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The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Voting in (and after) the Revolution

Janet Heine Barnett (Colorado State University – Pueblo)

And what of the two methods of voting that were respectively championed by Borda and Condorcet in their pre-revolutionary works?

Clearly, elections of all kinds were being held in France during the Revolution. And, within those elections, the use of ballots to select from a slate of three or more candidates was not unknown. For example, delegates to the First National Convention were chosen by some local assemblies through a two-stage balloting process similar to what is known today as the Plurality with Elimination Method of Voting: if a candidate received a majority of the votes in the first ballot, that individual served as the elected delegate; otherwise, the top two candidates in the first round of voting were placed on a second ballot and the winner of that round became the elected delegate.[75] In other local assemblies, however, voting by voice (rather than ballot) served as an expedient way to select delegates that avoided counting ballots and was generally justified by a revolutionary-spirited appeal to its openness [Crook 1996, p. 88].

The use of ranked preference ballots that would require a more elaborate algorithm for determining the election outcome, such as those proposed by Borda and Condorcet, thus did not find its way into popular political elections. Not even in his rejected draft Constitution of 1792, which included a complicated electoral system for delegates to the Convention, did Condorcet mention the Pairwise Comparison Method of Voting that he concluded was the most likely to produce correct results in his 1785 Essai. Importantly for the later rediscovery of their works, however, the discussion of voting theory as it was developed by Borda and Condorcet did not die with the publication of their works in the 1780s. Within the Academy of Sciences, the outcomes of its various elections—which in turn depended on the method of election used—continued to have important consequences, both for the individual members of the Academy and the scientific community in general. Within this context, the voting methods proposed by both Borda and Condorcet did garner attention, both during and in the immediate aftermath of the French Revolution. Indeed, the Academy served as “a living laboratory where[in] to test, in the small, the virtues and the vices of different voting rules” [Barberà et al. 2020, p. 2].

Interestingly, the “voting laboratory” of the Academy of Sciences adopted the Borda Count Method of Voting for membership elections beginning in 1795, at the recommendation of author, historian and French statesman Pierre Daunou (1761–1840). In fact, this was the method in use when Napoléon Bonaparte[76] was granted membership in the National Institute[77] that replaced all Royal Academies between 1795 and 1815. Napoléon also served for a brief time as the Institute’s president, during which time his sole intervention in its running appears to have been his March 1800 request for a review of the balloting system—the same system by which he himself had been elected. The requested review was subsequently conducted by Daunou—the same person whose recommendation of Borda’s method led to its 1795 adoption.

In Daunou’s report on elections by ballot (presented in July 1801 and published in 1803), he reversed directions and expressed serious reservations about the point system proposed by Borda. In fact, his report was quite critical not only of the method itself, but also of Borda’s presentation of it in his 1784 “Memoire.”[78] After recounting Condorcet’s observations on the defects of Borda’s proposed method, Daunou wrote:

But it suffices to attentively read the memoir of Borda himself, to be convinced that his method is based on inaccurate, incomplete observations, which lead to very false results.[79]

He also offered the following comparison of Borda’s method with the method proposed by Condorcet.

Indeed, the results of this addition with regard to two subjects are variable according to the number of their other competitors, and according to the various ranks that these others obtain before, after or between the . . . two [given] subjects; while the comparisons between two candidates, or the preference given to one over the other, is, in the thought of each elector as in the general thought, a simple, constant, determined comparison, independent of all other relations.[80]

While Daunou’s representation of Condorcet’s Pairwise Comparison Method of Voting must certainly have appealed to its members’ quest for uniformity, constancy and rationality, the Institute nevertheless continued to use the Borda Count Method of Voting for some (but not all) types of elections. The fact that it began to use a variety of methods in its elections does, however, show that one of Daunou’s recommendations was heeded; namely, that different voting methods are more or less suitable for different types of elections. This latter point is one that is well worth raising with students, alongside a discussion of the advantages and disadvantages of the various methods of voting. In the next (and final) section of this article, we consider how the eighteenth-century French connection to voting theory might be used to promote student understanding of and interest in these and other issues related to collective decision making.


[75] The technical details of today’s Plurality with Elimination Method of Voting are different in that today’s method eliminates only the candidate with the least number of first-place votes from the ballot in each round. The use of ranked preference ballot as part of the Plurality with Elimination Method of Voting also avoids the need to cast additional ballots: the eliminated candidate in each round is simply removed from the list of each voter’s original preference ballot and first-place votes are re-counted until a majority candidate emerges.

[76] Although not a scientist himself, Napoléon was a strong supporter of mathematics and science; he is famously quoted, for example, to have said “The advancement and perfection of mathematics are intimately connected with the prosperity of the State.” The letter from which this quotation is taken was written by Bonaparte on August 1, 1812, to thank Laplace for providing him with a copy of his 1812 Théorie analytique de probabilités (Analytic probability theory). It appears as letter 19028 in the Correspondance de Napoléon, t. 24 (1868), p. 112, and reads in full (in English translation) as follows:

Monsieur le Comte Laplace, I have received with pleasure your treatise on the calculation of probabilities. There is a time when I would have read it with interest; today I must confine myself to testifying to you the satisfaction I experience every time I see you producing new works which are perfected and extended by this first of the sciences. They contribute to the illustriousness of the nation. The advancement and perfection of mathematics are intimately connected with the prosperity of the State.

Initially celebrated as a “Man of the Revolution” due to the military victories that he helped France to achieve as commander of the Army of Italy, Napoléon assumed the hereditary title of First Consul of France in 1799 before declaring himself Emperor in 1804 (until his abdication of that title in 1815).  It was during his time as First Consul that Napoléon was elected to the National Institute.

[77] The Academy of Sciences corresponded to the First Class of the National Institute during the Revolutionary and Napoléonic periods.

[78] To be fair, we note that, based on his description of Borda’s “Memoir,” it appears Daunou did not correctly interpret the details of Borda’s analysis of the “special election” method (the camouflaged version of the Borda Count Method of Voting in which head-to-head comparisons play a role). Daunou’s criticisms of the basic Borda Count Method of Voting were appropriate, however. Nor were Condorcet and Daunou the only Academicians to offer criticisms of Borda’s point system. In response to the observation that his system allowed voters to strategically manipulate election outcomes simply by insincerely ranking the strongest competitor to their preferred candidate in last place, even if that competitor was not honestly their last-place choice, Borda is famously reported to have remarked: “My scheme is intended only for honest men” (as quoted in [Mascart 1919, p. 130]). We recall that both Borda and Condorcet were dead by the time that Napoléon  requested a review of the Institute’s voting system, however, so that neither could have issued a rebuttal to Daunou’s report.

[79] Daunou 1803, p. 46.

[80] Daunou 1803, p. 48.


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