Stochastic user equilibrium with a bounded choice model

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Stochastic user equilibrium with a bounded choice model

Watling, David Paul; Rasmussen, Thomas Kjær; Prato, Carlo Giacomo; Nielsen, Otto Anker

Published in:

Transportation Research. Part B: Methodological

Link to article, DOI:

10.1016/j.trb.2018.05.004

Publication date:

2018

Document Version

Publisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):

Watling, D. P., Rasmussen, T. K., Prato, C. G., & Nielsen, O. A. (2018). Stochastic user equilibrium with a

bounded choice model. Transportation Research. Part B: Methodological, 114, 254-280. DOI:

ContentslistsavailableatScienceDirect

Transportation

Research

Part

B

journalhomepage:www.elsevier.com/locate/trb

Stochastic

user

equilibrium

with

a

bounded

choice

model

David Paul Watling

a,∗

_{, Thomas Kjær Rasmussen}

b

_{, Carlo Giacomo Prato}

c

_,

Otto Anker Nielsen

b

a Institute for Transport Studies, University of Leeds, 36-40 University Road, Leeds LS2 9JT, United Kingdom

b Department of Management Engineering, Technical University of Denmark, Bygningstorvet 116B, 2800 Kgs. Lyngby, Denmark c School of Civil Engineering, The University of Queensland, Brisbane 4072, Queensland, Australia

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 5 May 2017 Revised 4 April 2018 Accepted 4 May 2018 Keywords:

Stochastic user equilibrium Route choice

Equilibrated choice sets Bounds in choice models Random utility

a

b

s

t

r

a

c

t

Stochastic User Equilibrium (SUE) models allow the representation of the perceptual and preferential differences that exist when drivers compare alternative routes through a trans- portation network. However, as an effect of the used choice models, conventional applica- tions of SUE are based on the assumption that all available routes have a positive prob- ability of being chosen, however unattractive. In this paper, a novel choice model, the Bounded Choice Model (BCM), is presented along with network conditions for a corre- sponding Bounded SUE. The model integrates an exogenously-defined bound on the ran- dom utility of the set of paths that are used at equilibrium, within a Random Utility Theory (RUT) framework. The model predicts which routes are used and unused (the choice sets are equilibrated), while still ensuring that the distribution of flows on used routes accords to a Discrete Choice Model. Importantly, conditions to guarantee existence and uniqueness of the Bounded SUE are shown. Also, a corresponding solution algorithm is proposed and numerical results are reported by applying this to the Sioux Falls network.

© 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/)

1. Introductionandmotivation

Formanydecades,thetwo dominantapproachesformodellingtransport networkequilibriumhavebeenDeterministic UserEquilibrium(DUE;Wardrop,1952)andStochasticUserEquilibrium(SUE;Daganzo&Sheffi,1977),includingvariantsof thebasicmodelstohandleissuessuchasroutecorrelation,time-dependentcongestion,multipleclassesoftraveller/vehicle, risk-relatedphenomenaandnon-additivetravelcosts. Inspiteofthesemanyadvances,thebasicassumptions thatremain provideachoicebetweentwoextremecases,namelyi)thatonlyrouteswithminimumcost areusedinDUE,andii)that allpossibleroutesareusedinSUEregardlessoftheircosts.

Inarecentpaper(Watlingetal.,2015),weillustratedtheimplausibilityoftheseextremeassumptionsbyconsideringthe unusedroutesinahighly-convergedDUEsolutionoftheSiouxFallsnetwork(LeBlancetal.,1975).Foreachdistinctunused route,wecalculatedtherelativetravelcostwithrespecttotheDUEtravelcostonthatroute’sODmovement.Thefrequency distributionoftherelativetravelcostsforunusedroutes,acrossallODmovements,ispresentedinFig.1.

WeparticularlyhighlighttwofeaturesfromFig.1thatmotivateourstudy:

∗ _{Corresponding author.}

E-mail address: [email protected] (D.P. Watling).

https://doi.org/10.1016/j.trb.2018.05.004

0191-2615/© 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

Fig. 1. Frequency distribution of relative travel costs of unused DUE paths for Sioux Falls network (route cost on route r for OD-relation m, c mr , relative to

cost on minimum cost route for OD-relation m ( c m, min )).

i) therearemanyunusedrouteswithatravelcostonlyslightlyhigherthanthecostoftheusedroutes,i.e.witharelative travelcostneartoone;

ii)there are many unused routeswith a travel cost morethan twice ashigh asthe cost ofthe used routes, i.e.with a relativetravelcostgreaterthantwo.

Pointi)exemplifies theimplausibilityoftheassumptioninDUEthatnosuch pathwouldactuallybe used.Thisseems unreasonablegivenimperfectionsindrivers’knowledge(evenifobtainedthroughcontemporaryinformationsystems),inthe modeller’sknowledgeofthefactorsthat motivate driverpreferences,andgiventhenaturalvariationsinreal-lifesystems. Conceptualunderstandingoftheunderlyingbehaviouralprocessessupportsthiscriticism:travellershavelimitationsinthe consideration ofalternative routes prior to choosingtheir preferredroute (Bovyand Stern,1990; Bovy, 2009) and, most relevantly, they considerspatio-temporal constraintsfor limitingthe considerationset (Papinski etal., 2009; Kaplan and Prato,2012).Moreover, empirical evidence supports thiscriticism: onlya fraction ofcommuters were observedto select the shortestpath (by distanceor travel time) in Copenhagen(Nielsen, 1996,2004), Lexington (Jan et al., 2000), Nagoya (Morikawaetal.,2005),Boston(Bekhoretal.,2006),Turin(PratoandBekhor,2006),andMinneapolis(Zhu,2011).

Therehavebeenseveralapproachesproposedintheliteratureforaddressingdeficiencyi);however,theyall(inourview) haveundesirable consequences.Withtheaimofclarifyingwhattheseconsequencesare,andatthesametimemotivating ourcurrentstudy,weneedfirsttoanalyseeachoftheseapproachesindetail.Wegrouptheapproachesintofourcategories: (I) ConventionalSUEmodels:Inthefirstclassofapproaches,deficiencyi)isaddressedbyadoptinganSUEmodel,based on conventional Random Utility Theory (RUT) distributions withunbounded errorterms (such asthat underlying thelogitandprobitfamiliesofmodels, forexample).This,however,then raisespoint ii), sinceall routeswill then attractsomeflow,regardlessofhowimplausibletheyare.Thisisatoddswithourunderstandingofthebehavioural processesdrivers arecapable ofadopting,astravellers havespatiotemporal constraintsthat limit the consideration ofalternativeroutes(Papinski etal., 2009;KaplanandPrato,2012)aswellaslimitationstotheircognitivecapacity (BovyandStern,1990;Bovy, 2009;Gao etal., 2011).It isalsoatodds withempiricalevidence: commutershavea limitinthe excessdistanceortraveltime withrespectto theminimumcostpath, withheterogeneous preferences goingfromlarge variationsinCopenhagen,Lexington andNagoya(Janetal., 2000; Nielsen,2004;Morikawa etal., 2005)to smallvariationsinMinneapolisandTurin (Zhu,2011; KaplanandPrato,2012).Thisis alsoexemplifiedin

Fig.2,whichisbasedon16,618GPSobservationscollectedamongcartravellersintheGreaterCopenhagenAreaover an extendedperiod oftime (Rasmussen etal., 2017).The figure illustrates thecumulative shareof observationsas afunction ofthe ratiobetweenthecost on theobserved path(path obtainedfromGPS data) andthe coston the minimumcost pathbetweenthe corresponding locations. As can be seen,only 1.8%of thetrips are madeusing a paththat is morethan 50% morecostly than the corresponding shortest. Alsonote that approximately50% of the tripsusetheshortestpath.Itshould alsoberemarkedthatwhileitistruethatmanynumericalsolutionalgorithms forsolving SUEwill, after a finite numberof iterations, only identify a subset of the available routes, thisis not

Fig. 2. Cumulative share of observations as a function of the ratio between the cost on the observed path r, c r , and the cost on the corresponding min- imum cost path c r, min . The figure is based on 16,618 observed path choices obtained from a large GPS dataset collected among car drivers in the Greater

Copenhagen Area during 2011. Path costs of the observed/corresponding shortest paths are based on equilibrated network travel costs obtained from the Danish National Transport Model.

intendedasabehaviouralmechanisminherentinthemodel; whichroutesare identifiedwillvaryaccordingtothe implementationofthealgorithmused,initialconditions,orderoflink-andOD-cellcoding,andnumberofiterations, ratherthandependingonanythingfundamentalinthemodel(see,e.g.,Bekhoretal.,2008).

(II) Choicesetpre-generationmethods:InordertoaddresstheproblemthatconventionalSUEmodelsidentifytoolargea subsetofusedroutes,asecondclassofapproacheshasbeentoapplytheSUEmodeltoonlyasubsetoftheavailable routes.ThissubsetisidentifiedinadvanceofrunninganSUEsolutionalgorithm,throughtheuseofanexplicitchoice setgenerationmethod(e.g.,Friedrichetal.,2001;PratoandBekhor,2006;Bovy,2009;Frejingeretal.,2009).While it addressesdeficienciesi) andii)above, itdoes soat asignificant price, leadingto newdisadvantages, including: inconsistency,namely thelevel-of-service values (e.g.,travel times) neededasinput to the pathgeneration will be differentfromthosethatariseafterrunningtheequilibriumalgorithmandthebehaviouralassumptionsofthechoice setgenerationmightdifferfromthoseunderlyingthechoicemodel;andpolicyinsensitivity,namelypolicy measures testedwiththeequilibriummodelmaymakeattractivesomenewoptionsthatwereunattractive(andsonot inthe pre-definedrouteset)withoutthepolicymeasure.

(III) Boundedly Rational models: In the field of route choice, the theory of bounded rationality has been interpreted as establishing theexistence of ‘indifferencebands’ onthe excesscost of aroute relative to the minimumcost route (Mahmassani and Chang, 1987; Hu and Mahmassani, 1997; Jayakrishnan et al., 1994; Mahmassani andLiu, 1999; SrinivasanandMahmassani,1999). ABoundedlyRationalUser Equilibrium(BRUE) hasbeenproposed asaspaceof flowsolutionsrepresentingdrivers’inertiatoroute-switching,withinwhichdriversare indifferenttoroutecost dif-ferenceswithintheirindifferencebands(MahmassaniandChang,1987;GuoandLiu,2011;Louetal.,2010;Dietal., 2013;DiandLiu,2016).TheresultingBRUEsolutionsetwilltypicallycontainflowsolutionsinwhichdemandis di-videdbetweenusedandunusedroutes(thusaddressingissueii)above).InasensetheBRUEapproachalsoaddresses issuei),sinceitadmitsflowsolutionsinwhichsub-optimalroutescouldappear,whilenotrulingout DUEsolutions. Thus, BRUEcan beseen asakindofweakeningofthe strictrequirementsofDUE.The pricepaidforthis weaken-ing,however,isthatwenolongerobtainpointestimatesofequilibrium,butwhatmightbecalledintervalestimates, though(unlikewithstatisticalintervalestimates)themodelgivesnohigherweighttoanypossibilitywithin the in-terval.Thisdeliberatenon-uniquenessresultsinwhatwebelievetobesignificantdifficultiesinusingthismodelfor trafficforecasting,policytestingandcost-benefitanalysis.

(IV) Endogenouschoicesetrestrictionmethods:ReturningtoSUE-typemodels,onerecentstreamofresearchhasconsidered addingadditionalrestrictionsand/or constraintstoaconventionalSUEformulation,inordertoarriveatamodelin whichitispossiblethatnotallroutesareused.Inourownearlierresearch,weproposedaformof‘restricted’SUEin whichthechoicesetofusedroutesisdeterminedbysome explicitconstraintthatisdependent ontheequilibrium solution(Watlingetal.,2015;Rasmussenetal.,2015).Werecentlyfurtherextendedthisapproachtoincludeasecond restriction,determinedpartlyby anexogenously definedcostthreshold(Rasmussenetal., 2017).Ina parallelpiece ofresearchinasimilarspirit,PelandChaniotakis(2017)recentlyproposedamodifiedformofSUEinwhichallused

routesmusthaveaflowaboveauser-definedflow-threshold.Thedisadvantagewithall theseapproachesisthatwe areunabletoguaranteethatequilibriumsolutionsevenexistinallcases,asindeeditispossibletodevelopexamples wheretheydonot.Evenincaseswherewe haveexistence,wehavenoguaranteeoftheuniquenessofthesolution. Again, thislack of theoretical guarantee of existence and uniqueness is a major priceto pay when one considers thetypicaluseofsuch modelsinpolicytesting,meaningthat wecannotguaranteetoattribute auniqueforecasted benefittoanytestedmeasure.

Takingtogetheralloftheperspectivesdiscussedabove,wesetoutthefollowingrequirementsfortheequilibrium mod-ellingapproachthatwepresentinthispieceofresearch:

• Itshouldneitherexcludeallsub-optimalroutes(asinDUE)norincludeallavailablealternatives(asintheconventional SUEmodels).

• Itshouldprovideapointestimateofequilibrium,akintoDUEandSUE,unliketheintervalestimateapproachofBRUE.1 • It should take the attractive feature of DUEin its ability to address choice set equilibration simultaneously with the

flow equilibrium, andconsistentlywiththedeterminationof potentialchoice sets.Thatisto say,it shouldprovidean equilibriumpredictionoftheused/unusedroutesthatisconsistentwiththetravelcosts.

• Like SUE, it should draw on RUT to represent travellers’ misperceptions in travel costs and the significant empirical evidenceofdrivers’responsivenesstofactorssuchastraveltimethattypicallymotivateroutechoice.

• Itspropertiesshouldallowtheestablishmentoftheoreticalconditionstoguaranteebothexistenceanduniquenessofthe resultingequilibriumsolution.

Accordingly,wefirstintroduceanovelclosed-formchoicemodelthatincorporatesboundswithinadiscretechoicemodel, wethendefineequilibriumconditionscorrespondingtothischoicemodel,andlastweproveexistenceanduniquenessofthe solution.Theequilibriumconditionsleadustoasolutionalgorithmfortheconsistentequilibrationofboththeflowandthe choicesets.

Thestructureofthepaperfollowsthepresentationofourmaincontributions.WeintroduceinSection2thefundamental componentofourformulation,namelywereviewexistingliteratureconcerningthresholdsinRandomUtilityModels(RUM) beforedescribing thedetails ofournovelchoice modelupon whichourwork isbased.InSection 3we develop network equilibriumconditionsforthenewlyintroducedchoicemodel,andgoontoprovetheexistenceanduniquenessproperties ofthemodel.WepresentinSection4numericalresultsofapplyingasolutionalgorithmforthismodeltoasmallnetwork, aswell astheSiouxFalls network.Withtheseexperimentswe highlightsomecharacteristics ofthenewmodelproposed aswell asgain insights intothe sensitivityofthe modelto input parameters.Finally, we presentconclusionsandfuture researchdirectionsinSection5.

2. Theoreticalapproachtochoicemodellingwiththresholdsorbounds

In the present section we introduce a fundamentalelement of our study, namely how the notion of a threshold or boundmightbeincorporatedwithinadiscretechoicemodel.Throughoutthispaperthefocusisonclosed-formmodels.In

Section2.1,webeginwiththesimplestcaseofaproblemwithtwoalternatives,andpresentanddiscusstherelativemerits ofalternative formulationsoflogit-inspired modelsincorporating thresholdsor bounds.From thisdiscussion, we identify themostpromisingapproachtotakeforward,andinSection2.2wegeneralisethisapproachtothecaseofanynumberof choicealternatives.Notably,Section2alsoservesanimportantroleofallowingustopositionourworkinexistingliterature, andtodistinguishournovelmodelfrompreviouslyproposedones.

2.1. Problemswithtwoalternatives 2.1.1. Thresholdindifferencemodel

One of the most relevant existing studies, which motivated some of our own work below, is that reported by

Krishnan(1977),who introduced the concept ofa minimumperceivable differencewithin a binarylogit formulation, in-spiredbyideasfromboundedrationality.KrishnanreferredtothisasIncorporatingThresholdsofIndifferenceinProbabilistic ChoiceModels,andhenceforbrevitywenamethistheThresholdindifferencemodel.Particularfeaturesofthisapproachare that:

• thethresholdisappliedtothedifferenceinrandomutilities;

• it isassumedthatindividualsdisplay differentkindsofbehaviour dependingon thethreshold:namelythey are indif-ferenttochoicesthathavearandom utilitydifferencewithin the“indifferenceband” definedby somegiventhreshold tolerance, whereas they behave accordingto a logit choice model when the random utility difference isoutside this indifferenceband;and

• an additionalparameteris introducedtorepresentthe conditionalprobability ofchoosingone ofthetwoalternatives, giventhatthechoiceshavea(random)utilitydifferencewithintheindifferenceband.

1 We recognise that interval estimates are also valuable. For models of the type we shall describe, these could also be developed from point equilibrium

Therefore,inaddition tothe logitutility parameters,two newparameters areintroduced (thatcan be estimatedfrom data),namelyoneforthethresholdthatdefinestheindifferenceband,andone fortheconditionalprobabilityofchoosing alternativeA1whentherandomutilitydifferenceiswithintheindifferenceband.

Letthe alternatives be denotedA1 andA2,the randomutilities U1 andU2,the systematicutilities V1 and V2,andlet

δ

denotewhatKrishnancallstheminimumperceivabledifference(MPD). Thesymbol denotes“is preferredto” andthe symbol∼ denotes“isequallypreferredas”.Thepreferencemodelisthenpostulatedas:

(i) A1  A2ifU1>U2+

δ

; (ii) A2  A1ifU2>U1+

δ

; (iii) A1 ∼ A2if|U1 –U2|≤

δ

.

BasedonaRUM,wecanthendefinetheclassificationprobabilitiesintothestates(i),(ii)and(iii)aboveas

π

1=Pr

(

A1 A2

)

=Pr

(

U1>U2+

δ

)

π

2=Pr

(

A2 A1

)

=Pr

(

U2>U1+

δ

)

π

12=Pr

(

A1∼ A2

)

=Pr

(

|

U1− U2

|

≤

δ

)

Forthecaseofabinarylogitmodelwithscaleparameter

θ

,wethenhave:

π

1=

(

1+exp

(

θ

(

δ

+V2− V1

)

)

)

−1

π

2=

(

1+exp

(

θ

(

δ

+V1− V2

)

)

)

−1

π

12=1−

π

1−

π

2

Itisfurtherassumedthatwheneverthealternativesareintheindifferenceband,althoughthereisnoutility-based pref-erence,achoiceisstillmade(butthechoicedoesnotdependontheutilities).Thisisdefinedbyincorporatinganadditional parameter

α

whichisequaltotheconditional probabilityofchoosingalternativeA1 giventhattheutilitiesarewithinthe indifferenceband.Thepostulatedmodelisthenoperationalisedbytranslatingthepreferencesintochoiceprobabilitiesas:

(i) ifA1  A2thenA1willbechosenwithprobability1; (ii) ifA2  A1thenA2willbechosenwithprobability1;

(iii) ifA1 ∼ A2thenA1 andA2 willbechosenwithprobabilities

α

and1–

α

respectively.

ThefinalprobabilitiesofchoosingalternativesA1andA2,denotedrespectivelyby

π

1∗and

π

2∗,arethengivenby:

π

∗

1 =

π

1+

π

12

α

π

∗

2 =

π

2+

π

12

(

1−

α

)

(

=1−

π

1∗

)

Combiningthesetogether,whatweeffectivelycomeupwithisan alternativewayofmappingthechoiceprobabilityof alternativeA1tothedifferenceinsystematicutility:

π

∗

1 =f1

(

V1− V2;

θ

,

δ

,

α

)

wherethesmoothfunctionf1 isgivenby

f1

(

x;

θ

,

δ

,

α

)

=

(

1+exp

(

θ

(

δ

− x

)

)

)

−1+

α



1−

(

1+exp

(

θ

(

δ

− x

)

)

)

−1₋

₍

_1+exp

₍

_θ

₍

_δ

_+x

₎

₎

₎

−1



_.

Itshouldbenotedthat wewritethefunctionaboveasoperatingonthesingledummyargumentxinordertosimplify theexpressionandtohighlightthekeyelementsofthefunction,ratherthanwhatissubstitutedintothefunction. 2.1.2. Thresholdchoicesetmodel

Krishan’sapproach,asdescribedabove,doesindeedincorporatetheconceptofathresholdintoaRUMformulation,but whatitendsupwithissomewhatdifferentfromtheoriginalobjectivesofourstudy.AsexplainedinSection1,our motiva-tionistoconsiderformulationsthatdonothavethepropertyofassigningnon-zeroprobabilitytoallavailablealternatives whateverthesystematicutilities.Inaddition,its motivationismorecloselyrelatedtoboundedrationalitythanours is,in that Krishan’smodelpostulatesthat individualsareratherindifferenttothe systematicutilities ofthealternatives within thethreshold,whereas ourinterest isinstill usinga conventional RUMtoassign choiceprobabilities toalternatives. The wayinwhichKrishnanpresentsthemodelassumptionsisappealing,however,andinthissub-sectionandthefollowingwe proposetwoalternativemodelsusingasimilarpresentationstyle.

• athresholdisappliedtothedifferenceinsystematicutilities;

• an option willonlybe inthe choice setofconsidered alternatives ifitis nomorethan a thresholdamountworse in systematicutility thantheother alternative,otherwiseit iscompletelydominatedby theotheralternative (writtenA1  A2 orA2  A1);

• if both options are in the choice set (sothe alternatives are competitive, which we denote A1A2), then individuals choosebasedonaregularRUM.

Thepostulatedpreferencemodelisthus: (i) A1 A2ifV1>V2+

δ

;

(ii) A2 A1ifV2>V1+

δ

; (iii) A1A2if|V1–V2|≤

δ

.

Theclassificationintothethreestatesi),ii)andiii)abovediffersfromKrishnan’sinthatitisbasedonsystematicutility ratherthanrandomutility,andsoitisnotprobabilistic.Thusinourcaseweobtainclassificationprobabilitiesof

π

1=Pr

(

A1 A2

)

=



1 ifV1>V2+

δ

0 otherwise

π

2=Pr

(

A2 A1

)

=



1 ifV2>V1+

δ

0 otherwise

π

12=Pr

(

A1A2

)

=1−

π

1−

π

2.

Sincethethresholdisappliedtothesystematicutilitydifference,itisstillconsistenttoassumeunboundedrandom util-itydifferences,andsowemaysimplyadoptabinarylogitmodelfortheA1A2 case,toobtainthefinalchoiceprobabilities of:

π

∗

1=

π

1+

π

12

(

1+exp

(

−

θ

(

V1− V2

)

)

)

−1

π

∗

2=1−

π

1∗

whichwhenallcasesarecombinedgives:

π

∗ 1=



_{1 ifV} 1>V2+

δ

0 ifV2>V1+

δ

(

1+exp

(

−

θ

(

V1− V2

)

)

)

−1 otherwise

π

∗ 2=1−

π

1∗.

AswiththemodelinSection2.1.1.,wecanthereforecapturethismodelthroughthemappingofthechoiceprobability ofalternativeA1tothedifferenceinsystematicutility(withoneparameterfewerinthiscase):

π

∗ 1= f2

(

V1− V2;

θ

,

δ

)

where: f2

(

x;

θ

,

δ

)

=



_{0 ifx<−}

_δ

(

1+exp

(

−

θ

x

)

)

−1 if

|

x

|

≤

δ

1 ifx>

δ

.

2.1.3. Boundedchoicemodel

Oneof thekey differencesbetweenKrishnan’sapproach inSection 2.1.1and our approach in Section 2.1.2is that the formerconsiders therandom utilitiesforanythresholdoperation whereas thelatterconsiders thesystematicutilities. In the presentsection, we return tothe behavioural principlesof 2.1.2, butinstead develop a modelbased on the random utilitiesasin2.1.1.Thusunlike2.1.1,wemodelthefactthatsomealternativeshavezerochoiceprobability,butunlike2.1.2 we derive the modelfrom RandomUtility Theory(RUT) principles. This isequivalent, aswe see below,to bounding the errordistributionofrandomutilitydifferences(betweenan alternativeandareferencealternative),andforthisreasonwe shallrefertoitasaBoundedChoicemodel.Neitherofthemodelspresentedearlierpossessthisproperty,sincethatin2.1.1 adoptsanunboundeddistribution,andthatin2.1.2isnotderivedfromRUT.Thus,ourterminologyforthemodeldescribed belowistorefertoa‘bound’ratherthana‘threshold’.Wealsobelievethisterminologytobeappropriatefromafunctional perspective,inthat athresholdisgenerallyusedasa(weaker)termtodistinguishpointsatwhichthebehaviour ofsome model/function changes(possiblyin a discontinuousway),whereas our boundingapproachcan be viewedas adaptinga singlemathematicalformbyuseoflowerandupperboundingfunctionsatchoiceprobabilitiesof0and1.

• aboundisappliedtothedifferenceinrandomutilitybetweeneachgivenalternativeA_i(withrandomutilityU_i)andan imaginary“referencealternative” A∗withrandomutilityU∗(weshallfirstderiveageneralcase,2 _{thenaspecialcaseof}

thisreferencealternative);

• anoptionwillonlybe inthechoicesetofconsideredalternatives ifthedifferencebetweenitsrandomutility andthe randomutilityofthereferencealternativeiswithinthebound;asaresult,eitherbothoptionswillbeinthechoiceset, orotherwisethatoptionwillbecompletelydominatedbytheother (real)alternative(i.e.inthe‘otherwise’case,either A1  A2 ifoption2isoutsidethebound,orA2 A1 ifoption1isoutsidethebound);

• ifbothoptionsare inthechoiceset(A1  A2),then individualschoose accordingto theoddsassociatedwiththetwo binarychoiceprobabilitiesofA1versusA∗andA2versusA∗.

AkeytechnicaldifferencefromthemodelinSection 2.1.2isthatinordertoimplementthechoicefortheA1A2 case, weneedtoconsidertheconditionaldistributionofrandomutilitydifferences,giventhattheboundingconditionismet(in 2.1.2,the conditionalandunconditionaldistributions coincide,sincethethresholdapplies onlytothesystematicutilities). Thistechnicalfeatureisconsistentwithabehaviouralassumptionthattravellersmakedecisionsintwostages.Firstlythey compare eachalternative ina pairwise manner tothe referencealternative,which throughthe boundingonthe random utility difference provides an implicitmechanism for choice set generation. This first stage corresponds to the first two firstbulletpointsinthe‘features’listedabove.Secondly,andcorrespondingtothethirdbulletpoint above,theycompare thealternatives inthechoicesetfromthefirststageto maketheirchoice,whereimportantlytheproportional useofthe alternativesismodelledusingthelog-oddsfromthefirststage.Thisuseofthelog-oddsinthiswaymeansthatchoiceset generationandchoicearehandledconsistentlybythesameunderlyingbehaviouralmodel.

Thepostulatedpreferencemodelisthus: (i) A1  A2ifU∗>U2+

δ

andU∗≤ U1+

δ

; (ii) A2  A1ifU∗>U1+

δ

andU∗≤ U2+

δ

; (iii) A1A2ifU∗≤ U2+

δ

andU∗≤ U1+

δ

.

Inordertooperationalisesuchapreferencemodel,wefirstpresentaderivationofthestandardbinarylogitmodelthat canthensubsequentlybeadaptedtodealwiththeboundingcondition.

Now,letussupposeaspecificationofrandomutilities Uj=

θ

Vj+

ε

j

(

j=1,2;

θ

>0

)

.

AccordingtoRUT,thechoiceprobabilitiesaregivenby

Pr

(

chooseA1 from

{

A1,A2

}

)

=Pr

(

U1≥ U2

)

=Pr

(

θ

V1+

ε

1≥

θ

V2+

ε

2

)

=Pr

(

ε

2−

ε

1≤

θ

(

V1− V2

)

)

=F

(

θ

(

V1− V2

)

)

where F(·) denotes the cumulative distribution function (c.d.f.) of the random variable

ε

2−

ε

1.Thus, our random utility approachprovidesamodelthatultimatelymapsthedifferenceinsystematicutilityV1− V2tothechoiceprobabilitythrough thec.d.f.oftherandomutilitydifferences.Inthespecialcasethat(

ε

1,

ε

2)arei.i.d.Gumbelrandomvariables,itiswellknown thatweobtainthefunctionF(

θ

x)≡ f0(x;

θ

)givenbythebinarylogitmodel:

f0

(

x;

θ

)

=

(

1+exp

(

−

θ

x

)

)

−1

(

−∞<x<∞

)

.

Wenowpresentanalternativewayofderivingthissame(standardlogit)choicefunction,theadvantageofthisderivation being that it may be readily modifiedto incorporate a bound. This alternative formulationis derived by introducing an imaginary“referencealternative” A∗withrandomutilityU∗givenby:

U∗=

θ

V∗+

ε

∗

wherewenowassumethat(

ε

1,

ε

2,

ε

∗)arei.i.d.Gumbelrandomvariables.Then,bythesameargumentasabove,foreachj inturn: Pr



chooseAjfrom



Aj,A∗



=f0



Vj− V∗;

θ



(

j=1,2

)

. Tomakesomenotationalsimplification,letusnowdefine:

x=V1− V2 y=V1− V∗ z=V2− V∗

whereclearlyx₌y_{− z}.Thelog-odds(logoftheoddsratio)

η

_jforalternative A_jversusA∗isthen(j₌1,2):

η

1=ln



f0

(

y;

θ

)

1− f0

(

y;

θ

)

=ln



(

1+exp

(

−

θ

y

)

)

−1 1−

(

1+exp

(

−

θ

y

)

)

−1

=

θ

y

η

2=ln



f0

(

z;

θ

)

1− f0

(

z;

θ

)

=

θ

z.

2 We suppose always that the imaginary alternative is defined so that at least one alternative is always in the choice set (so we can neglect the final pos-

sibility below that U ∗_{> U}

1 + δand U ∗> U 2 + δ). This condition is automatically satisfied when we suppose (as we shall subsequently) that the imaginary

ThechoiceprobabilityforalternativeA1 from{A1,A2}isthengivenby: exp

(

η

1

)

exp

(

η

1

)

+exp

(

η

2

)

=

exp

(

θ

y

)

exp

(

θ

y

)

+exp

(

θ

z

)

=

(

1+exp

(

−

θ

(

y− z

)

)

)

−1

=f0

(

y− z;

θ

)

=f0

(

x;

θ

)

(

sincex=y− z

)

.

Inthiscase,ascouldhavebeenforecastedfromthewell-knownIIA property,thechoiceprobabilitiesbetween alterna-tivesA1 andA2areindependentoftheutilityofthereferencealternative.

Thisalternativederivationofthestandardlogitchoicefunctionisnowmodifiedtoincorporateabound.Againwe intro-duceanimaginaryreferencealternativeA∗.ThekeydifferenceisthatthechoiceprobabilityforalternativeA1 from{A1,A∗} isnowsupposedtobegivenby

Pr

(

chooseA1from

{

A1,A∗

}

)

=g

(

y;

θ

,

δ

)

=

⎧

⎨

⎩

0 ify≤ −

δ

f0

(

y;

θ

)

− f0

(

−

δ

;

θ

)

1− f0

(

−

δ

;

θ

)

ify>−

δ

.

Thelogic ofthisfunction isthat, since y measuresthe extentto whichalternative A1 isworse than thereference al-ternative,thebreak-pointy=−δ correspondsto acasewhereA1isperceived tobeexactly

δ

≥ 0utility unitsworse than thereferencealternative.Fory≤ −δ,ourproposedboundingconditionassertsthat alternativeA1 willbeunused. Fory> −δ,on theother hand,the proportional sharebetweenA1 andA∗ is givenby a lineartransformationof thelogit choice function.Thislineartransformationistriviallyderivedfromthebasis ofRUT,inwhichgisacumulativedistribution func-tionofrandomutilitydifferences, andthetransformationsimplyscales thelogitchoicefunctionsothatitisaproperc.d.f. (i.e.itsums toone, notablyunlike themodel proposedin Section2.1.2).The second branchofthe choicefunction isthe conditionaldistributionofutilitydifferences,giventhaty>−δ.Aswell aspreservingthetheoreticallinktoRUT,wemay alsonotetheimportantproperty(forlateranalysis)thatgiscontinuous,evenaty₌−δ.

Similarly,thechoice probabilityforalternativeA2 from{A2,A∗}issupposed tobeg(z;

θ

,

δ

).Underthe boundedmodel, thelog-odds

λ

jforalternative AjversusA∗isthengivenby(forj=1,2):

λ

1=

⎧

⎨

⎩

−∞ ify≤ −

δ

ln



g

(

y;

θ

,

δ

)

1− g

(

y;

θ

,

δ

)

ify>−

δ

λ

2=

⎧

⎨

⎩

−∞ ifz≤ −

δ

ln



g

(

z;

θ

,

δ

)

1− g

(

z;

θ

,

δ

)

ifz>−

δ

. Now,fory>−δ: g

(

y;

θ

,

δ

)

1− g

(

y;

θ

,

δ

)

= f0

(

y;

θ

)

− f0

(

−

δ

;

θ

)

1− f0

(

−

δ

;

θ

)

1− f0

(

y;

θ

)

− f0

(

−

δ

;

θ

)

1− f0

(

−

δ

;

θ

)

= f0

(

y;

θ

)

− f0

(

−

δ

;

θ

)

1− f0

(

y;

θ

)

.

Underthisboundedmodel,thechoiceprobabilityforalternativeA1from{A1,A2}isthusgivenby:

exp

(

λ

1

)

exp

(

λ

1

)

+exp

(

λ

2

)

=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

0 ify≤ −

δ

f0

(

y;

θ

)

− f0

(

−

δ

;

θ

)

1− f0

(

y;

θ

)

f0

(

y;

θ

)

− f0

(

−

δ

;

θ

)

1− f0

(

y;

θ

)

+ f0

(

z;

θ

)

− f0

(

−

δ

;

θ

)

1− f0

(

z;

θ

)

ify>−

δ

andz>−

δ

=h

(

y,z;

θ

,

δ

)

1 ifz≤ −

δ

Now, f0

(

y;

θ

)

− f0

(

−

δ

;

θ

)

1− f0

(

y;

θ

)

= f0

(

y;

θ

)

1− f0

(

y;

θ

)

−

(

1+exp

(

θδ

)

)

−1 1−

(

1+exp

(

−

θ

y

)

)

−1 andaspreviouslynoted:

f0

(

y;

θ

)

1− f0

(

y;

θ

)

=

exp

(

θ

y

)

. Asforthesecondtermabove,

(

1+exp

(

θδ

)

)

−1 1−

(

1+exp

(

−

θ

y

)

)

−1 =

(

1+exp

(

θδ

)

)

−1

(

1+exp

(

θ

y

)

)

−1 = 1+exp

(

θ

y

)

1+exp

(

θδ

)

. Hence: f0

(

y;

θ

)

− f0

(

−

δ

;

θ

)

1− f0

(

y;

θ

)

= exp

(

θ

y

)

−1+exp

(

θ

y

)

1+exp

(

θδ

)

= exp

(

θ

(

y+

δ

)

)

− 1 1+exp

(

θδ

)

.

ThereforethegeneralchoiceprobabilitymodelforalternativeA1 versusA2givenreferencealternativeA∗whichisbetter thanA1 andA2byyandzutilityunitsrespectively,andgivenbound

δ

>0,is:

h

(

y,z;

θ

,

δ

)

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

0 ify≤ −

δ

exp

(

θ

(

y+

δ

)

)

− 1

(

exp

(

θ

(

y+

δ

)

)

− 1

)

+

(

exp

(

θ

(

z+

δ

)

)

− 1

)

ify>−

δ

andz>−

δ

1 ifz≤ −

δ

.

Again,itisnotablethathiscontinuous.Weavoidfurthersimplifyingtheexpressionabove,i.eweavoiddividing through-outbyexp(

θ

(y+

δ

))− 1),sinceaty=−δthistermwouldbezero,inorderthatintheformgivenwemayequallyconsider theboundarycaseswithinthesecondterm(e.g.foreasiercomparisonwithearlier-presentedmodels,aswellasmakingthe continuitymoreevident)as:

h

(

y,z;

θ

,

δ

)

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

0 ify<−

δ

exp

(

θ

(

y+

δ

)

)

− 1

(

exp

(

θ

(

y+

δ

)

)

− 1

)

+

(

exp

(

θ

(

z+

δ

)

)

− 1

)

ify≥ −

δ

andz≥ −

δ

1 ifz<−

δ

.

Theaboveformistheonethatweshallhenceforthtakeforwardasthemainfunctionalformforh.

We may alsoremark that thisgeneral model isclearly not independent ofthe utility of the referencealternative; in particular,itcannotbewrittenasafunctionofy− z(=x)only,asinthestandardbinarylogitmodel.

Withaslightrearrangement,wehaveathirdpossiblewayofexpressingthisfunction

h

(

y,z;

θ

,

δ

)

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

0 ify<−

δ

exp

(

θ

y

)

− exp

(

−

θδ

)

(

exp

(

θ

y

)

− exp

(

−

θδ

)

)

+

(

exp

(

θ

z

)

− exp

(

−

θδ

)

)

ify≥ −

δ

andz≥ −

δ

1 ifz<−

δ

andinthisformitisevidentthat,as

δ

→∞,soexp(−

θδ

)→0for

θ

>0,andsointhelimitwerecoverthestandardbinary logitmodel,asmightbeexpected.

Now,weshallconsideraspecialcaseofthismodelinwhichthereferencealternativehassystematicutility: V∗=max

(

V1,V2

)

.

Wenowconsidertwopossiblecases:

i) WhenV1 ≥ V2 (i.e.whenx≥ 0),itfollowsthatV∗=V1.Hencey=V1− V∗=0andz=V2− V∗=V2− V1=−x.Thatis:≥ 0⇒y=0andz=−x.

ii) WhenV2 ≥ V1 (i.e.whenx ≤ 0),it followsthat V∗=V2 . Hencey=V1− V∗=V1− V2=xandz=V2− V∗=0.Thatis: ≤ 0⇒y=xandz=0.

Nowincasei),withy=0,wecanneverhavey<−δsatisfiedif(asitisassumed)

δ

≥ 0,andsointhiscaseweobtain (withy₌0andz_=−xforx_{≥ 0}substitutedinthe‘mainfunctionalform’forhnotedabove):

h

(

0,−x;

θ

,

δ

)

=

⎧

⎨

⎩

exp

(

θδ

)

− 1

(

exp

(

θδ

)

− 1

)

+

(

exp

(

θ

(

−x+

δ

)

)

− 1

)

if0≤ x≤

δ

1 ifx>

δ

.

Incaseii),sincez=0,wecanneverhavez<−δ satisfiedwith

δ

≥ 0,andsointhiscaseweobtain(withy=xandz=0 forx≤ 0): h

(

x,0;

θ

,

δ

)

=

⎧

⎨

⎩

0 ifx<−

δ

exp

(

θ

(

x+

δ

)

)

− 1

(

exp

(

θ

(

x+

δ

)

)

− 1

)

+

(

exp

(

θδ

)

− 1

)

if −

δ

≤ x≤ 0 .

Combining thesecaseswe obtaina choiceprobability modelforalternative A1 versusA2 whichis independentofthe referencealternativeA∗,dependingonlyontheutilitydifferencexbetweenA1 andA2 andonthebound

δ

>0relativeto

themaximumutilityalternative: f3

(

x;

θ

,

δ

)

=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

0 ifx<−

δ

exp

(

θ

(

x+

δ

)

)

− 1

(

exp

(

θ

(

x+

δ

)

)

− 1

)

+

(

exp

(

θδ

)

− 1

)

if −

δ

≤ x≤ 0 exp

(

θδ

)

− 1

(

exp

(

θδ

)

− 1

)

+

(

exp

(

θ

(

−x+

δ

)

)

− 1

)

if0≤ x≤

δ

1 ifx>

δ

.

Aratherobvious,butimportant,remarktomakeisthatwhilethegeneralformofourboundedmodelhpresentedearlier isnotexpressibleonlyasafunctionoftheutilitydifferencex,clearlythespecialcaseconsideredtoderivef3aboveis. 2.1.4. Comparisonandevaluationofcandidateapproaches

FromtheanalysisinSections2.1.1–2.1.3,wehavethreecandidateapproachestoconsider,alongsidethe‘null’choiceofa standardbinarylogitmodel:

BinaryLogitModel:

f0

(

x;

θ

)

=

(

1+exp

(

−

θ

x

)

)

−1

(

−∞<x<∞

)

ThresholdIndifferenceModel:

f1

(

x;

θ

,

δ

,

α

)

=

(

1+exp

(

θ

(

δ

− x

)

)

)

−1+

α



1−

(

1+exp

(

θ

(

δ

− x

)

)

)

−1₋

₍

_1+exp

₍

_θ

₍

_δ

_+x

₎

₎

₎

−1



ThresholdChoiceSetModel:

f2

(

x;

θ

,

δ

)

=

⎧

⎨

⎩

0 ifx<−

δ

(

1+exp

(

−

θ

x

)

)

−1 if

|

x

|

≤

δ

1 ifx>

δ

BoundedChoiceModel:

f3

(

x;

θ

,

δ

)

=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

0 ifx<−

δ

exp

(

θ

(

x+

δ

)

)

− 1

(

exp

(

θ

(

x+

δ

)

)

− 1

)

+

(

exp

(

θδ

)

− 1

)

if −

δ

≤ x≤ 0 exp

(

θδ

)

− 1

(

exp

(

θδ

)

− 1

)

+

(

exp

(

θ

(

−x+

δ

)

)

− 1

)

if0≤ x≤

δ

1 ifx>

δ

.

ThesefourdifferentapproachesareillustratedinFig.3,foranexamplewith

θ

₌0.5,

δ

₌4and

α

₌0.5.

Anobviousfeatureoff1 isthat— likef0 — itissmoothandasymptotically approaches0atits lowerendand1atits upperend;thus,bothalternativesalwayshaveanon-zeroprobabilityofbeingchosen.Modelf2isnotonlynon-smoothbut itisdiscontinuousatx_{= ±}

δ

.Modelf3,whilenon-smooth,isacontinuousmappingoveritsrange.As shownlater,these willbeimportantpropertiesinoursubsequentanalysis.

Modelsf2 andf3are appealinginthecontext oftheissuesidentified intheIntroduction,since theyallowforthe pos-sibilityofan alternativetobecompletelyunused(i.e.tohavezeroprobability).Modelf1sharesmoreincommonwiththe conceptofboundedrationality,asimplementedinBRUEmodelsforexample(see§1),sinceitpositsthatindividualsare rel-ativelyindifferenttochoiceswithutilitiesthatdifferbylessthanthethreshold(inthecaseoff1,‘relatively’meansrelative tothechoiceprobabilitiessuggestedbyabinarylogitmodel).Inpractice,thisisachievedbyaprobability-weightingofthe randomutility beinginsideoroutsidethethreshold,providingasmooth ‘flattening’oftheoriginal binarylogitmodel.In comparisonwithastandardbinarylogitmodel,theparticularfeatureofmodelf1thereforeistomakeaspecialtreatment ofalternativeslyingwithinthethreshold.Modelf2,ontheotherhand,isconcernedwithdefiningapoint(ontheutility dif-ferencescale)atwhichachoicebecomessounattractiverelativetoitscompetitor(s)thatthereiszeroprobabilityofitsuse; inthissenseits contributionismoreconcerned withwhat happensoutsidethethreshold.Modelf3issimilarto f2inthis respect,butadditionallyadjustschoiceprobabilitiesastheboundisapproachedinordertoprovideacontinuousmapping. Intuitively,then,itmaybebettertothinkofthemodelsoperatingonthresholdsorboundsthatdiffermarkedlyinmeaning andscale:f1 isa“smallthreshold” model,interestedinidentifyingrelativelyindistinguishableoptions,whilef2 andf3 are “largethreshold/bound” models,interestedinidentifyingoptionsthataresorelativelypoorcomparedwithalternativesthat (formodellingpurposes)wemightneglectthattheymaybechosen.

Asidefromthedifferentbehaviouralobjectivesofthesemodels,theyallprovideopportunitiesforimplementationwithin anetworkequilibriumcontext.However,basedontherequirementsidentifiedinSection1,weshallselectonlyoneofthem forfurtherdevelopment:

Fig. 3. Choice probability of choosing alternative 1 as function of difference in systematic utility between two alternatives, various approaches.

• Threshold Indifference Model: Krishnan’s model f1 is seemingly attractive to explore, as a kindof RUM counterpart to BRUE.The factthatthemappingf1 iscontinuousshould meanthat,understandard assumptionsonother elementsof the model,such equilibriacan be theoretically guaranteed to exist.However, some morework would be requiredon findinganefficient,tractableimplementationofthisapproachformorethantwoalternatives.Lioukas(1984)providesa formulationforthreealternatives,butthisisalreadyrathercomplexgiventhatitinvolvesacombinatorialproblemdue tothedifferentorderingsofthealternativesthatcanoccurwithintheindifferenceband.Weshallnot,however,further studyKrishnan’smodelinthe presentpaper,since itdoesnot accord withourrequirementinSection 1 toeliminate somepotentialalternativesfromthechoiceset.

• ThresholdChoiceSet Model: Modelf2 isattractive inthat it satisfiestherequirement todistinguish aroute asunused, whileadopting RUT. Thismodel is infact the basis ofan approach that wasgeneralised andimplemented within an equilibriumframeworkinRasmussen etal., (2017).Themajordisadvantage totheapproachisthediscontinuityofthe choicemodel;examplescanbeconstructedtoshowthatthisleadstonon-existenceofequilibriainsomecircumstances. Therefore thisdoesnot accord with therequirement that the approach should haveguaranteed existence/uniqueness properties,andsoitisnotconsideredfurther.

• BoundedChoiceModel:Thenewlyproposedmodelf3 isthemostattractiveinthatitsatisfiestherequirementto distin-guisharouteasunused,whileatthesametimeprovidingacontinuousmappingofthechoicemodel.Thusitprovides promiseatleastthatexistenceofequilibriamaybeproven.

In conclusion, then, the Bounded ChoiceModel is selected asthe approach to consider further in thecontext ofthe presentstudy,giventherequirementssetoutinSection1.

2.2. Problemswithmultiplealternatives

In the present section, we propose a generalisation of the two-alternative Bounded Choice Model presented in

Section 2.1.3 to the caseof n ≥ 2 alternatives. This is a relatively straightforward extension of the binarycase, and we willre-usemanyofthederivationsandintermediatefunctionsdefinedthere.

Themodelfeaturesarethat:

• aboundisappliedtothedifferenceinrandomutilitybetweenagivenalternativeandanimaginaryreferencealternative A∗;

• an optionwillonlybeinthechoice setofconsideredalternativesifthedifferencebetweenits randomutilityandthe randomutilityofthereferencealternativeiswithinthebound;

• for the subset ofconsidered alternatives, indexed by J ⊆

{

1,2,....,n

}

,the choice probabilities are given by theodds associatedwiththebinarychoiceprobabilitiesofA_jversusA∗for j_∈J.

Supposeaspecificationofrandomutilitiesfortherealandimaginaryalternativesof: Uj=

θ

Vj+

ε

j

(

j=1,2...,n

)

U∗=

θ

V∗+

ε

∗

where(

ε

1,

ε

2,…,

ε

n,

ε

∗)arei.i.d.Gumbel,3anddefine: yj=Vj− V∗

(

j=1,2,...,n

)

.

Then,combiningRUTwithaboundingconditiongivesus(followingthesamerationaleasforthebinarycase):

Pr



chooseAj from



Aj,A∗



=g



yj;

θ

,

δ



=

⎧

⎨

⎩

0 ifyj≤ −

δ

f0



yj;

θ



− f0

(

−

δ

;

θ

)

1− f0

(

−

δ

;

θ

)

ifyj>−

δ

(

j=1,2,...,n

)

. withthelog-odds

λ

_jforalternative A_jversusA∗of:

λ

j=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

−∞ ifyj≤ −

δ

ln



g



yj;

θ

,

δ



1− g



yj;

θ

,

δ





ifyj>−

δ

(

j=1,2,...n

)

. Inthebinarycaseweshowedthatwecanwrite:

g

(

y;

θ

,

δ

)

1− g

(

y;

θ

,

δ

)

=

exp

(

θ

(

y+

δ

)

)

− 1

1+exp

(

θδ

)

(

y>−

δ

)

andsotheoddsexp(

λ

j)foralternative AjversusA∗is: exp

(

λ

j

)

= 1 1+exp

(

θδ

)



0 ifyj≤ −

δ

exp



θ



yj+

δ



− 1 ifyj>−

δ

(

j=1,2,...n

)

whichcanalsobewritten:

exp

(

λ

j

)

= 1 1+exp

(

θδ

)



exp



θ



yj+

δ



− 1



₊

(

j=1,2,...n

)

. wherethe+operatoraboveissuchthat(x)₊=max(x,0).

Wethus obtaina choiceprobability modelforalternativeAjfrom{A1,A2,…An},givenreferencealternative A∗ whichis

betterthanA_kbyy_kutilityunits(k₌1,2,…,n),andgivenbound

δ

_>0: hj

(

y;

θ

,

δ

)

= exp



λ

j



n k=1exp

(

λ

k

)

=



exp



θ



yj+

δ



− 1



₊ n k=1

(

exp

(

θ

(

yk+

δ

)

)

− 1

)

₊

(

j=1,2,...,n

)

whereyisthen-vector(y1,y2,…,yn).

Thefinal stepis, asinthe binarycase, wherewe select a particular(logical) specificationofthe imaginaryreference alternative,whereby:

V∗=max

(

V1,V2,...,Vn

)

and then resubstituting for yj=Vj− V∗(j=1, 2, …, n) we obtain a choice probability model for alternative Aj from

{A1,A2,…An}givensystematicutilitiesV=(V1,V2,…,Vn)andbound

δ

>0: pj

(

V;

θ

,

δ

)

=



exp



θ



Vj− max

(

V1,V2,...,Vn

)

+

δ



− 1



+ n k=1

(

exp

(

θ

(

Vk− max

(

V1,V2,...,Vn

)

+

δ

)

)

− 1

)

₊

(

j=1,2,...,n

)

.

Asinthebinarycase, withaslightre-writingoftheexpression,itiseasilyseenthat,inthelimitas

δ

_→∞,thismodel approachestheconventionallogitmodel,aswouldbeexpected:

pj

(

V;

θ

,

δ

)

→ exp



θ

Vj



n k=1exp

(

θ

Vk

)

as

δ

→∞.

Inthe other limit as

δ

→0,the minimumcost routes becomedominant androutesonly slightlymore costly become unused(the dominationoftheminimumcostroutescanalsobe obtainedby increasing

θ

asinatraditionalMNLmodel, howeverinsuch casesallrouteswithacostbelowthecostbound wouldstill beused). Namely,we havea choicemodel thatifusedforroutechoicesintheirextremecasesapproacheseitheraSUEwitha probabilitythat allroutesarechosen no-matterhowcircuitoustheymightbe,oraDUEwhereonlytheminimumcostroutesarechosen.

3 Note that the bounded choice model we shall derive does not itself have Gumbel error terms, due to the transformation implied by the bounding

3. BoundedSUE:Definition,existenceanduniqueness

WenowmoveontoconsiderhowtheBoundedChoiceModelintroducedinSection2maybeimplementedinanetwork equilibriumcontextand,asafirststage,wesetupthebasicnotation.

Weconsideranetworkasadirectedgraphconsistingoflinksindexedbya (a=1,2,…,A)andorigin-destination(OD) pairs indexedby m (m=1,2,…,M).We define thedemand dmforOD-pair m composinga non-negativeM-dimensional

vectord,theindexsetRmofall simplepaths (withoutcycles)foreach OD-pairm,thenumberNmofpathsinRmandthe

unionRofthesetsRmoverallOD-pairs.TherouteindexsetsareconstructedsothatR={1,2,..,N},whereN= M



m=1 Nm.

Denotetheflowonpathr∈RmbetweenOD-pairmasxmrandletxbetheN-dimensionalflow-vectorontheuniversal

choicesetacrossallMOD-pairs,sothatthenotationxmr referstoelementnumberr+ m_−1

k=1

Nk intheN-dimensionalvector

x.Denotetheflowonlink a(a=1,2,…,A)asfa andlet f=(f1,f2,…,fa,…,fA)be theA-dimensionallinkflow-vectorwherefa

referstoelementnumberainf.TheconvexsetGofdemand-feasiblenon-negativepathflowsisgivenby:

G=



x∈RN +: Nm r=1xmr=dm, m=1,2,...,M



(1) whereRN

+denotesN-dimensional,non-negativeEuclideanspace.Next,define

δ

amr equalto1iflinka ispartofpathrfor

OD-pairmandzerootherwise.Thentheconvexsetofdemand-feasiblelinkflowsis: F=



f∈RA₊: fa= M  m=1 Nm  r=1

δ

amrxmr, a=1,2,...,A,x∈G



(2)

In vector/matrix notation, let x andf be column vectors, and define



as the A× N-dimensional link-path incidence matrix.Then therelationship betweenlink andpathflows maybewritten asf=



x.We supposethat thetravel coston pathrforOD-pairmisadditiveinthelinktravelcostsoftheutilisedlinks:

cmr

(

x

)

= A

a=1

δ

amr· ta

(



x

)

(

r∈Rm; m=1,2,...,M; x∈G

)

(3) Definet(f) (t: RA

+→RA+) asthevectorofgeneralised linktravelcost functions,andc(x) (c:RN+→RN+) asthe vectorof generalisedroutetravelcostfunctions.Thentherelationshipsbetweenpathandlinkflows,andbetweenlinkandpathcosts, maybesuccinctlywrittenas:

f=



xandc

(

x

)

=



T_t

₍

_

_x

₎

_{. (4)}

On the demand side, for each OD movement m=1, 2, …,M, we let Pmr(c) denote the proportionof drivers on that

movementthatwouldchoosepathr_∈Rmwhenthevectorofpathcostsisc∈RN.

Havingintroduced thenetwork notation, we nowturn topropose a newequilibriummodel.We dothisby usingthe BoundedChoiceModeldefinedinSection2.2within aconventionalSUEframework,settingthesystematicutilityusedin

Section2.2equaltothenegativeofthetravelcost.

Inordertoaccommodatethepossibilityofeitherassuminganabsoluteorrelativebound,weconsidertwocases: (1) RelativeCostModel:Variablebounds

δ

(c)=(

δ

1(c),

δ

2(c),…,

δ

M(c))where

δ

m

(

c

)

=

(

τ

− 1

)

min

{

cms:s∈ Rm

}

(

m=1,2,...,M;

τ

>1

)

.

(2) AbsoluteCostModel:ConstantOD-specificbounds,

δ

(c)≡

δ

=(

δ

1,

δ

2,…,

δ

M)where

δ

m>0form=1,2,…,M.

Thisyieldsthefollowingdefinition:

Definition1. BoundedStochasticUserEquilibrium(Bounded(

δ

)SUE) Therouteflowx∈GisaBoundedSUEifandonlyifitisbothanSUE:

xmr=dmPmr

(

c

(

x

)

)

(

r∈Rm; m=1,2,...,M

)

andthechoicemodelisgivenbytheBoundedChoiceModelformbasedonbounds

δ

(c): Pmr

(

c

)

=

(

exp

(

−

θ

(

cmr− min

(

cms: s∈Rm

)

−

δ

m

(

c

)

)

)

− 1

)

₊ 

t∈Rm

(

exp

(

−

θ

(

cmt− min

(

cms: s∈Rm

)

−

δ

m

(

c

)

)

)

− 1

)

+

(

r∈Rm; m=1,2,...,M

)

.

WefirstconsidertheissueofwhetheranyflowallocationmightexistsatisfyingtheBoundedSUEconditions.

Proposition1. ExistenceofBoundedSUESolutions

Supposethatforeachm=1,2,…,M,theODdemanddmispositiveandfinite,andthatthefeasibleroutesetRm =∅.Suppose

thatc(·)iscontinuousandboundedovertheconvexsetG(definedbyEq.(1))andthat

δ

(·)iscontinuousandboundedoverc(G).