between

betweencustomers

MirtaB.Gordon(1),Jean-PierreNadal(2),DenisPhan(3)andJeanVannimenus(2)

(1)LaboratoireLeibniz-IMAG,

46,Ave.F´elixViallet,38031GrenobleCedex1,France

(mirta.gordon@imag.fr,

http://www-leibniz.imag.fr/Apprentissage/Membres/Gordon/index.html)(2)LaboratoiredePhysiqueStatistique,EcoleNormaleSup´erieure,

24rueLhomond,75231Pariscedex05,France(nadal@lps.ens.fr,http://www.lps.ens.fr/˜nadal)(3)CREM,Universit´edeRennes1,France

(denis.phan@univ-rennes1.fr,http://perso.univ-rennes1.fr/denis.phan/)

Abstract

Inthispaperweconsideradiscretechoicemodelwhereheterogeneousagentsaresubjecttomutualinfluences.Weexploresomeconsequencesonthemarket’sbehaviour,inthesimplestcaseofauniformwillingnesstopaydistribution.Weexhibitafirstorderphasetransitionintheprofitoptimisationbythemonopolist:ifthesocialinfluenceisstrongenough,thereisaregimewhere,ifthemeanwillingnesstopayincreases,oriftheproductioncostsdecrease,theoptimalsolutionforthemonopolistjumpsfromasolutionwithahighpriceandasmallnumberofbuyers,toasolutionwithalowpriceandalargenumberofbuyers.Dependingonthepathofpricesadjustmentsbythemonopolist,simulationsshowhystereticeffectsonthefractionofbuyers.

1Introduction

Inthispaperweexploretheeffectsofsocialinteractionsonthepropertiesofasimplemarketmodel,inwhichtheindividualshavetomakeabinarychoice(whethertobuyornotasingleunitofagood)givenapricefixedbyasingleseller(amonopolist).Eachindividualhasareservationprice,i.e.themaximumpriceheisreadytopayforthegood,whichisthesumoftwoterms:anidiosyncraticwillingness-to-pay(IWP),andasocialcomponentproportionaltothefractionofhisneighboursthatbuy.Thislastterm,knownintheeconomicslitteratureasanexternality,istheresultofmutualinteractionsbetweencustomers.Asaconsequence,themarketmaypresentcomplexbehaviours[1,2].Asweshowinthefollowing,theseinteractionsintroducemultiplesolutionsinthedemandfunctionandareresponsiblefortheexistenceofatransitionintheoptimalstrategyofthemonopolist.

ThereisastraightforwardanalogybetweenthecustomersdescriptionandtheIsingmodel,whichhasbeenpointedoutinrecentpapersineconomics[3,4,5,6,7].DependingonthenatureoftheIWPs,theanalogycorrespondstotwodifferentfamiliesofmodelsinstatisticalmechanics:eithertheIWPsarerandomlychosenandremainfixed,ortheypresentindependenttemporalfluctuationsaroundafixed(homogeneous)value.TheformercasecorrespondstoaRandomFieldIsingModelmodel(quencheddisorder).Ifthedistributionofthetemporalfluctuationsinthelatterislogistic,itcorrespondstoanannealed,thatisthermal,disorder.

Onthesupplyside,weassumethatthemonopolistdoesnotknowtheIWPofeachcustomer,butisawareofitsdistributionamongthepopulation.Hedeterminesthepricethatoptimizeshisprofit.Sincethedemandmaybeamultiplevaluedfunctionoftheprice,themonopolist’ssituationisrisky.

1

InthispaperweconsiderthecaseofquencheddisorderinwhichthedistributionoftheIWPsisuniformandthesocialinfluenceisglobal.Thelatterassumptionisequivalenttothemeanfieldapproximation,andallowsustoobtainanalyticresults.Wedeterminethesupplyanddemandcurves,andthephasediagram,asafunctionoftheaverageIWPofthepopulation,andofthesocialinfluencestrength.

2Simplemodelsofdiscretechoicewithsocialinfluence

WeconsiderasetΩNofNagentswithaclassicallinearIWPfunction[8].Eachagenti∈ΩNeitherbuys(ωi=1)ornot(ωi=0)oneunitofthesinglegivengoodinthemarket.ArationalagentchoosesωiinordertomaximizehissurplusfunctionVi:

󰀂

Jikωk−P),(1)maxVi=maxωi(Hi+

ωi∈{0,1}

ωi∈{0,1}

k∈ϑi

wherePisthepriceofoneunitandHirepresentstheidiosyncraticpreferencecomponent.

Someotheragentsk,withinasubsetϑi⊂ΩN,suchthatk∈ϑi,hereaftercalledneighboursofi,influenceagenti’spreferencesthroughtheirownchoicesωk.Thissocialinfluenceisrepresentedherebyaweightedsumofthesechoices.LetusdenoteJikthecorrespondingweighti.e.themarginalsocialinfluenceonagenti,ofthedecisionofagentk∈ϑi.Whenthissocialinfluenceisassumedtobepositive(Jik>0),itispossible,followingDurlauf[4],toidentifythisexternaleffectasastrategiccomplementarityinagents’choices[9].

Forsimplicityweconsiderhereonlythecaseofhomogeneousinfluences,thatis,identicalpositiveweightsJik=Jϑandidenticalneighbourhoodstructuresϑofsizen,foralltheagents.Thatis,

Jik=Jϑ≡J/n>0∀i∈ΩN,k∈ϑi,(2)

2.1Psychologicalversuseconomicpointsofview

DependingonthenatureoftheidiosyncratictermHi,thediscretechoicemodel(1)may

representtwoquitedifferentsituations.FollowingthetypologyproposedbyAndersonetal.[10],wedistinguishapsychologicalandaneconomicapproachtoindividualchoices.Withinthepsychologicalperspective(Thurstone[11]),theutilityhasastochasticaspectbecause“therearesomequalitativefluctuationsfromoneoccasiontothenext...foragivenstimulus”.Inthiscase,theIWPspresentindependenttemporalfluctuationsaroundafixed(homogeneous)value(thispointofviewwillbereferredtohereafterastheTP-model).Ifthedistributionofthesetemporalfluctuationsislogistic,itcorrespondstoanannealeddisorder,thatis,tofinitetemperature[7,12]

Onthecontrary,withintheeconomicperspectiveofMcFadden[13](seeAndersonetal.[10]),eachagenthasawillingnesstopaythatisinvariableintime,atleastduringtheperiodunderconsideration,butmaydifferfromoneagenttotheother(wecallhereafterthisperspectivetheMcFmodel).ThissituationisknowninthePhysicslitteratureasamodelwithquencheddisorder.EvenifasellerknowsthestatisticaldistributionoftheIWPoverthepopulation,hecannotobserveeachspecificindividualIWP.Inthelangagueofinteractivedecisiontheory,thissellerisinaemriskysituation.

Thus,thesetwoperspectives,whichdifferinthenatureoftheindividualwillingnesstopay,correspondtoverydifferenttheoreticalmodels.

IntheTPmodel,theidiosyncraticpreferencehastwosub-components:aconstantde-terministictermH(thesameforalltheagents),andatime-andagent-dependentadditiveterm󰀦i(t)(Hi=H+󰀦i).The󰀦i(t)arei.i.d.randomvariablesofzeromean;inthesimula-tionstheyarerefreshedateachtimestep(asynchronousupdating).Agentidecidestobuy

2

accordingtotheconditionalprobability

P(ωi=1|zi(P,H))=P(󰀦i>zi(P,H))=1−F(zi(P,H)),

with

zi(P,H)=P−H−Jϑ

󰀂

ωk,

(3)(4)

k∈ϑi

whereF(zi)=P(󰀦i≤zi)isthecumulativedistributionoftherandomvariables󰀦i.InthestandardTPmodel,theagentsmakerepeatedchoices,andthetimevaryingcomponents󰀦i(t)aredrawnateachtimetfromalogisticdistributionwithzeromean,andvarianceσ2=π2/(3β2),FL(z)=(1+exp(−βz))−1.

IntheMcFmodel,theprivateidiosyncratictermsHiarerandomlydistributedovertheagents,butremainfixedduringtheperiodunderconsideration.Therearenotemporalvariations:the󰀦iarestrictlyzero.InanalogywiththeTPmodel,itisusefultointroducethefollowingnotation:Hi=H+θi.Inthelimitofaverylargenumberofcustomers,thisimplies:

󰀂1

Hi=H.(5)lim

N→∞NiThecustomer’sbehaviour,giventhepriceandthechoicesinhisneighbourhoodϑi,isde-terministic.Agentibuysif:󰀂

ωk.(6)θi>P−H−Jϑ

k∈ϑi

Iftheθiarelogisticallydistributedwithzeromeanandvarianceσ2=π2/(3β2)overthe

population,thenthecorrespondencebetweentheTPandtheMcFmodelsisbetterthelargerthenumberofagents,butitisactuallystrictonlyinthelimitofaninfinitepopulation.Noticehoweverthat,althoughformulatedoriginallyforalogisticIWPdistribution,theMcFmodelmaybegeneralizedtoanydistribution.InthepresentpaperwerestrictourinvestigationtotheMcFmodelandweillustrateitsbehaviourintheparticularcasewheretheIWPsareuniformlydistributedoverthepopulation.

2.2Staticversusdynamicpointsofview

Hereafter,weconcentrateontheMcFmodelinthe“global”externalitycase,consideringhomogeneousinteractionsandfullconnectivity,whichisequivalenttothemeanfieldmodelatzerotemperatureinphysics.

Withinthisgeneralframework,weareinterestedintwodifferentaspects.Firstweconsiderastaticpointofviewcomputingthesetofpossibleeconomicequilibria,solvingfortheequalitybetweendemandandsupply.Thiswillallowustoanalyseinsection4theoptimalstrategyofthemonopolist,asafunctionofthemodelparameters.

Weconsidernextthemarket’sdynamicsassumingtheusualparallelMonteCarloupdat-ingrules,whichinthemodelsofeconomicagentscorrespondtomyopiclearningwithfullinformation(i.e.basedonthelastiteration,withoutmemory):basedontheobservationofthebehaviouroftheotheragentsattimet−1,eachagentdecidesattimettobuyornottobuy.Weshowthat,ingeneral,themarketconvergestowardsthestaticequilibriaoftheprecedingsection,exceptforapreciserangeoftheparametervalueswhereinterestingstaticaswellasdynamicfeaturesareobserved.

InPhysics,thesetwokindsofanalysiscorrespondtothestudyof,respectively,thethermalequilibriumpropertieswithinthestatisticalensembleframeworkononeside,andtheoutofequilibriumdynamics(which,inmostcases,approachesthestaticequilibriumthrougharelaxationprocess)ontheotherside.

3

a21h-p0-1η=10<η<1 -a-2-3-4-501η=02345jFigure1:

B j678Customer’sphasediagramintheplane(j,h−p):Theregionwithdiagonalstripescorrespondsto

parametersforwhichallthecustomersarebuyers(η=1).Thedarkgreyregioncorrespondstoparametervalueshinderingadoption(η=0).ForjjBthelattersolutiondoesnotexist,andthereisarangeofvaluesofh−pforwhichthetwosolutions(η=1andη=0)coexist.3AggregatedemandAsdiscussedintheprecedingsection,weconsiderthefullconnectivitycaseinthelimitofaverylargenumberofagents.Thepenetrationrateηdefinedasthefractionofcustomers󰀁Nthatchoosetobuyatagivenprice,η≡limN→∞k=1ωk/N,can󰀁beapproximatedbythesocialinfluencetermoftheagents’surplusfunction(eq.(1)):η≈k∈ϑωk/(N−1).InthelargeNlimit,equation(6)maythusbereplacedbyθi>P−H−Jη,

(7)

whereθiisarandomvariable.

Forthefollowingdiscussionitisconvenienttodividebothsidesoftheprecedingequationbyσ,thevarianceofthedistribution,andconsidernormalizedvariables

xi≡θi/σ,p≡P/σ,h≡H/σ,j≡J/σ.

Then,thedistributionofxihaszeromeanandunitaryvariance.Intheillustrationsweconsidertheuniformdistributiondefinedby

󰀄

1/2aif−a≤x≤a

f(x)=(9)

0otherwise

√

witha=

(8)

wherez,definedbyequation(10),dependsonp,h,jandη.

Sinceforagivenp,equation(11)definesthepenetrationrateηasafixed-point,inversionofthisequationgivesaninversedemandfunction:

pd(η)=h+jη+G(η)

whereG(η)istheinverseofthecomplementarydistributionfunction;itsatisfies:

󰀃∞

f(x)dx=η.

G(η)

(12)

(13)

Thus,G(η)isanonincreasingfunctionofη.NoticethatitsderivativesatisfiesG󰀈(η)=−1/f(G(η)).Givenvaluesofjandh,formostvaluesofp,(11)hasauniquesolutionη(p).Thesearethevaluesofηthatsatisfyp=pd(η)wherepd(η)isgivenbyequation(12).

However,forjlargerthanacriticalvaluejB,thatdependsonthespecificdistributionf(x)thereisarangeofpricessuchthat,foranyvalueofpwithinthisrange,(11)hasmultiplesolutions.Moreprecisely,iff(x)ismonomodal(likeinthepresentcase),therearetwostablesolutionsandanunstableone,andjB=1/fMax,wherefMaxisthemaximumvalueoff(x).Theunstablesolutionsarethosewithpositivederivativeofpd(η)(theycorrespondtostateswherethefractionofbuyerswouldincreaseuponincreasingtheprice).

Theupperandlowervalues,p1andp2,oftherangepresentingmultipledemandsolutions,

p1(j,h)(14)

areobtainedfromtheconditionthateq.(12)hasasingledegeneratesolutionη(p).Fordifferentiablepdfs,itis:

η=1−F(z),and

d(1−F(z))

321 32a-j

j=3ah=-3a/2210 p(η)s p(η)-ca/2

d0j=a-1h=a/2-20.00 p(η)s p(η)-c-a

0.250.500.751.00d-1-20.00-a0.250.500.751.00Figure2:

Inversesupplyanddemandcurvesfordifferentvaluesofhandj.

4Supplyside

Onthesupplyside,weconsideramonopolistfacingheterogeneouscustomersinariskysituationwherethemonopolisthasperfectknowledgeofthefunctionalformoftheagents’surplusfunctionsandtheirmaximisationbehaviour(1).Healsoknowsthestatisticaldis-tributionoftheidiosyncraticpartofthereservationprices,h+xi.However,themonopolistcannotobserveanyindividualreservationprice.Heonlyobservestheaggregateresultoftheindividualchoices(tobuyornottobuy),thatis,thefractionofcustomersη,whoseexpectedvalueforagivenpriceisgivenbyequation(11).Noticethat,astheinteractionsareglobal,inthelimitofalargenumberofcustomers(rigorouslyforN→∞)thisisthesamequantityastheonethatentersinthetermofsocialinfluenceamongcustomers,equation(7).

4.1Profitmaximisation

Letc≡C/σbethemonopolist’scostinunitsofσ(thevarianceofthedistributionoftheIWP)foreachunitsold,sothatp−cishis(normalized)profitperunit.

Sinceeachcustomerbuysasingleunitofthegood,themonopolist’stotalexpectedprofitis(p−c)Nη.Heisleftwiththefollowingmaximisationproblem:

pM=argmaxΠ(p),

p

(16)

whereNΠ(p)istheexpectedprofit,with:

Π(p)≡(p−c)η(p),

(17)

andη(p)isthesolutiontotheimplicitequation(11).Ifthereisnodiscontinuityinthedemandcurveη(p)(henceforj≤jB),pMsatisfiesdΠ(p)/dp=0,whichgivesdη/dp=−η/patp=pM.Usingtheimplicitequation(11)tocalculatethederivative,weobtainatp=pM:

f(z)

p,

(18)

wherezhastobetakenatp=pM.

Becausethemonopolistobservesthedemandlevelη,wecanuseequation(11)toreplace1−F(z)byη.Aftersomemanipulations,equation(18)givesthemonopolist’spriceasafunctionofthedemand,whichmaybeinterpretedasan(effective)inversesupplyfunctionps(η):

ps(η)=c−η[G󰀈(η)+j].(19)

6

1,0ηj=3a j=1.5a 8p-cp1(j=0)M8Π/N6p1(j=a)4p1(j=1.5a)p1(j=2a)p1(j=3a)MMMM6j=1.5aj=3aj=0j=a 0,5 j=a j=042 pηM20,0-6-4-202468-aa3a0-6-4-2h-c-a024a3a680-6-4-202468h-c-aa3ah-cFigure3:Fractionofbuyersη,optimalpricepMandmonopolist’sprofitΠM,asafunctionoftheaveragewillingnesstopayh−cfordifferentvaluesofthesocialinfluenceweightj.Left:Forj=3a>jBthefractionofbuyershastwosolutions(η=0andη=1forarangeofvaluesofh−c.Center:ThelinepMηshowstheoptimalpricesforj(20)

Themonopolist’ssupplypriceisthesolutionof(20)whichmaximizeshisprofit.Iff(x)isdifferentiable,themaximumsatisfies

d2Π

63a4j-23ac=h-h-ca20η=10<η<1η=00123-a-2-4h-c=a-jj45B j678Phasediagramintheplane(j,h−c):Theregionwithdiagonalstripescorrespondstooptimalpricesforwhichallthecustomersbuytheproduct.ForjjBandh−cadoption(η=1).Surprisingly,fora−jjBthefractionofbuyersismultivalued,andfora−ja−j.Otherwise,η=0.However,eveniftheinequalityissatisfied,thequestionofwhetherthecustomerswillactuallybuyornotisacoordinationproblem,whoseissuedependsonthedynamicsoftheadoptionprocess.

Thefractionofbuyers,theoptimalpriceandthecorrespondingmonopolist’sprofitarerepresentedasafunctionofh−configures3,fordifferentvaluesofj.

Wedeterminedthephasediagraminthecaseasmoothdistribution,namely,thelogis-tic[12].Itpresentsaricherstructure,butwithsimilarfeaturesastheuniformdistributionanalyzedhere.

5Dynamicalfeatures

Weshortlydiscussheresomedynamicalaspects,consideringamarketparalleldynamicswithmyopiccustomers:alltheagentsmakethirdecisionsattimetbasedontheobservationofthebehaviouroftheotheragentsattimet−1.Theadoptionbyasingleagentinthepopulation(a“directadopter”)maythenleadtoasignificantchangeinthewholepopulationthroughachainreactionof“indirectadopters”[8].WithintheMcFmodel,thedynamicsforthefractionofadoptersinthelargeNlimitisthengivenby

η(t)=1−F(p−h−jη(t−1))

(25)

andη(t)convergestoasolutionofthefixedpointequation(11).Aswehaveseen,given

u

,handp,twostableandoneunstablefixedpointsappearin(11)forvaluesofj≡J/σj>jB

largeenough(largeJorsmallσ).Thestablesolutionscorrespondtothetwopossiblelevelsofηatagivenprice.Varyingthepricesmoothly,atransitionmaybeobservedbetweenthesephases.Thejumpinthenumberofbuyersoccursatdifferentpricevaluesaccording

8

1η0.80.60.40.201.55h = 0 j = 3.3 4N = 10 1.61.651.71.75increasing pdecreasing p1η0.80.60.40.2p1.8increasing pdecreasing pη1increasing pdecreasing p0.80.6h = 0 j = 3.41 4N = 10 p1.651.71.751.80.40.20h = 0 j = 3.5 N = 104 p1.651.71.751.80u

Behaviourofthedemandasafunctionoftheprice,forsocialweightvaluesatjBandvaluesslightly

smallerandlarger,showingthathystereticeffectsinthecaseoffinitesizesystemsdonotdisappearpreciselyat

u

j=jB,incontrastwiththetheoreticalpredictionsobtainedinthemeanfieldlimitN→∞.

Figure5:

towhetherthepriceincreasesordecreases,leadingtohysteresisloops.Insomecases,thenumberofcustomersevolvesthroughaseriesofclusteredflips(betweenωi=1andωi=0),thatwecallavalanches(noticethattheseavalanchesaredynamicalfeaturesthatariseduringtheupdatingprocess,andarenottheavalanchesreferredtoin[15],leadingtometastablestates.ForsmallvaluesofjthereisasinglefixedpointforallvaluesofP,andnohysteresisatall[8].

ThecurvesinFigure5,representthefractionofcustomersηasafunctionoftheprice,obtainedthroughasimulationofthewholedemandsystem.Theresultscorrespondtoupstreamanddownstreamtrajectories,obtainedupondecreasing(increasing)thepricesstepwise.Weobserveahysteresisphenomenonwithdiscontinuoustransitionsbetweenthelowandhighadoptionregimes.Typically,alongthedownstreamtrajectory(withincreasingprices)theexternalityeffectinducesastrongresistanceofthedemandsystemhinderingthenumberofcustomersfromdecreasing.Inallthesecases,avalanchesatwhichlargefractionsofcustomerschangetheirstateoccurattheso-called“firstorderphasetransition”.

6Conclusion

Inthispaper,wehavecomparedtwoextremespecialcasesofdiscretechoicemodels,theMc-Fadden(McF)andtheThurstone(TP)models,inwhichtheindividualsbearalocalpositivesocialinfluenceontheirwillingnesstopay,andhaverandomheterogeneousidiosyncraticpreferences.IntheMcFmodelthelatterremainfixed,andgiveraisetoacomplexmar-ketorganisationduetothecorrespondingquencheddisorder:theMcFmodelbelongstotheclassofRandomFieldIsingmodels(RFIM).IntheTPmodel,alltheagentsshareahomogeneouspartofwillingnesstopaybuthaveanadditive,timevarying,idiosyncraticcharacteristic.Whenthelatterisdrawnfromalogisticdistribution,thismodelcorrespondstoafinitetemperatureIsingmodel,thatis,withanannealed(thermal)disorder.Thus,theMcFandTPmodels,whichareconsideredasequivalentintheeconomicsliterature,havequitedifferentpropertiesfromthephysicist’spointofview.Inthispaperwehavediscussedsomeofthem,andtheirconsequencesonthemarket’sbehaviour.

Consideringthatthemonopolistoptimiseshisownprofit,wehaveexhibitedanewfirstorderphasetransition:whenthesocialinfluenceisstrongenough,thereisaregimewhere,uponincreasingthemeanwillingnesstopay,ordecreasingtheproductioncosts,theoptimalmonopolist’ssolutionjumpsfromonewithahighpriceandasmallnumberofbuyers,toonewithalowpriceandalargenumberofbuyers.

Weillustratedthegeneralconclusiononasimpleexamplewheretheidiosyncraticpartofthewillingnesstopayisuniformlydistributedamongthecustomers.

Wehaveonlyconsideredfullyconnectedsystems:thetheoreticalanalysisofsystemswithfiniteconnectivityismoreinvolved,andrequiresnumericalsimulations.Thesimplestconfigurationisonewhereeachcustomerhasonlytwoneighbours,oneoneachside.The

9

correspondingnetwork,whichhasthetopologyofaring,hasbeenanalysednumericallybyPhanetal.[8])whoshowthattheoptimalmonopolist’spriceincreasesbothwiththedegreeoftheconnectivitygraphandtherangeoftheinteractions(inparticular,inthecaseofasmallworld).Buyers’clustersofdifferentsizesmayform,sothatitisnolongerpossibletodescribetheexternalitywithasingleparameter,likeinthemeanfieldcase.Amoresystematicstudyofthesepropertieswillbethesubjectoffuturework.

Acknowledgements

TheauthorsacknowledgesupportfromtheACIComplexSystemsinSocialandHumanSciences(CNRSandFrenchMinistryofEducation),projectELICCIR.

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