©CopyrightJASSS

JamesMillington,TimButlerandChrisHamnett(2014)

Aspiration,AttainmentandSuccess:AnAgent-BasedModelofDistance-BasedSchoolAllocation

JournalofArtificialSocietiesandSocialSimulation 17(1)10<http://jasss.soc.surrey.ac.uk/17/1/10.html>

Received:03-Nov-2012Accepted:17-Jun-2013Published:31-Jan-2014

Abstract

Inrecentyears,UKgovernmentshaveimplementedpoliciesthatemphasisetheabilityofparentstochoosewhichschooltheywishtheirchildtoattend.Inherentlyspatialschool-placeallocationrulesinmanyareashaveproducedageographyofinequalitybetweenparentsthatsucceedandfailtogettheirchildintopreferredschoolsbaseduponwheretheylive.Wepresentanagent-basedsimulationmodeldevelopedtoinvestigatetheimplicationsofdistance-basedschool-placeallocationpolicies.Weshowhowasimple,abstractmodelcangeneratepatternsofschoolpopularity,performanceandspatialdistributionofpupilswhicharesimilartothoseobservedinlocaleducationauthoritiesinLondon,UK.Themodelrepresents'school'and'parent'agents.Parental'aspiration'tosendtheirchildtothebestperformingschool(asopposedtoothercriteria)isaprimaryparentagentattributeinthemodel.Thisaspirationattributeisusedasameanstoconstrainthelocationandmovementofparentagentswithinthemodelledenvironment.Resultsindicatethattheselocationandmovementconstraintsareneededtogenerateempiricalpatterns,andthatpatternsaregeneratedmostcloselyandconsistentlywhenschoolsagentsdifferintheirabilitytoincreasepupilattainment.Analysisofmodeloutputforsimulationsusingthesemechanismsshowshowparentagentswithabove-average–butnotveryhigh–aspirationfailtogettheirchildaplaceattheirpreferredschoolmorefrequentlythanotherparentagents.Wehighlightthekindsofalternativeschool-placeallocationrulesandeducationsystempoliciesthemodelcanbeusedtoinvestigate.

Keywords:Education,Inequality,Aspiration,Schools,School-PlaceAllocation,ParentalChoice

Introduction

1.1 Ithaslongbeenrecognisedthatspaceisanimportantconstraintonaccesstohighqualityeducationandhealthcareservices(e.g.,Bradleyetal.1978;McLafferty1982).IntheUK,provokedbydemandstoimproveeducationalattainmentandinlightofthe'wideningchoice'agendaforwardedbybothNewLabourandConservative-ledCoalitiongovernments,therehasrecentlybeenmuchinterestinthegeographyofinequalityineducationprovisionandattainment(e.g.,ButlerandHamnett2007;BurgessandBriggs2010;HarrisandJohnston2008;Gudson2011).Inrecentyears,bothNewLabourandConservativeshaveimplementedpoliciesthatemphasisetheabilityofparentstochoosewhichschooltosendtheirchildto,inpartwiththeintentionofdrivingupeducationalstandards(HamnettandButler2011).Despitethis,evidencesuggeststhattherehasbeenlittlechangeinschoolintakecomposition(AllenandVignoles2007;GibbonsandTelhaj2007).

1.2 Althoughworkcontinuestoinvestigatethecausesandconsequencesofeducationalpolicyusingtraditionalquantitative(e.g.,Allenatal.2013)andqualitative(e.g.,ButlerandHamnett2012)methods,thereisclearscopeforapplyingnewtoolssuchasagent-basedsimulationtoinvestigatetheseissues(Tangetal.2007;Maroulisetal.2010a;HarlandandHeppenstall2012).However,verylittleworkinthissubjectareahasbeenpursuedusingagent-basedmodelling(ABM).Maroulisetal.(2010b)examinedtheimpactsofchoice-basedreformsinChicagoPublicSchoolsusinganagent-basedframeworktoshowhowvariationinindividuals'emphasesinachievementledtoconstraintsonthenumberofnewschoolsthatcouldsurviveinagivendistrict.HarlandandHeppenstall(2012)showedhowsimplerulesallowedanagent-basedmodeltoreproduceempiricalschoolallocationdataforaregionofnorthernEngland.Toourknowledge,thesestudiesarethecurrentextentoftheliteratureusingagent-basedsimulationtoinvestigateeducationsystems.

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1.3 Oneoftheprimaryadvantagesofanagent-basedapproachforexaminingspatialmovementsofindividuals,comparedtousinganalyticspatialmodelssuchasgravityorradiationmodels(e.g.,Siminietal.2012),isthedisaggregatedandheterogeneousrepresentationofsystemelementsitallows.Whereasanalyticmodelsassumethesystemelementsbeingrepresentedarehomogenous,agent-basedapproachescanrepresentheterogeneousindividualswhichallowsuserstoexaminetheimportanceofdifferencesbetweenindividualsforsystem-leveloutcomes,andalsotheconsequencesofsystem-levelpropertiesforparticularindividualsystemelements.Furthermore,agent-basedsimulationmodelsrepresentenvironmentally-situatedentitiesthatarecapableofflexibleautonomousactiontomeetdesiredobjectives(i.e.,agentsthatcanactindifferentwaysdependingontheirenvironmentalcontext;O'Sullivan2008).Thesesimulationframeworkscanrepresentspatially-explicitprocesseswhentheirmodelassumptionsmeanthattherelativespatiallocationofheterogeneousindividualsinfluencesthecircumstances,andthereforebehaviour,ofothersimulatedindividuals(e.g.,Millingtonetal.2012).Thedegreetowhichdatainformstherepresentationoftheworldinthesemodelscanrangefromsimple,abstractmodelsusedasthoughtexperiments,throughlocallyspecificmodelsthataimtounderstandhowgeneralsocio-economicprocessesplayoutinparticularsettings,tohighlydetailedsimulationsthatrepresentverylarge,multi-dimensionalsystems(O'Sullivan2008).Ourapproachhereisatthesimple,abstractendofthisspectrumandis'generative'(Epstein1999,2006)inthatweseektoexplorehowthelocalinteractionofsimulatedheterogeneous,autonomousagentscanresultintheemergenceofmacroscopic(societal)regularities.Subsequently,wecanexaminetheimplicationsoftheindividualinteractionsthatproducesocialregularitiesfordifferentgroupsofindividuals(e.g.,withsimilarattributes).

1.4 Herewepresenttheinitialdevelopmentofanagent-basedsimulationmodelforinvestigatingtheimplicationsofUKlocaleducationauthorityschool-placeallocationpolicy.Webeginwithabriefoverviewoftheempiricalmacroscopic(i.e.,school-level)relationshipsweaimtoreproduce,beforethenpresentingmodelstructureandthelocalinteractionsitrepresents(i.e.,individualparents'attributesanddecision-making).Weexploredifferentsetsofrulesforinteractionsbetweenagents(schoolsandparents)andexaminetheirimpactonthereproductionofschool-levelrelationshipsandtheconsequencesforgroupsofparentswithsimilarattributes.Finally,wediscussourresultsandhighlightpotentiallyusefulwaysforwardforusingthismodellingapproachtoexaminesocialandpolicy-relatedquestions.

ThegeographyofinequalityinUKstateschooling

2.1 ThestatesecondaryschoolallocationprocessinEnglandandWalesisoperatedbylocaleducationalauthorities(LEAs).WithinanLEA,parentscanchoosetoapplytoasetnumberofschoolsfortheirchildtoattend,whichtheyrankintermsofpreference.Usingtheseapplicationsandrankings,LEAsthenallocateplacestoschools.Withtheexceptionofselective(e.g.,faith)schools,theallocationprocessisinherentlyspatialasplacesatover-subscribed(popular)schoolsareallocatedaccordingtothedistance

afamilylivesfromtheschool(nearestbeingallocatedfirst)[1].Thecloserafamilylivestoapopularschool,thebetterchanceofsecuringaplaceattheschoolforthechild.Thisformof'choice',thatbothfostersandaimstoaccommodateaspirationsofparentsbutwhichrequiresarationingmechanismtobalancethesupplyanddemandofpopularschools,producesageographyofinequalitywithwinnersandlosersthatsucceedorfailtogettheirchildintopreferredschools.

2.2 Thepatternsproducedbythiseducationalgeographyofinequalitycanbeseeninempiricaldataonschoolperformance,popularityandtraveldistances(HamnettandButler2011).SchoolperformanceismeasuredbythepercentageofstudentsachievingfiveormoreGeneralCertificateofSecondaryEducation(GCSE)gradesofA*–C(whichwerefertoasGCSE-5+).Schoolpopularitycanbemeasuredusingtheratioofparentapplicationstoavailableplaces(A:P)andtraveldistancesbythemaximumdistancewhichchildrenattendingtheschoollivefromit(MaxDist).ForempiricaldataonschoolsinsevenEastLondonLEAsin2007/08,HamnettandButler(2011)showhowtheapplicationandallocationcriteriadescribedaboveresultinapositiverelationshipbetweenA:PandGSCE-5+(seeTableVinHamnettandButler2011),anegativerelationshipbetweenA:PandMaxDist(seeTableVIIinHamnettandButler2011),andsmallerMaxDistformorepopularschoolscomparedtolesspopularschools(seeFigure5inHamnettandButler2011).

2.3 Todefinethesepatternsmorequantitativelysothatmodeloutputcanbebettercomparedweusepubliclyavailabledata[2]fortheBarkingandDagenhamLEAaveragedacrossfiveyears(2007-2011).Wefitlinearregressionmodelsandcalculatethe

coefficientofdeterminationr2forrelationshipsbetweenGCSE-5+andA:P,betweenA:PandMaxDist,andbetweenGCSE-5+andMaxDist(Figure1).AsHamnettandButler(2011)foundformultipleEastLondonLEAs,theserelationshipsshowthatschoolswithhigherpercentagesofstudentsachievingfiveormoreGCSEsatgradesA*–Caremorepopular(i.e.,havegreaterapplicationstoplacesratios,Figure1a),thatstudentsatmorepopularschoolsonaverageliveclosertotheschoolcomparedtolesspopularschools(Figure1b),andinturn,thatstudentsatthepoorestperformingschoolstravelonaveragefarthertoschoolthanthoseatthebestperformingschools(Figure1c).

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Figure1.Empiricalschool-levelrelationshipsforschoolsinBarkingandDagenham2007–2011.Foreachplotm=regressioncoefficient,p=p-valueofregression,r2=coefficientofdetermination.

2.4 Theseempiricalpatternshighlighttheimportanceofthelocationwhereparentsliveforthechancesofgettingtheirchildintopopular,generallyhigherperforming,schools.However,interviewswithparentsinEastLondonhaveshownarangeofattitudestowardstheimportanceofeducation,fromlargelyindifferenttobeingthehighestpriority(ButlerandHamnett2011).Forexample,contrastthefollowingtwostatementsfromtwoparentsinthesameareaofEastLondon:

"Ithink[educationis]reasonablyimportant.Iwouldn'tputitupthereasreallytoprankedjustbecause,youknow,Ithinkthere'smoreimportantthingsinachild'slife."(ButlerandHamnett2011,p.105)

"Oh,thesinglemostimportantthingthataparentcangivetotheirchildreninlifeisafirst-classeducation"(ButlerandHamnett2011,p.98).

2.5 Withtheintentionofproducingasimple,abstractagent-basedsimulationmodelthatcangeneratethegeneralempiricalpatternsdescribedabove,wefocusontheover-archingconceptofparental'aspiration'tosendachildtothebestperformingschool(measuredbyexamresults).Weusethemodeltorepresentarangeof'aspiration'regardingeducationalattainment,fromlargelyindifferenttohighpriority.Furthermore,weusethisnotionalmeasureofaspirationasameanstoconstrainthelocationandmovementofparentswithinamodelledenvironment.Weexaminedifferentmechanisms(i.e.,modelrulesets)toidentifyhowtheyinfluencegeneratedpatternsandexaminethesensitivityofseveralkeymodelvariables.Firstwedescribethegeneralmodelstructure,beforethenpresentingresultsandanalysisofthedifferentmodelrule-andparametersets.

ModelStructure

3.1 OurdescriptionofmodelstructureusesselectedpartsoftheOverview,Designconcepts,andDetails(ODD)protocol.Thefull

ODDdescriptioncanbefoundonlinewiththemodelcodeatopenABM.org[3].

Purpose

3.2 ThepurposeofthismodelistoinvestigatemechanismsunderlyingthegeographyofeducationalinequalityintheUKandtheconsequencesofthesemechanismsforindividualswithvaryingattributesandmobility.

Entities,statevariablesandscales

3.3 Twotypesofagentsarerepresented;parentsandschools.Oneiterationofthemodelisassumedtobeequivalenttoasingleyear.Althoughparentsandschoolshaveexplicitspatiallocations,nospacescalesareimpliedorassumedandthemodelenvironmentisatorus.Atorusisappropriateinoursimple,abstractmodeltosuppressboundaryeffectssothatobservedvariationinattainmentandaccesstoschoolscanreliablybeattributedtostraight-linedistancesbetweenschoolsandparents(similartoapproachesusedinurbansegregationmodels;e.g.,LaurieandJaggi2003,FossettandDietrich2009).Themostimportantparentattributeistheiraspirationtosendtheirchildtothebestperformingschool.Aspirationtakesavaluefrom1to100andvariesbetweenparents.Thisrangeofvaluesrepresentsvariationinparents'attitudestowardswhethereducationalattainment(intermsofGCSEs)istheprimarycriteriaforselectingaschool(highvalues)oralessimportantcriteriainselectingaschool(lowvalues).Itreflectshowsomeparentswillseektomaximisethepossibilityoftheirchildattaininghighgradeswhereasotherwillbesatisfiedwithless(inthelightofotherprioritiesnotrepresentedinthismodel).Parents'aspirationvaluesaresetwhentheparentiscreatedanddonotchangethroughtime.Parentsareassumedtohaveasinglechild(notrepresentedasanindividualagent,butimplicitlyasanattributeoftheparent).Eachparenthastwochildattributes;child-ageandchild-attainment.Child-ageismeasuredinyearsandchild-attainmenttakesavaluefrom1to100.Bothchild-ageandchild-attainmentchange

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throughtime.Initially,parents'child-attainmentisequaltotheiraspirationandchild-agehasavalueof9.Parentshaveanexplicitanduniquelocation(i.e.,parentscannotsharealocationwithotherparentsoraschool).Parentscanpotentiallychangelocationoncethroughtime.

3.4 Schoolshaveafinitenumberofplacesavailableforallocationtoparentseachyear.Schoolshavefiveacademicyearcohorts(i.e.,grades)ofparents(pupils)andhaveaGCSE-scoreattributewhichiscalculatedasthemeanofchild-attainmentvaluesofparentswithchild-age=15allocatedtotheschool(i.e.,GCSE-receivingfinal-yearstudents[4]).GCSE-scorecantakeavaluefrom0to100.Asbothallocatedparentsandchild-attainmentcanchangethroughtime,soGCSE-scorecanvarybetweenschoolsandthroughtime.Schoolshaveavalue-addedattributewhichisassignedatmodelinitializationandcanvarybetweenschoolsbutdoesnotchangethroughtime.Value-addedcantakeavaluebetween0.0and1.0.Eachyearschoolsrecordtheparentsallocatedaplaceandtheparentswhichappliedtotheschool.Schoolshaveanexplicitanduniquelocationwhichcannotchangethroughtime.Schoollocationscanberandomacrossthemodelenvironment,orwithequaldistancebetweeneachschool(i.e.,onagrid).

ProcessOverviewandscheduling

3.5 Eachyearexistingparentsintheenvironmentincreasetheirchild-agebyavalueof1.Schoolacademicyearcohortsarealsoaged(e.g.,year8parentsbecomeyear9parents)andparentsallocatedaschoolplaceinthepreviousyear(whentheirchildwasage10,nowage11)becomeyear7.Afterthisincrease,parentswithchild-age=16areremovedfromthemodelenvironment,astheirchildrenareassumedtohavereceivedtheirGCSEsandleftschool.Thisassumptionrepresentsthefactthathouseholdsthatnolongerhavechildrenatschoolwillnotbecompetingforplacesatschools(andnotbeoccupyingplacesatthoseschools)andensuresspaceisavailableintheenvironmentforparentswithyoungerchildrenapplyingforschoolplaces.Creatingthisspaceisimportantsothatthereproductionofeducationalinequalitycanbeexaminedthroughtimeandsharessimilaritieswithsimilarlysimple,abstractmodelsofresidentialsegregationthatassumeafixedpercentageofagentsleavethemodelenvironmentinagiventimestep(e.g.,Portugali2000;O'Sullivan2009).

3.6 Newparentsarethenaddedtothemodelenvironment.ThenumberofnewparentsaddedisgivenbyFamilies*Number-of-Schools.Thevaluesofthesevariablesarespecifiedbytheuseratmodelinitializationanddonotchangethroughtime.Newparentsareassignedtounoccupiedlocationsinthemodelenvironment.Locationassignmentcanbespatiallyrandomorconstrainedbyaspiration.Ifconstrained,themeanaspirationofparentsintheMooreneighbourhood(i.e.,8surroundinglocations)ofeachunoccupiedlocationiscalculated(knownaslocation-value;ifagivenlocationhasnoparentsinitsMooreneighbourhooditslocation-valueissettothemeanaspirationofallparentsinthemodelenvironment).Newparentsareassignedtheunoccupiedlocationwithgreatestlocation-valuewhichisalsolessthanthatparent'saspiration.Ifnolocationmatchesthesecriteria(i.e.,allunoccupiedlocationshavelocation-value>newparentaspiration)thenewparentisassignedthelocationwiththesmallestlocation-value.

3.7 Parentswhichhavenotyetbeenallocatedaschool(i.e.,thosewithchild-age=9orchild-age=10)thenassessschools.Theseparentsassesswhethertheybelievetheyarewithinthe'catchment'ofeachschool.Eachyearthemeanspatialdistanceofallallocatedparentsataschooliscalculated.Parentsassumetheyareinaschoolcatchmentiftheirdistancetotheschoolislessthanthesmallestmeandistanceforthelastparent-memoryyears.Theparent-memoryparameterissetbytheuseratmodelinitialization,doesnotvaryintimeorbetweenparents,andcantakeavaluefrom1to5.Unallocatedparentsalsoassesswhichschooltheyconsidersatisfactorytosendtheirchildto.Satisfactoryschoolsarethosewith:

(1)

Finally,theseunallocatedparentsassesswhich'poor'schoolstheywanttoavoidsendingtheirchild.Theseschoolstobeavoidedarethosewith:

(2)

Theavoided-thresholdparameterisaglobalparameterwhichdoesnotvaryintimeorbetweenparentsandcantakeavaluefrom0to1.

3.8 Parentswithchild-age=9(i.e.,oneyearbeforetheywillbeallocatedtoaschool)thencheckiftheywanttomovefromtheircurrentlocationtotrytoincreasetheirchanceofhavingtheirchildallocatedtoasatisfactoryschool(inthenextyear)bylivingclosertothatschool.ParentsrankthetopNumber-of-Rankschoolsusingoneofeightstrategies,dependingontheircircumstances(seeTable1).Theythenchecklocationsincatchmentsoftheseschoolsinrankorder(startingwithtoprank)untilalocationisfoundorallschoolcatchmentshavebeenchecked.Parentswillnotbeabletomoveiftherearenounoccupiedlocationsinacatchmentorifavailabilityoflocationsisconstrainedbylocation-value(seeabove).Theyearinwhichachild-age=9istheonlyoneinwhichparentscanmove.Ultimately,thisisnotanaccuraterepresentationofrealityasparentscouldmovemultipletimesbeforeandafterschoolallocation.However,thissingle-moveassumptioniscrediblegiventhataspectsthatmayinfluencehousemovesotherthanschooling(e.g.,changesinfamilyincomeorsize)arenotrepresentedinthissimple,abstractmodel.

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Table1:Idealisedparentrankingstrategiesformoving.SchoolsconsideredbyparentstobesatisfactoryortobeavoidedaredeterminedbyEq.1andEq.2,respectively.

Strategy Criteria Response1 Believedtobeinnoschool

catchmentNoschoolsconsideredsatisfactoryNoschoolsavoided

RankallschoolsbyGCSE-scoredescending

2 BelievedtobeinnoschoolcatchmentNoschoolsconsideredsatisfactoryAtleastoneschoolavoided

Rankallschools–exceptthoseavoided–byGCSE-scoredescending

3 BelievedtobeinnoschoolcatchmentAtleastoneschoolconsideredsatisfactoryNoschoolsavoided

RankschoolsconsideredsatisfactorybyGCSEscoredescending,thenallotherschoolsbyGCSEscoredescending

4 BelievedtobeinnoschoolcatchmentAtleastoneschoolconsideredsatisfactoryAtleastoneschoolavoided

RankschoolsconsideredsatisfactorybyGCSEscoredescending,thenallotherschools–exceptthoseavoided–byGCSEscoredescending

5 BelievedtobeinoneormoreschoolcatchmentNoschoolsconsideredsatisfactoryNoschoolsavoided

RankschoolsconsideredsatisfactorybyGCSEscoredescending,thenallotherschoolsbyGCSEscoredescending

6 BelievedtobeinatleastoneschoolcatchmentNoschoolsconsideredsatisfactoryAtleastoneschoolavoided

RankschoolsconsideredsatisfactorybyGCSEscoredescending,thenallotherschools–exceptthoseavoided–byGCSEscoredescending

7 BelievedtobeinatleastoneschoolcatchmentAtleastoneschoolconsideredsatisfactoryNoschoolsavoided

Donottrytomove

8 BelievedtobeinatleastoneschoolcatchmentAtleastoneschoolconsideredsatisfactoryAtleastoneschoolavoided

RankschoolsconsideredsatisfactorybyGCSEscoredescending,thenallotherschools–exceptthoseavoided–byGCSEscoredescending

3.9 Parentswithchild-age=10rankthetopNumber-of-Rankschoolsthattheywillapplytosendtheirchildtousingoneofeightstrategiesdependingontheircircumstances(seeTable2).Todeterminewhichstrategytouse,parentscheckwhichschool

catchment(s)theybelievetheyarelocatedwithinandwhetherthereareschoolstheydeemsatisfactorytosendtheirchild[5].

Table2:Idealisedparentrankingstrategiesforschoolapplication.SchoolsconsideredbyparentstobesatisfactoryortobeavoidedaredeterminedbyEq.1andEq.2,respectively.

Strategy Criteria Response1 Believedtobeinno

schoolcatchmentNoschoolsconsideredsatisfactoryNoschoolsavoided

Rankallschoolsbydistanceascending

2 BelievedtobeinnoschoolcatchmentNoschoolsconsideredsatisfactoryAtleastoneschoolavoided

Rankallschools–exceptthoseavoided–bydistanceascending

3 Believedtobeinnoschoolcatchment

Rankschoolsconsideredsatisfactorybydistanceascending,thenallotherschoolsbydistanceascending

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AtleastoneschoolconsideredsatisfactoryNoschoolsavoided

4 BelievedtobeinnoschoolcatchmentAtleastoneschoolconsideredsatisfactoryAtleastoneschoolavoided

Rankschoolsconsideredsatisfactorybydistanceascending,thenallotherschools–exceptthoseavoided–bydistanceascending

5 BelievedtobeinoneormoreschoolcatchmentNoschoolsconsideredsatisfactoryNoschoolsavoided

RankthoseschoolsbelievedtobeinthecatchmentofbyGCSEscoredescending,thenallotherschoolsbydistanceascending

6 BelievedtobeinatleastoneschoolcatchmentNoschoolsconsideredsatisfactoryAtleastoneschoolavoided

Exceptforthoseschoolsavoided,rankschoolsbelievedtobeinthecatchmentofbyGCSEscoredescendingfollowedbyallotherschoolsbydistanceascending

7 BelievedtobeinatleastoneschoolcatchmentAtleastoneschoolconsideredsatisfactoryNoschoolsavoided

RankthoseschoolsbelievedtobeinthecatchmentofbyGCSEscoredescending,thenthoseschoolsconsideredsatisfactorybydistanceascending,thenallothersschoolsbydistanceascending

8 BelievedtobeinatleastoneschoolcatchmentAtleastoneschoolconsideredsatisfactoryAtleastoneschoolavoided

Exceptforthoseschoolsavoided,rankthoseschoolsbelievedtobeinthecatchmentofbyGCSEscoredescending,thenthoseschoolsconsideredsatisfactorybydistanceascending,thenallother(non-avoided)schoolsbydistanceascending

3.10 Schoolsthenallocateplaces[6]toparentswithchild-age=10readyfortheirchildtobecomeapupiloftheschoolthenextyear.Schoolsallocateapplicantsthatrankedthemhighestfirst(startingwiththeclosestparentandallocatinginascendingorderofdistance).Onceallschoolshaveallocatedthesetop-rankingparents,ifplacesremainunallocated(i.e.,iftherewerelessparentsrankingthemfirstthantotalplacesavailable)schoolsthenallocateapplicantsthatrankedthemsecond(again,allocatingbydistanceascending).Thisprocesscontinues(thirdranks,fourthranks,etc.)untilallparents'rankingshavebeenchecked.Schoolsthathaveremainingplacesafterallrankedpreferenceshavebeenallocated,thenallocateremainingplacestounallocatedparentsondistance(closestallocatedfirstthenbydistanceascending).Thisapproachtoschool-placeallocationimplementstheGale-Shapley(1962)method(withrankingdeterminedbydistancetoschool)asusedbyLEAsinEngland(Allenetal.2010).

3.11 Finally,alsoinpreparationforthefollowingyear,existingparentsalreadyallocatedaplaceataschoolupdatetheirchild-attainmentusingtheidealisedrelationship:

(3)

whereCAischild-attainment,SVAisthevalue-addedoftheschoolattended,SPEisSchool-Peer-Effect,PEisParent-Effect,SCAisthemeanchild-attainmentofallparentsallocatedaplaceattheschoolandtdenotesthetimestep.Multiplefactorsarebelievedtoinfluencechangesinpupilattainmentduringtheirtimeatsecondaryschool,attributabletoindividualpupils'backgrounds(e.g.,ethnicityandclass;Connolly2006;Hamnettetal.2007)andschool-levelfactors(e.g.,schoolcompositionandpeer-effects;Thruppetal.2002;Willms2010).ThestructureandcompositionofEq.3allowstherelativeimportanceofseveralfactorstobeexamined.Specifically,School-Peer-EffectandParent-Effectcantakevaluesfrom0(noeffectonpupilattainmentthroughtime)to0.5(largeeffectonpupilattainmentthroughtime)andreflecttheinfluenceoftheattainmentofpupils'peersandtheaspirationsoftheirpupils'parentsonattainment.Thepossibleinfluenceoffactorsbeyondpupilsandparentsthemselves(i.e.,facilities,teachersetcoftheschoolitself)isreflectedbyvalue-added.Wheneachofthesethreevariablesiszerothereisnochangeinpupilattainmentthroughtime.SchoolsthenupdatetheirGCSE-scoretobethemeanofchild-attainmentofallocatedparentswithchild-age=15(i.e.,year11pupils).

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ModelTestingandEvaluation

StateVariables

4.1 Toevaluateandtestthemodelweconsiderstatevariablesatthreedifferentlevelsofaggregationandscale.TomeasurespatialautocorrelationofparentaspirationattheleveloftheentiresimulatedenvironmentweuseMoran'sI,calculatingthestatisticandtheprobabilitythatthestatisticisstatisticallysignificantlydifferentfromzero(p)foreachtimestep.WecannotestimateMoran'sIforempiricaldataandsodonotconsiderthisanempiricalpatternbywhichtoassessthegenerativepropertiesofthemodelrules.However,thismeasureisausefulmeanstoidentifyiflocalinteractionsrulesresultinsystem-levelspatialpatterns(i.e.,spatialautocorrelationofparentagentattributes).AttheschoollevelwemeasureGCSE-score(correspondingtoGCSE-5+),A:PandMaxDistandfitlinearregressionmodels(withcoefficientandsignificancevaluesmandp,respectively)andcalculatethecoefficientofdetermination(r2)foreachcombinationofpairsofthesevariablesineachtimestep(aswedidforempiricaldatain

Figure1).Forregressionmodelsfrommodeloutputwecalculatethemeanmodelcoefficient,meanr2andthemeannumberof

timestepsduringamodelruninwhichregressionp>0.05.Weusevaluesform,pandr2asindicatorstocomparehowresultsfordifferentmodelrulessetsreproducethegeneralpatternsoutlinedabove,usingthequantifiedvaluesforourempirical

regressionsasaguide.Thus,weassumethatmodelresultswithlargerabsolutevaluesofm,higherr2valuesandlowerpvaluesindicatestrongerreproductionofgeneralempiricalpatterns.Attheparentlevelweconsiderparentstrategyforschoolapplication(seeTable2),applicationsuccess,distancetoallocatedschool,aspiration,child-attainmentchange,andwhethertheparentmovedornotpriortoallocation.Parentapplicationsuccessisevaluatedbycomparingwhetherachildwasallocatedaplaceattheirtoprankedschool(successifso,otherwisenot).Changeinchild-attainmentisthedifferencebetweenchild-attainmentwhenchild-age=10(i.e.,priortoenteringschool)andwhenchild-age=15(i.e.,whenreceivingGCSEresultsandleavingschool).Wedonotcurrentlyhaveempiricalequivalentswithwhichtocomparethesemeasures.

Methods

4.2 Totestandevaluatethemodelweconsiderwhichmodelrulesand/orconditionsarenecessaryforthemodeltogeneratetheschool-levelempiricalpatternsdescribedabove(Section2).Hence,theapproachis'generative'(Epstein1999).Wealsoexaminehowdifferentrulesandmodelconditionsinfluencespatialautocorrelationofparentaspirationacrosstheentiremodelenvironment.Weevaluatetheimportanceofthespatialdistributionofschools,locationconstraints,andschoolvalue-added(Table3).Weevaluatetheimportanceofthreemodelruleoptions:i)thespatialdistributionofschools,byeitherlocatingthemrandomlyacrossthemodelenvironmentorregularlyspaced(onagrid);ii)theimportanceofparentallocationconstraints,byrunningthemodelwithandwithouttheconstraintoflocation-value;andiii)theimportanceofschoolvalue-added,byeithersettingallschools'value-addedtozeroorallowingvaluestovaryrandomly(between0and1).Foreachoftheeightrule-setsthecombinationsoftheseruleoptionsproduce(Table3),werunthemodel25timesfor100timesteps(withparametervaluesasshowninTable4).Foranalysisweuseonlythelast80timestepsofeachmodelrunasrandominitializationmeansthatittakesatleast10timestepsbeforeschoolshavehadasinglecohortofstudentspassthroughtheschoolwithchild-attainmentcorrectlycalculated(Eq.3,andseeinitialvariationinvariablesinFigure2).

Table3:Combinationsofrulesformodeltesting.

RuleSet RandomSchools LocationConstraints SchoolValue-Added1 No Yes Yes2 No No Yes3 Yes Yes Yes4 Yes No Yes5 No Yes No6 No No No7 Yes Yes No8 Yes No No

Table4:Defaultparametervaluesusedinmodelruns(unlessotherwisespecified).

Parameter ValueFamilies 100Parent-Memory 5Number-of-Schools 9Number-of-Ranks 4

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Initial-School-GCSE-Distribution Uniform,1Parent-Aspiration-Distribution Gaussian,μ=50σ=20Aspiration-Mean 50School-Value-Added-Distribution Gaussian,μ=0σ=0.1Avoided-Threshold 0.5School-Peer-Effect 0.25Parent-Effect 0.25

4.3 Forthemodelthatbestgeneratesempiricalpatternswealsoperformsensitivityanalyses(forparametersandvaluesasshowninTable5).

Table5:Parametervaluesusedinsensitivityanalyses.

ParameterSet Parameter Value9 NumberofRanks 210 NumberofRanks 611 AvoidedThreshold 0.0512 AvoidedThreshold 0.2513 AvoidedThreshold 0.7514 AvoidedThreshold 0.9515 SchoolPeerEffect,ParentEffect 0.00,0.0016 SchoolPeerEffect,ParentEffect 0.25,0.0017 SchoolPeerEffect,ParentEffect 0.50,0.0018 SchoolPeerEffect,ParentEffect 0.00,0.2519 SchoolPeerEffect,ParentEffect 0.50,0.2520 SchoolPeerEffect,ParentEffect 0.00,0.5021 SchoolPeerEffect,ParentEffect 0.25,0.5022 SchoolPeerEffect,ParentEffect 0.50,0.50

Results:Rulesets

4.4 Relationshipsbetweenstatevariables(Table6)indicatethatthemodelproducesresultswhichgenerateempiricalpatternsmost

closelyandconsistently(i.e.,largem,highr2,lowp)whenparentsareconstrainedbywheretheycanlive,whenschoolsdifferentiallyaddvaluetopupils'attainmentandwhenschoolsarenotrandomlylocated(i.e.,RuleSet1,Table3).ResultsindicatethatatleastoneoftheLocationConstraintsorSchoolValue-Addedrulesisneededtogeneratetherelationshipbetweenschoolperformanceandpopularity(GCSE-5+vs.A:P).Ifneitherispresent(e.g.,RuleSets6and8),therangeofschoolperformanceisverylow(i.e.,littledifferencebetweenmaximumandminimumschoolGCSE-5+,Figure2),andthereforenoclearpreferencesbetweenschoolsarise.TogeneratestrongspatialautocorrelationinparentaspirationtheLocationConstraintsruleisneeded(RuleSets1,3,5,7)astheseconstraintsproduce'neighbourhoods'ofaspiration(e.g.,Figure3).Inturn,whencombinedwithnon-randomlocationofschools(i.e.,RuleSets1and5),constrainingthelocationofparentsgeneratesgoodrelationshipsatschoollevel.Thus,patterninbothlocationofparentsandschoolsisrequiredtogenerateempiricalrelationships,andthisisenhancedwhenschoolsdifferinthevaluetheyaddtopupils'attainment.

Table6:Measuresofrelationshipsbetweenkeyempiricalvariablesfordifferentmodelrulesets[7]

Rule GCSE-5+vsA:P GCSE-5+vsMaxDist A:PvsMaxDist Moran'sISet m r2 p m r2 p m r2 p Stat. p

1 7.89 0.96 0.00 -0.78 0.70 5.58 -0.09 0.65 10.00 0.92 0.002 6.88 0.85 0.68 -0.18 0.38 47.20 -0.02 0.36 50.92 0.01 32.503 8.53 0.94 0.12 -0.41 0.23 70.16 -0.04 0.20 71.16 0.91 0.004 6.51 0.76 4.84 -0.01 0.13 74.68 0.00 0.14 74.44 0.01 31.505 6.98 0.97 0.00 -0.67 0.68 8.60 -0.09 0.65 10.96 0.92 0.006 1.79 0.30 56.68 0.00 0.12 76.16 0.00 0.11 76.88 0.01 46.007 7.36 0.92 2.60 -0.26 0.18 71.92 -0.03 0.16 74.96 0.91 0.00

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8 1.37 0.26 61.40 0.01 0.15 74.44 0.00 0.17 72.56 0.01 54.00

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Figure2.TimeseriesofGCSEandrangeandA:Prange.Solidlinesarethemeanof25modelrunsforthegivenparameterset.Shadedareasarethe95%confidenceintervalaroundthecorrespondingmean,calculatedfromthestandarderrorof25model

runs.

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Figure3.Examplefinalmodelstatemaps.Mapsareofparentaspiration(lightershadesarehigheraspiration).Schoolsareshownashouseicons(lightershadesindicatehigherGCSE-score).NumbersindicateRuleSets(assummarisedinTable3).

Results:Sensitivityanalysis

4.5 Whencomparedtoresultsforthebestmodelruleset(RuleSet1),sensitivityanalysisresultsindicatethatmodelledschool-levelrelationshipsarelargelyinsensitivetovariationinparametersinfluencingparents'schoolrankingandchild-attainmentchange(Table7).Theexceptionisforhighervaluesofavoided-threshold(Eq.2).Whenavoided-threshold=0.95(ParameterSet6)relationshipsbetweenGCSE-5+andMaxDistandbetweenA:PandMaxDistbreakdown(i.e.,thereisnorelationship)andforavoided-threshold=0.75(ParameterSet5)therelationshipsbecomeweaker(i.e.,norelationshipmorefrequently).

Table7:Measuresofrelationshipsbetweenkeyempiricalvariablesforsensitivityanalyses[8]

Parm GCSE-5+vsA:P GCSE-5+vsMaxDist A:PvsMaxDist Moran'sISet m r2 p m r2 p m r2 p Stat. p

1 11.21 0.93 0.00 -0.70 0.65 8.76 -0.06 0.61 12.16 0.93 0.002 7.66 0.90 0.00 -0.79 0.69 6.08 -0.09 0.60 12.68 0.91 0.003 8.35 0.96 0.00 -0.80 0.71 5.00 -0.09 0.67 8.04 0.92 0.004 8.09 0.96 0.00 -0.80 0.70 4.12 -0.09 0.66 7.08 0.92 0.005 5.39 0.93 0.04 -0.68 0.59 20.56 -0.11 0.53 25.80 0.92 0.006 6.12 0.88 0.40 -0.04 0.13 76.52 0.00 0.11 78.12 0.93 0.007 9.67 0.95 0.00 -1.06 0.69 7.60 -0.11 0.67 9.12 0.92 0.008 10.69 0.93 0.00 -1.18 0.71 5.64 -0.11 0.69 6.44 0.92 0.009 10.80 0.93 0.00 -1.13 0.68 9.60 -0.10 0.66 12.40 0.92 0.0010 8.11 0.95 0.00 -0.84 0.70 5.52 -0.10 0.65 10.28 0.92 0.0011 7.39 0.96 0.00 -0.71 0.68 7.76 -0.09 0.63 12.32 0.92 0.0012 7.37 0.96 0.00 -0.72 0.64 10.16 -0.09 0.60 17.76 0.91 0.0013 7.16 0.96 0.00 -0.68 0.67 8.28 -0.09 0.63 12.92 0.91 0.0014 7.04 0.97 0.00 -0.66 0.66 9.72 -0.09 0.62 13.12 0.92 0.00

Results:Parent-levelanalysis

4.6 Attheparentlevel,plotsofproportionsofparentsinclassesofonestatevariablesagainstclassesofotherstatevariablesareusefultoidentifyrelationshipsbetweenthosevariables(e.g.,Figure4).Someoftheserelationshipsareappropriateforverifyingmodelfunctiongivenmodelstructure,butothersareinterestingtounderstandwhatthemodelstructureimpliesforparentswithdifferentattributes.

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Figure4.Relationshipsbetweenvariablesattheparent-levelofanalysis(forRuleSet1).

4.7 Asexpected,givendistance-basedschoolallocationrulesinthemodel,resultsshowthatparentslivingclosertotheirallocatedschoolaremorelikelytohaverankedthatschoolastheirtoppreference(Figure4a).Furthermore,ofparentsthatsuccessfullygettheirchildintotheirtoprankedschool,morehavepositivechild-attainmentchange(bluecoloursinFigure4a)thannegative(redcolours).However,itisnottheparentsintheclosestdistanceclass(distance<10)thathavegreatestpositivechild-attainmentchange.Rather,onaverageitisparentsintheseconddistanceclass(10–20)thatachievegreatestchild-attainmentincreases.Forexample,agreaterproportionofparentsintheseconddistanceclasshavepositivechild-attainmentthanthoseintheclosest(82%comparedwith56%)andoverallmeanchild-attainmentchangeisgreaterforthesecondclass(+6.36)thantheclosest(+1.76).Incontrast,thevastmajority(92%)ofparentsinthefourthdistanceclassandgreater(i.e.,allocateddistance≥30)havenegativechild-attainmentchange.Ofparentsinthesedistanceclasses,only1%weresuccessfulingettingtheirchildintotheirtop-rankedschool,againhighlightingtheimportanceofdistanceallocationrulesandtheirrelationshipwithschool(andpupil)performance.

4.8 Whenweconsiderrelationshipsbetweenparentaspiration,allocateddistanceandwhetherparentshavemovedornot(Figure4b)weobservethatasaspirationincreasesparentsaremorelikelytomove.Parentsthatmovearemorelikelytoliveintheclosestallocatedschooldistanceclass(distance<10).Noparentsinthefarthestallocatedschooldistanceclasses(i.e.,distance≥30)movedtobeinthatposition,andtheseparentsarenotthosewithlowestaspiration.Rather,parentswithhigherthanaverage,butnotveryhigh,aspiration(i.e.,thosewithaspiration60–70)aremostlikelytobeallocatedtoaschoolwithdistance≥30.

4.9 Parentsinthismedium-highaspirationclass(aspiration60–70)alsofailtogettheirchildintotheirpreferredschoolmoreoftenthanparentsinotheraspirationclasses(Figure4c).Parentswithaspiration≥70aremorelikelytobeinaschoolcatchmentwhenapplyingtoaschool(Figure4c,increasedproportionsofstrategies5–8intheseclasses).Furthermore,asaspirationincreases,thelikelihoodofsucceedingusingstrategy3torankschoolsforallocationdecreases(Figure4c;i.e.,thedangerofnotbeingintheschoolcatchmentofasatisfactoryschoolisgreaterforthosewithhigheraspiration).

Discussion

ModelStructure

5.1 Ourmodeltestingandevaluationshowshowdifferencesbetweenschools'performance(andthereforeparents'schoolpreferences)combinedwithdistance-basedschool-placeallocationrules,areneededtoreproduceempiricallyobservedschool-levelpatterns.Theapproachwehaveusedis'generative'(Epstein1999,2006),seekingtoexplaintheemergenceofmacroscopic(societal)regularitiesarisingfromthelocalinteractionofsimulatedheterogeneous,autonomousagents.Usinganagent-basedsimulationmodelwehavegeneratedobservedmacroscopicregularities(i.e.,relationshipsbetweenschoolperformance,popularityandallocation)fromthe'bottomup'bysimulatingindividualparents'aspirationsregardingeducationalattainmentandtheireffortstodosointhefaceofdistance-basedschoolallocationrules.

5.2 Startingwithrandomlocationsofparentsandschoolswithidenticalperformance(GCSE-5+),thisgenerativeapproachallowsustoshowthateitheri)differencesintheabilitiesofparentstomovetolocationsnearpreferredschools,orii)variationintheincreasesinattainmentschoolscanprovidetopupilsareneededtogeneratetheempiricalrelationshipbetweenschoolexam(GCSE)performanceandschoolpopularity.Ifneitheroftheserulesispresentinthemodel,variationinschoolperformanceisnotproduced(Figure2)meaningthatparents'schoolpreferencesareinconsistentacrossthemodelledenvironment(i.e.,itisnotclearwhichschoolsarebetterthanothers).Differencesintheabilitiesofparentstomoveareimportantforcreatingneighbourhoods(groupings)ofparentswithsimilaraspiration(Figure3;Moran'sIinTable6).Inturn,theseneighbourhoodsmeanthatchildrenwithsimilarinitialattainment(becauseinitialchild-attainmentisequaltoparentaspiration)aremorelikelytoattendthesameschool,reinforcingimprovementsinschoolperformance(viaSchool-Peer-Effect).Variationintheimprovementthatschoolscancontributetopupils'attainmentalsoproducesvariationinschoolperformance,buttoalesserdegreethantheneighbourhoodsofaspirationeffect.Wefoundthatincreasingtherangeofimprovementthatschoolscontributetopupilattainment(i.e.,fromσ=0.1forSchool-Value-Added-Distributiontoσ=0.5,Table4)doesincreasethiseffect,butstilldoesnotproduceasconsistentlysignificantrelationshipsasforparentlocationconstraints(e.g.,pforGCSE-5+vs.MaxDistof22.48comparedwith5.58and8.60forRuleSets1and5respectively,Table6).

5.3 Anothermodelrule-setweinvestigatedwastherandomlocationofschools(i.e.,RuleSet3,Table3).Whenschoolsarerandomlylocatedspatiallyacrossthemodelenvironment,theempirically-observeddistancerelationshipscollapse.Thisis

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becauseoftheabilitiesofparentstomoveintodesiredschoolcatchmentsandbecauseoftheagentlogicusedinthemodel.Withrandomschoollocations,schoolsareoftenclosetogetherandsoparentscanbelocatedinmorethanasingleschoolcatchment(i.e.,schoolcatchmentsoverlap),asituationwhichdoesn'toccurwhenschoolsareregularlyspaced.Consequently,agreaternumberofparentsareinschoolcatchmentswhenitcomestoparentsrankingforallocationandschoolsallocating,asshownbygreaterproportionsofparentsindistanceclasses10–20and20–30usingrankingstrategy7than3(compareFigures5aand5b).Theoverlappingofschoolcatchmentsalsomeansthatparentsaremorelikelytosucceedingettingtheirchildrenintotheirdesiredschoolswhenfartherfromthem(compareFigures5aand5b)asparentsneartheschoolmayhavesenttheirchildtoadifferent,butnearby,school.Thissituationimpliesthemodelisnotusefulforconsideringsituationswhereschoolsarenotspacedequally.However,initscurrentformwewouldnotexpectthemodeltobeusefulinthissituationasparents'rankinglogicformovingonlyconsidersthesinglebestschool(anditscatchment)anddoesnottakeaccountofwhichlocationsintheenvironmentwouldallowthemtobeinmultiple(good)schoolcatchments.Consequently,asthemodellogiccurrentlystands,themodelisbestusedwithregularlyspacedschoolssothatparentsareveryunlikelytobeinmorethanoneschoolcatchment,astheagentlogicassumes.

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Figure5.Relationshipbetweendistanceandstrategyatparent-levelforrandomlylocatedschoolsfora)RuleSet1andb)RuleSet3(specifiedinTable3).

5.4 Alsointhecurrentmodelstructure,threemechanismsexistwherebyfeedbacksbetweenaspirationandchild-attainmentcanoccur.Thesearetheassumptionsthat:

i. initialchild-attainmentisequaltotheaspirationoftheparents(Att=Asp);ii. throughtimeatschoolchild-attainmentisinfluencedbytheaspirationoftheparent(PEinEq3);andiii. throughtimeatschoolchild-attainmentisinfluencedbythemeanchild-attainmentofschoolpeers(SPEinEq3).

5.5 Tochecktheimportanceoftheseassumptionsforthegenerationoftheschool-levelempiricalrelationships(e.g.,Figure1),weranthemodelforthedifferentcombinationsoftheseassumptionsbeingpresentinthemodelornot(seeTable8,while

maintainingotherassumptionsofRuleSet1[9]).ThesetestsindicatethatiftheAtt=Aspassumptionisabsent,themodeldoesnotgeneratetheschool-levelempiricalrelationshipswhenPEisalsonotpresent(i.e.,RuleSets10and12),butperformsbetterwhenPEispresent(i.e.,RuleSets14and15).Furthermore,theSPEeffectseemstobetheleastimportantofthethreeassumptionshighlightedabove,aswhenthisassumptionsisabsentbuttheothertwoassumptionsarepresentschool-levelrelationshipsarenotaffected(i.e.,compareresultsforRuleSet13withRuleSet1).Neighbourhoodsofaspirationareproducedinallcombinationsofassumptions(indicatedbyhighMoran'sIvalues).Theseresultsmakesenseasalthoughparentsofsimilaraspirationstillclustertogether,ifthereisnolinkbetweenparentaspirationandchild-attainment(viaeitherAtt=AsporPE)littlevariationinschoolperformanceisproducedbyspatialvariationinparentaspiration.AlthoughtheAtt=Aspassumptionisuseful,theseresultsimplythataperfectcorrelationbetweenaspirationandinitialchild-attainmentisnotnecessaryforthemodeltogenerateempiricalrelationships.

Table8:Measuresofrelationshipsbetweenkeyempiricalvariablesforaspirationandchild-attainmentmechanisms.

RuleSet PE SPE Att=Asp GCSE-5+vsA:P GCSE-5+vsMaxDist A:PvsMaxDist Moran'sI

r2 p r2 p r2 p Stat. p

9 F F T 0.93 0.00 0.60 17.24 0.59 17.68 0.93 0.0010 F F F 0.90 0.00 0.42 39.52 0.41 43.80 0.92 0.0011 F T T 0.93 0.00 0.61 17.36 0.61 16.76 0.93 0.0012 F T F 0.88 0.00 0.44 38.12 0.46 37.36 0.93 0.0013 T F T 0.95 0.00 0.70 5.52 0.67 8.48 0.92 0.0014 T F F 0.94 0.00 0.67 11.48 0.64 14.60 0.92 0.0015 T T F 0.94 0.00 0.68 10.48 0.65 13.88 0.92 0.00

Parent-levelpatterns

5.6 Ourgenerativeapproachtomodellinghasshownthattheconsequencesofourassumptionsaboutthesystemattheindividual,parent,levelcangeneratetheempiricallyobservedrelationshipsandpatternsatthehigher,school,level.Althoughthismodelisahighlysimplifiedconceptualisation,itallowsustoexaminerelationshipsbetweenentitiesatthelowerlevelandbetweenupperandlowerlevelsthatwouldnotbepossible(orattheleast,verydifficult)intherealworld.Forexample,ourparent-levelresults(forRuleSet1,Table3)showthatingeneralthoseintheseconddistanceclass(distance10–20)achievegreatest child-attainmentincreases,andnotthoseintheclosestdistanceclass(distance<10,Figure4a).Thisisbecausethoseparentsthatliveintheclosestdistanceclasshaveonaveragegreateraspirationthanthoseintheseconddistanceclassandthereforehavegreatestchild-attainmentinitially.Consequently,thechild-attainmentoftheseclosestparentsisonaveragemorelikelytodecreasethanincrease.

5.7 Anotherinterestingfindingfromourparent-levelanalysisisthatthoseparentswithaspiration60–70failtogettheirchildintotheirpreferredschoolmoreoftenthanotherparents(closelyfollowedbythosewithaspiration50–60,Figure4c).Asnotedintheresults,thelikelihoodoffailingtogetintoapreferredschoolusingstrategy3(rankschoolsconsideredsatisfactorybydistanceascending,thenallotherschoolsbydistanceascending,Table2)increasesasaspirationincreases(Figure4c).Althoughagreaterproportionofparentswithaspiration70–80failwhenusingstrategy3comparedtoparentswithaspiration60–70,parentsinthisloweraspirationclasshaveagreaterproportionofparentsusingthisstrategyoverall(parentswithhigheraspirationaremorelikelytobeinaschoolcatchmentandthereforeusestrategies5–8).Parentswithaspiration60–70arenolesslikelytofindthemselvesoutsideaschoolcatchmentthanparentswithloweraspiration(Figure4c)butbecausetheiraspirationishighertheyconsideronlybetterschoolssatisfactoryfortheirchild.Thismeanstheyhavefewerschoolstorank(sodistancetothoseschoolsislikelytobegreater),andeachofthoseschoolsismorelikelytohavegreaternumbersofparentsdeemingthemsatisfactorytosendtheirchildto(andsotheseschoolshavemanyparentsrankingthemasmostpreferred).Incontrast,parentswithloweraspiration(e.g.,aspiration<50)aremorelikelytogetintotheirpreferredschooleventhoughnotinanyschool'scatchment,bothbecausethedistancetothenearestschoolislikelytobesmaller(becausetherearemoreschoolsdeemedsatisfactory)and

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becausetherearefewerotherapplicantsrankingthatschoolasmostpreferred(becauseotherparentsaremorelikelytoavoidit).Thefailureofparentswithaspiration60–70togetintotheirpreferred(i.e.,topranked)schoolisreflectedintheirgreaterallocationdistancesthananyotheraspirationclass(Figure4b).Furthermore,onlyapproximately3%ofparentswithaspiration60–70move,andmanyremainstuckinthepositionofhavinghigherthanaverageaspirationbutnotbeing'intherightplace'(spatially)whentheyinitiallyarriveinthemodelenvironment(becauseoflocationconstraints).Theseparentshaveaspirations'toohigh'relativetotheirabilitytomoveintopreferredschoolcatchments.

5.8 Thequestionthenarises;howmightschoolallocationrulesorpoliciesbemodifiedtohelpthoseparentswithaboveaverage,butnotveryhigh,aspiration(andthereforemobility)getintobetterschools(oratleastschoolstheywant)?Onewaymightbetoincreasethestandardsofschoolssothatagreaternumbermeettheaspirationsofparents.Insodoing,thenumberofschoolsthatparentswithaboveaveragebutnotveryhighaspirationdeemsatisfactorytosendtheirchildtowillincreaseandthedangerofnotbeinginaschoolcatchmentshoulddecrease.ToinvestigatethisweexamineascenarioinwhichwerunthemodelasforRuleSet1(Table3)butwithagreatermeanschoolvalue-addedof0.2(althoughwiththesamestandarddeviationaspreviouslyof0.1).Resultsforthis'improvedschoolstandards'scenarioindicatethatincreasedabilityofschoolstoraisechild-attainmentproduceschangesinstrategiesforparentswithhigheraspirationanddecreasestheproportionofparentswithaspiration60–70thatfailtogettheirchildintotheirpreferredschool(Figure6,comparetoFigure4c).Furthermore,thisincreaseinmeanschoolvalue-addedincreasestheproportionsofparentsinotheraspirationclassesthatfailtogettheirchildintoapreferredschool,resultinginamoreevendistributionoffailureacrosstheaspirationclasses.

Figure6.Relationshipbetweenaspirationandstrategyatparent-levelfor'improvedschoolstandards'scenario.

Prospectsforfuturework

5.9 The'improvedschoolstandards'scenarioisjustoneexampleofthekindsofscenarioswecanexaminewiththemodel.Themodelcouldalsobeusedtoexplorealternativeschoolallocationrulesandpolicies,whichmightincluderandomlotteriesforschoolallocation(e.g.,Allenetal.2013),opening'free'schoolsthatmayuseaptitudeasaselectioncriterion(e.g.,Hatcher2011),ortheclosureofunder-performingschools.FuturechangestothemodelmightextendittoenablerepresentationofothercriteriausedinUKstateschoolallocation(e.g.,religiousfaith,attendanceofsiblings).

5.10 Themodelpresentedhereusesonlyasingleparentagentvariable(aspirationforhigheducationalattainment)tosimultaneouslyrepresentthegoalsofparentsandtheconstraintsontheirabilitytomeetthosegoals.However,therearemanyfactorsunderlyingwherefamilieswantand/orareabletoliveandwhichschoolstheyperceiveasdesirablefortheirchildtoattend.Forexample,educationalaspirationvariesbyclassandethnicity(ButlerandHamnett2011,2012)andtheabilitytomovehousetoachievetheseaspirationsisaneconomicquestioninfluencedbythehousingmarket.Therepresentationofagentsandtheirenvironmentwithmultipleattributesthatmoreaccuratelyreflectmotivationsandconstraintsisneeded.Thereisnoreasonwhyaspirationforeducationalattainmentandeconomicwealthshouldbecorrelatedandfuturemodellingmayexplorehowvariationsindistributionsofthesefactorsresultindifferentwinnersandlosersthroughtimeandacrossspace.Improvingthisrepresentation

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willrequireindividual-leveldataonattributes,preferencesandallocations.Theseimprovementsinrepresentationanddatawouldalsoallowaninvestigationofmotivationsforschoolchoicebeyondexamresultsalone,allowingagentstoidentifypreferencesbaseduponschoolethnicandsocio-economiccompositionandtheattributesofotherparentsthatsendtheirchildtoaschool(althoughthatisinsomewaysrepresentedherethroughtheAtt=Aspassumption).

5.11 Whatwehavebeenabletoshowhereusingasimple,abstractagent-basedmodelthatrepresentsindividualparentsindisaggregatedmanner,andwhichwasnotimmediatelyapparentattheoutset,ishowconstraintsonindividuals'movements,whencombinedwithdistanceallocationrules,producewinnersandlosersthatarenotdirectlycorrelatedtotheindividuals'attributes.Thatis,itisnotagentswithlowestaspirationthatareleastsatisfiedwiththeirschoolallocationoutcomes,andinsteaditisparentagentswithaboveaverage,butnotveryhigh,aspirationthatfailtogettheirchildintotheirpreferredschoolmorefrequentlythanotherparents.Usingdisaggregated,agent-basedsimulationapproacheslikethisallowsinvestigationofindividual-leveloutcomesofsystemlevelpolicies.Wheninformedmoredirectlybyindividual-leveldata,andusedincombinationwithscenariosofdifferenteducationpolicies,thismodellingapproachwillallowustomorerigorouslyinvestigatetheconsequencesofthosepoliciesforeducationinequalitiesacrossspaceandthroughtime.

Acknowledgements

JMisgratefultotheLeverhulmeTrustforanEarlyCareerFellowshipheldduringthetimeoftheresearchpresentedhere.WearealsogratefulforthecommentsfromthreereviewerswhichhelpedtoimprovethemanuscriptandtoDavidDemerittwhoseideasinitiatedthiswork.TBandCHalsowishtoacknowledgethesupportoftheEconomicandSocialResearchCouncil(ESRC)whichfundedtheproject'Gentrification,ethnicityandeducationinEastLondon'(RES-000-23_0793)aswellasthecontributionofProfessorRichardWebberandDrMarkRamsdenandDrSadiqMirtotheoriginalprojectwhichgaverisetothefindingsthatinspiredthiscollaborationwithJM.

Notes

1Othercriteriasuchastheattendanceofsiblingsataschoolandspecialeducationalneedsarealsoconsideredbutinfluenceaveryminorproportionofallapplicants.

2Datafrom:DepartmentforEducation.SecondarySchoolGCSEPerformanceTables2010:BarkingandDagenham.HMSO.2011.URL:http://www.education.gov.uk/schools/performance/archive/schools_10/pdf_10/301.pdf.Accessed:2012-10-18.(ArchivedbyWebCite®athttp://www.webcitation.org/6BVNtJTe9);DepartmentforEducation.SecondarySchoolGCSEPerformanceTables2011:BarkingandDagenham.HMSO.2012.URL:http://www.education.gov.uk/schools/performance/2011/download/pdf/301_ks4.pdf.Accessed:2012-10-18.(ArchivedbyWebCite®athttp://www.webcitation.org/6BVNziv1Z);LondonBoroughofBarkingandDagenham.TheRightSecondarySchool:Informationforparentsaboutmovingtosecondaryschoolsin2013.LondonBoroughofBarkingandDagenham.2012.URL:http://www.lbbd.gov.uk/Education/Admissions/Documents/RSS2013.pdf.Accessed:2012-10-18.(ArchivedbyWebCite®athttp://www.webcitation.org/6BVOM2pTi);LondonBoroughofBarkingandDagenham.TheRightSecondarySchool:Informationforparentsaboutmovingtosecondaryschoolsin2012.LondonBoroughofBarkingandDagenham.2011.URL:http://www.lbbd.gov.uk/Education/Admissions/Documents/RSS2012.pdf.Accessed:2012-10-18.(ArchivedbyWebCite®athttp://www.webcitation.org/6BVOUK84I)

3http://www.openabm.org/model/3364/version/1/

4Inrealschools,pupilsinyear11maybeaged15or16dependingontheirbirthdate.However,thetemporalresolutionofthemodelisoneyearandchildagesareupdatedsimultaneouslysoweassumepupilsareaged11duringschoolyear7,12duringyear8,etc.untilbeingage15duringyear11.

5Notethatrankingstrategiesforbothmovingandapplicationincludesituationsinwhichparentsdonotconsideranyschoolssatisfactorytosendtheirchildto.Inthisunsatisfactorysituationintherealworld,parentsmayhavethemeanstomovetoalocationoutsidetheircurrentLEAwheretheythinktheytheirchildwillgetaplaceatasatisfactorystateschool.Alternatively,iftheyhavethemeanstheymayremovetheirchildfromthestateschoolsystemandsendthemintoprivateschooling.Neitheroftheseoptionsisrepresentedbythecurrentmodelstructure,whichisessentiallyaclosedsystem.

6Inreality,schoolplacesareallocatedbytheLocalEducationAuthority(LEA)andnotbyindividualschools.However,theschool-drivenallocationprocedureusedinthemodelhereisconsistentwiththelogicusedbyanLEAanddoesnotrequiretheuseofancillarymodelobjectsotherthanschoolsandparents.

7Inthetablemismeanregressioncoefficientofalinearregressionbetweenthetwovariables,pisthemeannumberof

timestepsinwhichp>0.05fortherelationshipbetweenthevariables,andr2isthemeancoefficientofdeterminationforthelinearregressionmodel.Allvaluesarefor80timestepsin25modelreplicates.

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8Inthetablemismeanregressioncoefficientofalinearregressionbetweenthetwovariables,pisthemeannumberof

timestepsinwhichp>0.05fortherelationshipbetweenthevariables,andr2isthemeancoefficientofdeterminationforthelinearregressionmodel.Allvaluesarefor80timestepsin25modelreplicates.

9Notethatwepresentonlysevencombinations,asthecombinationwithallthreeassumptionstrueisequivalenttoRuleSet1inTable3.

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