1 0 2 10p1p2 20p1p2 12p1p2 21p1p2 Figure1MarkovchainmodelofthecustomersswitchingbehaviourFigure2FromstandardMarkovchainanalysisthesteadystateprobabilitiesforeachofthethreestatesgr ID: 402426
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Q:=0@�2 10(p1;p2)�(10(p1;p2)+12(p1;p2))12(p1;p2)20(p1;p2)21(p1;p2)�(20(p1;p2)+21(p1;p2))1A:Figure2:InnitesimalgeneratoroftheMarkovchainmeansthatthecustomeriswithprovider1,state2thatheiswithprovider2andstate0thathedoesnotuseanyservice.Theparameter isaconstantrateindependentofprices.TheresultinginnitesimalgeneratorQisgivenin 1 0 2 10(p1;p2) 20(p1;p2) 12(p1;p2) 21(p1;p2) Figure1:Markovchainmodelofthecustomer'sswitchingbehaviourFigure2.FromstandardMarkovchainanalysis,thesteady-stateprobabilitiesforeachofthethreestatesgroupedintherowvector=(i)i=0;:::;2existandaregivenbythesolutionofequationsQ=0;2Xi=0i=1:Ifc= (212(p1;p2)+221(p1;p2)+10(p1;p2)+20(p1;p2))+10(p1;p2)21(p1;p2)+20(p1;p2)12(p1;p2)+10(p1;p2)20(p1;p2);wehave0=10(p1;p2)21(p1;p2)+10(p1;p2)20(p1;p2) c+12(p1;p2)20(p1;p2) c1=(20(p1;p2)+212(p1;p2)) c2= (10(p1;p2)+221(p1;p2)) c:III.NON-COOPERATIVEGAMEFROMTHEPROVIDERS'SIDETheprevioussectiondescribesthebehaviourofacustomerasafunctionofpricessetbyproviders.Thequestionisnowtodenethebestpricingstrategyforeachproviderknowingthatbehaviour.Wethereforehaveaso-calledStackelberggame[9],withleaders(theproviders)choosingtheirpricesknowingtheconsequencestheywouldhaveonusers'behaviour,andthefollowers(theusers)whosereactionisadirectconsequenceofproviders'prices.Thismeansthatprovidersplayrst,butusingbackwardinduction,theyanticipatetheresultingstrategyofenduserswhoactuallymakethelastmove.Itisimportanttostressthatourmodel,consideringasinglecustomerinfrontoftwoproviders,issufcientifassumingthateachuserhasabehaviourindependentofothers'.ThecaseofNuserscanthenindeedbeeasilyderivedbymultiplyingtheexpectedrevenuebyN(thankstotheindependence).IntherststepoftheStackelberggame,eachprovidertriestomaximizeitsrevenue.Thereisatrade-offtobeanalyzedbetweenthefactthatincreasingthepricewillincreasetherevenuepercustomer,butontheotherhandpotentiallyreducethenumberofcustomers(i.e.,theprobabilityofhavingtheuserascustomerinourcase).TherevenuepercustomerRiforprovideri2f1;2gisthereforeexpressedformallyasthepricechargedmultipliedbytheprobabilitythatthiscustomersisindeedwithprovideri,i.e.,Ri=pii8i2f1;2g,ormoreexactlyusingtheexpressionsforthesteady-stateprobabilitiesoftheMarkovchain:R1(p1;p2)=p1 (20(p1;p2)+212(p1;p2)) cR2(p1;p2)=p2 (10(p1;p2)+221(p1;p2)) c:Fromthoseexpressions,itisclearthattherevenueofaproviderdependsonthepricestrategyoftheconcurrent.Indeed,steady-stateprobabilitiesarefunctionsofrateswhichthemselvesdependonbothprices.Asaconsequence,thiststheframeworkofnon-cooperativegametheory[9].Eachproviderstrivestonditsbeststrategy,i.e.,itspricemaximiz-ingitsrevenue,whichcanbemodiedbythestrategyofthecompetitor.ThesolutionconceptisthatofaNashequilibrium:aNashequilibriumisapriceprole(p1;p2)suchthatnoprovidercanunilaterallyincreaseitsrevenue,i.e.,R1(p1;p2)=maxp10R1(p1;p2)andR2(p1;p2)=maxp20R2(p1;p2):IngeneraltheexistenceofaNashequilibriumcannotbeen-suredwithoutassumptions,noritsuniquenesswhenexistenceisshown.Inthecasewhereratefunctionsaresimpleenoughintermsofprices,wemayndtheformoftheNashequilibriaanalytically(seenextsection).Otherwise,thecomputationscanbeperformednumericallyusingthefollowingalgorithm.Wedenethebestresponseofeachproviderasafunctionof providedthosevaluesarepositive.AtmostoneNashequilib-riumwithstrictlypositivepricesispossibleinthatcase.V.NUMERICALRESULTSA.ChurnratesandpricesintheliteratureInSectionIIthetransitionratesthatmarkthepassagefromaprovidertotheotherareshowntodependonthepricesofferedbytheprovidersandSectionIVillustratesthat,inverysimpliedcases,theNashequilibriumcanbeobtainedanalytically,thoughnoteasily.Inordertoadoptamodelascloseaspossibletoreality,webrieyreviewtherelatedliteratureonthemathematicalrelationshipbetweenchurnratesandpricesinthissub-section,thatwhichwillbeadoptedduringournumericalanalysis.Signicanteffortshavebeenspenttoidentifythemostrelevantfactorsindeterminingchurn(oftennamedchurndeterminants).Inordertomodeltherelationshipbetweenpricesandchurnratesinaquantitativefashionbothparametricandnon-parametricapproacheshavebeenproposedintheliterature.Amongthenon-parametricapproacheswecancite[10],whereneuralnetworksanddecisiontreesareemployed,and[11]whereanovelevolutionarylearningalgorithmisproposed.Sinceweneedaclosedformrelationshipherewearemoreinterestedinparametricapproaches.Themostwidespreadmodeladoptedintheliteraturetorepresentthatrelationshipisthelogitmodel,whichemploysalogisticprobabilitydistributionfunction[12][13][14].Theargumentofthelogisticfunctionisalinearfunctionofanumberofchurndeterminants.Themostgeneralexpressionoftheprobabilitythatauserchurnsinthenextperiod(e.g.,ayearasin[12])isthenPchurn=1 1+e�I;(5)whereIisthelogitfactor,inturngivenbyI=nXi=1iXi;(6)whereXi,i=1;:::;naretheexplanatoryvariables(churndeterminants)andi,i=1;:::;n,arethecoefcientsrepresentingtherelativeimportanceofthosedeterminants.Inthispaperwehavefocussedonthepricefactor,sothatwecangrouptheimpactoftheotherchurndeterminantsintheoverallterm ,arrivingatthesimplerexpressionPchurn=1 1+ e�p1=p2(7)fortheprobabilitythatinthespeciedperiodtheuserswitchesfromProvider1toProvider2.Wemayemploythatexpressionforatimeperiodofanyduration,sowecanadoptitintheMarkovchainmodeldescribedinSectionII.Wenotethat,accordingtoexpression(7),thereisanonzeroprobability,namelyPchurn=1 1+ ,thattheuserswitchesproviderduetotheensembleofotherdissatisfactionfactors,evenwhentheserviceofferedbythelosingproviderisfree.B.TransitionratesThetransitionratesthatweconsidernowarechosentoreecttheconclusionsoftheprevioussubsection.However,thedataandconclusionsdrawnfromtheliteraturedonotprovideuswithacompletedescriptionofallthetransitionratesweneed,inparticularthestatewhereusersdonotsubscribetotheserviceisnotencompassedinpreviousresults.Moreover,theliteratureconsidersdiscretizedtime,whereaswefocushereonacontinuous-timemodel.ThatlatterdifcultyisaddressedherebyassumingthattimeperiodsconsideredinSubsectionV-A,areshortwithrespecttothemeansojourntimeinagivenstate.Thisimpliesthatthediscretetimetransitionprobabilitiesareapproximatelythecontinuous-timetransitionratesmultipliedbytheperiodduration.Consequently,wewouldliketoconsidertransitionratesfromstatei2f1;2gtostatej2f1;2gnfigoftheform 1+ ie�pi=pj;where�0representstheinverseoftheperiodduration.Sincerepresentstheusersensitivitytoprices,weconsideritisthesameforthedifferentstatesofthemodel.However,suchanexpressionwouldimplythatalltransitionratesbeinaninterval[=(1+ i);]regardlessofthepricevalues.Thisisnotrealistic,sinceitwouldimplythataprovidercouldensureanarbitrarilylargerevenuebysettingaverylargeprice.Wethereforeneedthatthetransitionratestoaprovideritendto0and/orthattheratesfromprovideritendto1,whenpi!1.Tothatend,weslightlymodifythepreviousexpression,andtaketransitionratesoftheformij(pi;pj)= ie�pi=pj= iepi=pj;(8)Weintroduceasymmetryamongprovidersthroughthepa-rameter i:asexplainedbefore,thisparameterencompassesthereasonsotherthanprice(e.g.,QualityofService,reputa-tion,...),whyausershouldleavestatei.Weproposetoaddresstheformerdifculty(nohintsregardingthetransitionsto/fromourstate0)byconsideringthatbeinginstate0correspondstoperceivingacostp0,thatreectstheinconveniencefornotbenettingfromtheservice.Wethereforetreatstate0asthetwootherstates,butconsideringp0asaxedvalueinsteadofastrategicvariable.Asaresult,thetransitionrateweconsiderfromanystatei2f0;1;2gtostatej2f0;1;2gnfigisgivenby(8).Themodelparametersthatweconsiderarethen:theusersensitivitytoprice,thelikeliness itostayincurrentstatei,i=0;1;2,theinverseofperiodduration(thisparametershouldnotplayaroleinourmodel,sincebyatimeunitchangewecanassume=1),theuserperceivedcostp0fornotbenettingfromtheservice.C.NumericalanalysisofthegameInthissubsection,wesuggesttostudythegamewhileconsideringthepreviousexpressionsofthetransitionrates.Thedependenceofthoseratesonproviderpricesaretoocomplicatedtosolvetheproblemanalytically,thereforewe 0 1 2 3 4 5 0 1 2 3 4 5 Provider1pricep1 Provider2pricep2 BR1(p2) BR2(p1) Figure7:Best-replycurvesofbothproviderswhen 2=7(otherparametervaluestakenfromS).verybeginningofthosetechnologies.Ontheotherhand,someservices/technologiescanlosevaluebecausetheygetabandonedorcanbereplacedbyotherones.Serviceswithhigherimportance/impactcanthereforebemodeledbyalargerp0Forthosereasons,weinvestigatenowtheeffectoftheno-servicecostp0ontheoutcomesofthegame.Weassumethatthevariationsofp0areonalongertimescalethanthegameonpricesanduserbehavior,sothatwestillcomputetheNashequilibriumofthepricinggameasdescribedintheprevioussubsection.Figure8plotstheNashequilibriumprices(p1;p2)versusp0,whilethecorrespondinguserrepartitionatsteady-stateisplottedinFigure9.Weremarkthatthesteady-statedistribu-tion(again,atNashequilibriumdependingonp0)isalmostconstantwhenp0varies,andequilibriumpricesincreaseclosetolinearlywithp0.Therefore,ifprovidersareawareofanincreaseinp0,theyshouldraisetheirpricescorrespondinglytobenetfromtheincreasedvalueoftheservice.Interestingly,thispriceincreasecompensatestheservicevalueincreasefromthepointofviewoftheusers,sincetheproportionofuserschoosingtheserviceornotisunchanged.E.InuenceofthepricesensitivityWestudyheretheeffectoftheparameter,thatrepre-sentedusers'sensitivitiestopricedifferencesbetweenthedifferentstatesoftheMarkovchain.Whenincreases,weexpectproviderstodecreasetheirpricessoastoattractmoreefcientlyamaximumofcustomers.TheNashequilibriumprices(p1;p2)whenvariesareshowninFigure10.WegivethecorrespondinguserrepartitionandproviderrevenuesinFigure11and12,respectively.Itappearsthatwhenthepricesensitivityexceedsagiventhreshold(thatisaround0:85),providersengageinapricewarthatmakestheirprice 0 20 40 60 80 100 0 50 100 150 200 250 300 Costp0 Serviceprices p1 p2 Figure8:Nashequilibriumpriceswhentheno-servicecostp0varies. 0 20 40 60 80 100 0 0:1 0:2 0:3 0:4 0:5 0:6 Costp0 Userrepartition d1 d2 d0 Figure9:UserrepartitionatNashequilibriumwhentheno-servicecostp0varies.tendto0,andtheynallygetnorevenuefromprovidingtheservice.Thisbenetsuserswhoallendupwithoneofthetwoproviders.Belowthatthreshold,anincreasedpricesensitivityalreadyimpliesaproviderpricereduction,andarevenuedecreaseforthem.Remarkalsothatprovider2,beingmoreattractivethanitsopponent(since 2� 1inourparametervalues),takesbenetofthatadvantagebysettinghigherprices.F.InuenceoftheasymmetrybetweenprovidersInourmodel,providersonlydifferfortheirparameter i,thatreectstheuserlikelinesstostaywithhim:from(8),alarger imeanslowertransitionratestotheotherstatesoftheMarkovchain.Thatasymmetrybetweenprovidersmayforexamplecomefromusersbeingmorereluctanttoleave