Price of Anarchy for the N-player Competitive Cascade Game with Submodular Activation Functions

14 Pages·2013·0.27 MB·English

Price of Anarchy for the N-player Competitive Cascade Game with
Submodular Activation Functions
XinranHe(cid:63)andDavidKempe(cid:63)(cid:63)
ComputerScienceDepartment,UniversityofSouthernCalifornia,
941BloomWalk,LosAngeles,CA,USA
Abstract. We study the Price of Anarchy (PoA) of the competitive cascade game following the
frameworkproposedbyGoyalandKearnsin[11].OurmaininsightisthatareductiontoaLinear
ThresholdModelinatime-expandedgraphestablishesthesubmodularityofthesocialutilityfunc-
tion.Fromthisobservation,wededucethatthegameisavalidutilitygame,whichinturnimplies
anupperboundof2onthe(coarse)PoA.Thiscleanerunderstandingofthemodelyieldsasimpler
proof of a much more general result than that established by Goyal and Kearns: for the N-player
competitivecascadegame,the(coarse)PoAisupper-boundedby2underanygraphstructure.We
alsoshowthatthisboundistight.
Keywords: Competitivecascadegame,PriceofAnarchy,Submodularity,Validutilitygame,Influ-
encemaximization
1 Introduction
The processes and dynamics by which information and behaviors spread through social networks have
longinterestedscientistswithinmanyareas[18].Understandingsuchprocesseshasthepotentialtoshed
lightonhumansocialstructure,andtoimpactthestrategiesusedtopromotebehaviorsorproducts.While
theinterestinthesubjectislong-standing,therecentincreasedavailabilityofsocialnetworkandinforma-
tiondiffusiondata(throughsitessuchasFacebookandLinkedIn)hasputintoreliefalgorithmicquestions
withinthearea,andledtowidespreadinterestinthetopicwithinthecomputersciencecommunity.
One particular application that has been receiving interest in enterprises is to use word-of-mouth
effects as a tool for viral marketing. Motivated by the marketing goal, mathematical formalizations of
influence maximization have been proposed and extensively studied by many researchers [9,14,17,23,
24,8,7,16]. Influence maximization is the problem of selecting a small set of seed nodes in a social
network,suchthattheiroverallinfluenceonothernodesinthenetwork—definedaccordingtoparticular
modelsofdiffusion—ismaximized.
Whenconsideringtheword-of-mouthmarketingapplication,itisnaturaltorealizethatmultiplecom-
panies,politicalmovements,orotherorganizationsmayusediffusioninasocialnetworktopromotetheir
productssimultaneously.Forexample,SamsungmaytrytopromotetheirnewGalaxyphone,whileAp-
pletriestoadvertisetheirnewiPhone.Companieswillnecessarilyendupincompetitionwitheachother,
soitbecomesessentialtounderstandtheoutcomeofcompetitivediffusionphenomenainthenetwork.
Motivatedbytheabovescenarios,severalmodelsforcompetitivediffusionhavebeenproposedand
studied [2,4,7,12,20,11,1,21]. Past work tends to follow one of two assumptions about the timing of
players’moves.Thefirstapproachistoassumethatallbutoneofthecompetitorshavealreadychosen
their strategies, and to study the algorithmic problem of finding the best response [2,4,7,12,5,6]. The
goalmaybemaximizingone’sowninfluence[2,4,7]orminimizingtheinfluenceofthecompetitors[12,
(cid:63)SupportedinpartbyNSFGrant0545855.
(cid:63)(cid:63)SupportedinpartbyNSFGrant0545855,anONRYoungInvestigatorAward,andaSloanRseearchFellowship.
2 XinranHeandDavidKempe
5].Theotherapproachistomodelthecompetitionasasimultaneousgame,inwhichallcompaniespick
theirstrategiesatthesametime[1,11,20,21].Thefinalinfluenceisdeterminedbytheinitialseedsetof
everycompanyandtheunderlyingdiffusionprocess.
In this paper, we follow the second approach. In the game, the players are companies (or other or-
ganizations)whotrytopromotetheircompetingproductsinthesocialnetworkthroughword-of-mouth
marketing. The players simultaneously allocate resources to individuals in the social network in order
toseedthemasinitialadoptersoftheirproducts.Theseresourcescouldbefreesamples,timespentex-
plainingtheadvantagesoftheproduct,ormonetaryrewards.Basedontheallocatedresources,thenodes
choose which of the products to adopt initially. Subsequently, the diffusion of the adoption of products
proceedsaccordingtothelocaladoptiondynamics.Thegoalforeachplayeristomaximizethecoverage
ofhis1ownproduct.
The local adoption dynamics play a vital role in determining properties of the game. In this paper,
we follow the framework proposed recently by Goyal and Kearns [11]. Their model decomposes the
localadoptiondecisionsintotwostages:switchingandselection.Intheswitchingstage,theuserdecides
whethertoadoptanyproductorcompanyatall.Thisdecisionisbasedonthesetofneighborswhohave
alreadyadoptedoneoftheproducts.Iftheuserdecidestoadoptaproduct,inthefollowingselectionstage,
shedecideswhichcompany’sproducttoadoptbasedonthefractionofneighborswhohaveadoptedthe
productfromeachcompany.
Forexample,assumethatiPhoneandGalaxyaretheonlytwosmartphonesavailable.Intheswitching
stage,auserdecideswhethertoadoptasmartphoneornot,basedonthefractionofherneighborswho
havealreadyboughtasmartphone.Ifshehasdecidedtoadoptasmartphone,intheselectionstage,she
decideswhethertochooseaniPhoneorGalaxybasedonthefractionofiPhoneusersandGalaxyusers
among her friends. The two stages are modeled using a switching function f (α +α ), which gives
v 1 2
the probability that the user adopts one of the products, and the selection function g (α ,α ), which
v 1 2
determinestheprobabilitythattheuserchoosestheproductofaspecificcompany.Hereα andα are
1 2
thefractionsoftheuser’sfriendswhohavealreadyadoptedtheproductfromthetwocompetitors.The
detailsofthemodelarepresentedinSection2.
Underthisframework,GoyalandKearnshavestudiedthePriceofAnarchy(PoA)ofthetwo-player
competitivecascadegame.Informally,thePoAisameasureofthemaximumpotentialinefficiencycre-
atedbynon-cooperativeactivity.(TheprecisedefinitionofthePoAisgiveninSection2.3.)Goyal and
KearnshaveshownthatthePoAundertheswitching-selectionmodelwithconcaveswitchingfunctions
andlinearselectionfunctionsisupper-boundedby4.2
Inthispaper,weshowthatastrongerPoAboundfortheGoyal-Kearnsmodelfollowsfromseveral
well-understood and general phenomena. The key observation is that by considering a time-expanded
graph, the Goyal-Kearns model can be considered an instance of a general threshold model. Then, the
result of Mossel and Roch [17] guarantees that the social utility function is submodular, and a simple
coupling argument establishes that players’ utility functions are competitive. With a submodular social
utilityfunction,thegameisavalidutilitygame.(ThistypeofproofwasusedpreviouslybyBharathiet
al.[2].)Finally,forvalidutilitygames,theresultsofVetta[22]andBlumetal.andRoughgarden[3,19]
establisha(coarse)PriceofAnarchyofatmost2.
Thankstotheaboveunderstanding,weobtainamuchmoregeneralresultwithamuchsimplerproof.
WeshowthatthePoAisupper-boundedby2forthecompetitivecascadegamewithanarbitrarynumber
of players and any graph structure with submodular activation functions f (·). We formally state this
v
resultinTheorem1.Moreover,byutilizingtheresultofRoughgardenin[19],weshowthatourbound
notonlyholdsforthePoAunderpureormixedNashequilibriabutalsoforthecoarsePoA.Wealsoshow
thattheproposedPoAboundistight.
1Throughoutthepaper,tosimplifythedistinctionofroles,weconsistentlyuse“she”todenoteindividualsinthe
socialnetworkand“he”todenotetheplayers,i.e.,thecompanies.
2Infact,theyprovedthatthePoAupperboundholdsinamoregeneralmodel,whichwewilldiscussinSection2.
PriceofAnarchyfortheN-playerCompetitiveCascadeGamewithSubmodularActivationFunctions 3
Theorem1. The coarse PoA is upper-bounded by 2 under the switching-selection model with concave
switchingfunctionsandlinearselectionfunctions.3
OurresultonthePoAboundholdsunderageneralizedversionoftheframeworkusedin[11].First,
and most importantly, our model allows for an arbitrary number of players. Second, we allow multiple
playerstotargetthesameindividualandalloweachplayertoputmultipleunitsofbudgetonthesamein-
dividual.4Thisgeneralizationenlargesthestrategyspacefromsetstomultisetsandsomewhatcomplicates
theanalysisofourmodel.Third,weassociateeachindividualinthenetworkwithaweightmeasuringthe
importanceofthenode.Fourth,wegeneralizetheadoptionfunctionsdefinedonthefractionofalready
adoptingneighborstoarbitrarysetfunctionsdefinedontheindividualswhohavepreviouslyadoptedthe
product.
1.1 RelatedWork
Ourworkismainlymotivatedby[11],lyingattheintersectionofinfluenceanalysisinsocialnetworks
and traditional game theory research. The model in [11] and the differences compared to our work are
discussedindetailaboveandinSection2.
Submodularity has been a recurring topic in the study of diffusion phenomena [17,14,12,2,4,5].
[14,17] have shown that influence coverage is submodular under local dynamics with submodularity.
The submodularity of global influence coverage can be utilized to design efficient algorithm for either
maximizing the influence [14] or minimizing the influence of the competitors [12]. Submodularity has
alsobeenappliedintheanalysisofacompetitiveinfluencegamebyBharathietal.[2].Bharathietal.use
asimilarapproachaswedointhispaper;theyalsoboundthePoAboundbyshowingthatthegameisa
validutilitygame.However,theyanalyzethecompetitivecascadegameunderasimplerdiffusionmodel.
Under their model, a node adopts the product from the neighbor who first succeeds in activating her; a
continuoustimingcomponentensuresthatthisnodeisuniquewithprobability1.
IntheproofforthePoAboundofthecompetitivecascadegame,wearedrawingheavilyonprevious
research on the PoA for valid utility games [22,19,3]. Vetta first showed that for a valid utility game,
the PoA for pure Nash equilibria is upper-bounded by 2 in [22]. Blum et al. and Roughgarden later
generalizedVetta’sresulttothecoarsePoAin[19,3].
Several other game-theoretic approaches have been considered for competitive diffusion in social
networks[21,1,20,10,6].[20]mainlyfocusesontheefficientcomputationoftheNashstrategyinstead
ofthetheoreticalboundofthePoA. [6]focusesonstudyingthealgorithmicproblemoffindingthebest
response. Though [1,21,10] studied the competitive cascade game from a game-theoretic perspective,
they mainly focused on the existence of pure Nash equilibria. [1] mainly focuses on the existence of
pureNashequilibriaunderadeterministicthresholdmodel.[10]alsotriestocharacterizethestructureof
thepureNashequilibriainthegame.ThePoAisstudiedin[21];however,theystudiedthePoAbound
ofpureNashequilibriaandusedadeterministicdiffusionmodelinsteadofthestochasticdynamicswe
useinourwork.Intheirmodel,thePoAisunboundedasintheGoyal-Kearnsmodelwithnon-concave
switchingfunctions.Asnotedby[4,5],smalldifferencesinthediffusionmodelcanleadtodramatically
differentbehaviorsofthemodel.
2 ModelsandPreliminaries
Inthissection,wedefinebasicnotation,presentthedifferentmodelsofdiffusionandtheN-playergame,
andincludeotherdefinitionsofconceptsusedinourproof.Inthegame,theplayersallocateresourcesto
3SimilartotheresultbyGoyalandKearns,ourPoAupperboundextendstoamoregeneralmodel.Wedefinethis
moregeneralmodelinSection2,andstateandprovethemoregeneralresultinSection3.
4ThemodelproposedbyGoyalandKearns[11]allowsformultipleunitsofbudgetonthesameindividual,butthe
proofdoesnotexplicitlycoverthisextension.
4 XinranHeandDavidKempe
thenodesinthegraphG = (V,E)towinthemasinitialadoptersoftheirproducts.Then,theadoption
of products propagates according to the local dynamics, described in detail in Section 2.1. The formal
definitionofthegameispresentedinSection2.3.
Throughout, we use the following conventions for notation. Players are typically denoted by i,j,k,
whilenodesareu,v,w.Forsets,functions,etc.,theidentityofaplayerisappliedasasuperscript,while
thatofanode(andtimestep)isappliedasasubscript.Vectorsarewritteninboldface,includingvectors
ofsets;inparticular,wefrequentlywriteS = (S1,...,SN)forthevectorofsetsofnodesbelongingto
thedifferentplayers.
2.1 Generaladoptionmodel
Thegeneraladoptionmodelisageneralizationoftheswitching-selectionmodeldescribedinSection1.
EachnodeinGisinoneoftheN+1states{0,1,...,N}.Anodevinstatei>0meansthatindividual
v has adopted the product of player i, while state 0 means that she has not adopted the product of any
player. In this case, we also say that v is inactive. Conversely, we say that node v is activated if she is
in one of the states i > 0. Initially, all nodes are inactive. The diffusion of the adoption of products is
aprocessdescribedbynodes’statechanges.Weassumethattheprocessisprogressive,meaningthata
nodecanchangeherstateatmostonce,from0tosomei>0,andmustremaininthatstatesubsequently.
Thediffusionprocessworksintwostages.WecallthefirststageSeedingandthesecondstageDif-
fusion.Inthefirststage,theinitialseedsofallplayersaredecidedbasedonthebudgetsthateachplayer
allocates to the nodes. The initial seeds are used as starting points for the diffusion stage. In the sec-
ond stage, the adoption propagates according to certain local dynamics based on the nodes who have
previouslyadoptedtheproducts.
Seedingstage: ThestrategyMi ofplayeriisamultisetofnodes.Wedefineαi asthenumberoftimes
v
thatv appearsinplayeri’smultiset.Foreachnodev ∈ V,if(cid:80)N αi = 0,theinitialstateofnodev is
i=1 v
0; otherwise, the initial state of node v is one of {1,2,...,N} with probabilities (α1v,...,αNv ), where
Zv Zv
Z = (cid:80)N αi is simply the normalizing constant. The decisions for different nodes are independent.
v i=1 v
Thus, if no player selects a node, the node remains inactive. Otherwise, the players win the node as an
initialadopterwithprobabilityproportionaltothenumberoftimestheyselectthenode.
Diffusionstage: Theimportantpartofdiffusionisthelocaldynamicsdecidingwhenanodegetsinflu-
enced,i.e.,changesherstatefrom0toi.LetSibethesetofnodesinstatei.Anodevwhoisstillinstate
0changesintostate1,...,N,0accordingtotheprobabilities
N
(cid:88)
(h1(S),...,hN(S),1− hi(S)).
v v v
i=1
We call hi(S1,...,SN) the adoption function of node v for product i. It gives the probability that
v
astillinactivenodev adoptsproductigiventhatSj isthecurrentsetofnodesinstatej.Theadoption
functionsmustsatisfythefollowingtwoconditions:
0≤hi(S)≤1, ∀v ∈V,i=1,...,N
v
(cid:80)N hi(S)≤1,∀v ∈V.
i=1 v
We call H (S) = (cid:80)N hi(S) the activation probability; it gives the probability that v adopts any
v i=1 v
productandchangesfromstate0toanystatei>0.
Equipped with the local dynamics of adoption, we still need to define in what order nodes’ states
are updated. In the general adoption model, we assume that an update schedule is given in advance to
PriceofAnarchyfortheN-playerCompetitiveCascadeGamewithSubmodularActivationFunctions 5
determine the order of updates. The update schedule is a finite sequence Q of nodes (cid:104)v ,...,v (cid:105), of
1 (cid:96)
length(cid:96).Anodecouldoccurmultipletimesinthesequence.
Nodes’ states are updated according to the order prescribed by the sequence. Let Si be the set of
t
nodesinstateiafterthefirsttupdates;Si istheseedsetofplayeriresultingfromtheseedingstage.In
0
eachroundt,thestateofnodev isupdatedaccordingtothelocaldynamicsofadoptionandpreviously
t
activated nodes, namely S = (S1 ,...,SN ). If node v is already in state i > 0, she remains in
t−1 t−1 t−1 t
statei.Otherwise,shechangesintostate1,...,N,0accordingtotheprobabilities
N
(cid:88)
(h1(S ),...,hN(S ),1− hi(S )).
v t−1 v t−1 v t−1
i=1
The states of all other nodes remain the same. The updates in different rounds are independent. The
diffusionstageendsafterthe(cid:96)updatesteps.Theprescribedupdatesequencemakesthismodeldifferent
fromthepreviouslystudiedIndependentCascadeandThresholdModels.Wediscussthedifferenceand
someimplicationsinmoredetailafterdefiningtheThresholdModelinSection2.4.
2.2 Usefulproperties
Wenextidentifythreeimportantpropertiesthatmakethemodelmoretractableanalytically:(1)additivity
oftheactivationprobabilityH ,(2)competitivenessoftheadoptionfunctionh and(3)submodularity
v v
oftheactivationfunctionf .
v
Definition1. The total activation probability H (S) = (cid:80)N h (S) is additive if and only if H can
v i=1 v v
be written as H (S) = f ((cid:83)N Si) for some monotone set function defined on V. We call f (S) the
v v i=1 v
activationfunctionforvwhenH isadditive.
v
Additivity implies that the probability for a node to adopt the product and change from inactive to
active only depends on the set of already activated nodes and not on which specific products they have
adopted. For example, the probability that one adopts a smartphone only depends on who has already
adoptedone,independentofwhoisusingiPhoneandwhoisusingGalaxy.
Tosimplifynotation,wedefineS−i =(cid:83) Sk,andS−i =(Sk) .
k(cid:54)=i k(cid:54)=i
Definition2. Theadoptionfunctionhi(S)forplayeriiscompetitiveifandonlyhi(S)≥hi(Sˆ)when-
v v v
everSˆi ⊆SiandS−i ⊆Sˆ−i.
Competitiveness means that the adoption function for player i is monotone increasing in the set of
nodesthathaveadoptedproductiandmonotonedecreasinginthesetofnodesthathaveadoptedsome
competitor’sproducts.5
Definition3. Theactivationfunctionf issubmodularifandonlyifforanytwosetS ⊆T ⊆V andany
v
nodeu∈V,
f (S∪{u})−f (S)≥f (T ∪{u})−f (T).
v v v v
5 Thisassumptionisreasonablewhenthereputationoftheproductisalreadywell-established.However,whena
newproductcomesout,thepresenceofcompetitorsmayhelppopularizetheproduct,byincreasingitsoverall
exposureorperceivedimportanceorrelevance.Theseeffectscouldleadtomorepurchasesevenforoneparticular
companyi.ThissubtledistinctionisdiscussedmoreinSection2.4.
6 XinranHeandDavidKempe
Submodularityofactivationfunctionsimpliesthattheoverallactivationprobabilityhasdiminishing
returns.Itintuitivelymeansthatthefirstfriendtobuyandrecommendasmartphonehasmoreinfluence
thanafriendwhorecommendsitaftermanyothers.
Goyal and Kearns have shown in [11] that the switching-selection model with concave switching
functionsandlinearselectionfunctionsisaspecialcaseofthegeneraladoptionmodelwithcompetitive
adoptionfunctionsandadditiveactivationprobabilities.Inaddition,duetotheconcavityoftheswitching
function f , the activation functions in the general adoption model are also submodular. Therefore, we
v
havethefollowinglemma:
Lemma1. Everyinstanceoftheswitching-selectionmodelwithconcaveswitchingfunctionsandlinear
selectionfunctionsisaninstanceofthegeneraladoptionmodelsatisfyingallthreeoftheaboveproper-
ties.
Lemma1allowsustoproveourPoAboundonlyforthegeneraladoptionmodel;itthenimpliesTheo-
rem1.
2.3 Thegame
ThecompetitivecascadegameisanN-playergameonagivengraphG = (V,E).Thestructureofthe
graph as well as all adoption functions are known to all the players. Each player i is a company. The
strategyforeachplayeriisamultisetMi ofnodes;weuseM =(M1,...,MN)todenotethestrategy
vectorforallplayersandαi forthenumberoftimesthatnodevappearsinMi.
v
Players’strategiesareconstrainedbytheirbudgetsBi,inthattheymustsatisfy|Mi|≤Bi.Wefurther
allownode-specificconstraintsrequiringthatαi ≤Ki forgivennode-specificbudgetsKi ≤Bi.These
v v v
may constrain players from investing a lot of resources into particularly hard-to-reach nodes; however,
thenode-specificconstraintsmostlyservetosimplifynotationinsomelaterproofs.Wesaythatastrategy
Miisfeasibleifalloftheaboveconditionsaresatisfied.
AllplayerssimultaneouslyallocatetheirbudgetstothenodesofG.Giventhechoicesthattheplayers
make,thepayoffsaredeterminedbythegeneraladoptionmodelasthecoverageoftheplayer’sproduct
amongtheindividualsinG.Eachnodevinthegraphisassociatedwithaweightω ≥0,measuringthe
v
importanceofthenode.Thepayofffunctionofplayeriisσi(M) = E[(cid:80) ω ],theexpectedsumof
v∈Si v
(cid:96)
weightsfromnodeshavingadoptedi’sproductafterall(cid:96)updatesteps.
Thesocialutilityγ(S )=(cid:80) σi(M)isthesumofweightsfromnodesadoptinganyoftheproducts.6
0 i
NoticethatwhentheactivationprobabilitiesH areadditive(Definition1),γ(·)onlydependsonS =
v 0
(cid:83)N Si,thesetofnodesactivatedaftertheseedingstage(butnotonwhichcompanytheychose).
i=1 0
Tosimplifynotation,wedefine(M−k,M˜k)=(M1,...,Mk−1,M˜k,Mk+1,...,MN),andinpar-
ticular(M−k,∅k)=(M1,...,Mk−1,∅,Mk+1,...,MN).
WesaythatastrategyprofileM isapureNashequilibriumifnoplayerhasanincentivetochange
hisstrategy.Namely,foranyplayeri,
σi(M)≥σi(M−i,M˜i) forallfeasibleM˜i.
LetOPTbeastrategyprofilemaximizingthesocialutilityfunction,andEQ thesetofallpureNash
pure
equilibria.ThepriceofanarchyofpureNashequilibriaisdefinedasfollows:
γ(OPT)
PurePriceofAnarchy= max .
M∈EQ γ(M)
pure
6Thisdefinitionimplicitlyassumesthattheproductcarriesavalueforthosewhoadoptit;thus,societyisbetteroff
whenmorepeopleadoptatleastoneproduct.
PriceofAnarchyfortheN-playerCompetitiveCascadeGamewithSubmodularActivationFunctions 7
However,thecompetitivecascadegamecouldhavenopureNashequilibrium[21].Thus,weextendour
analysis to more general equilibrium concepts. A coarse (correlated) equilibrium of a game is a joint
probabilitydistributionPwiththefollowingproperty[19]:ifM isarandomvariablewithdistribution
P,thenforeachplayeri,andallfeasibleMˆi:
E [σi(M)]≥E [σi(M−i,Mˆi)].
M∼P M−i∼P−i
SimilartothePoAforpureNashequilibria,thecoarsepriceofanarchyisdefinedas
γ(OPT)
CoarsePriceofAnarchy= max ,
P∈EQcoarse EM∼Pγ(M)
whereEQ isthesetofallcoarseequilibria.
coarse
2.4 TheThresholdModel
Ouranalysiswillbebasedonacarefulreductionofthegeneraladoptionmodeltothegeneralthreshold
modeldefinedin[14,15].Inthegeneralthresholdmodel(withN =1),everynodevinthenetworkhas
anassociatedactivationfunctionfˆ(·).Atthebeginningoftheprocess,eachnodedrawsathresholdθ
v v
independentlyanduniformlyfrom[0,1].StartingfromaninitiallyactivesetS ,anodebecomesactive
0
attimet(i.e.,isamemberofS )ifandonlyiffˆ(S )≥θ .Theprocessendswhenforoneround,no
t v t−1 v
newnodehasbecomeactive(whichisguaranteedtohappeninatmost|V|steps).Iftisthetimewhen
thishappens,theinfluenceoftheinitialsetS isdefinedasσ (S )=E[(cid:80) ω ].Inabeautifulpiece
0 ω 0 v∈St v
ofwork,MosselandRochestablishedthefollowingtheoremaboutthefunctionσ :
ω
Theorem2 (Mossel-Roch[17]).Iff ismonotoneandsubmodularforeverynodev inthegraph,then
v
σ ismonotoneandsubmodularunderthegeneralthresholdmodel.
ω
Given the apparent similarity between the general adoption model and the general threshold model
(say,forN = 1),itisilluminatingtoconsiderthewaysinwhichthemodelsdiffer,andtheimplications
forthecompetitivecascadegame.Inthegeneraladoptionmodel,asequenceofnodestoupdateisgiven,
andnodesonlyconsiderchangingtheirstatewhentheyappearinthesequence.Bycontrast,inthegeneral
thresholdmodel,nodesconsiderchangingtheirstateineachround.
Soatfirst,itappearsasthoughasequencerepeating|V|timesapermutationofall|V|nodeswould
allow a reduction from the threshold model to the adoption model. However, note that in the adoption
model, each node makes an independent random choice whether to change her state in each round,
whereasinthethresholdmodel,therandomchoicesarecoupledviathethresholdθ ,whichstaysconstant
v
throughout the process. In particular, if f (S ) > 0, then a node appearing often enough in the update
v 0
sequencewilleventuallybeactivatedwithprobabilityconvergingto1,whereasthisneednotbethecase
inthegeneralthresholdmodel.
The increase in activation probability caused by multiple occurrences in the update sequence has
powerful implications for the competitive game. It allows us to establish rather straightforwardly the
competitiveness(Definition2)ofeachplayer’sobjectivefunction,andthesubmodularity(Definition3)
ofthesocialutility.Bycontrast,Borodinetal.[4]showthatbothpropertiesfailtoholdformostnatural
definitionsofcompetitivethresholdgames.Attheheartofthecounter-examplesin[4]liesthefollowing
kindofdynamic:Attime1,anodeurecommendstov theuseofaGalaxyphone,butfailstoconvince
v.Attime2,anothernodew recommendstov theuseofaniPhone.Ifv decidestoadoptasmartphone
attime2,mostnaturalversionsofathresholdmodel(aswellasunderthegeneraladoptionmodel)allow
for an adoption of a Galaxy phone as well. This “extra chance” results in synergistic effects between
competitors, and thus breaks competitiveness. Under the model of [11], this problem is side-stepped. v
will only consider adopting a smartphone in step 2 when she appears in the sequence at time 2; in that
case,adoptionofaGalaxyphoneinstep2willbeconsideredindependentlyofwhetherw hasadopted
aniPhone.ThisobservationfleshesoutthediscussionalludedtoinFootnote5.
8 XinranHeandDavidKempe
2.5 Validutilitygames
A valid utility game [22] is defined on a ground set V with social utility function γ defined on subsets
ofV.ThestrategiesofthegamearesetsSi ⊆ V (itispossiblethatnotallsetsareallowedasstrategies
0
for some or all players), and the payoff functions are σi for each player i. The social utility is defined
ontheunionofallplayers’sets,γ((cid:83)N Si).Thedefinitionrequiresthatthreeconditionshold:(1)The
i=1 0
social utility function γ(·) is submodular; (2) For each player i, σi(S ) ≥ γ(S ) − γ(S−i,∅i); (3)
0 0 0
(cid:80)N σi(S )≤γ(S ).
i=1 0 0
3 UpperBoundontheCoarsePriceofAnarchy
Inthissection,wepresentourmainresult:theupperboundonthecoarsePoAwithsubmodularactivation
functions.WeprovetheupperboundonthePoAbyshowingthatthecompetitivecascadegameisavalid
utility game. We note, however, that the strategy space of our competitive cascade game consists of
multisets,whereasthestandarddefinitionofutilitygameshasonlysetsasstrategies.Inordertodealwith
thissubtletechnicalissue,weusethefollowinglemma,whoseproofisgivenintheappendix.
Lemma2. LetG =(G,{hi},{Ki},{Bi},{ω },Q)beanarbitraryinstanceofthecompetitivecascade
v v v
game.Then,thereexistsaninstanceGˆ= (Gˆ,{hˆi},{Kˆi},{Bˆi},{ωˆ },Qˆ)withthesamesetofplayers,
v vˆ v
andthefollowingproperties:
1. (cid:80)N Kˆi ≤ 1forallv ∈ Vˆ.(Atmostoneplayerisallowedtotargetanode,andwithatmostone
i=1 vˆ
resource.)
2. Foreveryplayeri,therearemappingsµi,µˆimappingi’sstrategiesinGtohisstrategiesinGˆandvice
versa,respectively,satisfyingthefollowingproperty:Ifforalli,eitherµi(Mi) = Mˆi orµˆi(Mˆi) =
Mi,thenforalli,σi(M)=σˆi(Mˆ).
In particular, Lemma 2 implies that the social utility is also preserved between the two games, and
strategies Mi are best responses to Mj,j (cid:54)= i if and only if the Mˆi are best response to Mˆj,j (cid:54)= i.
(Otherwise,aplayercouldimprovehispayoffintheothergamebyswitchingtoµi(Mi)orµˆi(Mˆi).)In
this sense, Lemma 2 establishes that for every competitive cascade game instance, there is an “equiva-
lent”instanceinwhicheachnodecanbetargetedbyatmostoneplayer,andwithatmostoneresource.
Therefore, we will henceforth assume without loss of generality that the strategy space for each player
consistsonlyofsets.
Infittingthecompetitivecascadegameintothevalidutilitygameframework,thegroundsetofthe
game is V, and the payoff function of player i is σi(S) = E[(cid:80) ω ]: the sum of weights from the
v∈Si v
(cid:96)
nodesv ∈V instateiattheendoftheupdatingsequence.Becauseofadditivity,thesocialutilityfunction
dependsonlyonS .Thatis,thefollowingiswell-defined:γ(S )=γ(S )=(cid:80) σi(S ).Therefore,the
0 0 0 i 0
thirdconditionofavalidutilitygame(sumboundedness)issatisfiedtrivially.Below,wewillprovethe
followingtwolemmas:
Lemma3. Assumethatforeverynodev,thetotalactivationprobabilityH (S)isadditive,andtheacti-
v
vationfunctionf (S)issubmodular.Then,thesocialutilityfunctionγ(S )issubmodularandmonotone.
v 0
Lemma4. Ifforeverynodevandeverplayerk,H (·)isadditiveandhk(·)iscompetitive,thenforeach
v v
playeri,wehaveσi(S )≥γ(S )−γ(S−i,∅i).
0 0 0
Lemmas3and4togetherestablishthatthecompetitivecascadegameisavalidutilitygame.Example
1.4in[19]showsthatthecoarsePoAofvalidutilitygamesisatmost2(Vetta[22]establishesthesame
forthePoA),provingthefollowingmainresultofourpaper:
PriceofAnarchyfortheN-playerCompetitiveCascadeGamewithSubmodularActivationFunctions 9
Theorem3. Assumethatthefollowingconditionshold:
1. Foreverynodev,thetotalactivationprobabilityH (S)isadditive.
v
2. Foreverynodev,theactivationfunctionf (S)issubmodular.
v
3. Foreveryplayeriandnodevinthegraph,theadoptionfunctionhi(S)iscompetitive.
v
Then,theupperboundonthePoA(andcoarsePoA)is2inthecompetitivecascadegame.
By Lemma 1, the switching-selection model with concave switching functions and linear selection
functions is a special case of the general adoption model with competitive adoption functions, additive
activationprobabilitiesandsubmodularactivationfunctions.Therefore,Theorem1followsnaturallyasa
corollaryofTheorem3.
ProofofLemma3. Webuildaninstanceofthegeneralthresholdmodelwhoseinfluencecoveragefunc-
tion σ (S ) is exactly the same as γ(S ). The idea is that for additive functions, the social utility does
ω 0 0
not depend on which node chooses which company, so the game is reduced to the case of just a single
influence.Theupdatesequencecanbeemulatedwithatime-expandedlayeredgraph.
Thetime-expandedgraphG isdefinedasfollows.7 Foreachnodev oftheoriginalgraph,wehave
(cid:96)
(cid:96)+1nodesvˆ ,vˆ ,...,vˆ inG .WeuseL = {vˆ | v ∈ V}todenotethesetofnodesinlayert.The
0 1 (cid:96) (cid:96) t t
activationfunctionsaredefinedasfollows:
1. Inlayer0,fˆ ≡0foreverynodev ∈V.
vˆ0
2. In layer t, 1 ≤ t ≤ (cid:96), consider a node v with switching function f . If v is the tth element of the
v
updatingsequence(v =v ),weset
t
(cid:40)
1 ifvˆ ∈S
fˆ (S)= t−1
vˆt f ({u|uˆ ∈S}) otherwise;
v t−1
otherwiseweset
(cid:40)
1 ifvˆ ∈S
fˆ (S)= t−1
vˆt 0 otherwise.
Finally,thetotalinfluenceisdefinedasσ (S )=E[(cid:80) ω ],whereSˆisthesetofnodesactivated
ω 0 vˆ(cid:96)∈Sˆ v
inthethresholdmodeloncenomoreactivationsoccur.Intheinstance,eachlayerL emulatesoneupdate
t
intheoriginalupdatesequenceofthegeneraladoptionmodel.
For each node vˆ in the layered graph, f (S) is submodular and additive. The submodularity and
t vˆt
monotonicityforthe0-1activationfunctionsaretriviallysatisfied.Forthenodesintheupdatesequence,
submodularityholdsbecauseweassumedthef (S)tobesubmodular,andmonotonicityfollowsbecause
v
theH (S)areadditive.
v
Next,weshowthatγ(S )=σ (S ),byusingastraightforwardcouplingbetweenthegeneralthresh-
0 ω 0
old model and the general adoption model. According to the construction of G , the state changes for
(cid:96)
allnodesexceptvˆ (wherev isthetth elementoftheupdatingsequence)aredeterministic.Therefore,
t t
we only need to draw the thresholds Θ = (cid:104)θ ,θ ,...,θ (cid:105) for the (cid:96) nodes in the update sequence: they
1 2 (cid:96)
aredrawnindependentlyanduniformlyfrom[0,1].Inthegeneraladoptionmodel,whenupdatingthetth
nodev inthesequence,ifnodev isstillinactive,shebecomesactiveifandonlyiff (S )≥θ .Ifthe
t t v t−1 t
nodeisalreadyactivated,sheremainsactivatedinthesamestate.Byinductionont,v ∈S ifandonlyif
t
vˆ ∈ Sˆ∩L .Thus,theoutcomesofthetwoprocessesarethesamepointwiseoverthresholdvectorsΘ:
t t
γ(S |Θ)=σ (S |Θ).Inparticular,theirexpectationsarethusthesame.
0 ω 0
Finally,Theorem2establishesthemonotonicityandsubmodularityofσ (S ),andthusalsoγ(S ).
ω 0 0
7Allactivationinformationisencodedintheactivationfunctions.Therefore,thereisnoneedtoexplicitlyspecify
theedgesofG .
(cid:96)
10 XinranHeandDavidKempe
ProofofLemma4. Webeginbyshowingthatundertheassumptionsofthelemma,σi(S )≤σi(S−k,∅k)
0 0
forallplayersk,i,k (cid:54)= i.Todoso,weexhibitasimplecouplingofthegeneraladoptionprocessesfor
thetwoinitialstatesS and(S−k,∅k),essentiallyidenticaltooneusedintheproofofLemma1in[11].
0 0
Noticethattheactivationfunctionsareadditive;therefore,wecancombineallstatesk (cid:54)=iintoonestate,
whichwedenoteby−i.
Theactivationprocessisdefinedbythewayinwhichnodesdecidewhethertoupdatetheirstate,and
ifso,towhichnewstate.Anequivalentwayofdescribingthechoiceisasfollows:foreachsteptofthe
updatesequence,wedrawanindependentuniformlyrandomnumberz ∈[0,1].Instept,assumingthat
t
nodev isstillinstate0,shechangesherstateto:
t
– stateiifz ∈[0,hi (S )).
– state−iiftz ∈[hivt(St−1),f ((cid:83)N Sj )).
t vt t−1 vt j=1 t−1
– state0otherwise.
To couple the two random processes with starting conditions (S−k,∅k) and S , we simply choose
0 0
thesamevaluesz forboth.LetXj denotethesetofnodesinstatej,j ∈{i,−i,0}aftertupdateswith
t t
startingcondition(S−k,∅k).Yj isdefinedanalogously,withstartingconditionS .
0 t 0
Conditionedonanychoiceof(z ,...,z ),asimpleinductionproofusingcompetitivenessofthehi
1 (cid:96) v
andmonotonicityofthef showsthatforeachtimet,Xi ⊇ Yi,X0 ⊇ Y0,andthusalsoX−i ⊆ Y−i.
v t t t t t t
Therefore,attheendoftheupdatesequence,thedesiredinequalityholdspointwiseover(z ,...,z ),and
1 (cid:96)
inparticularinexpectation,asclaimed.Finally,havingestablishedthatσi(S )≤σi(S−k,∅k),weuseit
0 0
inthefollowingcalculations:
(cid:88)
γ(S )−γ(S−i,∅i)= (σk(S )−σk(S−i,∅i))
0 0 0 0
k
(cid:88)
=σi(S )+ (σk(S )−σk(S−i,∅i))
0 0 0
k(cid:54)=i
≤σi(S ).
0
4 TightnessofthePoAUpperBound
Wegiveaninstanceofthecompetitivecascadegameinthe(morerestrictive)switching-selectionmodel
toshowthatourupperboundof2forthePoAinTheorem3istight.
LetN bethenumberofplayers.ThegraphconsistsofastarwithonecenterandN leaves,aswellas
N isolatednodes.Eachplayerhasonlyoneunitofbudget,andtheupdatesequenceisanypermutation
ofthenodesinthestargraph.Theswitchingfunctionsaretheconstant1forallnodesinthegraph,which
impliesthatifanodehasanyneighborwhohasadoptedtheproduct,thenodealsoadoptstheproduct.
The selection functions are simply the fraction of neighbors who have adopted the product previously.
Under this instance, the unique Nash equilibrium has every player allocating his unit of budget to the
center node of the star graph. By placing the budget at the center node, the expected payoff for each
playeris N+1,whileplacingitonanyothernodeatmostleadstoapayoffof1.However,thestrategy
N
that optimizes the social utility is to place one unit of budget at the center node of the star graph while
placingallothersattheisolatednodes.Thus,thePoA(andalsoPriceofStability)is 2N .AsN goesto
N+1
infinity,thelowerboundonthePoAtendsto2.Therefore,wehaveprovedthefollowingproposition:
Proposition1. Theupperboundof2onthePoA(andthusalsocoarsePoA)istightforthecompetitive
cascadegameevenforthesimplerswitching-selectionmodel.