= s,thereexistsaplayerj suchthats 6= s0 andc (s0) ≥ c (s). Finally, foranygivenstates,wedefine
j j j j
thesocialcost c(s)tobethesumoftheplayers’ costsins,i.e.,c(s) = c (s). Astateminimizingthe
i∈P i
socialcostinagameiscalledasocialoptimum.
P
§In a mixed Nash equilbrium, a player’s strategy is any probability distribution over available strategies, and no player can
improvehisexpectedcostbychoosinganotherdistribution.
¶SomedefinitionsofPareto-optimalityrequiretheretobenootherstateinwhichnoplayerhashighercostthanatsandatleast
oneplayerhaslowercost.Itiseasytocheckthatourresultscarryovertothisalternativedefinitionwithminormodificationstothe
proofs.
3
We will focus on the class of games known as congestion games, which are known to always possess
a pure Nash equilibrium [19]. In a congestion game, players’ costs are based on the shared usage of a
common set of resources R = {r ,...,r }. A player’s strategy set S ⊆ 2R is a collection of subsets
1 m i
of R; his strategy s ∈ S will thus be a subset of R. Each resource r ∈ R has an associated non-
i i
decreasing costor“latency”function` : {1,...,n} → N;iftplayersareusingresourcer,theyeachincur
r
a cost of ` (t). Thus in a state s = (s ,...,s ), the cost of player i is c (s) = ` (f (r)), where
r 1 n i r∈si r s
f (r) = |{j : r ∈ s }|isthenumberofplayersusingrunders.
s j
P
Of particular interest are congestion games where the latency functions are linear (` (t) = α t+β ),
r r r
polynomial (` (t)isadegree-k polynomial intwithnon-negative coefficients), orexponential (` (t) = αt
r r r
for 1 ≤ α ≤ α.) For simplicity of notation, we shall assume that ` (t) = t for all r in the linear case,
r r
` (t) = tk for all r in the polynomial case, and ` (t) = αt for all r in the exponential case. This will not
r r
affect ourlowerbounds, whicharebased onexplicit constructions ofthisrestricted form, anditisnothard
tocheck thatthe upper bounds gothrough aswell; forexample, itisstraightforward toincorporate general
non-negative coefficientsbyreplicating resources. Weomitthedetails, whicharetechnical butstandard.
Note that in our congestion games, Pareto-optimal and strong equilibria maynot exist, and gamesmay
havethefirstwithoutthesecond,aswenowshow.
Proposition 2.1 There exist asymmetric congestion games with linear, polynomial, and exponential laten-
cieswithintegerbase(i.e.,` (t) = αtforsomepositiveintegerα)thathavePareto-optimalNashequilibria
r
butnostrongNashequilibria.
Proof: Consider an n-player congestion gamewithlatency functions `(·)which has aNash equilibrium e
but no strong Nash equilibrium; for example, it is easy to construct a Prisoner’s Dilemma-style game with
thisproperty.
From this starting point, we construct a modified game that has a Pareto-optimal Nash equilibrium but
stillnostrongNashequilibrium. Wefirstreplaceeachresourceintheoriginalgamewithasetofmresources
inthemodifiedgameforaconstantm > `(2);thusmislargerthan2,2k andα2 forlinear,polynomial, and
exponentiallatenciesrespectively. Strategiesinthemodifiedgamecorrespondtothoseintheoriginalgame,
except that theformer include allm copies ofthe resources ofthe latter. Thishas the effect of multiplying
playercostsbyafactorofm,butdoesnotchangethesetofNashandPareto-optimal Nashequilibria.
We then add one more player, n + 1, to the modified game; this player has a single strategy s
n+1
consisting of new resources {rˆ : i = 1,...,n}. Also, for players 1,...,n, we append resource rˆ to
i i
every strategy of player i except for the equilibrium strategy e . Note that this makes the modified game
i
asymmetriceveniftheoriginaloneisnot. Thereisanobviousbijectionbetweenstatesoftheoriginalgame
andthose ofthemodified game,and weshall abuse notation byidentifying them. Wewritec (s)andc0(s)
i i
forthecosttoplayeriofstatesintheoriginalgameandinthemodifiedgamerespectively.
NowitiseasytoseethattheoriginalNashequilibriume(togetherwithstrategys forplayern+1)is
n+1
aPareto-optimalequilibriumforthemodifiedgame: plainlyitremainsaNashequilibrium,andanycoalition
moveresultsinacostincreaseforplayern+1.
It remains to show that the modified game does not have any strong Nash equilibria. We do this by
showing that if a state s is a strong Nash equilibrium in the modified game, it must correspond to a strong
Nash equilibrium in the original game. Consider an arbitrary coalition of players in state s in the original
game, and an arbitrary group move to some other strategies, resulting in a new state s0; we aim to show
that one of the coalition members has cost in s0 that is at least his cost in s. We do this by examining the
corresponding moveinthemodifiedgame. SincesisastrongNashequilibriumthere,wemusthaveatleast
one player, say player i, for whom c0(s0) ≥ c0(s). Since c0(s0) ≤ mc (s0)+`(2), and c0(s) ≥ mc (s), we
i i i i i i
obtainmc (s0)+`(2) ≥ mc (s),implying thatc (s0)+ `(2) ≥ c (s).
i i i m i
Since all player costs must be integral with the latency functions weconsider, and `(2) < 1, it must be
m
thatc (s0)≥ c (s),whichiswhatweneeded.
i i
Wenotethatwewilluseaverysimilarconstruction laterintheproofofTheorem4.2.
4
2.2 Efficiency ofequilibria
As is standard, wemeasure the relative efficiency loss for a specific type of equilibrium for a given family
ofgamesG asthemaximumpossibleratio,overallgamesinthefamily,ofthesocialcostofanequilibrium
stateeinthatgametothecostofasocialoptimumoofthesamegame,or
c(e)
sup .
c(o)
G,e
Thismeasureisknownasthepriceofanarchy(orcoordinationratio)inthecaseofNashequilibria, andthe
strong price of anarchy whendiscussing strong Nashequilibria. Inaddition tothese, wewillalso consider
thecaseofPareto-optimalNashequilibria,inwhichcasewecalltheaboveratiotheParetopriceofanarchy.
Clearly the strong price of anarchy is no larger than the Pareto price of anarchy, which in turn is no larger
thanthestandard (Nash)priceofanarchy.
3 Symmetric games
In this section we prove upper bounds on the strong and Pareto price of anarchy for symmetric congestion
gameswithpolynomialandexponentiallatencyfunctions. Weshallseethatthesearemuchsmallerthanthe
known values for the standard Nash price of anarchy. Thus for symmetric games increased coordination,
whenpossible, leadstogreaterefficiency.
3.1 The basicframework
The main vehicle for these proofs is a simple framework that allows us to bound the price of anarchy in
terms of the maximum ratio of the player costs at equilibrium and the maximum ratio of the player costs
at a social optimum. This is the content of the following theorem, which we note applies to all symmetric
games,notonlycongestion games.
Theorem3.1 Given a particular symmetric game with n players, let the state e be a Pareto-optimal Nash
equilibrium andsbeanyotherstate. Letρ bedefinedasmax c (e)/c (e)overallplayersi,j,andρ be
e i,j i j s
similarlydefinedasmax c (s)/c (s). Then
i,j i j
c(e)
≤ max{ρ ,ρ }.
e s
c(s)
Proof: By symmetry, we can assume without loss of generality that the players are ordered by cost in
both e and s: that is, c (e) ≤ ··· ≤ c (e) and c (s) ≤ ··· ≤ c (s). Thus c (e)/c (e) = ρ , and
1 n 1 n n 1 e
c (s)/c (s)= ρ . Westartfrome,andconsiderthehypothetical moveinwhicheveryplayerimovesfrom
n 1 s
e tohiscorresponding strategy s ins. SinceeisPareto-optimal, theremustexistsomeplayer j forwhom
i i
c (s)≥ c (e).
j j
We now upper bound the social cost of equilibrium, c(e) = c (e), and lower bound the social cost
i i
c(s) = c (s) of state s. Consider first the c (e) values. We have c (e) ≤ ··· ≤ c (e) ≤ ··· ≤
i i i P 1 j
c (e) = ρ c (e). The sum c (e) is therefore maximized when c (e) = c (e) = ··· = c (e) and
n Pe 1 i i 1 2 j
c (e) = ··· = ρ c (e), giving an upper bound of jc (e) + (n − j)ρ c (e). Similarly, for the c (s)
j+1 e 1 j e j i
P
values, we have c (s) ≤ ··· ≤ c (s) ≤ ··· ≤ ρ c (s). The sum c (s) is minimized when c (s) =
1 j s n i i 1
··· = c (s) = c (s)/ρ andc (s) = ··· = c (s),andistherefore atleast (j−1)cj(s) +(n−j +1)c (s).
j−1 j s j n P ρs j
Recallingthatc (s) ≥ c (e)andcombining thetwobounds, weobtain
j j
c (e) jc (s)+(n−j)ρ c (s) j +(n−j)ρ
i i ≤ j e j ≤ e . (1)
Pici(s) (j−1ρ)scj(s) +(n−j+1)cj(s) (jρ−s1) +(n−j +1)
P
5
Differentiating withrespecttoj,wefindthatthisexpression ismaximizedatj = 1orj = n. Intheformer
casethequotient isatmost 1+(n−1)ρe ≤ ρ ,whileinthelattercaseitisatmost n ≤ ρ .
n e (n−1)/ρs+1 s
Thus, for a family of symmetric games, if we can find ρ and ρ such that c (e)/c (e) ≤ ρ for all
e o i j e
Pareto-optimalequilibriae,andc (o)/c (o) ≤ ρ forallsocialoptimao,wecanboundthePareto(andhence
i j o
the strong) price of anarchy by max{ρ ,ρ }. We now proceed to do this for polynomial and exponential
e o
symmetriccongestion games.
3.2 Polynomiallatencies
For the case of polynomial latencies, where each resource r has latency function ` (t) = tk, we show the
r
following:
Theorem3.2 For symmetric congestion games with polynomial latencies of degree k, the Pareto price of
anarchy(andhencealsothestrongpriceofanarchy)isatmost2k+1.
Remarks: (i)Notethat the Paretoprice of anarchy is muchsmaller than the known value ofkk(1−o(1)) for
thestandard Nashpriceofanarchy[6,8,1]. (ii)ItisnothardtoverifythattheupperboundinTheorem3.2
is tight up to a constant factor. To see this, consider a 2-player symmetric game with m = 2k resources
{r ,...,r }, and the following four strategies: {r ,r }, {r ,r ,r }, {r ,r ,r }, and {r ,r ,...,r }.
1 m 1 2 1 3 5 2 4 6 3 4 m
Thesocialoptimumoccurswhenthetwoplayerschoose{r ,r ,r }and{r ,r ,r },foratotalcostof6. On
1 3 5 2 4 6
the other hand, the state in which the players choose {r ,r } and {r ,r ,...,r } is astrong equilibrium,
1 2 3 4 m
anditscostism. Thepriceofanarchyisthus m = 2k.
6 6
ProofofTheorem3.2: FollowingtheframeworkofTheorem3.1,itsufficestoderiveupperboundsonthe
ratiosofplayercostsbothatequilibrium andatasocialoptimum. Thiswedointhefollowingtwoclaims.
Claim3.3 Inthesituation ofTheorem3.2,wehavemax ci(e) ≤ 2k.
i,j∈P cj(e)
Proof: Consider any two players i and j atan equilibrium e, and the hypothetical movein which player i
switchesfromhiscurrentstrategye toj’sstrategye ,resulting inthenewstatee0.
i j
Wenowboundc (e0)intermsofc (e). Notethat
i j
c (e0)= f (r)k = (f (r)+1)k + f (r)k ≤ (f (r)+1)k.
i e0 e e e
rX∈ej r∈Xej\ei r∈Xej∩ei rX∈ej
(This captures the intuition that in switching to e , player i pays at most what player j would pay if there
j
wereonemoreplayerusingeachresource.) Fromthis,itfollowsthat
ci(e0) ≤ r∈ej(fe(r)+1)k ≤ max (fe(r)+1)k ≤ 2k.
cj(e) P r∈ej fe(r)k r∈ej fe(r)k
P
SinceeisaNashequilibrium, wemusthavec (e) ≤ c (e0),andthusc (e) ≤ 2kc (e).
i i i j
Claim3.4 Inthesituation ofTheorem3.2,wehavemax ci(o) ≤ 2k+1.
i,j∈P cj(o)
Proof: Asintheproof ofClaim 3.3, consider anytwoplayers iand j atasocial optimum o. Assume that
c (o) ≥ c (o)astheclaim isimmediately trueotherwise. Again, consider themoveinwhichimovesfrom
i j
hiscurrentstrategyo toj’sstrategyo ,resulting inthenewstateo0.
i j
6
Sinceoisasocialoptimum,thesocialcostofo0mustbeatleastthatofo;i.e., c (o0)− c (o) ≥ 0.
l l l l
Usingthefactthat c (s) = f (r)k+1 foranystates,wehave
l l r s P P
0 ≤ P f (r)k+P1− f (r)k+1
o0 o
r r
X X
= f (r)k+1− f (r)k+1
o0 o
r∈Xoi⊕oj r∈Xoi⊕oj
= (f (r)+1)k+1−f (r)k+1 − f (r)k+1−(f (r)−1)k+1 ,
o o o o
r∈Xoj\oi(cid:0) (cid:1) r∈Xoi\oj(cid:0) (cid:1)
wherethesecondlinefollowssincef (r)= f (r)forr 6∈ o ⊕o .
o0 o i j
Now observe that c (o) = f (r)k = f (r)k + f (r)k, and add this to both
i r∈oi o r∈oi\oj o r∈oi∩oj o
sidesoftheabovetoget
P P P
c (o) ≤ (f (r)+1)k+1−f (r)k+1 + f (r)k
i o o o
r∈Xoj\oi(cid:0) (cid:1) r∈Xoi∩oj
− f (r)k+1−(f (r)−1)k+1−f (r)k .
o o o
r∈Xoi\oj(cid:0) (cid:1)
Itisnothardtoverifythatthelastsummationhereisalwaysnon-negative.
Sincec (o) = f (r)k + f (r)k,wehave
j r∈oj\oi o r∈oi∩oj o
cPi(o) ≤ r∈oj\oPi((fo(r)+1)k+1−fo(r)k+1)+ r∈oi∩oj fo(r)k
c (o) f (r)k + f (r)k
j P r∈oj\oi o r∈oi∩oj Po
≤ r∈oj\oi((Pfo(r)+1)k+1−foP(r)k+1)
f (r)k
P r∈oj\oi o
(fo(Pr)+1)k+1−fo(r)k+1
≤ max ,
r∈oj\oi fo(r)k
wherethesecondlinefollowsbecausetheratioofthefirstsumsinthenumeratoranddenominatorisgreater
than1. Thislastquantity canbeseentobeatmost2k+1,provingtheclaim.
Finally,combining Claims3.3and3.4withTheorem3.1completestheproofofTheorem3.2.
3.3 Exponential latencies
For the case of exponential latencies, where each resource has latency function ` (t) = αt, we show the
r
followingupperboundontheParetoandstrongpricesofanarchy.
Theorem3.5 For symmetric congestion games with n players and exponential latencies, the Pareto price
ofanarchy(andhencealsothestrongpriceofanarchy)isatmostmax{α,n}.
ToproveTheorem3.5,weagainrelyontheframeworkofTheorem3.1andestablish thefollowing two
claims,whichboundtheratiosofplayercostsatequilibrium andatsocialoptimum.
Claim3.6 Inthesituation ofTheorem3.5,wehavemax ci(e) ≤ α.
i,j∈P cj(e)
Proof: Consider an equilibrium state e and a move by player i from his current strategy e to player j’s
i
strategy e , resulting in thenew state e0. Clearly foreach resource r ∈ e , player ipays atmost afactor of
j j
αmorethanj paysate. Hencec (e0)≤ αc (e). SinceeisaNashequilibrium, wemusthavec (e) ≤ c (e0)
i j i i
andtheclaimfollows.
7
Claim3.7 Inthesituation ofTheorem3.5,wehavemax ci(o) ≤ α(n+1).
i,j∈P cj(o)
Proof: Thisproof followsalong thesamelinesasthatofClaim 3.4. Consider asocial optimum o,andthe
move in which player i moves from o to player j’s strategy o . Since o is a social optimum, this cannot
i j
decrease theoverallsocialcost. Hence
0 ≤ fo0(r)αfo0(r)− fo(r)αfo(r)
r r
X X
= fo0(r)αfo0(r)− fo(r)αfo(r)
r∈Xoi⊕oj r∈Xoi⊕oj
= (f (r)+1)αfo(r)+1−f (r)αfo(r) − f (r)αfo(r)−(f (r)−1)αfo(r)−1 .
o o o o
r∈Xoj\oi(cid:0) (cid:1) r∈Xoi\oj(cid:0) (cid:1)
Writingc (o) = αfo(r)+ αfo(r) andadding thistobothsides, weobtain
i r∈oi\oj r∈oi∩oj
P P
c (o) ≤ (f (r)+1)αfo(r)+1−f (r)αfo(r) + αfo(r)
i o o
r∈Xoj\oi(cid:0) (cid:1) r∈Xoi∩oj
− (f (r)−1)(αfo(r)−αfo(r)−1),
o
r∈Xoi\oj
andagainwecanseethatthelastsummationisalwaysnon-negative.
Combiningthiswiththefactthatc (o) = αfo(r)+ αfo(r),weget
j oj\oi oi∩oj
P P
ci(o) ≤ r∈oj\oi((fo(r)+1)αfo(r)+1−fo(r)αfo(r))+ r∈oi∩oj αfo(r)
cj(o) P oj\oiαfo(r)+ oi∩oj αfo(r) P
≤ r∈oj\oi((fo(r)P+1)αfo(r)+1−fPo(r)αfo(r))
P oj\oiαfo(r)
≤ (f (r)+1)α.
o P
Sincef (r) ≤ n,thiscompletestheproofoftheclaim.
o
A direct application of Theorem 3.1 to the results of these claims gives an overall bound of α(n+1),
which we now improve to max{α,n}. We first follow the proof of Theorem 3.1 through inequality (1),
whereonecanseethattheratioontheright-hand sideismaximizedwhentheplayerj whosecostincreases
iseitherplayer1orplayern. Intheformercase,theresultingratioisatmostρ ,whichisαforexponential
e
latencies. In the latter case, the resulting ratio becomes nρo , or αn(n+1) for exponential latencies.
n−1+ρo n−1+α(n+1)
Thisisclearlyatmostn.
Incontrasttotheaboveupperbound, wenowshowthatthestandardNashpriceofanarchyinexponen-
tialcongestion gamesismuchlarger—indeed, exponential inn.
Proposition 3.8 For symmetric n-player congestion games with exponential latencies αt, the (standard
Nash)priceofanarchyisatleast(α)α(αα/2−−11)n.
2
Proof: Ourconstruction isbasedonthatofChristodoulou andKoutsoupias [8]forthecaseoflinearlaten-
cies. Ourgamecontainsmgroupsofresources,andn = mtplayers. Theplayersaredividedevenlyintom
equivalenceclasses,labeled{1,...,m},withtplayersperclass. Eachofthemgroupsofresourcesconsists
of m resources, each labeled with a different k-tuple of equivalence classes. The available strategies for
k
(cid:0) (cid:1)
8
all players are to take either (1) all resources in asingle group ofresources, or(2) for any iin {1,...,m},
allresources thatarelabeled withi.
Given the value of α, we choose m and k such that m ≤ 1+(m −1)α; for example, we can choose
k
k = α andm = α−1forintegerα ≥ 4. Itcanbeverifiedthat,withthesesettings, thestateinwhicheach
2
player takes all resources labeled with his equivalence class number is a Nash equilibrium, while the state
in which each player takes the group of resources corresponding to his equivalence class (i.e., a player in
classitakesallresources ingroupi)isasocialoptimum. Astraightforward calculation thenshowsthatthe
ratioofaplayer’s Nashcosttohiscostatsocialoptimum iskαt(k−1),whichis(α)α(αα/2−−11)n fortheabove
2
valuesofk andm.
Forcompleteness, weshowthatthissamepriceofanarchyforstandard Nashisupperbounded byαn:
Proposition 3.9 Forasymmetric (andhence also symmetric) congestion gameswithexponential latencies,
the(standard Nash)priceofanarchyisatmostαn.
Proof (sketch): As in the proofs of related results in [6, 8, 1], in a game with latency functions `(t), we
canproveanupperboundonthepriceofanarchybyfindingc ,c ≥ 0suchthattheinequalityy`(x+1)≤
1 2
c x`(x)+c y`(y) holds for all 0 ≤ x ≤ n,1 ≤ y ≤ n; this implies a price of anarchy of at most c2 .
1 2 1−c1
When`(t)= αt,thisclearlyholdswithc = 0andc = αx ≤ αn.
1 2
1
Remark: Withsomeextrawork,theboundinProposition 3.9canbeimprovedtoO(α(1−α)n).
4 Asymmetric games
In this section we extend the investigation of the previous section to asymmetric games, and find that the
situation isquite different. First wewillsee that, for asymmetric congestion games withpolynomial laten-
cies, thestrong (andtherefore alsothePareto)priceofanarchy isessentially thesameasthestandard Nash
price of anarchy. We willthen go on to consider exponential latencies, where wefind that the Pareto price
ofanarchyisthesameasstandardNash,butthestrongpriceofanarchyissignificantly smaller.
Webeginbyconsidering polynomial latencies.
Theorem4.1 For asymmetric congestion games with polynomial latencies tk, the strong price of anarchy
isatleastbΦ ck,whereΦ isthepositive solution of(x+1)k = xk+1.
k k
Remark: Φ is a generalization of the golden ratio (which is just Φ ); its value is k (1+o(1)). Hence
k 1 logk
the lower bound of Theorem 4.1 is of the form kk(1−o(1)), which is asymptotically the same value for the
Nashpriceofanarchyobtained in[6,8],andveryclosetotheexactvalueobtained byAlandetal.[1].
Proof: Ourlowerbound construction is based on that ofAland et al., extended so asto handle the stricter
requirementofastrongequilibrium. Consideran(asymmetric)gamewithnplayers(nassumedsufficiently
large). Each player i has exactly two possible strategies, e and o . There are n + m resources labeled
i i
{r ,...,r }, where m is a constant to be chosen later. For each player i, strategy o consists of the
1 n+m i
singleresourcer . (Weshallmodifythisslightlyforsomeoftheplayersshortly.) Strategye consistsofthe
i i
resources {r ,...,r }. (Thusformostplayerse consists ofexactlymresources.)
i+1 i+m i
Weclaimthatthestatee = (e ,...,e )isastrong Nashequilibrium. Toseethis, notethatunder ethe
1 n
costforplayeriisc (e) = i+m min{j−1,m}k.Ifnowplayerimovestohisalternativestrategyo ,re-
i j=i+1 i
sultinginanewstatee(i),hiscostbecomesc (e(i)) = min{i,m+1}k.ToshowthateisaNashequilibrium,
P i
weneedtoshowthatc (e(i)) ≥ c (e)foralli.
i i
Now note that, for all players i ≥ m+1, we have c (e(i))−c (e) = (m+1)k −mk+1. Thus if we
i i
choose m = bΦ c to be the smallest integer such that (m+1)k ≥ mk+1, we ensure that c (e(i)) ≥ c (e)
k i i
9