The Price of Stochastic Anarchy

15 Pages·2008·0.15 MB·English

The Price of Stochastic Anarchy
ChristineChung1,KatrinaLigett2?,KirkPruhs1??,andAaronRoth2
1 DepartmentofComputerScience
UniversityofPittsburgh
{chung,kirk}@cs.pitt.edu
2 DepartmentofComputerScience
CarnegieMellonUniversity
{katrina,alroth}@cs.cmu.edu
Abstract. Weconsiderthesolutionconceptofstochasticstability,andpropose
thepriceofstochastic anarchy asanalternativetothepriceof (Nash) anarchy
for quantifying the cost of selfishness and lack of coordination in games. As a
solutionconcept,theNashequilibriumhasdisadvantagesthatthesetofstochas-
ticallystablestatesofagameavoid:unlikeNashequilibria,stochasticallystable
statesaretheresultofnaturaldynamicsofcomputationallyboundedanddecen-
tralizedagents,andareresilienttosmallperturbationsfromidealplay.Theprice
of stochasticanarchycanbeviewed asasmoothed analysisofthepriceofan-
archy,distinguishingequilibriathatareresilienttonoisefromthosethatarenot.
Toillustratetheutilityofstochasticstability,westudytheloadbalancinggame
onunrelatedmachines.ThisgamehasanunboundedlylargepriceofNashanar-
chyevenwhenrestrictedtotwoplayersandtwomachines.Weshowthatinthe
twoplayercase,thepriceofstochasticanarchyis2,andthateveninthegeneral
case,thepriceofstochasticanarchyisbounded.Weconjecturethatthepriceof
stochasticanarchyisO(m),matchingthepriceofstrongNashanarchywithout
requiringplayercoordination.Weexpectthatstochasticstabilitywillbeusefulin
understanding therelativestabilityofNashequilibriainothergameswherethe
worstequilibriaseemtobeinherentlybrittle.
?Partiallysupported byan AT&TLabsGraduate Fellowshipand anNSFGraduate Research
Fellowship.
??Supported in part by NSF grants CNS-0325353, CCF-0448196, CCF-0514058 and IIS-
0534531.
1 Introduction
Quantifyingthepriceof(Nash)anarchyisoneofthemajorlinesofresearchinalgorith-
micgametheory.Indeed,onefourthoftheauthoritativealgorithmicgametheorytext
editedbyNisanetal.[20]iswhollydedicatedtothistopic.ButtheNashequilibrium
solutionconcepthasbeenwidelycriticized[15,4,9,10].First, itis a solutioncharac-
terizationwithoutaroadmapforhowplayersmightarriveatsuchasolution.Second,
at Nash equilibria, players are unrealistically assumed to be perfectly rational, fully
informed, and infallible. Third, computing Nash equilibria is PPAD-hard for even 2-
player,n-actiongames[6],anditisthereforeconsideredveryunlikelythatthereexists
apolynomialtimealgorithmtocomputeaNashequilibriumeveninacentralizedman-
ner.Thus,itisunrealistictoassumethatselfishagentsingeneralgameswillconverge
precisely to the Nash equilibriaof the game,or thatthey will necessarilyconvergeto
anythingatall.Inaddition,thepriceofNashanarchymetriccomeswithitsownweak-
nesses;itblindlyusestheworstcaseoverallNashequilibria,despitethefactthatsome
equilibriaaremoreresilientthanotherstoperturbationsinplay.
Consideringthese drawbacks, computerscientists have paid relatively little atten-
tiontoiforhowNashequilibriawillinfactbereached,andevenlesstothequestionof
whichNashequilibriaaremorelikelytobeplayedintheeventplayersdoconvergeto
Nashequilibria.Toaddresstheseissues,weemploythestochasticstabilityframework
fromevolutionarygametheorytostudysimpledynamicsofcomputationallyefficient,
imperfect agents. Rather than defining a-priori states such as Nash equilibria, which
mightnotbereachablebynaturaldynamics,thestochasticstabilityframeworkallows
ustodefineanaturaldynamic,andfromitderivethestablestates.Wedefinetheprice
of stochastic anarchy to be the ratio of the worst stochastically stable solution to the
optimalsolution.The stochastically stable states of a gamemay, butdo notnecessar-
ily,containallNashequilibriaofthegame,andsothepriceofstochasticanarchymay
bestrictlybetterthanthepriceofNashanarchy.Ingamesforwhichthestochastically
stablestatesareasubsetoftheNashequilibria,studyingtheratiooftheworststochas-
tically stable state to the optimal state can be viewed as a smoothed analysis of the
priceofanarchy,distinguishingNashequilibriathatarebrittletosmallperturbationsin
perfectplayfromthosethatareresilienttonoise.
The evolutionary game theory literature on stochastic stability studies n-player
gamesthatareplayedrepeatedly.Ineachround,eachplayerobservesheractionandits
outcome,andthenusessimplerulestoselectheractionforthenextroundbasedonly
on her size-restricted memory of the past rounds. In any round, players have a small
probabilityof deviatingfromtheir prescribeddecision rules. The state of the game is
thecontentsofthememoriesofalltheplayers.Thestochasticallystablestatesinsucha
gamearethestateswithnon-zeroprobabilityinthelimitofthisrandomprocess,asthe
probabilityof errorapproacheszero. The play dynamicswe employ in this paper are
theimitationdynamicsstudiedbyJosephsonandMatros[16].Underthesedynamics,
eachplayerimitatesthestrategythatwasmostsuccessfulforherinrecentmemory.
Toillustratetheutilityofstochasticstability,westudythepriceofstochasticanar-
chyoftheunrelatedloadbalancinggame[2,1,11].Toourknowledge,wearethefirst
to quantifythe loss of efficiencyin anysystem when the playersare in stochastically
stableequilibria.Intheloadbalancinggameonunrelatedmachines,evenwithonlytwo
playersand two machines,there are Nash equilibriawith arbitrarilyhigh cost, andso
the price of Nash anarchyis unbounded.We show thatthese equilibriaare inherently
1
brittle, and that for two players and two machines, the price of stochastic anarchy is
2. This result matches the strong price of anarchy [1] without requiring coordination
(atstrongNashequilibria,playershavetheabilitytocoordinatebyformingcoalitions).
Wefurthershowthatinthegeneraln-player,m-machinegame,thepriceofstochastic
anarchyisbounded.Morepreciselythepriceofstochasticanarchyisupperboundedby
thenmthn-stepFibonaccinumber.We alsoshowthatthepriceofstochasticanarchy
isatleastm+1.
Ourworkprovidesnewinsightintotheequilibriaoftheloadbalancinggame.Un-
like some previouswork on dynamics for games, our work does not seek to propose
practicaldynamicswithfastconvergence;rather,weusesimpledynamicsasatoolfor
understandingthe inherentrelativestability of equilibria.Insteadof relyingonplayer
coordinationtoavoidtheNashequilibriawithunboundedcost(asisdoneinthestudy
of strong equilibria), we show that these bad equilibria are inherently unstable in the
face of occasionaluncoordinatedmistakes. We conjecture that the price of stochastic
anarchyisclosertothelinearlowerbound,parallelingthepriceofstronganarchy.
In light of our results, we believe the techniques in this paper will be useful for
understandingtherelativestabilityofNashequilibriainothergamesforwhichtheworst
equilibria are brittle. Indeed,for a variety of gamesin the price of anarchyliterature,
theworstNashequilibriaofthelowerboundinstancesarenotstochasticallystable.
1.1 RelatedWork
Wegiveabriefsurveyofrelatedworkinthreeareas:alternativestoNashequilibriaas
asolutionconcept,stochasticstability,andtheunrelatedloadbalancinggame.
Recently,severalpapershavenotedthattheNashequilibriumisnotalwaysasuit-
ablesolutionconceptforcomputationallyboundedagentsplayinginarepeatedgame,
and have proposed alternatives. Goemans et al. [15] study players who sequentially
play myopic best responses, and quantify the price of sinking that results from such
play.FabrikantandPapadimitriou[9]proposeamodelinwhichagentsplayrestricted
finite automata. Blum et al. [4,3] assume only that players’ action histories satisfy a
propertycallednoregret,andshowthatformanygames,theresultingsocialcostsare
noworsethanthoseguaranteedbypriceofanarchyresults.
Although we believe this to be the first work studying stochastic stability in the
computer science literature, computer scientists have recently employed other tools
from evolutionary game theory. Fisher and Vo¨cking [13] show that under replicator
dynamicsin the routinggame studied by Roughgardenand Tardos[22], playerscon-
verge to Nash. Fisher et al. [12] went on to show that using a simultaneous adaptive
samplingmethod,playconvergesquicklytoaNashequilibrium.Forathoroughsurvey
of algorithmic results that have employed or studied other evolutionary game theory
techniquesandconcepts,seeSuri[23].
Stochasticstabilityanditsadaptivelearningmodelasstudiedinthispaperwerefirst
defined by Foster and Young [14], and differ from the standard game theorysolution
conceptofevolutionarilystablestrategies(ESS).ESSarearefinementofNashequilib-
ria, and so do notalways exist, and are notnecessarily associated with a naturalplay
dynamic.Incontrast,agamealwayshasstochasticallystablestatesthatresult(bycon-
struction)fromnaturaldynamics.Inaddition,ESSare resilientonlyto singleshocks,
whereasstochasticallystablestatesareresilienttopersistentnoise.
2
Stochastic stability has been widely studied in the economics literature (see, for
example, [24,17,19,5,7,21,16]). We discuss in Sect. 2 concepts from this body of
literaturethatarerelevanttoourresults.WerecommendYoung[25]foraninformative
andreadableintroductiontostochasticstability,itsadaptivelearningmodel,andsome
relatedresults.Ourworkdiffersfrompriorworkinstochasticstabilityinthatitisthe
first to quantifythe socialutility ofstochasticallystable states, the price ofstochastic
anarchy.
Wealsonoteaconnectionbetweenthestochasticallystablestatesofthegameand
thesinksofagame,recentlyintroducedbyGoemansetal.asanotherwayofstudying
thedynamicsofcomputationallyboundedagents.Inparticular,thestochasticallystable
states of a game under the play dynamics we consider correspond to a subset of the
sink equilibria, and so provide a frameworkfor identifying the stable sink equilibria.
In potential games, the stochastically stable states of the play dynamics we consider
correspondtoasubsetoftheNashequilibria,thusprovidingamethodforidentifying
whichoftheseequilibriaarestable.
Inthispaper,westudythepriceofstochasticanarchyinloadbalancing.Even-Dar
etal.[8]showthatwhenplayingtheloadbalancinggameonunrelatedmachines,any
turn-takingimprovementdynamicsconvergetoNash.Andelmanetal.[1]observethat
thepriceofNashanarchyinthisgameisunboundedandtheyshowthatthestrongprice
ofanarchyislinearinthenumberofmachines.Fiatetal.[11]tightentheirupperbound
tomatchtheirlowerboundatastrongpriceofanarchyofexactlym.
2 Model andBackground
We now formalize (from Young [24]) the adaptive play model and the definition of
stochastic stability. We then formalize the play dynamics that we consider. We also
provide in this section the results from the stochastic stability literature that we will
lateruseforourresults.
2.1 AdaptivePlayandStochasticStability
LetG=(X,π)beagamewithnplayers,whereX = n X representsthestrategy
j=1 i
setsX foreachplayeri,andπ = n π representsthepayofffunctionsπ :X →R
i j=1 i Q i
foreachplayer.Gisplayedrepeatedlyforsuccessivetimeperiodst = 1,2,...,andat
each time step t, player i plays soQme action st ∈ X . The collection of all players’
i i
actionsattimetdefinesaplayprofileSt = (St,St,...,St).Wewishtomodelcom-
1 2 n
putationallyefficientagents,andsoweimaginethateachagenthassomefinitememory
ofsize z,andthataftertime stept, allplayersremembera historyconsistingofa se-
quenceofplayprofilesht =(St−z+1,St−z+2,...,St)∈(X)z.
Weassumethateachplayerihassomeefficientlycomputablefunctionp :(X)z×
i
X → Rthat,givenaparticularhistory,inducesa sampleableprobabilitydistribution
i
over actions (for all players i and histories h, p (h,a) = 1). We write p for
a∈Xi i
p .Wewishtomodelimperfectagentswhomakemistakes,andsoweimaginethat
i i P
attimeteachplayeriplaysaccordingtop withprobability1−(cid:15),andwithprobability
i
(cid:15)QplayssomeactioninX uniformlyatrandom.3Thatis,forallplayersi,forallactions
i
3Themistakeprobabilitiesneednotbeuniformrandom—allthatwerequireisthatthedistri-
butionhassupportonallactionsinX .
i
3
a∈X ,Pr[st =a]=(1−(cid:15))p (ht,a)+ (cid:15) .Thedynamicswehavedescribeddefine
i i i |Xi|
aMarkovprocessPG,p,(cid:15) withfinitestatespaceH = (X)z correspondingtothefinite
histories.Fornotationalsimplicity,wewillwritetheMarkovprocessasP(cid:15) whenthere
isnoambiguity.
Thepotentialsuccessorsofahistorycanbeobtainedbyobservinganewplaypro-
file,and“forgetting”theleastrecentplayprofileinthecurrenthistory.
Definition2.1. For any S0 ∈ X, A history h0 = (St−z+2,St−z+3,...,St,S0) is a
successorofhistoryht =(St−z+1,St−z+2,...,St).
TheMarkovprocessP(cid:15) hastransitionprobabilityp(cid:15) ofmovingfromstateh =
h,h0
(S1,...,Sz)tostateh0 =(T1,...,Tz):
n (1−(cid:15))p (h,Tz)+ (cid:15) ifh0isasuccessorofh;
p(cid:15)h,h0 =(0Qi=1 i i |Xi| otherwise.
We will refer to P0 as the unperturbed Markov process. Note that for (cid:15) > 0,
p(cid:15) > 0 for every history h and successor h0, and that for any two histories h and
h,h0
hˆ notnecessarilyasuccessorofh,thereisaseriesofz historiesh ,...,h suchthat
1 z
h =h,h =hˆ,andforall1<i≤z,h isasuccessorofh .Thusthereispositive
1 z i i−1
probability of moving between any h and any hˆ in z steps, and so P(cid:15) is irreducible.
Similarly,thereisapositiveprobabilityofmovingbetweenanyh andanyhˆ inz +1
steps,andsoP(cid:15)isaperiodic.Therefore,P(cid:15)hasauniquestationarydistributionµ(cid:15).
The stochastically stable states of a particular game and player dynamics are the
stateswithnonzeroprobabilityinthelimitofthestationarydistribution.
Definition2.2 (Foster and Young [14]).A state h is stochastically stable relative to
P(cid:15)iflim µ(cid:15)(h)>0.
(cid:15)→0
Intuitively,weshouldexpectaprocessP(cid:15) tospendalmostallofitstimeatitsstochas-
ticallystablestateswhen(cid:15)issmall.
Whenaplayeriplaysatrandomratherthanaccordingtop ,wecallthisamistake.
i
Definition2.3 (Young [24]). Suppose h0 = (St−z+1,...,St) is a successor of h. A
mistakeinthetransitionbetweenhandh0 isanyelementSt suchthatp (h,St) = 0.
i i i
Notethatmistakesoccurwithprobability≤(cid:15).
We can characterize the number of mistakes required to get from one history to
another.
Definition2.4 (Young [24]). For any two states h,h0, the resistance r(h,h0) is the
minimumtotalnumberofmistakesinvolvedinthetransitionh→h0ifh0isasuccessor
ofh.Ifh0isnotasuccessorofh,thenr(h,h0)=∞.
Notethatthetransitionsofzeroresistanceareexactlythosethatoccurwithpositive
probabilityintheunperturbedMarkovprocessP0.
Definition2.5. WerefertothesinksofP0asrecurrentclasses.Inotherwords,arecur-
rentclassofP0isasetofstatesC ⊆H suchthatanystateinC isreachablefromany
otherstateinC andnostateoutsideC isaccessiblefromanystateinsideC.
4
WemayviewthestatespaceH asthevertexsetofadirectedgraph,withanedge
fromhtoh0ifh0isasuccessorofh,withedgeweightr(h,h0).
Observation2.6. WeobservethattherecurrentclassesH ,H ,...,whereeachH ⊆
1 2 i
H,havethefollowingproperties:
1. Fromeveryvertexh∈H,thereisapathofcost0tooneoftherecurrentclasses.
2. For each H and for every pair of vertices h,h0 ∈ H , there is a path of cost 0
i i
betweenhandh0.
3. ForeachH ,everyedge(h,h0)withh∈H ,h0 6∈H haspositivecost.
i i i
Let r denote the cost of the shortest path between H and H in the graph de-
i,j i j
scribedabove.WenowconsiderthecompletedirectedgraphGwithvertexset{H ,H ,...}
1 2
inwhichtheedge(H ,H )hasweightr .LetT beadirectedminimum-weightspan-
i j i,j i
ningin-treeofG rootedatvertexH .(Anin-treeisadirectedtreewhereeachedgeis
i
orientedtowardtheroot.)ThestochasticpotentialofH isdefinedtobethesumofthe
i
edgeweightsinT .
i
Youngprovesthefollowingtheoremcharacterizingstochasticallystablestates:
Theorem2.7 (Young[24]).Inanyn-playergameGwith finitestrategysetsandany
setofactiondistributionsp,thestochasticallystablestatesofPG,p,(cid:15) aretherecurrent
classesofminimumstochasticpotential.
2.2 ImitationDynamics
In this paper, we study agents who behave according to a slight modification of the
imitationdynamicsintroducedbyJosephsonandMatros[16].(Wenotethatthismodi-
ficationisofnoconsequencetotheresultsofJosephsonandMatros[16]thatwepresent
below.)Playeriusingimitationdynamicsparameterizedbyσ ∈ Nchooseshisaction
attimet+1accordingtothefollowingmechanism:
1. PlayeriselectsasetY ofσ playprofilesuniformlyatrandomfromthez profiles
inhistoryh .
t
2. ForeachplayprofileS ∈ Y,i recallsthepayoffπ (S)he obtainedfromplaying
i
actionS .
i
3. Player i playsthe actionamongthese that correspondsto hishighestpayoff;that
is,heplaystheith componentofargmax π (S).Inthecaseofties,heplaysa
S∈Y i
highest-payoffactionatrandom.
The value σ is a parameterof the dynamicsthat is taken to be n < σ ≤ z/2. These
dynamicscanbeinterpretedasmodelingasituationinwhichateachtimestep,players
arechosenatrandomfromapoolofidenticalplayers,whoeachplayedinasubsetofthe
lastz rounds.Theplayersarecomputationallysimple,andsodonotcounterspeculate
the actionsof their opponents,instead playingthe actionthathas workedthe bestfor
theminrecentmemory.
WewillsaythatahistoryhismonomorphicifthesameactionprofileS hasbeen
repeatedforthelastzrounds:h=(S,S,...,S).JosephsonandMatros[16]provethe
followingusefulfact:
Proposition2.8. A set of states is a recurrent class of the imitation dynamics if and
onlyifitisasingletonsetconsistingofamonomorphicstate.
5
Since the stochastically stable states are a subset of the recurrent classes, we can
associatewitheachstochasticallystablestateh =(S,...,S)theuniqueactionprofile
S itcontains.Thisallowsustonowdefinethepriceofstochasticanarchywithrespect
toimitationdynamics.Forbrevity,wewillrefertothisthroughoutthepaperassimply
thepriceofstochasticanarchy.
Definition2.9. GivenagameG = (X,π)withasocialcostfunctionγ : X → R,the
priceofstochasticanarchyofGisequaltomax γ(S) ,whereOPTistheplayprofile
γ(OPT)
thatminimizesγandthemaxistakenoverallplayprofilesSsuchthath=(S,...,S)
isstochasticallystable.
Given a game G, we define the better response graph of G: The set of vertices
correspondstothesetofactionprofilesofG,andthereisanedgebetweentwoaction
profilesS andS0 ifandonlyifthereexistsaplayerisuchthatS0 differsfromS only
inplayeri’saction,andplayeridoesnotdecreasehisutilitybyunilaterallydeviating
fromS toS0.JosephsonandMatros[16]provethefollowingrelationshipbetweenthis
i i
betterresponsegraphandthestochasticallystablestatesofagame:
Theorem2.10. IfVisthesetofstochasticallystablestatesunderimitationdynamics,
thenV ={S :(S,...,S)∈V}iseitherastronglyconnectedcomponentofthebetter
responsegraphofG,oraunionofstronglyconnectedcomponents.
Goemans et al. [15] introduce the notion of sink equilibria and a corresponding
notion of the “price of sinking”, which is the ratio of the social welfare of the worst
sink equilibrium to that of the social optimum. We note that the strongly connected
componentsofthebetterresponsegraphofGcorrespondtothesinkequilibria(under
sequentialbetter-responseplay)ofG,andsoTheorem2.10impliesthatthestochasti-
callystablestates underimitationdynamicscorrespondto asubsetofthesinksofthe
betterresponsegraphofG,andwegetthefollowingcorollary:
Corollary2.11. ThepriceofstochasticanarchyofagameGunderimitationdynamics
isatmostthepriceofsinkingofG.
3 Load Balancing:GameDefinitionand PriceofNashAnarchy
The load balancing game on unrelated machines models a set of agents who wish to
schedulecomputingjobsona setof machines.Themachineshavedifferentstrengths
andweaknesses(forexample,theymayhavedifferenttypesofprocessorsordiffering
amounts of memory), and so each job will take a differentamount of time to run on
each machine.Jobs on a single machineare executedin parallelsuch thatall jobs on
anygivenmachinefinishatthesametime.Thus,eachagentwhoscheduleshisjobon
machineM endurestheloadonmachineM ,wheretheloadisdefinedtobethesumof
i i
therunningtimesofalljobsscheduledonM .Agentswishtominimizethecompletion
i
timefortheirjobs,andsocialcostisdefinedtobethemakespan:themaximumloadon
anymachine.
Formally,aninstanceoftheloadbalancinggameonunrelatedmachinesisdefined
byasetofnplayersandmmachinesM ={M ,...,M }.Theactionspaceforeach
1 m
player is X = M. Each player i has some cost c on machine j. Denote the cost
i i,j
6
of machine Mj for action profile S by Cj(S) = is.t.Si=jci,j. Each player i has
utility function π (S) = −C (S). The social cost of an action profile S is γ(S) =
max C (S). iWe define OsPi T to be the actionPprofile that minimizes social cost:
j∈M j
OPT = argmin γ(S). Without loss of generality, we will always normalize so
S∈X
thatγ(OPT)=1.
The coordinationratio of a game (also known as the price of anarchy)was intro-
ducedby KoutsoupiasandPapadimitriou[18],and is intendedto quantifythe loss of
efficiencyduetoselfishnessandthelackofcoordinationamongrationalagents.Given
agameGandasocialcostfunctionγ,itissimpletoquantifytheOPTgamestateS:
OPT=argminγ(S).Itislessclearhowtomodelrationalselfishagents.Inmostprior
workithasbeenassumedthatselfishagentsplayaccordingtoaNashequilibrium,and
thepriceofanarchyhasbeendefinedastheratioofthecostoftheworst(purestrategy)
NashstatetoOPT.Inthispaper,werefertothismeasureasthepriceofNashanarchy,
todistinguishitfromthepriceofstochasticanarchy,whichwedefinedinSect.2.2.
Definition3.1. For a game G with a set of Nash equilibrium states E, the price of
(Nash)anarchyismax γ(S) .
S∈E γ(OPT)
Weshowherethatevenwithonlytwoplayersandtwomachines,theloadbalancing
game on unrelated machines has a price of Nash anarchy that is unbounded by any
functionofmandn.Considerthetwo-player,two-machinegamewithc =c =1
1,1 2,2
andc = c = 1/δ,forsome0 < δ < 1.ThentheplayprofileOPT = (M ,M )
1,2 2,1 1 2
is a Nash equilibriumwith cost1. However,observethatthe profile S∗ = (M ,M )
2 1
isalsoaNashequilibrium,withcost1/δ(sincebydeviating,playerscanonlyincrease
their cost from 1/δ to 1/δ +1). The price of anarchy of the load balancing game is
therefore1/δ,whichcanbeunboundedlylarge,althoughm=n=2.
4 Upper Bound onPriceofStochastic Anarchy
Theloadbalancinggameisanordinalpotentialgame[8],andsothesinksofthebetter-
responsegraphcorrespondto the purestrategyNash equilibria.We thereforehaveby
Corollary2.11thatthestochasticallystablestatesareasubsetofthepurestrategyNash
equilibriaofthegame,andthepriceofstochasticanarchyisatmostthepriceofanarchy.
Wehavenotedthateveninthetwo-person,two-machineloadbalancinggame,theprice
ofanarchyisunbounded(evenforpurestrategyequilibria).Therefore,asawarmup,we
boundthepriceofstochasticanarchyofthetwo-player,two-machinecase.
4.1 TwoPlayers,TwoMachines
Theorem4.1. Inthetwo-player,two-machineloadbalancinggameonunrelatedma-
chines,thepriceofstochasticanarchyis2.
Notethatthetwo-player,two-machineloadbalancinggamecan haveatmosttwo
strictpurestrategyNashequilibria.(Forbrevityweconsiderthecaseofstrictequilibria.
The argument for weak equilibria is similar). Note also that either there is a unique
Nashequilibriumat(M ,M )or(M ,M ),ortherearetwoatN = (M ,M )and
1 1 2 2 1 1 2
N =(M ,M ).
2 2 1
AnactionprofileN ParetodominatesN0ifforeachplayeri,CNi(N)≤CNi0(N0).
7
Lemma4.2. IftherearetwoNashequilibria,andN ParetodominatesN ,thenonly
1 2
N isstochasticallystable(andviceversa).
1
Proof. Note thatif N ParetodominatesN , then italso Paretodominates(M ,M )
1 2 1 1
and (M ,M ), since each is a unilateral deviation from a Nash equilibrium for both
2 2
players.Considerthemonomorphicstate(N ,...,N ).Ifbothplayersmakesimulta-
2 2
neous mistakes at time t to N , then by assumption, N will be the action profile in
1 1
h = (N ,...,N ,N ) with lowest cost for both players.Therefore,with positive
t+1 2 2 1
probability,bothplayerswilldrawsamplesoftheirhistoriescontainingtheactionpro-
fileN ,andthereforeplayit,untilh =(N ,...,N ).Therefore,thereisanedgein
1 t+z 1 1
G fromh={N ,...,N }toh0 ={N ,...,N }ofresistance2.However,thereisno
2 2 1 1
edgefromh0 toanyotherstateinG withresistance< σ.Recallourinitialobservation
thatinfact,N Paretodominatesallotheractionprofiles.Therefore,nosetofmistakes
1
willyieldanactionprofilewithhigherpayoffthanN foreitherplayer,andsotoleave
1
state h0 willrequireatleastσ mistakes(sothatsomeplayermaydrawa samplefrom
theirhistorythatcontainsnoinstanceofactionprofileN ).Therefore,givenanymin-
1
imumspanningin-treeofG rootedath,wemayaddanedge(h,h0)ofweight2,and
removetheoutgoingedgefromh0,whichwehaveshownmusthavecost≥σ.Thisisa
minimumspanningtreerootedath0 withstrictlylowercost.Wehavethereforeshown
that h0 has strictly lower stochastic potentialthan h, and so by Theorem2.7,h is not
stochasticallystable.SinceatleastoneNashequilibriummustbestochasticallystable,
h0 =(N ,...,N )istheuniquestochasticallystablestate. ut
1 1
Proof (ofTheorem4.1).IfthereisonlyoneNashequilibrium(M ,M )or(M ,M ),
1 1 2 2
thenitmustbetheonlystochasticallystablestate(sinceinpotentialgamesthesearea
nonemptysubsetofthepurestrategyNashequilibria),andmustalsobeOPT.Inthis
case,thepriceofanarchyisequaltothepriceofstochasticanarchy,andis1.Therefore,
wemayassumethattherearetwoNashequilibria,N andN .IfN Paretodominates
1 2 1
N ,thenN mustbeOPT(sinceloadbalancingisapotentialgame),andbyLemma
2 1
4.2,N istheonlystochasticallystablestate.Inthiscase,thepriceofstochasticanarchy
1
is1(strictlylessthanthe(possiblyunbounded)priceofanarchy).Asimilarargument
holdsif N Pareto dominatesN . Therefore,we may assume that neitherN nor N
2 1 1 2
Paretodominatetheother.
Withoutlossofgenerality,assumethatN isOPT, andthatinN = (M ,M ),
1 1 1 2
M is the maximallyloadedmachine.Suppose thatM is also the maximally loaded
2 2
machine in N . (The other case is similar.) Together with the fact that N does not
2 1
ParetodominateN ,thisgivesusthefollowing:
2
c ≤c
1,1 2,2
c ≤c
2,1 2,2
c ≥c
1,2 2,2
FromthefactthatbothN andN areNashequilibria,weget:
1 2
c +c ≥c
1,1 2,1 2,2
c +c ≥c
1,1 2,1 1,2
Inthiscase,thepriceofanarchyamongpurestrategyNashequilibriais:
c c +c c +c c
1,2 1,1 2,1 1,1 2,1 2,1
≤ ≤ =1+
c c c c
2,2 2,2 1,1 1,1
8
Similarly,wehave:
c c +c c +c c
1,2 1,1 2,1 1,1 2,1 1,1
≤ ≤ =1+
c c c c
2,2 2,2 2,1 2,1
Combining these two inequalities, we get that the price of Nash anarchy is at most
1+min(c /c ,c /c ) ≤ 2.Sincethepriceofstochasticanarchyisatmostthe
1,1 2,1 2,1 1,1
priceofanarchyoverpurestrategies,thiscompletestheproof. ut
4.2 GeneralCase:nPlayers,mMachines
Theorem4.3. The general load balancing game on unrelated machines has price of
stochasticanarchyboundedbyafunctionΨ dependingonlyonnandm,and
Ψ(n,m)≤m·F (nm+1),
(n)
whereF (i)denotestheithn-stepFibonaccinumber.4
(n)
Toprovethisupperbound,weshowthatanysolutionworsethanourupperbound
cannotbestochasticallystable.Toshowthisimpossibility,wetakeanyarbitrarysolu-
tionworsethanourupperboundandshowthattheremustalwaysbeaminimumcost
in-tree in G rootedat a differentsolution that has strictly less cost than the minimum
costin-treerootedatthatsolution.WethenapplyProposition2.8andTheorem2.7.The
proofproceedsbyaseriesoflemmas.
Definition4.4. ForanymonomorphicNashstateh = (S,...,S),lettheNashGraph
ofhbeadirectedgraphwithvertexsetM anddirectededges(M ,M )ifthereissome
i j
playeriwithS =M andOPT =M .LettheclosureM¯ ofmachineM ,betheset
i i i j i i
ofstatesreachablefromM byfollowing0ormoreedgesoftheNashgraph.
i
Lemma4.5. InanymonomorphicNashstateh=(S,...,S),ifthereisamachineM
i
suchthatC (S)>m,theneverymachineM ∈M¯ hascostC (S)>1.
i j i j
Proof. Supposethiswerenotthecase,andthereexistsanM ∈ M¯ withC (S) ≤ 1.
j i j
Since M ∈ M¯ , there exists a simple path (M = M ,M ,...,M = M ) with
j i i 1 2 k j
k ≤ m.SinceS isaNashequilibrium,itmustbethecasethatC (S) ≤ 2because
k−1
bythe definitionofthe Nash graph,the directededgefromM to M impliesthat
k−1 k
there is some player i with S = M , but OPT = M . Since 1 = γ(OPT) ≥
i k−1 i k
C (OPT) ≥ c ,ifplayerideviatedfromhisactioninNashprofileS toS0 = M ,
k i,k i k
he would experience cost C (S)+c ≤ 1+1 = 2. Since he cannot benefit from
k i,k
deviating(by definitionof Nash),it mustbe thathiscostin S, C (S) ≤ 2. By the
k−1
sameargument,itmustbethatC (S)≤3,andbyinduction,C (S)≤k ≤m. ut
k−2 1
Lemma4.6. ForanymonomorphicNash stateh = (S,...,S) ∈ G with γ(S) > m,
thereisanedgefromhtosomeg = (T,...,T)whereγ(T) ≤ mwithedgecost≤ n
inG.
1 ifi≤n;
4F (i)= F (i)∈o(2i)foranyfixedn.
(n) ( ij=i−nF(n)(j) otherwise. (n)
P
9