List - Multidimensional Welfare Aggregation

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Multidimensional Welfare AggregationAuthor(s): Christian ListReviewed work(s):Source: Public Choice, Vol. 119, No. 1/2 (Apr., 2004), pp. 119-142Published by: SpringerStable URL: http://www.jstor.org/stable/30025816 .

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Public Choice 119: 119-142, 2004. 119a 2004 KluwerAcademicPublishers. Printed in theNetherlands.

Multidimensional welfare aggregation *CHRISTIANLISTNuffieldCollege, OxfordOX1 INEFU.K.;e-mail: [email protected] 13 February 003Abstract. Most accounts of welfareaggregation n the traditionof Arrow's(1951/1963) andSen's (1970/1979) social-choice-theoreticrameworks epresenthe welfareof an individual ntermsof asingle welfareorderingora single scalar-valuedwelfare function.I developa multi-dimensionalgeneralizationof Arrow'sandSen's frameworks, epresentingndividualwelfarein terms of multiple personalwelfare functions,corresponding o multiple 'dimensions' ofwelfare.I show that,as in the one-dimensionalcase, the existence of attractiveaggregationproceduresdependson certain nformational ssumptions, pecificallyaboutthemeasurabilityof welfare and its comparabilitynot only across individuals but also across dimensions. Istate severalimpossibilityand possibility results. UnderArrow-typeconditions, insufficientcomparability crossindividuals eads to dictatorship f a single individual,while insufficientcomparabilityacross dimensions leads to dominance of a single dimension. Given sufficientcomparability oth across ndividualsandacrossdimensions,a rangeof possibilities emerges.I discuss the substantivemplicationsof the results.

1. IntroductionThe concern of this paperis the problemof aggregating he welfare of theindividualmembersof a groupinto the correspondingwelfareof the groupas a whole. Individualwelfare is typicallyassessed in terms of some normat-ively relevantevaluationstandard.Examplesof suchevaluation tandards, r'currenciesof welfare', aremoney,indices of resources,utility,or Rawlsianprimarygoods.By Arrow's theorem(1951/1963), thereexists no procedure or aggreg-ating individualwelfareorderingsover a set of alternatives nto collective,

* Previous versions of thispaperwerepresentedat the seminarof thePhilosophyProgramof the Research School of Social Sciences, AustralianNationalUniversity,Canberra,n June2000, at anNSF Workshopon Models of Individualand PublicChoice,held at the Universityof Californiaat Irvine, n July2000, andat the AnnualMeetingof the PublicChoice Societyin New Orleans,Louisiana,in March 1999. The author wishes to express his gratitudetothe seminar and conferenceparticipantsat these occasions and to A.B. Atkinson,GeoffreyBrennan,John Dryzek, RobertGoodin, Iain McLean, Kevin Roberts,John Weymark,andan anonymousreviewer for very helpful comments and discussion. Financialsupportof the(British)Economic andSocial ResearchCouncil,the GermanNationalMerit Foundationandthe GoodhartFund is gratefullyacknowledged.

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120or social, ones where the proceduresatisfies a set of minimal conditions(transitivityof social orderings,universaldomain, the weak Paretoprin-ciple, independence of irrelevant alternatives and non-dictatorship).Sen(1970/1979) has shown that Arrow'stheoremdependscruciallyon the as-sumptionthat individual welfare is not interpersonally omparable:giveninterpersonal omparability f welfare evels orunits,theimpossibilityresultdisappears.Thequestionof whetheror notinterpersonallyignificantnform-ation s accessibledependson thechosen 'currencyof welfare'. Thequestionabout the significantcontent,personaland interpersonal, f individualwel-fare information s called the question of measurabilityand interpersonalcomparability.Sen's (1970/1979) framework s more general than Arrow's in that itallows alternativeassumptionson measurabilityand interpersonal ompar-ability.But there is one assumptionthat both accounts share;namely theassumptionthat individual welfare can be expressed in terms of a singlewelfareorderingor a single scalar-valuedwelfare function.This assumptionrequires he existence of a single 'dimension' or a single 'currency'of wel-fare, in respectof which each individual'swelfare over a set of alternativescan be assessed.

This paper provides a multidimensionalgeneralizationof Arrow's andSen's frameworks.The welfare of each individualwill be expressed n termsof multiplepersonalwelfare functions,one for each relevant 'dimension'.Like a one-dimensional ramework, he multidimensional ramework aisesthe questionof measurability ndinterpersonal omparability.Unlike a one-dimensionalframework, t raises an additionalquestion. Is it possible tocompareanindividual'swelfare n one dimensionwith thatsame individual'swelfarein anotherdimension?This, roughly,will be called the questionofinterdimensionalomparability.1I will first identify a multidimensionalArrowproblem.In the absenceof sufficient nterpersonal omparability, ny aggregationprocedure atisfy-ing Arrow-typeconditionswill make one individualdictatorial.And in theabsence of sufficient interdimensional omparability,any such aggregationprocedurewill make one dimensiondominant.An Arrow-type mpossibilityresult can be avoidedif sufficientcomparabilityboth acrossindividualsandand acrossdimensions s admitted,and a rich set of possibilitiesemerges.

Although the literaturecontains some formally closely relatedresults,multidimensionalitys usuallynot developedas an interpretation. uchres-ults areRoberts's 1995) resultsontheaggregation f multipleopinionsaboutthe welfare of a groupof individuals nto a single social orderingand Kh-melnitskaya's(1999) and Khmelnitskayaand Weymark's 2000) results onsocial welfareorderings or differentscales of individualutilitymeasurement



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121in distinctpopulationsubgroups.Amongst the firstpaperson multidimen-sional welfare are Plott, Little, and Parks (1975) and Sen (1980/81), theformerproviding an Arrow-styletheoremfor aggregationacross multipledimensions within a single individual.Kelsey (1987) providesa relevantre-interpretationf the literature n informational ssumptions n social choice.Less closely relatedprecursorsare resultsby Fishburn 1971) and BatraandPattanaik 1972) on multi-stagemajoritydecisions,involvingnestedaggreg-ation overnestedsubprofilesof a givenprofileof preferenceorderingsacrossindividuals.Finally,a companionpaper o thispaper s concernedwith multi-dimensionalpreferenceaggregationwithoutanyforms of comparability, utwithintradimensionalingle-peakedness List,2002).Thepaper s in fivesections.InSection2, I will brieflydiscuss thepoliticaltheory backgroundof multidimensionalaggregationproblems.In Section 3,I will address the formalizationof measurability,nterpersonal omparabil-ity and interdimensional omparability.n Section 4, I will statethe results,and, in Section 5, I will draw some conclusions. The derivationof the mainimpossibilityresult will be stated n an appendix.

2. Multidimensionalitynd social choiceSen has argued hat the choice of a normatively elevantevaluationstandard,or 'currencyof welfare',maybe as consequentialas thechoice of an aggreg-ationprocedure tself. The locus classicus is his famouspaper"EqualityofWhat?' (Sen, 1982). People may agree thatequalityof some form mattersin the design of an aggregationprocedure,andyet disagreeon the question"Equalityof What?". ncomeegalitarians,marginal-utility galitarians,otal-utility egalitarians, ndRawlsianprimarygoods egalitariansmay all claim touse 'egalitarian'aggregationprocedures.But, in lightof theirdifferentviewson what 'currency'shouldbe equalized, they derivefundamentallydifferentconclusions as to whatsocial arrangementshouldbe pursued.Let me illustrate how the choice of a 'currencyof welfare' affects therecommended ocial arrangements.ncome egalitarians eek to equalizein-come across individuals.Thismayleadto anunequaldistribution f utility,asthe capacityto convert ncome into utility may differacrossindividuals.Forinstance,someone with a certainmedicalconditionmay need more incometo sustaina particularutility level than someone without thatmedical con-dition. Marginal-utility galitariansseek to equalize marginalutility acrossindividuals,which typically amountsto maximizingthe sum-totalof utilityacross individuals.This will not in general lead to an equal distributionofincome, nor to an equal distributionof utility. Those individuals who aremore efficient in converting ncome into utility may receive more income



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122underthis proposal,andthose who are less efficientmay receive less. Fur-ther, a distribution hat equalizes marginal utility across individuals (andtherebymaximizes total utility) is not typically one in which utility itselfis equallydistributed.Total-utility galitarians,by contrast,seek to equalizeutilityitself. This will not in generallead to a distributionwhich maximizestotal utility,nor to an equal distributionof income. Total-utility galitariansendorse ncomeinequalitiesbetweenindividuals f these incomeinequalitieslead to a moreequaldistribution f utility.Undertotal-utilityegalitarianism,someone with a medical condition as in the example above should receivemore income thansomeone without thatmedical condition.However,total-utility egalitarians ace the challenge of expensive tastes. If a person hasan expensive taste andrequirescaviar to achieve the same utility level thatothers achieveby consumingbread, hentotal-utilityegalitarianismeems toentail thatthatperson oughtto receive additional esources n order o affordcaviar.Manyfind this conclusionunacceptableand therefore eek to identifya 'currency' hat is less 'subjectivist' hanutility(so that it is immuneto theproblemof expensivetastes),butmore welfare-relevanthan income(so thatit is sensitiveto specialneeds, like medicalones). Whethera 'currency'withthe desiredproperties xists is a matterof philosophicaldebate,butRawlsianprimarygoods, as discussedbelow,are sometimesheld to be a plausiblesuchcurrency.Senuses the termcompetitive luralityto referto thisdiversity n views onwhat the relevant'currencyof welfare' is: differentproposalsstandas rivalsto each other.Accordingto the classical assumptionof one-dimensionality,one 'currencyof welfare' is to be selectedas the relevantone fromamongstthiscompetitiveplurality.Againstthis assumption,Sen (e.g., 1985, 1987),Walzer 1983) andothershaveargued hatthe relevant nformationormanyproblemsof welfare eval-uation s multidimensional.The effects of alternative ocial arrangementsnan individual's ncome, nutritionand shelter,health,educationalprospects,social status,and so on, may all be relevant.But it may be impossible torepresent his informationn termsof a single one-dimensional currencyofwelfare'.The use of multipleevaluationstandardsmay thereforebe warran-ted. Sen introduces he termconstitutivepluralityto refer to this view, thattherelevantevaluation tandards internallydiverse(e.g., 1987:2-3).

AlthoughSen and Walzer both defend constitutivelypluralconceptionsof welfare,their accountsare quite different.Sen's accountis motivatedbythe Aristotelianessentialist view that several human unctionings form anessentialpartof a 'good life' and are thus relevantdimensionsfor welfareevaluation(see also Nussbaum, 1992). Such functioningsmay range frombeing well nourished and being free from avoidable disease to being able



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123to participate n social life and having self-respect.Walzer'spluralityofspheres of goods, by contrast, s motivatedby a communitarian nd contex-tualist theoryof the good. The relevantspheresare determinedby context-specific social meanings in the relevantsociety. Examplesof such spheresmightbe health, education,employment,politicalinfluence,andmoney.Senacknowledges heneed foraggregationandcross-dimensionalndexing(e.g.,1991; 1997:sectionA.7.3), whereasWalzerdefends themutualseparateness,andarguably ncomparability, f differentdimensions.In particular,Walzerrequiresthat different spheres of goods be kept separateand that differ-ent (dimension/goods-specific)principlesof justice apply to different suchspheres.The evaluationstandardof Rawls's Theoryof Justice (1971), an indexof primarygoods, is also constitutivelyplural, ncludingrights,libertiesandopportunities,ncomeandwealth,and the social basesof self-respect Rawls,1971:60-65). However,Rawlsmakesthe tacitassumption hatsummarizinga pluralityof primarygoods into a single index (which is ordinallymeas-urable and interpersonally omparable) s possible, but is silent on how toconstruct uch anindex.Formally,Rawls'ssocial-choice-theoretic roposals,particularlyhedifferenceprinciple,arepresented n a one-dimensional orm.A numberof social choice theoristshave addressed heproblemof indexingprimarygoods andderived some Arrow-type mpossibilityresults,all basedon the assumption hat the comparabilitybetweendifferentprimarygoods islimited (for instance, Plott, 1978; Gibbard,1979; Blair, 1988; but see Sen,1991).At apracticalevel, the HumanDevelopment ndex(HDI),asemployedbythe UnitedNationsDevelopmentProgramme, s also a constitutivelyplural'currencyof welfare',albeit an aggregateone. The HumanDevelopmentIn-dex combinesthreecomponents,each of which is itself anaggregatemeasureacross people in a countryor region. The three componentsare longevity,knowledge, and standardof living. Longevity is measured n terms of lifeexpectancyat birth.Knowledge is measured n terms of a combinationofthe adultliteracyrateandenrolment n primary, econdary,andtertiaryedu-cation. Standardof living is measured n terms of GDP per capita (UNDP,2002).The presentapproachallows to formalizea constitutivelypluralconcep-tion of welfare at a micro-level: it allows the separaterepresentation f theeffects of eachalternative n eachindividual n each one of multiplerelevantdimensions.



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1243. Measurability,nterpersonal omparability nd interdimensionalcomparabilityLet N = {1, 2, ... , n} be a set of individuals,andX = {xl, x2, . . a set ofalternatives.Suppose,further, hatthere are k relevantdimensions,containedin K = {1, 2, ... k}. We will assume n > 1, IXI > 2, and, unless statedotherwise,k > 1.2To each individual e N, therecorrespondsa k-tupleWi :=< Wij >jEK= < Wi,i, W2, ... , Wik> of personalwelfarefunctions,containingone Wij:X - R for each dimensionj E K. For each individual E N and eachdimensionj E K, the functionWij assigns to each x e X a real numberWij x). HereWij x) representshewelfareof individual indimension underalternative .For example, let X be a set of employmentpolicy alternatives,and let1, 2, 3 e K be the dimensionsof income,health and social status.ThenWil,Wi2andWi3represent ndividual 's welfare over the alternativesn X fromthe perspectivesof income,healthand social status.A profileof k-tuplesofpersonalwelfare unctionsis ann-tuple{WiliENIW1, W2, ..., Wn} of such k-tuples,containingone k-tupleWi for each in-dividuali N. As a notationalconvention,we use { }-brackets o denoten-tuples across individualsand < >-brackets to denote k-tuples acrossdimensions.A multidimensional ocial welfarefunctional (MSWFL) s a functionFwhichmapseachprofileof k-tuplesof personalwelfarefunctions(in a givendomain),IWijiEN,to a corresponding ocial orderingR = F({Wi}iEN)n X,whereR is reflexive,transitiveandconnected.R inducesa strongorderingPand an indifferencerelationI, definedas follows: for all xl, x2 e X,

x1Px2 f andonly if xlRx2 andnot x2Rx1;xlIX2 if andonly f xlRx2 andx2Rx1.A one-dimensional social welfare functional (SWFL) is simply anMSWFL for the special case k = 1. How can we formalizeassumptionson measurability, interpersonal comparability and interdimensional

comparabilityof welfare? The formalizationto be developed is directlyanalogous to Sen's well known method of formalizing assumptionsonmeasurabilityand interpersonal omparability n a one-dimensionalsocialchoice framework e.g., Sen, 1970/1979; Sen, 1982: Ch. 11; see also List,2001). We firstexplainthe idea behindthe formalizationand then proceedto stating the formalizationproperly.Let us consider statementsof thefollowingforms:Level Comparisons (LC). Alternativex1 from the perspectiveof individual



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125ii in dimension i is at least as good as alternative 2 from the perspectiveofindividual 2 in dimension 2; formallyWi,j (x1) > Wi22X2).Unit Comparisons (UC). The ratio of [individual I's gain/lossin dimensionjl as aresultof switchingfrom alternative 1 tox2]to [individual2'sgain/lossin dimensionj2as a result of switchingfromalternative i to Y2]is X,wherekis a realnumber;ormally(Wij1 X2)Wi j1 (Xl )/(Wi2~2y2)Wi2j2(Y)) = k.The key idea is that differentassumptionson measurability, nterpersonalcomparability ndinterdimensional omparability f welfareimply differentconditions under which (LC)- and (UC)-statements are meaningful.Specifically,we have the following:- Ordinalmeasurability 0) impliesthat(LC)- statementsaremeaningfulwheniI = i2 and 1 = j2.- Cardinalmeasurability(C) implies that (LC)- statementsand (UC)-statementsaremeaningfulwhenil = i2 andjl = j2.- InterpersonalevelcomparabilityLper)mpliesthat(LC)-statements re

meaningfulwhenjI = j2 butiI i2.- Interpersonal nitcomparabilityUper)mpliesthat(UC)-statements remeaningfulwhenjl = j2 butil # i2.- Interdimensional level comparability (Ldim) implies that (LC)-statementsaremeaningfulwhen il = i2 butjl j2- Interdimensional nitcomparabilityUdim)mpliesthat(UC)-statementsaremeaningfulwhen i1 = i2 butjl # j2.We use the labels (Nper) and (Ndim) to referto, respectively,no interpersonalcomparability nd no interdimensional omparability.Once we haveassigneda profileof k-tuplesof personalwelfarefunctions{WiliEN o a set of individuals,we can of course make (LC)- and (UC)-statementsrelative to that profile. However, whether such statements aremeaningfuldependson how uniquethe profile {Wi}iN is. Suppose, for in-stance, that each Wij in the profile {Wili}N is uniqueonly up to a positivemonotonictransformation, ossibly a differenttransformation or differentindividualsanddimensions. Then interpersonal r interdimensionalevel orunit comparisonsare not well-defined and thus not meaningful, as thesecomparisonsarenot in general nvariantunder he specifiedtransformations.Suppose,moregenerally,we specify the class of transformations1 up towhich each profile {Wi}ijN is unique. We can then ask whether each kind of(LC)- or (UC)-statement s invariantunder all the transformationsn P. A



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126particularkind of (LC)- or (UC)-statement s said to be meaningful f andonly if it is invariantunderall transformationsn P.

We state assumptionson measurability, nterpersonal omparabilityandinterdimensional omparabilityby specifyingthe class of transformations)upto whicheachprofile{WiliENs takento be unique.The smaller his classof transformations,he more information s containedin a profile.Table 1lists severalalternative uch classes of transformations.3An example shows how to read Table 1. Consider OLdimNper, the as-sumptioncorrespondingo the row labelledLdimNpernd the column labelledO. Then OLdim Nper is the assumptionof ordinalmeasurability,nterdimen-sional level comparability,but no interpersonalcomparability.Accordingto OLdimNper, P is the class of all n-tuplesof k-tuplesof transformations,{< Oij >jEK}iEN such thateach 4ij : R -- R is a positivemonotonictransformation, nd,for each i e N, Oil = Oi2 = ... ==ik.Now supposethat each {WiJiEN is unique up to the transformationsn ci.We thenrequire hat a MSWFLbe invariant nder hese transformations. heideabehindthisrequirements this.Suppose{WiliENan be transformednto{W*}iiNby some transformationn D. Then {Wi}iEN and {[W}ieNaretakento contain exactly the same relevantinformation.Therefore our MSWFLshouldmap {Wi}iEN and {fW}iENo the same social ordering.Formally,wecan statethisrequirement s follows.Invariance assumptionwith respect to (D.For any IWiIiEN nd {W*}iENin the domain of F, if there exists {< Oij >jEKIiEN c such that,for eachi N and eachj K, W = Oij(Wij),henF({WidiEN) = F({W*}ieN).Thus each assumption n Table 1 definesa specificclass of transformations(D, and we can consider the corresponding nvarianceassumption withrespect to D. We use the name of each assumption n roundbrackets,forexample(OLdimNper),o denotethe correspondingnvarianceassumption.4Let me make one final remark.Suppose(Dand I aretwo classes of trans-formations uch that C 4. Forexample, Dcorresponds o OLdimNper,ndT correspondso CLdimUdimNper.ThenanyMSWFLsatisfyingthe invarianceassumptionwithrespectto ) will also satisfythe invarianceassumptionwithrespectto 4.

4. ResultsBefore we can present mpossibilityandpossibilityresults,we need to statemultidimensionalgeneralizationsof Arrow's conditions.The generalizationof universal domain, the weak Pareto principle and independence of



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127Table1. The class of transformations ' is the class of all n-tuplesof k-tuplesof transforma-tions, {< ij >jeK}ieN, with the following properties:

NdimNper

UdimNper

LdimNper

LdimUdimNperNdimUper

NdimLper

NdimLperUperLdimLperUdimUper

UdimLperUper

LdimUdimUper

LdimUdimLperUper

OOrdinalmeasurabilityEachij : R - R is apositivemonotonictransformation, nd...no furtherassumptionrequirednot applicable

for each i E N, oil =bi2 = ... = oiknot applicablenot applicable

foreachj E K,4lj =q2j = ... = njnot applicableall fij are identicalnot applicable

not applicable

not applicable

not applicable

CCardinalmeasurabilityEachoij : R - R is a positive affinetransformation, nd...

no furtherassumption equired

foreach i e N, Bai , ai2, ..., aik E R such thatoil + ail = i2 + ai2 = ... = oik + aiknot applicable

for each i E N, #il = oi2 = ... = ikfor eachj e K, Balj,a2j... anj e R such that4lj + alj = 42j + a2j= ... = Qnj+ anjnot applicable

for eachj e K, =lj = 2j = ... = njnot applicable3al ... alk, a21..., a2k,anl,...., ankE Rsuch that411 + all = ... = 'lk + alk =21 + a21= ... = 2k + a2k='nl + anl = ... = nk + ankfor each j e K, blj = =2j = ... = Pnjand3al, a2 ... ak E R suchthat,for each i EN, il + al = 'i2 + a2 = ...= 'ik + akforeach i EN, 'il = Oi2= ... = ikand3al, a2 .., an ER suchthat,for eachj e K, lj + al = 2j + a2 =...= 'nj + anall 'ij areidentical



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128irrelevantalternatives s straightforward.Given any profile of k-tuples ofpersonalwelfarefunctions,{WiliEN,we define R := F({WiiEN).Universal Domain (U). The domain of F is the set of all logicallypossibleprofilesof k-tuplesof personalwelfarefunctions.WeakPareto Principle (P). Let {WiliENbe any profile in the domain ofF. For any x1, x2 e X, we have x1Px2 whenever,for all i E N and allj E K, Wij(xl) > Wij(x2).Independence of Irrelevant Alternatives (I). Let {WiLiEN nd {W*}iENbe any profiles n the domainof F. Supposex1, x2 E X suchthat,forall i ENand allj e K,Wij(x1)- W (x1) andWij(x2)= W*(x2).ThenxlRx2 if andonly if x1R*X2.The generalizationof non-dictatorships less straightforwardsee also List,2002). In the multidimensionalframework,non-dictatorshipcorrespondsto two conditions:non-dictatorshipand non-dominance.Non-dictatorshiprequiresthe non-existence of a fixed single individual whose k-tuple ofpersonal welfare functions always determinesthe social ordering.Non-dominancerequires he non-existenceof a fixed single dimension such thatthe personalwelfare functions(acrossindividuals) n that dimensionalwaysdetermine he social ordering.Non-Dictatorship (D). There does not exist an individual i e N (adictatorfor F) such that, or all profiles IWiliENn the domain of F and allX1, X2 E X, f(< Wij(x1) >jeK) > f(< Wij(X2)>jEK) impliesX1PX2,wheref: Rk --+R is a strictly ncreasing unction.Non-Dominance (Dom). There does not exist a dimension j E K (adominantdimension orF) suchthat,for all profiles{WiliENn the domainofF and all x1,X2 E X, f({Wij(X1)}iEN)> f({Wij(X2)xiEN) impliesx1Px2,wheref: Rn-> R is a strictly ncreasing unction.Non-double-dictatorship, inally, requires the non-existence of a fixedsingle individualand a fixed single dimension such that this individual'spersonalwelfare function in that dimension always determinesthe socialordering.Non-Double-DictatorshipDD). Theredoes not exist an individual e N anda dimensionj E K such that,for all profiles{Wi}i~Nin the domainof F and



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129all x1, x2 E X, Wij(x1) > Wij(x2) implies1Px2.Under all these generalized non-dictatorship onditions, personal welfarefunctions other than the dictatorialor dominantones can act at most astie-breakers,namely in those cases in which the dictatorial or dominantfunctionsreflect indifference.In the absence of both interpersonal omparabilityand interdimensionalcomparability,Arrow's theorem(in the version of Sen, 1970/1979; see alsoList, 2002) impliesthe followingresult.Theorem1. There exists no MSWFLsatisfying (ONdimNper) or (CNdinNper),and(U), (P), (I) and(DD).A multidimensional aggregation problem with n individuals and kdimensions, with (ONdimNper) or (CNdimNper), is equivalent to a one-dimensional aggregation problem with nk individuals and withoutinterpersonal omparability.Theorem 1 thus follows directlyfrom Arrow'stheorem.4.1. Lexicographicdictatorshipsand lexicographichierarchiesofdimensionsIs interpersonalcomparability, n analogy to the one-dimensional case,sufficient or avoidingthe multidimensionalArrowproblem?The answertothisquestion s negative.If we assumeordinalor even cardinalmeasurability,with interpersonalcomparabilityof both levels and units, but withoutinterdimensional omparability, nyMSWFLwhich satisfies(U), (P) and(I)still violates (Dom).Theorem 2. There exists no MSWFL satisfying (ONdimLper) or(CNdimLperUper), and(U), (P), (I) and(Dom).The result can be seen as a reinterpretation f results by Roberts (1995,theorem 6) and Khmelnitskayaand Weymark (2000, theorem 2). SeeAppendix1.

Similarly, if we assume ordinal or even cardinal measurability,withinterdimensional comparability of both levels and units, but withoutinterpersonalcomparability,any MSWFL satisfying (U), (P) and (I) stillviolates (D).Theorem 3. There exists no MSWFL satisfying (OLdimNper) or(CLdimUdimNper),and(U), (P), (I) and(D).



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130

See also appendix1. Given comparabilityacross individualsbut not acrossdimensions (i.e. (ONdimLper) or (CNdimUper)), there are MSWFLs satisfying(U), (P), (I) and (D), but not (Dom). Examples are suitable lexicographichierarchies of dimensions, as defined below. Given comparabilityacrossdimensionsbut not across individuals i.e. (OLdimNper)or (CUdimNper)), thereare MSWFLssatisfying(U), (P), (I) and (Dom), but not (D). Examplesaresuitable exicographicdictatorships,also definedbelow.A MSWFL F is a lexicographichierarchyof dimensions f there exist kstrictly increasing functions fl, f2,.. fk : Rn --+ R (possibly different), onefor each dimension n K, and a permutation of K suchthat,for any {Wi}iENandanyx1, x2 EX,

x1Px2 if andonly iffocrj)({Witj)(X1)isN) > fej)({Wiaj)(X2)}ieN) for somej e Kand fa(h)({Wia(h)(X1)}iEN) = f(h)({Wia(h)(X2)ieN) for all h < j.

Under this definition,the dimensions are ranked n a fixed hierarchyofimportance.For each dimensionj, there exists a function fj (to be calledan intradimensional ggregation unction) for aggregating he dimension-j-specific welfare information or each alternativex across individuals nto asingle aggregate figure for that dimension. The alternatives n X are thenranked, exically, according o theaggregate igures or the dimensionswhichare first, second, third,and so on in the hierarchy.As noted above, a lex-icographichierarchyof dimensions need not violate (D). Given sufficientinterpersonal omparability, ny non-dictatorial ne-dimensionalSWFLcanbe the basis for the intradimensional aggregation functions fl, f2 ..., fk-Prominent xamplesare:

- if (at least) (CNdimUper) is satisfied:a 'utilitarian' ntradimensional ggregationunction:fj({Wij(X)}iEN) = XlWlj(x) + o2W2j(x) + ... + ,nWnj(x),where1, 2, . * > 0;

- if (at least) (ONdimLper)iS satisfied :a 'maximin' ntradimensional ggregation unction:fj({Wij(x)}ieN) := min(Wlj(x), W2j(x),..., Wnj(x)).

In a lexicographic hierarchyof dimensions, different intradimensionalaggregationfunctions can be chosen for different dimensions. Given in-terpersonalcomparabilitywithin each dimension,but no interdimensionalcomparability, lexicographichierarchyof dimensionsmay in some casesbe an attractiveMSWFL.In Theoryof Justice,Rawls contrasts he 'specialconception'with the 'generalconception'ofjustice.The 'special conception'



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131is intended o applyto socio-economicallywell developedsocieties, whereasthe 'general conception' is intended to apply to socio-economically lessdeveloped societies. The 'general conception' requiresthat "the differenceprinciple[be] appliedto all primarygoods includinglibertyand opportun-ity" (Rawls, 1971: 83; my italics).This presumably nvolves aggregation, naccordancewith (DOM),across all primarygoods. Accordingto theorem2,if the differentdimensions are constitutedby differentprimarygoods and ifthese differentprimarygoods aremutually ncommensurableas capturedby(ONdimLper)or (CNdimLperUper)),then Rawls's 'general conception' of justiceleads to an impossibilityresult. The 'special conception' of justice, on theotherhand,assigns lexical priority o some primarygoods (e.g., libertyandopportunity)over others.A lexicographichierarchyof dimensionscapturespreciselythis idea.A MSWFLF is a lexicographic dictatorship f there exist n strictlyin-creasing functions fl, f2,.. fn : Rk -- R (possibly different), one for eachindividual n N, and a permutation of N suchthat,for any {Wi}ieN andanyx1,X2 E X,

x1Px2 if andonly ifand fa(i)({Wcj(i)j(xi)}jGK) fo(i)(< WT(i)j(x2)>jEK)fa(h)(< Wa(h)j(X1) >jeK) = fa(h)(< We(h)j(X2) >jeK) for some i E Nfor all h < i.

Ina lexicographicdictatorship,he individualsarearrangedn a fixedhier-archyof decisiveness.For each individual , there exists a functionfi (to becalledanintrapersonal ggregation unction)foraggregatinghisindividual'sk-tupleof welfareinformation or each alternativex into a single aggregatefigure for that individual. The alternatives n X are then ranked,lexically,accordingto the aggregatefiguresfor the individualswho arefirst, second,third,and so on in the hierarchy.A lexicographicdictatorshipneed not vi-olate (Dom). Given sufficient interdimensional omparability, ntrapersonalaggregation unctions thatfocus on more thanone dimension are available,e.g.

- if (at least) (CUdimNper) is satisfied:a 'utilitarian' ntrapersonalaggregationunction:fi(< Wij(x) >jeK) := olWil(x) + X2Wi2(X) + . . . +kWik(X),where

.1,X2,.. ,Xk > 0;

- if (atleast) (OLdimNper) is satisfied:a 'maximin' ntrapersonalaggregationunction:fi(< Wij(x) >jeK) := min(Wil(x), Wi2(X),..., Wik(X)).

In a lexicographicdictatorship,different ntrapersonal ggregation unc-tions can be chosen for different individuals. However, unless there are



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132particular easons why certain individuals should be given lexical priorityover others, lexicographic dictatorshipsare hardlyattractive rom a liberalegalitarian iewpoint.4.2. StrongerpossibilityresultsComparabilityboth across individualsand across dimensions is required fwe regardboth(D) and (Dom) and the otherArrow-type onditionsas indis-pensable.To state strongerpossibilityresults,we must distinguishbetweendifferentways of aggregatingacross individualsand dimensions.Aggrega-tion may takeplace eitherin two steps or in one step. Two-stepaggregationcan mean one of two things: (1) for each individual,the k-tuple of wel-fare information or each alternative s first aggregatedacross dimensionsinto an aggregatefigurefor thatindividual,and these aggregatefiguresarethen aggregatedacross individuals;(2) for each separatedimension, thedimension-specificwelfare nformationor each alternatives firstaggregatedacross individuals nto an aggregate igurefor thatdimension,and these ag-gregatefiguresarethenaggregatedacross dimensions.One-stepaggregationmeans thatthe entireprofileof k-dimensionalpersonalwelfare functionsisaggregateddirectly nto an overallsocial ordering.54.3. Multidimensional tilitarianrulesProposition4. Thereexist MSWFLssatisfying(CUdimUper),U), (P), (I), (D)and(Dom).Given interdimensional comparability and interpersonal comparabilityof units, suitable multidimensionalutilitarianrules satisfy (U), (P), (I),(D) and (Dom). All multidimensionalutilitarianrules are expressible astwo-step aggregationmechanisms.We will distinguishbetween those thatprioritise ntrapersonal ggregationandthose thatprioritise ntradimensionalaggregation.A MSWFL F is a multidimensionalutilitarian rule that prioritisesintrapersonalaggregation f there exist(i) n strictly increasing ntrapersonal ggregation unctionsfl, f2, ... fn

Rk 4 R;(ii) all,a2, ... , an 0 (with= 1) (weights corresponding to the nindividuals)such that, for any {Wi}ieN and any xl, x2 e X.



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133

X1Rx2 if and only if CieNaifi(< Wij(x1) >jeK) 1 CiNaifi (< Wij(x2) >jeK).Possible choices for fl, f2, ... ,fn are, for example, 'utilitarian' ntraper-sonalaggregation unctionsor 'maximin' ntrapersonal ggregationunctions(requiringat least (CLdimUdimUper))as definedabove.A SWFL F is a multidimensional utilitarian rule that prioritisesintradimensional ggregation f there exist(i) k strictly increasing intradimensional aggregation functions

fl, f2, . , fk : Rn - R;(ii) 8l,P2, ..., Sk 0 (with lijeK = 1) (weights corresponding o the kdimensions)suchthat,for any {Wi}iEN andxl, x2 e X,

X1RX2f and only if jeK jfj {Wij(Xl)}iN) > LjeK jfj({Wij(x2)ieN).Possible choices for fl, f2,.. ,fk are, for example, 'utilitarian' intradi-mensionalaggregation unctions or 'maximin' ntradimensional ggregationfunctions (requiring at least (CUdimLperUper))as defined above.Multidimensionalutilitarian ules are versions of the classical utilitarianprinciple:make social choices so as to maximize the sum-totalof welfare. Inthe case of rules thatprioritise ntrapersonal ggregation,anaggregate igureis first determinedfor each individual(by intrapersonalaggregation),andthese aggregatefiguresare then addedup across individuals.In the case ofrules thatprioritise ntradimensional ggregation,an aggregate igureis firstdetermined or each dimension(by intradimensional ggregation),and theseaggregate iguresare then addedup acrossdimensions.The conceptof two-stepaggregation ombinedwith different ypes of in-trapersonal rintradimensionalggregationunctions(e.g., utilitarian nes aswell as maximinones) greatlyextends the varietyof possible utilitarian ulesas comparedwith the one-dimensionalcase. Suppose,for example,we haveinterdimensionalomparabilitynot only of units but also of levels. We mightthenuse maximin ntrapersonal ggregation unctionsfor each individualasthe basis for utilitarianwelfaresummationacross individuals.Foreach indi-vidual and each alternative,we find the dimension n which the individual sworstoff and define the individual'saggregate igureto be his or her welfarein thatdimension.Thesum-totalof welfare s thendeterminedby summationof theseaggregate iguresacross ndividuals.Toconsidera differentexample,suppose we have interpersonal omparabilitynot only of units but also oflevels. We might then use maximinintradimensional ggregation unctions



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134as the basis for utilitarianwelfare summationacross dimensions:for eachdimensionand eachalternative,we identifythe individualwho is worst off inthat dimensionand define the dimension'saggregate igureto be the welfareof the worst-off individual in that dimension.The sum-total of welfare isthen determinedby summationof these aggregate iguresacross dimensions.Othercombinationsarepossible.The first of the two rulescaptures he view that the overall welfare of anindividual houldbe identifiedwith the lowestcomponentof this individual'swelfare k-tuple,but that subsequentaggregationacross individuals shouldtake the form of utilitarian ummation.The second rulecaptures he view that- in a Rawlsianspirit the overall 'social' welfare in each dimensionshouldbe identifiedwith the welfare of the worst-offindividual n thatdimension,butthat- diverging rom Rawls- subsequentaggregationacross dimensionsshould take the form of utilitarian ummation.

4.4. MultidimensionaleximinrulesProposition 5. There exist MSWFLs satisfying (OLdimLper),U), (P), (I), (D)and(Dom).Given interdimensional comparability and interpersonal comparabilityof levels, suitable multidimensionalpositional rules, in particular eximinrules, satisfy (U), (P), (I), (D) and (Dom).6 We will distinguishbetweenthreetypes of multidimensional eximin rules;two-steprules thatprioritiseintrapersonalaggregation, two-step rules that prioritise intradimensionalaggregation,andone-steprules.A MSWFL F is a multidimensional leximin rule that prioritisesintrapersonalaggregation f thereexist(i) n strictly increasing ntrapersonal ggregation unctionsfl, f2,.. fn "Rk -+ R; and(ii) for each x e X, a permutation - [i] of N (dependingon x) suchthat f[l](< W[l]j(x) >jeK) I f[2](< W[2]j(X) >jcK) * ... < f[n](< Wtnljx) >jeK);7

such that, for any {Wi}iEN and any x1, X2 E X,

X1PX2f andonly iff[i](< W[ilj(x1) >jeK) > f[i](< W[ij(x2) >jeK) for some i E Nand f[h](< W[hbj(X1) >jeK) -= f[h](< W[hl(X2) >jeK) for all h < i.



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135Possible choices for fl, f2, ... , fn are, for example, 'maximin' intraper-sonal aggregationfunctions or 'utilitarian' ntrapersonal ggregationfunc-

tions (requiring at least (CLdimUdimLperUper))as defined above.A MSWFL F is a multidimensional leximin rule that prioritisesintradimensional ggregation f there exist(i) k strictly increasing intradimensional aggregation functions

fl, f2,....fk : Rn -+ R; and(ii) for each x e X, a permutation [j] of K (dependingon x) such thatf[l]({Wi[1](X)}iEN) < f[2]({Wi[2](X)}ieN) fmj({Wiuj(X2)lieN) for somej e Kand f[h]({Wi[h](Xl)}ieN) = f[h]({Wi[h](X2)1ieN)or all h < j.Possible choices for fl, f2, .. fk are, for example, 'maximin' intradi-mensionalaggregation unctionsor 'utilitarian' ntradimensionalggregation

functions (requiring at least (CLdimUdimLperUper))as defined above.A MSWFLF is a (one-step)multidimensionaleximinrule if there exists(i) for eachx e X, a bijectionh - [h] = (i,j) from{1, 2, ..., nk}to Nx K

(depending on x) such that W[l] x) < W[2](x) < ... W[h](X2) for some h e {1, 2, ..., nk}and Wyg](xl) = W[g](x2) for all g < h.

Multidimensional eximin rules are versions of Rawls's (lexicographic)differenceprinciple:makesocialchoices so as to maximizethelowest welfarelevels; if there are ties, maximize, in a lexicographichierarchy, he secondlowest, third lowest, ..., welfare levels. These welfare levels are overallpersonalwelfare evels inthecase of rulesthatprioritise ntrapersonalggreg-ationandoveralldimensionalwelfare evels in thecase of rulesthatprioritiseintradimensionalggregation.Here the concepts of one-step and two-step aggregationcombined withdifferenttypes of intrapersonalor intradimensionalaggregationfunctions(e.g., utilitarianones as well as maximin ones) also extend the varietyofpossible leximin rules as comparedwith the one-dimensional ase. Suppose,for example, we have interdimensionalcomparabilityof levels and units.



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136We might then use utilitarian ntrapersonal ggregation unctions for eachindividualas the basis for leximin aggregationacross individuals.For eachindividualand each alternative,we take the (possibly weighted)sum of thisindividual's welfare levels in all dimensions to be the aggregatefigureforthat individual.The leximin rule is then appliedto these aggregatefiguresacross individuals:make social choices so as to maximize the lowest suchaggregate iguresacrossindividuals; f there areties, maximize,in a lexico-graphichierarchy,he secondlowest,third owest, ..., suchaggregate igures.To considera different xample, supposewe haveinterpersonal omparabilityof levels and units. We mightthen use utilitarian ntradimensional ggrega-tion functionsfor eachdimensionas the basis for leximinaggregationacrossdimensions. For each dimension and each alternative,we take the (possiblyweighted)sum of dimension-specificwelfare levels across individuals o bethe aggregatefigurefor thatdimension. The leximin rule is now appliedtothese aggregate iguresacross dimensions:make social choices so as to max-imize the lowest such aggregate iguresacrossdimensions; f thereareties,maximize,in a lexicographichierarchy, he second lowest, third owest, ...suchaggregate igures.The first of the two rules capturesthe view that the welfare levels ofan individual should be identifiedwith the (possibly weighted) sum of thecomponentsof this individual'swelfarek-tuple,but thatsubsequentaggreg-ation across individualsshouldfocus, lexically,on thelowest, secondlowest,thirdlowest, ..., such individualwelfare levels. This is one version of theRawlsiandifferenceprinciple,where the primarygoods indexing problemfor each individual s solved by summationacross differentprimarygoods.The secondrulecaptures he view thatfor each dimension he overallwelfarein that dimensionshould be identifiedwith the (possibly weighted) sum ofdimension-specificwelfare across individuals,but thatsubsequentaggrega-tion across dimensions shouldfocus, lexically,on the lowest, secondlowest,third owest, ..., such dimensionalwelfare levels.A one-step multidimensional eximin rule, finally, treats the multidi-mensionalaggregationproblemas if a one-dimensional eximin rule wereappliedto a society of nk individuals,effectivelyconsideringeach of the nkindividual-dimension airs as a distinctindividual and focussing, lexically,on the lowest, second lowest, third owest, ..., welfare levels amongstthesenk individual-dimension airs.

5. Some implicationsWhat conclusions can be drawn from this? Table 2 summarizesthe basicresults of thispaper.



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137In short, if we assume comparabilityacross individuals,but not acrossdimensions, the only aggregation proceduressatisfying Arrow-typecon-

ditions are (possibly lexicographic)hierarchiesof dimensions. And if weassumecomparability crossdimensions,butnot acrossindividuals, he onlyaggregationproceduressatisfying Arrow-typeconditions are (possibly lex-icographic) dictatorshipsof individuals.If we assumecomparabilityacrossboth individualsanddimensions,a rangeof aggregationprocedures atisfy-ing non-dictatorship nd non-dominanceemerge, such as multidimensionalutilitarian ulesandmultidimensionaleximinrules.Is it plausibleto assume nterpersonal omparability nd interdimensionalcomparability f welfare?Letme firstturn o interpersonal omparability.Arrow himself excludes interpersonal omparisonsof welfare from hisframework,holding"that nterpersonal omparisonof utilities has no mean-ing and, in fact, thatthereis no meaningrelevant o welfarecomparisons nthe measurability f individualutility" Arrow,1951/1963:9).In responseto this view, two pointsmust be noted.First,it is not obviousthatinterpersonal omparisonsof utilityare indeedas meaninglessas Arrowclaims they are,but even if interpersonal omparisonsof utilitywere empir-ically meaningless,this wouldnot entail that there is no othernon-empiricalbutnonethelessnormatively ignificantandnon-arbitrary ayof makingsuchcomparisons.Second, as noted above, the questionof whetherinterpersonal ompar-isons are meaningful in a given sense depends crucially on what welfareevaluationstandardwe choose to compare.To givejust two examples,inter-personalcomparisonsof moneyare unproblematic leavingpractical ssuesaside),andinterpersonal omparisonsof the amountof health care or educa-tionapersonhasaccess to mayalso be unproblematicagain eavingpracticalissues aside). This means that, even if we concede Arrow's view that "inter-personalcomparisonof utilities has no meaning" (my italics), this entailsnot thatinterpersonal omparisonsare in principle mpossible,butonly thatsuch comparisonsarepossible only for certainwelfareevaluationstandardsother thanutility.And severalwelfare evaluationstandardswiththerequiredpropertieshavebeenproposed, ncludingRawls'sindex of primarygoods andSen's functionings.Comparability crossdimensions s a more seriousproblem.An importantintuitionunderlyingmany arguments or a constitutivelypluralconceptionof welfare is thatwelfarehas several,possibly mutually ncommensurable,aspects, and that those aspects cannot easily be combined into a singleoverall index. Accordingly,the results of the presentpaperraise a tensionthat needs to be takenseriously.In particular,hey raise a tension between



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138Table2.

SupposeF is a MSWFL atisfying I), (P),and(U).Do we haveinterdimensionalomparability?

Yes No

Do we haveinterpersonalcomparability?

Do we haveinterpersonalcomparability?

Yes No Yes No

Which ypeof interpersonalcomparabilitynd nterdimensionalcomparabilityo we have?

of units of levels

F can be amultidimensionalutilitarianule.F canbe amultidimensionalleximinrulerule.

iis a(possiblylexicographic)dictatorship.

F is a'(possiblylexicographic)hierarchy fdimensions.

F is a(possiblylexicographic)doubledictatorship.

the followingthree claims:(i) There exist severalnormativelyrelevantdimensions of welfare,which(for instance, as a result of their different social meanings) are not

mutuallycommensurable.(ii) There are decisions which affect different dimensions, but whichcome in 'packages', indivisible into separate dimension-specificsub-decisions.



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139(iii) In makingdecisions of the kinds describedby (ii), thereshouldbe nodominantdimension hathas lexicalpriorityndeterminingheoutcome.It seems that a violation of at least one of (i), (ii), or (iii) is inevitable.Prima acie, Walzer'sapproach o constitutivepluralityseems more vulner-able to this tension than Sen's or Rawls's. Walzer'sapproachrequiresthatdifferentdimensions(spheresof goods) be kept separate.As a result of thisrequirement,he approach annot accommodatecross-dimensionalndexingas easily as Sen's or Rawls's approaches.The most consistent Walzerianescape from the tension,presumably he one pursued n Spheresof Justice,would be to tryto avoidclaim(ii). Indeed,whenWalzer'sargues hatdifferentdistributiveprinciplesshould be appliedto differentdimensions(spheresofgoods), on the basis of the different social meanings of these dimensions(goods), Walzer'sproposalmight be interpretedas a version of the abovediscussed strategyof subdividingdecisions into severaldimension-specificsub-decisions and thus 'defining away' the multidimensionalaggregationproblem.But, as we have indicatedabove, while lexicographicdictatorshipsarehardlyattractive rom a liberal-egalitarian iewpoint,there are situations nwhich lexicographichierarchiesof dimensions(i.e. violations of claim (iii))are defensible. (Thus there is an asymmetrybetween the normativeappealof non-dictatorship nd the normativeappeal of non-dominance.)Rawls's'special conception' of justice explicitly assigns lexical priorityto someprimarygoods, such as liberty and opportunity,over others,thus in effectdefininga lexicographichierarchy f dimensions.If there s a tensionbetweenholding a constitutivelypluralconceptionof welfareand acceptinginterdi-mensionalcomparability,henRawls's defence of a lexicographichierarchyof differentprimarygoods in the 'specialconception'of justice is clearlythesocial-choice-theoreticallymost compellingsolution to thattension.

Notes1. Interpersonaland interdimensionalcomparabilitywill be treated in formally similarways. According to some philosophicalconventions,what I call comparabilityacrossdimensions is called commensurability cross dimensions. Forsimplicity,I will refer to

comparability n the case of bothpersonsanddimensions.2. Thequestionof how therelevantdimensionsareto be identified i.e., the questionof howthe set K is to be interpreted)s animportant hilosophicalmatter,but t will herebe takento lie outside the scope of social choice theory.Nussbaum'sargument or a specificlist ofbasic functionings(Nussbaum,1992) and Walzer'saccount of how social meaningsde-termine he relevantdistributive pheres Walzer,1983)aretwodifferentapproacheso theidentificationanddemarcation f relevantdimensions,corresponding o the two different



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140accountsof constitutivepluralitybrieflyintroducedabove (essentialistandcontextualist,respectively).3. A transformation4 : R -+ R is positive monotonicif, for all s, t in R, s < t implies4(s) < 4 (t).4 : R - R is positive affineif there exist a, b in R, with b > 0, such that,for all t in R, 4 (t) = a + bt.4. For a moredetaileddiscussion of the logical relationbetweenmeaningfulstatementsandclasses of admissibletransformationsn the one-dimensionalcontext, see Section 5 ofBossert andWeymark 1996).5. Note that all two-step aggregationmechanismscan also he interpretedas one-step ag-gregationmechanisms,but not all one-step aggregationmechanismsare expressibleastwo-stepaggregationmechanisms.6. For simplicity, I will here focus only on leximin rules. The results can, however,begeneralizedto otherpositionalrules. Further,he leximin rules discussedhere all satisfysome formof ananonymityrequirement,.e., invarianceunderpermutation f individualsand/or nvarianceunderpermutation f dimensions.For the discussionof non-anonymouspositionalrulesin the one-dimensionalcase, see Bossert andWeymark 1996).7. To define a leximax rule, we simply need to replace all the ""-symbols in the definitionof the permutation [i]. Analogous remarksapply to thedefinitionsbelow.

8. Robertsproves this result for a domain more restrictedthan a universaldomain (U),namely for an ordinalagreementunrestricteddomain(OAU) (hence U is statedhere insquarebrackets).He explains,however,that "OAUcan be changedto NPU or U to givethe same result as stated n Theorem6" (p. 164).9. Robertsuses the strongParetoprinciple,butpointsout that the weak Paretoconditionissufficient or the resultsof his paper(p. 144).10. Under the specified identificationof Roberts'sframeworkwith our framework,FC isequivalent o (CNdimLperUper).

ReferencesArrow,K. (1951/2nded. 1963). Social choice and individualvalues. New York:Wiley.d'Aspremont,C. (1985). Axioms for social welfareorderings. n L. Hurwicz,D. SchmeidlerandH. Sonnenschein Eds.), Social goods and social organization:Essays in memoryofElishaPazner,19-76. Cambridge:CambridgeUniversityPress.Batra,R. and Pattanaik,P.K. (1972). Transitivemulti-stage majoritydecisions with quasi-transitive ndividualpreferences.Econometrica40: 1121-1135.Blair, D.H. (1988). The primary-goods ndexationproblemin Rawls's 'Theoryof Justice'.Theoryand Decision 24: 239-252.Bossert,W. andWeymark, .A.(1996). Utilityin socialchoice. Forthcomingn S. Barbera,P.J.Hammondand C. Seidel (Eds.),Handbookof utilitytheory,Volume2. Boston:Kluwer.Fishburn,P.C.(1971). Thetheoryof representativemajoritydecision.Econometrica39: 273-284.Gibbard,A. (1979). Disparate goods and Rawls's difference principle: A social choicetheoretic reatment.TheoryandDecision 11:267-288.Kelsey, D. (1987). The role of information n social welfarejudgements.OxfordEconomic

Papers39: 301-317.



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141Khmelnitskaya,A.B. (1999). Social welfare orderingsfor differentsubgrouputility scales.Discussion paper#198, Center for Rationalityand InteractiveDecision Theory,Hebrew

Universityof Jerusalem.Khmelnitskaya,A.B. and Weymark,J.A. (2000). Social choice with independentsubgrouputilityscales. Social Choice and Welfare17:739-748.List,C. (2001). A note on introducing 'zero-line'of welfare as anescape-route romArrow'sTheorem.PacificEconomicReview6 (special section in honourof AmartyaSen): 223-238.List,C. (2002). Intradimensionalingle-peakedness nd the multidimensionalArrowProblem.Theoryand Decision 52: 287-301.Nussbaum,M. (1992). Human functionings and social justice: In defense of Aristotelianessentialism.Political Theory20: 202-246.Plott, C.R. (1978). Rawls's theoryof justice: An impossibilityresult.In H.W.GottingerandW. Leinfellner Eds.),Decision theoryand social ethics, 201-214. Dordrecht:D. Reidel.Plott, C.R., Little,J.T. andParks,R.P.(1975). Individualchoice when objectshave 'ordinal'values. Reviewof EconomicStudies42: 403-413.Rawls,J. (1971). A theoryof justice. Cambridge,MA: HarvardUniversityPress.Roberts,K. (1995). Valuedopinionsor opinionatedvalues: The doubleaggregationproblem.In K. Basu, P.K.PattanaikandK. Suzumura Eds.), Choice, welfareand development:AFestschrift nHonourof AmartyaSen, 141-165. Oxford:OxfordUniversityPress.Sen, A.K. (1970/79). Collectivechoice and social welfare.San Francisco:Holden-Day.Sen, A.K. (1980/81). Pluralutility.Proceedings of the AristotelianSociety81: 193-215.Sen, A.K. (1982). Choice,welfare,and measurement.Oxford:Blackwell.Sen, A.K. (1985). Commodities nd capabilities.Amsterdam:North-Holland.Sen, A.K. (1987). Thestandardof living.Cambridge:CambridgeUniversityPress.Sen, A.K. (1991). On indexingprimarygoods andcapabilities.Mimeo. HarvardUniversity.Sen, A.K. (1997). On economicinequality.Enlargeded. Oxford:OxfordUniversityPress.United NationsDevelopmentProgramme UNDP) (2002). Humandevelopmentreport2002.New YorkandOxford:OxfordUniversityPress.Walzer,M. (1983). Spheresofjustice. New York:Basic Books.



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142Appendix 1: Proof of Theorems 4 and 5Both theoremscan be deducedfrom theorem6 in Roberts 1995). A formallycloselyrelatedresult is Khmelnitskaya 1999, theorem2), also presentedwith a modifiedproofin Khmelnitskaya ndWeymark 2000, theorem2), as mentionedabove. Thisresultuses a continuitycondition n addition o conditionsequivalent o the onesusedhere. The result can also he restated or thepresentmultidimensional ramework."Theorem6. Let there be n opinions aboutthe well-being of r individuals.(Inparticular, = r is permitted.) f f satisfies[U]8,I, P9, andFC10, hen there exists anindividual anda strictlymonotonic functionW withr arguments uchthat,for allx, y X,

W(u(x, i)) > W(u(y, i)) = xPy;i.e., individual 's opinionsare dictatorial."We will explicitly deduce only theorem 2 from Roberts's theorem (hereaftersimplyreferred o as R); theorem3 can be deducedanalogously.Assume, for a contradiction, hat F is a MSWFL satisfying (CNdimLperUper),(U), (P), (I) and (Dom) (any MSWFL satisfying (ONdimLper) ill also satisfy(CNdimLperUper)).eidentifyn in R (thenumberof opinions)withk (thenumberofdimensions),r (the numberof individuals) n R with n (the numberof individuals),u(x, i, j) (theopinionof j aboutthe welfare of i underalternative ) in R withWij x)(thewelfare of individual in dimension underalternative ), andf in R withF. It iseasily seen thatf satisfiesR's conditionsU, I, P andFC, whence,by R, there existsa strictly ncreasingfunctionW with r (= n in ourframework)arguments uch that,for all x, y E X,

W(u(x, -, i)) > W(u(y, ., i)) =i xPy (in R),i.e., forall x, y E X,

f({Wij(x)}iEN)> f({Wij(y))iEN) xPy,wherethe functionW in R is identifiedwiththe function . This contradicts ondition(Dom) in our framework.To deducetheorem3, simplyidentifyn in R with n, r in R with k, u(x, j, i) in RwithWij x), andf in R with F. O

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