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FORMAL THEORIES OF
SOCIAL WELFARE

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.

I. INDIVIDUAL CHOICE AND VALUES

The historical development of the notion of social
welfare cannot be easily understood without reference
to the gradual evolution of a formal analysis of individ-
ual choice, which we briefly summarize. Three charac-

teristics of this history, which are shared with the
history of the concept of social welfare, are striking:
(1) the form of the basic problems were established
during the eighteenth century and display the charac-
teristic rationalism and optimism of the Enlightenment;
(2) the analysis retained its general form but underwent
systematic transformation under the impact of twen-
tieth-century epistemological currents; and (3) there
are strong historical links with the development of the
theory of probability and its applications, links which
are not easy to explain on purely logical grounds.

The first work to discuss individual choice system-
atically is that of Daniel Bernoulli in 1738. He was
concerned to explain phenomena of which insurance
was typical—that individuals would engage in bets
whose actuarial value was negative. Bernoulli's solution
was that what guided the individual's decisions to
accept or reject bets was not the money outcomes
themselves but their “moral values” as he judged them.
In later terminology, the individual attached utilities
to different amounts of money and accepted an un-
certainty if and only if it increased the expected value
of the utility. He also postulated that in general utility
increased by lesser and lesser amounts as the quantity
of money increased, an assumption now known as
diminishing marginal utility. Then the individual
would shy away from bets which were actuarially
favorable if they increased uncertainty in money terms
(in particular, if they involved a very small probability
of very high returns) and would accept insurance
policies if they reduced monetary uncertainty, for the
high returns offered in the one case had relatively little
additional utility, while the low returns avoided in the
second case imply large losses of utility. Bernoulli thus
required a cardinal utility (in this case, an interval
scale) for his explanation of human behavior under
uncertainty.

The idea that the drive of an individual to increase
some measure of satisfaction explained his behavior was
widespread, though rather vague, in the eighteenth
century; Galiani, Condillac, and Turgot argued that in
some measure the prices of commodities reflected the
utilities they presented to individuals, for individuals
were willing to pay more for those objects which
provided them more satisfaction. This particular doc-
trine, indeed ran into a difficulty that Adam Smith
noted, that water was surely more useful than diamonds
but commanded a much lower price. But the doctrine
that the increase of utility or happiness is the complete
explanation of individual behavior is most emphasized
by Jeremy Bentham in writings extending from 1776
to his death in 1832. Further, and even more impor-
tantly, Bentham introduced the doctrine of the paral-
lelism between the descriptive and the normative in
terpretations of utility; not only does an individual seek
happiness but he ought to do so, and society ought
to help him to this end. “Nature has placed mankind
under the governance of two sovereign masters, pain
and pleasure. It is for them alone to point out what
we ought to do, as well as to determine what we shall
do.... By the principle of utility is meant that princi-
ple which approves or disapproves of every action
whatsoever, according to the tendency which it ap-
pears to have to augment or diminish the happiness
of the party whose interest is in question” (Bentham,
1780; 1961). Bentham took it for granted that utility
was a measurable magnitude; he further elaborated in
various ways the factors which determine utility, such
as nearness in time and certainty, but at no point is
there a clearly defined procedure for measuring utility,
such as would be demanded by modern scientific phi-
losophy. The one suggestion he made was that suffi-
ciently small increments in wealth were not percep-
tible; therefore, a natural unit for measuring utility is
the minimum sensible, or just noticeable difference, as
psychophysicists were later to term it.

Although Bentham's notions were widely influential,
especially among English economists (as well as being
violently repudiated by the romantic thinkers of the
early nineteenth century), a further elaboration was not
achieved until about 1870 when Bentham's simple
hedonistic psychology proved to be of surprising use
in economic analysis. Smith's water-diamond paradox
was at last resolved; while water as a whole was more
valuable than diamonds, the relevant comparison was
between an additional increment of water and an
additional increment of diamonds, and since water was
so much more abundant, it was not surprising that the
incremental or marginal utility of water was much
lower. (Actually, Bentham had already shown Smith's
error but did not directly relate utilities to prices in
any form; in any case, Bentham's contribution was not
recognized.) This basic point was grasped simulta-
neously by Stanley Jevons in England, Léon Walras
in France, and Carl Menger in Austria, between 1871
and 1874; they had in fact been anticipated by Gossen
in Germany in 1854.

The further technical developments of the theory
of individual choice in economic contexts are not of
interest here, but the power of the utility concept led
among other things to an analysis of its meaning.
Already in his doctoral dissertation in 1892, Mathe-
matical Investigations in the Theory of Value and
Prices, the American economist Irving Fisher observed
that the assumption of the measurability of utility in
fact was inessential to economic theory. This point was
developed independently and taken up much further
by Vilfredo Pareto, from 1896 on. At any moment,

given the prices of various goods and his income, an
individual has available to him all bundles of goods
whose cost does not exceed his income. The “marginal
utility” theory stated that he chose among those bun-
dles the one with the highest utility. But all that was
necessary for the theoretical explanation was that the
individual have an ordering of different bundles; then
the individual is presumed to select that bundle among
those available which is highest on his ordering. Thus
only ordinal preferences matter; two utility functions
which implied the same ordinal preference compari-
sons would predict the same choice of commodity
bundles at given prices and income. But this meant
in turn that no set of observations on the individual's
purchasing behavior could distinguish one of these
utility functions from another. In fact, more generally,
no observation of the individual's choices from any set
of bundles could make this distinction. But then the
neo-positivist and operational epistemology, so char-
acteristic of this century, would insist that there was
no meaning to distinguishing one utility function from
another. It was the ordering itself that was meaningful,
and all utility functions which implied it were equally
valid or invalid.

The ordinalist position, defined above, only began
to spread widely in the 1930's and became orthodox,
ironically enough at a moment when the foundations
for a more sophisticated theory of cardinal utility had
already been laid. The general approach is to make
some additional hypotheses about the kind of choices
which an individual will or ought to make. Then it
is demonstrated that there is a way of assigning numer-
ical utilities to different possible bundles of goods or
other alternative decisions such that the utilities
assigned reflect the ordering (higher utility to preferred
alternatives) and that the function assigning utilities
to alternatives has some especially simple form. More
particularly, it is assumed that the different commodi-
ties can be divided into classes in such a way that the
preferences for commodities in one class are inde-
pendent of the amounts of the commodities in the other
classes. Then there is a way of assigning utilities to
bundles of commodities within each class and defining
the utility of the entire bundle as the sum of the utilities
over classes. Such a definition of utility can easily be
shown to be an interval scale. This process by which
utilities are simultaneously assigned within classes and
in total so as to satisfy an additivity property has
become known as conjoint measurement.

A particular case of conjoint measurement is of
special significance. An ordinalist position undermined
Bernoulli's theory of choice (described above) in risky
situations; if cardinal utility had no meaning, there was
no way of taking its mathematical expectation. But in
the case of risk-bearing, it is very natural to make an
appropriate independence assumption, and it is possi-
ble so to choose a utility function that an individual's
behavior in accepting or rejecting risks can be
described by saying that he is choosing the higher
expected utility. The philosopher Frank Ramsey made
this observation in a paper published posthumously in
1931, in the collection called The Foundations of
Mathematics and Other Essays (p. 156), but it made
no impact; the point was rediscovered by John von
Neumann and Oskar Morgenstern, as part of their great
work on the theory of games, in 1944. The cardinalist
position in this case is rehabilitated, but it has changed
its meaning. It is no longer a measure inherently asso-
ciated with an outcome; instead, the utility function
is precisely that which measures the individual's will-
ingness to take risks.

II. THE SOCIAL WELFARE FUNCTION

1. Bentham's Utilitarianism. To Bentham, the util-
ity of each individual was an objectively meaningful
magnitude; from the point of view of the community,
one man's utility is the same as another's, and therefore
it is the sum of the utilities of all individuals which
ought to determine social policy. Bentham is indeed
concerned strongly to argue that the actual measure-
ment of another's utility is apt to be very difficult, and
therefore it is best to let each individual decide as much
as possible for himself. In symbols, if U1,..., Un are
the utilities of the n individuals in the society, each
being affected by a social decision, the decision should
be made so as to make the sum, U1 + U2 +... + Un,
as large as possible. An expression of this form, which
defines a utility for social choices as a function of the
utilities of individuals, is usually termed a social welfare
function. Bentham's conclusion is really clearly enough
stated, but there are considerable gaps in the underly-
ing argument. The addition of utilities assumes an
objective or at least interpersonally valid common unit;
but no argument is given for the existence of one and
no procedure for determining it, except possibly the
view that the just noticeable difference is such a unit.
Even if the existence and meaningfulness of such a unit
is established, it is logically arbitrary to add the util-
ities instead of combining them in some other way.
The argument that all individuals should appear
alike in a social judgment leads only to the conclusion
that the social welfare function should be a symmetric
function of individual utilities, not that it should be
a sum.

The Bentham criterion was defended later by John
Stuart Mill, but his arguments bear mostly on the
propriety and meaning of basing social welfare judg-
ments on individual preferences and not at all on the

commensurability of different individuals' utilities or
on the form of social welfare function. Mill, like Henry
Sidgwick and others, considered the primary use of
Bentham's doctrines to be applicability to the legal
system of criminal justice; since the conclusions arrived
at were qualitative, not quantitative, in nature, vague-
ness on questions of measurability was not noticed.
After the spread of marginal utility theory, the
economist F. Y. Edgeworth expounded the notion of
utility much more systematically than Bentham had
done, with little originality in the foundations, though
with a great deal of depth in applications. In particular,
he applied the sum-of-utilities criterion to the choice
of taxation schemes. The implication is one of radical
egalitarianism, as indeed Bentham had already per-
ceived. If, as is usually assumed, the marginal utility
of money is decreasing, if all individuals have the same
utility function for money, and if a fixed sum of money
is to be distributed, then the sum of utilities is
maximized when money income is distributed equally.
(Here, “money” may be thought of as standing for all
types of desired goods.) Then the only argument against
complete equality of income is that any procedure to
accomplish it would also reduce total income, which
is the amount to be divided. The argument can be also
put this way; resources should be taken from the rich
and given to the poor, not because they are poorer
per se but because they place a higher value on a given
quantity of goods. If it were possible to differentiate
between equally wealthy individuals on the basis of
their sensitivities to income increments, it would be
proper to give more to the more sensitive.

Apart from Edgeworth, there was little interest in
applying the sum-of-utilities criterion to economic or
any other policy. Very possibly, the radically egali-
tarian implications were too unpalatable, as they
clearly were to Edgeworth. Subsequent work on “wel-
fare economics,” as the theory of economic policy is
usually known, tended to be very obscure on funda-
mentals (although very clarifying in other ways).

2. Ordinalist Views of the Social Welfare Function.
Pareto's rejection of cardinal utility rendered mean-
ingless a sum-of-utilities criterion. If utility for an
individual was not even measurable, one could hardly
proceed to adding utilities for different individuals.
Pareto recognized this problem.

First of all, he introduced a necessary condition for
social optimality, which has come to be known as
Pareto-optimality: a social decision is Pareto-optimal
if there is no alternative decision which could have
made everybody at least as well off and at least one
person better off. In this definition, each individual is
expressing a preference for one social alternative
against another, but no measurement of preference
intensity is required. Pareto-optimality is thus a purely
ordinal concept.

It is, however, a weak condition. It is possible to
compare two alternative social decisions only if there
is essential unanimity. To put the matter another way,
among any given set of alternatives there will usually
be many which would satisfy the definition. A mani-
festly unjust allocation, with vast wealth for a few and
poverty for many, will nevertheless be Pareto-optimal
if there is no way of improving the lot of the many
without injuring the few in some measure. Pareto
himself was very clear on this point.

Pareto-optimality is nevertheless a very useful con-
cept in clearing away a whole realm of possible deci-
sions which are not compatible with any reasonable
definition of social welfare. It might be argued that
every application of utilitarianism in practice, as to
law, has in fact used only the concept of Pareto-
optimality. In welfare economics, similarly, it has
turned also to be useful in characterizing sharply the
types of institutional arrangements which lead to
efficient solutions, making it possible to isolate the
debate on distributive problems which it cannot solve.

Pareto later (1913) went further. He suggested that
each individual in his judgments about social decisions
considers the effects on others as well as on himself.
The exposition is a bit obscure, but it appears to coin-
cide with that developed later and independently by
the economist, Abram Bergson (1938). Each individual
has his own evaluation of a social state, which is a
function of the utilities of all individuals: Wi(U1,...,
Un). Since the evaluation is done by a single individual,
this function has only ordinal significance. The Ui's
themselves may be thought of as an arbitrary numerical
scaling of the individuals' preferences; they also have
only ordinal significance, but this creates no conceptual
problem, since the choice of the social welfare func-
tion, Wi, for the ith individual, already takes account
of the particular numerical representation of individ-
uals' ordinal utilities.

Interpersonal comparisons of utility are indeed
made, but they are ethical judgments by an observer,
not factual judgments.

Pareto (but not Bergson) went one step further. The
“government” will form the social welfare function
which will guide it in its choices by a parallel
amalgamation of the social welfare functions of the
individuals, i.e., a function, V(W1,..., Wn). Pareto's
concept of a social welfare function remained un-
known, though the concept of Pareto-optimality be-
came widely known and influential beginning with the
1930's, as is clear in Bergson's work. The latter became
very influential and is accepted as a major landmark;
but in fact it has had little application.

Bergson accepted fully the ordinalist viewpoint, so
that the ethical judgments are always those of a single
individual. This approach loses, however, an important
feature of most thinking about social welfare, namely,
its impartiality among individuals, as stressed by
Bentham and given classic, if insufficiently precise,
expression in the categorical imperative of Kant. In
Bergson's theory, any individual's social welfare func-
tion may be what he wishes, and it is in no way
excluded that his own utility plays a disproportionate
role. Pareto, by his second-level social welfare function
for the government implicitly recognized the need for
social welfare judgments not tied to particular individ-
uals. But the ordinalist position seems to imply that
all preferences are acts of individuals, so that in fact
Pareto had no basis for the second level of judgment.

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.

III. SOCIAL WELFARE AND VOTING

1. The Theory of Elections in the Eighteenth and
Nineteenth Centuries. In a collective context, voting
provides the most obvious way by which individual

preferences are aggregated into a social choice. In a
voting context, the ordinalist-cardinalist controversy
becomes irrelevant, for voting is intrinsically an ordinal
comparison and no more. (Indeed, the failure of voting
to represent intensities of preference is frequently held
to be a major charge against it.) The theory of elections
thus forcibly faced the problems raised by ordinalism
long before it had been formulated in economic
thought.

The theoretical analysis of social welfare judgments
based on voting first appeared in the form of an exami-
nation of the merits of alternative election systems in
a paper of Jean-Charles de Borda, first read to the
French Academy of Sciences in 1770 and published
in 1784 (a translation by Alfred de Grazia is in Isis, 44
[1953], 42-51). Borda first demonstrated by example
that, when there are more than two candidates the
method of plurality voting can easily lead to choice
of a candidate who is opposed by a large majority.
He then proposed another method of voting, one
which has been subsequently named the rank-order
method (or, sometimes, the method of marks). Let each
voter rank all the candidates, giving rank one to the
most preferred, rank two to the second, and so forth.
Then assign to each candidate a score equal to the sum
of the ranks assigned to him by all the voters, and
choose the candidate for which the sum of ranks is
lowest.

Borda's procedure is ordinal, but the arguments
advanced for it were in effect cardinal. He held that,
for example, the candidate placed second by an indi-
vidual was known to be located in preference between
the first- and third-place candidates; in the absence of
any further information, it was reasonable to argue that
the preference for the second-place candidate was
located half-way between those of the other two. This
established an interval scale for each individual. He
then further asserted that the principle of equality of
the voters implied that the assignments of ranks by
different individuals should count equally.

Borda thus raised most of the issues which have
occupied subsequent analysis: (1) the basing of social
choice on the entire orderings of all individuals of the
available candidates, not merely the first choices; (2)
the measurability of individual utilities; and (3) the
interpersonal comparability of preference (Borda made
interpersonal comparability an ethical judgment of
equality, not an empirical judgment).

In 1785, Condorcet published a book on the theory
of elections, which raised important new issues.
Condorcet seems to have been somewhat aware of
Borda's work but had not seen any written version of
it when he wrote. Condorcet's aim was to use the
theory of probability to provide a basis for social
choice, and this program takes up most of the work,
though this aspect has had little subsequent influence.
Although he purports to apply the theory of proba-
bility to the theory of elections, in fact the latter is
developed in a different way.

The most important criterion which Condorcet laid
down is that, if there were one candidate who would
get a majority against any other in a two-candidate
race, he should be elected. The argument for this crite-
rion might be put this way. Let us agree that in a
two-candidate race majority voting is the correct
method. Now suppose, in an election with three candi-
dates, A, B, and C, that C, for example, is not chosen.
Then, so it is argued, it is reasonable to ask that the
result of the three-candidate race be the same as if
C never were a candidate. To put it another way, it
is regarded as undesirable that if A is chosen as against
B and C, and the voters are then told that in fact
C was not even eligible, that the election should then
fall on B. The Condorcet criterion is in the fullest
ordinalist spirit; it is consistent with the view that the
choice from any set of alternatives should use no infor-
mation about voters' preferences for candidates not
available. Condorcet himself noticed an objection; if
an individual judges A preferred to B and B to C, there
is some vague sense in which his preference for A
against C is stronger than his preference for A against
B. Indeed, as we have seen, this was the starting point
for Borda's defense of the rank-order method.

In fact, Condorcet used his criterion to examine
Borda's rank-order method. He showed that it did not
necessarily lead to choosing the pairwise majority can-
didate. Moreover, no modification of the rank-order
method which allowed for nonuniform ranks would
satisfy the Condorcet criterion.

Condorcet's second major achievement was to show
that his criterion had the possibility of paradoxical
consequences. It was perfectly possible that, with three
candidates, A be preferred to B by a majority, B to
C by a majority, and C to A by a majority. For exam-
ple, suppose that one-third of the voters preferred A
to B and B to C, one-third preferred B to C and C
to A, and one-third preferred C to A and A to B. This
possibility has become known in the literature as the
“paradox of voting,” or the Condorcet effect. The
paradox of voting, in generalized form, and the possi-
bility of its elimination have become the main themes
of recent literature.

In the terminology introduced at the beginning of
this article, (pairwise) majority voting defines a relation
which is connected (there must be a majority for one
or the other of two alternatives, if the number of voters
is odd) but need not be transitive.

Condorcet has a proposal for dealing with a case

of intransitivity, at least when there are three candi-
dates. Of the three statements of majority preference,
disregard the one with the smallest majority; if this
is the statement, C preferred to A by a majority, then
the choice is A, being preferred to B and “almost
preferred” to C. He extends this proposal to cases with
more than three candidates, but no one has been able
to understand the extension.

Like Bernoulli's work (1738; trans. 1954) on the
expected-utility criterion for choice under uncertainty,
the papers of Borda and Condorcet had few significant
direct successors, (Laplace however gave a more
rigorous version of Borda's probabilistic argument for
the rank-order method). Indeed the value of their work
only came to be appreciated when others came to the
problem independently, 160 years later. Since Con-
dorcet's work made use of the theory of probability,
it, like Bernoulli's, was recorded in various histories
of the theory of probability during the nineteenth
century; in the thorough and widely read history of
Todhunter (1865), Borda's and Condorcet's theories of
elections were included with the probabilistic theory.

The only significant published nineteenth-century
work on the theory of election that is known today
is that of the English mathematician E. J. Nanson,
published in 1882 in Australia, in Transactions and
Proceedings of the Royal Society of Victoria, 19 (1882),
197-240. Nanson makes no reference to Condorect, but
it is hard to believe that his work is independent. He
notes the paradox of voting, in a manner which suggests
that he regarded it as well known, and accepts fully
the Condorcet criterion. His work consists primarily
in showing that each of several voting methods that
have been proposed fail to satisfy the Condorcet crite-
rion, in that one could find a system of preference
orderings for individuals such that there exists a candi-
date who would get a majority against any other but
would not be chosen. He then proposes a method
which will satisfy the criterion: rank all candidates
according to the rank-order method. Then eliminate
all candidates for which the sum of ranks is above the
average. With the remaining candidates from the
rank-orders again, considering only those candidates,
and repeat the process until one candidate is selected.

Among the methods considered and found wanting
by Nanson was preferential voting, an adaptation of
the Hare system of proportional representation to the
election of a single candidate. In 1926 George Hallett,
a leading American advocate of proportional repre-
sentation, suggested a modification which met the
Condorcet criterion. He developed a procedure, the
details of which need not be repeated here, which,
starting with the orderings of all the candidates by all
the voters, picked out a candidate, A, and a set of
candidates, B1,..., Br, such that A is preferred by
a majority to each of B1,..., Br. Then the Bi's are
eliminated from further consideration; the orderings
of only the remaining candidates are now used, and
the process is repeated. It may be added that Hallett
is fully aware of the work of both Condorcet and
Nanson and refers to both of them.

Duncan Black has called attention to some contri-
butions of C. L. Dodgson (Lewis Carroll), printed but
not published, particularly one of 1876. Dodgson
accepted the Condorcet criterion and observed the
possibility of paradox of voting; he used the criterion,
as Nanson did a few years later, to criticize certain
voting methods. By implication rather than directly,
he suggested an ingenious solution for the cases of
paradox; choose that candidate who would have a
majority over all others if the original preference scales
of the voters were altered in a way which involved
the least possible number of interchanges of prefer-
ences. (When there are three candidates, this proposal
coincides with Nanson's.)

Dodgson raised one more conceptually interesting
point, that of the possibility of “no election.” His
discussion is inconsistent. At one point, he contends
that if the paradox occurs, there should be “no elec-
tion”; however, a little further on, he argues that if
“no election” is a possibility, then it should be entered
among the list of candidates and treated symmetrically
with them. In the context of elections themselves, the
possibility is uninteresting; but if we think of legislative
proposals, “no election” means the preservation of the
status quo. Dodgson is noting that legislative choice
processes do not take all the alternatives on a par but
give a special privileged status to one.

Dodgson made no reference to predecessors; how-
ever, his pamphlets were designed to influence the
conduct of Oxford elections, and scholarly footnoting
would have been inappropriate. Whether or not he
read Todhunter's passages on Borda and Condorcet
cannot now be determined. Of course, no subsequent
work was influenced by him.

2. Current Analysis of Social Welfare Based on
Rankings. After a long but exiguous history, the gen-
eral theory of elections suddenly became a lively
subject of research beginning with the papers of Black
published in 1948 and 1949 and Arrow's 1951 mono-
graph. Since then there has been an uninterrupted
spate of discussion, which is still continuing. It is per-
haps not easy to see exactly why the interest has
changed so markedly. Neither Black nor Arrow were
aware at the time they first wrote of any of the preced-
ing literature, though it is hard to exclude the possi-
bility that some of this knowledge was in a vague sense
common property. Arrow has noted (Social Choice and

Individual Values, p. 93) that when he first hit upon
the paradox of voting, he felt sure that it was known,
though he was unable to recall any source.

Both Black and Arrow are economists, and some
historical tendencies in economics, in addition to the
general theory of marginal utility, played their role.
(1) A number of marginal utility theorists, such as
Marshall and Wicksteed, had tried to demonstrate that
their theories were, as Bentham had originally held,
applicable in fields wider than the purely economic.
(2) In particular, economists in the field of public
finance were forced to recognize that public expendi-
tures, which are plainly a form of economic activity,
were in principle regulated by voters. A voter who
was also a taxpayer could usefully be thought of as
making a choice between public and private goods;
the actual outcome would depend upon the voting
process. Problems of this type were studied by Knut
Wicksell in 1896, Erik Lindahl in 1919, and Howard
Bowen in 1943. These works tend in a general way
to a combined theory of political-economic choice. (3)
Other economists, particularly Harold Hotelling in
1929, and Joseph Schumpeter in his 1942 book Social-
ism, Capitalism, and Democracy, suggested models of
the political process analogous to that of the economic
system, with voters taking the place of consumers and
politicians that of entrepreneurs. (4) Marginal utility
theorists, e.g., Edgeworth in 1881, and the Austrians,
Carl Menger and Eugen von Böhm-Bawerk, about the
same time, had been concerned with problems of
bargaining, where one buyer meets one seller, rather
than the more usual competitive assumptions of many
buyers and sellers. The development of game theory by
von Neumann and Morgenstern was intended to meet
this problem, but the formulation took on such general
proportions that it suggested the possibility of a very
general theory of social behavior based on the founda-
tion of individual behavior as governed by utility func-
tions. (5) The ideas of Pareto and Bergson were now
widespread and raised demands for clarification.

Most of these topics could be interpreted both
descriptively and normatively, and some of this duality
has persisted in the current literature. There are two
main themes in the literature, associated with the
names of Black and Arrow, respectively: (1) demon-
stration that if the preference scales of individuals are
not arbitrary but satisfy certain hypotheses, then ma-
jority voting is transitive; (2) formulation of sets of
reasonable conditions for aggregating individual pref-
erences through a kind of generalized voting and
examining the consequences; if the set of conditions
is strong enough, there can be no system of voting
consistent with all of them.

Suppose that all the alternative decisions can be
imagined arrayed in a certain order in such a way that
each individual's preferences are single-peaked, i.e., of
any two alternatives to the left of the most preferred
(by an individual), he prefers the one nearest to it, and
similarly with two alternatives to the right. This would
be the case if the “Left-Right” ordering of political
parties were a valid empirical description. Black dem-
onstrated that if preferences are single-peaked then no
paradox of voting can arise. Put another way, the
relation, “alternative preferred by a majority to alter-
native B,” is an ordering and in particular is transitive.

Current work, particularly that of Amartya Sen and
Gordon Tullock, has developed generalizations of the
single-peaked preference condition in different direc-
tions. The conditions are too technical for brief pres-
entation, but, like single-peakedness, they imply cer-
tain types of similarity among the preference scales
of all individuals.

Arrow stated formally a set of apparently reasonable
criteria for social choice and demonstrated that they
were mutually inconsistent. The study arose as an
attempt to give operational content to Bergson's con-
cept of a social welfare function. The conditions on
the social decision procedure follow: (1) for any possi-
ble set of individual preference orderings, there should
be defined a social preference ordering (connected and
transitive) which governs social choices; (2) if every-
body prefers alternative A to alternative B, then society
must have the same preference (Parento-optimality);
(3) the social choice made from any set of available
alternatives should depend only on the orderings of
individuals with respect to those alternatives; (4) the
social decision procedure should not be dictatorial, in
the sense that there is one whose preferences prevail
regardless of the preferences of all others.

Condition (3) in effect restricts social decision pro-
cedures (or social welfare criteria) to generalized forms
of voting; only preferences among the available candi-
dates are used in deciding an election. The inconsist-
ency of these conditions is in fact a generalized form
of the paradox of voting; no system of voting, no matter
how complicated, can avoid a form of the paradox.
As in the original Condorcet case of simple majority
voting, all that is meant by the paradox is that it could
arise for certain sets of individual preference orderings.
If individual preference orderings were restricted to
a set for which the conditions of Black, Sen, or Tullock
hold, then majority voting and many other methods
would satisfy conditions (2-4).

The evaluation of the Arrow paradox has led to
considerable controversy, still persisting.

In one version of Arrow's system, condition (2) was
replaced by another which, loosely speaking, stated
that a change of individuals' preferences in favor of

a particular alternative A would raise its social prefer-
ence, if possible. The existence of the paradox is not
altered by this substitution. Recent work by Kenneth
May and later Yasusuke Murakami showed that this
condition, together with condition (3), had powerful
implications for the nature of the social decision proc-
ess. Specifically, it followed that the choice from any
pair of alternatives is made by a sequence of majority
votes, where outcomes of the vote at one step can enter
as a vote at a later step. In general, some individuals
may vote more than once, and some votes may be
prescribed in advance. If however it is assumed in
addition that all individuals should enter symmetrically
into the procedure and also that the voting rule should
be the same for all pairs of alternatives, then the only
possible voting rule is pairwise majority decision, i.e.,
the Condorcet criterion.

BIBLIOGRAPHY

For histories of the theory of individual choice in eco-
nomics, see E. Kauder, A History of Marginal Utility Theory
(Princeton, 1965), and G. J. Stigler, “The Development of
Utility Theory,” Journal of Political Economy, 58 (1950),
307-27, 373-96. Bernoulli's paper originally appeared in
Comentarii Academiae Scientiarum Imperiales Petropoli-
tanae, 5 (1738), 175-92. It has been translated into English
in Econometrica, 12 (1954), 23-36. The quotation from J.
Bentham appears in his An Introduction to the Principles
of Morals and Legislation, reprinted in The Utilitarians
(Garden City, N.Y., 1961), p. 17. For a survey of the theory
of conjoint measurement, see P. C. Fishburn, “A Note on
Recent Developments in Additive Utility Theories for
Multiple-Factor Situations,” Operations Research, 14 (1966),
1143-48.

No adequate secondary sources exist for most of Part II.
See Bentham's work just cited; W. Stark, ed. Jeremy
Bentham's Economic Writings (London, 1954); M. P. Mack,
Jeremy Bentham: An Odyssey of Ideas, 1748-1792 (New
York, 1963); F. Y. Edgeworth, Mathematical Psychics
(London, 1881), and idem, “The Pure Theory of Taxation,”
in Papers Relating to Political Economy (London, 1925), II,
102; V. Pareto, The Mind and Society (New York, 1935),
4, 1459-74; A. Bergson, Essays in Normative Economics
(Cambridge, Mass., 1966), Part I; W. S. Vickrey, “Measuring
Marginal Utility by Reaction to Risk,” Econometrica, 13
(1945), 319-33, and idem, “Utility, Strategy, and Decision
Rules,” Quarterly Journal of Economics, 74 (1960), 507-35;
J. M. Fleming, “A Cardinal Concept of Welfare,” Quarterly
Journal of Economics, 64 (1952), 366-84; J. Harsanyi, “Car-
dinal Welfare, Individualistic Ethics, and Interpersonal
Comparisons of Utility,” Journal of Political Economy, 56
(1953), 309-21; W. E. Armstrong, “Utility and the Theory
of Welfare,” Oxford Economic Papers, New Series, 3 (1951),
259-71; L. Goodman and H. Markowitz, “Social Welfare
Functions Based on Individual Rankings,” American Journal
of Sociology, 58 (1952), 257-62; J. Rothenberg, The Meas
urement of Social Welfare (Englewood Cliffs, N.J., 1961);
and J. Rawls, “Distributive Justice,” in P. Laslett and
W. G. Runciman, eds. Philosophy, Politics, and Society,
Third Series (Oxford, 1967), pp. 58-82.

For the work discussed in Part III, see C. G. Hoag and
G. Hallett, Proportional Representation (New York, 1926),
for the work of Hallett, Hare, and others on proportional
representation and preferential voting. See also: J. Rothen-
berg, op. cit.; K. J. Arrow, Social Choice and Individual
Values, 2nd ed. (New York, 1963); D. Black, The Theory
of Committees and Elections (Cambridge, 1958); I.
Todhunter, A History of the Mathematical Theory of Proba-
bility from the Time of Pascal to that of Laplace (Cambridge
and London, 1865); A. K. Sen, “A Possibility Theorem on
Majority Decisions,” Econometrica, 34 (1966), 491-99;
G. Tullock, Toward a Mathematics of Politics (Ann Arbor,
1967), Ch. III; and Y. Murakami, Logic and Social Choice
(London and New York, 1968). The work of Condorcet is
discussed by Black, pp. 159-80; see also G. G. Granger, La
mathématique sociale du Marquis de Condorcet (Paris, 1956),
esp. pp. 94-129. Condorcet's study was entitled Essai sur
l'application de l'analyse à la probabilité des décisions
rendues à la pluralité des voix (Paris, 1785). For Laplace's
work on elections, see Black, pp. 180-83.

KENNETH J. ARROW

[See also Democracy; Economic History; Economic Theory
of Natural Liberty; Equality; Game Theory; Probability;
Rationality; Utilitarianism; Utility.]