The purpose of a theory of social welfare, or social
choice as it is sometimes revealingly termed, is to
provide a normative rationale for making social deci-
sions when the individual members of the society have
varying opinions about or interests in the alternatives
available. Any kind of decision, social or individual,
can be regarded as the interaction of the preferences
or desires of the decision-maker with the range of alter-
native decisions actually available to him, to be termed
the opportunity set. The latter may vary from time
to time because of changes in the wealth or technology
of the community. The usual formalism of social wel-
fare theory, derived from economic theory, is that
preferences (or tastes or values) are first expressed for
all logically possible alternatives. Then the most pre-
ferred is chosen from any given opportunity set.
As will be seen, there is serious and unresolved
dispute about the strength of the statements which it
is appropriate to make about preferences. One com-
mon demand is that preferences form an ordering of
the alternatives. In terms of formal logic, a preference
relation between pairs of alternatives is said to be
transitive if whenever alternative A is preferred to
alternative B and alternative B to alternative C, then
A is preferred to B; and it is said to be connected if,
for any two distinct alternatives, either A is preferred
to B or B to A. An ordering of the alternatives is a
preference relation which is both transitive and con-
nected; and it will be seen that this definition corre-
sponds to an everyday use of the term, “ordering.”
(In the economic literature, it has proved essential
to consider the possibility of indifference as well as
preference, between pairs of alternative social deci-
sions. For the purposes of this article, however, we
assume the absence of indifference, to simplify the
exposition.)
Still a stronger demand is that preferences be meas-
urable, that there exist a numerical representation
which correctly reflects preference (the more preferred
of two alternatives always has a higher number
associated with it). Such a numerical representa-
tion is usually termed a utility function. In the termi-
nology used by mathematical psychologists, a utility
function may constitute an interval scale, that is,
statements of the form, “the preference for A over
B is so many times the preference for C over D,” are
regarded as meaningful. In that case, the utility func-
tion is arbitrary as to the location of its zero point
and its unit of measurement, but otherwise uniquely
defined. A still stronger requirement is that the utility
function constitute a ratio scale, that is, statements of
the form, “the utility (or value) of A is so many times
as great as that of B.” Such statements imply a natural
zero; the utility function is unique up to a unit of
measurement. If it is assumed that no meaning can be
given to quantitative comparisons of preference but
only to the ordering of alternatives, it is customary
to speak of ordinal utility or preferences; if, on the
contrary, utility is considered to constitute an interval
or ratio scale, the term, cardinal utility or preferences,
is used.
The need for a theory of social welfare arises from
the need in the real world for social decisions. It is
simply a fact, as Hobbes pointed out, that there are
a great many decisions which by their nature must be
made collectively and without which all members of
the society would be much worse off—decisions on
legal systems, police, or certain economic activities
best conducted collectively, such as highways, educa-
tion, and the kind of insurance represented by public
assistance to disadvantaged groups.
A formal theory of social welfare then has the fol-
lowing form: given a representation of the preferences
of the individual members of the society in ordinal or
cardinal form, to aggregate them in some reasonable
manner to form a preference system for society as a
whole. Given the social preference system, and given
a particular opportunity set of alternatives, the choice
which society should make is that alternative highest
on the social preference system.
