YEW-KWANG NG
INTERPERSONAL
LEVEL COMPARABILITY
COMPARABILITY
OF UTILITY
IMPLIES
DIFFERENCES*
It is generally believed that the possibility of making interpersonal comparisons of utility levels (as required, e.g., in applying the Rawlsian maximin
criterion) does not require interpersonal comparison of utility differences
(which involve cardinal utilities). Thus, if we are confined to level comparability, it is believed that no cardinal utility need be involved. Here, I wish
to show that, with some very general conditions, the general possibility of
interpersonal comparison of utility levels implies interpersonal comparison
of utility differences (or unit comparability, with suitable scaling). Thus, the
sufficiency of ordinalism is a mistaken belief.
Interpersonal comparisons of utility levels involve saying whether one individual is better or worse off than (or the same as) another individual in the
same or a different social state, e.g., UJ(x) > UK(x) or UJ(x) = UK(y). Interpersonal comparisons of utility differences involve saying whether the utility
difference between two social states for one individual is higher/lower than,
or equal to, the utility difference between the same two or some other two
social states for another individual, e.g., Ua(x)- Ua(y)> U r ( x ) - U~:(y),
or UJ(x)- UJ(y)= Un(z)- UK(w). 1 Obviously to be able to make such
comparisons of utility differences, we have to stick to a particular set of
cardinal measures of utility (one cardinal measure for each individual). The
utility index for each individual has to be unique up to a proportionate transformation (i.e., ratio-scale measurability, more cardinal than the NeumannMorgenstern index which is unique up to a positive afine transformation).
But level comparability has been taken as necessitating purely ordinal utility
indices unique only up to a positive monotonic transformation, i.e., coordinal
transformation.
The general belief that the Rawtsian maximin criterion involves only ordinal
but not cardinal utilities is quite widespread. The following quotations from
two of the most prominent writers in the area suffice to illustrate the point.
Theoryand Decision 17 (1984) 141-147. 0040--5833/84/0172-0141500.70.
9 1984 by D. ReidelPublishingCompany.
142
Y E W - K W A N G NG
Unlike the sum-of-utilities approach, however, this [i.e., Rawls's maximin criterion] does
not require that the units in which different individuals' utilities are measured be comparable, only that we be able to rank different individuals according to some scale of
satisfaction . .. the maximin criterion requires only interpersonal ordinality, whereas the
classical view requires interpersonally comparable units. (Arrow, 1973, pp. 253-254,
italics added.)
Rawls's (1971) 'maximin' conception of justice . . . deliberately avoids comparisons of
[utility] gains and losses . . . basing social preference on the levels of individual welfare
without regard to cardinal measures that permit comparisons of gains and losses...
(Sen, 1974, p. 393.) . . . we do not need at all a cardinal measure of individual welfare
levels for the Rawls ordering (Sen, 1970, p. 393.)
I am not going to argue that the above quotations are wrong as I do need
two conditions to establish m y result that the general possibility of interpersonal comparison o f utility levels implies interpersonal comparison o f
utility differences. However, since b o t h conditions are very generally applicable, the above quotations, being written b y influential authors, may well be
very misleading to uncareful readers. (See also the subtle difference between
unit comparability and comparability o f utility differences, a distinction
I owe to Arrow, discussed in Remark 1 below.)
EXISTENCE OF TWO POTENTIALLY "OVERLAPPING" INDIVIDUALS
(POI). (a) Among all individuals and all social states, there exist (at least) two
individuals such that one is judged better off under some social state than
another under some (the same or different) social state and the reverse is true
for some other social state(s). (b) For at least one pair o f individuals satisfying
Condition (a) neither individual is indifferent between all social states.
If Condition (a) in POI does not hold, all individuals can be ranked first,
second . . . . . such that, for all n, the nth individual dominates the (n + 1)th
in the sense that he is better off in his worst social state than the other in his
best. This is obviously a very special case unlikely to apply for all practically
relevant cases. Condition (b) is almost certainly satisfied for all individuals,
not to mention just a pair. Thus, POI is a very general condition. Even if we
refrain from making interpersonal level comparison across different social
states, POI is still a very general condition. There may exist some individual
who is better off than another under any relevant social state. But this is
e x t r e m e l y unlikely to apply for all pairs o f individuals in a large society.
I N T E R P E R S O N A L LEVEL COMPARABILITY
143
SEMI-CONNECTEDNESS AND CONTINUITY. (a) Semi-connectedness of
social states: For each individual i (or at least for each of some pair of individuals satisfying POI), if xPiy, there exists another social state z such that
yliz and there exists a continuum of social states connecting x and z. (b) Continuity of individual preferences.
Semi-connectedness, though not perfectly general, is a considerably less
restrictive condition than the usual one of connectedness. Semi-connectedness
does not require all aspects (or dimensions) of (the space of) social states to
be divisible. Essentially, if there exists one (or several) divisible aspect (e.g.,
personal income) that is sufficiently important to .the individual, Semiconnectedness will be satisfied. As illustrated in Figure 1, if other aspects
are indivisible, the domain of social states consists of vertical lines (hyperspaces) disconnected from each other. But if the divisible aspect is sufficiently
important, then for any xPiy, if we reduce that aspect (his income) from x by
a sufficient amount, we will reach a social state z such that zIiy, assuming
the usual continuity of preference which is Part (b) of the above condition
requiring that individual preferences do not jump if the social state is changed
by an infinitesimal amount (in the divisable aspect). There may exist some
individual of high principle who prefers, say, social states without the use of
Personal
income
X
Z
--ui
Other aspect(s)
Fig. 1.
144
Y E W - K W A N G NG
torture irrespective of what happens to his income and the income levels of
all other individuals, such that Semi-connectedness does not apply to him.
But we require Semi-connectedness and Continuity to be satisfied only for a
certain pair of individuals satisfying POI. The condition may thus be expected
to apply for practically all relevant cases.
PROPOSITION: Given the Existence of two Potentially "Overlapping" Individuals (POI) and Semi-connectedness and Continuity, the general possibility
of interpersonal comparison of utility levels implies interpersonal comparison
of utility differences.
Proof. From POI, there exist two individuals J and K and social states
x, x' (may be the same or distinct) and y, y' such that :2
(1)
VJ(x) > V~'(x'),
(2)
uK(y) > uJ(y'),
and such that xPay ', and yPKx' (due to Condition (b) in POI).
From Semi-connectedness, there exists z such that zlJy ' and there exists a
continuum of social states connecting x and z. (For illustration, see Figure 2
which is not necessary for the proof.) Thus, as we move along (or within) this
Y
,
$
i
-uf
\u~
K's indifference map
J's indifference map
Fig. 2.
INTERPERSONAL
LEVEL COMPARABILITY
145
continuum from x towards z, U J is changed continuously from UJ(x) and
(at least eventually) reduced towards UJ(z) = UJ(y'). Similarly, there exists
w such that x'IKw and there exists a continuum of social states connecting
y and w. As we move within this continuum f r o m y towards w, U K is changed
continuously from U K ( y ) and eventually reduced towards UK(w) = UK(X').
Thus, there exist social states s and t (distinct or otherwise) such that
(3)
uJ(s) =
or so it is judged, and there exist social states q and r such that
(4)
UJ(s) > UJ(q) = ut((r) < uK(t).
This holds since the s and t in (3) can be chosen such that UJ(s) > UJ(z) and
UK(t) > UK(w).
From (3) and (4),
(5)
UJ(s) -- UJ(q) = uZ((t) -- ug(r),
which is interpersonal comparison of utility differences.
Q.E.D.
Remarks
(1) After establishing the above proposition, I concluded in the first draft of
this paper that, with our two reasonable conditions, the general possibility
of interpersonal comparison of utility levels implies interpersonal comparability of utility differences or units. However, following a helpful comment
from Arrow, we may distinguish a subtle difference between comparability
of utility differences and comparability of utility units. For a particular set
of scales for individual utility functions, we may have, say,
(5')
ug(s) - ug(q) = 20 utils = Un(t) -- UK(r) = 10 utils.
This does not imply that 2 utils for J always equal 1 util for K throughout
all social states. Even if this happens to be true for a particular set of utility
scales, it cannot remain so under arbitrary but coordinal transformation of
the utility scales. Thus, for some other social states, and/or for some other
set of utility scales, we may have
(6)
UJ(a) -- UJ(b) -- 5 utits = UK(c) -- UK(d) = 7 utils.
146
YEW-KWANG NG
and (5') and (6) need not be inconsistent. In this sense, a unit of utility is
non-comparable. However, what is important (for social choice anyway)
is interpersonal comparability of utility differences between social states.
Once we have such comparability, we can always choose a set of utility
scales representing individual preferences such that the units are also comparable, a
(2) It is not difficult to see that full interpersonal comparability of utility
differences or units (with suitable scaling) does not imply any level comparability. This is so since unit comparability is invariant with respect to
arbitrary addition to or subtraction of constants from individual utility
functions.
(3) In response to the first draft of this paper, Sen makes the observation
that utilitarianism is informationally more demanding in a w6rld of more
than two persons than in a two-person world. For example, one might be
able to say that
U' (x) -- Ul (y) > U2(y) --,V~, (x) > V3(y) -- U3(x) > O,
without being able to say whether Ei Ui(x) >, <, = F-,iUi(Y), though such
a comparison is possible for the two-person case. This is an interesting
observation that did not occur to me." However, it does not affect our conclusion. In the proof of our proposition, (5) is established as an equality,
not just an inequality, thanks to Semi-connectedness and Continuity. It
is true that, for a given pair of social states, the relevant comparison of
utility differences may be in the form of inequalities rather than equalities.
But if we can establish equalities of utility differences for other social states,
we could choose a set of utility scales such that we also have comparability
of utility units as remarked above. Then for any two utility differences
that are unequal, we do not just know which one is larger but also by
how many times one utility difference is larger than the other. This will
then provide one with sufficient information to practice utilitarianism if one
chooses to.
It is true that the actual comparison of utility differences may involve
many practical difficulties and will probably be very imprecise. (See, however, Ng (1975) on ways of practical comparison.) But our proposition
shows that, if this is so, the same must be true for the comparison of utility
levels.
INTERPERSONAL
I, E V E L C O M P A R A B I L I T Y
147
NOTES
* I am grateful to Kenneth Arrow and Amartya Sen for helpful comments.
i For a more detailed discussion of interpersonal comparison, see Sen (1979).
2 If we refrain from making interpersonal level comparison across different social states,
x a n d x ' are taken as the same social state, and so arey a n d y ' .
3 Basu (1982) makes the subtle distinction between comparability of utility differences
in the traditional framework (his Axiom 2) and that in the modern social choice framework (Axiom 2*) and shows that the former implies cardinal utility while the latter
implies cardinality with an additional assumption on the connectivity of u(x) which is
satisfied given our assumption of Weak Connectedness and Continuity.
REFERENCES
Arrow, KennethJ.: 1973, 'Some Ordinalist-Utilitarian Notes on Rawls's Theory of
Justice', Journal of Philosophy 7 1 , 2 4 5 - 2 6 3 .
Basu, Kaushik: 1982, 'Determinateness of the Utility Function: Revisiting a Controversy
of the Thirties', Review of Economic Studies 49, April, 307-311.
Ng, Yew-Kwang: 1975, 'Bentham or Bergson? Finite Sensibility, Utility Functions, and
Social Welfare Function', Review of Economic Studies 42, 5 4 5 - 5 7 0 .
Sen, Amartya K.: 1970, 'Interpersonal Aggregation and Partial Comparability', Econometrica 38, 393-409.
Sen, AmartyaK.: 1974, 'Informational Bases of Alternative Welfare Approaches',
Journal of Public Economics 3 , 3 8 6 - 4 0 3 .
Sen, AmartyaK.: 1979, 'Interpersonal Comparisons of Welfare', in M. Boskin (ed.),
Economics and Human Welfare:Essays in Honor of Itbor Scitovsky, Academic Press,
New York. Reprinted in Sen, Choice, Welfareand Measurement, BlackweU, Oxford,
1982.
Department o f Economies,
Monash University,
Clayton, Victoria 3168,
Australia