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Dictionary of the History of Ideas | ||

*1. The Theory of Elections in the Eighteenth and
Nineteenth Centuries.* In a collective context, voting

provides the most obvious way by which individual

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

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,..., B*r*, such that A is preferred by

a majority to each of B1,..., B*r*. Then the B*i*'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.

Dictionary of the History of Ideas | ||