1 For helpful discussion or comments, I am grateful to Paul Benacerraf, John Burgess,

Dick Jeffrey, Jim Joyce, John Roemer, and two anonymous referees for Nous.

1

Harsanyi’s ‘Utilitarian Theorem’ and Utilitarianism

Mathias Risse

Department of Philosophy, Yale University

forthcoming in “Nous”

1. Introduction

1.1 In 1955, John Harsanyi proved a remarkable theorem:1 Suppose n agents satisfy the assumptions

of von Neumann/Morgenstern (1947) expected utility theory, and so does the group as a whole (or

an observer). Suppose that, if each member of the group prefers option a to b, then so does the

group, or the observer (Pareto condition). Then the group’s utility function is a weighted sum of the

individual utility functions. Despite Harsanyi’s insistence that what he calls the Utilitarian Theorem

embeds utilitarianism into a theory of rationality, the theorem has fallen short of having the kind

of impact on the discussion of utilitarianism for which Harsanyi hoped. Yet how could the theorem

influence this discussion? Utilitarianism is as attractive to some as it is appalling to others. The

prospects for this dispute to be affected by a theorem seem dim. Yet a closer look shows how the

theorem could make a contribution. To fix ideas, I understand by utilitarianism the following claims:

(1) Consequentialism: Actions are evaluated in terms of their consequences only.

(2) Bayesianism: An agent's beliefs about possible outcomes are captured probabilistically.

(3) Welfarism: The judgement of the relative goodness of states of affairs is based

2 The formulations of Welfarism and Summation are taken from Sen (1979).

2

exclusively on, and an increasing function of, the individual utilities in these states.

(4) Summation: One collection of individual utilities is at least as good as another if and only

if it has at least as large a sum total.2

Bayesianism is normally not considered part of the definition of utilitarianism. However, for

utilitarianism to be an action-guiding theory, rather than a theory of right-making characteristics,

it must accommodate uncertainty. To account for the peculiar status of this condition, I refer to this

doctrine as Bayesian Utilitarianism. The Bayesian-utilitarian agent assesses the probability of all

possible outcomes, considers the utility of all relevant agents, forms the sum over the utilities for

each outcome, discounts each outcome with its probability and chooses an action with a maximal

probability-weighted sum over sums of utilities. This picture requires elaboration, but it allows us

to localize a conceptual place for Harsanyi’s theorem within utilitarianism. For now we see that the

latter, if it makes any contribution at all, makes it as an argument for Summation once the other

claims of utilitarianism have been granted.

Summation has always been central to utilitarianism. It is mostly taken for granted, rather

than defended. In his Introduction to the Principles of Morals and Legislation, Bentham writes:

“The community is a fictitious body, composed of the individual persons who are considered as

constituting as it were its members. The interest of the community is then, what? - the sum of the

interests of the several members who compose it” (chapter 1, iv). Mill’s Utilitarianism implicitly

assumes summation for the assessment of group welfare (cf. in particular the end of chapter iv). One

way of conceiving of utilitarianism even is as a theory that makes as much sense as possible of the

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

given circumstances, is objectively right, is that which will produce the greatest amount of

happiness on the whole; that is, taking into account all whose happiness is affected by the

conduct.” A little later: “[B]y greatest happiness is meant the greatest possible surplus of

pleasure over pain, the pain being conceived as balanced against an equal amount of pleasure, so

that the two contrasted amounts 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 objective 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 or wrongness of actions is determined by the goodness

and badness of their consequences. This general definition can be made more precise in various

ways.” For an interpretation of Mill as a utilitarian who rejects Summation, see Marshall (1982).

4 This utility version of the maximin principle and the summation principle have a

3

idea of the greatest amount of happiness (assessed by summation) for the greatest number of people.3

In such approaches, no demand for arguments for summation arises. Still, a philosophically

satisfactory utilitarianism must distinguish between the different claims of which that doctrine is

composed and explore the entailments among them. Clearly, Consequentialism, Bayesianism, and

Welfarism do not obviously imply any specific view about how to assess the group welfare. For

instance, a utility version of the Rawlsian maximin principle is available, too.4 Also, once these

distinguished status among group decision rules in a social choice framework. According to a

theorem by d'Aspremont and Gevers (1977) and Deschamps and Gevers (1978), these are the

two principles that remain after a number of reasonable assumptions have been made.

5 Consider two influential objections. First, recall Williams’ objection in terms of

integrity. By expecting the agent to sum up utilities, so the argument goes, utilitarianism does not

allow her to take seriously her own concerns, special obligations, etc. (see, e.g., Williams’

contribution to Smart/Williams (1982)). This objection is especially forceful when group

welfare is evaluated by 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 comes about through summation without thereby acting in too

self-centered a way. But if group welfare is assessed in terms of the utility maximin principle,

the same complainer can plausibly be charged with selfishness. After all, that principle aids those

who have least. Second, recall Nozick’s (1974) claim that utilitarianism gains its plausibility

from the idea that a group is a “big person”, and that this idea warrants the summation over

individual utilities. But since this idea no longer appeals to us, so Nozick argues, summation

loses its plausibility, and so does utilitarianism. A defense is to argue that summation is plausible

without conceiving of the group as a large person. This again underlines the importance of an

argument for Summation.

4

different claims are separated, it becomes clear that important criticisms of utilitarianism address

Summation, rather than other claims.5 Thus arguments for Summation are called for.

1.2 I submit that Harsanyi’s theorem does provide an argument for an advocate of Consequentialism,

5

Bayesianism, and Welfarism to endorse Summation. My argument for this claim will come with

qualifications. For our discussion will touch on major debates in moral theory and decision theory

too complex to be settled here. Moreover, no argument for Summation on the basis of the other three

claims is possible without theoretical commitments at various points. Still, I hope to show that in

this case, a theorem does indeed make a very substantial contribution to moral theory. This view is

bound to be controversial: many moral philosophers dislike the idea that formal results matter to

their discipline, while economic theorists writing on expected utility theory also reject the claim that

this theorem contributes to utilitarianism (in particular Sen (1976), (1977), (1986), Roemer (1996),

Weymark (1991)). I hope to convince the reader that such views are misguided.

Section 2 presents the theorem, which requires merely elementary formal notation. The main

argument of this study is in sections 3-5. The challenge is to show just what the connection is

between utilitarianism and von Neumann/Morgenstern (vN/M) expected utility theory. In particular,

we need to show that the utilitarian notion of utility is connected to the notion of utility used in

vN/M theory in an illuminating way. In section 6, I address what I take to be a version of the most

prominent argument against the usefulness of Harsanyi’s framework for utilitarianism. In section

7, we take stock of the qualifications made along the way and assess the argument we have

developed. I do not systematically discuss the assumptions of the theorem, that is, the claim that

collective preferences satisfy the vN/M axioms, and the Pareto condition. To bypass vexing doubts

about “group metaphysics,” I regard “collective preferences” as preferences of an observer

concerned about the well-being of the relevant group at least to the extent captured by the Pareto

principle. That principle itself is an intuitively immensely plausible idea. It expresses the idea that

the observer’s preferences should preserve universal agreements among the individual group

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 vN/M assumptions on collective preferences

by arguing that, when groups are concerned, at least as high standards should apply as when

only the individual is affected (cf. Harsanyi (1982) and Harsanyi (1975)). There is a literature

exploring what happens if the collective preferences fail to satisfy all vN/M axioms, cf. Epstein

and Segal (1992). Authors who reject the vN/M assumptions “for groups” tend to argue that

groups are 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

conception of outcomes; see the discussion of Diamond (1967) under “Independence” in section

4. Relevant for 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, where utilities and probabilities need to be aggregated (see Seidenfeld et al. (1989),

Mongin (1995), Hild et al. (1998), and Levi (1990)).

7 Harsanyi’s theorem has come in for a good deal of discussion over the years, both in

philosophy and in economics. Much of this will be mentioned in passing. I should emphasize

the particular importance of Broome (1991a) and Roemer (1996). (We will disregard Broome

6

members.6 I also take for granted that we understand for which groups it is reasonable to think of

an observer as being concerned with their well-being in the sense expressed by Pareto (which is, if

the observer is a version of the traditional “impartial observer”, the question of determining the

scope of moral considerations). I exclude such questions because they arise elsewhere as well and

should thus not distract us. The crucial and controversial matter to explore is the relationship

between vN/M theory and utilitarianism.7

(1987), since its ideas are taken up in Broome (1991a).) Broome explores many philosophical

issues pertaining to expected utility theory in general and to Harsanyi’s theorem in particular. He

has made a strong case for the philosophical relevance of the theorem. Yet he is not specifically

concerned with how this theorem could make a contribution to utilitarian theory. In that regard,

this study differs from his important work, and it will also disagree with him at many points

along the way. On the other hand, Broome also provides an important argument that we shall

enlist for our purposes in section 5. Roemer (1996) rejects, in a very sophisticated way, any

usefulness for utilitarianism of Harsanyi’s theorem. But although Roemer’s book is a reflection

of the state-of-the-art in the boundary area common to economic theory and ethics/political

philosophy, he largely ignores Broome (1991a). The argument in this study is a refutation of

Roemer’s argument and its relatives. This divergence 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.

Coulhon and Mongin (1989), Mongin (1994), Deschamps and Gevers (1979), Fishburn (1984),

and Hammond (1981). Cf. Fishburn (1982) for a formal development of expected utility theory.

7

2. Harsanyi’s Utilitarian Theorem

2.1. Let me begin by introducing the vN/M preference theory and its representation theorem.8 Let

M = (O1, ..., Om) be a set of outcomes, and L the set of probability distributions (“lotteries”) over

M. Let Pi, Ii, and Ri be the strict preference, indifference, and weak (i.e., “preferred-or-indifferent-

to”) preference relation for individual i (1#i#n), defined on L, and let P, I, and R be the

corresponding relations for the observer. So preferences in this theory are preferences over lotteries.

8

One interpretation of these lotteries is in terms of actions whose outcomes are known only

probabilistically. Elements of L can be written as (p1O1;...; pnOm), where pi is the probability of Oi.

A real-valued function u represents a relation S (or is a utility representation of S) if and only if for

any two elements p and q in the domain of S, pSq if and only if u(p)#u(q). It is only through this

notion of preference representation that the concept of utility occurs in this theory. A profile of

utility functions (u1, ..., un) is a set of n utility functions, one for each agent. A representation u is

called expectational if and only if u(p1O1; ....; pnOn) = p1u(O1)+...+pnu(On), where u(Oi) is the utility

value of the lottery that assigns probability 1 to outcome Oi and 0 to all other outcomes. The vN/M

representation theorem then says the following: If an agent’s preferences over lotteries satisfy

certain conditions, then (a) there exists an expectational utility representation of these preferences;

and (b) for any two such representations u and v, there exist a positive real number a and a real

number b such that u = av + b. That is, the expectational representation is “unique up to positive

affine transformations.” The assumptions on the preference relation differ among axiomatizations,

but they all have the same basic structure and include versions of the 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 any number a between 0

and 1, [ap + (1-a)r]Ri[aq + (1-a)r]. (“If q is preferred-or-indifferent to p, then any lottery that

involves q with probability a and some lottery r with probability (1-a) is preferred-or-

indifferent to a lottery that involves p with probability a and r with probability (1-a).”)

Continuity: For any three lotteries p, q, r if pPiq and qPir, then there exists a number a

between 0 and 1 such that q Ii [(ap + (1-a)r). (“If p is strictly preferred to q and q is strictly

9 It might not be straightforward why Strong Pareto is also a condition of the preservation

of universally existing agreements among the group members. But it is. Suppose a group M falls

into non-overlapping and non-empty groups O and P. Suppose that the members of O are

indifferent between lotteries p and q, but that the members of P have a strict preference for q.

According to Strong Pareto, the group should prefer q as well. This is reasonable, because the

members of O care about q and p equally much, so no harm is done to them by letting the

9

preferred to r, then there is a probability a such that q is indifferent between a gamble that

involves obtaining p with probability a and obtaining r with probability (1-a).”)

We discuss these axioms in section 4. For Harsanyi’s theorem, suppose we have n agents. We need

the following conditions for Harsanyi’s theorem:

Pareto Indifference: For all p, q 0 L, if pIiq for all i, 1#i#n, then pIq.

Semi-Strong Pareto: For all p, q 0 L, if pRiq for all i, then pRq.

Strong Pareto: For all p, q 0L, if pRiq for all i, then pRq, and if, furthermore, there exists an

i such that pPiq, then pPq.

Independent Prospects: For each i = 1, ...n, there exist pi and qi0L such that piIjqi for all i…j

and piPiqi.

The Pareto conditions demand that universally shared agreements about preferences among the

group members be preserved in the observer’s preferences. Independent Prospects requires that for

each agent there be a pair of lotteries between which she has a preference, but between which

everybody else is indifferent.9 Under these conditions, then, Harsanyi’s theorem shows that an

members of P have their way. The agreement among all those who have a preference is

preserved without disregarding 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 can be constructed from the others.

10

expectational representation of the observer’s preferences is a sum over expectational

representations of the preferences of the group members:

Proposition (Harsanyi’s Utilitarian Theorem): Suppose Ri, i=1, ..., n and R satisfy the vN/M

axioms and suppose that Pareto Indifference is satisfied. Let vi be an expectational

representation of Ri, and let v be an expectational representation of R. Then there exist

numbers ai and b such that for all p 0L

v(p) = 3aivi(p)+b

(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 the single-profile format: it treats only of one profile of utility

functions at a time. As opposed to this, Arrow’s (1951) Impossibility Theorem, for example, is in

the multi-profile format, addressing more than one profile of functions at a time. Theorems in the

single-profile format naturally apply only to one profile at a time. Harsanyi’s theorem, for instance,

implies the existence of certain coefficients for a given profile of utility functions, but for a different

profile, we obtain different coefficients. However, a complete formulation of utilitarianism requires

10 Cf. Rubinstein (1984) and Roberts (1980) for the distinction between the two formats

and Mongin (1994) for its relevance in the context of the utilitarian theorem. A multi-profile

model for Harsanyi’s theorem is available, but it is only the outcome of recent research. Cf.

Coulhon and 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 independence of irrelevant alternatives). The discussion about the single profile

versus the multi profile approach is not an issue for Harsanyi; in Harsanyi (1979) he claims that

his theorem could be applied to just any n-tuple of individual utility functions. Harsanyi’s

theorem is only one among a number of theorems deriving a utilitarian group choice function

11

the multi-profile format. For it is utilitarian doctrine that each person count equally. An explicit

formulation of this claim would stipulate that the aggregation be indifferent between two

distributions that only differ in terms of the distribution of the overall utility across persons. But

such a condition must compare and thus refer to several profiles at once and cannot be captured in

the single-profile format. Therefore, as discussed here, Harsanyi’s theorem cannot make a complete

case for utilitarian summation. So we should think of Harsanyi's theorem as providing an argument

for the summation method as such, while not implying anything about the weights given to the

individuals. An argument for equality must then come from elsewhere. Thus we are interested in

Harsanyi's theorem as an argument for the following condition, which does not imply anything about

the coefficients:

Summation': There is a set of weights (or coefficients) such that, for any two profiles of

utility functions, one profile is at least as good as the other if and only if it has at least as

large a weighted sum of individual utilities, weighted according to the given coefficients.10

from assumptions about individuals 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 appendix in Parfit (1984). For a discussion of the notion of utility within utilitarian theory,

see Haslett (1990).

12

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 as pleasure or satisfaction. Desire-satisfaction

theories account for well-being in terms of the realization of desires. Whereas mental-state accounts

ignore whether there is any ‘fit’ between mental states and states of the world, desire-satisfaction

accounts understand well-being precisely in terms of the extent of such a fit. Finally, objective-list

theories explicate well-being in terms of a list of properties that constitute a person’s well-being

regardless of both mental states and desires. Curiously, many utilitarians do not have much use for

the concept of “utility” when discussing well-being. Nevertheless, since these theories are the

current accounts of well-being, they are the current candidates for explicating the concept of "utility"

contained in the very notion of “utilitarianism”.11

However, listing these accounts fails to capture the complexities of developing a satisfactory

notion of well-being. That notion, no matter how we explicate it, must fill various roles in moral and

prudential deliberation and in our conception of and interaction with others, and those roles place

demands on any 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 account of the

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

nature of welfare, at least in the Anglo-American world.” An exemplary development of such a

view is to be found in Griffin (1986). However, since his account comes very close to an

13

domain of prudential value that it tries to cover; second, it must be what we want to use, for

purposes of moral judgement, as the basis for comparison between different persons; and

third, it has to lend itself to the sorts of measurement that moral deliberation needs (p 108).

In this study I adopt a desire-satisfaction account of well-being and assume that such an account can

be developed in a way that satisfies Griffin’s desiderata and possibly others. More specifically, I

adopt a version of this account that has been amended in at least two ways in response to common

objections. On the one hand, this account will be able to distinguish between desires whose

satisfaction contributes to a person’s well-being and those whose satisfaction fails to do so. (One

may desire that there be life in some remote solar system, but its existence would not contribute to

one’s well-being.) On the other hand, this account will focus not on actual desires (which may too

easily conflict with a person’s well-being), but desires that a person would have were she properly

informed, thinking clearly, without any prejudices and biases, etc. Such an account is an idealized-

desire-satisfaction (IDS-) account of well-being. I have little to say to develop this conception in

detail, but restrict myself to investigating how it can be connected to vN/M 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 for Harsanyi’s theorem. Given the prominence

of IDS-accounts, it seems reasonable, then, to adopt this conception for the sake of this discussion.12

objective-list account, 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 vN/M theory and utilitarianism use notions of utility that are not in any

evident way connected has been made before, cf. Broome (1991b) and Ellsberg (1954); but as

Broome points out, this ambiguity continues to be a source of confusion and misunderstandings.

Savage (1954), p 98 also points out that vN/M have a new notion of utility, and that there is

confusion only because they use the old word.

14

3.2 But how does the vN/M notion of utility bear on this notion of well-being? In vN/M, “utility”

refers to numbers representing preferences. Those values are, as Hampton (1994) put it, “just

numbers.” Apparently, vN/M theory only shares a word with utilitarianism, and one that is not even

used by many utilitarians. There are two ways of developing this claim into an objection to the

usefulness of the vN/M framework and thus of Harsanyi’s theorem for utilitarianism. I develop both

and argue that they fail. Nevertheless, they leave us with a challenge that needs to be met for vN/M

theory to have any bearing on utilitarianism, and providing a response to it is our main concern in

the remainder of this essay. Once it is met, it is straightforward to see the place of Harsanyi’s

theorem within utilitarian theory.13

The first objection states that vN/M theory and utilitarianism are divorced from each other

because the former has nothing to do with any of the current notions of well-being. While the vN/M

representation theorem shows that individuals’ preferences can be numerically represented in a

convenient form, Harsanyi’s theorem makes a statement about the representation of an observer’s

15

preferences. Nothing of substance follows about utilitarianism. Yet this objection is misguided. The

vN/M theorem is an analytical result that assumes a preference relation and derives numbers that

represent preferences. However, from the fact that utilities are “just numbers” in the vN/M model,

it does not follow that no conceptual connection exists between the vN/M theory and utilitarianism.

Claiming that it does is to confuse a logical relationship established within a specific model with

methodological, epistemological, or possibly ontological insights that such a model by itself cannot

provide. A connection between vN/M theory and utilitarianism could be demonstrated, for example,

by showing that a notion of utility as well-being explicates the meaning of preferring. A demand for

such an account arises as follows. The vN/M theory uses a preference relation as a syntactically

primitive symbol, which is explicated within in the model only through the assumptions made about

it. Formal results are then derived from these assumptions. But in addition to proving results within

this model, we also need to interpret them. In particular we need to ask about the meaning of

“preferring,” just as we need to ask about the meaning of the material implication or the meaning

of probability. It is through its possible semantic function vis-a-vis the preference relation that the

notion of utility as well-being might be connected to the notion of utility as preference

representation. The challenge is to provide an interpretation of preferring that is coherent with the

assumptions on the preference relation in the model.

3.3 Another objection arises now, which, if successful, would meet this challenge in a way that

undermines any attempt to find a useful conceptual connection between vN/M preference theory and

utilitarianism. Sure enough, one may say, the representation theorem fails to show that, only because

numerical utilities are derived from preferences in the vN/M model, there can be no connection of

14 For more discussion on this, cf. Joyce (1999), pp 19-23.

16

vN/M preference theory to utilitarianism. And sure enough, we need an interpretation of preferring.

However, the objector insists, preferring should be understood within the confines of a behavioristic

account of psychology. Such an account analyzes “desire” and “belief” (and other “mental”

vocabulary) in terms of observables. Preferences are observable in choices. On this picture, it only

makes sense to speak of well-being to the extent that it can be observed in choice, that is, only in

terms of preferences. No question about the meaning of preferring arises that cannot be answered

in terms of observables. So if there is room for “utility,” it must be derivative of preferences. No

questions about connections between two notions of utility emerge, since only 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 provided a congenial environment for

behaviorism. Attempts to derive “utility” and “probability” from “preferences” were motivated

precisely by such a psychology. Yet we have abandoned this picture largely because its costs are too

high: it forbids us from saying too much that we want and need to say, not just in practice, but also

in theory. So being committed to behaviorism is being committed in more daring ways than being

committed to a notion of individual well-being that accounts for the meaning of preferring in terms

other than observables.14

3.4 The second objection fails, and thus we must answer the challenge posed by the first in another

way. In general, once we drop the behavioristic account of psychology, the question of what

15 For an elaboration of this point, see Sumner (1996), chapter 5.1. For thoughts on the

interpretation of preferring, see Gibbard (1998).

16 Broome (1991a) also talks about a betterness-relation, but means a relation that is

entailed by a person’s goodness For a utilitarian, of course, those two notions coincide.

17

“preferring” means becomes urgent. For preferring, if not understood as the behaviorists would have

it, sits uneasily between choosing and desiring.15 Is there, then, an interpretation of preferring that

makes for a connection to utilitarianism and is coherent with the vN/M assumptions on the

preference relation? I submit that there is. To prepare the argument, note that, straightforwardly, the

concept of a person’s well-being entails a betterness-relation: outcome O1 is better than O2 if and

only if O1 is more conducive to her well-being than O2. This relation extends to lotteries. For when

an agent acts under circumstances of risk and can predict outcomes only probabilistically, she must

evaluate such probabilistic prospects from the point of view of her well-being. It might be clumsy

to speak of amounts of well-being pertaining to risky prospects, but the idea is clear and familiar

enough. If this betterness-relation satisfies the vN/M axioms, then there is a conceptual connection

between utilitarianism and vN/M preference theory. For then the vN/M representation theorem

shows that this relation can be represented by expectational utility functions. This result would be

an important milestone on our way to explore what contribution Harsanyi’s theorem makes to

utilitarianism. Our next task is to show that this betterness-relation does satisfy those axioms.16

4. Expectational Representations of Well-Being

4.1 There has been a great deal of controversy about the vN/M axioms. In light of the currency of

Humean views on human psychology, the extent of this controversy is unsurprising. According to

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 the utilitarian notion of utility and the decision-theoretic sense of utility to begin with, it

will be implausible that the notion of well-being is constrained by the axioms of expected utility

18

Humeans, there are two main kinds of psychological states, namely, beliefs and desires. Desires are

unlike beliefs 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 rectification of factual errors on which they

might be based. From this point of view, then, any constraints on preferences must seem dubious.

The IDS-account of well-being adopted in this study is at odds with the Humean view. For the

desires whose satisfaction is taken to be constitutive of the agent’s well-being emerge through

rational scrutiny and reflection. I submit that any development of the IDS-account should regard the

vN/M axioms as reasonable constraints on the betterness-relation.

At this stage, then, it matters that we have adopted the IDS-account, rather than any other

account of well-being. For that approach endorses rational constraints on desires. The arguments in

this section will not convince anybody with reservations about rational constraints on desires.

Rather, the arguments are, axiom by axiom, directed at somebody who endorses the IDS-account

of well-being in principle and is considering reasonable constraints on a detailed development of

that account. This approach to the vN/M axioms puts us in a better position to argue for them than

attempts to argue for them as general constraints on rational behavior (on which, of course, a lot of

ink has been spilled). For it is possible without too much stretching to construct scenarios in which

the one or the other axioms is violated while it is nevertheless unclear just why such behavior would

be straightforwardly “irrational.”17 This is in particular so on a Humean view, which supports only

theory. I argue that this view is wrong on the IDS-account of well-being.

19

a thin notion of rationality. By arguing for the axioms as reasonable constraints on the IDS-account

of well-being, we are getting some mileage out of the starting position. Although the axioms are

defined for lotteries, for the sake of simplicity I discuss them as if they were defined for outcomes

(lotteries that assign probability 1 to one outcome). The arguments generalize in a straightforward

way.

4.2 Let us begin with Independence. This axiom insists that outcomes be evaluated independently

of each other. To see how this might be problematic, consider a famous example due to Diamond

(1967). Suppose you can give a good A to one of two people. Suppose that, if only one gets A, it

does not matter which one: you are indifferent between (A, 0) and (0, A). Independence entails that

you are indifferent between a lottery resulting in (A, 0) and (0, A) with probability ½, respectively,

and a lottery resulting in (0, A) with probability ½ and once more (0, A) with probability ½, that is,

a lottery resulting in (0, A) for sure. Yet this seems unreasonable. It seems unfair for it to be a matter

of indifference whether one person gets A for sure (suppose A is a donated kidney!) or whether both

have an equal chance of obtaining A. However, it is no surprise that a fairness problem arises if

fairness is not considered when outcomes are individuated. The outcomes are underdescribed as (A,

0) and (0, A) if fairness is of importance. If fairness is included in the description of the outcomes,

the problem disappears. Let F denote a state of affairs in which A has been distributed fairly and -F

denotes a state of affairs in which it has 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 probability ½ each and (A, 0, F) for sure. This is not counter-intuitive. The reader

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 the above strategy when discussing Diamond. However, Roemer (1996) discusses

Diamond without considering this approach and is led to conclude that Diamond’s example is a

knock-down argument against any usefulness of Harsanyi’s theorem in ethics (p 140). Harsanyi

(1975) does not employ this strategy, but bites the bullet, presenting cases in which it allegedly

does not matter whether some benefit or burden is distributed with or without a lottery.

Economists in general do not 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 the Allais-paradox (see p 180), to which both Broome and Joyce also apply

20

may find it odd to include fairness in the description of outcomes. But why would (A, 0) be an

outcome while (A, 0, F) would not? As Savage (1954), put it: “A consequence is anything that may

happen to the person" (p 13), and surely, having received A in a fair way counts by this standard.

This response to Diamond is called “loading-up-the-consequences.” It is controversial

because one may worry that any example questioning an axiom can be dissolved in this way.

However, for consequentialism to be even plausible, there needs to be a theory of individuating

outcomes that captures all relevant considerations. If, in Diamond’s example, fairness it taken to be

relevant, it must appear in the description of the outcomes, and then no problem arises. Otherwise,

there should have been nothing problematic about the initial implication of Independence. One may

object that, if Independence only applies if the outcomes are described completely, it is trivially true.

It is not clear just why that would be a problem, but it is clear that Independence trivially fails and

that consequentialism is unattractive if outcomes are underdescribed.18

this strategy. Joyce refers to a lack of appreciation of the point made above as the most common

mistake regarding decision theory. The literature on utilitarianism also acknowledges that

consequentialism needs a notion 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 by it. One may be concerned that, if we apply the strategy of loading up

the consequences to Diamond, we already need to assume a notion of fairness prior to our

conquentialist theory. But that is not a problem. For consequentialism does not aim at reducing

all moral vocabulary to talk about outcomes. Worries about consequentialism raised from the

point of view of, say, fairness do not concern the possibility of defining these notions

consequentialistically. Rather, they are worries about how to make room for them within a

consequentialist framework, no matter how we define them.

19 See Anand (1987) for very strong criticism; Broome (1991a) ignores this axiom,

suggesting that it is implausible and submitting that his book is concerned with different

21

This discussion makes a general point about outcomes, but it also indicates how to argue that

the betterness-relation satisfies Independence. Once one outcome has come about, no other outcome

has come about or can come about. Therefore, this outcome should be evaluated on its own terms,

without any reference to other outcomes. This seems eminently reasonable if all considerations

relevant from the point of view of well-being have been considered in the individuation of outcomes.

Independence, then, is a plausible constraint on the betterness relation, given this understanding of

outcomes.

4.3 Although Completeness has found few supporters as an axiom of rational choice,19 one may

problems. Curiously, even though Broome needs to defend the view that the betterness-relation

entailed by a person’s goodness satisfies the axioms of expected utility theory, he presents a

substantial defense only of Independence. He rejects Completeness, thinks of Continuity as

merely a technical assumptions, and of Transitivity as true as a matter of logic. For general

issues about comparability and commensurability, see Chang (1997).

20 See Moore (1903), p 28. This principle has recently come in for a good deal of

discussion, cf. Kagan (1988), Hurka (1998), Lemos (1998), and references therein.

22

think that it is more plausible as a condition on our betterness-relation. After all, utilitarians are

criticized as simple-minded for advocating well-being as an overriding value; surely they should be

entitled to the theoretical benefits from what they are taken to task for and find it easy to argue that

the betterness-relation satisfies Completeness. However, championing any value as overriding does

not entail that the nature of that value allows for adjudication between any two outcomes. It is

unclear, in particular, that on the best versions of the IDS-account, there is always a synthesis of

possibly diverging and conflicting desires into an overall attitude towards any outcome, which then

would make it possible to compare any two outcomes. Yet the case for Completeness is not

hopeless. This is in particular so on an “organic” understanding of value as championed, for

instance, in Moore’s Principia Ethica. Moore states the idea of the principle of organic unities as

follows: “The value of a whole must not be assumed to be the same 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 of each aspect of the outcome whose conjunction constitutes its overall evaluation. For

no aspect can be evaluated without considering the presence of the others. Such a conception of

value and the derivative view on evaluating outcomes does by no means entail Completeness; but

21 There is a tendency in the literature to interpret Completeness as a requirement of

coherent extendibility: that is, although the preference relation need not completely order the

lotteries, there must be an extension of this relation that satisfies the vN/M axioms and does so

order the lotteries. See Joyce (1999), p 45, and references on that page. However, important

objections to Completeness still apply to this understanding of the axiom.

22 Cf. Elster (1979) p 27, and Rachels (1998) for a discussion of the better-than case.

23

it does undermine a strong source of intuitions for the implausibility of Completeness. That source

is the idea that we evaluate outcomes aspect by aspect, which makes it likely that 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 outcomes is immensely beneficial to an agent: it keeps her from being torn or

paralyzed. Surely this point carries some weight if idealized desires are central to well-being. These

two points together suggest that we should allow for the possibility that the best developments of

the IDS-account include a betterness-relation that satisfies Completeness. The case for Completeness

remains questionable, but surely not hopeless.21

4.4 Transitivity is intuitively appealing. To see how Transitivity could be problematic, consider the

following scenario, called improving oneself to death:22 Suppose my well-being depends on two

kinds of goods, and I prefer getting more of the one that I have less of, as long as the loss with

regard to the other is “small”. Suppose I start with 10 units of the first and 9 units of the second

good. Then in the following list, I prefer each member to its predecessor, 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 gain in one dimension may be acceptable at the expense of a loss in

23 Scenarios such as improving oneself to death suggest that our strong intuitions for the

transitivity of the relations preferred to or better than live on their apparent proximity to relations

such as larger than or taller than. The most comprehensive discussion of transitivity that I am

aware of is in Maher (1993). His strategy for arguing for transitivity is as follows: Transitivity

has a high prima facie plausibility: most people would endorse it and would be willing to correct

violations when those are pointed out. There arguments for and against transitivity, but they are

both wanting. So that leaves us with the prima facie plausibility. Maher thinks that this is as

good a case as we can have for substantive normative principles. In the end he takes a pragmatic

line – let us see what kind of theories we can build on the respective principles and judge them

from there. Although Broome (1991a) discusses a betterness relation entailed by a person’s

24

another. Repeated occurrences of that phenomenon undermine Transitivity. (If the strategy of

loading-up-the-consequences is accepted, outcomes will tend to have the 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 such as improving-oneself-to-death. A better response is an appeal to the “organic”

understanding of value. This theory of value denies that outcomes can be evaluated factor by factor;

but it is precisely this kind of evaluation that drives the above example. On an organic understanding

of value, comparing outcomes by preferring more of the good that one has less of is not merely odd,

but scenarios such as improving-oneself-to-death serve to illustrate what is wrong about evaluating

outcomes aspect by aspect. So Transitivity, like Completeness, seems plausible under the organic

understanding of value. And Transitivity, unlike Completeness, clearly has the pre-theoretical

intuitions on its side.23

goodness rather than her well-being, he would probably want to press the same point about logic

mentioned above with regard to well-being as well.

24 Without the continuity axiom, we still have the expected utility form, but the functions

are not scalar-valued, but vector-valued, where the vectors have lexicographically ordered

components, cf. Hausner (1954). In light of this result Harsanyi (1982) thinks that we do not

need to consider Continuity a rationality axiom. Hajek (1998) agrees. Broome (1991a) thinks of

continuity 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 I

rank O2 between O1 and Or. By Continuity, there is a number 0#p#1 such that the utility of Or

equals the sum of p times the utility of O1 and (1-p) times the utility of Or. Yet such a number p

does not exist, and thus Continuity fails.

25

4.5 Continuity is sometimes seen as a “technical” assumption.24 However, this classification is

dubious to begin with, and Hajek (1998) shows how one may question Continuity. Hajek argues for

the importance of infinite utilities, but Continuity forces utilities to be finite.25 He claims that there

is no good argument for Continuity, and that in particular an appeal to consistency in decision

making fails to support it. But be that as it may, a case can be made for imposing Continuity on the

betterness-relation. Continuity is plausible if the agent can compare and weigh amounts of well-

being derived from different outcomes. Continuity is an expression of the agent’s ability to do so.

Examples intended to show how Continuity fails frequently involve death. Would you really, so we

are asked, risk death (which in such examples must have a low negative, but finite utility), even with

a very small probability? We are expected 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 need coffee, but

26 Continuity is also contradicted by so-called lexicographic preferences, where one

factor is considered unconditionally more important than another (i.e., no amount of the latter

can make up for any loss of the former). However, such scenarios would be implausible

again under the organic understanding of value.

26

none is left. So I go and buy coffee, running a risk of being killed in traffic. In terms of Continuity:

there is a probability p such that I am indifferent between the outcome I have no coffee for sure and

a lottery involving the outcome I die in a traffic accident with probability p and the outcome I have

coffee with probability (1-p). Thus it is not as strange as it may seem to 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 1662 Port Royal Logic, one of the founding documents of expected utility

reasoning: “only infinite things such as eternity 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

the applicability of infinite utilities is rather restricted, and surely provides no reason to abandon

Continuity. In this spirit, Morgenstern (1976) points out that the vN/M theory compares to

Newtonian mechanics, which fails for objects traveling at a speed close to that of light, but otherwise

does just fine. So the case for arguing that the betterness-relation satisfies Continuity is fairly good.26

4.6 In conclusion, a case can be made for adopting Completeness, Transitivity, Continuity, and

Independence as constraints on the betterness -relation in a development of the IDS-account. There

is potential for disagreement, but at least on an “organic” theory of value our case does not look

unpromising. The critical axioms is, of course, Completeness. While registering these qualifications,

27 This is a long section, and a preview might be welcome. In 5.2, I use the analogy to the

measurement 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

27

I assume for the sake of the argument that the betterness-relation entailed by the IDS-account of

well-being does indeed satisfy these four axioms. The subsequent argument is only as good as this

claim, but let us see what we can make of it.

5. The Quantitative Structure of Well-Being

5.1 Section 4 does not yet show that Harsanyi’s theorem provides any useful insights about

utilitarianism. What we have shown is this. Suppose that the individual betterness-relations and the

observer’s betterness-relation satisfy the vN/M assumptions, and suppose that a Pareto condition

holds. If we choose any expectational representation of the individual betterness-relations and of the

observer’s betterness-relation, respectively, then the observer’s representation is a weighted sum

over those of the individuals. This is an interesting result about expectational representations, but

we would gain a deeper insight if we could show in addition that one of the expectational

representations actually measures the agent’s well-being (i.e., that well-being itself is

“expectational”). For then the following would be true: If the individual betterness-relations and the

observer’s betterness-relation are related by a Pareto condition, then the function that measures the

observer’s well-being is a weighted sum over the functions that measure the individuals’ well-being.

What I argue next, however, is not that one of those representations measures the agent’s well-being,

but that the family of expectational representations as such does so. Yet it will soon become clear

that Harsanyi’s theorem loses none of its status as a contribution to utilitarianism for this reason.27

more from the measurement of heat for the measurement of utility than just what it is for a

family of functions to measure something. In 5.3, I discuss their view, but suggest that the

prospects are dim for them to convince us that the family of expectational representations does

indeed measure well-being. Next I enlist an argument from Broome (1991a), which I think

shows that this family does indeed measure well-being. In 5.4, I discuss some worries about the

argument of this section.

28

5.2 We need to explain what it means for a family of functions to measure something. To this end,

and for its usefulness as an analogy, let me briefly discuss the measurement of heat. Von Neumann

and Morgenstern (1947) use this analogy, and so does Broome (1991a). When we try to measure

heat, any property of a substance or a device that changes when it is heated or cooled may serve as

the basis of a thermometer. For instance, we may define the change in temperature to be proportional

to the change in length of a column of liquid in a capillary tube. In order to calibrate a thermometer

we assign numerical values to the temperatures of two points (given constant pressure). The

position of liquid at these points is marked and the distance between them is divided into equal

intervals. For example, on the Celsius scale, there are 100 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

Fahrenheit scale, there are 180 intervals (with the freezing point set at 32 and the boiling point at

212). Fahrenheit and Celsius can be transformed into each other using the equations tf = 9/5 tc + 32

and tc = 5/9 (tf -32). In jargon, the two scales are positive affine transformations of each other. A

function f is a positive affine transformation of a function g if there exists a positive real number a

and real number b such that f = ag+b. If f is such a transformation of g, then the converse is true as

well. Any scale that is a positive affine transformation of the Celsius scale can be used as a

29

temperature scale in the sense outlined above, and vice versa. Each of these functions is as good a

measuring scale as any other. It is in this sense that it makes sense to say of this whole family of

functions, rather than of any specific function, that it measures heat.

The fact that this family of functions measures heat in this sense provides a characterization

of meaningful statements about temperature comparisons. Meaningful statements are those that are

invariant across all scales in that family. For instance, the statement that it is warmer in New Haven

at 11 am than at 10 am is meaningful because it is either true according to all such scales or false

according to all of them: inequalities between temperature differences are preserved across all

scales. But the statement that it is twice as warm at 11 am 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 much as the increase

between 2 pm and 3pm. For the ratio between temperature differences remains constant across

scales.

5.3 Von Neumann and Morgenstern did not merely introduce this analogy to illustrate what it is for

a family of functions to measure something. They thought that we learn more from it about

measuring utility. Although it has been 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 of it. They

took themselves to be contributing to an area of research where not much progress had been made,

namely, the measurement of utility. The analogy 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

intuitively clear feeling of one body feeling warmer than another, whereas nowadays we have

28 They use the expression is preferred to just as the phrases is warmer than and is to the

left of are used in their respective domains. Just as there is nothing in any physical body in

addition to heat that corresponds to a noun formed to the predicate is warmer than, so there is

nothing in an agent in addition to utility that corresponds to the noun preference. This usage of

the word “preference” deserves some attention.

30

thermometers to make statements about comparative strengths of temperature differences. Von

Neumann and Morgenstern conceive of utility as part of physically describable nature in the same

sense in which heat is. They believe that they have made a discovery that advances utility

measurement in the same way in which the thermometer advanced the measurement of heat.

They talk about “utility” in two ways: On the one hand, there are numerical utilities

analogous to temperature values. Yet on the other hand, there is utility analogous to heat: a physical

property to be measured.28 According to them, the key to utility measurement is the discovery of 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 “observationally reproducible.” The

additional natural operation (i.e., in addition to “preferring”) that von Neumann and Morgenstern

think they discovered is the concatenation of events with probabilities. If we can talk about events,

so they argue, we can talk about probabilistic concatenations of events. Since they regard events as

the location of utility, this operation applies to utility as well, and thus we have “discovered” that

utility itself is expectational. And then it only takes the axiomatic postulation of properties of the

concatenation operation to obtain a measurement of utility based on these two natural operations.

Those properties must be chosen such that the behavior of numerical utilities captures the

expectational nature of utility.“We have practically defined numerical utility as being that thing for

29 Strictly speaking, this is not quite correct. We must distinguish between two functions:

On the one hand, expectational representations are defined, like other representations, on the

domain of a preference relation (i.e., on lotteries). Suppose U is such a function, and suppose the

outcomes are (O1, ...., On). U induces a function u defined on the outcomes such that u(Oi) equals

the value of U applied to the lottery that delivers Oi for sure. The class of functions that the

representation theorem determines as being closed under positive affine transformations is the

class of those functions defined on the outcomes, and it is this family of functions that

corresponds to the family of functions that are positive affine transformations of the Celsius

scale in the case of heat. However, since in vN/M theory, for each such function U there is a

function u, and vice versa, we can talk loosely here, and in particular we can talk about the class

of expectational representations, as measuring utility. Note an important difference between the

heat case and the utility case: temperature scales are linear functions, but that need not be the

case for utility.

31

which the calculus of mathematical expectations is legitimate”, so they say (p 28). In conclusion,

they believe that, through their representation theorem, they can determine a family of functions

closed under positive affine transformations that measure utility just as positive affine

transformations of the Celsius scale measure heat. This family of functions, of course, is the family

of expectational representations.29

Developing this view involves problems in the philosophy of science. Do we really measure

anything? What do “intuitive clarity” and “observational reproducibility” amount to? Most critically,

what are we to make of von Neumann and Morgenstern’s claim that they “discovered” a new

operation, namely the concatenation of events with probabilities? Without having an appropriate

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 carry probabilities, while consequences carry utilities. Acts are ranked in terms of their

expectations. The vN/M model assigns both probabilities and utilities to outcomes and thus

cannot capture the idea that one state of the world leads to different consequences, depending on

what action was chosen. The same example can be reproduced in vN/M notation as well:

32

theory of what such a “discovery” amounts to, this claim has all the advantages of theft over honest

toil: it delivers the expectational nature of utility without further ado. Below I will have more to say

about the measurement-question. However, I suspect that these questions cannot be answered in

such a way that we would find it ultimately plausible, on this account, that the family of

expectational representations does indeed measure well-being. But since this 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 I think 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 and A2. If you choose A1 and outcome O1 occurs, you receive $100; if outcome O2

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 probability 1/3 and O2 with probability 2/3, A1 and A2

are equally good for you. That is, the prospect of obtaining $100 with probability 1/3 and $200 with

probability 2/3 is, as far as your well-being is concerned, on a par with the prospect of obtaining $20

with probability 1/3 and $320 with probability 2/3.30 Since we are assuming that the betterness-

outcomes would then have to be described in terms of their monetary reward and in terms of the

action chosen, with the probabilities suitably adjusted. But since this leads to clumsy notation, I

present Broome’s example in the Savage-style that he himself chose.

33

relation entailed by your well-being can be represented by an expectational function u, we obtain

the following equation:

1/3u($100) + 2/3 u($200) = 1/3u($20) + 2/3 u($320)

Simple algebraic transformations show that

{u($100)-u($20)}/{u($320) - u($200)} = 2

That is, the utility difference between the two amounts of money you could obtain were O1 to occur

is twice as big as the utility difference between the two amounts you could obtain were O2 to occur.

The fact that, in O1, you receive $100 rather than $20 if you choose A1 is a consideration in favor

of A1. Similarly, the fact that, in O2, you receive $320 rather than $200 if you choose A2 is a

consideration in favor of A2. So the second equation shows that the consideration in favor of A1

counts, as far as the overall well-being pertaining to those prospects is concerned, twice as much as

the consideration in favor of A2. More generally, the utility values tell us how much differences in

well-being count proportionately in the determination of the comparative overall well-being

pertaining to those prospects. Since we are dealing with a family of expectational representations,

the proportionality statements do not change across representations.

But if the difference in well-being between $100 and $20 counts for twice as much as the

difference between $320 and $200 in the determination of your overall well-being pertaining to a

lottery, it is plausible to infer that these differences measure genuine differences in well-being. As

Broome points out (p 147), the only way of denying this inference is to insist on a difference

31 This is so simply because any function that shares with all the functions in that family

what they have in common (i.e., ratios of utility differences) is a function in that family (i.e., is

itself a positive affine transformation of all the functions in that family).

32 To put the point differently: the vN/M axioms guarantee that preferences have a

sufficiently rich structure to allow for expectational representations. The present argument shows

that preferences 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

closed under positive affine transformations.

34

between amounts of well-being and the way they count towards overall well-being. Yet this seems

like an empty distinction, because plausibly, differences in well-being can count proportionately the

way they do only because they amount to genuine differences in well-being. If this is right, we have

shown that, in general, the ratio of differences in well-being is whatever the ratio of the

corresponding utility differences is. This entails that the agent’s well-being itself can be expressed

as a positive affine transformation of any expectational representations of the betterness-relation.31

That is, the agent’s well-being itself is expressed by a function in this family of expectational

representations. This does not imply that well-being is expressed or measured by any of those

functions rather than by any other. Instead, it means that the function measuring the agent’s well-

being is a member of the family of expectational representations, and just like in the case of heat,

the choice of any specific function as the measurement scale is arbitrary. So we are justified in

saying that the whole family of expectational representations of the agent’s betterness-relation

measures her well-being, just as the whole family of positive affine transformations of the Celsius

scale measures heat.32 Thus we have finally met the challenge posed in section 3, that is, to find a

35

conceptual connection between the vN/M notion of utility as preference representation and the

utilitarian notion of utility as well-being.

We can conclude then that Harsanyi’s theorem teaches us the following: If the individual

betterness-relations and the observer’s betterness-relation are related by a Pareto condition, then any

of the functions that measures the observer’s well-being is a weighted sum over any profile of

functions that measure the individuals’ well-being. So if one accepts Consequentialism, Welfarism,

and Bayesianism, one ought to accept Summation’ as well. For if (given those three conditions) one

tries to act like an impartial observer to the minimal extent that one accepts universal preference

agreements in the relevant group, then one cannot help but accept Summation’. For the remainder

of this section and in section 6, we will address worries about and objections to this argument and

the conclusion just drawn. In section 7 we will discuss the conclusion some more.

5.5 Let me address two worries about this argument. One may still wonder whether we should speak

of a measurement in this context. What made the measurement 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

of heat independently of subjective perception. Yet what counts as measurement is a vexing

question. On the one hand, there are paradigmatic cases of measurement, such as the measurement

of length, or of heat, that are remarkably different 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

our case, what would be measured is well-being according to the IDS-account, where the

measurement process would be idealized deliberation. Yet for our purposes, nothing depends on

using the term. No harm is done if the reader replaces occurrences of the verb “measure” with the

33According to the account in Luce et al. (1971), this proposal counts as a measurement

but Luce et al. take vV/M theory as a paradigm of what a measurement is. This, of course, just

moves the concerns one level higher up. Luce et al. (1971) present what they call a

representational approach to measurement, which hinges on the construction of a

homomorphism between what they call a relational structure and the real numbers, supplied

with certain relations (p 8ff). If there is such a homomorphism, there is measurement. In that

sense, von Neumann and Morgenstern do measure a property. For a critical discussion of Luce et

al. (1971), see Berka (1983), in particular p 64, p 114, p 153, and p 174. According to Berka, we

should not speak of measurements here. Note that there is a third worry that we can properly

address only in section 6, but that should be mentioned here. Recall from section 2 that

Harsanyi’s theorem only applies to one profile of representations at a time. So for any observer

representation and any profile of individual representations there exist coefficients such that the

observer representation is a weighted sum over the individual representations using those

coefficients. But if we now use another observer representation or other individual

representations, the coefficients would be different. In section 6 we develop the tools to respond

to this worry.

36

corresponding forms of “express” or “capture.”33

Another worry is that, if the argument of this study is correct, an agent should be risk-neutral

with regard to her well-being. Evidently, so an objector might say, this is empirically false, and it

is false that persons ought to be risk-neutral in this way. Before I address this objection, let me make

sure that it is not 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. To see why, suppose that all that matters

34 For a historical account of expected utility theory, see chapter 1 in Joyce (1999), in

particular on the St. Petersburg paradox. For space reasons alone, I have not discussed views that

37

to a person’s well-being is the money she receives in an outcome. Then the utility functions assign

to amounts of money numerical utilities capturing the well-being deriving from the money. Suppose

u is such a function, and suppose we are looking at an action that leads to outcomes O1, ..., On with

probabilities p1, ...., pn. If the betterness-relation satisfies the vN/M axioms, then

u(p1O1, ..., pnOn) = p1u(O1) + .... + pnu(On)

If u measures well-being, as I have argued, then this equation indeed expresses risk-neutrality with

regard to well-being. Risk-neutrality about money, however would be captured as follows, where

$Oi denotes the amount of money that the agent would receive in state Oi:

u(p1$O1, ..., pn$On) = u(p1$O1 + .... + pn$On)

This equation says that an agent attaches as much well-being to a monetary lottery as to the

expectation of this lottery. But that is an entirely different claim. The argument in this study does

not require u to satisfy the second equation. As readers familiar with expected utility theory know,

this response is as old as the notion of utility itself, and in fact, the desire to draw a distinction

between the points expressed by these two equations motivated the very introduction of the notion

of utility. But if this point is acknowledged, it should be clear that our intuitions about risk-attitudes

with regard to well-being are not as developed as our intuitions about risk-attitudes with regard to,

say, money, both empirically and normatively. The argument of this study, if correct, entails that a

development of the IDS-account of well-being should conceive of well-being in a way that entails

risk-neutrality with regard to well-being. Worries about that claim should be raised as objections to

some part of the argument.34

construct 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 that

much the motivation for these alternatives will disappear if (a) it is acknowledged that outcomes

need to be described completely for any consequentialist theory to work, and (b) once the vN/M

axioms are understood as constraints on the betterness-relation entailed by well-being rather than

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).

38

6. Interpersonal Comparisons of Utility

6.1 We now address an important objection to any usefulness of Harsanyi’s theorem to utilitarian

theory, presented by Roemer (1996).35 To this end we need to discuss interpersonal comparisons of

utility. Utility comparisons pose significant conceptual and practical difficulties. For that reason,

social choice theorists have chosen an axiomatic approach to both intrapersonal and interpersonal

comparisons. In such an approach, different senses of utility comparability can be characterized,

although we may not know how to make such comparisons. Different notions of utility

comparability are captured by invariance properties of profiles of representations. The idea is to say

for any profile (u1, ..., un) which profiles are equivalent to it, where equivalence amounts to

expressing the same information. Once we have introduced such an equivalence relation, we count

as meaningful precisely those statements about intrapersonal and interpersonal comparisons that

hold for all equivalent profiles. We used a version of this idea to characterize meaningful statements

about utility (and temperature) comparisons in section 5. Consider examples: In the simplest case,

39

all representations 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 than he prefers y to z”) fail to hold across all

representations. Next define a profile (u'1, ..., u'n) as equivalent to (u1, ..., un) if there are positive real

numbers ai and real numbers bi such that ui = aiu'i+bi. Then it is meaningful to say that i prefers an

outcome x to y more than he prefers z to a, since the relevant inequalities remain fixed among all

equivalent representations. However, it is not meaningful to say that individual i prefers x to z more

than individual j prefers y to a.

We need to make assumptions on comparability for summation over utilities to be

meaningful. To see why, suppose we take all profiles to be equivalent. Suppose we have profiles

(u1, ..., un) and (u'1, ..., u'n), and suppose that the functions u'i assign a much broader range of values.

We can obviously choose profiles in such a way that the new observer utility function generated by

summation not only assigns different utility values to lotteries, but does so in such a way that not

even the observer’s ranking is preserved. So how much utility comparability do we need to obtain

a meaningful notion of summation? Define a profile (v1, ..., vn) as equivalent to (u1, ..., un) if there

is a positive real number a such that ui=avi +bi for all i and real numbers bi. Then many kinds of

comparisons become meaningful. For instance, it makes sense 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)-ui(y))/(uj(z)-uj(a))] remain unchanged if ui is replaced with vi=aui+bi and uj is

replaced with vj=avj+bj. As is easy to check, this is indeed the case. Thus interpersonal comparisons

of utility differences make sense under this notion of equivalence. However, comparisons of utility

levels make no sense: it is not meaningful to say that agent i is “better off” than j. Still, if we use

36 Roemer (1996), chapter 1, presents a discussion of notions of utility comparability. In

particular, he shows that the notion of comparability discussed in the text is the weakest such

notion that 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

the assumption of completeness for the group or the observer implies the possibility of

interpersonal comparisons. For if an observer can compare any two outcomes, he can in

particular compare any two outcomes X and Y that only differ in terms of how agents i and j

fare. This argument is unconvincing. For nothing about group preferences requires outcomes to

be distinguished in terms of how good they are for specific members of the group. All that is

required about the connection of the group preferences with the individual preferences is a

Pareto condition. To require more 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 determine not only the preferences of the individuals, but also the

social preferences which they have arrived at, and can also determine the fact that they regard

those social preferences as representing 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 which are involved, i.e., we can perform interpersonal

comparisons of preferences." The idea is that the agents start with a group representation about

which they claim that it expresses an even-handed compromise, where each individual's function

is counted with coefficient 1. Harsanyi’s theorem then delivers n individual functions (uniquely

40

this notion of interpersonal comparability, group rankings are preserved under changes of

equivalent representations. And at least in that sense then summation is meaningful.36

determined). Since the agents agree that the group function expresses a compromise in which

they are all equally considered, they can use these functions 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 that Jeffrey is wrong.)

37 In Mongin's (1994) multi-profile version, the coefficients are constant acrossprofiles.

41

This notion of interpersonal comparability answers a worry about the argument in sections

3-5 that I omitted in section 5.5. Consider profiles (u1, ..., un) and (u'1, ..., u'n) of expectational

representations and an expectational representation u for the observer. Harsanyi’s theorem implies

the existence of coefficients ai and a constant b such that u = 3i aiui+ b, and of coefficients a'i and

b' such that u = 3i a’iu’i + b’. But we do not have in general ai=a'i.37 However, if those two profiles

are equivalent in the sense required by this notion of interpersonal comparability, it is true that

ai3i ai = a’i 3i a’i, 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’s theorem apply. Consider

the set C of all profiles of representations (u1, ..., un). Suppose we can make interpersonal

comparisons for utility summation to be meaningful. Thus there is a subset D of C containing

equivalent profiles over which it is meaningful to sum. Moreover, by the vN/M representation

theorem, there exists a subset E of C including all profiles consisting of expectational

representations. That is, E is the subset of D to which Harsanyi’s theorem applies. However, there

is no guarantee that D and E have any element in common: the profiles to which Harsanyi’s theorem

applies are not necessarily profiles over which it is meaningful to sum. Therefore, the summation

42

in Harsanyi’s theorem may not be meaningful from a utilitarian point of view and 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 satisfies the vN/M axioms. So we should

understand well-being in terms of an idealized-desire account constrained by the imposition of the

vN/M axioms on the betterness-relation. Thus there is no notion of well-being determining a set of

profiles of representations over which it is meaningful to sum, but which is disjoint from the set of

profiles that satisfy those axioms. Roemer’s argument is motivated by the idea that expectational

functions “merely represent” 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 empirical or statistical

inquiries is at odds with the expectational representations of an agent’s betterness-relation. But that

is not problematic. In the present approach, expectational utility functions represent the betterness

relation entailed by an IDS-notion of well-being. Interpersonal comparability of well-being must be

thought of as integrated into such an account. Empirical notions of well-being and empirical

methods to compare well-being can only be approximations of such notions. In light of this, what

Roemer shows is that such approximation 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 endorse Summation’.

Summation’ should be adopted because for any observer whose well-being is connected to the well-

being of the individual members of the relevant group in the minimal sense that his betterness-

43

relation preserves agreements among their preferences – for any such observer it is true that any

function that measures his well-being is a weighted sum over any profile of functions that measure

the individuals’ well-being. Put differently, if an advocate of Consequentialism, Welfarism, and

Bayesianism accepts the Pareto condition (which I think is not much to ask), then the argument of

this study entails that she cannot help but accept Summation’. The crucial segments of the argument

are to show that the vN/M axioms constrain the betterness-relation entailed by the IDS-account of

well-being, and that the family of expectational representations measures well-being. To make these

arguments, I have appealed to an idealized-desire account of well-being and to an organic theory

of value.

Recall the qualifications we adopted along the way. To begin with, in virtue of its single-

profile format, Harsanyi’s theorem cannot deliver an argument for equal consideration in the

summation. Therefore, Harsanyi’s argument only presents us with a partial defense of utilitarian

summation in the presence of Consequentialism, Welfarism, and Bayesianism (i.e., an argument for

Summation’, not for Summation).Yet it would be puzzling if those conditions with merely moderate

additions implied equal consideration for individuals. It is very surprising already that the addition

of Pareto suffices to entail Summation’. Additional conditions to obtain equality of consideration

must be formulated in the multi-profile context. But we should expect them to be close to

postulations of equal consideration; we should not except a surprising and illuminating derivation

of the sort that here provides an argument for Summation’. So although this qualification entails that

we have “only” a partial defense of utilitarian summation, this should not subtract from the insight

that the theorem provides.

Another qualification is that the vN/M version of expected utility theory is a limited theory

44

of its kind by not considering subjective probabilities. All individuals and the observer use the same

probabilities. Epistemic disagreements are ignored. This does not strike me as a serious restriction

under the observer-interpretation of “collective preferences.” Suppose the observer needs to compare

two lotteries (i.e., actions whose outcomes are known only probabilistically). If she accepts Pareto,

then she will do so in a way that involves summation over the individuals’ utility functions. But this

observer is an idealized observer who can therefore safely be assumed to have better information

instructing her epistemic judgements than the individuals. We only have reason to use the device

of such an observer if we also accept this idealizing assumption 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’s theorem is forthcoming if disagreements about probability are considered (see Mongin

(1995)).

In conclusion, then, in spite of these qualifications, Harsanyi’s theorem makes a substantial

contribution to utilitarian theory. The surprising nature of the result provided by Harsanyi’s theorem

should make it plausible why it would take theoretical commitments at certain points to adopt this

as an argument. I hope that, at the very least, this study helps illuminate connections between

certain views in ethics and decision theory.

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