Bentham or Nash?
On the Acceptable Form of Social Welfare
Functions*
YEW-KWANG N G
Monash University.
Clayton, Victoria 3168
The acceptability of the Nash Social Welfare Function is
questioned because a minute (perhaps hardy perceivable) werfare
change of;someone with a very low welfare level might overwhelm
enormous welfare changes of others. The axiomatic derivation of the
Nash S W F by Kaneko and Nakamura is analyzed. Arguments for
and against social welfare as separable in, linear in, and an
unweighted sum of individual werfares are critically discussed.
Separability follows from the individualistic ethics (Fleming,Sugden
and Weale). Linearity followsfrom the 'logic'of rational choice in the
face of risk (Harsanyi) . (Diamond's counter-example is rejected.)
Unweighted sum follows from either ( i ) informational restriction
precluding interpersonal comparison of welfare levels (D 'Aspremont
and Gevers, Maskin); or (ii) with finite sensibility. equation ofjustperceivable increments of werfare across persons (Edgeworth) or the
Weak Majority Preference Criterion ( N g ) .
policy on individual welfares. I also agree with
Friedman (1953, p.5) that much disagreement on
policy issues is due to different judgements about
the objective effects of different policy measures
instead of disagreement about the ultimate policy
objective. Nevertheless, there is also much scope
for disagreements of the latter type which must also
impinge on our agreement or the lack of it on the
desirable policies to pursue.
Even if there is a consensus on the policy effects,
agreement on the 'correct' form of SWF need not
settle all problems. This is so since different
individuals and different social groups may fight
for their self-interests rather than pursuing the
overall welfare even if everyone agrees, at an ethical
level, on the 'correct' social objective. These selfinterested social groups may even include
governments which typically d o not behave like a
benevolent despot maximizing social welfare.
Collective decisions are undertaken by ordinary
people like voters, politicians, and bureaucrats who
I Introduction
What should the social objective be? Should we
be maximizing the Benthamite SWF (social welfare
function) of the sum of individual welfaresl, or the
Nash SWF of the product of individual welfares, or
perhaps something else? Should only individual
welfares enter our SWF?
It is true that even if we have agreement on
answers to questions as posed above, we are far
from being in agreement in social policies, as we
may still disagree as to the effects of a particular
* I am grateful to Murray Kemp, Amartya Sen, Robert
Sugden and two anonymous referees for comments.
lThe welfare of an individual may differ from his
preference (or utility) due to a concern for others,
ignorance and imperfect foresight, or irrationality. (See
Ng, 1979, p. 7ff.) In this paper, I shall largely abstract
away this difference between welfare and preference.The
concept of SWF used here is Bergsonian instead of
Arrovian.
238
1981
BENTHAM OR NASH?
239
may base their decisions more on their own
interests. This was pointed out nearly a century ago
by Wicksell (1896) and repeatedly emphasized by
the Virginia School, especially Buchanan (e.g.
1975, 1978; Buchanan, er al., 1978). Despite
substantial works in this area (e.g. Downs, 1957;
Olson, 1965; Tullock, 1965, 1976; Niskanen, 1971;
Breton, 1974; Brennan and Buchanan, 1980;
Fiorina and Noll, 1978; see also surveys by
Ferejohn and Fiorina, 1975; Mueller, 1979;
Peacock and Wiseman, 1979), most studies in
welfare economics still concentrate on analyzing
what is socially optimal in some sense, as though
presuming that governments will pursue social
optimality. This is partly due to the complicated
nature of actual public choice. processes and partly
due to the fact that the simple social-optimaiity
approach can still usefully serve as a n ideal to aim
at, a standard to compare wit), and a foundation to
base. further analysis on. F o r example, an
important contribution in the recent public choice
literature concerns what constitutional limits will
best constrain self-interested government
bureaucrats to function (especially with respect to
the amount of government revenue raised) in a
socially desirable way (Brennan and Buchanan,
1980). But what a r e the socially optimal
constitutional limits depends also on the form of
SWF. Thus the abstract discussion on the 'ideal'
social objective is still of relevance.z Hence, while
believing that the contribution of the Virginia
School should be given more attention, I shall
concentrate here on the ideal form of SWF instead
of on how to implement it in practice.
Thirdly, it is true that questions of what is the
ideal SWF or what we ought to pursue necessarily
involve value judgements which cannot be
scientificalIy right or wrong. Nevertheless, one can
show what are the implications of certain value
judgements and thus convince their believers to
disclaim them, or vice versa. In other words, value
judgements may be shown to be non-basic (Sen,
1967, 1970 Ch. 5; Ng, 1979, Appendix IA). Thus,
there is much scope for logical argument in the
realm of ethics.
In the next section, I question the acceptability of
the Nash SWF. In Section 111, I critically discuss
arguments for and against the Bentham SWF at
different levels: separability, linearity, a n d
unweighted sum. The last section provides a brief
summary.
At the risk of repetition, it may be added that
nothing normative can be deduced from positive
premises alone (Hume's law). Hence, any attempt
to reject or accept any SWF must be based on some
value premises. For example, I explain in the next
section that the Nash SWF implies certain things I
believe most people r e g a r d as ethically
unacceptable, and that one of the conditions used
to derive the Nash SWF is also unacceptable.
However, the acceptance or rejection must
ultimately be based on one's ethicai intuition.
Something that is obviously ethically unacceptable
(or otherwise) to me need not necessarily appear
the same to others. Readers must make up their
own mind. (The Appendix addresses some
methodological issues on the rejection of axiomatic
theories.)
In a somewhat similar spirit, Dennis Mueller
suggested in a private conversation that, due to practical
difficulties of welfare measurement, the specific SWF I
derived (Ng, 1975) is difficult to use in specific social
decision, but could be used to adjudicate among different
voting methods (and, may I add, any other schemes,
principles etc.) depnding on which method tends to yield
results more consistent with the SWF.
'On the reasonableness of the Pareto principle, see Ng
(1979, Section 2.1). Sen (1979, 1981) however argues
strongly against welfarism (social welfare is a function of
and only of individual welfares) itself. See Ng (198 I ) for a
defence of welfarism.
4 1 skipover the technical point that there does not exist
a real-valued function for a lexicographic ordering;
representation by vectors of real numbers is still possible.
I1 A Crifique o/ the Nash SWF
It seems reasonable to start from a Paretian SWF
where social welfare is an increasing function of
and only of individual ~ e l f a r e s But
. ~ we still have
to choose from a large number of SWFs ranging
from theleximin on one extreme through the Nash,
the Bentham, etc., to the leximax on the
other e ~ t r e m e .T~h e leximin S W F is the
lexicographic extension of the maximin SWF
which maximizes the welfare of the worst-off
individual. The lexicographic extension specifies
that if the welfare of the worst-off individual is held
constant, then social welfare increases if the welfare
of the second worst-off individual increases, and so
on. The leximax is the lexicographic extension of
the maximax which maximizes the welfare of the
best-off. Both the maximin and the maximax
violate the Pareto principle but their lexicographic
extensions do not. But both leximin and leximax
240
THE ECONOMIC RECORD
are so patently unacceptable' that I will waste no
space arguing against them. In fact, but for the
surprising attention to Rawls in recent years, they
are not worth even a mention except as limiting
cases. In contrast, the Nash S W F or something
close to it seems to be more reasonable and seems to
command widespread support.6 I Shall thus
concentrate on it in this section.
First it must be noted that an individualistic,
Paretian SWF (Bergsonian) must be based on
interpersonal comparison of cardinal individual
utilities o r welfares. Kemp and Ng (1976) and
Parks (1976) establish the impossibility of
ordinality and Sen (1970, pp. 123-30), DeMeyer
and Plott (1971), Osborne (1976) and Kalai and
Schmeidler (1 977) establish the impossibility of
non-comparibility.' In fact, even if it were possible
to have a S W F based only on ordinal welfares, it
would be meaningless to distinguish between say
the Nash S W F (which maximizes the product of
individual welfares) and the Bentham SWF (which
maximizes the sum). If individual welfare indices
are just ordinal, we can always transform a Nash
S W F into a Bentham SWF and vice verso, and
similarly for many others, by suitable monotonic
transformation of individual welfare indices. For
e x a m p l e , M a x W = I l W' is equivalent t o
Max Y = C V i where Vi=log Wi. Thus we shall
discuss the problem in the framework of
interpersonally comparable cardinal individual
welfares.
The S a s h SWF maximizes the product of
individual welfares.* When any individual welfare
W'becomes smaller, the social welfare significance
of a given increase in others' welfares ( W'") also
'Why is an infinitesimal welfare improvement of the
worst-off worth an enormous welfare loss by others,
including the second, third. . . wont-oE? If one puts
oneself in the position of the worst-off individual, one will
almost certainly reject the leximin SWF resolutely. (See
Ng, 1975, p. 558.)
a Forexample, see Fair (1971). Kaneko and Nakamura
( 1979). In a recent survey, I asked a random sample of
economists in British universities whether they believe in
or are more inclined towards the Bentham or the Nash
SWF; over 50 per cent of the respondents chose Nash.
' S e e Pollak (1979) for some extensions of the KempNg-Parks results and Roberts (1980a. 1980b. 1980~)
which cover a number of related issues.
ONash (1950) was concerned with the more positive
quation ofbargaining where a status quo position is also
clearly defined. The Nash SWF is used, however, in a
normative context.
SEPT.
becomes smaller. When any W' becomes zero or
negative, the Nash SWF violates the Pareto
principle since any increase in (W+9 then leaves
social welfare unchanged or even decreases it. To
avoid this, the Nash SWF has to be confined to
positive individual welfares. Even so restricted, the
acceptability of the Nash SWF is still doubtful. For
example, consider the social states a, b, c, dwith the
indtvidual welfare indices as displayed below.
According to the Nash SWF, W ( a ) > W(b). But
many people find it ethically desirable to provide
happy life for 998 miserable individuals even if this
involves making a n equally miserable person 2 a
little bit worse off still. This is probably why some
people choose to give up their lives for the good of
other^.^ Also, according to the Nash SWF,
W(c) > W(d). But one may be prepared to sacrifice
2000 welfare units of the well-off person 1 if 998
miserable individuals can be made fairly happy,
especially if, a s a bonus, the worst-off Mr 2 can also
be made a little better o f f . ' O
a
b
1000
1
1000
1(j-3000
I
1000
0
0
b
I
0
1000
Recently, Kaneko and Nakamura (1979)
showed that an apparently reasonable set of
conditions uniquely implies the Nash SWF. How
d o we reconcile our rejection of the Nash SWF with
this recent result? A closer examination of their
conditions reveals an unacceptable one that in itself
comes very close to implying the Nash SWF, as
shown below.
The conditions are: (I) the Pareto principle, (11)
'Independence of Irrelevant Alternatives with
Neutral Property', (111) Anonymity, ( I V )
Continuity. I find all these conditions reasonable in
'Here the choice is not maximizing one's own welfare,
a divergence between preference (or utility) and welfare
due to a concern for the welfare of others discussed in Ng
( 1979, p. 78.).
'"As a finn believer of the Bentham SWF, I do not
regard this minute increase in M r 2's welfare as of much
significance.The 'bonus' is given to the Rawlsians.
1981
24 1
BENTHAM OR NASH?
the context except Condition 11. In themselves,
both Independence and Neutrality (properly
defined) are reasonable conditions, but I shall
argue that Condition I1 is defined in a way that
implies more than just Independence and
Neutrality p u t together, at least in the
interpersonally comparable cardinal utility
framework. (On the other hand, in the pure
ordering approach, Condition I1 can easily be
strengthened to rule out any SWF. See Kemp and
Ng, 1976, 1977.) To avoid using heavy notations, 1
shall give a verbal explanation instead of repeating
the formal statement of Condition 11.
~~
scale factor. It is convenient to adopt the
convention of choosing welfare origins such that
the worst social state x, corresponds to all
individuals having zero welfare. It can then be seen
that an individual's ordering of the first versus the
second prospect will be the same as that of the third
versus the fourth for all probability distributions a
and B if his weifare indices have the same
proportion between a, and a2 and between b, and
b,, i.e. Wl(ul)/(W'(a2)
= W1(b1)/W'(b2).'*
With this
understanding, the ethical implication of
Condition 11and its near equivalence with the Nash
S W F can be readily seen.
~
~~
01
02
b,
b,
~
~~
C1
Person I
0.001
1000
0.000 000 00 1
0.001
I000000
I
Person 2
1000'
0.001
I 000 000
I
0.000 000 00 1
0.00 I
First Kaneko and Nakamura define a worst
social state x, where all individuals are in their
Worst. This is called the origin, and social welfare is
evaluated by considering increases of individual
weffares from this .worst state. Now if we take any
two (say a,, az) out of the set of social states
(excluding those as bad as x,, the worst social
state), we can form a prospect (a,a,, azu2,a 3 x o )
where the a's (and similarly the ps below) are
probability distribution summed to unity. With the
same three states q , a,, x,, we may form a second
prospect by choosing a different probability
distribution &, 3/, &. An individual may be
indifferent between the two prospects or may prefer
one to the other. If we choose another pair of social
states b,, bz, we may similarly form two prospects
(a1619 a,b,, a
4 (the third), (P,bl, P A B 4 (the
fourth) using respectively the same two probability
distributions a and fl as above. Now Condition I1
says that, if for each and every individual, his
ordering of the first versus the second prospect is
the same as that of the third versus the fourth for all
probabiIity distributions a and j3, then social
ordering of social states uI versus a2 must also be
the same as that of b, versus b,.
Kaneko and Nakamura assume that each
individual has a weak ordering over social states
satisfying the Marschak (1950) axioms for expected
utility maximization. This ensures the existence of a
preference-representing utility or welfare function
with only two degrees of freedom, an origin and a
For illustrative purposes, consider first the case
of only two persons. After fixing the welfare
origins, we are still left with one degree of freedom
in individual welfare indices. Let us use this degree
of freedom to scale these indices such that they are
interpersonally comparable to the ethical observer.
Consider the welfare indices displayed in the
accompanying table. In the state u I , person 1 is very
close to his worst state while person 2 is fairly
happy. In a2, the reverse is true. If our SWF is
anonymous, we should have a, socially indifferent
to az.(This, however, is not an essential part of our
argument.) Now consider 6, and 6,. It seems
reasonable to regard 6, as preferable to b,. Though
person 1 is even closer to the worst state still, person
2 is very much more better off in b, than in b,. As
the welfare index of 0.001 is already very small, a
fvrther reduction to any (non-negative) figure
cannot mean much of a deterioration. This social
superiority of b, to b, can be made even more
convincing ifwe further consider cI and c2. Now, it
is person 1 who finds c1 much more preferable to
c2. Let us form the prospects P, = (fb,, fcI), P, =
(+b2,f c J Since individual preferences satisfy the
" I f a, .Wi(a,) + 22 .W'(aa,) + a, .O > , = , <
+ b2 .W'(QU,) + fi, .O, multiplying ' 9 ( a , )
andW(a,) by a constant clearly leaves the inequality or
equality unchanged. In fact, one may also show that
Condition I1 is satisfied only if the proportionality
requirement holds, but this is not relevant to our purposes
here.
fl, .W'(o,)
242
THE ECONOMC RECORD
Marschak axioms, both individuals choose PIover
P,. However, by construction, both b , , b2 and c,, cz
satisfy the precondition of Condition I1 with
respect to aI, a,. So, if we are socially indifferent
between a , and a2, Condition I1 demands us to be
socially indifferent between b1 and bZ and between
c, and c,. If W ( a , ) = W(a,) and if we transform a , ,
a, into b,, b2 by multiplying each individual welfare index by a constant (which may differ over
individuals), we must still have W(b,) = W(b,)
from Condition 11. This can be satisfied by a
continuous SWF if it is in the multiplicative form.
We also need the condition of Pareto principle to
ensure the correct sign of the multiplicative SWF
and the condition of Anonymity to dispense with
the need to enter each W‘ into the SWF with
differingpowers. Also, the condition of Continuity
rules out lexicographic orderings and hence ensures
representation by a continuous numerical function.
The above are of course just explanatory
remarks put in a non-rigorous form. The
contribution of Kaneko and Nakamura is to show
rigorously that their set of conditions uniquely
implies the Nash SWF. But the meaning of
Condition I1 has not been adequately explained.
To exemplify this last point further, consider the
welfare indices of the following case of four
persons.
It is clear that a, is preferred to a2 by the
Bentham, the Nash, the maximin, the leximin, the
leximax, or the naive ‘equity’ SWF (which ranks
social states inversely to the ratio of the highest to
the lowest individual welfares), and in fact by any
SWF satisfying Anonymity and the Pareto
principle. Now, by construction of the welfare
indices, b, and b, satisfy the precondition of
Condition I1 with respect to a, and a,. Thus, if we
prefer a , to a*, Condition I1 dictates us to prefer b,
to b, as well. But this latter social preference ( b ,
over b2)no longer follows from Anonymity and the
Pareto principle. In fact, according to the Bentham,
the maximin, the leximin, the maximax, the
leximax, or the naive ‘equity’ SWF, b, is preferred
to b , .
From the above, it is clear that the ‘neutrality’ of
SEPT.
Condition I1 contains much more than the usual
requirement of neutrality (between alternatives)
and in fact binds us almost to something equivalent
to the essence of the Nash SWF.
It may also be noted that, just as the Nash SWF
is not invariant with respect to the choice of an
origin, Condition 11is not invariant with respect to
the choice of the third state (x,,) in formulating the
prospects. If x,, is replaced by the bliss state or any
other state, the same Condition I1 then produces a
different result. Thus, why the ‘neutrality’ should
be applicable (if at all) only with respect to x,,
remains a mystery.
111 Steps to the Bentham SWFSeparability, Linearity and Unweighted Sum
The Bentham SWF maximizes the unweighted
sum of individual welfares. This entails that social
welfare is a separable and linear function of
individual welfares. We shall first consider
arguments supporting separability and linearity
before those supporting unweighted sum.
(A) Separability-Fleming (1952) and Sugden and
Weale (1979)
Fleming establishes separability, i.e., W =
X f ( W’) from a very reasonable set of axioms: the
existence of a SWFl2; the Pareto principle, and the
‘Elimination of Indifferent Individuals’ (strong
version). For some problems, the existence of a
SWF (which presupposes a social ordering) may be
a strong assumption to make. But this axiom
imposes no restriction at all on the question of
which SWF we should adopt. Taking also the
Pareto principle as acceptable, let us thus
concentrate on the crucial ‘Elimination of
Indifferent Individuals’ (Fleming’s Postulate E).
POSTULATE E (Elimination of Indifjerent
Individuals):
Given at least three individuals, suppose that
both individuals i and j are indifferent between x
and iand also indifferent between y and y’ but
individual i prefers x (or x’) to y (or y’) and
individualj prefers y (or y‘) to x (or x’). Suppose
also that all other individuals are indifferent
between x and y, and indifferent between Y and
y’ (but not necessarily indifferent between x and
x’and between y andy‘). Then social preferences
Fleming only lists postulates ensuring the existence
of a social ordering. With continuity, this ensures the
existence of a SWF as well. Continuity and the existence
of a S W F is implicitly assumed in Fleming’s denvation.
The existence of a social ordering alone is not sufficient for
Fleming’s purpose as a lexicographic social ordering
cannot be represented by a SWF at all.
1981
BENTHAM OR NASH?
must always go in the same way between x and y
as they d o between x‘ and y‘.
The ethics behind the postulate is the following.
In the choice between x and y , since all other
individuals are indifferent, social preference should
be decided by the preferences of individuals i andj.
Suppose the preference of individual i of x over y is
judged on some grounds (stronger, worthier, etc.)
to outweigh the dispreference of j such that x is
socially preferred toy. Then in the choice between
x‘ and y’, since all other individuals are again
indifferent for both i andj, x is indifferent to x‘ and
y is indifferent toy‘, so the preference of i ofx’ over
y‘ must also outweigh the dispreference ofj to give
X’ socially preferred to y’.
The strength and possible objections to the
postulate spring from the fact that, while all other
individuals are indifferent between x and y and
between x’ and y‘ they are not necessarily
indifferent between x (or y) and x’ (or y’). Thus
some people may want (probably based on some
misunderstanding, see below) to base the social
judgement as t o whether the preference of
individual i should outweigh the dispreference o f j
on the welfare levels of other individuals. One
possible reason for this is that welfare levels are
misunderstood to be something else such as income
levels. Then, even if individual i (and also]) has the
same income levels between x and X’ and between y
and y’, his welfare change between x and y may be
different from that between x’ and y‘ if the income
levels of other individuals differ and produce
external effects on i (and also j ) . However,
Postulate E does not refer to income levels but
holds individual i (and]) indifferent between x and
x’ and between y and y’, taking everything into
account. The objection to Postulate E requires in
fact the rejection of individualistic ethics. As
Harsanyi (1955) explains, Postulate E is a natural
requirement of individualistic ethics.
Fleming’s set of axioms ensures that social
welfare is a separable function of individual
welfares, i.e., W = Zff WQ.For any given set of the
J i functions, we may a d o p t m o n o t o n i c
transformations making W a sum of the
transformed individual welfare indices, as Fleming
does. This, however, is not permitted by the
cardinal welfare framework. As Fleming himself is
emphatic in pointing out, the transformed indices
need not bear direct relationship to say a cardinal
measure of happiness. Thus, in our interpersonal
comparable cardinal welfare framework, we must
view Fleming’s result as no more than separability.
243
A recent contribution in support of a separable
S W F is Sugden and Weale (1979). They adopt a set
of very compelling axioms within the
individualistic, contractarian framework which
implies Fleming’s Postulate E and the separability
in social welfare function. I find their argument so
persuasive that I do not find their following
conclusion presumptuous: ‘In so far as our
argument is valid, therefore, future formulations of
welfare economics should include this separability
requirement as a matter of course’. I do not believe
there is any escape from this separability satisfying
the individualistic ethics. Those who prefer a nonseparable SWF should have a hard time reconciling
with their simultaneous belief (as most of them
apparently do) in individualism.
However, separability may easily be regarded as
a stronger result than it really is. In fact, within the
confine of a separable SWF, enormous flexibility is
still possible. For example, if we write W =
Z( W’)l-m/(l-~)where a is some constant, we have a
Paretian and anonymous SWF which is separable.
But by varying the value of SI,we can accommodate
virtually all degrees of ‘egalitarianism’. With a = 0,
wehavetheBenthamSWF;witha = 1, wehave the
Nash SWF, with CL approaching infinity, we
approach the maximin SWF. Thus, the result on
linearity is an important advance over separability.
(B) Linearity-Harsanyi ( 1953, 1955)
Harsanyi shows that W=WW’(where each k’is
a constant) by assuming that social as well as
individual preferences satisfy the set of axioms for
expected utility maximization (Marschak 1950)
and that the society is indifferent between any two
prospects which are indifferent to every individual.
This appears to me to be very reasonable. But there
is an apparently very persuasive objection raised by
Diamond (1967). Consider a society of two
identical individuals, 1 and 2, facing a choice
between two alternatives x and y, with two equally
probable states of nature iz and /?; let the cardinal
and interpersonally comparable individual welfare
indices be as shown in the following table (ignoring
z for the moment):
244
THE ECONOMIC RECORD
Harsanyi's set of axioms implies that the society
is indifferent between x and y (by suitable scaling).
But many people may find y (strictly) preferable to
x. If we ask why is y preferable to x, there may be
two different answers. First, consider Diamond's
explanation: 'I a m willing to accept the sure-thing
principle for individual choice but not for social
choice, since it seems reasonable for the individual
to be concerned solely with final states while society
is also interested in the process of choice'
(Diamond, 1967, p. 766). But the society consists of
individuals. If all individuals are concerned with
final states, why should the society be concerned
with the process of choice? I can think of only two
reasons why the society may be concerned with the
process of choice that are consistent with
individualism. First, some individuals in the society
have views about the process of choice and will be
hurt or feel happy by the adoption of certain
processes. In this case, what needs to be done is just
to amend the pay-off matrices or to construct a
more complex one to take account of this feeling.
This is then not a valid ground against Harsanyi's
set of axioms and the implied SWF, provided the
feeling about the process of choice has been taken
into account in the welfare calculus. Secondly, even
if no individual in the society now has any
feeling about the process of choice a s such, it may
still be regarded as a legitimate concern since the
process of choice may affect the welfare of
individuals in the future who may not exist now. In
this case again, what needs to be done is just to take
account of the effects on individual welfares in the
future as well. Thus, I cannot find any ground
consistent with individualism that is a valid
objection to Harsanyi on the grounds of the
process of choice or the like. Let us turn to a
different objection.
Even if the feeling for the process of choice, the
welfare of future generations, etc., has been taken
into account or has been abstracted away, some
people may still regard y as preferable to x because
y gives individual 2 a 'fair shake' while x does not.
To be socially indifferent between x and y is thus
regarded as being unfair against 2. But I fail to see
why it is unfair if the society is also indifferent
between y and -7 (see the table above), as Harsanyi's
axioms dictate. Is it unfair against both individuals
(!) to be socially indifferent between x, y and z,
despite the fact that both individuals are indifferent
between y and a 50-50 chance of x and z? It is
probably difficult to convince those who still regard
SEPT.
y as preferable to both x and I,but I hope that the
following argument will persuade at least some of
them to have a second thought.
Let us accept the reasonable assumption that
social welfare is continuous in individual welfares
(an infinitesimal change in individual welfares does
not cause a big jump in social welfare). To prefer y
over x and z then implies that if we change the
welfare indices of y sufficiently slightly, then this
new alternative y* is still preferred to x. To see this
more intuitively, note that we have to change the
welfare indices by a finite amount (due to
continuity), say to y' (see the table below; the
precise figure is immaterial) to make it indifferent to
x. We can then increase the welfare indices back a
little toy" which will then be preferable to x and z . I
wish to show that the adoption of a S W F dictating
this preference is detrimental to the welfare of both
individuals and hence irrational at least from the
viewpoint of individualism, when individual
preferences satisfy the Marschak axioms.
We do not adopt a SWF to guide our social
choice for one particular known instance only.
Rather, once a SWF is adopted, it is used for all
instances until, for some reason, it has been
discarded in favour of another. Moreover, even if
we consider a once-and-for-all choice only, we do
not know which particular alternatives that choice
will involve at the 'constitutional stage' or the
'original position'. Thus, on the abstract problem
of the ideal SWF, we must consider all possible
instances of choice. For the particular instance of
the choice between x and y", the preference of y"
over x does not appear to be clearly unreasonable.
The expected welfare ( E W 9 of both individuals
being0.45 withy" (noting that OT = fi = ))and EW'
= I , E W z = 0 with x. Since 2 gains 0.45 and I
looses 0.55 in terms of expected welfare by the
choice ofy" over x, it does not necessarily appear to
be a social loss to the non-Benthamite. Similarly,
this is true for the preference ofy" over z. However,
ex nnle, we d o not know whether we will be
confronted with the choice between y" and x or
between y" and z; both must be regarded as equally
probable. I t then becomes clear that if individual
preferences satisfy the Marschak axioms, both
individuals would be in favour of the adoption of a
SWF preferring both x and z toy" rather than the
reverse, since the preference ofy" over x and z gives
E W = EW = 0.45, while the reverse preference
gives E W = EWL = 0.5. Individualism thus
clearly dictates the adoption of the linear SWF.
1981
245
BENTHAW OR NASH?
reject social welfare as a sum of individual welfares
must explain why either WE or EW (but not both)
is acceptable and the other is not. This is not an
easy task within the individualistic ethics, ignoring
incorrect probability estimates.
While Harsanyi's result of a linear SWF is a
strong one, it does not provide a system of weights
(for any particular set of scales for individual utility
indices, or a set of scales for any particular
system of weights), a difficulty emphasized by
Pattanaik (1968).'* Thus the result on unweighted
sum by the finite sensibility approach is a
significant advance. Before discussing this, let us
discuss another result on unweighted sum that is,
however, based on ussuming that this problem of
interpersonal comparison of welfare differences
has been solved somehow.
It may be objected that the above argument is
based on regarding (meta) social welfare as a
function of expected individual welfares, i.e.,
W E = W ( E W 1 ,. . ., E W 3 = W ( E j 0 ,
. . .,
Z, 0, WT) where 0, is the probability of a state of
nature times the probability of that choice instance.
For example, confining only to the two equally
probablechoice instances x against y" and z against
y", the probability of a prevailing in the choice
instance x againsty" is 4 x = In the presence
of risk, perhaps it is better to maximize expected
welfare as a function of ex-post individual welfares,
I.e.,EW= Z,O,W(C,. . ., W",).However,itcanbe
seen that, even using this latter criterion, the choice
of y" over x and z is still clearly undesirable since
E W W ) = tW(0.9, 0) + +W(O, 0.9) and EW(x, z )
= fW(1,O) + )W(O, I).ThusEW(x,z) > E W W )
i f W is increasing in individual welfares, i.e., if the
Pareto principle is satisfied.
v,
+ a.
r
U
I
2
B
I
I
0
0
U
1
0
Y"
Y'
Y
X
B
a
B
a
B
a
P
0
0
I
I
0
1
0.89
0
0
0.89
0.9
0
0.9
In the presence of risk, ignoring differences in (or
incorrect) probability estimates (on which see
Harris and Olewiler, 1979J,there seem to be good
grounds for either maxihizing social welfare as a
function of individual expected welfares or
maximizing expected social welfare as a function of
ex-post individual welfares. Within the framework
of individualism, neither objective can be rejected
as unreasonable. The reasonableness of both W E
and EW can in fact be used as an argument for a
linear SWF. If social welfare is a sum (unweighted
or weighted with constant individual weights k 3 of
individual welfares, then WE and EW are
equivalent. Thus W E = ZjkiXjO,Wj = C,O,CfkiWj
= E W . I f social welfare is not a sum of individual
welfares, the maximization of EWand WEwill not,
in general, yield the same r e ~ u 1 t . Thus
l ~ those who
l 3 If W E and EWalways yield the same result, we have
W E = W(Z,O,V. . . ., E,o,w:, = EW = z,e,wyj,
. . ., W?. Differentiation with respect to W) gives
dW/dEW'=c?W/dWj at any given situation. Such an
equality cannot hold through for every set of8's unless W
is a sum (unweighted or weighted with constant
individual weights) of its arguments.
0
(C) Unweightrd Sum-D'Aspremont
and Cevers
( I 977) and Maskin ( 1978)
While the derivations of the separability and
linearity in the S W F discussed above are fully
based on ethical postulates, the theorem on
utilitarianism by D'Aspremont and Gevers ( 1977,
p. 203) is partly based on informational restriction.
With the Pareto principle and Anonymity, a SWF
must assume the Benthamite form if interpersonal
comparison of (individual) welfare levels is not
possible though comparison of welfare drferences
is possible.15 Roughly, this result may be given the
"Pattanaik also raises some other objections though
questioning the popufar objection by Arrow ( I 963). Time
permitting, I hope to pursue these and related issues in
another paper. For Harsanyi's own reply to Diamond, see
Harsanyi (1975).
"Since D'Aspremont and Gevers are dealing with
social welfare functionals (Sen, 1970, p. 129). they need
the additional assumption of Independence (of irrelevant
alternatives) which is much weaker that Arrow's
correspondingcondition as Arrow's framework rules out
the relevance of' preference intensities. In fact, in the
framework of social welfare functionals where preference
246
THE ECONOMIC RECORD
following intuitive explanation. T h e Pareto
principle ensures that social welfare depends
positively on and only on individual welfares.
Anonymity requires that different individuals be
treated alike. Level non-comparability then leaves
only individual welfare differences to be treated on
the same footing. Moreover, invariance of social
choice with respect to individual welfare levels
(vertical shift in the whole individual welfare
function) also rules out any non-linear, though
anonymous function. This leaves the Bentham
SWF as the only choice.
Maskin (1 978) relaxes level non-comparability
by requiring what he calls ’full comparability’ (and
by assuming a weaker version of the Elimination of
Indifferent Individuals). This terminology which
originates with D’Aspremont and Gevers is
unfortunate, As Maskin himself (p. 94) observes,
‘full comparability’ still imposes some restriction.
In fxt, it rules out some form of interpersonal
comparison. Specifically it requires, among other
things, that social choice be invariant with respect
to a (vertical) shift in all individual welfare
functions by a common value. This is less restrictive
than the level non-comparability of D’Aspremont
and Gevers (1977, p. 201) which requires
invariance of social choice even in the presence of
unequal vertical shifts in individual welfare
functions. However, i t still imposes a real
restriction (though i t happens to be one that I am
personally ethically in favour of). For example,
consider the welfare indices in the following table.
I
x
I:
Y
I
200
100 100
100010
Y‘
~
100
10
1
100’00
100001
Personally, I find y clearly socially preferable to
x. But many people may prefer x t o y (as would be
the case for a Nash SWF), considering the welfare
gain of Mr 2 from 1 to 10 units as more important
than the loss of Mr 1 from 200 to 100 units. If we
intensity is a relevant consideration, Independence is a
compelling condition (Ng, 1979, pp. 142-4). In contrdst,
a (Bergson)SWF necessarily satisfies Independencesince
W(xJ depends only on W’(x), i = 1, . . ., n by
construction. It is my conjecture that every nonlexicographic(or at least every continuous) social welfare
functionalsatisfying Independence can be represented by
a Bergson SWF.
SEPT.
construct a new pair of social states x’ and y’ by
raising both individual welfare levels by 100 000,
these same people may now find y’ preferable to x’,
as is the case of a Nash SWF. Now. both
individuals being fairly well off, the gain of 9 units
may not be considered worth the loss of 100 units.
If we have truly fufl comparability, this kind of
ethics should be accepted as logical (though I find i t
unpalatable). But it is, in fact, ruled out by
Maskin‘s condition of ‘full comparability’.
(D) Unweighted Sum - the Finite Sensibility
Approach
Recognizing that human beings are not infinitely
sensitive, a just noticeable increment of happiness
may be used as an interpersonally valid unit of
welfare. This approach has a long tradition. The
concept was touched on as far back as 1781 by
Borda and in 1881 by Edgeworth. Edgeworth took
it as axiomatic or, in his words, ’a first principle
incapable of proof, that the ‘minimum sensibile’ or
the just perceivable increments of pleasure, of all
pleasures for all persons, a r e equatable
(Edgeworth, 1881, pp. 7ff., 60ff.). Armstrong
(1951), Goodman and Markowitz (1952), and
Rothenberg (1961) have also discussed the
problem. It has also been explored in its morr
positive aspects in decision theory and psychology,
using the term ‘just noticeable difference’. (See
Fishburn, 1970, for a survey.)
Assuming finite sensibility, Ng (1975) derives the
Bentham SWF based mainly on the following
postulate:
Weak Majority Preference Criterion (WMP):
For any two alternatives (or social states) .r and
y , if no individual prefers y to x and (1) if n, the
number of individuals, is even, at least n/?
individuals prefer .r to y ; (2) if n is odd, at least
(n-1)/2 individuals prefer x to y and at least
another individual’s welfare level is not lower in
.Y than in y , then social welfare is higher in x than
in y .
As Mueller observes,
WMP is obviously a combination of both the
Pareto principle and the majority rule principle
that is at once significantly weaker [hence more
acceptable as a sujicient condition for a social
improvement] than both. In contrast to the
Paretocriterion it requires a majority to be better
o f f ,rather than just one [individual], to justify a
move. And, in contrast to majority rule, it allows
the majority to be decisive only against an
1981
BENTHAM OR NASH?
indifferent minority. In spite of this apparent
weakness, the postulate nevertheless proves
strong enough to support a Benthamite social
welfare function [Mueller, 1979, pp. 181-21.
The reason why W M P leads us to the Bentham
SWF is not difficult to see. It requires that
individual welfare differences sufficient to give rise
to preferences of half of the population must be
regarded as socially more significant than
differences not sufficient to give rise to preferences
(or dispreferences) of another half. Since any group
of individuals comprising 50 per cent of the
population is an acceptable half, this effectively
makes a just perceivable increment of welfare of
any individual equivalent to that of any other
individual.
The ethical argument supporting WMP and the
Bentham SWF and replies to a number of possible
objections have been outlined in Ng (1975,
Section 5). Here, we consider the following
objection raised in a personal discussion by David
Friedman (cf: Sen, 1970, p. 94; Pattanaik, 1971,
p. 150). If individual welfare levels are continuous
(which is the assumption used in Ng, 1975), then
even an indifference may involve a welfare
difference. It is thus not clear that a just perceivable
increment of welfare should be regarded as
interpersonally equatable. For example, a person
poor in perception or reporting may fail to notice
or report a welfare difference more significant in
some subjective sense than one noticed or reported
by another more perceptive person. On the other
hand, if welfare levels are discontinuous so that
individual welfare does not change until an
increment of pleasure is perceived, then these
discontinuous little jumps in welfare may differ
across individuals in some sense and hence should
not be regarded as equivalent.
My reply to the above objection is several fold.
First, it may be noted that the above objection is
not against the principle of the Bentham SWF (i.e.
Was an unweighted sum of W')but just against the
way the various W' should be best measured or
interpersonally compared. In other words, if the
marginal preference of a person half as perceptive
(in terms of reporting a given amount of subjective
welfare) as another is counted twice as important,
the above objection will presumably no longer
apply. Thus, this objection is properly regarded
rather as pointing to the difficulty of making
interpersonal comparison of welfare than as an
objection against the Bentham SWF as such.
247
Secondly, if we shy away from making any
interpersonal comparison of cardinal individual
welfares, then a S W F is impossible. Thus, for the
question of which SWF to adopt, we must accept
some form of comparison. Most methods of
interpersonal welfare comparison are rough
estimates (if not guesstimates) perhaps partly
influenced by personal value beliefs. It seems to me
that the method of measuring the levels of just
perceivable increments of pleasure offers to date
the most promising objective way of making
interpersonal comparison. (On some practical
difficulties and ways to overcome them, see Ng,
1975, Section 9.) If we accept the above objection, it
just means that even this best available method is
not ideal. With finite sensibility (which is an
indisputable fact) and without some form of
interpersonal comparison, we cannot make a
reasonable social choice except in the unlikely cases
where every individual strictly prefers x to y . Even
the indifference of one individual is sufficient to
cloud the issue unless we are prepared to make the
interpersonal comparison that any welfare
difference associated with his indifference is not
strong enough to overshadow the preferences of
others. Thirdly, even if it is true that a just
perceivable increment of welfare differs across
individuals either due to different 'perceptive'
powers or due to different 'jumps' in welfare, how
are we to know whose just perceivable increment is
larger and whose smaller? In the absence of specific
knowledge regarding this, it seems best to regard
them as equal. (CJ Lerner, 1944, p. 29ff.f.; Sen,
1973.) If we adopt the convention of designating an
average just perceivable increment of welfare as
one, then the just perceivable increment of any
particular individual, in the absence of any specific
knowledge, may be expected to have a probability
distribution such as depicted in Figure 1, i.e., he is
just as likely (if a t all) t o have a lower as a higher
than average value. We can thus minimize our
mistake by taking his increment to be one, the
average value (Ng, 1980).
Our arguments for taking a just perceivable
increment of welfare as equatable across
individuals (at least in the absence of an
eudiamonometer) seem to be supported by the
following argument of Harsanyi:
The metaphysical problem would be present even
if we tried to compare the utilities enjoyed by
diererent persons with identical preferences and
with identical expressive reactions to any
situation. Even in this case, it would not be
THE ECONOMIC RECORD
FIGURE1
inconceivable that such persons should have
different susceptibilities to satisfaction and
should attach different utilities to identical
situations . . . identical expressive reactions may
well indicatedifferent mentalstates withdifferent
people. At the same time, under these conditions
this logical possibility of different susceptibilities
to satisfaction would ,hardly be more than a
metaphysical curiosity. If two objects or human
beings show similar behaviour in all their
relevant aspects open to observation, the
assumption of some unobservable hidden
difference between them must be regarded as a
completely gratuitous hypothesis and one
contrary to sound scientific method. (This
principle may be called the ‘principle of
unwarranted differentiation’.) In the last
analysis, it is on the basis of this principle that we
ascribe mental states to other human beings at
all: the denial of this principle would at once lead
us to solipsism. Thus in the case of persons with
similar preferences and expressive reactions we
are fully entitled to assume that they derive the
same utilities from similar situations [Harsanyi
1955. p. 317.
Many people are, however, reluctant to regard a
unit increase in the welfare of the better-off as
equivalent to B unit increase in that of the worse-off.
First, this may be due to what I called ‘utility
illusion’ or the double-discounting of the incomes
of the rich (Ng, 1975, Section 2). Psychological
studies show that, apart from extremal values, the
just noticeable increment to any stimulus value is a
constant proportion of that value, or Sensation =
k log Stimulus. Thus, by equating just perceivable
increments of welfare, we are already discounting
the incomes of the rich very heavily. To discount
further their welfares involves double-discoun ting.
SEPT.
Secondly, many people may have in mind the
undesirable effects of a high degree of inequality
apart from the diminishing marginal welfare of
income, due to external effects, social conflicts, etc.
These are, however, all taken into account by a
(long-run) Bentham SWF. Thirdly, we may have
‘pure egalitarianism’ which is apparently very
appealing but is foreign to my ethical intuition.
Suppose I have two children, one very happy and
one not very happy. If I have only one serve of icecream, I will have no hesitation a t all to give it to the
one who enjoys ice-cream more, whether i t happens
to be the happy child o r the less happy child,
assuming no envy, etc. In other words, I will
consider only the amount of happiness (whichever
child enjoying it) 1 can increase rather than the
amounts of happiness they already possess.
IY Brief Summary
After an introductory section justifying the
discussion of the acceptable form of SWFs, the
acceptability of the Nash SWF (and by extension,
similar ‘egalitarian’ SWFs) is questioned because a
minute (perhaps hardly perceivable ) welfare
change of an individual with a very low welfare
level might overwhelm enormous welfare changes
of other individuals. The axiomatic derivation of
the Nash SWF by Kaneko and Nakamura (1979) IS
analyzed and the acceptability of one of their
axioms is questioned.
Arguments leading to the Bentham SWF are
discussed at several levels. First, ‘social welfare
should be a separable function of individual
welfares’. The arguments of Fleming (1952) and
Sugden and Weale (1979) are reviewed and shown
t o be based on nothing m o r e t h a n the
individualistic ethics. Separability is thus a
compelling requirement for any S W F consistent
with individualism. Secondly, Harsanyi (1953,
1955) shows further that our S W F must be linear in
(a weighted sum of) individual welfares by
assuming that social as well as individual
preferences satisfy the Marschak axioms for
expected utility maximization and that two
prospects are socially indifferent if they are
indifferent to all individuals. It is argued that
Diamond’s (1967) famous ‘counterexample’ need
not affect the acceptability of Harsanyi’s axioms
as a SWF is not chosen just for one particular
known instance only. Thirdly, social welfare must
be an unweighted sum of individual welfares if,
with some reasonable ethical requirements such as
the Pareto principle and Anonymity, informational
1981
249
BENTHAM OR BASH?
restriction precludes interpersonal comparison of
welfare levels in some Senses (D'Aspremont and
Gevers, 1977; Maskin, 1978). W i t h full
comparability, we still have the Bentham result if,
recognizing that human beings are not infinitely
sensitive, the just perceivable increments of welfare of
all persons are taken as equatable (Edgeworth, 1881)
or if the Weak Majority Preference Criterion (which
is at once significantly weaker than the Pareto
principle and the majority rule principle) is accepted
(Ng, 1975). An objection to this is replied to in some
detail.
APPENDIX
On the Rejection of Axiomatic Theories
It might be reasoned that, since I reject the Nash SWF
on the unacceptability of some of its implications, so
o t h e r s can reject t h e Bentham S W F on the
unacceptability of some of its implications. So why waste
valuable pages going through sets of axioms that imply
the Bentham SWF?
Logical arguments may, establish ' A implies B'. If B
contains something ethical, so must A. If one is certain
that B is not acceptable, one should reject B a n d hence a
fortiorireject A . But if 'A implies B'holds, one should also
explain why A is unacceptable, to complete the job. But
quite often one is not absolutely certain that B is or is not
acceptable. If A can be shown to be acceptable (usually A
is broken down toa number ofsimple parts called axioms;
an axiom is not necessarily acceptable though a good
axiom is simple, straightforward, and acceptable), then
one can be persuaded that B should also be acceptable. I
have shown ( I ) that the Nash SWF implies something
dejnitely not acceptable in my view and hence I can reject
the Nash SWF on this ground. Moreover, I have also
shown (11) that a set of axioms (Kaneko and Nakamura)
used to derive the Nash SWF contains something
unacceptable. If one definitely agrees with (I), one must
reject the Nash SWF. If one does not agree or is uncertain
about (I) but agrees with (II), he should reject Kaneko
and Nakamura but should suspend judgement about the
Nash S W F before he has further arguments. This is so
because if A implies B. the fact that A is not acceptable
does not imply that B is also unacceptable (though it is
possible that B is unacceptable). 'No person should be
hanged' ( X ) implies that 'Innocent persons should not be
hanged' ( Y).That Xis Unacceptable (to some people) does
not imply that Y must also be unacceptable. Since I want
toreject the NashSWF(notjust suspendingjudgement), I
must not only show that theset ofaxiomsused toderiveit
contains some unacceptable parts, I must also show that
the Nash S W F itself is or that it implies something
definitely unacceptable.
On the other hand, I believe in the Bentham SWF
which many people do not accept. One way to convince
them is t o argue in its favour directly. Another way is to
show that a set of reasonable axioms ( A ) implies the
Bentham SWF (B). If this implications holds, those still
rejecting B should explain why A is unacceptable. If they
cannot, further reflections on their part may convince
them that B is acceptable after all. There may exist some
people who will definitely reject the Bentham SWF and
any set of axioms sufficient to imply it, just as I reject the
Nash SWF, no matter how thoroughly they reflect. Then
they diKer from me in their basic ethical intuition which
may not be removable by logical arguments.
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