The Schelling Model of Ethnic Residential Dynamics

©CopyrightJASSS

ErezHatnaandItzhakBenenson(2012)

TheSchellingModelofEthnicResidentialDynamics:BeyondtheIntegrated-SegregatedDichotomyofPatterns

JournalofArtificialSocietiesandSocialSimulation 15(1)6<http://jasss.soc.surrey.ac.uk/15/1/6.html>

Received:25-Dec-2010Accepted:31-Aug-2011Published:31-Jan-2012

Abstract

TheSchellingmodelofsegregationisanagent-basedmodelthatillustrateshowindividualtendenciesregardingneighborscanleadtosegregation.Themodelisespeciallyusefulforthestudyofresidentialsegregationofethnicgroupswhereagentsrepresenthouseholderswhorelocateinthecity.Inthemodel,eachagentbelongstooneoftwogroupsandaimstoresidewithinaneighborhoodwherethefractionof'friends'issufficientlyhigh:aboveapredefinedtolerancethresholdvalueF.ItisknownthatdependingonF,forgroupsofequalsize,Schelling'sresidentialpatternconvergestoeithercompleteintegration(arandom-likepattern)orsegregation.Thestudyofhigh-resolutionethnicresidentialpatternsofIsraelicitiesrevealsthatrealityismorecomplicatedthanthissimpleintegration-segregationdichotomy:someneighborhoodsareethnicallyhomogeneouswhileothersarepopulatedbybothgroupsinvaryingratios.Inthisstudy,weexplorewhethertheSchellingmodelcanreproducesuchpatterns.Weinvestigatethemodel'sdynamicsintermsofdependenceongroup-specifictolerancethresholdsandontheratioofthesizeofthetwogroups.Werevealnewtypeofmodelpatterninwhichaportionofonegroupsegregateswhileanotherportionremainsintegratedwiththesecondgroup.Wecomparethecharacteristicsofthesenewpatternstothepatternofrealcitiesanddiscussthedifferences.

Keywords:SchellingModel,EthnicSegregation,Minority-MajorityRelations

Introduction

1.1 TheSchellingmodelofsegregation(Schelling1971,1978)isoneoftheearliestagent-basedmodelsofsocialscience.ThemodelwasintroducedbyThomasSchellingtoillustratehowindividualincentivesandindividualperceptionsofdifferencecanleadcollectivelytosegregation(Schelling1978p.148).Whilethemodelisindicativeofavarietyofphenomenawhereindividualstendtorelocateaccordingtotheshareofsimilarneighbors,itwasfoundespeciallyusefulforthestudyofresidentialsegregation.

Theoriginalmodel

1.2 IntheSchellingmodel,agentsoccupycellsofrectangularspace.Acellcanbeoccupiedbyasingleagentonly.Agentsbelongtooneoftwogroupsandareabletorelocateaccordingtothefractionoffriends(i.e.,agentsoftheirowngroup)withinaneighborhoodaroundtheirlocation.Themodel'sbasicassumptionisasfollows:anagent,locatedinthecenterofaneighborhoodwherethefractionoffriendsfislessthanapredefinedtolerancethresholdF(i.e.,f<F),willtrytorelocatetoaneighborhoodforwhichthefractionoffriendsisatleastf(i.e.,f≥F)(Schelling1978p.148).NotethatahighthresholdvalueofFcorrespondstoalowagent'stolerancetothepresenceofstrangerswithintheneighborhood.

1.3 SchellingstudiedthedynamicsgeneratedbythissimplerelocationruleforthecaseoftwogroupsofequalsizeandacommontolerancethresholdFforbothgroups.ThestudyrevealedtheexistenceofacriticaltolerancethresholdFsegr≈1/3suchas,forF<Fsgrinitiallyrandompatternremains,intime,random-like,whileforF≥Fsgrinitiallyrandompatternconvergestoasegregatepattern(Figure1).

Figure1.(a)initialconditionofoneofSchelling'sexperiments;(b)stablesegregatedpatternobtainedinseveraliterations(Schelling1974)

1.4 TworesultsofSchelling'sstudyareregardedascentral:(a)thevalueofFsgrisessentiallylowerthantheintuitivelyexpectedvalueof1/2;(b)thechangeofthelimitpatternfrompseudo-randomtosegregatedasFpassedthevalueofFsgrisabrupt,thusthemodeldoesnotproduceintermediatepatterns.

Schellingmodelversusreal-worldresidentialpatterns

1.5 Real-worldresidentialdynamicsismainlygovernedbyeconomicfactorsand,thus,Schellingmodelpatternsareusuallyconsideredashavinggreatertheoreticalthanappliedimportance.However,insomecasesdirectcorrespondencebetweenthemodelpatternsandrealresidentialdistributionscanbeestablished.Ourstudyismotivatedbyoneofthesecases:theJewish-ArabethnicresidentialdistributioninIsraelicities.BasedontheIsraelipopulationcensusof1995(BenensonandOmer2003),wewereabletoconstructpatternsforethnicallymixedIsraelicitiesattheresolutionofindividualhouses,whichisequivalenttotheresolutionoftheSchellingmodel.

1.6 EmpiricalevidencesupportsSchelling-likeviewsoftheresidentialinteractionsbetweentheJewishandArabsfamiliesinIsrael;bothtendtoresideinneighborhoodsthathavesufficientnumbersofresidentsoftheirethnicgroup(Omer1996).Thus,wecanconsiderresidentialpatternsinthemixedIsraelicitiesasa"stylized"outcomeoftheresidentialchoiceofSchelling'sresidentialagents.

1.7 WeconsidertheresidentialpatternofArabMuslims,ArabChristiansandJewsinthecitiesofYaffoandRamle,bothofwhicharelocatedincentralIsrael.YaffoisactuallythesouthernpartofthecityofTelAviv-YaffowhileRamleislocated15kmsouth-eastofYaffo.TheethnicpatternofYaffoin1995ispresentedinFigure2.ThecitycontainssegregatedArabandJewishareas;inAjami,aneighborhoodinthecenter(AreaA),ChristiansandMuslimslivetogether,andinYaffo-Daled,aneighborhoodinthesouth(AreaB),thepopulationisJewish.SomeoftheboundariesbetweenJewishandArabareasinYaffoaresharp(AreaD),whilesomeareextended(AreaC).

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1.8 TheethnicresidentialpatternofthecityofRamleispresentedinFigure3andismorecomplexthanYaffo.LargeareasofthecityarepopulatedalmostsolelybyJews(AreaA),whilesomearepopulatedalmostexclusivelybyMuslims(AreaB).Thecitycontainsintegratedareas(AreaC),which,unlikeYaffo,areremotefromthesegregatedArabareas.ThecityalsocontainsareaspopulatedbyMuslimsandChristians(AreaD)andanareawhereChristiansandJewscoexist(AreaE).

1.9 YaffoandRamlepatternsarethusessentiallymorevariablethantheintegratedandsegregatedpatternscharacteristicoftheSchellingmodel:thepatterncontainshomogeneouspatchesofArabandJewishpopulationsandseveralintegratedareaswithdifferentfractionsofArabandJewishpopulationsineach.Yaffo'sintegratedpatchesareattheboundarybetweenthesegregatepatches;inRamlesegregatedandaggregatedareasarenotnecessarilyadjacentandcanbeseparatedbythenon-populatedareas.

Figure2.EthnicresidentialdistributionofYaffoin1995

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Figure3.EthnicresidentialdistributionofRamlein1995

1.10 SeveralexplanationsforthisqualitativedifferencebetweentheobservedpatternsandSchelling'stheoreticalpatternscanbeproposed.First,ignoringeconomics,YaffoandRamlepatternsarestillinadevelopmentalstage,thussomeofthepatternsmaybetemporary.Second,low-incomeJewsandArabsmaynotbeabletoavoidlivingtogether,whilethericherJewsandArabsareabletosegregate.Third,contrarytothepreviousexplanation,poorJewsandArabsmaybeintolerantofeachotherandthussegregate,whilewealthyhouseholdersmaybetolerantandthusstaytogether.Eachoftheseexplanationscouldbetestedifthedataonmigrationactivityandindividualwealthwereavailable.However,takingYaffoandRamlepatternsasasourceofinspiration,cantheobservedpatternsstillbeexplainedwithinthenon-economicSchellingframework?

1.11 TosimplifyletusconsidertheMuslimandChristianArabsasasinglegroup.Conceptually,intheJewish-ArabresidentialpatternsofYaffoandRamle,wecanspecifythreequalitativelydifferentconfigurations(Figure4):

Figure4.ThreequalitativelydifferentpatternsinYaffoandRamleandthecorrespondingSchelling-likepatterns:(a)twosegregatedpatches;(b)segregatedpatchadjacenttointegratedpatch;(c)integratedpatchesadjacenttotwosegregatedpatches

1.12 Inthefirstpattern(Figure4a),twosegregatedgroupsareseparatedbyanarrowboundaryoftransition.Inthesecondpattern(Figure4b),themajorityofmembersofoneofthegroupsaresegregatedwhilesomeofthemembersareintegratedwiththemembersoftheothergroup.Thethirdandmostcomplexpattern(Figure4c)presentstwosegregatedgroups,eachadjacenttotheintegratedarea.

1.13 Inthispaper,wedemonstratethatthevarietyofSchellingmodelpatternsisgreaterthantherandom-segregateddichotomyandthatthepatternsinFigures4aand4b,canbegeneratedbythemodelifthenumbersoftwogroupsorgrouptolerancethresholdsaredifferent.However,theSchellingmodeldoesnotgeneratepersistentpatternsthatcontainsegregatedpatchesofbothgroupstogetherwithintegratedpatches,asinFigure4c.

1.14 Thestructureofthepaperisasfollows:Section2presentsthestate-of-the-artofSchellingmodelstudiesanddescribesthekindofdynamicsthatgeneratesthepatternsthatareneithersegregatednorintegrated(wecallthemmixed).Section3formallydescribesthemodelandthemethodologyofthestudy.TheresultsofthestudyarepresentedinSection4andarefollowedbythediscussioninSection5.

Schellingmodelstudies

Irregularpartitionofspace

2.1 StudiesoftheSchellingmodelfocusonconditionsenablingsegregationandonthecharacteristicsofthepatternsproducedbythemodel.Ascanbeexpected,themodelresultsarequalitativelyrobusttothestructureofresidentialspace.Irregularpartitionoftheplaneintopolygonalunitsandneighborhooddefinitionbasedonpolygonadjacencydonotchangethemainresultsofthepattern'sdichotomyandtheabrupttransitionfromintegratedtosegregatedlimitpattern(FlacheandHegselmann2001).LaurieandJaggi(2002,2003)supplementedthisfindingbysystematicallyinvestigatingSchellingmodeldynamicsasdependingonthesizeoftheneighborhoodsandrevealed,asshouldbeexpected,thatthesizeofhomogeneouspatchesincreaseswiththegrowthoftheneighborhoodradiuswhilethenumberofhomogeneouspatchesinpatternsdecreases.

Asymmetricrelations,morethantwogroups

2.2 Portugalietal.(1994,1995)showedthatSchellingmodelpatternsconvergetowardsegregationevenifonlyagentsofonegroupreacttothefractionoffriendswhileagentsoftheothergroupareindifferent.However,inthatcasethetolerancethresholdthatseparatessystemsconvergingtowardintegratedasopposedtosegregatedpatternsisessentiallyhigherthanthevalueof1/3,characteristicforthesymmetriccase.Benenson(1998)investigatedtheSchellingmodelforthecaseofseveralgroupsofagentscharacterizedbyseveralbinarycharacteristics.Inadditiontointegratedandsegregatedstablepatterns,herevealedunstablebutpersistentregimesinwhichhomogeneouspatternsoftheagentsofoneormoregroupsarerepeatedlyself-organizingandvanishing.

Accountingforeconomicdifferencesbetweenagents

2.3 Severalstudies(BenardandWiller2007;Benenson1999;Fossett2006b;Portugali2000;BenensonandHatna2009)consideragentscharacterizedbycontinuous"economicstatus,"whichcanbecomparedwiththecells'price.Theagentssearchforneighborhoodsthatareboth"friendly"and"sufficientlywealthy."Asmightbeexpected,systemdynamicsbecomemorevariableinthiscase,dependingonhowtheagents'residentialpreferencesdependontheneighbors'groupidentityandstatus.

Agentsthatexchangeplaces

2.4 Aqualitativelydifferentwayofmodelingagents'relocationistoassumethatunsatisfiedagentsexchangetheirplacesratherthansearchforavacancyoverallunoccupiedcells.Thisviewattractedtheattentionofseveralresearchers(PollicottandWeiss2001;Zhang2004),whoalsoassumethatthepreferencesofagentsarenon-monotonous,andtheymightpreferneighborhoodswithalowfractionofforeignerstothoseoccupiedexclusivelybyfriends.Thedynamicsofthemodelpatternsinthiscasearealsomorecomplexthantheintegrated-segregateddichotomy.However,themathematicalproperties

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ofthesemodelsdifferqualitativelyfromthestandardsettingsandwethusconsiderthisformulationasdifferentfromSchelling'soriginalmodel.

The"BoundedNeighborhood"model

2.5 AnothervariantoftheSchellingmodelisthe"BoundedNeighborhood"modelthatwasalsopresentedbySchelling(1978,p.155)butislesspopularthantheversiondiscussedabove.Inthe"BoundedNeighborhood"version,insteadofaregulargrid,thegridisdividedintoblocks,whichareessentiallylargerthantheneighborhoodsofthepopularversionofthemodel.Nomatterwheretheagentislocatedwithintheblock,itreactstothefractionoffriendsalloverit,andleavestheblockifthefractionoffriendsisbelowthetolerancethresholdF.Differentagents,however,canreacttodifferentthresholdfractionsoffriendswithintheblock,andthemainparameterofthe"BoundedNeighborhood"modelisthedistributionoftheagents'F-values.

2.6 TheanalyticinvestigationofthisversionofthemodelwasstartedbySchelling(1978)andcontinuedbyClark(1991,1993,2002,2006).Theyhavedemonstratedthatbothgroupscanpersistinablockincaseagentsofeachtypeessentiallyvaryintheirtolerancetostrangers.However,theequilibriumisnotgloballystableand,dependingoninitialconditions,block'spopulationcanconvergetoasegregatedoraggregatedstateforthesamedistributionsoftolerance.

2.7 Aspatialversionofthe"BoundedNeighborhood"model,inwhichtheagents'populationconsistsofmorethantwoethnicgroupsandresidentialagentsreacttothepopulationstructureandmigrateamongadjacentblocksincaseofunsatisfieddemands,hasbeeninvestigatedindepthinaseriesofrecentsimulationsbyFossett(FossettandWaren2005;Fossett2006a;Fossett2006b),whoalsodirectlyrelatesthemodel'sresultstoresidentialpatternsinthereal-worldcities.FossettaimedatreflectingthesituationintheUnitedStatesandconsideredthreeresidentialgroups—White,BlackandHispanic—whoalsodifferintheireconomicstatus.Theaggregateunitsusedare7×7cellblocksandthecityisrepresentedbythe12×12gridofblocks.Fossett(2006a)investigatedtherelationshipbetweenethnicandstatussegregationandconcludedthatthisinteractionmightlimitourabilitytointegratetheethnicgroups.Specifically,hepointedoutthatreductioninhousingdiscriminationbyracemaynotnecessarilyleadtolargedeclinesinethnicsegregation.

ApplicationoftheSchellingmodelforrealcities

2.8 SeveralattemptstoapplytheSchellingmodeltothereal-worldsituation(Benensonetal.2002;KoehlerandSkvoretz2002;BruchandMare2006)providelikelihoodapproximationsofthesegregatedormorecomplexresidentialdistributionsobservedincities.Inwhatfollowswelimitourselvestoabstractmodelsanddonotdelveintogreaterdetailsoftheseimplementations.

"Liquid"and"Solid"dynamics

2.9 AgeneralinsightintotherelationbetweentherulesandthepatternsengenderedbytheSchellingmodelwassuggestedbyVinkovicandKirman( 2006),whoconsideracontinuousanalogoftheSchellingmodel.Namely,theystartwithrepresentingeightcellsofthe3×3neighborhoodaseightequalsectors,andthenweakenthediscreterepresentationofspacebyconsideringeachsector'sangleasacontinuousvariable.Thecontinuousrepresentationmakesitpossibletodescribethemodeldynamicsbymeansofdifferentialequationsandtofurtherinvestigatethedynamicsoftheboundariesofthehomogeneousclusters.

2.10 TheanalysisofthecontinuousmodelresultsinadifferentiationbetweentwofundamentaltypesofrelocationrulesfortheSchellingmodel:rulesthatspecifySchelling'ssystemasrepresentingsolid-likematterandrulesthatspecifyitasrepresentingliquid-likematter(VinkovicandKirman2006).TheSchellingmodelrepresents"solidmatter"iftherulesenablerelocationtoabetterlocationonly.Inthiscase,thesystemstallswhenconvergingtoapatterninwhichnoneoftheagentscanimprovetheirstate.Usually,thesolidSchellingpatternstallsafterabouttentimesteps,withanessentialfractionofagentswhoaredissatisfiedwiththeirlocationbutunabletofindabetterone.Therulesthatallowrelocationtocellsofthesameutilityresultinliquid-likedynamics.TheliquidSchellingsystemdoesnotstallevenifalltheagentsaresatisfiedwiththeirlocationand,eventually,reachesapersistentstatewherethecharacteristicsofthepattern(suchaslevelofsegregation)donotchange,vastmajorityofagentsaresatisfiedwiththeirlocationandraremigrationsresultinslowandnon-directedevolutionofpatchboundaries.

2.11 VinkovicandKirman(2006)demonstratethatthepersistentpatternsofthemodelthatallowsrelocationtobetterlocationsonlyarenumerousanddependonthedetailsofthemodelrules,whilethepersistentpatternsofthe"liquid"viewofrelocationarejusttwo:integratedandsegregated.Theyalsoclaimthatthemodeldynamicsarerobusttovariationsintherules'detailsinsofarasthesetofrulesresultin"solid"or"liquid"patterndynamics.

2.12 Thesolid-liquiddichotomyisrelativelynewandthestudiesoftheSchellingmodelweareawareofdonotspecifyexplicitlywhetherthemodelrulespermitagentstorelocatebetweencellsofthesameutility.However,whenapplyingSchelling'sviewtotherealworld,itseemsnecessarytoenablerelocationbetweenresidencesofthesameutilityinordertoreflectnumerousreasonsofhouseholdmigrationsthatarenotcapturedbytheneighborhoodandethnicbasedview,say,distancetofacilitiesorjobs.

2.13 FollowingVinkovicandKirman's(2006)view,weinvestigatedasetofrulesthatontheonehanddirectlyinterpretsSchelling'sideaofresident'sreactiontothefractionoffriendsintheneighborhoodasadeterminantoftheresidentialchoice,andontheotherhandenablesmigrationbetweencellsofthesameutility,i.e.,entailsaliquid-likedynamics(BenensonandHatna2011).Thestudyofthemodelrevealsqualitativelynewresults:thevarietyofitspatternsisgreaterthanrandom-segregateddichotomy.Namely,forasufficientlywiderangeofparameters,themodelpatternconvergestoamixedstate,inwhichanessentialportionofonegroupsegregateswhiletherestofitsmembersremainintegratedwiththeothergroup.

Themixedpatterns

2.14 Themixedpatternsarequalitativelydifferentfromthewell-knownsegregatedandintegratedones.Toillustrate,letusconsideracitywitha0.2:0.8ratioofgroupsizes.Inthiscase,forvaluesofFwithintheintervalof˜(0.08,0.17),aninitiallyrandompatternconverges,intime,toastateinwhichpartoftheareaisoccupiedbythemajorityexclusively,whiletherestoftheareaisaggregated(Figure5).Aswehavedemonstratednumericallyandsupportedanalytically(BenensonandHatna2009;BenensonandHatna2011),thevaluesofFthatgeneratemixedpatternsarelowerthanFsgr.

Figure5.(a)integrated(b)mixedand(c)segregatedpersistentpatternsproducedbytheSchellingmodelforthe0.2:0.8Blue-to-Greensizeratio

2.15 MixedpatternsfurtherunderminetheviewoftheSchellingmodelasrobusttovariationsofassumptions.Thequalitativelydifferentpatternsobtainedfortheunequalgroupnumbersmanifestthepossibilitythattheemergenceofreal-worldresidentialpatterns,asthosepresentedinFigures2-4,canstillbequalitativelyexplainedbytheSchellingmodel.Thedeviationfromthe1:1ratioofthegroups'sizesisonlyoneofseveralpossiblealterationsofthe"commonlyaccepted"settings.Thesesettings,however,arejustatraditionandarenotrelatedtothebasicSchellingassumptionofagentswhoaimatresidingwithinfriendlyneighborhoods.Thereareseveralotherunnecessarysettings,suchastheassumptionsofequaltolerancethresholdofmembersofbothgroups,identicalreactionofagentsofagivengrouptoneighborsandagents'identicalviewoftheneighborhoodsizeandform.

2.16 Inthispaper,weinvestigatetwoalterationsofthestandardassumptions:westudySchellingmodeldynamicsinthecaseofnon-equalgroupsandnon-equaltolerancethresholdsofthegroups(whicharestillidenticalforallagentswithinthegroup).Wedemonstratethatthemixedpatternsaretypicalforapopulationofhighlybutnotabsolutelytolerantagentsanddiscusstheextenttowhichthisformalresultcanexplainthereal-worldresidentialpatternsasobservedinYaffoandRamle.

Formalizationofthemodelandthemethodologyofinvestigation

3.1 Inthispaper,weapplythemodelrulesdefinedinBenensonandHatna(2011).

Formalrepresentationofthemodelrules

3.2 Letusdenoteanagentasa,acellashandtheneighborhoodofh,excludinghitself,asU(h).WeconsiderU(h)tobeaMoore'ssquaren×nneighborhood(Moore1970)anddenotethefractionofa's"friends"(i.e.,agentsbelongingtoa'sgroup)amongallagentslocatedwithintheneighborhoodU(h)asfa(h).NotethatweignoreemptycellswithinU(h).

3.3 Cityspace:N×Ngridofcellsonatorus,thelatterusedtoavoidboundaryeffects.Acellcannotbeoccupiedbymorethanasingleagent.Thereisnoemigrationorimmigration.

3.4 Therearetwogroupsofresidentialagents:Blue(B)andGreen(G).

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3.5 Time:discretetime,asynchronousupdating(Cornforthetal.2005).WefollowSchelling'sviewofresidentialbehaviorandassumethatagentsobservethesystem'schangesimmediatelyaftertheyoccur.Ateachtimestep,everyagentdecideswhetherandwheretorelocate.Agentsareconsideredinrandomorder,whichisestablishedanewateachtimestep.

3.6 Agent'satolerancethresholdFa:followingSchelling'sview,weassumethatanagentalocatedathissatisfiedifthefractionoffriendswithinU(h)isFaorhigher.AsthetolerancethresholdiscommonforallagentsoftheBlueorGreengroup,wedenoteitasFBorFGrespectively.

3.7 Neighborhood'sU(h)utilityfora:wedefinetheutilityua(h)oftheneighborhoodU(h)foraas:

ua(h)=min(fa(h),Fa)/FaifFa>0 (1)

andassumethatua(h)=1ifFa=0(absolutelytolerantagenta)(Figure6)

Figure6.Theutilityoflocationhforanagentaasafunctionofthefractionoffriends,fa,withinU(h):(a)absolutelytolerantagent(Fa=0),(b)agentseekingfor50%offriends(Fa=0.5),(c)absolutelyintolerantagent(Fa=1)

3.8 Agent'sresidentialbehavior:ateachtimestept,anagentalocatedathdecideswhethertorelocateorremainsath.Whenrelocating,anagentaconsiderswvacantlocationsandcomparestheirutilitytotheutilityofitscurrentlocationh.Theawarenessofaaboutvacanciesdoesnotdependonthevacancies'distancetoh.

3.9 Agentaperformsitsrelocationdecisionintwosteps:

Step1:Decideswhethertorelocate:-Generatesarandomnumberp,uniformlydistributedon(0,1).-If(ua(h)<1or(ua(h)=1andp<m))thentriestorelocate,otherwisestaysath.

Step2:Ifthedecisionis"torelocate,"thensearchesforanewlocationanddecideswhethertomovethere:-Memorizestheutilityua(h)ofthecurrentlocationh.-ConstructsasetV(a)ofrelocationopportunitiesbyrandomlyselectingwvacanciesfromallcellsthatarevacantatthatmoment.-Estimatesutilityua(v)ofeachv∈V(a)fora,andselectsoneofthehighestutilityua(vbest).IfthereareseveralbestvacanciesinV(a),choosesoneofthemrandomly.-Movestovbestifeitherua(h)<1andua(vbest)>ua(h),theutilityofvbestishigherthanthatofh,orua(h)=1andua(vbest)=1,theutilityofhishighandthemovetovbestisunrelatedtothenumberoffriendsintheneighborhood,-Otherwisestaysath.

Form>0,theabovemodelrulesentailliquid-likedynamics.Note,however,thatonlyagentswhoaresatisfiedwiththeirlocationareallowedtorelocatetoavacancyofequalutility;unsatisfiedagentscanmovetohigherutilityvacanciesonly.

3.10 Itisimportanttoemphasizethemannerinwhichagentschoosevacancies.Accordingtothedefinitionofaneighborhood'sutility(Equation1),agentsaresatisficers(Simon1982)whoregardalocationhassatisfactoryaslongasthefractionoffriendsfawithinU(h)isequalorhigherthantheirtolerancethresholdFa.Allsatisfactorylocationsareofthesameutility1,andagentsdonotdistinguishbetweenthem.

Themethodologyofmodelinvestigation

3.11 Weinvestigatethepersistentmodelpatternsbasedona50×50torus,assumingthat2%ofthecellsareempty(d=2%).Weassumethattherateofspontaneousrelocationattemptsis1%periteration(m=0.01)andanagentconsidersatmost30vacancieswhentryingtorelocate(w=30).

3.12 Indiscretecellspace,thenumberofcellsintheneighborhoodexcludingthecentralcelldeterminestheseriesofpossibletolerancethresholdsF.Forfullyoccupied3×3Mooreneighborhoodofeightcells,thisserieswouldbeF=0/8,1/8,…7/8,8/8.Schelling(1971)employeda3×3Mooreneighborhood.However,ninepossiblevaluesofFareinsufficientforunderstandingthemodeldynamics.Wethususea5×5Mooreneighborhood,whichresultsintheseriesof25F-values(0/24,1/24,…,24/24),sufficientforrepresentingthemodelphenomenainfull.

3.13 Inwhatfollows,weinvestigatemodelpatternsasdependingonthe:

FractionβofBlueagents.TolerancethresholdFBofBlueagents.TolerancethresholdFGofGreenagents.

3.15 WeassumethattheBlueagentsareminorityandinvestigatemodelpatternsfortheseriesofβ-values:0.05,0.15,…,0.5.Asexplainedabove,theseriesofvaluesforFBandFGare0/24,1/24,…,24/24.

3.16 Figure7presentsthe3D"half-cube"oftheinvestigatedparameters'space.Webeginthestudyofmodelpatternswiththeparameter'svaluesonthecube'ssurfaceandthencombinetheresultsinordertodescribemodelpatternsforallvaluesofβ,FBandFG.

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Figure7.ParameterspaceoftheSchellingmodelasinvestigatedinthispaper

3.17 Toensurethatthemodelpatternsarenotdependentoninitialconditions,allmodelrunsarerepeatedstartingwithtwoinitialpatterns:randomandfullysegregated(Figure8).

Figure8.Modelinitialpatterns(a)random,(b)fullysegregated

Characterizationofmodelpatterns

TheuseofMoran'sIforrecognizingintegratedandsegregatedpatterns

3.18 InwhatfollowswecharacterizetheglobalpropertiesoftheBlueandGreenagents'patternsusingtheMoran'sindexIofspatialassociation(Anselin1995;GetisandOrd1992;ZhangandLinb2007)appliedtobinarydata(Lee2001;Griffith2010).

3.19 TheMoran'sIstatisticiscalculatedforthespatialvariablexhovertheoccupiedcellsaccordingto(2).Thevariablexhisdefinedasfollows:xh=0ifhisoccupiedbyBlueagentandxh=1ifhisoccupiedbyGreenagent.

(2)

wherexi,xjdenotethevaluesofxhincellsi,j;M=Round((1-d)×N2)istheoverallnumberofoccupiedcells,xbaristhemeanvalueofxhovertheoccupiedcells,wij=1ifj∈U(i),andwij=0,otherwise.

3.20 AvalueofMoran'sIcloseto0representsarandom(integrated)patternwhileavaluecloseto1representscompletesegregation(seeZhangandLinb2007forareview).Basedonthepermutationtestforthe50×50toruscity,criticalvaluesofMoran'sIequals0.002atp=0.01.Inwhatfollows,weconsideracitypatternasrandomifMoran'sIisbelow0.002.Werunsimulationsfor50,000timesteps,whicharesufficientforthepatterntoreachapersistentstate,ifexist,orexhibitfluctuations(seebelow).Thecharacteristicsofthepatternswerecalculatedfor10,000timeintervalbetweentheiteration40,000and50,000.

Threepossiblepartsofthemixedpattern

3.21 TheMoran'sIcalculatedovertheentirecityareaisusefulfordistinguishingbetweenfullyintegrated(random)patterns,forwhichMoran'sIisclosetozero,segregatedpatterns,forwhichIiscloseto1andother(mixed)patterns,forwhichMoran'sIishigherthan0butlowerthan1.However,thevalueofMoran'sIcannotdistinguishbetweenthetwotypesofmixedpatternspresentedinFigures4band4c;forbothofthesethevalueofMoran'sIwouldbeintermediate.Todistinguishbetweenthesetypesofpatterns,weemploythefollowingalgorithmthatrecognizesthetwosegregatedandthe"residual"partsofthecityaspresentedinFigures4a-c(thesizeofalltheneighborhoodsbeloware5×5):

1. Identifythesegregated(BlueandGreen)parts:LoopbyalloccupiedcellsandidentifytheBluecellsthathaveonlyBlueneighborsandtheGreencellsthathaveonlyGreenneighbors.DenotethesetwocellsetsasB^andG^respectively.

2. Identifytheboundaryofthesegregatedparts:LoopbyalloccupiedcellsthatdonotbelongtoB^orG^,andidentifythecellswhichneighborhoodshavenon-emptyoverlapwithG^orB^.DenotethissetofcellsasE1^.

3. Identifytheboundaryoftheresidualpartofthecityandcombineitwiththeboundaryofthesegregatedparts:LoopbyalloccupiedcellsthatdonotbelongtoB^,G^orE1^,andidentifythose,neighborhoodswhichhavenon-emptyoverlapwithE1^.DenotethesecellsasE2^anduniteE1^andE2^intoE^.

4. DenotethecellsthatdonotbelongtoB^,G^orE^asM^.

Figure9presentsthealgorithmstepsforthepatternsinFigures4band4c.Asstatedabove,themodelcityisatorus.

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Figure9.Thealgorithmforidentifyingsegregated(B^,G^),boundary(E^)andresidual(M^)parts,appliedtothepattern(b)and(c)inFigure4b

3.22 BelowwedistinguishbetweenthepatternsinFigures4band4cbasedonthesizeofthesegregatedBluec(B^),segregatedGreenc(G^)andresidualc(M^)partsinthecity.Namely,wedefinetheC-indexas:

C=min(c(B^),c(G^),c(M^))/N2 (3)

Thereal-worldandabstractpatternsinFigure4containeithertwoorallthreepossibleparts,andtherelativeareaofeachpartishigh.WethusclassifyapatternasoftheFigure4ctypeifthevalueofCisatleast10%(C≥10%).HighC-valuethresholdof10%guaranteesthateachofthethreepartscouldbevisuallyrecognizedinthepattern,ascharacteristicofthereal-worldpatternsinFigure4.

Persistentandfluctuatingpatterns

3.23 OursettingoftheSchellingmodelallowsforspontaneousrelocationand,thus,themodelresidentialpatternneverstalls.However,forthemajorityofβ,FBandFGvaluesthepatternconvergestopersistentstate,inwhichitssegregationcharacteristicsdonotchangesignificantly.Whenitisclearfromthecontext,weuse"modelpattern"whenreferringto"modelpersistentpattern."

3.24 Forsomecombinationsofparametersthepatternsdonotstabilize.AnextremeexampleisthecaseofFG=3/24,FB=7/24,β=0.5.AscanbeseenfromFigure10a,theMoran'sIforthissetofparametersfluctuates,intime,between0.35-0.70andsodoestheC-index,whichfluctuatesbetween0and15%.Figure10bpresentstwoextremepatternsgeneratedforthissetofparameters:onesegregated,qualitativelysimilartothepatternpresentedinFigure4a,andtheothermixed,qualitativelysimilartothepatternpresentedinFigure4c.TheleftpatternischaracterizedbytherelativelyhighvalueofMoran'sIandlowvalueoftheC-indexwhiletherightpatternischaracterizedbytherelativelylowvalueofMoran'sIandhighvalueoftheC-index.

Figure10.ThefluctuatingpatternofFG=3/24,FB=7/24,β=0.5:(a)ThedynamicsofMoran'sIandC-indexforthefirst200,000iterations(b)Typicalmixedandsegregatedpatternsobserved.

Resultsofthemodelinvestigation

4.1 Inwhatfollows,webrieflyrepeattheclassicmodelresults(§4.2)andproceedwithinvestigatingthemodelbehaviorasdependentonallthreeparametersβ,FBandFG.§4.6-§4.11dealwithspecialsubsetsofthe(β,FBandFG)parameterspace.§4.18onwardsconsidersthegeneralsituation.

Basiccase:equalgroupsofequaltolerancethresholds(β=0.5,FG=FB=F)

4.2 ToverifySchelling'sconclusionsregardingthecaseofgroupsofequalsizeandtolerancethresholds(Figure11),weperformed25modelrunsforwhichFG=FB=F=0/24,1/24,…,24/24,and,inaddition,ninerunswithFG=FB=F=0/8,1/8,…,8/8thataimedtoinvestigatetheinfluenceoftheneighborhoodsizeonmodelpatterns.

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Figure11.Theinvestigatedsetofparametersβ=0.5,FG=FB

4.3 TheMoran'sIvaluesasdependentonFarepresentedinFigure12.

Figure12.Moran'sIformodelpersistentpatternsasdependentonFfor(a)3×3(b)5×5neighborhoods

4.4 Qualitatively,ourresultsarefullyconsistentwithSchelling'sfindings.Namely,theintervaloftheF-valuesconsistsoftwosubsets:F≤4/24andF≥5/24.ForF≤4/24,thepatternsarerandom(integrated),whileforF≥5/24thepatternsaresegregated.Thetransitionbetweenthetwosubsetsisabrupt.Inthecaseofthe3×3neighborhood,theresultsarequalitativelythesamebutlessprecise:randompatternsarecharacteristicofF≤1/8=3/24,whileforF≥2/8=6/24thepatternsaresegregated.

4.5 Thepatternsforthecaseofthe5×5neighborhoodarepresentedinFigure13.TheunsmoothboundaryofthesegregatedpatternofF=5/24seemstocorrespondtotheconceptualpatternofFigure4a.Forthesegregatedpatterns(F≥5/24),thehigherF,thesmoothertheboundarybetweentheclustersofagentsofthesamecolor.ThisexplainstheslowgrowthofMoran'sIwiththegrowthofFinFigure12.

Figure13.PersistentmodelpatternsforthecaseofequalgroupsasdependingonF

Persistentpatternsforthe2D-subsetsofparameters

4.6 Inwhatfollows,weinvestigatethemodelpersistentpatternsforthecaseofGreenmajorityversusBlueminorityandvaluesofparameterslimitedtothree2Dsubspacesofthe3Dhalf-cube(Figure14):

β∈(0,0.5],FB=FG:equaltoleranceofbothgroups(Figure14a);β∈(0,0.5],FG=0,FB∈[0,1]:absolutelytolerantGreenmajority(Figure14b);β∈(0,0.5],FG∈[0,1],FB=0:absolutelytolerantBlueminority(Figure14c).

Figure14.Threeinvestigated2D-subsetsofparameters(a)β∈(0,0.5],FB=FG,(b)β∈(0,0.5],FG=0,FB∈[0,1],(c)β∈(0,0.5],FG∈[0,1],FB=0

4.7 FollowingBenensonandHatna(2011),wedenote,foragivenβ,Frnd,βasthehighestvalueofFthatgeneratesrandom(integrated)persistentpattern(characterizedbyclosetozerovalueoftheMoran'sI)andFsgr,βasthelowestvalueofFthatentailsasegregatedpattern(characterizedbyclosetounitvalueoftheMoran'sI).Asshownabove,forβ=0.5,thevaluesofFrnd,0.5andFsgr,0.5areadjacent,i.e.,Frnd,0.5=4/24,andFsrg,0.5=5/24.

BlueminorityversusGreenmajority,equaltolerancethresholdinbothgroups(β∈(0,0.5],FB=FG=F)

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4.8 Aspresentedin§2.14,thecaseofunequalgroupsexhibitsmixedpatterns.Toillustrate,letusconsiderthecaseofβ=0.2.ThevaluesofMoran'sIforthepersistentpatternasdependentonF=FB=FGandthepatternsforselectedvaluesofFarepresentedinFigure15.

4.9 AscanbeseeninFigure15a,thedependenceofthemodelpatternonFincaseofβ=0.2isdifferentthanforthecaseofβ=0.5.Namely,forF=2/24,3/24and4/24thevaluesofMoran'sIarenon-zero,butyetlowerthanthatcharacteristicofthesegregatedpatterns.AscanbeobservedinFigure15b,thepatternsforthesethreevaluesofFareneitherrandomnorsegregated.AccordingtoFigure15,Frnd,0.2=1/24,whileFsgr,0.2=5/24(notethatthevaluesofFsgr,0.2andFsgr,0.5arethesame,andequalto5/24).ForthevaluesofF∈(Frnd,0.2,Fsgr,0.2),partofthecityareaisoccupiedbytheGreenmajorityexclusivelywhiletherestoftheareaisintegrated.AsitisshowninBenensonandHatna(2009,2011),withthegrowthofF,theintegratedpartoftheintegratedregionbecomesmorecompact,thusentailingagrowthoftheMoran'sI.ThecityabruptlysegregateswhenFincreasesfrom4/24to5/24.ForF≥5/24,theMoran'sIvalueslowlyincreaseswithanincreaseinFandtheboundarybetweentheGreenandBlueclustersbecomessmoother,justasforβ=0.5.

Figure15.(a)Moran'sIand(b)persistentpatternsasdependentonF,forβ=0.2

4.10 ThecompleteviewofMoran'sIdependenceonFandβispresentedinFigure16a.Notethat(a)forβ>0.05thepatternabruptlysegregateswhenFpassesthethresholdvalueofFsgr,β=5/25,whileforβ=0.05,Fsgr,0.05=6/25;(b)thevalueofFrnd,βdecreaseslinearlyandthewidthoftheintervaloftheF-valuesgeneratingthemixedpatternsincreaseswiththedecreaseinβ.ThedependenceoftheC-indexonFandβispresentedinFigure16b.ItindicatesthatforallF≠5/24thepatternsconsistoftwopartsonly,eitherofsegregatedBlueandGreenorofsegregatedGreenandmixedparts.ForF=5/25,themaximalvaluesofCarepositivebutfarbelowthe10%levelweacceptedascharacteristicofthecomplexpatterninFigure4c.Thenon-zeroC-indexobtainedforF=5/24characterizesessentialnon-smoothnessoftheboundarybetweenthesegregatedBlueandGreenareas.Thetypeofpatternasdependentonβ,FB,FG,isshowninFigure16c.Inatheoreticalstudy(BenensonandHatna2009)wehavedemonstratedthatforF∈(Frnd,β,Fsgr,β),incaseofthemixedpattern,thedistributionoftheGreenandBlueagentswithintheintegratedpartisrandom.Itisimportanttonotethatthefractionofminoritywithinthesepatchesisalwayshigherthanβ,growswiththegrowthofβandreaches0.5whenFapproachesFsgr,β.ThelatterexplainstheabruptchangefrommixedtosegregatedpatternwhenFpassesthethresholdvalueofFsgr,β.

Figure16.Thedependenceof(a)Moran'sIand(b)C-indexonFforβ=0.05,0.15,…,0.5andFB=FG=Fand(c)thesetsofparameterscharacteristicoftherandom,mixedandsegregatedpatternsinthehalfcube

Absolutelytolerantgroup(β∈(0,0.5],FG=0orFB=0)

4.11 Inthissection,weexaminethecaseinwhichagentsofonegroupsarecompletelytolerant.Westartbydescribingthecaseofequalsizegroups(β=0.5)andcompletelytolerantGreenagents(FG=0)and

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thendescribethecasewherethetolerantgroupiseitheramajorityorminority.

4.12 Thecaseofequalgroups(β=0.5)andcompletelytolerantagentsofoneofthegroupsisbrieflymentionedbyPortugalietal.(1997),whonotedthatdespitetheabsoluteneutralityofmembersofonegroup,theresidentialpatternsegregateswhenthetolerancethresholdoftheothergroupissufficientlyhigh.Oursimulationsessentiallyextendthisresult.Asabove,forgivenβ,wedenoteasFB,rnd,βthehighestvalueofFBthatentailsarandompatternandasFB,sgr,βthelowestvalueofFBthatentailsasegregatedpattern.

4.13 Figure17presentsthedependenceofMoran'sIonFBforthecaseofβ=0.5andFG=0andthepatternsforselectedvaluesofF.AccordingtoFigure17aFB,rnd,0.5=4/24andMoran'sI,valueincreaseslinearlywithfurtherincreaseofFBuntilFBreachesFB,sgr,0.5=13/24.UnlikethecaseofFG=FB,thereisnoabruptchangeinthelevelofsegregationwhenFBapproachesFB,sgr,0.5.ForFB∈[5/24,12/24]thepatternismixed.

4.14 ThecorrespondingpersistentpatternsareshowninFigure17b.PartoftheareaofthemixedpatternsisoccupiedexclusivelybythefullytolerantGreenagents,whilefewGreenagentsresidewithintheintegratedpartofthepatterns.TheBlueagentsavoidneighborhoodswheretheirfractionisbelowFB,and,thusthefractionoffullytolerantGreensresidingwithintheBlue-dominatedareasdecreaseswiththeincreaseinFB.ForFB=10/24,11/24and12/23,thepatternsslightlyfluctuateintime,becauseofthefewGreenagentsthatresidewithintheBluearea.However,theC-indexalwaysremainsbelow3%.

Figure17.CharacteristicsofmodelpersistentpatternforFG=0,β=0.5:(a)Moran'sIasdependentonFB;(b)persistentpatternsforselectedvaluesofFB

4.15 ThedependencesofMoran'sIandConβandFBforthecaseofabsolutelytolerantmajorityorminorityarepresentedinFigure18.Foratolerantmajority(Figure18c),adecreaseinβentailsadecreaseinFB,rnd,βandFB,sgr,β,butatdifferentratesandwiththelengthofthe(FB,rnd,β,FB,sgr,β)intervalthusincreasing.ThedependenceofMoran'sIonFGandβforthecaseofabsolutelytolerantBlueminorityispresentedinFigure18a.Inthiscase,thedecreaseinβresultsinthegrowthofbothFG,rnd,βandFG,sgr,βandareductionoftheintervalofmixedpatternstozerowhenβdecreasesto0.15.ThethreedomainsofparametersforbothcasesarepresentedinFigure18e.TheC-index(Figure18b,18d)isnon-zeroforthevaluesofFjustbelowthethresholdthatguaranteesfullsegregation,namelyFB,FG=10/2411/24and12/23.ThemodelpatternsforthesevaluesofFslightlyfluctuateintime.

Figure18.Moran'sIincaseofabsolutelytolerantmajorityorabsolutelytolerantminority:(a)absolutelytolerantBlueminority(FB=0);(b)absolutelytolerantGreenmajority(FG=0);(c)parameterscharacteristicoftherandom,mixedandsegregatedpatterns

4.16 PersistentpatternsforthecaseofabsolutelytolerantGreenmajorityandβ=0.3arepresentedinFigure19a(FB,rnd,0.2=3/24,FB,sgr,0.2=13/24).Justasforβ=0.5,withthegrowthofFBwithin(FB,rnd,0.2,FB,sgr,0.2),thefractionofabsolutelytolerantGreenagentsresidingwithintheBlue-dominatedareasdecreaseswiththegrowthofF,until,atFB=13/24,completesegregationisreached.Figure19bpresentstheseriesofpersistentpatternsforthecaseofabsolutelytolerantBlueminorityβ=0.3.Justasinthecaseofabsolutelytolerantmajority,thefractionofminoritywithinthemajority-dominatedareadecreaseswithanincreaseinFG,and,inparallel,thesizeoftheminoritypatchesincreases.

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Figure19.Model'spersistentpatentsforβ=0.3:(a)completelytolerantGreenmajority(FG=0);(b)completelytolerantBlueminority(FB=0)

4.17 Overall,thepersistentpatternsincaseofabsolutelytolerantminoritydonotdifferfromthoseobtainedforthecaseofabsolutelytolerantmajority.TheintervaloftheF-values,forwhichthesepatternsareproduced,becomesnarrowerandshiftstowardthehighervaluesofFwiththedecreaseinβ.

Generalcase

4.18 Tocompletethedescriptionofthemodelpersistentpatterns,letusconsidertheirdependenceonallthreeparameters:β,FBandFG.Wedothatforfourhorizontalcross-sectionsofthehalf-cubeatβ=0.05,0.2,0.35,and0.5.ThefoursurfacesofMoran'sIandCforFG≤17/24andFB≤17/24andcorresponding3Dillustrationofthefull(FB,FG,β)domainsproducingrandom,mixedandsegregatedpatternsarepresentedinFigure20.

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Figure20.(a)Moran'sIand(b)C-indexasdependenton(FG,FB)forFG≤17/24,FB≤17/24(forlargerFGorFBthepatternsarealwayssegregated)andβ=0.05,0.2,0.35and0.5(thecasesofFB=FGaremarked);(c)the3Dpresentationofthe(FB,FG,β)domainsproducingrandom,mixed,andsegregatedpatterns

4.19 Withtheincreaseinβ,thedomainof(FB,FG)valuesproducingmixedpatternsintheleft-topincreases.Thisdomainrepresentsthecaseofatolerantminoritygroupwhichbecomemorenumerousasβincreasesandthuscontributesmoretotheemergenceofmixedpatterns.Correspondingly,withtheincreaseinβthedomainof(FB,FG)valuesproducingmixedpatternsintheright-buttondecreasesasthenumberoftolerantmajorityagentsdecreases.The(FB,FG)domainbecomessymmetricacrosstheFB=FGwhenβreaches0.5.Thedomainofthe(FB,FG)valuesproducingtheintegratedpatternsislargestforβ=0.5andshrinkswiththedecreaseinβ;themixedpatternareagrowsatitsexpense.

4.20 Allthepersistentpatternsproducedinthe(β,FB,FG)domainarecharacterizedbyzeroorverylowvalueoftheC-index(figure20b).Thenon-zerovaluesofCareobtainedforvaluesofparametersthatlieontheboundaryofthedomainthatguaranteessegregation,namely,forthevaluesofFBandFGthatsatisfytheconditionFB+FG=10/24,11/24,12/24.ForthesevaluesofFBandFG,thepatternsfluctuateintime,andtheamplitudeofthefluctuationsoftheMoran'sIandC-indexdoesnotdependonβ.Forfew(FB,FG)pairsonthisboundary,theamplitudeofthefluctuationsreachesthelevelofFG=3/24,FB=7/24,β=0.5(Figure10).Wedeferinvestigationofthisphenomenatofuturework.

Discussion

5.1 OurversionoftheSchellingmodelisbasedontwoqualitativeassumptions:first,agentsaresatisficers,i.e.,theydonotdistinguishbetweenlocationswherethenumberoffriendsisabovetheirtolerancethreshold;second,therelocationrulesallowsatisfiedagentstomigratebetweenvacanciesofthesameutility.ThesatisficingprincipleisconsistentwithSchelling'soriginalformulation(Schelling1978),whiletherelocationrulesentailliquid-likedynamicsthatpreventpatternsfromstallinginastatewheresomeoftheagentsaredissatisfiedbecausenosatisfactoryvacanciesareavailable.

5.2 Wedemonstratedthatsatisficers,whocanrelocatebetweenneighborhoodsofthesameutility,remainrandomlydistributedinspaceiftheirdemandsforhavingfriendswithinneighborhoodsislow,segregateiftheirdemandsarehighandproducemixedpatternswhentheirdemandsforfriendswithintheneighborhoodareintermediate.Inthelattercase,thepopulationpatterndependsontheinvestigatedparameters:thefractionofminorityβandtheleveloftoleranceFBandFGoftheagentsofeachgroup.

5.3 Canwerelatetheseresultstothereal-worldpatternsasobservedinRamleandYaffo(Figures2-4)?Basically,yes:ArabhouseholdersareminoritiesinYaffoandRamleandempiricalevidencesuggeststhatmembersofArabminoritieslivinginmixedcitiesinIsraelaresomewhatindifferenttothepresenceofJewishneighborsintheirbuildingorinneighboringbuildings(Benensonetal.2002).WecanalsoassumethatIsraeliArabsandJewsaresatisficersinregardtotheethnicstructureoftheirresidentialneighborhoodsandprobablyuseadditionalcriteria,besidesethnicityofneighbors,whensearchingforadwelling.Ourresearchdemonstratesthatifthesetendenciescharacterizetheentirepopulationgroups,theresidentialdynamicsentailmixedpatternsthatareindeedobservedoverpartsofYaffoandRamle.

5.4 ThemodelgeneratedpersistentpatternsthatareconsistentwithtwoofthethreepatternspresentedinFigure4:segregatedpatternssuchastheonepresentedinFigure4a,andmixedpatternssuchastheoneinFigure4b.However,YaffoandRamlepatternsaremorecomplexthanthoseobtainedinthemodelscenariosinvestigatedinthispaper:theybothcontainmixedpartssidebysidewithsegregatedones,asisschematicallyrepresentedinFigure4c.Aswehaveshownabove,theSchellingmodeleveninextended,three-dimensional,parameterspacedoesnotproducesuchpersistentpatterns.Statedformally,withinthemodelframeworkthatweinvestigatedinthispaper,nomatterwhatthefractionofminorityortolerancethresholdsofthetwogroups,segregatedandmixedareascannotpermanentlycoexist.

5.5 ThepatternsinwhichsegregatedandmixedareascoexistemergeanddissolveperiodicallyforvaluesofFthatlieontheboundarybetweenthedomainsofparametersthatentailpersistentlysegregatedandpersistentmixedpatterns.Thesefluctuatingpatternsshouldbefurtherinvestigatedinordertoassesstheirrobustnesstothevariationinthemodelrules,especiallyifwefurthergeneralizethemodelandassumethattheleveloftoleranceamongGreenandBlueagentscanvary.

5.6 Apreliminaryinvestigationdemonstratesthatmorecomplexpatternsthanthoseobtainedinthispapercanindeedbeobtainedinthiscase,andthesepatternscontainoneintegratedandtwosegregatedregionsforarangeofparametersthatiswiderthanthatrevealedinthispaper.Webelievethatsuchageneralizationtakesthemodelclosertothereality.Indeed,theassumptionthatmembersofanethnicgroupareidenticalintheirattitudetomembersofanotherethnicgroupseemsartificialnomatterwhichgroupsareunderdiscussion.However,theonlyresultinthisrespectregardingArabsandJewsistheongoingempiricalstudybyOmeretal.(2012),whodemonstratethatJewsandArabsinYaffoessentiallyvaryintheirresidentialtolerancetothemembersoftheothergroup.

Acknowledgements

Theauthorswouldliketothanktheeditorandtwoanonymousreviewersfortheirhelpfulcomments.

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