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.
