_CONCERNING THE SELECTIONS AND THE DECISIONS OF ASSEMBLIES._
The probability of the decisions of an assembly depends upon the
plurality of votes, the intelligence and the impartiality of the members
who compose it. So many passions and particular interests so often add
their influence that it is impossible to submit this probability to
calculus. There are, however, some general results dictated by simple
common sense and confirmed by calculus. If, for example, the assembly is
poorly informed about the subject submitted to its decision, if this
subject requires delicate considerations, or if the truth on this point
is contrary to established prejudices, so that it would be a bet of more
than one against one that each voter will err; then the decision of the
majority will be probably wrong, and the fear of it will be the better
based as the assembly is more numerous. It is important then, in public
affairs, that assemblies should have to pass upon subjects within reach
of the greatest number; it is important for them that information be
generally diffused and that good works founded upon reason and
experience should enlighten those who are called to decide the lot of
their fellows or to govern them, and should forewarn them against false
ideas and the prejudices of ignorance. Scholars have had frequent
occasion to remark that first conceptions often deceive and that the
truth is not always probable.
It is difficult to understand and to define the desire of an assembly in
the midst of a variety of opinions of its members. Let us attempt to
give some rules in regard to this matter by considering the two most
ordinary cases: the election among several candidates, and that among
several propositions relative to the same subject.
When an assembly has to choose among several candidates who present
themselves for one or for several places of the same kind, that which
appears simplest is to have each voter write upon a ticket the names of
all the candidates according to the order of merit that he attributes to
them. Supposing that he classifies them in good faith, the inspection of
these tickets will give the results of the elections in such a manner
that the candidates may be compared among themselves; so that new
elections can give nothing more in this regard. It is a question now to
conclude the order of preference which the tickets establish among the
candidates. Let us imagine that one gives to each voter an urn which
contains an infinity of balls by means of which he is able to shade all
the degrees of merit of the candidates; let us conceive again that he
draws from his urn a number of balls proportional to the merit of each
candidate, and let us suppose this number written upon a ticket at the
side of the name of the candidate. It is clear that by making a sum of
all the numbers relative to each candidate upon each ticket, that one of
all the candidates who shall have the largest sum will be the candidate
whom the assembly prefers; and that in general the order of preference
of the candidates will be that of the sums relative to each of them. But
the tickets do not mark at all the number of balls which each voter
gives to the candidates; they indicate solely that the first has more of
them than the second, the second more than the third, and so on. In
supposing then at first upon a given ticket a certain number of balls
all the combinations of the inferior numbers which fulfil the preceding
conditions are equally admissible; and one will have the number of balls
relative to each candidate by making a sum of all the numbers which each
combination gives him and dividing it by the entire number of
combinations. A very simple analysis shows that the numbers which must
be written upon each ticket at the side of the last name, of the one
before the last, etc., are proportional to the terms of the arithmetical
progression 1, 2, 3, etc. Writing then thus upon each ticket the terms
of this progression, and adding the terms relative to each candidate
upon these tickets, the divers sums will indicate by their magnitude the
order of their preference which ought to be established among the
candidates. Such is the mode of election which The Theory of
Probabilities indicates. Without doubt it would be better if each voter
should write upon his ticket the names of the candidates in the order of
merit which he attributes to them. But particular interests and many
strange considerations of merit would affect this order and place
sometimes in the last rank the candidate most formidable to that one
whom one prefers, which gives too great an advantage to the candidates
of mediocre merit. Likewise experience has caused the abandonment of
this mode of election in the societies which had adopted it.
The election by the absolute majority of the suffrages unites to the
certainty of not admitting any one of the candidates whom this majority
rejects, the advantage of expressing most often the desire of the
assembly. It always coincides with the preceding mode when there are
only two candidates. Indeed it exposes an assembly to the inconvenience
of rendering elections interminable. But experience has shown that this
inconvenience is nil, and that the general desire to put an end to
elections soon unites the majority of the suffrages upon one of the
candidates.
The choice among several propositions relative to the same object ought
to be subjected, seemingly, to the same rules as the election among
several candidates. But there exists between the two cases this
difference, namely, that the merit of a candidate does not exclude that
of his competitors; but if it is necessary to choose among propositions
which are contrary, the truth of the one excludes the truth of the
others. Let us see how one ought then to view this question.
Let us give to each voter an urn which contains an infinite number of
balls, and let us suppose that he distributes them upon the divers
propositions according to the respective probabilities which he
attributes to them. It is clear that the total number of balls
expressing certainty, and the voter being by the hypothesis assured that
one of the propositions ought to be true, he will distribute this number
at length upon the propositions. The problem is reduced then to this,
namely, to determine the combinations in which the balls will be
distributed in such a manner that there may be more of them upon the
first proposition of the ticket than upon the second, more upon the
second than upon the third, etc.; to make the sums of all the numbers of
balls relative to each proposition in the divers combinations, and to
divide this sum by the number of combinations; the quotients will be the
numbers of balls that one ought to attribute to the propositions upon a
certain ticket. One finds by analysis that in going from the last
proposition these quotients are among themselves as the following
quantities: first, unity divided by the number of propositions; second,
the preceding quantity, augmented by unity, divided by the number of
propositions less one; third, this second quantity, augmented by unity,
divided by the number of propositions less two, and so on for the
others. One will write then upon each ticket these quantities at the
side of the corresponding propositions, and adding the relative
quantities to each proposition upon the divers tickets the sums will
indicate by their magnitude the order of preference which the assembly
gives to these propositions.
Let us speak a word about the manner of renewing assemblies which should
change in totality in a definite number of years. Ought the renewal to
be made at one time, or is it advantageous to divide it among these
years? According to the last method the assembly would be formed under
the influence of the divers opinions dominant during the time of its
renewal; the opinion which obtained then would be probably the mean of
all these opinions. The assembly would receive thus at the time the same
advantage that is given to it by the extension of the elections of its
members to all parts of the territory which it represents. Now if one
considers what experience has only too clearly taught, namely, that
elections are always directed in the greatest degree by dominant
opinions, one will feel how useful it is to temper these opinions, the
ones by the others, by means of a partial renewal.