You are here

The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Voting Theory in Condorcet’s Essai

Janet Heine Barnett (Colorado State University – Pueblo)

Borda described the purpose of his brief “Memoire” simply as an illustration that, in the case of elections with three or more candidates, the common belief that a plurality of (first-place) votes “always indicates the wish of the electorate” is mistaken [Borda 1784, p. 657]. In contrast, Condorcet began his 1785 book with a 191-page Discours Préliminaire,[28] or Preface, in which he laid out the motivation behind and the general layout of his 304-page Essai. Voting theory itself occupied only a small part of Condorcet’s massive (495-page!) book [Condorcet 1785]. Nevertheless, his name is also connected to standard topics within an undergraduate treatment of voting theory. In fact, three related concepts are named in his honor: Condorcet Paradox, Condorcet Candidate and Condorcet Fairness Criterion. These ideas are related to a voting method that involves head-to-head comparisons of each pair of candidates, a procedure that Borda also considered in his treatment of “special elections.” In this and the next two sections, we examine excerpts from the Preface to Condorcet’s Essai that justify the attachment of his name (and not Borda’s) to these ideas.

Condorcet’s analysis of elections involving more than two candidates appeared relatively late in the Preface of his Essai, following several examples related to jury trials and legislative voting that illustrated his concerns about the Plurality Method of Voting.[29] His reasons for seeking an alternative to the Plurality Method of Voting were, in fact, quite similar to those presented by Borda:

The method used in ordinary elections is defective. In effect, each voter is limited to naming the one that he prefers: thus in the example of three Candidates, someone that votes for A does not announce his view on the preference between B & C, and similarly for the others. However, there may result from this manner of voting a decision that is actually contrary to the plurality [choice].[30]

Like Borda, Condorcet also illustrated how a plurality result could be “actually contrary to the plurality [choice]” with a specific example:

Suppose, for example, 60 Voters, of whom 23 are in favor of A, 19 are in favor of B, & 18 are in favor of C; suppose next that the 23 voters for A would have unanimously decided that C is better than B; that the 19 Voters for B would have decided that C is better than A; and finally that of the 18 Voters for C, 16 would have decided that B is better than A, & only 2 that A is better than B.

One would have therefore, 1° 35 votes for the proposition B is better than A, & 25 for the contradictory proposition.[31]

2° 37 votes for the proposition C is better than A, & 23 for the contradictory proposition.

3° 41 votes for the proposition C is better than B, & 19 for the contradictory proposition.

We would therefore have the system of the three propositions that have the plurality, formed of the three propositions.

B is better than A,
C is better than A,
C is better than B,

            which implies a vote in favor of C.[32]

In other words, A would win the election under the Plurality Method of Voting, since A has more first-place votes than the other two candidates, but in reality C should win, since C is preferred in every head-to-head comparison against the other candidates. Today, such a candidate is called a Condorcet Candidate, and the assertion that a Condorcet Candidate (when one exists) should win the election is known as the Condorcet Fairness Criterion.

Either of the two examples that we have shared so far (from Borda and Condorcet respectively) suffice to show that the Plurality Method of Voting violates the Condorcet Fairness Criterion: in both cases, there is a Condorcet Candidate who loses the election under the Plurality Method of Voting.[33] Like Borda, Condorcet also used his example (the first of several) to set the stage for proposing an alternative to the Plurality Method of Voting. He continued his discussion of this particular example as follows:

What’s more, we would have two propositions that form a vote in favor of C.

C is better than A,
C is better than B,

one decided by a plurality of 37 against 23, the other by a plurality of 41 against 19.

The two propositions that form a vote in favor of B,

B is better than A,
B is better than C,

one decided by a plurality of 35 against 25, the other by a minority of 19 against 41.

Finally the two propositions that form a vote in favor of A.

A is better than B,
A is better than C,

decided by a minority, the one of 25 against 35, the other of 23 against 37.

So the one among these Candidates who would really have the preference of the plurality, would be precisely the one that, in following the ordinary method, would have the least numbers of votes.

Thus it is that A who, following the ordinary form, would have the most votes, is to the contrary found in reality to be the one who to be the furthest from being the wish of a plurality [of the voters].[34]

Condorcet’s move to a consideration of propositions (rather than simply the head-to-head vote counts themselves) may seem an unnecessary complication,[35] until one recalls the title of his text: Essay on the Application of the Analysis of Probabilities to Decisions Rendered by a Plurality of Votes. As previously noted, this work was part of a broader investigation into the problem of how individuals within a society could be provided with sufficient assurance that the decisions rendered by groups of individuals (e.g., trial juries, legislators, voters) are, in fact, correct. Naturally, the appraisals of these individuals could be mistaken concerning the issue at hand, even in the best of circumstances.[36] By considering the likelihood that each individual vote is correct, Condorcet was able to use the relatively new mathematical field of probability[37] as a tool for analyzing collective decision-making procedures under various conditions (e.g., the number of jurors or voters, the competence of the individual decision makers), with the goal of maximizing the probability that the final outcome is correct, or true.[38] In the case of elections, formulating ballot results as propositions (each of which has a truth value), allowed Condorcet to consider the probability that a given proposition (e.g., A is better than candidate B) is actually true, as well as the probability that a compound system of such propositions is true. Condorcet was especially interested in the probabilities of the compound systems, as these represented an aggregate opinion of all voters from which the election winner could be chosen. In the previous example, for instance, the opinion “B is better than A; C is better than A; C is better than B” is the one with the highest probability of being correct, which in turn implies that C is the candidate who should be elected.

Since the connection to probability theory is not part of a standard undergraduate treatment of voting theory, we will not examine the details of how probability did (and did not) enter in Condorcet’s analysis of elections.[39] Instead, we turn to the conclusions that Condorcet drew from the example presented above, which are relevant to share with students.


[28] Condorcet began his Preface with a declaration of his Enlightenment beliefs in the possibility of human progress and the power of rationality to promote it, framed as a dedication to his friend and mentor Turgot [Condorcet 1785, p. ii]:

A great man, whose lessons, examples, and especially friendship I will always regret [losing], perceived that the truths of the moral and political sciences are susceptible of the same certainty as those which form the system of physical sciences, and even the branches of those Sciences which, like Astronomy, seem to approach mathematical certainty.

This opinion was dear to him, because it led to the consoling hope that the human species will necessarily progress towards happiness and perfection, as it has done in the knowledge of the truth.

It was for him that I have undertaken this work, where by submitting to the Calculus [of probabilities] questions of interest to the common utility, I have tried to prove, at least by an example, this opinion which he wanted to share with all those who love the truth: he saw with difficulty several who, persuaded that one could not hope to deal with them, in questions of this kind, disdained, for this reason alone, to occupy themselves with the most important matters.

[29] In speaking of the origins and initial motive for using the Plurality Method of Voting to decide among alternatives, Condorcet again revealed his Enlightenment sentiments [Condorcet 1785, pp. ii–iii]:

When the custom of submitting all individuals to the will of the greatest number, introduced itself in societies, and men agreed to regard the decision of the plurality as the common will of all, they did not adopt this method as a means to avoid error and to conduct oneself according to decisions based on truth: rather they found that for the sake of peace and general utility, it was necessary to place the authority where the force was, & that, since it was necessary to let oneself be guided by a single will, it was the will of the small number which naturally had to sacrifice itself to that of the greater.

. . .

Among us, on the contrary, affairs are most often decided by the voice of a body of Representatives or Officers, either of the Nation or of the Prince. It is therefore in the interest of those who dispense the public force, to use that force only to support decisions conforming to the truth, and to give to the Representatives, who are responsible for pronouncing them, rules which answer to the rightness of their decisions.

[30] Condorcet 1785, p. lviii.

[31] Condorcet had earlier defined the “contradictory proposition” (in French: “proposition contradictoire”) to be the negation of the given proposition.

[32] Condorcet 1785, p. lviii.

[33] With regard to Condorcet’s first example and the Borda Count Method of Voting, it is straightforward to check that the Condorcet Candidate would have been chosen as the election winner; this is also the case for the example provided by Borda and the Borda Count Method of Voting. We come back to the question of whether this is always the case later in this section.

[34] Condorcet 1785, pp. lviii–lix.

[35] . . . and, for students in a liberal arts course or high school setting, bringing probability into the picture is most certainly going to be an unnecessary complication!

[36] Condorcet carried out much of the analysis in his Essai under the assumption that the voting circumstances were quite good:

We will first suppose that the assemblies are composed of Voters who are equally right-minded and with equal insight: we suppose that none of the Voters has any influence on the voices of the others, and that all opine in good faith. [Condorcet 1785, p. xxj].

Today’s readers may find these assumptions to be rather naïve. However, those familiar with the practice of mathematics will quickly see how the suppositions of equal right-mindedness and insight greatly simplified Condorcet’s initial analysis, by allowing the same probability of a correct decision to be assigned to each individual voter. As for the idea that individual voters might be motivated not solely by their own personal preferences, but by their honest opinion of who (or what) would be in the best interest of the general welfare of the society at-large (e.g., setting the innocent, but not the guilty, free; legislating laws that would not unnecessarily restrain individual liberties; electing the candidate who would best serve the needs of the organization or state), Condorcet was indeed an optimist who believed strongly in the possibility of individual enlightenment and social progress. Nevertheless, he was also well-acquainted with the realities of voting (in the Academy and elsewhere), and dedicated one of the later sections of his Essai to an examination of voting under non-ideal conditions such as “inequality of insight of right-mindedness of the voters, the supposition that the probability of their votes is not constant, the influence that some could have over the others, the bad faith of some, . . .”, commenting further that “these latter researches were necessary in order to be able to apply the theory in practice” [Condorcet 1785, p. xxiij]. This aspect of Condorcet’s Essai thus makes it an excellent model of how mathematicians tackle complex problems to share with students.

[37] The beginnings of mathematical probability reside in the study of games of chance that took place in the sixteenth and seventeenth centuries, including the 1564 book Liber de ludo aleae (The Book on Games of Dice) by Girolamo Cardano (1501–1557) and the famous exchange of letters between Pierre de Fermat (1601–1665) and Blaise Pascal (1623–1662) on “the problem of points.” The first published work on mathematical probability, De Rationiis in Ludo Aleae (On Reasoning in Games of Dice), was written by Christian Huygens (1629–1695) and published in 1675. The early eighteenth century then witnessed two significant publications in the field: The Doctrine of Chance by Abraham de Moivre (1667–1754) in 1711 and Ars Conjectandi (Arts of Prediction) by Jacob Bernoulli (1654–1705) in 1713. These two works inspired wider interest in probability among eighteenth-century mathematicians, including Buffon, Pierre-Simon de Laplace (1749–1827), Thomas Bayes (1767–1701), and, of course, Condorcet himself. Condorcet's first publication on probability was a 1784 memoir that appeared in the Academy’s proceedings. He also wrote a second full-length book on the applications of probability, Élémens du Calcul des Probablities et Son Application aux Jeux de Hasard, à la Loterie, et aux Jugemens des Hommes (Elements of the Calculus of Probability and its Application to Games of Chance, to the Lottery, and to the Judgements of Men), which was published posthumously in 1805. For a study of how the theory of probability was shaped by the Enlightenment concerns of the eighteenth century, and vice versa, see Lorraine Daston’s award-winning book Classical Probability in the Enlightenment, which Daston herself describes as “a study of a mathematical theory, but a mathematical theory about rationality in an age intoxicated by reason” [Daston 1988, p. xvii].

[38] In the case of a jury trial, the notion of a “correct decision” is clear: guilty individuals should be judged to be guilty, while the innocent should go free. In the case of voters in a legislative assembly or popular election, the notion of a “correct decision” was understood by Condorcet as the selection of the law or candidate that would best serve the common good.

[39] The interested reader may wish to consult [Young 1988] for a contemporary interpretation of Condorcet’s analysis. Or, better yet, one can find Condorcet’s treatment of these questions on pp. lxi–lxvii of the Preface and pp. 118–136 of Part 2 of his Essai [Condorcet 1785].

Janet Heine Barnett (Colorado State University – Pueblo), "The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Voting Theory in Condorcet’s Essai ," Convergence (September 2020)