3. Conjoint Measurement and Additive Social Wel-
fare Functions. In the field of social choice, as in that
of individual choice, the methods of conjoint measure-
ment have led to cardinal utilities which are consistent
with the general operational spirit of ordinalism.
William S. Vickrey, in 1945, suggested that the von
Neumann-Morgenstern theory of utility for risk-
bearing was applicable to the Bergson social welfare
function. The criterion of impartiality was interpreted
to mean that the ethical judge should consider himself
equally likely to have any position in society. He then
would prefer one decision to another if the expected
utility of the first is higher. The utility function used
is his von Neumann-Morgenstern utility function, i.e.,
that utility function which explains his behavior in risk-
bearing. Since all positions are assumed to be equally
likely, the expected utility is the same as the average
utility of all individuals. In turn, making the average
utility as large as possible is equivalent to maximizing
the sum of utilities, so that Vickrey's very ingenious
argument is a resuscitation, in a way, of Benthamite
utilitarianism.
Though Vickrey's criterion is impartial with respect
to individual's positions, it is not impartial with respect
to their tastes; the maker of the social welfare judgment
is implicitly ascribing his own tastes to others. Further
it has the somewhat peculiar property that social
choices among decisions where there may be no
uncertainty are governed by attitudes towards risk-
bearing.
Fleming, in 1952, took another direction, which has
not been followed up but which is worthy of note.
Suppose that an ethical judge is capable of making
social welfare judgments for part of the society inde-
pendently of the remainder. More precisely, suppose
that for any social decision which changes the utilities
of some individuals but not of others, the judge can
specify his preferences without knowing the utility
level of those unaffected by the decision. Then it can
be shown that there are cardinal utility functions for
the individuals and a cardinal social welfare function,
such that, W = U1 +... + Un. W and U1,..., Un
are interval scales, but the units of measurement must
be common. Again there is additivity of utility, but
note now that the measurements for individual utility
and for social welfare are implied by the social welfare
preferences and do not serve as independent bases for
them.
Harsanyi in 1955 in effect synthesized the points of
view of Vickrey and of Fleming. His argument was
that each individual has a von Neumann-Morgenstern
utility function, expressing his attitude toward risk, and
society, if it were rational, must also have a von
Neumann-Morgenstern utility function. It is then easy
to demonstrate that society's utility function must be
a weighted sum of the individuals' utilities, i.e.,
W = a1U1 +... + anUn. Since each individual utility
is an interval scale, we can choose the units so that
all the coefficients ai, are 1. This result differs from
Vickrey's in that the utility function of the ith individ-
ual is used to evaluate his position, rather than the
utility function of the judge.
Distantly related to these analyses is the revival, by
W. E. Armstrong, and by Leo Goodman and Harry
Markowitz, of Bentham's use of the just noticeable
difference as an interpersonally valid unit of utility.
It has proved remarkably difficult to formulate theories
of this type without logical contradiction or at least
paradoxical implications.
So far all these results have led to a sum-of-utilities
form, though with varying interpretations. As remarked
earlier, the notion of impartiality requires symmetry
but not necessarily additivity. John Rawls in 1958
proposed an alternative form for the social welfare
criterion, to maximize the minimum utility in the
society. This formulation presupposes an ordinal inter-
personal comparison of utilities. He shares with
Vickrey and Harsanyi a hypothetical concept of an
original position in which no individual knows who
he is going to be in the society whose principles are
being formulated. However, he does not regard this
ignorance as being adequately formulated by equal
probabilities of different positions; in view of the
permanence of the (hypothetical) choice being made,
he argues that a more conservative criterion, such as
maximizing the minimum, is more appropriate than
maximizing the expected value.
