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Harsanyi’s ‘Utilitarian Theorem’and Utilitarianism

Mathias RisseKennedy School of Government, Harvard University

1. Introduction

1.1 In 1955, John Harsanyi proved a remarkable theorem:1 Supposen agentssatisfy the assumptions of von Neumann0Morgenstern~1947! expected utilitytheory, and so does the group as a whole~or an observer!. Suppose that, ifeach member of the group prefers optiona to b, then so does the group, orthe observer~Pareto condition!. Then the group’s utility function is a weightedsum of the individual utility functions. Despite Harsanyi’s insistence that whathe calls theUtilitarian Theoremembeds utilitarianism into a theory of rational-ity, the theorem has fallen short of having the kind of impact on the discus-sion of utilitarianism for which Harsanyi hoped. Yet howcould the theoreminfluence this discussion? Utilitarianism is as attractive to some as it is appall-ing to others. The prospects for this dispute to be affectedby a theoremseemdim. Yet a closer look shows how the theorem could make a contribution. Tofix ideas, I understand by utilitarianism the following claims:

~1! Consequentialism: Actions are evaluated in terms of their consequencesonly.~2! Bayesianism: An agent’s beliefs about possible outcomes are capturedprobabilistically.~3! Welfarism: The judgement of the relative goodness of states of affairsis based exclusively on, and an increasing function of, the individual util-ities in these states.~4! Summation: One collection of individual utilities is at least as good asanother if and only if it has at least as large a sum total.2

Bayesianism is normally not considered part of the definition of utilitarian-ism. However, for utilitarianism to be an action-guiding theory, rather than a

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© 2002 Blackwell Publishing Inc., 350 Main Street, Malden, MA 02148, USA,and 108 Cowley Road, Oxford OX4 1JF, UK.

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theory of right-making characteristics, it must accommodate uncertainty. Toaccount for the peculiar status of this condition, I refer to this doctrine asBayesian Utilitarianism. The Bayesian-utilitarian agent assesses the probabil-ity of all possible outcomes, considers the utility of all relevant agents, formsthe sum over the utilities for each outcome, discounts each outcome with itsprobability and chooses an action with a maximal probability-weighted sumover sums of utilities. This picture requires elaboration, but it allows us tolocalize a conceptual place for Harsanyi’s theorem within utilitarianism. Fornow we see that the latter, if it makes any contribution at all, makes it as anargument for Summation once the other claims of utilitarianism have beengranted.

Summation has always been central to utilitarianism. It is mostly taken forgranted, rather than defended. In hisIntroduction to the Principles of Moralsand Legislation, Bentham writes: “The community is a fictitious body, com-posed of the individual persons who are considered as constituting as it wereits members. The interest of the community is then, what?—the sum of theinterests of the several members who compose it”~chapter 1, iv!. Mill’s Utili-tarianism implicitly assumes summation for the assessment of group welfare~cf. in particular the end of chapter iv!. One way of conceiving of utilitarian-ism is as a theory that makes as much sense as possible of the idea of thegreatest amount of happiness~assessed by summation! for the greatest num-ber of people.3 In such approaches, no demand for arguments for summationarises. Still, a philosophically satisfactory utilitarianism must distinguishbetween the different claims of which that doctrine is composed and explorethe entailments among them. Clearly, Consequentialism, Bayesianism, and Wel-farism do not obviously imply any specific view about how to assess thegroup welfare. For instance, a utility version of the Rawlsian maximin princi-ple is available, too.4 Also, once these different claims are separated, it becomesclear that important criticisms of utilitarianism address Summation, rather thanother claims.5 Thus arguments for Summation are called for.

1.2 I submit that Harsanyi’s theorem does provide an argument for anadvocate of Consequentialism, Bayesianism, and Welfarism to endorse Sum-mation. My argument for this claim will come with qualifications. For ourdiscussion will touch on major debates in moral theory and decision theorytoo complex to be settled here. Moreover, no argument for Summation on thebasis of the other three claims is possible without theoretical commitments atvarious points. Still, I hope to show that in this case, a theorem does indeedmake a very substantial contribution to moral theory. This view is bound tobe controversial: many moral philosophers dislike the idea that formal resultsmatter to their discipline, while economic theorists writing on expected utilitytheory also reject the claim that this theorem contributes to utilitarianism~inparticular Sen~1976!, ~1977!, ~1986!, Roemer~1996!, Weymark ~1991!!. Ihope to convince the reader that such views are misguided.

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Section 2 presents the theorem, which requires merely elementary formalnotation. The main argument of this study is in sections 3–5. The challenge isto show just what the connection is between utilitarianism and von Neumann0Morgenstern~vN0M ! expected utility theory. In particular, we need to showthat the utilitarian notion of utility is connected to the notion of utility used invN0M theory in an illuminating way. In section 6, I address what I take to bea version of the most prominent argument against the usefulness of Harsanyi’sframework for utilitarianism. In section 7, we take stock of the qualificationsmade along the way and assess the argument we have developed. I do notsystematically discuss the assumptions of the theorem, that is, the claim thatcollective preferences satisfy the vN0M axioms, and the Pareto condition. Tobypass vexing doubts about “group metaphysics,” I regard “collective prefer-ences” as preferences of an observer concerned about the well-being of therelevant group at least to the extent captured by the Pareto principle. Thatprinciple itself is an intuitively immensely plausible idea. It expresses theidea that the observer’s preferences should preserve universal agreements amongthe individual group members.6 I also take for granted that we understand forwhich groups it is reasonable to think of an observer as being concerned withtheir well-being in the sense expressed by Pareto~which is, if the observer isa version of the traditional “impartial observer”, the question of determiningthe scope of moral considerations!. I exclude such questions because theyarise elsewhere as well and should thus not distract us. The crucial and contro-versial matter to explore is the relationship between vN0M theory andutilitarianism.7

2. Harsanyi’s Utilitarian Theorem

2.1 Let me begin by introducing the vN0M preference theory and its represen-tation theorem.8 Let M 5 ~O1, ... , Om! be a set of outcomes, and L the set ofprobability distributions~“lotteries”! over M. Let Pi , Ii , and Ri be the strictpreference, indifference, and weak~i.e., “preferred-or-indifferent-to”! prefer-ence relation for individual i~1 # i # n!, defined on L, and let P, I, and R bethe corresponding relations for the observer. So preferences in this theory arepreferences overlotteries. One interpretation of these lotteries is in terms ofactions whose outcomes are known only probabilistically. Elements of L canbe written as~p1O1;... ; pnOm!, where pi is the probability of Oi . A real-valuedfunction u representsa relation S~or is a utility representationof S! if andonly if for any two elementsp and q in the domain of S, pSq if and only ifu~p! # u~q!. It is only through this notion of preference representation thatthe concept of utility occurs in this theory. Aprofile of utility functions~u1, ... , un! is a set ofn utility functions, one for each agent. A representa-tion u is called expectational if and only if u~p1O1; . ...; pnOn! 5p1u~O1!1...1pnu~On!, where u~Oi ! is the utility value of the lottery that assignsprobability 1 to outcome Oi and 0 to all other outcomes. The vN0M represen-

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tation theoremthen says the following: If an agent’s preferences over lotteriessatisfy certain conditions, then~a! there exists an expectational utility represen-tation of these preferences; and~b! for any two such representationsu andv,there exist a positive real numbera and a real numberb such that u5 av 1 b.That is, the expectational representation is “unique up to positive affine trans-formations.” The assumptions on the preference relation differ among axioma-tizations, but they all have the same basic structure and include versions ofthe following axioms:

Completeness: For any two lotteries p and q, either pRiq or qRip.Transitivity: For any three lotteries p, q, and r, if pRiq and qRir, then pRir.Independence:For any three lotteries p, q, and r, if pRiq, then for anynumber a between 0 and 1,@ap 1 ~12a!r#Ri @aq 1 ~12a!r# . ~“If q ispreferred-or-indifferent to p, then any lottery that involves q with proba-bility a and some lottery r with probability~12a! is preferred-or-indifferentto a lottery that involves p with probabilitya and r with probability~12a!.”!Continuity:For any three lotteries p, q, r if pPiq and qPir, then there existsa numbera between 0 and 1 such that q Ii @~ap1 ~12a!r!. ~“If p is strictlypreferred to q and q is strictly preferred to r, then there is a probabilityasuch that q is indifferent between a gamble that involves obtaining p withprobability a and obtaining r with probability~12a!.”!

We discuss these axioms in section 4. For Harsanyi’s theorem, suppose we haven agents. We need the following conditions for Harsanyi’s theorem:

Pareto Indifference: For all p, q[ L, if pI iq for all i, 1 # i # n, then pIq.Semi-Strong Pareto: For all p, q[ L, if pRiq for all i, then pRq.Strong Pareto: For all p, q[L, if pRiq for all i, then pRq, and if, further-more, there exists an i such that pPiq, then pPq.Independent Prospects: For each i5 1, ... n, there exist pi and qi[L suchthat piIjqi for all iÞj and piPiqi .

The Pareto conditions demand that universally shared agreements about pref-erences among the group members be preserved in the observer’s preferences.Independent Prospects requires that for each agent there be a pair of lotteriesbetween which she has a preference, but between which everybody else is indif-ferent.9 Under these conditions, then, Harsanyi’s theorem shows that an expec-tational representation of the observer’s preferences is a sum over expectationalrepresentations of the preferences of the group members:

Proposition~Harsanyi’s Utilitarian Theorem!: Suppose Ri , i51, ... , n andR satisfy the vN0M axioms and suppose that Pareto Indifference is satis-fied. Let vi be an expectational representation of Ri , and let v be an expec-


tational representation of R. Then there exist numbers ai and b such thatfor all p [L

v~p! 5 Saivi ~p!1b

~a! Suppose Semi-Strong Pareto is satisfied. Then the ai are non-negative.~b! Suppose Strong Pareto is satisfied. Then the ai are positive.~c! The ai are unique if and only if Independent Prospects is satisfied.

2.2 The theorem is in what is called thesingle-profileformat: it treats only ofone profile of utility functions at a time. As opposed to this, Arrow’s~1951!Impossibility Theorem, for example, is in themulti-profile format, addressingmore than one profile of functions at a time. Theorems in the single-profileformat naturally apply only to one profile at a time. Harsanyi’s theorem, forinstance, implies the existence of certain coefficientsfor a given profileofutility functions, but for a different profile, we obtain different coefficients.However, a complete formulation of utilitarianism requires the multi-profileformat. For it is utilitarian doctrine that each personcount equally. An explicitformulation of this claim would stipulate that the aggregation be indifferentbetween two distributions that only differ in terms of the distribution of theoverall utility across persons. But such a condition must compare and thusrefer to severalprofiles at once and cannot be captured in the single-profileformat. Therefore, as discussed here, Harsanyi’s theorem cannot make acom-plete case for utilitarian summation. So we should think of Harsanyi’s theo-rem as providing an argument for the summation method as such, while notimplying anything about the weights given to the individuals. An argumentfor equality must then come from elsewhere. Thus we are interested inHarsanyi’s theorem as an argument for the following condition, which doesnot imply anything about the coefficients:

Summation': There is a set of weights~or coefficients! such that, for anytwo profiles of utility functions, one profile is at least as good as the otherif and only if it has at least as large a weighted sum of individual utili-ties, weighted according to the given coefficients.10

3. Two Notions of Utility

3.1 Contemporary utilitarianism acknowledges three accounts of well-being.Mental-state accounts explicate well-being in terms of mental states such aspleasure or satisfaction. Desire-satisfaction theories account for well-being interms of the realization of desires. Whereas mental-state accounts ignorewhether there is any ‘fit’ between mental states and states of the world, desire-satisfaction accounts understand well-being precisely in terms of the extent ofsuch a fit. Finally, objective-list theories explicate well-being in terms of a

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list of properties that constitute a person’s well-being regardless of both men-tal states and desires. Curiously, many utilitarians do not have much use forthe concept of “utility” when discussing well-being. Nevertheless, since thesetheories are the current accounts of well-being, they are the current candi-dates for explicating the concept of “utility” contained in the very notion of“utilitarianism”.11

However, listing these accounts fails to capture the complexities of develop-ing a satisfactory notion of well-being. That notion, no matter how we expli-cate it, must fill various roles in moral and prudential deliberation and in ourconception of and interaction with others, and those roles place demands onany theory of well-being. Griffin~1986! puts the point as follows:

First, we need the account of well-being that we adopt~...! to be a plausible accountof the domain of prudential value that it tries to cover; second, it must be whatwe want to use, for purposes of moral judgement, as the basis for comparisonbetween different persons; and third, it has to lend itself to the sorts of measure-ment that moral deliberation needs~p. 108!.

In this study I adopt a desire-satisfaction account of well-being and assumethat such an account can be developed in a way that satisfies Griffin’s desider-ata and possibly others. More specifically, I adopt a version of this accountthat has been amended in at least two ways in response to common objec-tions. On the one hand, this account will be able to distinguish between desireswhose satisfaction contributes to a person’s well-being and those whose satis-faction fails to do so.~One may desire that there be life in some remote solarsystem, but its existence would not contribute to one’s well-being.! On the otherhand, this account will focus not on actual desires~which may too easily con-flict with a person’s well-being!, but desires that a person would have wereshe properly informed, thinking clearly, without any prejudices and biases, etc.Such an account is anidealized-desire-satisfaction~IDS-! account of well-being. I have little to say to develop this conception in detail, but restrict myselfto investigating how it can be connected to vN0M expected utility theory.Clearly, if utilitarianism cannot provide us with a satisfactory account of well-being, it has bigger problems than the inability to find a conceptual place forHarsanyi’s theorem. Given the prominence of IDS-accounts, it seems reason-able, then, to adopt this conception for the sake of this discussion.12

3.2 But how does the vN0M notion of utility bear on this notion of well-being? In vN0M, “utility” refers to numbers representing preferences. Thosevalues are, as Hampton~1994! put it, “just numbers.” Apparently, vN0M theoryonly shares a word with utilitarianism, and one that is not even used by manyutilitarians. There are two ways of developing this claim into an objection tothe usefulness of the vN0M framework and thus of Harsanyi’s theorem for util-itarianism. I develop both and argue that they fail. Nevertheless, they leave us


with a challenge that needs to be met for vN0M theory to have any bearingon utilitarianism, and providing a response to it is our main concern in theremainder of this essay. Once it is met, it is straightforward to see the placeof Harsanyi’s theorem within utilitarian theory.13

The first objection states that vN0M theory and utilitarianism are divorcedfrom each other because the former has nothing to do with any of the currentnotions of well-being. While the vN0M representation theorem shows that indi-viduals’ preferences can be numerically represented in a convenient form, Har-sanyi’s theorem makes a statement about the representation of an observer’spreferences. Nothing of substance follows about utilitarianism. Yet this objec-tion is misguided. The vN0M theorem is ananalytical result thatassumesapreference relation andderivesnumbers that represent preferences. However,from the fact that utilities are “just numbers” in the vN0M model, it does notfollow that no conceptual connection exists between the vN0M theory and util-itarianism. Claiming that it does is to confuse a logical relationship estab-lished within a specific model with methodological, epistemological, or possiblyontological insights that such a model by itself cannot provide. A connectionbetween vN0M theory and utilitarianism could be demonstrated, for example,by showing that a notion of utility as well-being explicates themeaningofpreferring. A demand for such an account arises as follows. The vN0M theoryuses a preference relation as a syntactically primitive symbol, which is expli-cated within in the model only through the assumptions made about it. For-mal results are then derived from these assumptions. But in addition to provingresults within this model, we also need to interpret them. In particular we needto ask about the meaning of “preferring,” just as we need to ask about themeaning of the material implication or the meaning of probability. It is throughits possible semantic function vis-à-vis the preference relation that the notionof utility as well-being might be connected to the notion of utility as prefer-ence representation. The challenge is to provide an interpretation of prefer-ring that is coherent with the assumptions on the preference relation in themodel.

3.3 Another objection arises now, which, if successful, would meet this chal-lenge in a way that undermines any attempt to find a useful conceptual connec-tion between vN0M preference theory and utilitarianism. Sure enough, one maysay, the representation theorem fails to show that, only because numerical util-ities are derived from preferences in the vN0M model, there can be no connec-tion of vN0M preference theory to utilitarianism. And sure enough, we needan interpretation of preferring. However, the objector insists, preferring shouldbe understood within the confines of abehavioristic account of psychology.Such an account analyzes “desire” and “belief”~and other “mental” vocabu-lary! in terms of observables. Preferences are observable in choices. On thispicture, it only makes sense to speak of well-being to the extent that it can beobserved in choice, that is, only in terms of preferences. No question about

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the meaning of preferring arises that cannot be answered in terms of observ-ables. So if there is room for “utility,” it must be derivative of preferences.No questions about connections between two notions of utility emerge, sinceonly one of them is meaningful to begin with.

Yet this account of psychology has become discredited. As is well-known,decision theory originated during the heyday of logical positivism, which pro-vided a congenial environment for behaviorism. Attempts to derive “utility”and “probability” from “preferences” were motivated precisely by such a psy-chology. Yet we have abandoned this picture largely because its costs are toohigh: it forbids us from saying too much that we want and need to say, notjust in practice, but also in theory. So being committed to behaviorism is beingcommitted in more daring ways than being committed to a notion of individ-ual well-being that accounts for the meaning of preferring in terms other thanobservables.14

3.4 The second objection fails, and thus we must answer the challenge posedby the first in another way. In general, once we drop the behavioristic accountof psychology, the question of what “preferring” means becomes urgent. Forpreferring, if not understood as the behaviorists would have it, sits uneasilybetween choosing and desiring.15 Is there, then, an interpretation of preferringthat makes for a connection to utilitarianism and is coherent with the vN0Massumptions on the preference relation? I submit that there is. To prepare theargument, note that, straightforwardly, the concept of a person’s well-beingentails abetterness-relation: outcome O1 is better than O2 if and only if O1 ismore conducive to her well-being than O2. This relation extends to lotteries.For when an agent acts under circumstances of risk and can predict outcomesonly probabilistically, she must evaluate such probabilistic prospects from thepoint of view of her well-being. It might be clumsy to speak of amounts ofwell-being pertaining to risky prospects, but the idea is clear and familiarenough. If this betterness-relation satisfies the vN0M axioms, then there is aconceptual connection between utilitarianism and vN0M preference theory. Forthen the vN0M representation theorem shows that this relation can be repre-sented by expectational utility functions. This result would be an importantmilestone on our way to explore what contribution Harsanyi’s theorem makesto utilitarianism. Our next task is to show that this betterness-relation does sat-isfy those axioms.16

4. Expectational Representations of Well-Being

4.1 There has been a great deal of controversy about the vN0M axioms. Inlight of the currency of Humean views on human psychology, the extent ofthis controversy is unsurprising. According to Humeans, there are two mainkinds of psychological states, namely, beliefs and desires. Desires are unlikebeliefs in that they do not purport to represent the world the way it is, and


thus, for Humeans, desires are not subject to rational scrutiny beyond the rec-tification of factual errors on which they might be based. From this point ofview, then, any constraints on preferences must seem dubious. The IDS-accountof well-being adopted in this study is at odds with the Humean view. For thedesires whose satisfaction is taken to be constitutive of the agent’s well-beingemerge through rational scrutiny and reflection. I submit that any develop-ment of the IDS-account should regard the vN0M axioms as reasonable con-straints on the betterness-relation.

At this stage, then, it matters that we have adopted the IDS-account, ratherthan any other account of well-being. For that approach endorses rationalconstraints on desires. The arguments in this section will not convince any-body with reservations about rational constraints on desires. Rather, thearguments are, axiom by axiom, directed at somebody who endorses the IDS-account of well-being in principle and is considering reasonable constraintson a detailed development of that account. This approach to the vN0M axi-oms puts us in a better position to argue for them than attempts to argue forthem as general constraints on rational behavior~on which, of course, a lot ofink has been spilled!. For it is possible without too much stretching to con-struct scenarios in which the one or the other axioms is violated while it isnevertheless unclearjust whysuch behavior would be straightforwardly “irratio-nal.”17 This is in particular so on a Humean view, which supports only a thinnotion of rationality. By arguing for the axioms as reasonable constraints onthe IDS-account of well-being, we are getting some mileage out of the start-ing position. Although the axioms are defined for lotteries, for the sake ofsimplicity I discuss them as if they were defined for outcomes~lotteries thatassign probability 1 to one outcome!. The arguments generalize in a straight-forward way.

4.2 Let us begin withIndependence. This axiom insists that outcomes be eval-uated independently of each other. To see how this might be problematic, con-sider a famous example due to Diamond~1967!. Suppose you can give a goodA to one of two people. Suppose that, if only one gets A, it does not matterwhich one: you are indifferent between~A, 0! and~0, A!. Independence entailsthat you are indifferent between a lottery resulting in~A, 0! and ~0, A! withprobability 1

2_, respectively, and a lottery resulting in~0, A! with probability 1

2_

and once more~0, A! with probability 12_, that is, a lottery resulting in~0, A!

for sure. Yet this seems unreasonable. It seemsunfair for it to be a matter ofindifference whether one person gets A for sure~suppose A is a donated kid-ney!! or whether both have an equal chance of obtaining A. However, it is nosurprise that a fairness problem arises if fairness is not considered when out-comes are individuated. The outcomes areunderdescribedas ~A, 0! and ~0,A! if fairness is of importance. If fairness is included in the description of theoutcomes, the problem disappears. Let F denote a state of affairs in which Ahas been distributed fairly and -F denotes a state of affairs in which it has

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not. Distinguish then the outcomes~A, 0, F!, ~A, 0, -F!, ~0, A, F!, and~0, A, -F!. Independence entails that you are indifferent between~A, 0, F! and~0, A, F! occurring with probability1

2_ each and~A, 0, F! for sure. This is not

counter-intuitive. The reader may find it odd to include fairness in the descrip-tion of outcomes. But why would~A, 0! be an outcome while~A, 0, F! wouldnot? As Savage~1954!, put it: “A consequence is anything that may happento the person”~p. 13!, and surely, having received A in a fair way counts bythis standard.

This response to Diamond is called “loading-up-the-consequences.” It is con-troversial because one may worry that any example questioning an axiom canbe dissolved in this way. However, for consequentialism to be even plausible,there needs to be a theory of individuating outcomes that captures all relevantconsiderations. If, in Diamond’s example, fairness it taken to be relevant, itmust appear in the description of the outcomes, and then no problem arises.Otherwise, there should have been nothing problematic about the initial impli-cation of Independence. One may object that, if Independence only applies ifthe outcomes are described completely, it is trivially true. It is not clear justwhy that would be a problem, but itis clear that Independence trivially failsand that consequentialism is unattractive if outcomes are underdescribed.18

This discussion makes a general point about outcomes, but it also indicateshow to argue that the betterness-relation satisfies Independence. Once one out-come has come about, noother outcome has come about or can come about.Therefore, this outcome should be evaluated on its own terms, without anyreference to other outcomes. This seems eminently reasonable if all consider-ations relevant from the point of view of well-being have been considered inthe individuation of outcomes. Independence, then, is a plausible constraint onthe betterness relation, given this understanding of outcomes.

4.3 AlthoughCompletenesshas found few supporters as an axiom of rationalchoice,19 one may think that it is more plausible as a condition on ourbetterness-relation. After all, utilitarians are criticized as simple-minded foradvocating well-being as an overriding value; surely they should be entitledto the theoretical benefits from what they are taken to task for and find iteasy to argue that the betterness-relation satisfies Completeness. However,championing any value as overriding does not entail that the nature of thatvalue allows for adjudication between any two outcomes. It is unclear, inparticular, that on the best versions of the IDS-account, there is always asynthesis of possibly diverging and conflicting desires into an overall attitudetowards any outcome, which then would make it possible to compare any twooutcomes. Yet the case for Completeness is not hopeless. This is in particularso on an “organic” understanding of value as championed, for instance, inMoore’s Principia Ethica. Moore states the idea of theprinciple of organicunities as follows: “The value of a whole must not be assumed to be thesame as the sum of the values of its parts.”20 On such an understanding of


value, the evaluation of outcomes does not fall into separate assessments ofeach aspect of the outcome whose conjunction constitutes its overall evalua-tion. For no aspect can be evaluated without considering the presence of theothers. Such a conception of value and the derivative view on evaluatingoutcomes does by no meansentail Completeness; but it does undermine astrong source of intuitions for the implausibility of Completeness. That sourceis the idea that we evaluate outcomes aspect by aspect, which makes it likelythat in many cases one outcome is superior to another in some aspects,but not in others. Consider also that the ability to compare any two outcomesis immensely beneficial to an agent: it keeps her from being torn or para-lyzed. Surely this point carries some weight ifidealized desiresare central towell-being. These two points together suggest that we should allow for thepossibility that the best developments of the IDS-account include a betterness-relation that satisfies Completeness. The case for Completeness remains ques-tionable, but surely not hopeless.21

4.4 Transitivity is intuitively appealing. To see how Transitivity could be prob-lematic, consider the following scenario, calledimproving oneself to death:22

Suppose my well-being depends on two kinds of goods, and I prefer gettingmore of the one that I have less of, as long as the loss with regard to theother is “small”. Suppose I start with 10 units of the first and 9 units of thesecond good. Then in the following list, I prefer each member to its predeces-sor, but Transitivity is implausible:~10, 9!, ~8, 10!, ~9, 8!, ~7, 9!, ~8, 7!,~6, 8!, ~7, 6!, etc. So when multi-dimensional outcomes are compared, a gainin one dimension may be acceptable at the expense of a loss in another.Repeated occurrences of that phenomenon undermine Transitivity.~If the strat-egy of loading-up-the-consequences is accepted, outcomes will tend to havethe kind of complex structure that facilitates such examples.! Broome~1991a!,pp. 11–12, insists that the betterness-relation is transitive as a matter of logic.However, it is hard to see how an appeal to logic could handle examples suchas improving-oneself-to-death. A better response is an appeal to the “organic”understanding of value. This theory of value denies that outcomes can beevaluated factor by factor; but it is precisely this kind of evaluation thatdrives the above example. On an organic understanding of value, comparingoutcomes by preferring more of the good that one has less of is not merelyodd, but scenarios such as improving-oneself-to-death serve to illustrate whatis wrong about evaluating outcomes aspect by aspect. So Transitivity, likeCompleteness, seems plausible under the organic understanding of value. AndTransitivity, unlike Completeness, clearly has the pre-theoretical intuitions onits side.23

4.5 Continuity is sometimes seen as a “technical” assumption.24 However, thisclassification is dubious to begin with, and Hajek~1998! shows how one mayquestion Continuity. Hajek argues for the importance of infinite utilities, but

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Continuity forces utilities to be finite.25 He claims that there is no good argu-ment for Continuity, and that in particular an appeal to consistency in deci-sion making fails to support it. But be that as it may, a case can be made forimposing Continuity on the betterness-relation. Continuity is plausible if theagent can compare and weigh amounts of well-being derived from differentoutcomes. Continuity is an expression of the agent’s ability to do so. Exam-ples intended to show how Continuity fails frequently involve death. Wouldyou really, so we are asked, risk death~which in such examples must have alow negative, but finite utility!, even with a very small probability? We areexpected to reject such an idea, thereby contradicting Continuity. But, in fact,we run such risks all the time: Suppose I get up in the morning, badly needcoffee, but none is left. So I go and buy coffee, running a risk of being killedin traffic. In terms of Continuity: there is a probability p such that I am indif-ferent between the outcomeI have no coffeefor sure and a lottery involvingthe outcomeI die in a traffic accidentwith probability p and the outcomeIhave coffeewith probability ~12p!. Thus it is not as strange as it may seemto assign death a finitely negatively value and to ponder it against others.

Surely, for eternal salvation and condemnation, such weighing may not work.As was already pointed out in the 1662Port Royal Logic, one of the found-ing documents of expected utility reasoning: “only infinite things such as eter-nity and salvation cannot be equaled by any temporal benefit”~Arnauld ~1996!,p. 275!. But it is hard to think of many other cases of this sort. Therefore theapplicability of infinite utilities is rather restricted, and surely provides no rea-son to abandon Continuity. In this spirit, Morgenstern~1976! points out thatthe vN0M theory compares to Newtonian mechanics, which fails for objectstraveling at a speed close to that of light, but otherwise does just fine. So thecase for arguing that the betterness-relation satisfies Continuity is fairly good.26

4.6 In conclusion, a case can be made for adopting Completeness, Transitiv-ity, Continuity, and Independence as constraints on the betterness -relation ina development of the IDS-account. There is potential for disagreement, but atleast on an “organic” theory of value our case does not look unpromising. Thecritical axioms is, of course, Completeness. While registering these qualifica-tions, I assume for the sake of the argument that the betterness-relation entailedby the IDS-account of well-being does indeed satisfy these four axioms. Thesubsequent argument is only as good as this claim, but let us see what we canmake of it.

5. The Quantitative Structure of Well-Being

5.1 Section 4 does not yet show that Harsanyi’s theorem provides any usefulinsights about utilitarianism. What we have shown is this. Suppose that theindividual betterness-relations and the observer’s betterness-relation satisfy thevN0M assumptions, and suppose that a Pareto condition holds. If we choose


any expectational representation of the individual betterness-relations and ofthe observer’s betterness-relation, respectively, then the observer’s representa-tion is a weighted sum over those of the individuals. This is an interestingresult about expectational representations, but we would gain a deeper insightif we could show in addition that one of the expectational representations actu-ally measuresthe agent’s well-being~i.e., that well-being itself is “expectation-al”!. For then the following would be true: If the individual betterness-relationsand the observer’s betterness-relation are related by a Pareto condition, thenthe function that measures the observer’s well-being is a weighted sum overthe functions that measure the individuals’ well-being. What I argue next, how-ever, is not thatoneof those representations measures the agent’s well-being,but that thefamily of expectational representationsas such does so. Yet it willsoon become clear that Harsanyi’s theorem loses none of its status as a contri-bution to utilitarianism for this reason.27

5.2 We need to explain what it means for a family of functions to measuresomething. To this end, and for its usefulness as an analogy, let me brieflydiscuss the measurement ofheat. Von Neumann and Morgenstern~1947! usethis analogy, and so does Broome~1991a!. When we try to measure heat, anyproperty of a substance or a device that changes when it is heated or cooledmay serve as the basis of athermometer. For instance, we may define thechange in temperature to be proportional to the change in length of a columnof liquid in a capillary tube. In order to calibrate a thermometer we assignnumerical values to the temperatures of two points~given constant pressure!.The position of liquid at these points is marked and the distance between themis divided into equal intervals. For example, on the Celsius scale, there are100 intervals between the freezing point of water at 1 atm pressure~set at 0!and its boiling point at 1 atm pressure~set at 100!, whereas on the Fahrenheitscale, there are 180 intervals~with the freezing point set at 32 and the boilingpoint at 212!. Fahrenheit and Celsius can be transformed into each other usingthe equations tf 5 905 tc 1 32 and tc 5 509 ~tf -32!. In jargon, the two scalesarepositive affine transformationsof each other. A functionf is a positive affinetransformation of a functiong if there exists a positive real numbera and realnumberb such that f5 ag1b. If f is such a transformation ofg, then the con-verse is true as well. Any scale that is a positive affine transformation of theCelsius scale can be used as a temperature scale in the sense outlined above,and vice versa. Each of these functions is as good a measuring scale as anyother. It is in this sense that it makes sense to say of this whole family offunctions, rather than of any specific function, thatit measures heat.

The fact that this family of functions measures heat in this sense providesa characterization of meaningful statements about temperature comparisons.Meaningful statements are those that are invariant across all scales in that fam-ily. For instance, the statement that it is warmer in New Haven at 11 am thanat 10 am is meaningful because it is either true according to all such scales or

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false according to all of them: inequalities between temperature differences arepreserved across all scales. But the statement that it is twice as warm at 11am as it was at 10 am is meaningless. For if this is true on the Celsius scale,it will be false on the Fahrenheit scale. What does make sense to say, though,is that the increase in heat between 10 am and 11 am is three times as muchas the increase between 2 pm and 3 pm. For the ratio between temperaturedifferences remains constant across scales.

5.3 Von Neumann and Morgenstern did not merely introduce this analogy toillustrate what it is for a family of functions to measure something. Theythought that we learn more from it about measuring utility. Although it hasbeen emphasized that their notion of utility is different from the utilitarian one,little attention has been paid to the fact that this is not how they conceived ofit. They took themselves to be contributing to an area of research where notmuch progress had been made, namely, the measurement of utility. The anal-ogy to heat, which they use extensively, is the key to understanding their idea.Prior to the development of a theory of heat, so they say, we only had an intu-itively clear feeling of one body feeling warmer than another, whereas nowa-days we have thermometers to make statements about comparative strengthsof temperature differences. Von Neumann and Morgenstern conceive of utilityas part of physically describable nature in the same sense in which heat is.They believe that they have made a discovery that advances utility measure-ment in the same way in which the thermometer advanced the measurementof heat.

They talk about “utility” in two ways: On the one hand, there are numeri-cal utilities analogous to temperature values. Yet on the other hand, there isutility analogous to heat: a physical property be measured.28 According tothem, the key to utility measurement is thediscoveryof a “natural operation”in the realm of utility. A natural operation is one that is “intuitively clear”~e.g., “warmer than” for temperature, “harder than” for minerals!, and “obser-vationally reproducible.” The additional natural operation~i.e., in addition to“preferring”! that von Neumann and Morgenstern think they discovered is theconcatenation of events with probabilities. If we can talk about events, sothey argue, we can talk about probabilistic concatenations of events. Sincethey regard events as the location of utility, this operation applies to utility aswell, and thus we have “discovered” that utility itself is expectational. Andthen it only takes the axiomatic postulation of properties of the concatenationoperation to obtain a measurement of utility based on these two natural opera-tions. Those properties must be chosen such that the behavior of numericalutilities captures the expectational nature of utility. “We have practically definednumerical utility as being that thing for which the calculus of mathematicalexpectations is legitimate”, so they say~p. 28!. In conclusion, they believethat, through their representation theorem, they can determine a family offunctions closed under positive affine transformations that measure utility just


as positive affine transformations of the Celsius scale measure heat. This fam-ily of functions, of course, is the family of expectational representations.29

Developing this view involves problems in the philosophy of science. Do wereally measureanything? What do “intuitive clarity” and “observational repro-ducibility” amount to? Most critically, what are we to make of von Neumannand Morgenstern’s claim that they “discovered” a new operation, namely theconcatenation of events with probabilities? Without having an appropriatetheory of what such a “discovery” amounts to, this claim has all the advan-tages of theft over honest toil: it delivers the expectational nature of utilitywithout further ado. Below I will have more to say about the measurement-question. However, I suspect that these questions cannot be answered in sucha way that we would find it ultimately plausible, onthis account, that the fam-ily of expectational representations does indeed measure well-being. But sincethis is what I am trying to argue, we should look elsewhere for support.

5.4 To make progress, I enlist an argument due to Broome~1991a!, which Ithink shows that the family of expectational representations of the betterness-relation measures well-being. To get the argument started, Broome~pp. 146–148! introduces the following scenario. You are comparing two actions, A1 andA2. If you choose A1 and outcome O1 occurs, you receive $100; if outcomeO2 occurs, you receive $200. If you choose A2 and O1 occurs, you receive$20, and if O2 occurs, you receive $320. Suppose that if O1 occurs with prob-ability 103 and O2 with probability 203, A1 and A2 are equally good for you.That is, the prospect of obtaining $100 with probability 103 and $200 withprobability 203 is, as far as your well-being is concerned, on a par with theprospect of obtaining $20 with probability 103 and $320 with probability 203.30

Since we are assuming that the betterness-relation entailed by your well-beingcan be represented by an expectational functionu, we obtain the followingequation:

103u~$100! 1 203 u~$200! 5 103u~$20! 1 203 u~$320!

Simple algebraic transformations show that

$u~$100!-u~$20!%0$u~$320! 2 u~$200!% 5 2

That is, the utility difference between the two amounts of money you couldobtain were O1 to occur is twice as big as the utility difference between thetwo amounts you could obtain were O2 to occur. The fact that, in O1, youreceive $100 rather than $20 if you choose A1 is a consideration in favor ofA1. Similarly, the fact that, in O2, you receive $320 rather than $200 if youchoose A2 is a consideration in favor of A2. So the second equation showsthat the consideration in favor of A1 counts, as far as the overall well-beingpertaining to those prospects is concerned, twice as much as the considerationin favor of A2. More generally, the utility values tell us how much differences

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in well-being count proportionatelyin the determination of the comparativeoverall well-being pertaining to those prospects. Since we are dealing with afamily of expectational representations, the proportionality statements do notchange across representations.

But if the difference in well-being between $100 and $20countsfor twiceas much as the difference between $320 and $200 in the determination ofyour overall well-being pertaining to a lottery, it is plausible to infer thatthese differences measuregenuine differencesin well-being. As Broome pointsout ~p. 147!, the only way of denying this inference is to insist on a differ-ence between amounts of well-being and the way theycount towards overallwell-being. Yet this seems like an empty distinction, because plausibly, differ-ences in well-being can count proportionately the way they doonly becausethey amount to genuine differences in well-being. If this is right, we haveshown that, in general, the ratio of differences in well-being is whatever theratio of the corresponding utility differences is. This entails that the agent’swell-being itself can be expressed as a positive affine transformation of anyexpectational representations of the betterness-relation.31 That is, the agent’swell-being itself is expressed by a function in this family of expectationalrepresentations. This does not imply that well-being is expressed or measuredby any of those functionsrather than by any other. Instead, it means that thefunction measuring the agent’s well-being is a member of the family of expec-tational representations, and just like in the case of heat, the choice of anyspecific function as the measurement scale is arbitrary. So we are justified insaying that the whole family of expectational representations of the agent’sbetterness-relation measures her well-being, just as the whole family of posi-tive affine transformations of the Celsius scale measures heat.32 Thus we havefinally met the challenge posed in section 3, that is, to find a conceptualconnection between the vN0M notion of utility as preference representationand the utilitarian notion of utility as well-being.

We can conclude then that Harsanyi’s theorem teaches us the following: Ifthe individual betterness-relations and the observer’s betterness-relation arerelated by a Pareto condition, then any of the functions that measures theobserver’s well-being is a weighted sum over any profile of functions that mea-sure the individuals’ well-being. So if one accepts Consequentialism, Welfar-ism, and Bayesianism, one ought to accept Summation' as well. For if~giventhose three conditions! one tries to act like an impartial observer to the mini-mal extent that one accepts universal preference agreements in the relevantgroup, then one cannot help but accept Summation'. For the remainder of thissection and in section 6, we will address worries about and objections to thisargument and the conclusion just drawn. In section 7 we will discuss the con-clusion some more.

5.5 Let me address two worries about this argument. One may still wonderwhether we should speak of ameasurementin this context. What made themeasurement of heat “a measurement,” one might say, was the availability of


a device such as mercury rising in a capillary tube that captures a feature ofheat independently of subjective perception. Yet what counts as measurementis a vexing question. On the one hand, there are paradigmatic cases of measure-ment, such as the measurement of length, or of heat, that are remarkably dif-ferent from what I propose as a measurement of well-being. On the other hand,what counts as measurement must depend on the respective domain. In ourcase, what would be measured is well-being according to the IDS-account,where the measurement process would be idealized deliberation. Yet for ourpurposes, nothing depends on using the term. No harm is done if the readerreplaces occurrences of the verb “measure” with the corresponding forms of“express” or “capture.”33

Another worry is that, if the argument of this study is correct, an agentshould be risk-neutral with regard to her well-being. Evidently, so an objectormight say, this is empirically false, and it is false that personsought tobe risk-neutral in this way. Before I address this objection, let me make sure that it isnot based on a mistake. It is a widely accepted claim that people are risk-averse about money. But nothing in my argument contradicts that claim. Tosee why, suppose that all that matters to a person’s well-being is the moneyshe receives in an outcome. Then the utility functions assign to amounts ofmoney numerical utilities capturing the well-being deriving from the money.Supposeu is such a function, and suppose we are looking at an action thatleads to outcomes O1, ... , On with probabilities p1, ...., pn. If the betterness-relation satisfies the vN0M axioms, then

u~p1O1, ... , pnOn! 5 p1u~O1! 1 .... 1 pnu~On!

If u measures well-being, as I have argued, then this equation indeed expressesrisk-neutrality with regard to well-being. Risk-neutrality about money, how-ever would be captured as follows, where $Oi denotes the amount of moneythat the agent would receive in state Oi :

u~p1$O1, ... , pn$On! 5 u~p1$O1 1 .... 1 pn$On!

This equation says that an agent attaches as much well-being to a monetarylottery as to the expectation of this lottery. But that is an entirely differentclaim. The argument in this study does not requireu to satisfy the second equa-tion. As readers familiar with expected utility theory know, this response is asold as the notion of utility itself, and in fact, the desire to draw a distinctionbetween the points expressed by these two equations motivated the very intro-duction of the notion of utility. But if this point is acknowledged, it should beclear that our intuitions about risk-attitudes with regard to well-being are notas developed as our intuitions about risk-attitudes with regard to, say, money,both empirically and normatively. The argument of this study, if correct,entailsthat a development of the IDS-account of well-being should conceive of well-

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being in a way that entails risk-neutrality with regard to well-being. Worriesabout that claim should be raised as objections to some part of the argument.34

6. Interpersonal Comparisons of Utility

6.1 We now address an important objection to any usefulness of Harsanyi’s theo-rem to utilitarian theory, presented by Roemer~1996!.35 To this end we needto discuss interpersonal comparisons of utility. Utility comparisons pose signif-icant conceptual and practical difficulties. For that reason, social choice theo-rists have chosen anaxiomaticapproach to both intrapersonal and interpersonalcomparisons. In such an approach, different senses of utility comparability canbe characterized, although we may not know how tomakesuch comparisons.Different notions of utility comparability are captured byinvariance proper-ties of profiles of representations. The idea is to say for any profile~u1, ..., un!which profiles areequivalentto it, where equivalence amounts to expressingthe same information. Once we have introduced such an equivalence relation,we count as meaningful precisely those statements about intrapersonal and inter-personal comparisons that hold forall equivalent profiles. We used a versionof this idea to characterize meaningful statements about utility~and tempera-ture! comparisons in section 5. Consider examples: In the simplest case, all rep-resentations are equivalent, that is, regarded as expressing the same information.But then it only makes sense to say that an agent prefers an outcome to another.Other statements~such as “Agent 1 prefers option x to option y more thanhe prefers y to z”! fail to hold across all representations. Next define a profile~u1' , ..., un

' ! as equivalent to~u1, ..., un! if there are positive real numbers ai andreal numbers bi such that ui 5 aiui

'1bi . Then it is meaningful to say that i pre-fers an outcome x to y more than he prefers z to a, since the relevant inequali-ties remain fixed among all equivalent representations. However, it is notmeaningful to say that individual i prefers x to z more than individual j prefersy to a.

We need to make assumptions on comparability for summation over utili-ties to be meaningful. To see why, suppose we take all profiles to be equiva-lent. Suppose we have profiles~u1, ..., un! and ~u1

' , ..., un' !, and suppose that

the functions ui' assign a much broader range of values. We can obviouslychoose profiles in such a way that the new observer utility function generatedby summation not only assigns different utility values to lotteries, but does soin such a way that not even the observer’sranking is preserved. So howmuch utility comparability do we need to obtain a meaningful notion of sum-mation? Define a profile~v1, ..., vn! as equivalent to~u1, ..., un! if there is apositive real numbera such that ui5avi 1bi for all i and real numbers bi .Then many kinds of comparisons become meaningful. For instance, it makessense to say that individual i prefers x to y twice as much as j prefers z to a.For such statements to make sense, it is sufficient that ratios of the kind@~ui ~x!2ui ~y!!0~uj ~z!-uj ~a!!# remain unchanged if ui is replaced with vi5aui1bi


and uj is replaced with vj5avj1bj . As is easy to check, this is indeed thecase. Thus interpersonal comparisons of utilitydifferencesmake sense underthis notion of equivalence. However, comparisons of utilitylevels make nosense: it is not meaningful to say that agent i is “better off” than j. Still, if weuse this notion of interpersonal comparability, group rankings are preservedunder changes of equivalent representations. And at least in that sense thensummation is meaningful.36

This notion of interpersonal comparability answers a worry about the argu-ment in sections 3–5 that I omitted in section 5.5. Consider profiles~u1, ..., un!and ~u1

' , ..., un' ! of expectational representations and an expectational repre-

sentationu for the observer. Harsanyi’s theorem implies the existence of coef-ficients ai and a constantb such that u5 Si aiui1 b, and of coefficients ai' andb' such that u5 Si ai

'ui' 1 b'. But we do not have in general ai5ai

'.37 However,if those two profiles are equivalent in the sense required by this notion of inter-personal comparability, it is true that

ai Si ai 5 ai' Si ai

' , for any 1# i # n

which means that individual weights are kept fixed.

6.2 We can now state the objection. Suppose the assumptions of Harsanyi’stheorem apply. Consider the set C of all profiles of representations~u1, ..., un!.Suppose we can make interpersonal comparisons for utility summation to bemeaningful. Thus there is a subset D of C containing equivalent profiles overwhich it is meaningful to sum. Moreover, by the vN0M representation theo-rem, there exists a subset E of C including all profiles consisting of expecta-tional representations. That is, E is the subset of D to which Harsanyi’s theoremapplies. However, there is no guarantee that D and E have any element in com-mon: the profiles to which Harsanyi’s theorem applies are not necessarily pro-files over which it is meaningful to sum. Therefore, the summation inHarsanyi’s theorem may not be meaningful from a utilitarian point of viewand thus the theorem cannot bear on utilitarianism.

If the argument in sections 3–5 is correct, the response is straightforward.I have argued that the betterness-relation entailed by the IDS-account satisfiesthe vN0M axioms. So we should understand well-being in terms of anidealized-desire account constrained by the imposition of the vN0M axiomson the betterness-relation. Thus there is no notion of well-being determining aset of profiles of representations over which it is meaningful to sum, butwhich is disjoint from the set of profiles that satisfy those axioms. Roemer’sargument is motivated by the idea that expectational functions “merely repre-sent” preferences. The argument in sections 3–5 entails that this view is wrong.What could happen, of course, is that a notion of well-being used for empiri-cal or statistical inquiries is at odds with the expectational representations ofan agent’s betterness-relation. But that is not problematic. In the present

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approach, expectational utility functions represent the betterness relation entailedby an IDS-notion of well-being. Interpersonal comparability of well-beingmust be thought of as integrated into such an account. Empirical notions ofwell-being and empirical methods to compare well-being can only be approx-imations of such notions. In light of this, what Roemer shows is that suchapproximation can be odds with the idealized notion of well-being, but again,that is not problematic, and it is not surprising.

7. Conclusion

A defender of Consequentialism, Welfarism, and Bayesianism should endorseSummation'. Summation' should be adopted because for any observer whosewell-being is connected to the well-being of the individual members of therelevant group in the minimal sense that his betterness-relation preserves agree-ments among their preferences—for any such observer it is true that anyfunction that measures his well-being is a weighted sum over any profile offunctions that measure the individuals’ well-being. Put differently, if an advo-cate of Consequentialism, Welfarism, and Bayesianism accepts the Pareto con-dition ~which I think is not much to ask!, then the argument of this studyentails that she cannot help but accept Summation'. The crucial segments ofthe argument are to show that the vN0M axioms constrain the betterness-relation entailed by the IDS-account of well-being, and that the family ofexpectational representations measures well-being. To make these arguments,I have appealed to an idealized-desire account of well-being and to an organictheory of value.

Recall the qualifications we adopted along the way. To begin with, in vir-tue of its single-profile format, Harsanyi’s theorem cannot deliver an argu-ment for equal consideration in the summation. Therefore, Harsanyi’s argumentonly presents us with apartial defenseof utilitarian summation in the pres-ence of Consequentialism, Welfarism, and Bayesianism~i.e., an argument forSummation', not for Summation!.Yet it would be puzzling if those conditionswith merely moderate additions implied equal consideration for individuals. Itis very surprising already that the addition of Pareto suffices to entail Summa-tion'. Additional conditions to obtain equality of consideration must be formu-lated in the multi-profile context. But we should expect them to be close topostulations of equal consideration; we should not expect a surprising and illu-minating derivation of the sort that here provides an argument for Summa-tion'. So although this qualification entails that we have “only” a partial defenseof utilitarian summation, this should not subtract from the insight that the theo-rem provides.

Another qualification is that the vN0M version of expected utility theory isa limited theory of its kind by not considering subjective probabilities. Allindividuals and the observer use the same probabilities. Epistemic disagree-ments are ignored. This does not strike me as a serious restriction under the


observer-interpretation of “collective preferences.” Suppose the observer needsto compare two lotteries~i.e., actions whose outcomes are known only proba-bilistically!. If she accepts Pareto, then she will do so in a way that involvessummation over the individuals’ utility functions. But this observer is anide-alized observer who can therefore safely be assumed to have better informa-tion instructing her epistemic judgements than the individuals. We only havereason to use the device of such an observer if we also accept this idealizingassumption as a part of her description. There seems to be no problem here,and it does not seem to be very surprising that no analogue of Harsanyi’stheorem is forthcoming if disagreements about probability are considered~seeMongin ~1995!!.

In conclusion, then, in spite of these qualifications, Harsanyi’s theoremmakes a substantial contribution to utilitarian theory. The surprising nature ofthe result provided by Harsanyi’s theorem should make it plausible why itwould take theoretical commitments at certain points to adopt this as an argu-ment. I hope that, at the very least, this study helps illuminate connectionsbetween certain views in ethics and decision theory.

Notes

1 For helpful discussion or comments, I am grateful to Paul Benacerraf, John Burgess, DickJeffrey, Jim Joyce, John Roemer, and two anonymous referees forNoûs.

2 The formulations ofWelfarismandSummationare taken from Sen~1979!.3 Cf. the following definitions of utilitarianism: Sidgwick~1890!, Book IV, chapter 1, par. 1

says: “By utilitarianism is here meant the ethical theory, that the conduct which, under any givencircumstances, is objectively right, is that which will produce the greatest amount of happinesson the whole; that is, taking into account all whose happiness is affected by the conduct.” A littlelater: “@B#y greatest happiness is meant the greatest possible surplus of pleasure over pain, thepain being conceived as balanced against an equal amount of pleasure, so that the two contrastedamounts annihilate each other for purposes of ethical calculation.” By way of contrast, Brandt~1992! writes: “Utilitarianism is the thesis that the moral predicates of an act—at least its objec-tive rightness or wrongness, and sometimes also its moral praise-worthiness or blameworthiness—are functions in some way, direct or indirect, of consequences for the welfare of sentient creatures,and of nothing else.”~p. 111! On that definition, no summation is included. Smart~1967! writes:“Utilitarianism can most generally be described as a doctrine which states that the rightness orwrongness of actions is determined by the goodness and badness of their consequences. This gen-eral definition can be made more precise in various ways.” For an interpretation of Mill as a util-itarian whorejectsSummation, see Marshall~1982!.

4 This utility version of the maximin principle and the summation principle have a distin-guished status among group decision rules in a social choice framework. According to a theoremby d’Aspremont and Gevers~1977! and Deschamps and Gevers~1978!, these are the two princi-ples that remain after a number of reasonable assumptions have been made.

5 Consider two influential objections. First, recall Williams’ objection in terms ofintegrity.By expecting the agent to sum up utilities, so the argument goes, utilitarianism does not allowher to take seriously her own concerns, special obligations, etc.~see, e.g., Williams’ contributionto Smart0Williams ~1982!!. This objection is especially forceful when group welfare is evaluatedby summation, but much weaker when it is assessed in terms of the utility maximin principle.For an agent may complain that his projects are not adequately acknowledged in a choice that

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comes about through summation without thereby acting in too self-centered a way. But if groupwelfare is assessed in terms of the utility maximin principle, the same complainer can plausiblybe charged with selfishness. After all, that principle aids those who have least. Second, recallNozick’s ~1974! claim that utilitarianism gains its plausibility from the idea that a group is a “bigperson”, and that this idea warrants the summation over individual utilities. But since this ideano longer appeals to us, so Nozick argues, summation loses its plausibility, and so does utilitarian-ism. A defense is to argue that summation is plausible without conceiving of the group as a largeperson. This again underlines the importance of an argument for Summation.

6 Harsanyi himself tends to think of collective preference in terms of an observer, cf. e.g.,Harsanyi~1977!. He justifies the imposition of the vN0M assumptions on collective preferencesby arguing that, when groups are concerned, at least as high standards should apply as when onlythe individual is affected~cf. Harsanyi~1982! and Harsanyi~1975!!. There is a literature explor-ing what happens if the collective preferences fail to satisfy all vN0M axioms, cf. Epstein andSegal~1992!. Authors who reject the vN0M assumptions “for groups” tend to argue that groupsare bound to make unfair decisions if they decide in an outcome-oriented way~see Sen~1976!,~1977!, and Roemer~1996!!. However, such reasoning tends to rest on an insufficient conceptionof outcomes; see the discussion of Diamond~1967! under “Independence” in section 4. Relevantfor the discussion of Pareto are Parfit~1984!, part 4, Temkin~1993!, chapter 9, and Gibbard~1987!;the Pareto condition becomes more problematic in the context of Bayesian aggregation, whereutilities and probabilities need to be aggregated~see Seidenfeld et al.~1989!, Mongin ~1995!, Hildet al. ~1998!, and Levi~1990!!.

7 Harsanyi’s theorem has come in for a good deal of discussion over the years, both in philos-ophy and in economics. Much of this will be mentioned in passing. I should emphasize the partic-ular importance of Broome~1991a! and Roemer~1996!. ~We will disregard Broome~1987!, sinceits ideas are taken up in Broome~1991a!.! Broome explores many philosophical issues pertainingto expected utility theory in general and to Harsanyi’s theorem in particular. He has made a strongcase for the philosophical relevance of the theorem. Yet he is not specifically concerned with howthis theorem could make a contribution to utilitarian theory. In that regard, this study differs fromhis important work, and it will also disagree with him at many points along the way. On the otherhand, Broome also provides an important argument that we shall enlist for our purposes in sec-tion 5. Roemer~1996! rejects, in a very sophisticated way, any usefulness for utilitarianism ofHarsanyi’s theorem. But although Roemer’s book is a reflection of the state-of-the-art in the bound-ary area common to economic theory and ethics0political philosophy, he largely ignores Broome~1991a!. The argument in this study is a refutation of Roemer’s argument and its relatives. Thisdivergence of views shows that the discussion of Harsanyi’s theorem is far from closed.

8 I present Harsanyi’s theorem following Weymark~1991!. For other recent proofs, cf. Coulhonand Mongin~1989!, Mongin ~1994!, Deschamps and Gevers~1979!, Fishburn~1984!, and Ham-mond~1981!. Cf. Fishburn~1982! for a formal development of expected utility theory.

9 It might not be straightforward why Strong Pareto is also a condition of the preservation ofuniversally existing agreements among the group members. But it is. Suppose a group M fallsinto non-overlapping and non-empty groups O and P. Suppose that the members of O are indiffer-ent between lotteries p and q, but that the members of P have a strict preference for q. Accordingto Strong Pareto, the group should prefer q as well. This is reasonable, because the members ofO care about q and p equally much, so no harm is done to them by letting the members of Phave their way. The agreement among all thosewho have a preferenceis preserved without disre-garding anybody else. This condition is equivalent to the affine independence of the functions,cf. Coulhon and Mongin~1989!. Roughly speaking, this means that none of those functions canbe constructed from the others.

10 Cf. Rubinstein~1984! and Roberts~1980! for the distinction between the two formats andMongin ~1994! for its relevance in the context of the utilitarian theorem. A multi-profile modelfor Harsanyi’s theorem is available, but it is only the outcome of recent research. Cf. Coulhonand Mongin~1989! and especially Mongin~1994!. This model, however, is more complex than


the present one and also comes with problems of its own~e.g., it uses an assumption of indepen-dence of irrelevant alternatives!. The discussion about the single profile versus the multi profileapproach is not an issue for Harsanyi; in Harsanyi~1979! he claims that his theorem could beapplied to just any n-tuple of individual utility functions. Harsanyi’s theorem is only one amonga number of theorems deriving a utilitarian group choice function from assumptions about individ-uals in certain formal settings. Cf. Mongin and d’Aspremont~1998!, and Roemer~1996!, chapter 4.

11 Cf. Shaw ~1999! for an introduction to utilitarianism and to those accounts in particular;on well-being cf., e.g., Griffin~1986!, Sumner~1996!, Kagan~1998!, pp. 29–41, and the appen-dix in Parfit ~1984!. For a discussion of the notion of utility within utilitarian theory, see Haslett~1990!.

12 Shaw suggests this is the most commonly held theory of well-being. Sumner~1996!, p. 122~who rejects it!! says: “Versions of the desire theory now define the orthodox view of the natureof welfare, at least in the Anglo-American world.” An exemplary development of such a view isto be found in Griffin ~1986!. However, since his account comes very close to an objective-listaccount, he has recently pointed out that he may be mis-characterized as an advocate of a desire-satisfaction account~see Griffin~2000!!.

13 The point that vN0M theory and utilitarianism use notions of utility that are not in anyevident way connected has been made before, cf. Broome~1991b! and Ellsberg~1954!; but asBroome points out, this ambiguity continues to be a source of confusion and misunderstandings.Savage~1954!, p. 98 also points out that vN0M have a new notion of utility, and that there isconfusion only because they use the old word.

14 For more discussion on this, cf. Joyce~1999!, pp. 19–23.15 For an elaboration of this point, see Sumner~1996!, chapter 5.1. For thoughts on the inter-

pretation of preferring, see Gibbard~1998!.16 Broome~1991a! also talks about a betterness-relation, but means a relation that is entailed

by a person’sgoodness. For a utilitarian, of course, those two notions coincide.17 To see this, cf. in particular the discussion of Continuity. It has been argued in particular

by Sen~1976!, ~1977!, ~1986!, and Weymark~1991! that if there is any connection between theutilitarian notion of utility and the decision-theoretic sense of utility to begin with, it will be implau-sible that the notion of well-being is constrained by the axioms of expected utility theory. I arguethat this view is wrong on the IDS-account of well-being.

18 A theory of individuating outcomes is crucial to expected utility theory; cf. Broome~1991a!,Broome ~1993!, and Joyce~1999!; see also Sosa~1993!. Both Broome and Joyce employ theabove strategy when discussing Diamond. However, Roemer~1996! discusses Diamond withoutconsidering this approach and is led to conclude that Diamond’s example is a knock-down argu-ment against any usefulness of Harsanyi’s theorem in ethics~p. 140!. Harsanyi~1975! does notemploy this strategy, but bites the bullet, presenting cases in which it allegedly does not matterwhether some benefit or burden is distributed with or without a lottery. Economists in general donot seem to like this strategy; cf. Mas-Collel et al.~1995!, which is a major text-book on micro-economic theory and which does not even mention this strategy as a possible response to theAllais-paradox~see p. 180!, to which both Broome and Joyce also apply this strategy. Joycerefers to a lack of appreciation of the point made above as the most common mistake regardingdecision theory. The literature on utilitarianism also acknowledges that consequentialism needs anotion of outcome that is broader than pre-theoretical intuitions have it; e.g., Shaw~1999!, p. 13–14, emphasizes that outcomes do not have to come after an act and do not have to be caused byit. One may be concerned that, if we apply the strategy of loading up the consequences toDiamond, we already need to assume a notion of fairness prior to our conquentialist theory. Butthat is not a problem. For consequentialism does not aim at reducing all moral vocabulary to talkabout outcomes. Worries about consequentialism raised from the point of view of, say, fairnessdo not concern the possibility ofdefining these notions consequentialistically. Rather, they areworries about how to make room for them within a consequentialist framework, no matter howwe define them.

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19 See Anand~1987! for very strong criticism; Broome~1991a! ignores this axiom, suggest-ing that it is implausible and submitting that his book is concerned with different problems. Curi-ously, even though Broome needs to defend the view that the betterness-relation entailed by aperson’s goodness satisfies the axioms of expected utility theory, he presents a substantial defenseonly of Independence. He rejects Completeness, thinks of Continuity as merely a technical assump-tion, and of Transitivity as true as a matter of logic. For general issues about comparability andcommensurability, see Chang~1997!.

20 See Moore~1903!, p. 28. This principle has recently come in for a good deal of discus-sion, cf. Kagan~1988!, Hurka ~1998!, Lemos~1998!, and references therein.

21 There is a tendency in the literature to interpret Completeness as a requirement of coherentextendibility: that is, although the preference relation need not completely order the lotteries, theremust be an extension of this relation that satisfies the vN0M axioms and does so order the lotter-ies. See Joyce~1999!, p. 45, and references on that page. However, important objections to Com-pleteness still apply to this understanding of the axiom.

22 Cf. Elster~1979!, p. 27, and Rachels~1998! for a discussion of thebetter-thancase.23 Scenarios such as improving oneself to death suggest that our strong intuitions for the tran-

sitivity of the relationspreferred toor better thanlive on their apparent proximity to relationssuch aslarger than or taller than. The most comprehensive discussion of transitivity that I amaware of is in Maher~1993!. His strategy for arguing for transitivity is as follows: Transitivityhas a high prima facie plausibility: most people would endorse it and would be willing to correctviolations when those are pointed out. There are arguments for and against transitivity, but theyare both wanting. So that leaves us with the prima facie plausibility. Maher thinks that this is asgood a case as we can have for substantive normative principles. In the end he takes a pragmaticline—let us see what kind of theories we can build on the respective principles and judge themfrom there. Although Broome~1991a! discusses a betterness relation entailed by a person’s good-ness rather than her well-being, he would probably want to press the same point about logic men-tioned above with regard to well-being as well.

24 Without the continuity axiom, we still have the expected utility form, but the functions arenot scalar-valued, but vector-valued, where the vectors have lexicographically ordered compo-nents, cf. Hausner~1954!. In light of this result Harsanyi~1982! thinks that we do not need toconsider Continuity a rationality axiom. Hajek~1998! agrees. Broome~1991a! thinks of continu-ity as a merely technical assumption.

25 This is easy to see: Suppose I assign infinite value to outcome O1 and 0 to Or, and that Irank O2 between O1 and Or. By Continuity, there is a number 0# p # 1 such that the utility ofOr equals the sum of p times the utility of O1 and ~12p! times the utility of Or. Yet such a num-ber p does not exist, and thus Continuity fails.

26 Continuity is also contradicted by so-called lexicographic preferences, where one factor isconsidered unconditionally more important than another~i.e., no amount of the latter can makeup for any loss of the former!. However, such scenarios would be implausible again under theorganic understanding of value.

27 This is a long section, and a preview might be welcome. In 5.2, I use the analogy to themeasurement of heat to explain how a family of functions can be taken to measure something.Von Neumann and Morgenstern~1947!, who introduce this analogy, think that we can learn morefrom the measurement of heat for the measurement of utility than just what it is for a family offunctions to measure something. In 5.3, I discuss their view, but suggest that the prospects aredim for them to convince us that the family of expectational representations does indeed measurewell-being. Next I enlist an argument from Broome~1991a!, which I think shows that this familydoes indeed measure well-being. In 5.4, I discuss some worries about the argument of thissection.

28 They use the expressionis preferred tojust as the phrasesis warmer thanand is to the leftof are used in their respective domains. Just as there is nothing in any physical bodyin additionto heat that corresponds to a noun formed to the predicateis warmer than, so there is nothing in


an agent in addition to utility that corresponds to the nounpreference. This usage of the word“preference” deserves some attention.

29 Strictly speaking, this is not quite correct. We must distinguish between two functions: Onthe one hand, expectational representations are defined, like other representations, on the domainof a preference relation~i.e., on lotteries!. Suppose U is such a function, and suppose the out-comes are~O1, ...., On!. U induces a function u defined on the outcomes such that u~Oi ! equalsthe value of U applied to the lottery that delivers Oi for sure. The class of functions that the rep-resentation theorem determines as being closed under positive affine transformations is the classof those functions defined on the outcomes, and it is this family of functions that corresponds tothe family of functions that are positive affine transformations of the Celsius scale in the case ofheat. However, since in vN0M theory, for each such function U there is a function u, and viceversa, we can talk loosely here, and in particular we can talk about the class of expectationalrepresentations, as measuring utility. Note an important difference between the heat case and theutility case: temperature scales are linear functions, but that need not be the case for utility.

30 This example is developed in the Savage~1954! version of expected utility theory. That is,acts are functions that assign a consequence to each state of the world. States of the world carryprobabilities, while consequences carry utilities. Acts are ranked in terms of their expectations.The vN0M model assigns both probabilities and utilities to outcomes and thus cannot capture theidea that one state of the world leads to different consequences, depending on what action waschosen. The same example can be reproduced in vN0M notation as well: outcomes would thenhave to be described in terms of their monetary rewardand in terms of the action chosen, withthe probabilities suitably adjusted. But since this leads to clumsy notation, I present Broome’sexample in the Savage-style that he himself chose.

31 This is so simply because any function that shares with all the functions in that family whatthey have in common~i.e., ratios of utility differences! is a function in that family~i.e., is itself apositive affine transformation of all the functions in that family!.

32 To put the point differently: the vN0M axioms guarantee that preferences have a suffi-ciently rich structure to allow for expectational representations. The present argument shows thatpreferences in terms of the agent’s betterness-relation can have such a rich structure only if well-being itself has a sufficiently rich structure that can be captured by a family of functions closedunder positive affine transformations.

33 According to the account in Luce et al.~1971!, this proposal counts as a measurement butLuce et al. take vV0M theory as aparadigmof what a measurement is. This, of course, just movesthe concerns one level higher up. Luce et al.~1971! present what they call a representationalapproach to measurement, which hinges on the construction of a homomorphism between whatthey call a relational structure and the real numbers, supplied with certain relations~p. 8ff !. Ifthere is such a homomorphism, there is measurement. In that sense, von Neumann and Morgen-stern do measure a property. For a critical discussion of Luce et al.~1971!, see Berka~1983!, inparticular p. 64, p. 114, p. 153, and p. 174. According to Berka, we should not speak of measure-ments here. Note that there is a third worry that we can properly address only in section 6, butthat should be mentioned here. Recall from section 2 that Harsanyi’s theorem only applies to oneprofile of representations at a time. So for any observer representation and any profile of individ-ual representations there exist coefficients such that the observer representation is a weighted sumover the individual representations using those coefficients. But if we now use another observerrepresentation or other individual representations, the coefficients would be different. In section 6we develop the tools to respond to this worry.

34 For a historical account of expected utility theory, see chapter 1 in Joyce~1999!, in particu-lar on the St. Petersburg paradox. For space reasons alone, I have not discussed views that con-struct a theory around their disagreement with the claim that well-being is expectational~cf.Machina~1990! for a brief, and by now somewhat dated, overview!. But it seems fair to say thatmuch of the motivation for these alternatives will disappear~a! if it is acknowledged that out-comes need to be described completely for any consequentialist theory to work, and~b! once the

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vN0M axioms are understood as constraints on the betterness-relation entailed by well-being ratherthan as characterizations of rational behavior without being embedded into such a theory.

35 A similar criticism has also been pressed by Sen~1976!, ~1977!, ~1986!; see also Weymark~1991!.

36 Roemer~1996!, chapter 1, presents a discussion of notions of utility comparability. In par-ticular, he shows that the notion of comparability discussed in the text is the weakest such notionthat renders utilitarianism coherent~i.e., preserves the observer’s ranking!; cf. also Sen~1970!,Weymark ~1991!. Two remarks are in order: First, Broome~1991!, p. 219–220 argues that theassumption ofcompletenessfor the group or the observer implies the possibility of interpersonalcomparisons. For if an observer can compareany two outcomes, he can in particular compareany two outcomes X and Y that only differ in terms of how agents i and j fare. This argument isunconvincing. For nothing about group preferencesrequiresoutcomes to be distinguished in termsof how good they are for specific members of the group. All thatis required about the connec-tion of the group preferences with the individual preferences is a Pareto condition. To requiremore deprives the theorem of its generality and strength. The second point concerns Jeffrey’s~1974!treatment of Harsanyi’s theorem. Jeffrey writes~p. 115!: “I take it that we can sometimes deter-mine not only the preferences of the individuals, but also the social preferences which they havearrived at,and can also determine the fact that they regard those social preferences as represent-ing an even-handed compromise between their conflicting personal preferences. In such a case,the ~...! results allow us to find commensurate unit intervals for the personal utility scales whichare involved, i.e., we can perform interpersonal comparisons of preferences.” The idea is that theagents start with a group representation about which they claim that it expresses an even-handedcompromise, where each individual’s function is counted with coefficient 1. Harsanyi’s theoremthen delivers n individual functions~uniquely determined!. Since the agents agree that the groupfunction expresses a compromise in which they are allequally considered, they can use thesefunctions for interpersonal comparisons. This approach to Harsanyi’s theorem is consistent with~but different from! my project. ~Broome ~1987!, p. 417 in a footnote laconically remarks thatJeffrey is wrong.!

37 In Mongin’s ~1994! multi-profile version, the coefficients are constant across profiles.

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