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On Ordinal Utility, Cardinal Utility, and Random Utility This is a repository copy of On ordinal utility, cardinal utility, and random utility . White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/2544/ Article: Batley, R. (2007) On ordinal utility, cardinal utility, and random utility. Theory and Decision, Online. ISSN 1573-7187 https://doi.org/10.1007/s11238-007-9046-2 Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ White Rose Research Online http://eprints.whiterose.ac.uk/ Institute of Transport Studies University of Leeds This is an author produced version of a paper published in Theory and Decision Journal. It has been peer reviewed but does not include the final publishers formatting and pagination. White Rose Repository URL for this paper: http://eprints.whiterose.ac.uk/2544/ Published paper Batley, Richard (2007) On ordinal utility, cardinal utility, and random utility. Theory and Decision, Online First, May 2007 White Rose Consortium ePrints Repository [email protected] Onordinalutility,cardinalutility,andrandomutility RichardBatley InstituteforTransportStudies,UniversityofLeeds,UK [email protected] ABSTRACT ThoughtheRandomUtilityModel(RUM)wasconceivedentirelyintermsof ordinal utility, the apparatus through which it is widely practised exhibits properties of cardinal utility.The adoption of cardinal utility as a working operation of ordinal is perfectly valid, provided interpretations drawnfrom thatoperationremainfaithfultoordinalutility.Thepaperconsiderswhether the latter requirement holds true for several measurements commonly derivedfromRUM.Inparticularitisfoundthatmeasurementsofconsumer surplus change may depart from ordinal utility, and exploit the cardinality inherentinthepracticalapparatus. Keywords: ordinal utility, cardinal utility, Random Utility Model, log sum, ruleofahalf 1 JEL Classification: B41 Economic Methodology; D01 Microeconomic Behaviour,UnderlyingPrinciples 1.INTRODUCTION The Random Utility Model (RUM) was conceived by Marschak (1960) and Block and Marschak (1960) as a probabilistic representation of the Neo Classical theory of individual choice, with probability deriving from variability in the individual’s preferenceswhen faced with repeated choices from the same finite choice set.Subsequent to its theoretical conception, Marschaketal.(1963)introducedmuchoftheapparatusbywhichRUMcould be translated to practice; this exploited an analogy with psychophysical modelsofjudgementandchoice,inparticularFechner’s(1859)modelandthe derivative models of Thurstone (1927) and Luce (1959).It was left to McFadden (1968, but unpublished until 1975) to complete the translation, which was achieved through a shift in the perspective of the presentation. McFadden reconstituted RUM from a model of an individual engaged in repeated choices, to one of the choices of apopulation of individuals.This was clearly motivated by pragmatism; the revised presentation was more amenable to the needs of practising economists, with interests more in 2 marketsthanindividuals,anddatatomatch.McFadden’sadjustmenttothe presentationthoughsomewhatinnocuousintheoreticaltermsopenedthe floodgates to practical RUM analysis.There began forty years of intensive application,culminatinginMcFadden’sawardingofthe2000NobelLaureate in Economics, and with it the securing of RUM’s place in the history of economicmethodology. Since the unwary might, quite understandably, draw confidence from such accolade, it would seem timely to illuminate a troublesomeand indeed potentiallydamningpropertyofRUMthatpersists.Thismaybeseenasa problem of compatibility arising from the analogy between NeoClassical theoryandpsychophysicalmodels.Morespecifically,themetricofthelatter carriescardinalproperties,incontrasttotheordinalityofutilitywithinNeo Classicaltheory.Severalpreviousauthorshavealludedtothisproperty(e.g. Goodwin and Hensher, 1978; Daly, 1978), and many have considered its empiricalconsequencesusuallyunderthelabelofthe‘scalefactorproblem’ (e.g. Swait and Louviere, 1993).But one might ask: ‘why are RUM practitioners so preoccupied with matters of utility scale, when utility is supposedlyordinal,andordinalutilityisrobusttoareasonablygeneralclass of transformations?’The purpose of the present paper is to rationalise the answertothisquestionineconomictheory. 3 ThepaperbeginsbyrehearsingthederivationofRUMfromthefundamental axioms of deterministic choice under certainty, thereby revealing the theoretical origins of RUM in terms of ordinal utility.The paper then proceedstodemonstratethat,inexploitingthevehicleoftheFechner(1859) model,theimplementationofRUMimpliestheadoptionofcardinalutility. Thisisnotinitselfaproblem;cardinalutilitycanbeusedquitedefensiblyasa working representation of ordinal utility, provided any such representation does not actually depart from the ordinality on which it is founded.The paperconsiderswhetherpracticeisadherenttothelatterrequirement. 2.ORDINALUTILITYANDDETERMINISTICCHOICE This section rehearses the theory of deterministic choice under certainty. Whilstthegeneraltheorywillbewellknowntomanyreaders,thefollowing attends to a particular interest in finite choice sets, and this may be less familiar to some.The purpose of the rehearsal is to establish a systematic basisforsection3,whichseekstoilluminatethetheoreticaloriginsofRUMin termsofordinalutility.Theinformedreadermayhoweverwishtoomitthis section,andmovedirectlytosection3. 4 Before proceeding, and following RUM convention, deference is made to Lancaster’s (1966) representation of goodsor in present parlance ‘alternatives’in terms of their constituent attributes.Formally, define an alternativetobeavector x = (x1 ,..., xK ),wherexk isthequantityofattributek for k = 1,..., K .Inwords,analternativerepresentsapointinafiniteattribute space.Againfollowingusualpractice,definewithinthisspaceafiniteset T ofalternativesthatconstitutesthe‘choiceset’ofN alternativesavailableto anindividual:T = {}x1 ,...,x N . The purpose of the theory of deterministic choice under certainty is to representanindividual’spreferencesintermsofanumericalutilityfunction. Though there is reasonable agreement as to the means of achieving this, whichistoimposeanaxiomsystemontheindividual’spreferences,several alternative presentations have been proposed.As is often the case, this variety has been motivated, in particular, by an interest in the possible generalityofpresentation.TheapproachemployedbelowfollowsBlockand Marschak (1960), although the reader is referred to Luce and Suppes (1965) fordiscussionofalternativepresentations. 5 Thetheoryrestsfundamentallyontwoaxioms1,asfollows: Axiom of Completeness: For every pair x n ,x m ∈T , either x n f x m , or x m f x n , or both x n f x m and x m f x n . Axiom of Transitivity: For every triad x n ,,x m ,xl ∈T if x n f x m and x m f xl , then x n f xl These two axioms, when taken together, establish a complete (weak) preferenceordering on TT.Since is finite, each alternativex n ∈T can be assigned a ‘rank’ on the basis of preference, where a rank is an integer rn , 1 ≤ rn ≤ N ,suchthatforeverypairwithintheranking: rn ≤ rm if x n f x m Now define the vector r = (r1 ,..., rN ), which is an integervalued function representingthecompleteorderingof T .Anystrictlydecreasingmonotone 1 Noting that relaxation of the assumption of a finite choice set necessitates an additional axiomimposingcontinuityofpreference.Continuityisdiscussedfurtherinsection7.1,albeit inadifferentregard. 6 functiononr willinducearealvaluedfunctionU onT thatpreservesthe preferenceordering,specifically: U n ≥ U m iff x n f x m (1) Hence we arrive at the ‘ordinal’ utility function U , wherein values are assignedtothefunctionsimplyonthebasisthattheU valueofapreferable alternativewillbegreaterthanthatofaninferioralternative.Otherwisethe scale ofU is arbitrary, and any transformation that preserves the ordering (i.e.theclassofstrictlymonotonictransformations)is,intheseterms,entirely equivalent.Thus(1)mayberewrittenequivalently: ˆ ˆ U n ≥ U m iff x n f x m (2) where Uˆ = f ()U , and f is a strict monotone of U Itisimportanttobeclearthattheabovetheoryprovidesnobasisfordrawing inferencesintermsofcardinalutility;i.e.intermsofdifferencesinutility(or any ratio thereof) across alternatives.The ordinal utility function simply ensures that an individual prefers alternatives with greater utility to those withless;i.e.thattheindividualactssoastomaximiseutility. 7 3.ORDINALUTILITYANDPROBABILISTICCHOICE Let us now generalise the analysis to accommodate a probabilistic representation of choice, again under certainty.This representation is motivated by the conjecture, emanating principally from the discipline of psychology, that the choices of an individual may not necessarily be ‘consistent’.Inconsistencyis,inthissense,usuallydefinedintermsofeither (orboth)oftwophenomena;first,violationsoftheAxiomofTransitivity;and second,theobservationthatanindividual,whenfacedwitharepeatedchoice task,maynotalwaysmakethesamedecisions.Havingintroducedthetheory ofprobabilisticchoice,attentionnowfocusesonRUM,whichconstitutesone particularformalisationofthetheory;interestedreadersmaywishtoreferto Luce and Suppes (1965), who summarise a fuller range of formalisations. Withreferencetotheaforementionedphenomena,RUMismotivatedbyan aspirationtoaccommodatethelatter,i.e.variationsinindividualpreferences
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