The Impartial Observer Theorem of Social Ethics

Abstract

I would like to dedicate this paper to the memory of Louis-André Gérard-Varet. His premature death in January 2001 deprived us all of a genuinely broad theoretical economist with deep foundational interests. He was one of the few to be conversant with the intricacies of the theory of subjective probability, and he himself co-authored a formal reconstruction of the Impartial Observer Theorem (d'Aspremont and Gérard-Varet, 1991). At the time of publishing this alternative reconstruction, I cannot refrain from remembering his exceptionally active intelligence, as well as the fruitful interchanges we had.

Full text

THE IMPARTIAL OBSERVER
THEOREM OF SOCIAL ETHICS
PHILIPPE MONGIN
Ecole Polytechnique & Centre National
de la Recherche Scientifique
I would like to dedicate this paper to the memory of Louis-Andre
ÂGe
Ârard-
Varet. His premature death in January 2001 deprived us all of a genuinely
broad theoretical economist with deep foundational interests. He was one
of the few to be conversant with the intricacies of the theory of subjective
probability, and he himself co-authored a formal reconstruction of the
Impartial Observer Theorem (d'Aspremont and Ge
Ârard-Varet, 1991). At the
time of publishing this alternative reconstruction, I cannot refrain from
remembering his exceptionally active intelligence, as well as the fruitful
interchanges we had.
1. GENERAL
Vickrey (1945) and Harsanyi (1953) are credited for having indepen-
dently introduced the following argument. One should compare income
distribution vectors from the viewpoint of an observer who, by assump-
tion, knows the income values, but does not know who has what, and in
particular does not know his own income. It is also assumed that this
observer gives an equal chance to the outcome of landing in each
possible position. Then, applying the von Neumann±Morgenstern theory
of risk to this special informational context, one concludes that income
distribution vectors must be ranked according to the mean rule of
utilitarianism. The argument as a whole is often referred to as the
I acknowledge helpful discussions with Claude d'Aspremont on the subject of this paper,
and I am grateful to Richard Bradley, John Broome, Marc Fleurbaey, Philippe Fontaine,
Francesco Guala, Edi Karni, Serge Kolm, Robert Leonard, Isaac Levi, Wlodek Rabinowicz,
and John Weymark for detailed comments on earlier drafts. I have also benefitted from
remarks made during a seminar and two conference sessions in Cergy-Pontoise, Montre
Âal
and Lund.
147
Economics and Philosophy,17 (2001) 147±179 Copyright #Cambridge University Press

Impartial Observer Theorem. It has no well-agreed formulation. Vick-
rey's supposedly seminal paper has just one paragraph on it, and it is
quite informal (1945, in 1994, pp. 24± 5). So is the slightly expanded
restatement in Vickrey (1960). Harsanyi's 1953 contribution has only two
pages without any symbolism, and his 1955 restatement is again very
terse. It is only later that Harsanyi (1977a) came to restate his insights
more formally ± though, as we will see, not satisfactorily. At the same
time, he put them more generally, dealing with abstract social states
rather than just income distribution vectors. When we contrast Vickrey's
and Harsanyi's versions in this paper, the latter will always mean
Harsanyi's mature version.
We provide a reconstruction of the `theorem', not in order to turn it
into a piece of mathematics, which it cannot be, but to precisely identify
all of the assumptions which have to be defended if it is to be regarded
as a serious ethical argument. Our reconstruction is, like Harsanyi's in
(1977a), based on the `extended preference' framework of social choice
theory, but differs from his in several respects. We emphasize the need to
assume uniformity of extended preferences among individual observers;
otherwise, the ordinary utilitarian formula cannot be derived. We argue
that uniformity of extended preferences is undefended in Harsanyi's
framework, and we are thus led to investigate weaker variants of the
conclusion in which each ethical observer adopts a utilitarian formula of
his own, with utility representations of the others' preferences de-
pending on the particular observer.
We also depart from the historical versions of the `theorem' in
considering subjective probability assessments instead of Vickrey's and
Harsanyi's equiprobable lotteries. Laplace's principle has raised innu-
merable objections, and this provides a serious, if only negative, reason
for considering a Bayesian variant in which subjective priors can differ.
Following this heuristics, we offer a novel and more sophisticated
formalization of the Impartial Observer. Conceptually, this variant will
be seen to have an effect similar to the previous one, that is, it entails
observer-dependent additive formulas and thus falls short of the
utilitarian objective. An advantage of this variant, however, is that it
provides a partial answer to an objection classically raised against the
use of the von Neumann±Morgenstern theorem in social ethics, that is,
that there is no conceptual ± in the sense of preference-based ± reason
for selecting the specific utility representations provided by the
theorem.
Another line of argument we examine is to take utility assessments
rather than preference judgements as primitive data for the axiomatic
construction. Following this direction, one can reach an observer-
independent formula by virtue of what we call the causal account of
interpersonal comparisons. But as we explain, this formula is of the
148 PHILIPPE MONGIN

`generalized' utilitarian sort ± that is, it does not entail identical weights
for the individuals. Also, the causal account does not deliver a theorem
in a real sense. It is rather an addition of claims that are philosophically
debatable. Those who accept this objective account would be better off in
taking a more direct ethical approach than that of the Impartial Observer
Theorem.
So the simplest message of the paper is this. There is no way in
which the Impartial Observer Theorem can bridge the whole gap from
impartiality to utilitarianism, even making generous allowance for
technical assumptions. But it is possible to conclude that at least in the
subjective version explained here, the reasoning proves something ± even
if the result is a long way from the official objective.
There is another point of general significance. Broadly speaking,
impartiality amounts to disregarding what is irrelevant in the peculia-
rities of a case when making a judgement on this case. In the present
context, impartiality has received a more determinate meaning. Vickrey
and Harsanyi equate `disregarding' with `not knowing', that is, they
interpret impartial judgements as being those made in a situation of
hypothetical ignorance. This is the seminal idea underlying the Impartial
Observer Theorem; it then leads to the surprising application of decision
theory to an ethical context. At the same time, Vickrey's and Harsanyi's
commentators understand them as also employing a commonsensical
notion of impartiality that recommends equal treatment of the indivi-
duals. Our analysis will reveal a tension between the two notions of
impartiality. We will argue that the Impartial Observer Theorem is best
understood by starting only with the first, and then examining whether
or not the second can be derived. We do not mean to suggest that
impartiality as equal treatment lacks normative warrant. Quite the
contrary. The point is that to assume it at the same time as the other
concept takes the edge off the argument. In order to make the best of the
Impartial Observer Theorem it seems methodologically sound to mini-
mize the number of purely ethical postulates.
The paper is organized as follows. Section 2 briefly relates the
Impartial Observer Theorem to the philosophical tradition of basing
ethical judgements on impartiality. Section 3 provides a relatively literal
reconstruction of Vickrey's version, and Section 4 moves to Harsanyi's (it
is a minor contribution of this paper to clarify the differences between
Vickrey and Harsanyi. As will be explained in Section 4, Harsanyi's
version needs three axioms besides the VNM one, to be called here,
Equal Chance,Consideration of Others, and Uniform Extended Preference.
Section 5 discusses the last two axioms, and Section 6 discusses the first
while explaining our subjective probability reconstruction. Section 7
provides a summary assessment of the Impartial Observer Theorem. The
technical details are covered in the appendix.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 149

2. THE ETHICS OF IMPARTIALITY
There is an important philosophical tradition which emphasizes imparti-
ality as a distinctive origin of moral judgements on collective life ±
notably, but not exclusively, in matters of justice. Allegedly, these
judgements should not depend on individuals' identities and other
particular circumstances. It is also argued that the symmetry or
interchangeability requirements implied by impartiality go a long way
towards determining not only the form, but the content of moral
judgements. As far as we can see, this broad impartiality tradition
borrows from two major sources ± that is, the eighteenth-century Scottish
writers, especially Adam Smith in The Theory of Moral Sentiments (1759),
and Kantianism rather than Kant himself (because it uses only edulco-
rated versions of the Categorical Imperative and the Universalization
Maxim). Encompassing though it is, the impartiality approach must be
kept distinct from that of state-of-nature (or contractarian) theories. This
difference is not sufficiently well reflected in today's textbook compar-
ison between Rawls and Harsanyi, which treats them both as if they
unproblematically belonged to the impartiality tradition.
1
We eschew the
task of arguing for these broad claims and focus instead on the
philosophical background of the Impartial Observer Theorem.
In 1953 Harsanyi identified the foundations of morality with `none-
goistic impersonal judgements of preference' ± a statement reiterated in
all his subsequent work. For instance, he claimed in (1977a, p. 49) that
`the moral point of view is essentially the point of view of a sympathetic
but impartial observer'. Notice that in Harsanyi's mature formulation,
impersonality or impartiality has become compounded with sympathy.
The passage just quoted only refers to Adam Smith, but in a 1958 paper
Harsanyi explicitly endorsed a version of Kant's universalization maxim.
His work thus reproduces the combination of Scottish and Kantian
elements that is typical of the impartiality tradition. It is also likely,
though not entirely clear-cut, that Harsanyi recognizes the difference
between the mental experiment involved in his observer construction,
and the hypothetical histories that underlie the state-of-nature approach.
Finally, even if the 1953 paper is ostensibly concerned with income
distribution, Harsanyi appears to strive towards a complete system of
ethics rather than just a theory of economic justice.
2
All in all, Harsanyi,
if perhaps not Vickrey, should count as a major representative of the
ethics of impartiality among twentieth-century writers.
1
Rawls's theory, not Harsanyi's, is the problem here. It does not belong solely to the
impartiality tradition. Hampton (1980) has discussed the sense in which it also belongs to
the social contract tradition.
2
Witness the distinctions he makes between ethics and other forms of rational behaviour in
(1977a).
150 PHILIPPE MONGIN

We aim at analysing the specific contribution of these authors'
`theorem' to the impartiality tradition, and specifically, at clarifying the
stark contrast between the weak philosophical premiss that philosophi-
cally motivates the reasoning, and its strong and questionable conclu-
sion. How does one proceed from impartiality, possibly compounded
with sympathy, to the utilitarian mean rule? It must be the case that the
assumptions of the `theorem' are more than just a formal dressing of the
intuition of the sympathetic-but-impartial-observer. The analytical task,
then, is first to delineate the added logical content, and second, to assess
its conceptual significance. We have already pointed out that the added
assumptions include that of von Neumann±Morgenstern (VNM) ration-
ality in order to model the observer's judgements. Both in Vickrey and
Harsanyi this assumption drives the additive form of the social evalua-
tion rule.
3
Critics of VNM rationality will then dismiss the utilitarian
looking conclusion of the `theorem' as being irrelevant. We will put aside
this sweeping criticism, and despite well recognized difficulties, assume
that the VNM axioms provide a satisfactory construal of rational
preference under risk. We will assume that the related set of axioms
introduced by Anscombe and Aumann's (1963) for the uncertainty case
is equally acceptable. There are further and less obvious difficulties in
the way of the argument, as will soon become clear.
3. VICKREY AND HARSANYI CONTRASTED
4
Vickrey gives no reason why the observer should give an equal chance to
each position in the society. In Harsanyi this Equal Chance (EC) principle
is to some extent argued for. Harsanyi (1953) claims that an impartial or
impersonal observer's judgements can be reproduced as any indivi-
dual's judgements of preference in a situation of
complete ignorance of what his own position, and the position of those
near to his heart, would be within the system chosen. (1976, p. 4)
He then adds that this state of ignorance
would be the case if he had exactly the same chance of obtaining the first
position . . . or the second or the third, etc., up the last position. (ibid.)
This is as much as Harsanyi is willing to say in order to defend the EC
principle. It seems clear that there are two steps in the argument, one
3
Of the two, only Harsanyi exploits the fact that the additive conclusion can be phrased in
terms of mean utilitarianism. Vickrey has in mind the Benthamite sum rule rather than the
mean rule.
4
What is said of Vickrey in this section is meant to apply to both his 1945 discussion, and
the `potential immigrant' discussion of his 1960 paper (in 1994, pp. 44± 5). As we read it,
the latter is but a brilliant illustrative restatement of the 1945 argument.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 151

from impartiality to ignorance, and the other from ignorance to EC itself.
Neither step is logically compelling, but the second one is specially easy
to criticize. It amounts to Laplace's application of the `principle of
insufficient reason': complete ignorance should be modelled as equi-
probability. There are famous objections against it. Rawls (1971, Section
28) endorses them, albeit in passing, because his conclusion is much
more drastically that one should not use probability at all in order to
model the observer's ignorance. We emphasize the following inter-
mediate possibility: to reject Laplace's application of the `principle of
insufficient reason', while remaining within the confines of a probabil-
istic ('Bayesian') decision theory. This line of argument amounts to
accepting the first step (from impartiality to ignorance) while rejecting
the second (from ignorance to EC). It is pursued in Section 6.
Both Vickrey and Harsanyi adhere to the von Neumann±
Morgenstern theory of preference under risk. This VNM assumption is
of course distinct from EC. One may accept Laplace's principle, while
disbelieving that lotteries should be evaluated in the linear way implied
by the von Neumann±Morgenstern axioms. Rawlsian critics, who reject
all the Vickrey±Harsanyi assumptions at once, should attend to these
obvious distinctions ± they do not always make them. Supposing now
that equiprobable distributions and VNM preferences are relevant in
modelling the impartial observer's judgements, one gets different
formalizations, as well as significantly different ethical implications,
depending on how one draws the line between what the observer is
supposed to know and not to know. In sum, the contrast between a
Rawlsian `fair' observer and a Harsanyian `impersonal' observer will
eventually depend on both the analytical treatment and the factual
content of ignorance.
5
In the case of Harsanyi versus Vickrey the
distinction simply boils down to the content of ignorance, analysed in
one and the same way.
The Vickreyan observer chooses an income distribution as would
any individual, `were he asked which of various variants of the economy
he would like to become a member of, assuming that once he selects a
given economy with a given distribution of income, he has an equal chance
of landing in the shoes of each member of it' (1945, p. 329; own
emphasis). Thus, Vickrey merely requires that members of society ignore
their position on the income distribution ladder.
6
In 1953, Harsanyi's
5
A point well recognized by Levi (1977) in his comparison of ignorance in Rawls and
Harsanyi.
6
Vickrey (1960, in 1994, pp. 44±5) compares the observer with a prospective immigrant
who contemplates various communities to migrate to, and is uncertain as to what income
he will receive in each of these. As before, income is the only variable of interest to the
observer.
152 PHILIPPE MONGIN

delineation of the content of ignorance was not clearly different from
Vickrey's, but, as early as 1955, he turned in a different direction.
In 1955, Harsanyi summarized the content of his earlier paper,
adding the following essential footnote: [Impersonality requires that the
observer have an equal chance of] `being put in the place of any
individual member of the society, with regard not only to his objective
social (and economic) conditions, but also to his subjective attitudes and
tastes. In other words, he ought to judge the utility of another
individual's position not in terms of his own attitudes and tastes but
rather in terms of the attitudes and tastes of the individual actually
holding this position' (1955, fn. 16, in 1976, p. 22; own emphasis). The
point was reiterated, though somewhat differently, in (1977a, p. 52).
Following these very clear suggestions, we conclude that, for Harsanyi,
the observer's ignorance must extend to i's subjective features, including
his preferences, and not only to the usual objects of individual preference
comparisons, such as money incomes or consumption levels.
Enough has now been said to formally separate Vickrey's version
from Harsanyi's. To model the Vickreyan observer let us suppose, very
simply, that each individual i1;...;ncompares, in terms of his actual
preference ordering, the distribution vector xwith an equal chance of
receiving any of the ncomponents of x, and the vector ywith an equal
chance of receiving any of the components of y. Formally, consider a
finite (the assumption is for convenience) set of possible income levels A
and construct the set Xof conceivable income distributions by assigning
an income level to each individual in all possible ways. Define Ato
be the set of all lotteries (i.e., probability measures) on A. We single out
for consideration the equiprobable lotteries, that is, for any
xx1;...xn2X,
Lx1=nx1;...;1=nxn:
Each individual i1;...;nis endowed with von Neumann±Mor-
genstern (VNM) preferences ion A, so that L
x
can be evaluated as:
viLx1=nv
ix1...1=nv
ixn
for some VNM utility representation v
i
of the individual preference.
7
We
routinely identify outcomes in Awith sure lotteries, so that we may use
the same symbol v
i
on the right- and left-hand sides. The expression
`VNM utility function' will refer sometimes to the utility representation
on the given lottery set, sometimes to its implied restriction to the
outcome set. By assumption, each individual is also endowed with a
7
For a precise statement of von Neumann and Morgenstern's axioms and the ensuing
representation theorem, the reader may consult Fishburn (1970). His treatment also
provides background material for the subjective probability variant of Section 6.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 153

moral preference * ion X; it will give rise to utility representations w
i
.
Now, Vickrey's version of Equal Chance is:
(*) for all i, and all x,yin X,x*iyiff L
x
iL
y
We also need a notion of actual preference Aion income distribution
vectors. We introduce it here by following Vickrey's implicit assumption
that the individual only cares about his own component in each
distribution vector:
(**) for all i, and all x,yin X,xAiyiff x
i
iy
i
.
Then:
Vickrey's Impartial Observer Theorem: If all individuals ihave
identical VNM preferences on the lottery set A, there exist common
utility representations uand wof, respectively, the individuals' actual
and moral preferences on X, such that for all x2X,
wx1=nn
i1ux:
This is indeed a utilitarian formula, but it has been obtained, quite
trivially, by assuming the individual von Neumann±Morgenstern prefer-
ences to be uniform. As Pattanaik (1968) has emphasized in discussing
Vickrey (1960), the assumption is hard to accept. Individuals actually
entertain varying risk attitudes, and moving to the normative side, there
appears to be no reason why they should not. Harsanyi's version avoids
this restriction, though at a price, as we will shortly see.
There is another, perhaps deeper, reason for taking leave of Vickrey's
version and moving to Harsanyi's. It is dubious that an individual who
retains his own preference to assess social states ± even not knowing his
position in the social state ± manifests impartiality or impersonality to an
extent sufficient to ground a moral judgement. The Impartial Observer
should step outside himself. He should take account of his own interests
no more and no less than if he were another individual. It follows that
one's actual VNM preferences must not be used directly to assess social
states morally. Axiom (*) has no ethical grounding. In Harsanyi's deeper
construction, a new preference concept ± extended preference ± will
mediate between moral preferences and actual VNM preferences. The
critical point against (*) somehow gets lost because of Vickrey's
uniformity assumption. Had he not made this unpalatable restriction, it
would have become obvious that his axiom was too blunt. Vickrey's
confusion between actual and ethically relevant assessments is probably
facilitated by the fact that under standard economic assumptions,
8
a
utilitarian formula with identical utility functions in the sum automati-
cally recommends equality of income; so that it does not matter for the
8
I. e., diminishing marginal utility and a well-behaved domain for the utility function.
154 PHILIPPE MONGIN

conclusion what utility function is chosen. This independence of the rule
from the specific utility function is limited to a highly particular case.
9
4. EXTENDED PREFERENCE AND HARSANYI'S IMPARTIAL OBSERVER
THEOREM
To formalize Harsanyi's version, one might want to borrow from the
following, independently developed construction of social choice theory.
After Arrow (1963, pp. 114±15), various writers in social choice theory
have discussed `extended sympathy', that is to say, judgements of the
following sort: Alternative xis better (or worse) for individual ithan
alternative yfor individual j. Arguing that it is possible and meaningful
to make such judgements, Suppes (1966), Sen (1970, Chapter 9), Kolm
(1972), Arrow (1977), Suzumura (1983) and others have formally
elaborated the extended preference approach, which will be extensively
employed here. This approach endows each member of the society iwith
both an actual preference relation defined on some set Xof alternatives
(social states), and a relation defined on suitably modified alternatives
(x, j), to be interpreted as `to be in social state xand in the position of j'.
That ican rank these `extended alternatives' may be a light or heavy
assumption, depending on how the individual's `position' is construed.
If it refers to the individual's position in the income distribution, or such
similar objective features, there is perhaps nothing very problematic
about it. At the other extreme, the `position' may be construed so as to
include all subjective features of the individual. Reading (x, j) in the
latter way, extended preference allows for numerous interpersonal
comparisons ± to wit, iis able not only to compare what it means for jto
be in xand to be in y, but even to compare what it means for jto be in x
and what it means for kto be in y. To illustrate the wide range of
meanings of `extended preference', recall Sen's (1970, pp. 149±50)
example in which jis a devout Muslim and kis a devout Hindu, while x
and yare the states in which the individual, whoever he is, eats pork or
beef, respectively. Sen's discussion was literally concerned with Suppes's
principle of justice, which involves only the weak construal of the word
`position'. However, this well-known example can also be ± and has been
± read in accordance with the strong construal. Then, it says in effect that
9
In view of the criticisms in this section, it seems permissible to disagree with the editors of
Vickrey's Selected Papers when they claim that, `as early as 1945 he sketched the basis of
modern utilitarianism, later developed by Harsanyi' (1994, p. 5). Without the crucial step
of endowing the observer with a special preference concept, Vickrey's VNM analysis has
no ethical significance. Vickrey's editors also credit him for having `not only the germ but
the whole idea' of Rawls's original position (1994, p. 14). Again, this seems to be an
overstatement.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 155

ishould be able to decide whether it is better or worse to break Hindu
law while participating in the personal features of a Hindu (hence,
adhering to Hindu law) than to break the Muslim law while partici-
pating in the features of a Muslim (hence, accepting Muslim law).
Once extended preference is introduced in a social choice theory, it
must somehow be connected with actual preference. The usual linkage
in the literature is that extended preference judgements should conform
with actual preference judgements whenever this is possible, that is, any
time it comes to comparing alternatives of the form (x, j) and (y, j) for
some given j. We call this principle Consideration Of Others (CO). In the
existing literature it is sometimes called the `acceptance principle', and it
is typically defended on the normative grounds of either `nonpatern-
alism' or `consumer's sovereignty'.
10
Extended preference is a natural tool for modelling the impartial
observer. Harsanyi's strong assumption on the content of the observer's
ignorance calls for a correspondingly strong interpretation of this
observer's `position'. This still leaves open various theoretical possibi-
lities, which we may classify as follows: (i) the second variable in (x, j)is
the name of individual j; (ii) it refers to properties that j's preferences
satisfy; (iii) it refers to causal factors which determine j's preferences.
The distinction between the first interpretation and the other two should
be clear: it is not the same to name `John' and to list properties or factors
bearing on John. The latter might provide a way of referring to John, but
it would be a roundabout one, and arguably, an unsatisfactory one.
There is a sense in which John's identity exceeds any description of John
in terms of abstract features. The distinction between the last two
interpretations is more elusive, if only because preference theory is in an
unsettled state and specialists do not always agree on causal imputations
in this area. But this distinction can at least be exemplified. As a relevant
property of j's preferences, take his index of risk-aversion; as a possible
cause for the value of his index, consider his wealth (since, according to
standard theory, wealth influences risk-attitudes).
There may be a further problem with the proper way to understand
(iii).
11
On the first reading, the second variable refers to those causal
factors which bring it about that there is an individual jhaving the
preferences he has. On the second reading, it refers to those causal
factors which bring it about that jhas the preferences he has, and it is
then implied that jcould have had other preferences while still being j.
We take the second reading as allowing also for the possibility that k
(different from j) could be subjected to the same causal factors as j, and
therefore have the same preferences as j, while still being k. The second
10
See, e.g., Sen (1970, p. 156).
11
A problem of this sort was pointed out by Isaac Levi in correspondence.
156 PHILIPPE MONGIN

reading offers more flexibility than the first. Essentially, it assumes that
j's and k's identities can be defined independently of what their
preferences are. We will take it for granted that it provides a suitable
interpretation for (iii).
As we read it, the bulk of the extended preference literature is
concerned with extended alternatives in the sense of either (i) or (ii), and
sometimes ambiguously so.
12
It is conceivable to reconstruct the
Impartial Observer Theorem on the basis of these particular interpreta-
tions.
13
But neither of them is Harsanyi's. In (1977a, pp. 58 ± 9), he clearly
differentiates between an individual's `subjective attitudes (including his
preferences)' and `all the objective causal variables needed to explain
these subjective attitudes', and he goes on to suggest that extended
preference is best understood in terms of the latter. In keeping with this
important comment, Harsanyi's (1977a, pp. 53±5) formal notion of
extended preference, as well as the corresponding restatement of the
`theorem', should be read with interpretation (iii) in mind. That is, for
each observer i, to make an extended preference judgement amounts to
comparing, `to be in social state xand under the influence of the factors
determining j's preferences', with `to be social state yand under the
influence of the factors determining k's preferences'.
Let us call this version of extended preference the causal one.
Harsanyi propounds it because it seems to convey effectively the notion
that interpersonal comparisons of preferences can be objective in nature.
The latter claim is recurrent in his work. He made it forcefully long
before he thought of employing the extended preference apparatus. In
his 1955 paper, where for the first time he introduced the famous
distinction between two kinds of preferences (`subjective' and `ethical',
to be later called `empirical' and `moral'), Harsanyi discussed at some
length interpersonal comparisons of utility. He claimed that they could
be predicted from earlier psychological data :
If two individuals have opposite preferences between two situations, we
try to find out the psychological differences responsible for this disagree-
ment and on the basis of our general knowledge of human psychology, try
to judge to what extent these psychological differences are likely to increase
or decrease their satisfaction derived from each situation. (1955, in 1976,
p. 17)
14
12
Compare this interpretation with Suzumura's (1983, pp. 133±6).
13
As perhaps Sen (1970, p. 150) had meant to suggest.
14
There is an interesting and little noticed connection between Harsanyi's long-standing
view that interpersonal comparisons are objective and his construction of the type
concept in his pioneering work on games of incomplete information, see Mongin and
d'Aspremont (1998). The game-theoretic variant of the Impartial Observer Theorem
proposed by d'Aspremont and Ge
Ârard-Varet (1991) makes direct use of the concept of a
player's type.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 157

The only significant difference between this early formulation and the
later ones in Harsanyi (1977a) is that the former does and the latter do
not employ the language of extended preferences. Clearly, Harsanyi
came to think that this notion would help him to convey his long-
standing conviction more strongly and convincingly, especially among
economists.
Having motivated the principles, we now introduce them axiomati-
cally. Denote the initial alternative set Xand the set of all individuals by
Nf1;...;ng. Then, XNis the set of extended alternatives. We also
introduce the set XNof extended lotteries, among which are the
extended equiprobable lotteries, that is: For any x2X,
Lx1=nx;1;...;1=nx;n:
Each individual i1;...;nis endowed with extended preferences E i
on XNsatisfying the VNM axioms, so that L
x
can be evaluated as:
viLx1=nv
ix...1=nv
ix
for some VNM utility representation v
i
of i's extended preference.
Now, each individual is also endowed with an actual and a moral
preference relation on X, respectively, iand * i. (Since there is no
possible confusion now, we drop the index A in the symbol of actual
preference Ai:Utility representations for these preferences will be
denoted by u
i
and w
i
, respectively. Harsanyi's Equal Chance principle
connects the moral with the extended preference, and is formally stated
like Vickrey's:
EC for all i, and all x,yin X,x*iyiff L
x
EiL
y.
Harsanyi's Consideration of Others principle connects extended with
actual preference:
CO for all iand j, and all x,yin X,(x, j)Ei(y, j) iff xjy.
Finally, we introduce a principle ± Uniformity Of Extended Preference
(UEP) ± which was not part of the earlier motivations. It says that
extended preferences are the same from one observer to the other:
UEP for all i,j, and any two lotteries l,l' in XN,lEil' iff l
Ejl'.
Harsanyi's Impartial Observer Theorem: Assume that for each
individual, the extended preference relation satisfies the VNM axioms,
and that EC, CO and UEP hold. Then, there exists a common utility
representation wof the individuals' moral preferences on X, such that:
for all x2X,
wx1=nn
j1ujx;
158 PHILIPPE MONGIN

where u1;...;unare utility representations of the individuals' actual
preferences.
The present statement is mathematically crude, but it strips Harsa-
nyi's `theorem' to its logical bare bones. It serves especially to highlight
the role of UEP. Without it, the conclusion does not hold, and we only get
that for each i, there exists a moral utility function 1=nn
j1uij, where u
ij
is a utility representation for j's preference depending on the particular
observer i. Harsanyi's own formal exposition (1977a, pp. 54 ± 5) may
suggest that the other three assumptions are sufficient to derive a set of
utility representations obeying the mean rule of utilitarianism. This is
not the case. Relevant details are provided in the appendix.
This (easy) logical point being granted, one should ask whether or
not UEP is conceptually justified. Is it not the case that extended
preferences can unproblematically be taken to be uniform from indivi-
dual to individual? Before moving on to this and related discussions in
Section 5, we record another problem that should be raised in connection
with the above formalization. The statements just given of Vickrey's and
Harsanyi's `theorems' are purely existential. They do not include any
uniqueness restriction on the derived utility representations wand u
j
,
j1;...;m. In fact, a glance at the proof in the appendix shows that there
are other choices than these representations that are compatible with the
axioms, but do not deliver the desired additive representation. The
functions wand u
j
,j1;...;m, have been selected among many to make
the utilitarian-looking formula come right. Prima facie, this casts a doubt
on the ethical significance of the Impartial Observer Theorem. This point
has been made clearly and forcefully by Weymark (1991) in connection
with his own formalization of the Impartial Observer Theorem.
15
More specifically, we may consider replacing the VNM indexes u
j
by
nonlinear transforms of them, thus destroying the additive form of the
social rule. Sen (1986) had argued that since this can always be done,
Harsanyi's two theorems, namely, the Impartial Observer Theorem and
the no less famous 1955 Aggregation Theorem, are only superficially
connected with utilitarianism. A good deal of Weymark's (1991)
reexamination of Harsanyi's work amounts to endorsing this objection. It
is true that the preference relations of traditional theory are purely
ordinal; so, if they are taken to be the only primitives, there can be no
conceptual reason for selecting VNM indexes rather than any ordinal
transforms of them. If, however, the notion of preference is understood
in a different sense, the primitives may include a further relation to
represent the intensity of preferences. Then adding suitable axioms, it
becomes possible to restrict the range of permissible utility representa-
15
See Weymark's (1991) Theorem 9 and the comments following it.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 159

tions to the set of VNM indexes, and the Sen±Weymark objection can be
countered, at least at the logical level. We have discussed this line of
analysis elsewhere and will not pursue it here.
16
Even granting that an argument can be made for VNM indexes, the
choice of the particular VNM index clearly influences the symmetric
form of the additive rule. A non-uniform linear rescaling of the u
j
would
still deliver a VNM representation for each individual, with all the
axioms being satisfied, but with weights other than 1/n. Hence, even if
the Sen±Weymark objection is answered along the lines we suggested,
there remains a significant arbitrariness problem. This further problem
can be addressed within the confines of the present paper. In Section 6
we show how subjective probability theory can be put to use in order to
fix the weights in a non-arbitrary way. The resulting formula, however,
does not involve equal weights.
5. THE CAUSAL ACCOUNT AND UNIFORM EXTENDED PREFERENCE
Following the causal interpretation of extended preferences, (x, j)Ei
(y, k) should be read as:
(*) `individual iprefers to be in social state xand under the influence
of the factors determining j's preferences than to be in social state yand
under the influence of the factors determining k's preferences'. Compare
this reading with the more standard one in which jand kdirectly refer to
individuals:
(**) `individual iprefers to be jin social state xthan to be kin social
state y'.
Preference judgements of the form (**) are typically analysed by
saying that the observer iidentifies himself with j's personality, or at
least, manages to reproduce in himself j's preference attitudes. The
words `sympathy' and `empathy' have been used to cover these
psychological experiences. Something of this sort seems to be needed to
explicate (**). By contrast, the causal account underlying (*) does not
require the observer's entering a particular mental state. Here, the
process of identifying oneself with the other gives way to the process of
deducing what one's own preference would be under certain ideal
conditions. Both the causal and the more standard account in terms of
identification must be contrasted with the notion that i records j's choices
16
In Mongin (1994) and, in more detail, in Mongin and d'Aspremont (1998) the possibility
of cardinalizing the preference relation constitutes the gist of an argument in favour of
Harsanyi's two theorems. Recently, Harvey (1999) has made this possiblity more concrete
by formally reconstructing the Aggregation Theorem in terms of a cardinal preference
relation. Note carefully that this is not the same reconstruction as that which makes
cardinal utilities the primitives in Harsanyi's analysis.
160 PHILIPPE MONGIN

in his extended preferences. A choice-based interpretation along this line
could perhaps be all right as long as (x,j) and (y,j) are compared, but
there is no way of extending it to comparisons between (x,j) and (y,k).
Most writers on extended preferences use sympathy and empathy
interchangeably. If one is concerned with distinguishing between the
two,
17
Harsanyi's account must definitely be put on the empathy side.
Roughly speaking, sympathy has to do with the observer's ability to be
affected in his own welfare by another's situation. Empathy is rather an
ability to understand another person, possibly by swapping places with
him and reproducing his experiences in oneself, but possibly also by
purely deductive means. Empathy does not imply benevolence, just
interest. The outcome of the empathetic exercise is a piece of knowledge
that the observer (say, the historian, but it may also be the man in the
street) can use for any purpose of his own, and not necessarily for the
benefit of the observed individual. On this account, and barring several
textual vacillations, Harsanyi would exemplify the intellectual extreme
of the empathy concept. The only feature of sympathy that is important
for him, as for any other extended preference theorist, is that by CO,
the observer positively correlates some of his preference judgements
with those of the observed person. In this limited sense, the impartial
observer of social ethics, including Harsanyi's, is also sympathetic.
How the correlation between two sets of judgements is obtained is a
matter different from, and conceptually prior to, the direction of
correlation. To reiterate, in Harsanyi it is an intellectual empathizing
process that brings about the positive correlation. To decide whether it
is better to be rich while being under the influence of expensive tastes
or to be poorer while being under the influence of more economical
tastes, the observer will use his knowledge of psychological laws and
deductive ability.
A definite advantage of the causal account is that it may save the
impartial observer theory from hazardous discussions of personal
identity. Typically, those who understand extended comparisons in
terms of one's identification with another's attitudes have stumbled on
the following problem: how much of the observer's identity is preserved
by sympathetic identification or by empathetic identification of the non-
deductive sort? Is there enough left, as it were, to warrant the claim that
it is the observer who makes the preference judgement? The point has
been put forward that if imust effectively enter j's or k's mental state to
make extended preference judgements, it cannot be i, after all, who
17
As Fontaine (1997) usefully does; we are indebted to him here. A majority of writers on
the Impartial Observer Theorem do not attempt to distinguish between `sympathy' and
`empathy'.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 161

makes them. There would be something self-destructive in the way
identification works.
18
A thought-provoking claim like this depends on a
detailed conception of personal identity, and not just on the way personal
identity relates to preference maps. Luckily, Harsanyi does not have to
delve into these deep waters of metaphysics. Following formula (*), the
observer's preferences are plainly this observer 's preferences. Harsanyi
would simply have to make it clear that he understands preferences
broadly enough. The observer's preferences are considered preferences,
and they are defined over objects which in the particular instance are
states of affairs, not objects of choice ± a clear departure from `revealed
preference' theory.
If in (*) we take j=k, we get the following class of extended
preference judgements:
'individual iprefers to be in social state xand under the influence of
the factors determining j's preferences than to be in social state yand
under the influence of the factors determining j's preferences'. It is the
class to which CO applies. The axiom states that the previous statement
must hold if and only if:
'individual jactually prefers to be in social state xthan to be in social
state y'. This equivalence can only be a matter of stipulation, not of
logical necessity. Harsanyi, who calls CO `the principle of acceptance' or
of `consumer's sovereignty' defends it as follows: `it requires us to accept
each individual's own personal preferences as the basic criterion for
assessing the utility (personal welfare) that he will derive from any given
situation' (1977a, p. 52). Essentially, CO plays the same role as the Pareto
principle does in standard welfare economics and in Harsanyi's
Aggregation Theorem. Even granting the `qualifications' that he is
willing to introduce in respect of both CO and the Pareto principle, both
are surrounded with normative difficulties. These difficulties are neither
new nor specific to the causal interpretation adopted here for extended
preferences. So we gloss over them.
The benefits that Harsanyi (1977a, p. 58) hopes to reap from his
causal account have to do exclusively with axiom UEP. Using the
language of utility functions rather than preferences, he claims that the
causal account implies an identical utility function for all observers. The
rest of this section is devoted to examining this claim. We will argue that
extended preferences cannot be independent of the particular observer.
We will salvage a version of the claim in which utilities stand for
themselves, without representing preferences.
18
This claim is Rothenberg's (1961) main objection against Harsanyi, and it also arises in
Kaneko (1984) and McKay (1986). Pattanaik (1968) expresses his disagreement with
Rothenberg but does not really provide an alternative account.
162 PHILIPPE MONGIN

To clarify the locus of the disagreement, our objection against
Harsanyi does not relate to his extraordinary assumption that there exist
psychological laws sufficiently precise to deliver predictions on the
individuals' dispositions, and that all observers both know these laws
and can make the correct inferences from them. Harsanyi himself is
willing to admit that his assumption is, indeed, far-fetched. But as we
read him, he would consider a failure of this assumption to be the only
reason why extended preferences might after all differ from one observer
to another. Differences among extended preferences would be comple-
tely explained by the primitive stage of psychological knowledge. They
would not count as differences in principle. But we will argue that there
are other reasons for doubting the conceptual validity of UEP, and these
reasons are of a permanent sort.
19
Take two observers iand h. On the face of it, the statement that:
x;jEiy;kif and only if Ehy;k
follows automatically neither in interpretation (*) nor in interpretation
(**) ± except, of course, whenever j=kand CO is granted. Think of a rich
man with expensive tastes and a poor man with economical tastes.
Suppose that they are clever and knowledgeable so that they fully
understand what it means to be rich with expensive tastes and poor with
economical tastes. There is nothing in the theory of extended preference
to exclude that these two individuals wish to swap their positions and
subjective features. In a (doctored) version of La Fontaine's famous fable,
the Banker would prefer to be poor and lighthearted, like the Shoemaker,
and the Shoemaker would prefer to be rich and worried, like the Banker.
Or take a Londoner who would like to live in Paris with the cultural
tastes of a Parisian, while the Parisian would like to live in London, with
the taste for success of a Londoner. There is nothing contradictory in
these imagined situations.
But perhaps we are relying here on the ordinary interpretation (*),
and the causal interpretation (**) would deliver a different conclusion.
Let us try and give it another chance. Underlying the causal account is
the thesis that all ordinary preference judgements are causally deter-
mined and all observers make full use of their supposedly complete
knowledge of causes. Accordingly, the following claim is also part of the
causal account:
19
Compare with Broome (1993) who also argues against Harsanyi (and a few others, like
Kolm, 1972) that extended preferences cannot be taken to be uniform from one individual
to another. Broome reaches his conclusion by emphasizing the distinction between an
object and a cause of preference, and we will proceed differently here, emphasizing
instead the distinction between preference judgements and utility amounts. Broome's
argument has apparently failed to convince Kolm (see his 1994 answer). Hopefully, the
present argument will.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 163

`All individuals ican predict which of xand yindividual jwill prefer'.
With some effort, one can devise an analogue of this claim for extended
comparisons involving different observed individuals jand k, as would
be required in order to defend UEP:
(+) `All individuals ican predict whether xsatisfies j's preferences
more than, less than, or equally as ysatisfies k's preferences'.
We discard the difficulties introduced by comparisons of the degree,
or the extent, to which the preferences of distinct individuals can be
satisfied ± a notion needed in order to state (+) ± since there is a simpler
critical point to make. Predictions relative to preferences, be they for one
observed individual or for two, do not normally have the form of
preference judgements. It would be fanciful to claim that (*) and (+) are
synonymous. Still, some readers might overlook the difference between
a preference judgement and a prediction of a preference judgement, and
incorrectly conclude that the preference judgement in (*) is unanimous,
starting from the correct understanding that in the causal account,
predictions of (+) are unanimous.
Let us try to exploit the causal account in yet another way. Statement
(*) refers to a preference judgement which, by the causal account or a
plausible extension of it, must also be causally determined. It follows
that any two observers, say land m, will be able to predict (*) identically
from their prior knowledge of the causes of this judgement. So the
following holds:
'All individuals lcan predict that iprefers to be in social state xand
under the influence of the factors determining j's preferences than to be
in social state yand under the influence of the factors determining k's
preferences'. However, identical predictions made on the same observer
iby different meta-observers are not at all what we were after.
Supposing that we could deduce it, a more relevant fact would be this:
all meta-observers make the same prediction for iand for any other
observer h. But the causal account does not warrant any conclusion of
the sort. It can be stretched to the point of saying that there are causal
factors underlying i's and j's extended preferences, not that these
factors are the same.
The previous hints are all that we could think of to defend UEP on
the basis of the causal account. But what about Harsanyi's own
defence? We do not think that he is confused to the point of reading a
preference into a prediction. The following important point has gone
generally unnoticed among commentators.
20
When Harsanyi really
20
An exception is Weymark (1991) at several points in his thorough commentary. In his
Theorem 10, Weymark exploits this observation to reconstruct the Impartial Observer
Theorem in terms of cardinal utilities taken as primitive concepts. He does not consider
its implications for the uniformity of moral assessments, as we do here.
164 PHILIPPE MONGIN

argues about extended preferences, as opposed to just mentioning them
in passing, or gesturing towards them, he replaces them with extended
utility assessments. It is in this language that he makes an argument for
uniformity:
the extended utility function v
i
should really be written as v
i
=v
i
(x,R
j
)...
Written in this form, the utility function indicates the utility that individual
iwould assign to the objective position xif the causal variables
determining his preferences were R
j
. Because the mathematical form of this
function is defined by the basic psychological laws governing people's
choice behavior, this function must be the same for all individuals. (1977a,
p. 58; we have adapted the notation)
Thus, utility values measure the individual's preference satisfaction,
given the objective variables (e.g., income) and subjective variables (the
individual's preference parameters) influencing it. By assumption,
subjective variables act causally in a stable and recognizable way. It
follows that the mapping from objective variables to numerical degrees
of satisfaction is perceived in the same way by all observers. So far so
good for this utility-based version of the causal account. But it does not
deliver uniformity of extended preferences, only uniformity of the
considered utility amounts. The statement that
vix;Rjviy;Rk
might well hold for all iwithout representing any observer's preferences.
Why should it? More utility can be obtained in xunder causal
circumstances R
j
than in yunder causal circumstances R
k
. This does not
imply that there is anybody who prefers, or should prefer, the first
extended alternative to the second.
But is it not possible to improve on Harsanyi's utility-based
argument? There is an obvious connection between utility and
preference which we have not yet exploited, namely, the utility amounts
v
i
(x,R
j
) and v
i
(y,R
k
) represent, respectively, the degree to which x
satisfies j's preferences, and the degree to which ysatisfies k's
preferences. (There are difficulties with any conception which is
concerned with social ethics, and takes utility numbers to represent
degrees of satisfaction rather than well-being. But we are keeping as
close as possible to Harsanyi's own argument, and thus we by-pass these
difficulties here.) Accordingly, we may couch the statement that
vix;Rjviy;Rkhold for all i
in the language of preference, and reach the statement that:
(+) `All individuals ican predict whether xsatisfies j's preferences
more, less, or equally than ysatisfies k's preferences'. We have already
encountered this statement. It leads nowhere. We conclude that there is
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 165

no way of bridging the gap between Harsanyi's argument in terms of
utilities and axiom UEP.
This paper is concerned with the Impartial Observer Theorem, not
with extended preference and extended utility theories in general, but it
might help to reinforce the critique of this section if we briefly discuss
writers other than Harsanyi.
21
Consider again Arrow's definition of
`extended sympathy' in 1963 (pp. 114±15). Literally understood, it does
not deal with preference judgements, but with comparisons of indivi-
duals' well-being. It also seems clear that Kolm (1972) was concerned with
well-being across individuals.
22
Let us formulate the well-being interpreta-
tion of extended comparisons explicitly, in order to contrast it with (*), (**),
and (+):
(***) `jin social state xis better-off than kin social state y'. If extended
comparisons are explicated by this statement, it is possible to take the
further step that they are uniform from one observer to the other.
Essentially, the argument for this conclusion will consist in saying that
well-being is an objective concept and devising a causal account for it.
Although the objective perspective is only sketched in Arrow, Kolm
and others, it is the most promising line for these writers to pursue. They
sometimes come closer to making a different argument, which bears a
definite resemblance to Harsanyi's and fails for the same general reason.
They would no longer regard (***) as providing a definition of extended
comparisons. Rather, they would construe them as being preference
comparisons of some sort, and they would conclude from the alleged
uniformity of these extended comparisons that well-being comparisons
like (***) make good sense. The argument differs from Harsanyi's version
of the `theorem' in several respects. For one, Arrow, Kolm and most
writers on extended comparisons take these comparisons to be only
ordinal, whereas Harsanyi needs cardinal comparisons for his utilitarian
conclusion to hold. For another, these writers are not so much interested
in deriving a fully-fledged social rule as in justifying a relevant class of
interpersonal comparisons. Finally, they are trying to reach well-being
conclusions, as we have just said, whereas Harsanyi is best interpreted
as a preference utilitarian. This said, the argument falls prey to the same
objections as Harsanyi's version of the `theorem'. It would be nice if
comparisons of well-being could be deduced from some universally
shared set of preferences, but there is no argument available to provide
the desired set of preferences.
To return now to the Impartial Observer Theorem. Taking the notion
of an extended preference at its face value, and because (UEP) is
21
Compare again with Broome (1993).
22
Kolm confirmed this to us. His answer (1994) to Broome is also rather explicit on this
score.
166 PHILIPPE MONGIN

indefensible on this reading, one is left with the following weaker
version:
The `Many Impartial Observers' Theorem: Assume that for each
individual, the extended preference relation satisfies the VNM axioms,
and that EC and CO hold. Then, for each i, there exists a representation
w
i
of his moral preferences on X, such that: for all x2X,
wix1=nn
j1uijx;
where uil;...;uin are utility representations of the individuals' actual
preferences.
6. EQUAL CHANCE
On a preliminary reading, Equal Chance is the axiom which in the
Impartial Observer Theorem, corresponds to the informal idea of
impartiality, while Consideration of Others would correspond to sym-
pathy. This reading has become popular among Harsanyi's followers. In
our interpretation there is no such simple pairing of informal ideas with
axioms. If sympathy is anywhere, it is in CO, but the requisite of
impartiality permeates the whole set of axioms, the arrangement of
which is determined by the following powerful philosophical idea:
Impartiality must be analysed as a mode of ignorance. Viewed in this light,
EC is not a direct rendering of impartiality, but a step in an argument
about impartiality. It is the connecting link between the informal notion
of ignorance and the application of expected utility theory that
eventually drives the conclusion. Accordingly, EC should not be seen as
an equal treatment ± that is, normative ± axiom. It is an epistemic axiom ±
in effect, Laplace's principle of dealing with complete uncertainty in
terms of equiprobable states of nature. Once this is accepted, important
consequences follow, as this section will try to make clear.
Bayesian writers have always been divided on the significance of
Laplace's principle. In case of ignorance, some ± especially in econo-
metrics ± take `diffuse priors' to be the starting point for Bayesian
updating. But others are content with saying that priors are whatever
they are. Neither the Dutch Book argument a
Ála de Finetti, nor the more
sophisticated axiomatic constructions by Savage (1954), Anscombe and
Aumann (1963) or Jeffrey (1983), imply restrictions on the particular
probability measures they derive. They just imply that there is a
probability measure and that it is unique in some technically well-
defined sense. When it comes to a multi-agent context of ignorance, as in
the theory of games of incomplete information, Bayesians disagree on
the question of whether or not there is a `common prior'.
23
Foundational
23
In a classic series of papers Harsanyi (1967±8) proposed to base the very notion of a game
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 167

issues of this sort should arise in any discussion of the Impartial
Observer or related notions of the `original position', as Rawls (1971,
pp. 171±2) pointed out with respect to Laplace's principle. Without
pursuing these complex issues in detail, we may stress that Harsanyi
cannot avail himself of Bayesianism tout court, and that there is a
plausible alternative within the confines of the doctrine, which replaces
EC with the assumption of subjective probabilities, one for each observer
i, whatever these probability measures are. In keeping with the spirit of
axiomatic Bayesianism these measures should be inferred from ante-
cedent preference conditions rather than taken for granted, and they
should be uniquely determined by the inference process. The following
construction is based on the Anscombe±Aumann version of axiomatic
Bayesianism which turns out to be specially easy to relate to the
Impartial Observer Theorem.
24
We will introduce, and actually need for the proof, more structure
than in the previous, more elementary versions. Let us replace the
nondescript set of alternatives Xby X1...Xn, where the X
i
are
(finite) personalized sets of outcomes. Corresponding to each of these,
there is a personalized lottery set YiXi;i1;...;n.Asocial state,
which will be the object of moral evaluation, is defined to be a mapping
f:N!Y1...Ynsuch that for all i,fi2Yi. That is to say, a social
state is an assignment of a personalized lottery to each individual; denote
by f(x
i
,i) the probability value given by fto the event of i's getting
outcome x
i
. Thus, there are two added structural features compared with
our earlier framework: outcomes are now individual-specific, and
society assigns to each individual chances of getting these outcomes
rather than a single outcome. One may, if one wishes, cancel the first
feature while preserving the second, which is the technically important
of the two. Then, take X1... XnA, for example, a set of mfeasible
income levels, as in the above formalization of Vickrey. The set Lof social
states can be redescribed as the following set of vectors:
Lff2Rmnjx2Afx;i1;8i2Ng:
For each individual iwe introduce a subjective preference relation Si
over L. It is this relation ± instead of the extended preference relation
Ei± that we will now make the basis of the individual's moral
preferences. Hidden behind subjective preference judgements, there is
one subjective probability measure for each i, which it is the purpose of
the construction to deliver explicitly. These measures will replace the
of incomplete information on the `common prior assumption'. Some game theorists have
come to question this familiar assumption.
24
This technical observation was made, but not yet put to use in Mongin and d'Aspremont
(1998, p. 455). Meanwhile, it has been exploited by Karni (1998) in an interesting way; see
the appendix.
168 PHILIPPE MONGIN

uniform measure of the more standard version. Accordingly, EC will be
substituted with the following Subjective Probability (SP) axiom:
SP for all i, and all f,gin L,f*igiff fSig.
Notice that Lis a convex set. As this is mathematically possible, we
submit the relations Si(or equivalently, * i) to the VNM axioms.
The notion of a conditional subjective preference (Si)
j
will play an
important role in the sequel. Intuitively, it is a restriction of i's subjective
preference relation to individual j's personalized lotteries, disregarding
what happens to the other individuals. Formally, it is defined from Si
as follows. For all iand j,
f(Si)
j
giff f' Sig' for all f',g' such that f'(j)=f(j) and g'(j)=g(j),
and fand gcoincide with each other outside j.
We need to connect subjective with actual preferences in the same
way as we previously connected extended with actual preferences.
Hence the following version of Consideration Of Others (CO'):
CO' for all iand j, and all fand gin L,f(Si)
j
giff fjg.
In the present variant, Consideration of Others has become a direct
implication of the Strong Pareto principle of social choice theory. It says
that if two social states differ only by the outcomes of some individual,
one is subjectively preferred to the other if and only if it is preferred by
that individual.
25
Given CO' our previous VNM assumption guarantees that the actual
preference relations ialso satisfy the VNM axioms. The further
condition we impose on the iis the mild one that they are nontrivial:
NT for all i, there exist f
i
,g
i
in Lsuch that fi>ig
i
(where as usual, >iis the strict preference relation derived from i).
We still need a notion of extended preferences. The relations Ei
are now defined on the set ' of all lotteries on extended alternatives of
the form (x
j
,j) where xj2Xjand j2N. As before, extended preferences
will be assumed to satisfy VNM. Conceptually, they serve a different
purpose than before. The Eirelations now provide a quantitative
benchmark for identifying the subjective probability measures p
i
that
underlie the Si. The idea of comparing preference assessments of
lotteries whose probability values are numerically given with preference
assessments of prospects which do not involve preassigned probability
numbers constitutes the distinctive feature of the Anscombe±Aumann
approach to subjective probability (as opposed, say, to Savage's). We
implement this heuristics by connecting Siwith Eiin the Probabil-
25
Given the equivalence mentioned in this statement, preference can be understood either
weakly or strongly.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 169

istic Consistency (PC) axiom below.
26
Essentially, it says that subjective
preferences conditional on jare identical with extended preferences
conditional on the same individual j, where the latter are defined in the
natural way (i.e., by disregarding what concerns the individuals other
than j).
More formally, we define conditional extended preferences from Eias
follows. For all iand j, and all land l' in ',
l(Ei)
j
l' iff Ei' for all ,' such that xj;jlxj;j,
0xj;jl0xj;jfor all x
j
, and and ' coincide with each other outside
the (x
j
,j) values. Then, Probabilistic Consistency states that:
PC for all iand j, and all l,l' in ' such that the marginal probability
distributions xj lxj;:and xj l0xj;:have full support,
lEijl0iff Hl
SijHl0;
where H(l)=l(., j)/
xj
l(x
j
,j) and H(l')=l'(., j)/
xj
l'(x
j
,j). The role of
the H mapping here is to turn the l(., j) and l'(., j) functions into
probability measures by normalizing them. Since the two conditional
preferences (Ei)
j
and (Si)
j
are not defined on the same mathem-
atical objects, we need this transformation in order to relate them to
each other.
Axiom PC says in words that whenever attention is restricted to
individual j's personalized outcomes, it does not matter whether the
observer assesses the chances of getting these outcomes in terms of his
extended preferences or of his subjective preferences. When the observer
disregards the possibility of being anyone other than j, the probability
numbers assigned by extended lotteries to the different individuals do
not matter any more, so there is no reason to expect comparisons based
on Eiand Sito be different. This is roughly the normative basis of
the PC axiom.
We can now state:
The Impartial Observer Theorem (Subjective Probability Version):
Assume that for each individual i, the subjective preference relation and
the extended preference relation satisfy VNM, and the actual preference
relation satisfies NT. Assume that SP, CO', and PC hold. Then, for each i,
there exist a representation w
i
of his moral preferences, a representation
v
i
of his extended preferences, representations u
i1
(on X
1
), ...,u
in
(on X
n
)
of the individuals' actual preferences, and a (full support) subjective
probability measure p
i
on N, such that for all social states f, and all
extended lotteries l:
26
This axiom originates in an early construction due to Karni and Schmeidler (1981). Karni
and Mongin (2000) have recently revived it. The interested reader is referred to the latter
paper for further philosophical motivations as well as technical details. See also
Schervish, Seidenfeld and Kadane (1991).
170 PHILIPPE MONGIN

(#) wifj2Nxj2Xj pijuijxjfxj;j
and
(##) bilj2Nxj2Xj uijxjlxj;j
Any alternative set of functions w
i;v
i;u
i1;...;u
in;p
isatisfying proper-
ties (#) and (##) must be such that for some a
i
>0 and (b
ij
)
j2N
u
ijj2Naiuij j2Nbij j2N;
p
ipi;
and w
iand v
iare positive affine transforms of w
i
and v
i
with a
i
being the
multiplicative factor of the transformations. (For a sketch of the proof,
see the appendix.)
Compare the utility representation derived for i's moral preference
with that of the `Many Impartial Observers' Theorem of Section 5, taking
account of the uniqueness properties stated here. The added value of the
present variant is clear. First, we have avoided EC by embodying a
subjective probability scheme into the ethical construction. Recall the
overall motivation: impartiality is to be interpreted epistemically. When
this is understood, EC appears to be doubtful, and gives way to SP and
PC. As to CO', it is, like CO, a purely ethical axiom. Notice that PC and
CO' together imply CO. The conclusion now is that the ethical observer i
assesses social states in terms of some probability measure p
i
. Following
the Bayesian tenet, it does not matter what values p
i
takes on. The only
important thing is that this probability is uniquely identifiable from j's
two sets of related preferences, as is spelled out in the uniqueness part of
the result.
Second, there is is a welcome by-product of the construction. Again
because of the uniqueness part, it overcomes the arbitrariness in the
choice of the u
ij
representations that plagued the previous versions of the
Impartial Observer Theorem. The functions u
ij
mentioned in the present
statement are nearly uniquely (i.e., up to common positive affine
transforms) those which `reveal' the subjective probability p
i
. They
cannot be rescaled arbitrarily without destroying this subjective prob-
ability, and thus the possibility of a Bayesian interpretation of the
individual's two preferences. The uniqueness of p
i
also means that the
individual weights in the utilitarian looking formula for w
i
are fixed once
and for all. Whether they are equal or not depends on the observer's
subjective beliefs, as summarized by p
i
. Thus, we have an answer to the
earlier objection (at the end of Section 4) that the choice of VNM
representations, or equivalently of individual weights, was an arbitrary
one. Weights are typically unequal, but there are reasons for that.
Returning to the point made at the beginning of this section, it is
common among Harsanyi's interpreters to match sympathy with a
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 171

version of CO, and impartiality with a version of EC. A reconstruction
along this line trades on the following notion of impartiality: to be
impartial is to treat the individuals equally. This is a normative, not an
epistemic notion of impartiality. It is coarse but commonsensical enough.
The reader would misunderstand us if he believed that by promoting an
epistemic reading of impartiality, we are dismissing impartiality as equal
treatment. Rather, we are dispensing with it. The trouble with axiom EC
interpreted normatively is not that it is wrong ± just that it begs too much
of the conclusion of the `theorem'. We end up with a reasoning that sounds
hardly like an argument for utilitarianism. We already knew that the
VNM assumptions went a long way towards begging the linear form of
the rule. We now see that EC begs the equal weights in the sum. True,
there remains at least one non-obvious step in the reasoning, which is to
base moral judgements on extended comparisons and CO. But extended
comparisons can only be justified in terms of ignorance. The ordinary
interpretation is saddled with the task of defending its own ignorance
scenario and making it plausible that ignorance and impartiality
(normatively understood) can be assumed for the same observer.
In sum, it seems both more rewarding and more consistent to derive
as much as possible from defining impartiality in terms of ignorance.
Once this line is taken, it quite naturally leads to unequal weights, and a
form of inequality appears to be justified in retrospect. The conclusion
may strike one as surprising or even unpalatable. At least, we do have an
argument now.
The Subjective Probability Version also runs counter to the desired
result that moral assessments are uniform. For the sake of theoretical
experimentation, we may superimpose UEP on the other axioms. The
result would be a set of representations of moral preferences of the form:
wifj2Nxj2Xjpiju0
jxjfxj;j;
where p
i
is i's uniquely defined subjective probability and u0
1;...;u0
nare
essentially unique utility representations of the individuals' actual
preferences that do not depend on i. This would be a `Many Impartial
Observers' formula again, the reason now being that the many observers'
subjective probabilities, not their views of the others' utilities, are
diverse.
There is an alternative plausible departure from EC that we may
eventually consider. It starts by arguing that objective frequencies are the
probabilistic data to rely on. This line takes us outside the confines of
standard Bayesianism. However, there is also a minority school of
`objective' Bayesianism, and such a label will perhaps not be unsuitable
for Harsanyi, given his double commitment to Bayesianism and an
objective construal of extended preference. According to the causal
account, the observer's state of ignorance is far from complete. The
172 PHILIPPE MONGIN

observer does not know his own preference features but is endowed
with all the required nomological knowledge about preferences. This
knowledge bears on the causal factors, like tastes or wealth, that
influence preferences. It is a further step, but one which is in line with
the causal account, to assume that the observer also knows something
about the empirical frequencies of the prevailing factors. These should
induce the probability values to be taken into account in the expected
utility formula. One can guess what the Impartial Observer Theorem
is that corresponds to this alternative assumption. If (UEP) holds, it
will lead to a utilitarian looking social rule independent from the
particular observer, but in which individuals have generally non-
uniform weights:
wxj2Nju0
jx
In this formula jis the (objective) probability that the causal factors
determining j's preferences obtain.
There is no reason to expect the objective probability values jto be
the same for each j. So once again, equal treatment does not emerge from
the analysis, and impartiality viewed epistemically diverges from
impartiality as it is more commonly envisaged. If we leave things at this
unelaborate stage, the earlier objection that the u'
j
are arbitrary will
destructively apply. To supersede it, we should embody `objective
Bayesianism' into a `revelation' subjective probability scheme in the style
of the previous one. Conceivably, this can be done, although we do not
know of any axiomatic work proceeding exactly along these lines.
7. WHAT IS LEFT OF THE IMPARTIAL OBSERVER THEOREM?
The Impartial Observer Theorem is surrounded with conceptual difficul-
ties. They can be put in the summary form of a succession of dilemmas.
Either the Impartial Observer Theorem is stated in Vickrey's limited
form, and it has no ethical interest. Or it is stated in Harsanyi's more
relevant form, and then it involves the difficulties of extended prefer-
ences. One version of extended preferences, which is not Harsanyi's,
requires one's identifying with the others, and leads to the puzzling
difficulties of personal identity. These difficulties may or may not be
insuperable, but the version under discussion cannot ensure the identity
of extended preferences. In the other account, which is Harsanyi's, and
which we called the causal account, one must be careful to decide
whether extended preference comparisons are preference comparisons
in the precise sense, or whether they are just a name for direct
comparisons of utility amounts by an observer. In the literal interpreta-
tion in terms of preference, Harsanyi's causal account is worthless as
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 173

regards delivering the conclusion that extended comparisons are
uniform. In the non-literal interpretation in terms of utility, the causal
account may lead to a claim that extended comparisons are uniform, but
comparisons are now of utility amounts, and this is not what the
Impartial Observer Theorem is concerned with.
The probabilistic discussion of the Impartial Observer Theorem
reinforces the negative conclusion that a genuinely utilitarian formula,
that is, with equal weights and no observer-dependence, is out of reach,
given the chosen primitives and premisses. In our main probabilistic
model, observer-dependence follows from the observer 's implicit reli-
ance on a subjective probability and the weights are unequal because
they are determined by this subjective probability. In another probabil-
istic model which was only sketched, observer-dependence is still
threatening on the utility side, and the weights are unequal, this time, for
objective reasons. The first model is that of standard Bayesianism; the
second model corresponds to an unelaborated theoretical possibility,
namely, `objective Bayesianism'. All and all, the probabilistic analysis
confirms Rawls's initial intuition that `there seems to be no objective
grounds in the initial situation for assuming that one has an equal
chance of turning out to be anybody. That is, this assumption is not
founded upon known features of one's society' (1971, p. 168).
Given this list of negative results, the argument can take two
different directions ± and, it would seem, only two. The first line consists
in taking the `theorem' to be about what it is overtly, namely, preferences,
and attempts to live with the fact that preference judgements are by their
very nature subjective, hence non-uniform. Then comes an interesting
argument due to Pattanaik (1968). By a reasoning of his own,
27
he comes
to the conclusion that the `theorem' could only lead to observer-
dependent social rules. Pattanaik's version appears to coincide with the
`Many Impartial Observers' Theorem (Section 5). At this juncture, he
suggests applying Harsanyi's other theorem, that is, the 1955 Aggregation
Theorem, to the many moral utility functions, in order to construct the
society's moral utility function. With the benefit of hindsight, Pattanaik's
suggestion strikes one as unconvincing. Since the Pareto principle is the
driving force in Harsanyi's other theorem, it means that the comparison
between diverse moral rules within the society is settled by unanimity
considerations. There are philosophical reasons for doubting that the
Pareto principle should apply to moral judgements in this highly direct
way. Besides, there are technical difficulties with Pattanaik's resolution
as soon as subjective probability enters the framework of the Impartial
27
As mentioned in Section 3, Pattanaik (1968) stresses the diversity of risk attitudes, hence
of VNM extended preferences, across individual observers. So his reasoning is identical
neither with Broome's (1993) nor with ours.
174 PHILIPPE MONGIN

Observer Theorem. In the Subjective Probability Version (Section 6), we
have obtained moral expected utility functions which can differ from
one observer to another both in terms of these observers' utility functions
and probability measures. But it has been demonstrated that Harsanyi's
Aggregation Theorem does not extend to the case of differing subjective
probabilities, even for weak forms of the Pareto principle.
28
So Patta-
naik's solution is blocked in this relevant version of the `theorem' ± a
difficulty that he could not foreshadow at the time of his comment.
The lesson of all this is that if the primitives are taken to be
preferences, there is no way out of observer-dependence. This is a
disappointing conclusion, but we should emphasize that along the
present lines, at least something can be proved. The Subjective Probability
Version of the Impartial Observer Theorem is neither entirely trivial nor
void of ethical content.
The second line of analysis is to draw the ultimate consequences of
Harsanyi's causal conception of interpersonal comparisons. We said that
the causal conception provided him with a reason to claim that extended
utility assessments are uniform across observers. Harsanyi's mistake was
to confound this statement with the non-equivalent one that extended
preferences are uniform. If the utility numbers are indicative of well-
being, one gets a roughly plausible idea of both extended judgements
and why they are uniform. One may wonder, however, if the present line
of analysis can deliver a theorem in the technical sense of a formal proof.
In the formalism, the common numerical assessments, rather than the
observers' extended preference relations, would be the primitives. Then,
the `theorem' could consist only in applying the expected utility formula
directly to these utility assessements, given some conceptually defensible
choice of probability values; objective frequencies would typically be
resorted to. The final result would be an observer-independent general-
ized utilitarian rule (i.e., with unequal weights). However, the argument
leading to it can only be philosophical, not mathematical. We now have
an observer who tries to assess social states in terms of the chances of
well-being that each of these states implies for him. Granting that well-
being can be quantified in the first place, a case must be made for using
objective frequencies as the relevant probability concept to model this
observer's ignorance. And a further argument is called for to defend
one's use of the expected utility formula, since it is not possible to invoke
the (preference-based!) VNM representation theorem. The second line of
analysis does not deliver an Impartial Observer Theorem in the genuine
sense of the word `theorem'.
28
For the finite state space of this paper, see Seidenfeld, Schervish and Kadane (1989) and
Mongin (1998).
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 175

APPENDIX
Vickrey's version of the Impartial Observer Theorem is mathematically
trivial, and so is Harsanyi's, but we give a formal proof of the latter to
highlight the role of the various assumptions.
Proof of Harsanyi's Impartial Observer Theorem. Take a VNM represen-
tation v
1
of Ei. For each i, CO implies that the restriction v
1
(., i)toXis
a utility representation of ion X. We put u0
i:v1:; ifor i1;...;n.
From UEP v
1
can also serve as a VNM representation of Eifor
i2;...;n. That is to say:
(#) for i1;...;n;v1LxviLyiff L
x
EiL
y
hence from EC:
for i1;...;n;v1Lxv1Lyiff x
*
iy.
Putting v
1
(L
x
)=w(x) for all x, we obtain the common representation of
moral preferences w. By the VNM property of v
1
, it satisfies that:
v1Lxwx1=nv
1x;1...1=nv
1x;n1=nu
0
1x...1=nu
0
nx;
as required.
If UEP is not part of the assumptions, step (#) is not permissible, and
the proof must be changed as follows. For each iand each j, CO implies
that the restriction v
i
(., j)toXis a utility representation of jon X.We
put u0
ij:vi:; jfor i,j1;...;n. Applying EC to each v
i
separately we
get an i-dependent representation of moral preferences:
viLxwix1=nu
0
i1x...1=nu
0
inx:
This is the `Many Impartial Observers' Theorem in Section 5.
In Weymark (1991, Theorem 9) and Karni and Weymark (1998), a
more assertive conclusion follows from stronger premisses. There, Xis
defined to be a lottery set, and the individual preference relations are
assumed to be VNM, with the result that the u'
i
in the additive
representation above are VNM functions. This ingredient is unneces-
sary to the formalization as long as the latter does not aim at
highlighting the cardinality issues underlying the choice of the u'
i
.
Notice that the `theorem' could receive an even more economical
formalization than the present one. By the VNM the extended
preferences are defined and satisfy the VNM axioms on the set of all
extended lotteries. This is too strong an assumption given the actual
use made of the VNM representation theorem in the proof. Restricted
domains may do, as Karni and Weymark (1998) have made clear in
their own framework.
It is clear that the proof above involves an element of arbitrariness,
that is, the choice of the VNM representation v
1
(., i) to represent all the
176 PHILIPPE MONGIN

relevant preference relations. Instead of putting u0
i:v1:; ifor
i1;...;n, we could put u0
i:fiov
1:; ifor i1;...;n, for any choice
of strictly increasing functions, and the axioms would still be satisfied.
The result would not be an additive representation, but only the
following additively separable representation:
wxf1ÿ1u0
1x  ...fnÿ1u0
nx:
The arbitrariness involved here is the essence of what we called the Sen±
Weymark objection in Section 4.
Here is a sketch of a proof for the Subjective Probability Variant of
Section 6. As a preliminary fact, observe that the set Lof social states is
convex because of the usual definitions of the sum of functions and the
multiplication of a function by a scalar, in terms of point values.
Sketch of the proof. Applying a variant of the VNM theorem due to
Fishburn (1970, Theorem 13.1), Sion Lcan be represented by:
(1) j2Nu
ijfj  j2Nxj2Xju
ijxjfxj;j;
where the u*
ij
are VNM representations unique up to positive affine
transformations involving a common multiplicative factor.
It follows that for all j,u*
i
j(f(j) represents (Si)
j
and that this
function is constant if and only if (Si)
j
is trivial. From PC, CO' and NT
this possibility is excluded.
A direct application of the VNM theorem to Eion ' entails that
this preference relation can be represented by:
(2) j2Nxj2Xjuixj;jlxj;j;
where u
i
is unique up to a positive affine transformation.
This implies that for each j,xj2Xjuixj;jlxj;jis a VNM representa-
tion of (Ei)
j
.
Now, apply PC to the VNM representations of (Si)
j
and (Ei)
j
,
respectively, that have just been obtained. It follows that for each j, there
are numbers a
ij
>0and b
ij
such that for all xj2Xj
u
ijxjaij uixj;jbij .
Renormalize the u*
ij
(x
j
) to set b
ij
= 0 for all j. Put pijaij=j2Naij and
uijxjuixj;jj2Naij . Then, given the uniqueness conditions for (1),
(#) wifj2Nxj2Xjpijuijxjfxj;j
represents Si, and given the uniqueness condition for (2),
(##) vilj2Nxj2Xjuijxjlxj;j
represents Ei.
THE IMPARTIAL OBSERVER THEOREM OF SOCIAL ETHICS 177

As to the uniqueness part of the Subjective Probability variant, it
follows from the uniqueness conditions for (1) and (2).
29
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29
It is instructive to compare the present variant with Karni's (1998) subjective probability
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observer. However, the major difference is that Karni's variant includes a special axiom
to the effect that the observer will give equal probability to each individual, so that the
social rule will have equal weights. Karni's own interpretation is on the normative rather
than the epistemic side, but one could also see his axiom as a preference analogue in the
subjective probability framework of Laplace's principle.
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