cardinality versus ordinality: a suggested compromise · cardinality versus ordinality: a suggested...

Cardinality versus Ordinality: A Suggested Compromise

By MICHAEL MANDLER*

By taking sets of utility functions as primitive, we define an ordering over assump-tions on utility functions that gauges their measurement requirements. Cardinal andordinal assumptions constitute two levels of measurability, but other assumptions liebetween these extremes. We apply the ordering to explanations of why preferencesshould be convex. The assumption that utility is concave qualifies as a compromisebetween cardinality and ordinality, while the Arrow-Koopmans explanation, sup-posedly an ordinal theory, relies on utilities in the cardinal measurement class. Insocial choice theory, a concavity compromise between ordinality and cardinality isalso possible and rationalizes the core utilitarian policies. (JEL D01)

Ordinal utility theory permits only those as-sumptions on utility functions that are preservedunder increasing transformations. The rationalefor this rule is that any assumption P not pre-served under increasing transformations cannotbe verified with observations of choice behav-ior: if a utility u satisfies P, there will existanother utility u� that does not satisfy P but thatrepresents the same preferences as u. Nonordi-nal assumptions therefore seem to be needlesslyrestrictive: given a nonordinal assumption, onemay always make a weaker assumption with thesame implications for choice behavior. Accord-ing to this view, the only role for cardinal utilityis the normative one of representing interper-sonal comparisons of utility.

Ordinalism’s first targets were diminishingmarginal utility (DMU) and concavity, whichhad long served as arguments for why consumerpreferences should be convex. Neither DMU norconcavity is preserved by increasing transforma-tions and hence neither is an ordinal assumption.The pioneer ordinalists such as Kenneth Arrow(1951) claimed that diminishing marginal utility istantamount to assuming that utility is cardinal.Arrow’s position remains predominant: either anassumption on utility is ordinal or it is cardinal(see, e.g., Andreu Mas-Colell et al., 1995, chap. 1).

This paper will take sets of utility functions—which we call psychologies—as primitive; thiswill define a finer gradation of properties ofutility that allows for intermediate standards ofmeasurement. In this framework, ordinal utilitytheory takes the psychology consisting of allincreasing transformations of any given utilityfunction as primitive and lies at one end of themeasurement spectrum.1 Cardinal utility theorytakes the psychology consisting of all increas-ing affine transformations of a given utility asprimitive; since this is a smaller set of utilityfunctions, cardinality is a stronger (more restric-tive) theory than ordinality. Ratio scales whichtake still smaller psychologies as primitive (thefunctions generated by all increasing lineartransformations) are also common, particularlyoutside of economics. But in addition to thesewell-known measurement scales, there is an in-finity of intermediate cases. Diminishing mar-ginal utility and concavity lie precisely in themiddle ground between cardinality and ordinal-ity. The psychology consisting of all concaveutility representations of a given preference re-lation is larger than any cardinal psychology itintersects, but smaller than any ordinal psychol-ogy it intersects. Concavity thus presupposes anintermediate standard of measurement and doesnot deserve its strongly nonordinalist reputa-tion. Compared to a cardinal psychology, a con-cave psychology is easier to assemble in that anagent does not have to make as many utility

* Department of Economics, Royal Holloway College,University of London, Egham, Surrey TW20 0EX, UnitedKingdom (e-mail: [email protected]). I thank NickBaigent, Douglas Bernheim, Juan Dubra, Birgit Grodal,Duncan Luce, Roy Radner, Herbert Scarf, Phil Steitz, JohnWeymark, and an anonymous referee for many helpfulcomments, corrections, and suggestions.

1 We follow the convention of letting “the transformation fof a function u” refer to the composition f � u rather f itself.

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judgments. And concave psychologies are a nat-ural primitive: they pinpoint the neoclassicalprinciple that the value agents place on con-sumption increments progressively diminishes.

In preference theory, the main advantage ofendowing agents with nonordinal properties ofutility is that they can provide rationales forassumptions on preferences. Concavity and DMUform the leading examples and offer explanationsof why and when preferences should be convex.To declare by fiat that preferences are convex(or that utility is quasiconcave) in contrast pro-vides no rationale. That DMU lives on in text-books as a motivation for convexity reflects theneed for justification; but by relying on DMU,the textbook stories leave the measurement sta-tus of consumer theory up in the air. Textbooksin fact often suggest that DMU in one good doesnot affect the marginal utility of other goods,which amounts to an assumption of additiveseparability and hence of cardinality. In con-trast, a concavity explanation of preference con-vexity incurs relatively small measurement cost.

In normative economics as well, a concavitycompromise is possible. If interpersonal compar-isons of utility are modeled with a set of concaveutility functions, then one may still derive theplausible core of utilitarianism. Indeed, the poli-cies backed by a concave psychology betterapproximate commonly held utilitarian intui-tions than the policies derived from cardinalversions of utilitarianism. Moreover, due to theircompromise status, concave interpersonal com-parisons are less demanding than the cardinalcomparisons that underlie traditional utilitarian-ism: fewer utility comparisons need to be made.We thus take issue with both prongs of the ordi-nalism-is-for-positive-theory versus cardinalism-is-for-normative-theory dichotomy.

We argue that concavity can be tested obser-vationally, although of course not using choicedata alone. While economists once saw the We-ber-Fechner law—that the marginal sensitivityto physical stimuli diminishes with stimulusintensity—as empirical support for DMU, thelink to Weber-Fechner remained little morethan an analogy. More recently, cognitive neu-roscience and neuroeconomics have opened anew window on the physiological correlates ofconsumption decisions that suggests we may beable to compare a consumer’s gains in satisfac-tion relative to a reference consumption level.As we will see, such a gauge would make it

possible to test experimentally for DMU in con-sumption. Here, too, the existence of compro-mise measurement classes between ordinalityand cardinality is relevant; the neural correlatesof consumption decisions provide nonordinalinformation that may be insufficient to test forcardinality. Perhaps because of the cardinal ver-sus ordinal gulf, neuroeconomics has not ad-dressed such traditional economic topics as themeasurement of utility. Compromise measure-ment categories may be able to reintegrate thestudy of consumer demand into a physiologyand psychology of consumption.

The literature on self-reported happinesslevels (e.g., Richard A. Easterlin, 1974; DavidG. Blanchflower and Andrew J. Oswald, 2004)would also gain from scrutiny of its measure-ment presuppositions. When the current litera-ture judges a regression’s goodness-of-fit as thesum of differences between reported and pre-dicted contentment levels, it equates units ofcontentment and thereby treats well-being asa cardinal scale. One could instead judge fit by acompromise measurement standard in which alldeviations of a subject’s ranking of satisfactionincrements from a predicted ranking would betreated as equal goodness-of-fit failures.

By taking sets of utility functions as primi-tive, we formalize what a nonordinal assumptionon utility entails and judge just how nonordinal itis. To say simply that some agent’s utility functionsatisfies diminishing marginal utility, for example,leaves open the question of exactly which utilityfunctions accurately describe the agent’s con-sumption experience. Cardinal understandings ofdiminishing marginal utility prior to the ordinalrevolution would take a utility function satisfyingDMU and all of its increasing affine transforma-tions as primitive; all functions in this set wouldbe taken as equivalent in every meaningful re-spect. But if one wishes to impose DMU andnothing more, then any utility that both exhibitsDMU and represents the same preferences shouldbe regarded as equally accurate. We thereforecharacterize the assumption of DMU using pre-cisely this set of utility functions; in the samesense a cardinalist would, we take all these func-tions to be equivalent. This characterization allowsus to gauge the extent of DMU’s nonordinalityand conclude that the assumption imposes rela-tively little measurement cost.

We begin the paper by showing that essentiallyany transitive preference relation � can be repre-

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sented by a psychology (even when � is incom-plete). Then we order psychologies according toset inclusion. An ordering of assumptions on util-ity functions by set inclusion of the psychologiesthat satisfy those assumptions would not take usvery far; it would apply only to trivial cases whereone assumption implies another. But by requiringfor a pair of assumptions that set inclusion holdsonly when a psychology has utilities that satisfyboth assumptions simultaneously, we can avoiddebilitating incompleteness, and measurementclasses of assumptions can be calibrated finely.Claims that may appear nonsensical—e.g.,“concavity” is a weaker assumption than “addi-tive separability”—can be stated rigorously.

We then illustrate the ordering by compar-ing concavity with another celebrated ratio-nale for the convexity of preferences, Arrow’s(1951) argument (following Tjalling Koop-mans) that an agent’s leeway to determine theprecise timing of consumption implies thatpreferences must be convex. Arrow reasonedthat this rationale for convexity, unlike dimin-ishing marginal utility, was free from anytaint of cardinality. We show, however, thatthe utility theory that lies behind the Arrow-Koopmans position is cardinal. The old neoclas-sical explanation, diminishing marginal utility orconcavity, thus rests on less demanding measure-ment assumptions.

Next, we compare cardinality and concavityas a basis for utilitarianism. If a psychologyconsisting of concave interpersonally compara-ble utility functions is taken as primitive we canstill derive the utilitarian recommendation thatwealth be redistributed from rich to poor. The“concave utilitarianism” that results allows atight derivation of commonly held utilitarianconvictions, a goal that cardinal utilitarianismhas notably failed to achieve. For example, con-cave utilitarianism characterizes the famous Pi-gou-Dalton principle, which adds egalitarianconstraints to a sum-of-cardinal-utilities order-ing. Even Francis Y. Edgeworth (1897), thefounding economic utilitarian, was suspiciousof policy conclusions that relied on the cardinaldetails of a utilitarian social welfare functionrather than on its concavity alone. Concave util-itarianism’s better fit makes sense: it utilizesonly the psychological premises that underlieneoclassical social welfare intuitions.

The standard cardinal formulation of utilitar-ianism requires an intimidating array of utility

comparisons: a planner must post an exact rateat which gains in utility for one agent translateinto gains for other agents. In contrast, for aconcave utilitarian to dispense policy advice, allthe planner must do is answer two questionsaffirmatively. First, is j’s gain judged to beordinally greater than i’s loss? (The exact ratioof j’s gain to i’s loss is now irrelevant.) Second,is i’s welfare level greater than j’s? This recipefor concave utilitarian policies is similar to“poverty orderings” of income distributions,but via the device of least concave utility therecipe is applicable to agents who consumemultiple goods. Concave utilitarianism’s reli-ance on a more limited set of utility compar-isons may be able to dampen the skepticismthat surrounds social welfare judgments. Thecardinal view demands hopelessly many in-terpersonal comparisons, encouraging thesuspicion that those comparisons are whollyarbitrary and subjective.

We close by looking at how neural correlatesof consumption can be used to test for concav-ity, and at how the present paper fits with earlierwork on measurement.

I. Psychologies and Orderings of Psychologies

An agent will be represented by a set ofutility functions U, called a psychology, whereeach u � U is defined on the same set X, whichwe call the domain of U. A psychology lists theutility functions that as a set characterize theagent’s psychological experience of consumingthe bundles in X. Only those properties theutility functions in U have in common describethe agent’s consumption experience. For ex-ample, suppose an agent has a psychology Ucontaining every continuous function that rep-resents some nontrivial preference relation on�n. This agent would then experience happinessor satisfaction as a continuous sensation, but anattribute of one of the utilities in U that is notshared by every other utility in U would notreflect the agent’s attitudes toward consump-tion. One function in U might be concave, forexample, but since there would then be noncon-cave functions in U as well, this agent would notbe characterized by the property of concavity.Psychologies can be assembled from severalsources: choice behavior, physiological and psy-chological measurement (see Section V), agents’testimony (Section V), and introspection.

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Preference relations emerge straightforwardlyfrom psychologies. We define �U, the preferencesinduced by U, by

x �U yN u�x� � u�y� for all u � U.

We will refer interchangeably to �U satisfying anassumption on preferences and the underlyingpsychology U satisfying the same assumption.

Induced preferences can satisfy the assump-tions usually imposed on preferences in eco-nomics. For instance, �U is complete if andonly if, for all u, v � U and x, y � A, u(x) �u(y)N v(x) � v(y). For any psychology U, �Umust be transitive.

Since psychologies consist of sets of utilityfunctions, they provide a more flexible tool forrepresenting preferences than solitary utility func-tions. In fact, psychologies can induce almost anypreference relation: except for transitivity and atechnical requirement, any preference relation canbe induced by some psychology, as Theorem 1reports. We do not pursue the details, but we couldgeneralize psychologies to cover even intransitivepreferences by dropping the requirement that allfunctions in U have a common domain.

A binary relation � on X is countably order-dense if there exists a countable Y � X such thatfor all x, z � X with x � z (i.e., x � z and not z� x), there exists a y � Y such that x � y � z.

THEOREM 1: If � is reflexive, transitive, andcountably order-dense, there is a psychologythat induces �.

Proofs can be found in the Appendix. Reflex-ivity, transitivity, and countable order-density arestandard assumptions that underlie the existenceof a utility representation for preferences. Indeed,the only difference between Theorem 1 and thestandard representation result is that Theorem 1does not suppose � is complete. Theorem 1 thusmakes do with weaker conditions than the Efe A.Ok (2002) result on representing an incompletepreference relation � with a (possibly infinite) setof utility functions. Juan Dubra et al. (2001) showthat an expected utility representation of possiblyincomplete preferences on lotteries also requiresno extra assumptions beyond the standard.

We turn to the ordering of the strength ofpsychologies. To keep things simple, we hence-forth require all psychologies to be complete.So every u � U is a utility representation for �U.

DEFINITION 1: A psychology U is no strongerthan a psychology V if and only if U � V.

Since set inclusion is transitive, the “no stron-ger than” relation on psychologies is transitiveas well. But when X has more than one element,the “no stronger than” relation is incomplete(e.g., if U consists of all utilities that representsome preference relation � and V consists of allutilities that represent some �� , then U �V � A). We define a “weaker than” relation onpsychologies as the asymmetric part of the “nostronger than” relation: U is weaker than V ifand only if U is no stronger than V and it is notthe case that V is no stronger than U.

To illustrate, consider three psychologies: Uconsists of every utility representation of somepreference relation �, V consists of some rep-resentation u of � and all increasing affinetransformations of u, and W contains only thesingle function u. Then U is weaker than V, andV is weaker than W (for the former conclusionwe need to assume that u has at least threedistinct points in its range). As this exampleindicates, weaker psychologies make fewer as-sertions about the utility experiences of agentsand should therefore be easier for agents toassemble. To put the matter in reverse, aspsychologies become stronger or smaller, theutility experiences described become moreand more limited. An agent with the psychol-ogy U makes only ordinal preference judg-ments, while an agent with the psychology Vmakes the same ordinal judgments as the Uagent, but in addition makes cardinal com-parisons of utility. That is, for any ( x, y, z, w)with u( z) � u(w), the V agent experiences aspecific ratio of utility differences

u�x� � u�y�

u�z� � u�w�.

Given (x, y, z, w), all v � V maintain the samevalue for this ratio. Finally an agent with psychol-ogy W not only makes the ordinal judgments ofthe U agent and the cardinal comparisons of the Vagent, but can in addition assign an exact utilitynumber to every consumption experience. Single-ton psychologies such as W are so implausiblystrong that they have never been employed ineconomics or measurement theory. Appropriately,therefore, our ordering of psychologies places sin-gleton psychologies at a polar extreme: a singleton


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psychology can never be weaker than any otherpsychology.

II. Properties of Utility and Orderingsof Properties

We now use the ordering over psychologiesto generate orderings over properties of utilityfunctions. Formally, a property P is simply a setof functions into the real line, where the do-mains of the functions can differ, and a utilityfunction u: A 3 � satisfies property P if andonly if u � P. We mostly consider familiar prop-erties that are routinely imposed on utility func-tions. For instance, one can define the property ofcontinuity as the set of all continuous functionsfrom arbitrary subsets of �n into the real line.

DEFINITION 2: A psychology U maximallysatisfies property P if and only if for each u �U, u satisfies P, and there does not exist apsychology V �

/U such that each v � V satis-

fies P.

In words, a psychology maximally satisfies Pif it is largest among psychologies whose con-stituent utility functions all satisfy P. So, forexample, a U with domain ��

n would maxi-mally satisfy the property of continuity if andonly if U consists of all functions u: ��

n 3 �that are continuous and that represent the samepreference relation. The domain of a psychol-ogy may determine whether it maximally satis-fies a property. For instance, if X � ��

n is finite,any psychology consisting of all utility func-tions that represent some � on X maximallysatisfies continuity, but this is certainly not thecase if, say, X � ��

n .The earlier ordering of psychologies suggests

the following ordering of properties.

DEFINITION 3:

(i) Property P is no stronger than property Q,or P �NS Q, if and only if for all U thatmaximally satisfy P and all V that maxi-mally satisfy Q, U � V � A implies U � V.

(ii) Property P is weaker than property Q, orP �W Q, if and only if P �NS Q and notQ �NS P.

In words, P is no weaker than Q if, wheneverP and Q are comparable (some utility satisfies

both simultaneously), the psychology maxi-mally satisfying P is no stronger than the psy-chology maximally satisfying Q. The relations�NS and �W need not be transitive or complete.Since our ordering of psychologies itself is notcomplete, the incompleteness is to be expected.Still, the incompleteness of �W can be extensivesince P and Q are unranked by �W if P and Qare inconsistent, that is, if P � Q � A. Forinstance, let P and Q denote, respectively, theproperties of strict concavity and strict convex-ity on nonsingleton convex subsets of �n.Then, since whenever a U maximally satisfies Pand a V maximally satisfies Q, it is always thecase that U � V � A, we have both P �NS Qand Q �NS P and therefore neither P �W Q norQ �W P.

The intransitivity of �NS and �W may comeas more of a surprise. We consider this subjectin separate papers (Mandler, 2001, 2003) whichshow that there are natural domains on which�W, and related orderings are transitive oracyclic.

An alternative “weaker than” ordering will beuseful later. First, we say a property P intersectsproperty Q when P � Q � A.

DEFINITION 4: Property P is strictly weakerthan property Q, or P �SW Q, if and only if (1)P intersects Q and (2) whenever U maximallysatisfies P, V maximally satisfies Q, and U �V � A, then U �

/V.

In contrast, property P is weaker than Q if it ismerely the case that P is no stronger than Q andthere is some U that maximally satisfies P andsome V that maximally satisfies Q such that U isweaker (in the ordering of psychologies) than V.

DEFINITION 5 (Ordinality): A psychology Uis ordinal if and only if whenever u � U then (v� U N there exists an increasing transforma-tion g such that v � g � u).2

If we say that two functions u and v agree when,for all x and y in X, u(x) � u(y)N v(x) � v(y),then a complete psychology U is ordinal if and

2 A function g: E3 �, where E � �, is (a) an increasingtransformation if and only if, when x, y � E and x � y, theng(x) � g(y); and is (b) an increasing affine transformationif and only if there exist a � 0 and b such that, for all x � E,g(x) � ax � b.


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only if u � U implies that when v agrees with uthen v � U.

DEFINITION 6 (Cardinality): A psychology Uis cardinal if and only if whenever u � U then (v� U N there exists an increasing affine trans-formation g such that v � g � u).

Keep in mind that utility functions are theprimitive. Consequently, theories that are some-times called cardinal, such as the von Neumann–Morgenstern theory of preferences over lotter-ies, do not necessarily qualify as cardinal in thecurrent schema. For instance, let the domain bethe set of lotteries over n prizes, X � {p � ��

n :¥i�1

n pi � 1}. While it is true that an agent withvNM preferences � over X will have as one ofits utilities a function v of the form v(p) � ¥i�1

n

pivi, the psychology of the agent is a distinctentity. The agent could have (a) an ordinalpsychology consisting of all increasing transfor-mations of v, (b) a cardinal psychology consist-ing of all increasing affine transformations of v,(c) some other cardinal psychology consistingof all increasing affine transformation of someutility that does not take the expected utilityform, or (d) a psychology that is neither ordinalnor cardinal. It may be plausible that vNMagents will make cardinal comparisons of utilitythat are representable by their vNM utility func-tions, but the vNM axioms do not entail thisconclusion.

We now define properties of utility as ordinalor cardinal and consider a couple examples.

DEFINITION 7: A property P is ordinal (resp.cardinal) if and only if any U that maximallysatisfies P is ordinal (resp. cardinal).

To illustrate how these definitions work, con-sider the ordinal property of quasiconcavity.Define a function u: Z3 � to be quasiconcaveif Z is a convex set and, for all x, y � Z and � �[0, 1], u(�x � (1 � �)y) � min{u(x), u(y)}. Toconfirm that quasiconcavity is ordinal, let Umaximally satisfy quasiconcavity and let u bean arbitrary element of U. Then, if u and vagree, there is an increasing transformation f:Range u 3 � such that f � u � v; since f isincreasing, for all x, y � Z and all � � [0, 1],v(�x � (1 � �)y) � min{v(x), v(y)}. Hence vsatisfies quasiconcavity and so, by Definition 2,v � U.

Additive separability will serve as a usefulexample of a cardinal property.

DEFINITION 8: A function u: A3 � satisfiesadditive separability if and only if, for someinteger n � 2, there exist component spaces Ai,i � 1, ... , n, such that A � A1 ... An andfunctions ui: Ai3 �, i � 1, ... , n, such that (1)for each x � A, u(x) � ¥i�1

n ui(xi), (2) for eachcomponent i, Range ui is an interval, and (3) atleast two of these intervals have nonempty in-terior.

The property of additive separability consists ofthe set of all functions that satisfy additiveseparability.

THEOREM 2: The property of additive sepa-rability is cardinal.

For a proof, see David H. Krantz et al. (1971),or for a more general cardinality result for ad-ditively separable functions, Mandler (2001).Theorem 2 provides the basis for Theorem 5,which establishes the cardinality of the Arrow-Koopmans theory of convex preferences.

III. Convexity of Preferences

We consider two rationales for convexity andassess their measurement implications.

A. Concavity as a Primitive

The preferences � defined on the domainX � �n are convex if and only if, for all y � X,the set {x � X: x � y} is convex. A function u:Z3 � satisfies concavity if and only if Z � �n

is a convex set and, for all x, y � Z and all � �[0, 1], u(�x � (1 � �)y) � �u(x) � (1 ��)u(y). As is well known, if the preferences �have a utility representation that satisfies con-cavity, then � is convex. So, if a psychology Umaximally satisfies concavity, then �U is convex.

Concavity, which means that agents experi-ence diminishing marginal increments of satis-faction as they progressively increase theirconsumption along any line, has long served asan informal rationale for why preferencesshould be convex. But since assumptions onutility are usually labeled as either ordinal orcardinal, concavity has appeared to be cardinal,thus adding to concavity’s illegitimacy. The


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next theorem shows, however, that concavity isin fact weaker than cardinality. Recall that prop-erty P intersects property Q if P � Q � A.

DEFINITION 9: Property P has range � n ifand only if, for each u � P, �Range u� � n.

THEOREM 3:

(i) Any ordinal property is no stronger thanconcavity, and concavity is no strongerthan any cardinal property.

(ii) Concavity is strictly weaker than any car-dinal property with range � 2 that inter-sects concavity.

(iii) Any ordinal property with range � 1 thatintersects the property of being concaveand continuous is strictly weaker than theproperty of being concave and continuous.

Remark. Due to the fact that a concave func-tion need not be continuous on the boundary ofits domain, there are nontrivial preferences—even preferences that exhibit strict preferencebetween arbitrarily many pairs of consumptionbundles—whose only utility representations areconcave. It follows that there are ordinal prop-erties with arbitrarily large range that are notweaker than concavity.

Theorem 3 (ii) reports an important fact aboutconcavity. Along a line, concavity as a psychol-ogy assumes that an agent experiences eachsuccessive unit of consumption as delivering asmaller increment of utility or satisfaction. Butconcavity does not require that each utility in-crement is a specific fraction of the previousincrement: agents experience diminishing mar-ginal utility but no additional extra-ordinal pre-cision. Cardinality, in contrast, requires thatagents can make cardinal comparisons of utility.Hence, the ratio of utility increments for somegiven pair of consumption changes can assumeonly one value: a cardinal agent who satisfiesconcavity would be able to say that an addi-tional unit of consumption delivers additionalutility equal to some specific percentage of theprevious unit of consumption. Cardinality thusrequires an agent to make an implausibly exten-sive set of psychological judgments.

Concavity can be ranked relative to someother classical assumptions in utility theory. If Xis a convex nonempty open subset of �n, theproperty of continuity on X is weaker than con-

cavity on X, and the property of being continu-ous and nonconstant on X is strictly weaker thanconcavity on X. If Y � �n contains a nonemptyopen set, then any ordinal property that is non-constant on Y is strictly weaker than continuityon Y. These assertions follow from the fact thatany concave function on an open set is contin-uous, but not vice versa, and the fact that anycontinuous increasing transformation preservescontinuity, but noncontinuous increasing trans-formations do not preserve continuity. We omitthe details, which vary only slightly from theproof of Theorem 3.

The properties of continuity and concavity-with-continuity are each associated with a set ofutility transformations, respectively increasingcontinuous and increasing concave functionsfrom � to �. This feature is by no means sharedby all properties of utility. Moreover, even inthese cases, the properties should not be con-fused with their associated transformations: thetransformation must be applied to a functionthat satisfies the property in question. For in-stance, an increasing concave transformation ofa nonconcave utility need not be concave.

B. The Arrow-Koopmans Theory

Arrow (1951), following unpublished re-marks by Koopmans, argued that if an agentholds a consumption bundle for a period oftime, say the interval [0, T], and can decide onthe timing of how that bundle is consumed,then the agent’s preferences must be convex.W. M. Gorman (1957) later proposed a similartheory. Arrow implicitly supposed that theagent’s total utility is the integral of the instan-taneous utility achieved at each moment from 0to T. An agent holding consumption bundle x ��

n chooses a function � from [0, T] to ��n that

maximizes this integral subject to the constraintthat � integrates to no more than x. The infor-mal argument for convexity is that an agentholding the bundle �x � (1 � �)y, � � [0, 1],could consume �x over �T units of time and(1 � �)y during the remaining (1 � �)T timeunits. If the agent’s utility at each instant isindependent of the consumption at other in-stants, �x consumed over �T time units shoulddeliver utility equal to � times the utility of xconsumed over T time units, and similarly for(1 � �)y. So, if x and y are indifferent, con-suming �x for �T units of time and (1 � �)y for


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(1 � �)T time units will leave the agent exactlyas well off as with x or y. But the agent may finda better temporal consumption pattern whenendowed with �x � (1 � �)y and thus be betteroff than with x or y. Hence indifference curvesare convex. Arrow argued that this account ofconvexity, unlike the supposedly cardinalist sto-ries that rely on diminishing marginal utility, isa purely ordinal explanation.

We follow Birgit Grodal’s (1974) formaliza-tion of the Arrow-Koopmans theory. An agenthas a utility function V: ��

n 3 � that takes theform

V�x� � sup��L �0

T

u��t�, t� d��t�

s.t. �0

T

�i�t� d��t� � xi , i � 1, ... , n,

where u: ��n [0, T] 3 �, � is Lebesgue

measure, t � u(�(t), t) is integrable, x � ��n ,

and L is the set of Lebesgue measurable func-tions �: [0, T]3 �n such that �i(t) � 0 for i �1, ... , n and a.e. t � [0, T]. Let �AK denote thepreference relation on ��

n that V is a utilityrepresentation of.

THEOREM 4 (Grodal, 1974): The utility func-tion V is concave and therefore �AK is convex.

We turn to the measurement requirements ofthe utility 0

T u(�(t), t) d�(t) in the maximizationproblem that defines V but now seen as a func-tion from L to �.

DEFINITION 10: A function U: L 3 � satis-fies utility integrability if and only if, for somepositive integer n, (i) there exists a u: ��

n [0,T] 3 � such that t � u(�(t), t) is integrableand U(�) � 0

T u(�(t), t) d�(t) for all � � L; (ii)�Range U� � 1.

Condition (ii) implies that Range U is a non-trivial interval (see the proof of Theorem 5).

For a psychology U to maximally satisfyutility integrability, U must be the largest psy-chology such that each function U � U satisfiesutility integrability (cf. Definition 2). For theproperty of utility integrability to qualify as

cardinal, every psychology U that maximallysatisfies utility integrability must consist of, andonly of, those functions that are the affine trans-formations of any function in U.

THEOREM 5: Utility integrability is a cardi-nal property.

Theorems 5 and 3 together imply that theArrow-Koopmans theory imposes stricter mea-surement requirements on agents than does con-cavity. Utility integrability has range � 2 andintersects concavity; so Theorem 3 implies thatconcavity is strictly weaker than utility integra-bility. Thus, concavity, despite its preeminentplace in preordinal utility theory, is nearer toordinalist standards of measurability. Theseconclusions address only the measurement sta-tus of utility integrability; the Arrow-Koopmanstheory is ingenious and despite its cardinalityoffers an intriguing explanation of why prefer-ences should be convex.

IV. Concave Utilitarianism: Between Cardinaland Ordinal Interpersonal Comparisons

It is common to think that even if positivepreference theory should be based on purelyordinal assumptions, utilitarianism is necessar-ily a cardinal enterprise. But we will see that theutilitarian recommendations that command thebroadest consensus rely on a measurement scalethat is weaker than cardinality: concave psychol-ogies reproduce the key utilitarian recommenda-tion that income be redistributed from high-utilityagents with low marginal utilities of income tolow-utility agents with high marginal utilities ofincome. Not every utilitarian conclusion can bederived from concavity, but the missing results arethe anti-egalitarian policies that harm low-utilityagents. Even many utilitarians have viewed suchrecommendations as ethically suspect.

The formal theory of social choice concen-trates on the standard measurement classes: or-dinal scales, interval or cardinal scales, andoccasionally ratio scales. Opening up the terrainbetween ordinal and cardinal measurementbrings out new possibilities. Any version ofutilitarianism based on a compromise measure-ment standard, such as the concavity proposalwe lay out, has the advantage that it is easier toconstruct (requires fewer interpersonal judg-ments) than the traditional cardinal doctrine.


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Consider a society of I agents who mustchoose from a set of social choices X. To keepa concrete example in mind, one we discusslater, X � ��

I might be a set of feasible incomeprofiles for the I agents.

Interpersonal welfare comparisons will berepresented by social utility functions u, whichare standard utility functions except that they aredefined on � X, where � denotes the set of agents{1, ... , I}. We interpret u(i, x) � u(j, y) to meanthat agent i with social choice x is at least as welloff as agent j with social choice y. So, in particular,u(i, � ) gauges agent i’s well-being. When combinedwith an aggregation rule, e.g., ¥i�� u(i, x) or mini��

u(i, x), a social utility u generates an ordering of X.To assess the measurement implications of

different ways of making interpersonal welfarecomparisons, we again consider psychologies ofutility functions. To keep the distinction be-tween social and individual choice in mind, calla psychology U of social utility functions asocial psychology. One may think of the inter-personal judgments represented by a social psy-chology as arising from the preferences of anindividual contemplating what it would be liketo be various agents under various social out-comes. A social psychology is then interpretedin the same way as an individual psychology:the psychology lists the utility functions thataccurately compare various consumption expe-riences. It just so happens that for social psy-chologies the set of possible consumptionexperiences is � X rather than an arbitrary set.

As before, we consider only social psycholo-gies U that are complete: U will contain only,though not necessarily all, increasing transforma-tions of some u � U. In the language of socialchoice theory, completeness implies that the min-imum measurement standard that any resultingsocial welfare function satisfies is ordinal levelcomparability (in the terminology of Kevin Rob-erts, 1980b). Since we do not require U to containall ordinally equivalent social utilities, our socialpsychologies can obey a stronger measurementstandard.3

We again compare cardinal, ordinal, and con-cave psychologies. Definitions 5 and 6 of ordi-nality and cardinality apply unchanged to socialpsychologies; we now use UO and UC to denote,respectively, generic ordinal and cardinal so-cial psychologies. But we must redefine con-cavity, since, due to �, the domain of a socialutility is not a convex set. We call a socialutility function u coordinately concave if andonly if X is convex and, for each i � �, u(i, � )is concave and continuous. The property ofcoordinate concavity is then simply the set ofall coordinately concave functions, and a con-cave social psychology is therefore a socialpsychology that maximally satisfies coordi-nate concavity. It is thus a largest social psy-chology that preserves the concavity of eachagent’s utility and where each social utilityrepresents the same complete ordering on � X. UCC will denote a generic concave socialpsychology.

We consider social welfare rankings that or-der the social choices X according to the sum ofthe utilities ¥i�� u(i, x). Given a social psychol-ogy U, the ordering �U

s over X is defined by

x �Us yN �

i��

v�i, x� � �i��

v�i, y� for all v � U.

So x is ranked above y when all the socialutilities in U rank x above y by a sum-of-utilities test. The superscript s distinguishes�U

s , an ordering over the social choices, from�U, the ordering over � X that any u � Urepresents. While �U is complete, �U

s neednot be: U might not specify enough socialwelfare judgments for utilitarian aggregationalways to be decisive.

Different social psychologies evidently gen-erate different “utilitarian” rankings. Call �UC

s ,�UO

s , and �UCC

s cardinal, ordinal, and concaveutilitarianism, respectively. Cardinal utilitarian-ism is the usual utilitarianism; any u in UCgenerates the same sum-of-utilities ordering ofsocial choices. But cardinal utilitarianism restson a demanding measurement standard; speci-fying a cardinal social psychology requiresmaking cardinal comparisons of utility acrossagents. Ordinal utilitarianism, at the other ex-treme, relies on a much weaker measurementstandard. Since ordinal utilitarianism requiresany ranking of x over y to pass a larger set ofsum-of-utilities tests, it ranks fewer social

3 Some social choice theories consider utility transfor-mations that vary as a function of the agent i, e.g., invari-ance with respects to individual origins of utility, used byClaude d’Aspremont and Louis Gevers (1977) to axioma-tize utilitarianism, which admits affine transformations ofthe form au(i, � ) � bi. In our framework, these transforma-tions translate into incomplete social psychologies.


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choices than cardinal utilitarianism. Indeed, wewill see that ordinal utilitarianism incompletenesscan be so extensive that it makes trivially fewrankings. Since a concave social psychology UCCcontains the utilities generated by the affine trans-formations of any u � UCC, and since the set of allordinal representations of u contains any concaverepresentation, concave utilitarianism producesmore rankings than ordinal utilitarianism, butfewer rankings than cardinal utilitarianism. Wewrite this down as a theorem.

THEOREM 6: Let the social psychologies UCbe cardinal, UO be ordinal, and UCC be coor-dinately concave. If UC � UCC � A, then UC �UCC and �UC

s � �UCC

s . If UCC � UO � A, thenUCC � UO and �UCC

s � �UO

s .

Concave utilitarianism relaxes the stringentmeasurement requirements of cardinal utilitari-anism, and, as in the case of preference theory,concavity as a psychology pinpoints the char-acteristically neoclassical views of economicpsychology. But does concave utilitarianism re-tain the egalitarianism and enough of cardinalutilitarianism’s decisiveness? Specifically, doesconcave utilitarianism recommend redistribu-tions from rich to poor and does it rank suffi-ciently many social choices?

An ordering �Us and a particular u in U in-

duce an ordering over utility vectors in �I thatwill help characterize �U

s . Given a social psy-chology U and u � U, we define Ru�U, anordering over vectors in �I (distinguished sub-sequently by letters in bold) as follows.

DEFINITION 11: Given a social psychologyU, v Ru�U w N there exist x, y � X with v �(u(1, x), ... , u(I, x)), w � (u(1, y), ... , u(I, y))and x �U

s y.

We say that u is feasible for a social utility uif and only if

u � F�u� � �u � �I: there exists x � X

such that u � �u�1, x�, ... , u�I, x��.

Not surprisingly, v is ranked higher than waccording to any cardinal utilitarian orderingRu�UC

if and only if the sum of the coordinatesin v is at least as large as the sum in w. Ordinalutilitarianism is also easily characterized. For

any vector v, let v* denote the vector formed byplacing the coordinates v1 , ... , vI in increasingorder.

THEOREM 7:

(i) If UC is cardinal, u � UC, and v, w arefeasible for u, then v Ru�UC

wN ¥i�� vi �¥i�� wi.

(ii) If UO is ordinal, u � UO, and v, w arefeasible for u, then v Ru�UO

wN v* � w*.

Result (ii) says that ordinal utilitarianismjudges x �UO

s y if and only if, according to anyu � UO, the ith best-off agent under x is at leastas well off as the ith best off under y for all i ��. Thus, modulo utilitarianism’s anonymity re-quirement (which implies that if u and v merelyrearrange indices without changing the utilitylevel of the ith best-off agent for any i, then uand v are tied according to Ru�UO

), ordinalutilitarianism recommends only Pareto im-provements. We will say that u is an anonymousPareto improvement over v if u* � v* (some-times this is called a Suppes-Sen improvement).Ordinal utilitarianism lacks the traditional redis-tributive conclusions of utilitarianism: it willnever recommend a small transfer of wealthfrom a high-utility agent to a low-utility agent.Cardinal utilitarianism of course recommendssuch transfers if the low-utility agent has ahigher marginal utility of wealth (according toany given u � UC) than the high-utility agent.

To characterize concave utilitarianism, firstrecall the definition of a least concave function.Let V denote an individual psychology consist-ing of all the continuous and concave utilityfunctions, defined on some convex domain X ��n, that represent some fixed preference rela-tion � on X. The function v� � V is leastconcave if and only if for every v � V thereexists an increasing concave transformation g:Range v� 3 � such that v � g � v�. GerardDebreu (1976) proved that a least concave v�exists. Social psychologies do not have convexdomains, however, and so the definition of leastconcavity needs amendment. Let u�(i, X) denotethe range of the function u�(i, � ).

DEFINITION 12: If UCC is a concave socialpsychology, then u� � UCC is least coordinatelyconcave or lcc if and only if for all u � UCC


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there exists an increasing transformation g:Range u� 3 � such that u � g � u� and, for alli � �, g is concave when restricted to u�(i, X).

Debreu’s least concavity theorem does not itselfimply that each UCC contains an lcc utility func-tion, but the proof extends easily. Let con Sdenote the convex hull of a set S.

THEOREM 8: If UCC is coordinately concave,u� � UCC is lcc, and v, w are feasible for u� , then

v � con�u � F�u��: u* � w*�

fv Ru��UCCw.

So if (u�(1, x), ... , u�(I, x))* is a convex combi-nation of Pareto improvements of (u�(1, y), ... ,u�(I, y))* then x �UCC

s y.The upper contour sets of the three utilitari-

anisms clarify what Theorem 8 says. SupposeI � 2 and that UC, UO, UCC have a commonelement u� that is lcc for UCC. To keep thingssimple, suppose F(u�) equals ��

I . Fixing an ar-bitrary w, the upper contour sets {v: v R w} forthe three utilitarian orderings R � {Ru��UC

,Ru��UO

, Ru��UCC} are pictured in Figures 1

through 3. In this two-dimensional case, {v:v Ru��UCC

w} exactly coincides with the convexhull given in Theorem 8 (see Theorem 9 below).

Figures 2 and 3 show that concave utilitari-anism ranks a much richer set of utility vectorsthan ordinal utilitarianism. Not only are the

anonymous Pareto improvements weakly supe-rior to w, but any policy that benefits a utility-poor agent at the expense of a utility-rich agentwith no net loss of least concave utility is alsosuperior. On the other hand, in contrast to car-dinal utilitarianism, concave utilitarianism doesnot declare any utility vector that lowers thewelfare of the worse-off agent to be superior tow. Indeed, for pairs of agents, changes from wthat harm the worse-off agent are the only or-derings made by Ru��UC

but not by Ru��UCC. Con-

cave utilitarianism thus stakes out an egalitariancompromise between standard cardinal utilitar-ianism and the narrow Paretian judgments madeby ordinal utilitarianism.

w

u_

1

u_

2

FIGURE 1. CARDINAL UTILITARIAN UPPER CONTOUR SET

w

u_

1

u_

2

FIGURE 2. ORDINAL UTILITARIAN UPPER CONTOUR SET

w

u_

1

u_

2

FIGURE 3. CONCAVE UTILITARIAN UPPER CONTOUR SET


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Cardinal utilitarianism has long been criti-cized for ignoring welfare levels and in partic-ular for recommending that low-utility agentsshould undergo arbitrarily large utility losseswhenever those losses lead to greater utilitygains for high-utility agents. Concave utilitari-anism does not suffer from this defect. More-over, concave utilitarianism does not arrive atits prohibition on harming the least well-off byinvoking an equity axiom (as in Hammond,1976). Rather, the egalitarianism stems directlyfrom the psychological content of concavity.

We can characterize this egalitarianism byturning to the second-order stochastic domi-nance and poverty ordering literatures (see G.Hanoch and Haim Levy, 1969; Serge-ChristopheKolm, 1969; Vijay S. Bawa, 1975; Charles Black-orby and David Donaldson, 1977; Anthony F.Shorrocks, 1983; James E. Foster and Shorrocks,1998a, 1988b; and the discussion of Roberts,1980b, below). In the literature on poverty order-ings, more equal distributions of income will, ce-teris paribus, raise the sum of utilities, no matterwhat concave function maps individual income tosocial welfare. In the current setting, the utilitylevels given by the least coordinately concavesocial utility functions play the role of incomes,and thus more equal distributions of least concaveutility with no net loss will be a concave-utilitarianimprovement, no matter how concave individualutility functions are. Thus, even when X specifiesconsumption of many commodities, agents’ wel-fare can be compared using a single-dimensionalvariable—least concave utility—analogous to in-come. In effect then, we give a measurementtheory and multicommodity rationale for the or-dering of utility distributions or income distribu-tions cited above.

Our characterization relies on the historicallyimportant Pigou-Dalton principle (see, e.g.,Herve Moulin, 1988).

Given a social utility u, an ordering R overutility vectors satisfies the Pigou-Dalton princi-ple if, for any j, k � � and any v, w feasible foru with vm � wm for m � { j, k}:

vj � vk � wj � wk and min�vj, vk� � min�wj,wk�

(PD) f v R w.

In words, R satisfies Pigou-Dalton if a switch toa new vector of utilities that affects only a pair

of agents is recommended when the sum ofutilities is unchanged, and the lower-utilityagent after the switch is no worse off than thelower-utility agent before the switch.

Figure 3 suggests that concave utilitarianismshould satisfy the Pigou-Dalton principle. Sub-ject to one caveat, the implication in (PD) infact goes both ways if we replace the “�” in(PD) with “�,” as Theorem 9 reports. In thissense, Pigou-Dalton characterizes concave util-itarianism. The caveat arises when a social util-ity function assigns ranges for the individualutility functions that do not overlap. For exam-ple, suppose two agents a and b must split a pieand some concave social utility function gives aand b sets of achievable utilities that are dis-joint. One may then multiply either a’s or b’sutility by any positive constant and thereby ar-rive at another concave social utility. So a con-cave social psychology could not judge eitheragent’s marginal utility of consumption to behigher or lower than the other’s, thereby emp-tying concave utilitarianism of its egalitariancontent. To exclude this wrinkle, it is sufficientto assume for any u � UCC that �i�� Int u(i, X)is an interval (where ‘Int’ denotes interior). Therange of each agent’s utility then does directlyor indirectly overlap the range of any otheragent’s utility; arbitrary rescalings of a singleindividual’s utility function are thereforeblocked, and nontrivial interpersonal compari-sons can be made.

THEOREM 9: Let u� be lcc for UCC and sup-pose �i�� Int u�(i, X) is an interval. If v, w arefeasible for u� and identical in all but two coor-dinates j and k, then

vj � vk � wj � wk and min�vj, vk� � min�wj, wk�

N v Ru��UCCw.

Thus we arrive at a “poverty ordering” of utilitydistributions despite beginning with preferencesdefined over an X that can specify consumptionof an arbitrary number of goods.

A contrasting result of Avinash K. Dixit andJesus Seade (1979) pins down the content ofTheorem 9. Dixit and Seade show in effect thatthere are preferences � on �n and x � �n, with� representable by a smooth increasing con-cave utility function, such that for any such


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representation u there is a y near x withu( y) � u( x) and Du( y) Du( x). If therewere two agents, each with the preferences �,but one endowed with x and the other en-dowed with y, then a planner who assigned uto both agents and ranked allocations accord-ing to the sum of utilities would always rec-ommend at least some transfer of goods fromthe agent with the low utility level (the “xagent”) to the agent with the high utility level(the “y agent”). Theorem 9 does not contra-dict this result. Dixit and Seade let y vary asa function of the concave representation u:given u, there exists a utility-rich y agent towhom a transfer from the x-agent would raisethe sum of the u’s. A concave utilitarian incontrast would require any transfer to raisethe sum of utilities calculated with all con-cave u’s, and any transfer from some specificy agent to the x agent would be rejected by asufficiently concave u.

To illustrate cardinal and concave utilitarian-ism’s common ground, and the paucity of ordi-nal utilitarian rankings, consider the canonicalproblem of ranking distributions of a singlegood.

Example.—A policymaker must choose adistribution of the aggregate output � 0, andso the set of social choices X is {x � ��

I : ¥i��

xi � }. We assume that the social psychol-ogies UC, UO, and UCC have a common usuch that each u(i, � ) is increasing, differen-tiable, and strictly concave in xi (the ith co-ordinate of x, i’s income) and constant in theother coordinates. Let u(i, � ) henceforth de-note social utility as a function of xi alone.Suppose u� is lcc for UCC and that �i�� Intu� (i, X) is an interval.

First, let all individuals be identical: u(i, � ) �u( j, � ) for any i, j � �. Although the cardinal andconcave utilitarian orderings �UC

s and �UCC

s arenot identical, they both rank e � ((1/I), ... ,(1/I)) above any other distribution in X. For�UO

s , in contrast, any x � X that varies by agentis unranked relative to e, and no pair of points inX is strictly ranked.

Next, allow u(i, � ) � u( j, � ) and let Du(k, xk)denote the derivative of u(k, � ), evaluated at xk.As in the identical-agent case, �UC

s ranks an xsuch that Du(i, xi) � Du( j, xj) to be superior toany other point in X. While �UCC

s no longer hasto make this ranking, Theorem 9 implies, given

some base distribution x, that some transfer ofincome from agent j to agent i will be superioraccording to �UCC

s if and only if

Du� �i, xi � � Du� � j, xj �

and u��i, xi� � u�� j, xj�.

So, rather than the equal derivative condition,any x such that

�Du��i, xi� � Du�� j, xj��u��i, xi� � u�� j, xj�� 0

for all (i, j) is undominated according to�UCC

s : agents’ utility levels and their marginalutilities of income (both calculated using u� )must be comonotonic. Ordinal utilitarianismagain makes very few orderings: given somex � X, no transfer from a lower utility agentto a higher utility agent is ranked relative to x,nor is any small transfer from a higher utilityagent to a lower utility agent. And any pointin X that leads to equal utility levels is alwaysunranked relative to all other points in X.

Notice how easy it is in this example for aconcave utilitarian to make policy decisions. Apolicymaker does not have to posit exact ratiosof different agents’ gains in utility. To judge asmall transfer, the policymaker need only de-cide if a transfer from i to j leads to a biggergain for j than i’s loss and if i’s ex ante welfarelevel is higher than j’s.

Although the cardinal and concave utilitarianorderings are not the same, it is concave ratherthan cardinal utilitarianism that provides thebetter rationale for the positions endorsed by theoriginal neoclassical utilitarians. The connec-tion to the Pigou-Dalton principle is revealing.Arthur C. Pigou, the architect of neoclassicalwelfare economics, considered himself both acardinalist and a utilitarian: at least in theory,each individual in society has a cardinal andinterpersonally comparable utility function,and policies should be evaluated by summingthe utility numbers that they lead to. ButPigou (1932) recognized that it is not easy tocome by cardinal utility information, andtherefore argued that, as a practical matter,welfare economics can validate only policiesthat do not rely on that information. Even


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Edgeworth (1897), the arch-utilitarian, didnot approve of policies whose justificationdepends on the specific choice of a cardinalutility function rather than on DMU alone(Mandler, 1999, chap. 6). In Pigou’s view, apolicy that lowers the income of a low-utilityindividual by $1 and raises the income of ahigh-utility individual by $x cannot be backedunambiguously, no matter how large x is,since the marginal utility of income might bevery small at high utility levels. On the otherhand, if the low-utility agent gains by $y, y �1, and the high-utility agent loses by $1, thenany sum-of-utilities test will recommend the pol-icy change if agents share a concave utility ofincome. Concave utilitarianism backs thesame recommendations, but it elevates thetheoretical status of not having cardinal infor-mation about individuals. According to con-cavity as a social psychology, there is nounknown but nevertheless real cardinal utilityfunction lurking out there—the most detailedinformation that could in principle be knownabout interpersonally comparable utility isthat it is concave. The close match betweenthe theory of concave utilitarianism and Pi-gou’s practical position is unsurprising: con-cave utilitarianism imposes exactly thepsychological content that informed Pigou’saccount of human happiness.

Finally, we can characterize a concave-utilitarian comparison of utility vectors thathave arbitrarily many coordinate changes by:

v Ru��UCCwN �

i � 1

n

v*i � �i � 1

n

w*i for all n � �,

where v, w � F(u� ) and where we againassume the same range condition that �i�� Intu� (i, X) is an interval obtains. Kevin Roberts(1980b) essentially hypothesized this equiva-lence (in different terminology) and suggestedHanoch and Levy (1969) and least concaveutilities as the appropriate tools. Roberts didnot recognize that social utilities cannot sat-isfy the traditional definition of least concav-ity, an omission that obscured the need for therange condition. We should point out thatRoberts did not take sets of concave utilityfunctions as primitive, but argued, like Pigou,that such a set can be assumed to contain theone “true” cardinal family of utilities.

V. Observable Tests of Concavity

Cardinal utility theory was rejected not be-cause it invoked mental relationships above andbeyond choice behavior; psychology routinelydoes just this without tarnishing its scientificstanding. Cardinality ran into trouble because,first, there was no convincing evidence thatsatisfaction is a cardinal quantity and, second,cardinal utility seemingly had no role to play inconsumer theory. Although these criticisms aretelling, they do not apply to all forms of non-ordinal utility. With regard to the secondcharge, we have seen that concave psychologiesdo serve an important purpose even in positivetheory—they provide a concise and only mildlynonordinal rationale for the convexity ofpreferences.

As for the first charge, the MRI revolution incognitive neuroscience has identified neuralcorrelates of consumption decisions that cantest whether an agent has a concave psychology.As we will see, a concavity test requires onlythat neural correlates have two properties: theyshould map into an agent’s ordinal preferenceranking of alternatives and they should order thegains or losses in well-being that an agent ex-periences relative to well-being at the agent’sreference consumption level. Such evidencecannot always be used to test for cardinality,which can require more extensive data.

Neuroscientists have uncovered various indi-cators of choice decisions. Brian Knutson et al.(2001), Martin P. Paulus et al. (2001), and HansC. Breiter et al. (2001) show the extent of ac-tivity in several brain areas (the nucleus accum-bens, the parietal cortex, and the sublenticularextended amygdala, among others) that corre-late with the magnitude of monetary rewardsthat subjects receive.4 Agents’ money gainscome on top of their base or reference consump-tion levels: background income and consump-tion are fixed and the experiments establishbenchmark levels of neural response. Also, thecorrelates of economic gains turn out to occurlargely at one set of neural sites, while corre-lates of losses occur at others (Breiter et al.,2001). While this fact is consistent with subjects

4 The recent special issue on neuroeconomics in Gamesand Economic Behavior (Aldo Rustichini, 2005) gives anoverview of the neuroeconomics literature. See also Paul W.Glimcher and Rustichini (2004).


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having stable ordinal preferences, it underscoresthat neural indicators give evidence of changesrelative to reference consumption. If currentprogress continues, this literature may well un-cover neural indicators that reliably track theconsumption increments that agents prefer andchoose.

We suppose therefore that an agent who firstconsumes a base or reference consumption bun-dle c and then adds an increment y to c displaysan identifiable neural response, which we labelN(c, y). (It may help to think of c and c � y asflows over a brief period.) Each N(c, y) maywell lie on a unidimensional or multidimen-sional scale, but the range of N could alsoconsist of qualitative events with no numericalstructure. We suppose the agent has a strictpreference relation � defined over consumptionvectors that is a weak order (asymmetric andnegatively transitive). The relation � will re-main the same for all base bundles c: for any cthe agent always chooses y over y� when c �y � c � y�. As usual, � is confirmable fromchoice data. The assumption that N correlateswith preference means that (a) strictly preferredbundles lead to distinct neural observations and(b) whenever the same pair of observationsarises, the same direction of preference obtains.Formally these properties for N mean that for allreference bundles c, c�, and increments y, y�, z, z�:

(1) c � y � c � y� f N�c, y� N�c, y��,

and

(2) c � y � c � y�, N�c�, z�� N�c, y�,

and N�c�, z�� N�c, y��

f c� � z � c� � z�.

An agent exhibits a benchmark neural re-sponse if some null reaction occurs whenevery � 0, i.e., there is an element 0 of Range Nsuch that, for all c, N(c, 0) � 0. We do nothave to assume explicitly that a benchmarkexists, but its presence would fit with ourinterpretation of N. The plausibility of (1) and(2) depends on how many increments y maybe added to c; if y is a continuous variable, anagent may not display the required neurolog-ical discrimination.

The neural indicators can be ordered by theagent’s ordinal ranking of consumption bun-dles: we define the relation P0 on Range N by� P0 �� if and only if there exist c, y, y� suchthat � � N(c, y), �� N(c, y�), and c � y � c �y�. Let P be the transitive closure of P0; weassume that P is a weak order. Given P, equiv-alence classes of neural indicators in Range Nare defined by � I �� N not � P �� and not ��P �.5 Notice that the ordering P can be deducedsolely from the observables � and N.

If P can be represented by a “utility” function , this function orders neural indicators accord-ing to whether the underlying consumption in-crements are more or less preferred: (�) � (��) if and only if � P ��. Call any such admissible.

Even if it turned out that N maps into asingle-dimensional variable that monotonicallyindicates which consumption increments are morepreferred, it would not follow that the changes insatisfaction from increments to consumption areproportional to, or an affine transformation ofN(c, � ). (Indeed it would not matter if N were alinear function of consumption, as Knutson andPeterson (2005) claim is the case.) Changes insatisfaction could well be formed by a different—and unobservable—psychological process andcould as easily be the cube or exponential of N(c,� ) as an affine transformation of N(c, � ). So, for thesame reason that traditional ordinal utilities arearbitrary up to an increasing transformation, if (N(c, � )) represents an agent’s ordering of con-sumption increments given c, then any increasingtransformation of (N(c, � )) represents the sameordering equally well.

But to test for concavity, the arbitrariness of is irrelevant; all that matters is the underlyingN. Let us say that observational DMU obtainsif, for any c and y and any admissible ,

�N�c � y, y�� N�c, y��.

If we assume that any equivalence class ofneural indicators indicates a comparable subjec-tive experience of well-being at any referenceconsumption bundle, then observational DMU

5 Formally, P is defined by � P �� if and only if � P 0��or there is finite set {�1, ... , �n} with � P0 �1 P0

... P0 �n P0

��. To derive the conclusion that P is a weak order, it issufficient to assume that P0 is acyclic and that if � P �� I ��or � I �� P �� then � P �� .


Fn5

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means that a second increment of y leads to asmaller psychic gain than the first increment ofy. Since every admissible represents the sameordering of neural indicators, if the inequalityabove holds for one admissible , then it holdsfor all admissible .

For an example of what observational DMUasserts, let there be just one consumption good,consumed at integer levels, and let the agent’spreference relation � always agree with �(more is better). The neural responses are one-dimensional. Specifically, we assume that forany c the neural responses are linear in y: forany c there is a kc � 0 such that N(c, y) � kcyfor all y. Evidently P is the order � (restrictedto the integers). If in addition we assume thatc � c� implies kc � kc�, then observationalDMU obtains. Notice that the linearity of eachN(c, � ) is consistent with observational DMU.Indeed observational DMU can be consistentwith each N(c, � ) having an increasing firstderivative.

Given our interpretation that N gives compa-rable information about changes in well-beingor utility at all c bundles, a psychology U for theagent should contain only those functions thatrepresent those changes in utility. That is, if (N(c, y)) � (N(c�, y�)) then a u � U shouldsatisfy u(c � y) � u(c) � u(c� � y�) � u(c�). Soif observational DMU holds and (c � y, y) and(c, y) are in the domain of N, a u � U shouldsatisfy

u�c � 2y� � u�c � y�

u�c � y� � u�c�� 1,

or, in other words, u is midpoint concave on itsdomain. (If u satisfies midpoint concavity on aconvex domain and is continuous, then u isconcave.)

Concavity of a psychology therefore leads totestable empirical claims. In some limitingcases, cardinality is testable too. By comparingN(c, y) to N(c�, y�), we may see if an agentregards a move from c to c � y as leading to abigger or smaller change in satisfaction or util-ity than a move from c� to c� � y�. Hence, ifevery feasible increment y is observable for allc, we can assemble from N a complete orderingof how an agent compares changes in satisfac-tion. If in addition there is some utility u thatrepresents these changes whose range is an in-

terval (e.g., when the consumption set is con-nected and u is continuous) then the ordering ofchanges in satisfaction is sufficient to pin downa cardinal family of utilities (see Kaushik Basu,1982). Determining whether a subject’s neuralresponses satisfy the conditions we have placedon N and whether some u that represents theresulting ordering of changes in utility has aninterval range would therefore constitute a testof cardinality. Not surprisingly, such a test ismore demanding than a test for concavity. Ifonly a discrete set of (c, y) are in the domain ofN (as in our earlier example), then one cannotconfirm or falsify the range condition. And ifthe variables c and y can be varied continuouslyit may be that the conditions (1) and (2) on N nolonger hold, again preventing a cardinality test.

Although we have so far used neural indica-tors to rank the satisfaction of consumptionincrements, the same theory applies to agents’verbal assessments of their satisfaction orhappiness: simply interpret N(c, y) as a one-dimensional variable that reports an agent’s as-sessment of the value of increment y given thebase c. When N(c, y) � N(c�, y�), the agent saysin effect that the utility gain of increment y atbase c is greater than the utility gain of incre-ment y� at base c�. The neuroscience literaturein fact finds that self-reported satisfaction andneural indicators correlate (see, e.g., Knutson etal., 2001). With regard to tests of cardinality,notice that the surveys of reported happinesslevels allow agents to report only a few possiblesatisfaction responses; thus the interval rangecondition could not be even approximatelyverified.

When faced with neurological correlates ofsatisfaction or surveys of self-reported happi-ness, economists sometimes view these data ascardinal scales (e.g., Ng, 1997). I have deviatedfrom this practice first by letting the set ofneural indicators be arbitrary rather than a uni-dimensional variable. But even if N is a singlevariable that correlates perfectly with happi-ness, our framework allows an agent’s subjec-tive sense of satisfaction to vary nonlinearlywith N (even when the interval range conditionis satisfied and cardinality is testable). Regres-sions on survey data of self-reported happinessoften import cardinality assumptions: if a re-gression treats equal differences between pre-dicted and reported happiness as an equal lossof goodness of fit, the survey responses are


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being interpreted as cardinal scales. The use ofcompromise measurement scales would permitweaker and more plausible interpretations onsurvey data. When subjects compare incrementsof satisfaction, one could judge any deviationof a subject’s ranking of satisfaction incre-ments from a predicted ranking to be an equalgoodness-of-fit failure. By making measure-ment assumptions explicit, it should be possibleto free both physiological and self-reported sat-isfaction data from cardinality entanglements.

VI. Related Literature on Measurement

Our ordering of properties of utility draws ontwo measurement literatures. The first is mea-surement theory proper (see, e.g., Krantz et al.,1971; Roberts, 1979; and, originally, Stanley S.Stevens, 1946), which identifies measurementclasses with sets of transformations. Ratioscales are defined by the set of increasing lineartransformations, interval or cardinal scales bythe set of increasing affine transformations, andordinal scales by the set of all increasing trans-formations. Measurement theory implicitly or-ders measurement classes by set inclusion; inthe cases mentioned the sets of transformationsare nested. A set inclusion ordering of transfor-mations will sometimes track the ordering ofproperties we consider, but it covers only a fewspecial cases and cannot define a sufficientlyrich array of measurement classes. The secondis the social choice literature that associates setsof transformations, now applied to multi-agentprofiles of utility functions, with a category ofinterpersonal comparability (Amartya Sen,1970; d’Aspremont and Gevers, 1977; Roberts,1980b; Walter Bossert and John A. Weymark,1996). Applying a smaller set of transforma-tions imposes a tighter interpersonal compara-bility requirement.

The drawback of both literatures is that theyeither work directly with transformations in-stead of utility functions or, when sets of utili-ties are primitive, concentrate on those sets ofutilities that are generated by applying a set(indeed a group) of transformations to an arbi-trary utility function. At first glance, this prac-tice may seem to be an advantage: any utilityfunction can then be a member of any of thestandard measurement classes. But taking arbi-trary psychologies as primitive admits a greatervariety of measurement standards and proves

more flexible. For instance, the set of continu-ous utility functions defines a measurementstandard that cannot be characterized by the setof continuous transformations (since an increas-ing continuous transformation applied to a non-continuous utility function will not generate acontinuous utility). Just as importantly, it isonly by taking psychologies as primitive thatwe can identify the measurement requirementsof assumptions on utility functions and hence tocompare the measurement requirements of dif-ferent assumptions.

Basu (1980) has also explored room for com-promise between ordinal and cardinal utilitytheory. In a spirit similar to the present paper,Basu contends that DMU resides in this middleground and remarks on the advantages of takingnonordinal assumptions as primitive. But Basusticks to the method of characterizing measure-ment classes via utility transformations. Fur-thermore, as Basu (1982) shows, the middleground that Basu (1980) linked to DMU ends upbeing equivalent to full-scale cardinality forcontinuous utilities defined on classical com-modity spaces. Basu concludes that utility the-ory prior to the ordinal revolution usedassumptions that were tantamount to cardinality(even when, as in Lange’s case, they were at-tempting to rid themselves of cardinal assump-tions). We have come to a different conclusion:by using sets of utility functions to comparemeasurement standards, compromises betweencardinality and ordinality will persist even withstandard economic commodities.

In contrast to the present view of measure-ment (and to the social choice view), the hard-line ordinalist position has had trouble figuringout what to do with cardinal utility theories. Alot of work (e.g., Debreu, 1960; Krantz et al.,1971) has gone into finding axioms on prefer-ences that ensure that preferences can be repre-sented by functions that are unique up toincreasing affine transformations. But from theordinal vantage point, the significance of theserepresentation results remains limited. Sincepreferences are the real primitive, the cardinalutility whose existence is established amountsonly to a convenience. A von Neumann–Mor-genstern utility function, for instance, simplyallows the ease of a function that is linear in theprobabilities. But other utility representations ofthe same preferences—e.g., the cube of a vNMutility—carry the same information about choice


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behavior. When psychologies are primitive, on theother hand, the cardinality of an assumption de-livers the important message that the assumptionentails a sizable departure from ordinality. Othernonordinal properties, such as concavity or conti-nuity, imply a smaller departure.

VII. Conclusion

The ordering of properties in this papergauges the measurement requirements of as-sumptions on utility functions. To illustrate howthis gauge can be applied, we argued in sectionIII that the Arrow-Koopmans explanation ofconvexity, despite its billing as an ordinal the-ory, relies on a demanding cardinal standard ofmeasurement rather than the weaker standardthat concavity entails. Consider the followingrebuttal, however. Although the Arrow-Koop-mans theory begins with a class of integralutility functions, one could object that thoseintegral utilities have no special significance;other utility functions that do not have the inte-gral form represent the same preference relationas the integral utilities, and we could includethem in the psychology that describes the agent.Furthermore, one can impose assumptions onpreference relations that imply the existence ofan integral utility representation (Grodal andJean-Francois Mertens, 1968; Vind, 1969)—just as there are axioms on preferences overfinite numbers of goods that imply that addi-tively separable utility representations exist(Debreu, 1960). So perhaps one could claim thatthe Arrow-Koopmans theory does give an ordi-nal explanation of the convexity of preferences.But this argument hinges on whether such anordinal axiomatization stands as a convincingprimitive. The key assumption needed for theexistence of an integral utility representation isan independence postulate, just as a similar in-dependence condition underlies the existence ofadditively separable utility representations. Inthe setting of time-dated goods, independenceasserts that preferences over any subset of thetime-dated goods are not affected by the con-sumption level of goods with different dates.Although independence is posed as an ordinalaxiom, it rests on deeper psychological founda-tions: independence relies on a claim that con-sumption at one date does not affect thesatisfaction of consumption at other dates. In-deed, this would appear to be the only rationale

that could make independence plausible. Butthen only those functions where the utility ex-perienced at one date is independent of theconsumption at other dates will represent thefull gamut of psychological claims asserted inan independence-based theory. As we know, theutility functions with this property form a car-dinal psychology of additively separable func-tions (or in the infinite case, integral functions);the other ordinally equivalent functions do notexpress the psychological presuppositions thatjustify independence. This paper takes sets ofutilities as primitive partly in order to give psy-chological rationales for ordinal axioms. Regard-ing convexity, it would beg the question to replaceconvexity with a different set of ordinal axiomsthat lead to the Arrow-Koopmans theory; onewould then have to give rationales for those axi-oms. And the psychologies that accompany therationales for some of those axions would becardinal.

Orderings of properties of utility can shedlight on the measurement requirements of ra-tionales for other assumptions on preferencesbesides convexity. Consider, for example, con-tinuity as an assumption on preferences (i.e., theassumption that strict upper and lower contoursets are open). The obvious justification for thecontinuity of preferences is to argue that satis-faction is a continuous psychological quantity.Although not quite ordinal, continuity of utilityis weaker in the measurement sense than severalother assumptions we have considered (e.g.,additive separability, concavity on open sets).Thus, as intuition suggests, the continuity ofpreferences can be justified using a measure-ment standard only slightly stronger than ordi-nal measurement. Once again, an ordinalistmight object that there is no need to assume thatutility functions are continuous in order to givea rationale for preference continuity; the conti-nuity of preferences already stands as an ordinalaxiom. But the psychology that motivates anassumption that preferences are continuous isnonordinal; it turns on the claim that satisfactionis a continuous quantity.

If any doubt lingers that nonordinal contentcan motivate an axiom on preferences, considerthe following trivial ordinalization of propertiesof utility. For any property P, define the prop-erty PO to consist of all functions that ordinallyagree with some u that satisfies P. Although POevidently must be ordinal, the psychological


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theory that leads one to believe that P obtainsevidently need not be. Since the psychologicaltheory behind independence or preference con-tinuity also leads to a nonordinal psychology,the rebuttals we have been considering involvejust this sort of trivial ordinalization.

APPENDIX

PROOF OF THEOREM 1:If � is complete in addition to transitive and

order-dense, the proof that there exists a U with�U � � is the standard existence theorem forutility functions. So assume that � is not com-plete. Let X/� denote the indifference classes of�: X/� is the partition of X such that, for all x �X and y � X, {x, y} � I � X/� if and only ifx � y and y � x. Define � on X/� by I � J if andonly if I � J and x � y for some x � I and y �J. For any z � X, let I(z) denote the indifferenceclass that z belongs to. For any x, y � X suchthat neither x � y nor y � x obtains, define twostrict partial orders �x and �y on X/� by � �{(I(x), I(y))} and � � {(I(y), I(x))}, respec-tively. Let �x

t and �yt denote the transitive clo-

sures of �x and �y, respectively.Given a binary relation P on domain Z, we

say T � Z is countably order-dense for P if T iscountable, and for all I, J � Z � T with I P J,there is a K � T such that I P K P J. Then, since� is countably order-dense, there is a W � X/�that is countably order-dense for �: one takes acountable Y � X given by the countable order-density of � and sets I � W if and only if I �Y � �. Let W� � W � {I(x), I(y)}. To showthat W� is countably order-dense for both �x

t and�y

t , suppose, to take the case of �xt , that I, J �

X/��W� satisfy I �xt J. If I �x J then I � J and

so, since W is countably order-dense for �,there is a K � W� such that I � K � J. If not I�x J, then by the definition of a transitive clo-sure, there exists a finite set of indifferenceclasses {I1, ... , In} such that I �x I1 �x

... �x In�x J. If neither I(x) nor I(y) is in {I1, ... , In},then the transitivity of � implies I � J, contra-dicting not I �x J. If one of I(x) or I(y) appearsin {I1, ... , In}, then the transitivity of � implieseither I � I(x) � J or I � I(y) � J and both I(x)and I(y) are in W�. If both I(x) and I(y) are in{I1, ... , In}, then the transitivity of � implies (1)I � I (x) � J, (2) I � I (y) � J, (3) I � I(y) �xI(x) � J, or (4) I � I(x) �x I(y) � J. Cases (1)and (2) have already been covered, (3) cannot

obtain since not y � x, while in case (4) I � I(x)�x

t J. We may therefore apply Peter C. Fishburn(1979, Theorem 3.2) or Marcel K. Richter(1966) to conclude there is a utility function ux,yon X/� such that L �x

t M implies u(L) � u(M).Or, to argue directly, simply let �: W� 3 � beone-to-one and define ux,y by

ux,y �I� � �J�W�:I �x

t J

1

2�� J�

for all I � X/� (letting a sum over the emptyset equal 0). Similarly, there exists a uy, x onX/� such that L �y

t M implies u(L) � u(M).Define the utility functions vx,y and vy, x on Xby setting vx,y( z) � ux,y(I( z)) and vy, x( z) �uy, x(I( z)) for all z � X.

Let U be defined by v � U if and only if v �{vx,y, vy,x} for some x, y � X such that not x �y and not y � x. Since vx,y(x) � vx,y(y) andvy,x(y) � vy,x(x), we have not x �U y and not y�U x. Since in addition it is easy to verify thatw � z if and only if u(w) � u(z) for all u � U,we have �U � �.

PROOF OF THEOREM 3:To show that concavity is no stronger than

any cardinal property, let UC be cardinal, letUCV maximally satisfy concavity, and supposethere exists u � UC � UCV. Since u � UC, forany u� � UC there exists an increasing affinetransformation g such that g � u � u�. Since anincreasing affine transformation of a concavefunction is concave, u� � UCV, and so UCV �UC. We omit the equally elementary proof thatany ordinal property is no stronger than concav-ity (or indeed no stronger than any property).For result (ii), let the cardinal property nowhave range � 2, and suppose v � UC � UCV. If f:� 3 � is increasing and strictly concave, wehave, since v � UCV, f � v � UCV. Since �Range v�� 2, there exist v1 , v2 , v3 � Range v and � � (0,1) with v1 � v2 � v3 and v2 � �v1 � (1 � �)v3.Since f is strictly concave, f (v2) � �f (v1) � (1 ��) f (v3). So f �Range v is not affine and therefore f� v � UC. Hence concavity is strictly weaker thanany cardinal property with range � 2 that inter-sects concavity.

To establish (iii), let UO maximally satisfy anordinal property with range � 1, let UCVC max-imally satisfy concavity and continuity, andsuppose that UO � UCVC � A. For v � UO �


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UCVC, Range v is a nontrivial interval. So let x1,x2 � X satisfy v(x1) � v(x2) and define D �{w � X: w � �x1 � (1 � �) x2 for some � �[0, 1]}. Since v is concave there is a C � Dthat is connected and such that v is monotoneon C. So there are z1, z2, z3 � C that satisfyz3 � (1/2)z1 � (1/2)z3 and v(z1) � v(z2) �v(z3). Let g: � 3 � be an increasing transfor-mation such that g(v(z2)) � (1/2)g(v(z1)) �(1/2) g(v(z3)). We then have g � v �/ UCVC, butsince g is increasing, g � v � UO.

PROOF OF THEOREM 4:See Grodal (1974).

PROOF OF THEOREM 5:Let U with domain L maximally satisfy util-

ity integrability and let U be an arbitrary ele-ment of U. Since it is plain that if W: L3 � issuch that there exists an increasing affine trans-formation g that satisfies g � U � W, then Wsatisfies utility integrability, we need only showthat for any W � U there exists an increasingaffine transformation g such that W � g � U. Itis sufficient to show that if g is an increasingtransformation and W � g � U satisfies utilityintegrability (i.e., W � U), then g is affine.

We establish this via Theorem 2. First, ob-serve that since �Range U� � 1, there exist �, �� L such that U(�) � U(��), and hence therealso exists, for any � � 0, a measurable C1 �[0, T] such that

0 � �C1

�u��t�, t� � u��t�, t�� d��t� � �,

where, for U � U, u: ��n [0, T]3 � is one

of the functions satisfying Definition 10 (i).(This u is fixed throughout.) By setting �sufficiently small, we can partition [0, T] intosets C1 and C2 such that in addition

�C2

�u��t�, t� � u��t�, t�� d��t� � 0.

For i � 1, 2, let Li be the restriction of L toCi—that is, the set of functions from Ci to �n

defined by �i � Li if and only if there exists� � L such that �i(t) � �(t) for all t � Ci.

Let Ui: Li 3 � be defined by Ui(�i) �Ci

u(�i(t), t) d�(t). We have L � L1 L2.Using (�1 , �2) to denote the � � L such that�(t) � �i(t) for t � Ci and i � {1, 2}, wehave U(�1, �2) � U1(�1) � U2(�2) for all� � L. Since, for i � {1, 2}, Ui(�i) �Ui(��i), �Range Ui� � 1.

To show that the Ui have ranges equal tonontrivial intervals, allowing us to apply Theo-rem 2, we use the following result from thetheory of integration of correspondences, some-times called Lyapunov’s theorem.

LYAPUNOV’S THEOREM: Given an atom-less measure space (�, F, �) and correspon-dence P: � * �, the set � P d� � {� p d�:p is integrable and p() � P() for a.e. � �}is convex.

By Lyapunov’s theorem, Range U, RangeU1, and Range U2 are convex sets and thereforeintervals. For example, for the case of RangeU1, the measure space is Lebesgue measure onC1 and the correspondence P would be definedby P(t) � {u(�1(t), t) � �: �1 � L1} for eacht � C1. Since Range U1 � C1

P d�, weconclude that Range U1 is convex and hence aninterval. Note that since �Range U1� � 1 and�Range U2� � 1, Range U1 and Range U2 havenonempty interiors.

Since W � U, there exists a w: ��n [0, T]3

�, where t � w(�(t), t) is integrable and W(�) �g(U(�)) � 0

T v(�(t), t) d�(t) for all � � L. Justas we argued for the case of U, U1, and U2, wehave W(�1, �2) � W1(�1) � W2(�2) for all (�1 ,�2) � L, where Wi(�i) � Ci

w(�i(t), t) d�(t),i � {1, 2}. Hence g(U1(�1) � U2(�2)) �W1(�1) � W2(�2) for all (�1 , �2) � L. So applyTheorem 2 to conclude that g is affine.

PROOF OF THEOREM 6:In text. See also the proof of Theorem 3.

PROOF OF THEOREM 7:We omit the details: 7 (i) is obvious and 7

(ii) is a slight variant on common argumentsin the social choice literature (see, e.g., Mou-lin, 1988).

PROOF OF THEOREM 8:If v � con{u � F(u�): u* � w*} then there

exist �1, ... , �m � [0, 1] and z1, ... , zm � F(u�)such that ¥k�1

m �k � 1, v � ¥k�1m �kzk, and


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zk* � w* for k � 1, ... , m. For any increasingg: Range u� 3 � such that g�u�(i, X) is concavefor each i � �,

�i��

g�vi� � �i��

g� �k � 1

m

�kzik� � �

i��

� �k � 1

m

�kg(zik)�

� �k�1

m

�k��i��

g(zik)� � �

k � 1

m

�k��i��

g(zik*)�

� �k � 1

m

�k� �i��

g(w*i )��

k � 1

m

�k� �i��

g(wi)��

i��

� �k � 1

m

�kg(wi)� � �i��

g�wi�,

where the two inequalities follow respectivelyfrom the concavity and increasingness of eachg�u�(i, X). Since v, w � F(u�), there exist x, y �X such that v � (u�(1, x), ... , u�(I, x)) and w �(u�(1, y), ... , u�(I, y)). Since u� is lcc, for each u �UCC there is a g meeting the above assumptionssuch that u � g � u� . Hence ¥i�� u(i, x) � ¥i��

u(i, y) for all u � UCC and therefore x �UCC

s yand vRu��UCC

w.

PROOF OF THEOREM 9:For y � �I, let ymax denote max{yj, yk} and

ymin denote min{yj, yk}.Assume vj � vk � wj � wk and vmin � wmin.

If both vj � vk � wj � wk and vmin � wmin thenv* � w* and the conclusion vRu��UCC

w is im-mediate. So assume at least one inequality isstrict. Define z1 � �I by zj

1 � wmin, zk1 � vj �

vk � wmin, and zi1 � vi for i � { j, k} and let z2

equal z1 with the j and k coordinates inter-changed. Since vj � vk � wj � wk, we concludethat zk

1 � wmax and therefore z1* � w* andz2* � w*. Since either vj � vk � wj � wk orvmin � wmin, the equality vj � �wmin � (1 � �)(vj � vk � wmin) defines a � in [0, 1] and this �also solves vk � �(vj � vk � wmin) � (1 � �)wmin.Hence v � �z1 � (1 � �)z2.

Let g be increasing and concave on Range u�and extend g to any r such that r � s for all s �Range u� by setting g(r) � sup{g(s): s � Rangeu�}. Since g remains concave and weakly in-creasing, the indented equalities and inequali-ties in the proof of Theorem 8 apply to such a g.To conclude that vRu��UCC

w, therefore, it is suf-ficient to show that for any u � UCC there is ag: Range u� 3 � that is increasing and concaveon Range u� such that u � g � u� . Since u� is lccthere is an increasing h: Range u� 3 � with u �h � u� such that, for all i � �, h�u�(i, X) is concave.Since u(i, X) and u�(i, X) are both intervals and his increasing, h�u�(i, X) is continuous. Also ob-serve that since �i�� Int u�(i, X) is an interval,Range u� is an interval. Now for any y � Rangeu�, there is an i and nontrivial interval E with y �E and E � Int u�(i, X). Since h is concave andcontinuous on E, the left and right derivatives of hare both nonincreasing on E and the right deriva-tive at any point in E is less than or equal to the leftderivative. These properties of the left and rightderivatives therefore hold at every point in Rangeu� which, together with the fact that Range u� is aninterval, implies that h is concave.

Now assume vRu��UCCw. The argument in this

direction does not require that �i�� Int u�(i, X)be an interval. If vj � vk � wj � wk, then ¥i��

vi � ¥i�� wi, contradicting vRu��UCCw. Hence

vj � vk � wj � wk. Suppose, contrary to thetheorem, that vmin � wmin. Consider transfor-mations g: con(Range u�) 3 � that are piece-wise-linear, increasing, concave, and whoseonly nondifferentiability occurs at wmin. Aslong as Dg(u�) � Dg(u�) � 0 for u� � wmin andu� � wmin, g will be concave and increasing, butotherwise these two derivatives are arbitrary. Sofix some Dg(u�) � 0 and choose Dg(u�) � 0small enough that g(vmax) � g(wmax). (Note thatvmax � wmin since vmin � wmin and vj � vk �wj � wk.) Hence g(vj) � g(vk) � g(wj) � g(wk)and so ¥i�� g(vi) � ¥i�� g(wi), again contra-dicting vRu��UCC

w.

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