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.
