# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

Voting theory has become a standard topic in the undergraduate mathematics curriculum. Its connection to important issues within a democratic society and the accessibility of its methods make a unit on voting theory especially well-suited for students in liberal studies program, as well as for students at the high school level. The pièce de resistance of such a unit is a somewhat startling theorem due to economist and Nobel Prize laureate Kenneth Arrow (1921–2017): there is no fair voting system.

First formulated in Arrow’s doctoral dissertation (published as the monograph Social Choice and Individual Values [Arrow 1951], Arrow’s Impossibility Theory can be stated as follows:

When voters have three or more alternatives, there is no voting method that can convert the ranked preferences of individuals into a community-wide transitive ranking of those alternatives, while also meeting a pre-specified set of fairness conditions in every election.

In other words, no matter how we decide to combine the individual preferences of a group in order to select a single winner from a slate of three or more candidates, something will sometimes go wrong with the election results in a way that can be characterized as either “irrational” (due to the failure of transitivity) or “unfair”. Unpacking what this means by exploring the relationship between different methods for determining election results (called Methods of Voting) and different notions of fairness (called Fairness Criteria) is the primary objective of the standard undergraduate treatment of voting theory.

The study of specific voting methods and their drawbacks actually dates back well before Arrow’s twentieth-century work. Indeed, Iain McLean has remarked that “the theory of voting has in fact been discovered four times and lost three times” [McLean 1990, p. 99]. Arrow, of course, was responsible for the fourth discovery. McLean’s 1990 article examines the first discovery, made at the hands of two medieval thinkers, Ramon Lull (c. 1235–1315) and Nicolas of Cusa (1401–1464), within the context of ecclesiastical elections. More recently, McLean [2019] has written about the third discovery by Charles Dodgson (1832–1898), the British mathematician more widely known as Lewis Carroll, who was motivated to write on the topic as a result of certain election decisions made by the faculty at Christ Church, Oxford. Yet, as those familiar with today’s treatment of voting theory will know, none of the names of Lull, Cusa or Dodgson/Carroll are generally associated with the topic.

In contrast, the second time that this discovery was made involved two late eighteenth-century French mathematicians for whom certain key ideas of voting theory are now named: Jean Charles, Chevalier de Borda (1733–1799) and Marie-Jean-Antoine-Nicolas de Caritat, Marquis de Condorcet (1743–1794). In this article, we examine the temporarily “lost” texts that explain the attachment of their names to those ideas:

 Figure 1: Portrait of Borda. Wikicommons. Figure 2: Portrait of Condorcet. Convergence Portrait Gallery.

We begin in the next section by considering the historical context in which these original works were written, together with biographical sketches of the professional lives of their authors in the period prior to the French Revolution, the start of which corresponds roughly to the publication dates of the two texts. We next provide an overview of the technical contents of the two texts as they pertain to the treatment of voting theory in today’s undergraduate curriculum. In the two penultimate sections of the article, we complete our biographical treatment of both men by sketching in a few details from their later lives, and also take a look at how their works were taken up (or not) by their contemporaries in France.

In closing, the final section of this article considers the role that Borda’s and Condorcet’s original texts and the related history could play within today’s classroom. For those interested in bringing their voices directly into the classroom conversation, we also offer a student project based on the original writings of Borda and Condorcet. That project, also entitled “The French Connection: Borda, Condorcet and the Mathematics of Voting Theory” (pdf), includes an appendix that further offers instructors the option of having students read more about the historical context in which Borda and Condorcet lived and worked, perhaps as part of an interdisciplinary unit with colleagues from history or the social sciences. For instructors who wish to dive even deeper into the project’s context or content during its classroom implementation, this article provides more detailed information, both historical and mathematical, than appears in either the project or its appendix. Let’s begin!

# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Borda and Condorcet in Pre-Revolutionary France

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

Like all human beings, Borda and Condorcet were products of the place and time in which they lived, as were their texts on voting theory.[1] Both men were born in France in the first half of the eighteenth century: Borda in 1733 and Condorcet in 1743. This situates their lives and works firmly within the period of the French Enlightenment, and the political revolutions which that movement helped to spur in the latter part of the eighteenth century. In this section, we focus on events related to their public lives, particularly the publication of their works on voting, in the period leading up to the 1789 start of the French Revolution. We begin with some general historical context.

While the starting date of the Enlightenment itself is debated by historians, its epicenter is universally recognized to be Paris. It was in the City of Lights that the artists, writers, politicians, scientists, and philosophers of the day began to regularly meet in intellectual circles, or salons, held in the homes of the (generally well-to-do) women, or salonnières, who organized and directed these gatherings.[2] Among the philosophes, as these thinkers came to be called, science and its successes over the preceding two centuries served as an important model of how knowledge can be generated through human reason and the evidence of our senses. Indeed, the name of the Enlightenment period (Siècle des lumières in French) itself grew out of the metaphor of bringing light to the dark ages that predated the Scientific Revolution. During the Enlightenment, human reason came to be seen not only as a means to shed light on our understanding of the universe in which we live, but as a means to improve our living conditions as social beings inhabiting that universe. This belief in the power of human reason as the primary source of authority led in turn to widespread criticism of existing political, religious and social institutions and the limitations they imposed on individual liberty and happiness.[3]

Figure 3: A Reading in the Salon of Mme Geoffrin, 1755.
By Anicet Charles Gabriel Lemonnier - Web Gallery of Art. Public Domain.

Unsurprisingly, one institution that escaped the criticism of the Enlightenment thinkers was the French Royal Academy of Sciences.[4] Founded in 1666 as one of the earliest learned societies in Europe, the Academy was initially organized around six sections (mathematics, mechanics, astronomy, chemistry, botany, and anatomy); members of the Academy belonged to one particular section based on their recognized accomplishments within that field of study. Importantly for the story we tell in this article, membership in the Academy was determined by election. The rules and regulations governing these elections were originally quite complicated, and Academy commissions were appointed to study the problem of reaching majority consensus in Academy elections as early as 1699 [Urken, 2009]. In the period immediately preceding the French Revolution, the usual process was for the current membership to produce, via voting, a list of ranked recommendations for new members that was forwarded to the King for his consideration; the King himself was not bound to accept the recommended ranking, or to even choose new members from the Academy’s list. The Academy itself, on the other hand, was expected to remain neutral in terms of politics, religion and social issues.

Both of our authors were elected to the Academy while still in their twenties, Borda in 1756 and Condorcet in 1769, based on the quality of their early works in mathematics. They also both began their mathematical educations under the Jesuits, Borda at the college in La Flèche and Condorcet first privately and then at the college in Reims. Despite the explicitly religious nature of the Society of Jesus itself, the goals behind the educational opportunities that it provided extended to the secular in that the Society sought to provide boys with the skills necessary to “take their place in that world even more strongly” and “be in a position to serve the common good” [Grendler 2014, p. 9]. Many of the boys who attended Jesuit colleges were of lower- and middle-class backgrounds, and the education they received was important simply for the sake of earning a living. Others—including both Borda and Condorcet—came from families with connections to the nobility[5] who were interested in seeing their sons trained for their future careers and to assume leadership positions within society.

In keeping with the job-oriented training that a Jesuit education sought to provide its students, mathematics was often taught in Jesuit colleges with an emphasis on its practical applications. The Jesuit college at La Flèche offered a course of study that was especially well-suited for the training of military engineers. Upon completing his studies at La Flèche in 1748, Borda entered the French army as a military mathematician. During this period of his military career, he wrote his first mathematical papers: a 1753 paper on geometry that brought Borda to the attention of Jean le Rond d’Alembert[6] (1717–1783), and a paper on the theory of projectile motion that secured his 1756 election to the Academy. Soon thereafter, in 1758, Borda enrolled at the School of Engineering at Mézières, taking just one year to complete the two-year course of study. He then entered the French navy, where he served in a variety of leadership capacities. These included participation in maritime campaigns during the American Revolutionary War between 1777 and 1778, appointment as the Major General of the Naval Army in 1781, and an appointment as the French Inspector of Naval Shipbuilding in 1784. In 1782, Borda was briefly held by the English after being captured while returning to France from Martinique, but he was soon released on his word of honor, perhaps due to his reputation as a scientist.

Ironically, Borda’s family had wanted him to pursue a career as a magistrate, while Condorcet’s family had wanted him to pursue a military career. In fact, both families had a long tradition of military service. Just days after his birth, Condorcet’s cavalry captain father was killed during an Austrian attack on the town of Neuf-Brisach.[7] Yet Condorcet himself rejected a military life, and instead transferred from the Jesuit Collège in Reims to the prestigious Collège de Navarre in Paris in 1758. He then studied at the equally prestigious College de Mazarin, also in Paris. At the age of 16, he successfully defended a dissertation on mathematical analysis (written in Latin) to a committee whose members included d’Alembert. Following this, Condorcet established himself as an independent scholar, living in Paris on a small allowance provided by his mother. As an independent scholar, Condorcet continued his study of mathematics, writing several works, including his Essai sur le calcul intégral [Condorcet 1765], which helped to secure his 1769 election to the Academy of Sciences.

Condorcet also fully immersed himself in Parisian Enlightenment society, attending the most important salons of the period and establishing friendships with d’Alembert[8] and other important philosophes. One philosophe friend in particular influenced Condorcet to become increasingly involved in public life: the reform economist Jacques Turgot (1727–1781). Turgot served as the national Minister of Finance under Louis XVI from 1774 to 1776, during which time he promoted policies aimed at mitigating the economic condition of the French people. After appointing Condorcet as the General Inspector of the French Mint in 1775, Turgot later persuaded him to remain in that post even after Turgot’s own dismissal from all governmental duties by Louis XVI. Condorcet remained head of the French Mint until 1790, serving for a time under Turgot’s political nemesis, Jacques Necker (1732–1804).

At least on the surface, then, our two authors shared a number of similarities: noble family backgrounds (both with long military traditions), initial training in the Jesuit tradition (but with no religious commitment of their own), early recognition of their mathematical talents (for which each earned the praise of d’Alembert), and prolonged service to their nation (Borda as a military man and Condorcet as an administrator). There were, however, marked differences between the two.

With regard to academic orientation, Borda’s interests were quite pragmatic, as one might expect of an engineer and military man. In addition to his study of applied topics such as projectile motion and fluid mechanics, he developed a number of utilitarian instruments, most notably the repeating circle, a celestial navigational device that employed a rotating telescope and a system of repeated observations to greatly reduce measurement error.[9] Borda’s other contributions in the area of marine navigation and cartography included development of a series of trigonometric tables, conduct of chronometer tests in the Caribbean, and creation of maritime charts of the Azores and Canary Islands.

Figure 4: Borda measuring the Peak of Ténériffe, Canary Islands.
Painting by Pierre Ozanne, 1775–1776. Wikimedia Commons.

Condorcet’s commitment to Enlightenment ideals, on the other hand, significantly influenced the direction of his scholarly work towards questions related to social issues. His interest in probability in the early 1780s, for example, was motivated in part by his vision of the role that probability could play in understanding and ameliorating social and economic conditions. Increasingly, however, Condorcet’s energies and interests were diverted away from considerations of how mathematics could promote social change, and towards active involvement in social reform efforts themselves. The extent to which this had occurred even before the start of the Revolution is highlighted by the author of Condorcet’s entry in the Dictionary of Scientific Biography, who remarked that “after 1787 Condorcet’s life is scarcely of interest to the historian of science” [Granger 1970–1990].

Borda and Condorcet also differed with regard to their general views on the nature of science and mathematics. Borda considered himself a disciple of Georges-Louis Leclerc, count de Buffon (1707–1788), a fellow Academician whose views about mathematics and science often came into conflict with those of d’Alembert.[10] As d’Alembert’s protégé, Condorcet held views about mathematics, science and its correct conduct that thus frequently ran opposite to those of Borda and Buffon. Within the political realm, the differences in their ideological beliefs were even more pronounced. A staunch monarchist, Borda supported the traditionalist views of Necker, Turgot’s political rival. Condorcet’s alignment with Turgot’s politics, on the other hand, placed him among the “modernists” who believed there was a need for constitutional restructuring in France.

Taken together, these differences were sufficiently divisive that Borda and Condorcet may have been seen as (and possibly were) rivals within the Academy. While Condorcet (in his letters to Turgot and elsewhere) was openly critical of many of Borda’s ideas, we will see later in this article that the two did work together on at least one occasion. And, as the economist Duncan Black (1908–1991) has noted, “during Borda’s prolonged absences from Paris [the two] corresponded on scientific matters” [Black 1958, p. 169].[11]

Whatever the personal relationship between the two men, Borda’s military and scientific travels did indeed often take him away from Paris and the day-to-day workings of the Academy. In contrast, Condorcet became intimately involved in the administrative work of the Academy relatively early, serving first as its Assistant Secretary (1773–1776) and then as its Permanent Secretary[12] (1776–1793). As Permanent Secretary, he was responsible for assembling the Academy committees that prepared responses to government requests for scientific studies related to proposed public works (e.g., prisons, canals) and composed reports on evaluations of new scientific discoveries.[13] His administrative role also placed Condorcet in the position of helping to organize the Academy’s various elections, including a vote to allow the publication his own Essai on voting theory [Condorcet 1785] by the Imprimerie Royale.

In the next section, we turn our attention to the mathematical details of both Borda’s and Condorcet’s texts on the topic of elections, a topic that was quite naturally of interest to all members of the Academy.

[1] Biographical information for Borda in this and later sections of this article is drawn from [Fairclough 2018], [Gillmour 1970–1990], [Mascart 1919], [Noguès n.d.], [O'Connor and Robertson 2003] and [Tietz 2017]; and, for Condorcet, from [Acton 2020], [Encyclopedia Britannica 1911], [Fonseca n.d.], [Granger 1970–1990], [Landes 2018] and [O'Connor and Robertson 1996]. Additional information about the relationship between the two men is drawn from [Black 1958], [Brian 2008] and [Young 1998].

[2] References consulted in connection with the Enlightenment and Parisian salon culture include [Brewer 2020], [Bristow 2017], [Haag 2020] and [Lougee 2020].

[3] One hears clear echoes of this, for example, in the Declaration of Independence of the United States, whose writers were strongly influenced by Enlightenment ideals.

[4] After the French Revolution, this was known simply as the French Academy of Sciences.

[5] Recall that Cordorcet’s actual name was Marie-Jean-Antoine-Nicolas de Caritat. The Caritats were an ancient family that had long resided in the town of Condorcet (located not far from Marseilles), from which he derived his title “Marquis de Condorcet.” This was a courtesy title (versus a title associated with land ownership) which had no legal significance, but served to indicate that Condorcet’s parents were members of the nobility. Borda came from a family of lesser untitled nobility that resided in the town of Dax, in the southwest region of France. His father held the rank of Lord (in French: Seigneur), while Borda himself held the rank of Knight (in French: Chevalier).

[6] In addition to his work in mathematics and physics, d’Alembert was a contributor to and co-editor (with the philosophe Denis Diderot (1713–1784)) of the Encyclopédie, the famous Enlightenment project aimed at spreading light across French society by way of educating its citizens. (For a student project based on one of d’Alembert’s contributions to the Encyclopédie, and more about his biography, see the Convergence article, “Investigations Into d'Alembert's Definition of Limit: A Mini-Primary Source Project for Students of Real Analysis and Calculus 2,” by Dave Ruch.)

[7] Neuf-Brisach is a fortified town in Alsace, near the Rhine River and the German border. It was designed by Sébastien Le Prestre de Vauban (1633–1707), a French military engineer whose principles for applying rational and scientific methods in fortification design were widely used for nearly a century. Neuf-Brisach is considered to be Vauban’s masterpiece and is now a UNESCO World Heritage site.

[8] Through his friendship with d’Alembert, Condorcet was recruited to write mathematical articles for supplements to Diderot’s famous Enlightenment project, the Encyclopédie. Their friendship was so close that, upon his death in 1783, d’Alembert left his property to Condorcet, thereby ending the financial difficulties that Condorcet faced up to that time. Soon thereafter, in 1786, Condorcet married the influential salonnière Sophie de Grouchy, whose salon at the Hôtel des Monnaies (which also housed the offices of the national Mint) was attended by foreign dignitaries such as Thomas Jefferson, who served as the American Minister to France from 1785 to 1789. The intellectual partnership between Condorcet and de Grouchy played a critical role in the social reform efforts in which Condorcet engaged during the last decade of his life, which we discuss in a later section of this article. We note here that, in contrast to Condorcet, Borda neither married nor participated in salon culture.

[9] Borda also published, in 1778, a pre-chronometer method for computing the longitude using lunar distances that is regarded as the best of several similar mathematical procedures that were available at that time. It was used, for example, by the 18031806 Lewis and Clark Expedition.

[10] See, for example, the discussion of the views held by Buffon and d’Alembert regarding rigor in mathematical proof in [Richards 2009].

[11] Black opened this sentence by claiming that “throughout their lives, the two were close friends” [Black 1958, p.179]. However, this claim is not repeated (or even mentioned) by any other historian that the author of this article has consulted, and it seems unlikely given the vast differences in their personal values and beliefs. Black provided little evidence for this assertion beyond providing a reference to a letter that is described and quoted in part by Borda’s biographer, French astronomer and mathematician Jean Mascart (1872–1935). According to Mascart [1919, p. 96], Borda wrote this letter to Condorcet as a response to the latter’s criticisms of the former’s work on fluid dynamics, and concluded it by writing (again, as quoted by Mascart):

Here is a response to your profession of faith: I am angry that you are not of my belief, but we will not both go to Paradise, except to argue about fluids. I embrace you with all my heart.

Whether the closing salutation [in French: Je vous embrasse de tout coeur] was merely a standard courtesy of the day or a genuine expression of fondness, the existence of the correspondence itself does suggest that relations between the two men were of a sufficiently cordial professional nature, even if not as personally close as Black suggested.

[12] Although his personal friendship with Turgot led to preferments (e.g., Turgot’s earmarking of a significant portion of the government’s funding for the Academy specifically to support Condorcet's research) that complicated Condorcet’s early relations with other Academicians, his 1776 election as the Academy’s Permanent Secretary was unanimous.

[13] Another of Condorcet’s responsibilities as Permanent Secretary was the preparation of official eulogies for Academicians and other important mathematicians and scientists. In all, he composed over fifty such eulogies, including those of his friend d'Alembert and his enemy Buffon, often inserting comments about mathematics, science and their role in society that were reflective of his own views, rather than those held by the subject of the eulogy.

# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – The Technical Side of Borda’s and Condorcet’s Works

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

As noted in the introduction, the two works on the mathematics of voting theory written by Borda and Condorcet, respectively, were essentially forgotten until the twentieth century. In this section, we consider the purely mathematical aspects of their works. We first briefly comment on their relative publication dates.

At first glance, Borda’s work had priority, by more than a decade, over that of Condorcet. Borda’s “Mémoire sur les élections au scrutin” (“Memoir on elections by ballot”) appeared in the Annual Proceedings of the French Academy of Sciences for the year 1781, together with an “Analysis” of that memoir that was likely written by Condorcet (in an editorial capacity derived from his position as Permanent Secretary). In a footnote to the title of his paper, Borda further asserted that he had already presented the ideas it contained to the Academy some 14 years earlier, on June 16, 1770—a claim repeated by the author of the “Analysis” of Borda’s “Mémoire” (again, likely Condorcet himself), but for which no archival evidence has yet emerged.[14] In fact, based on the available archival documentation, Brian [2008, p. 2] has shown that:

The date of 1781 is a fake. The story really begins the 14th of July of 1784, exactly five years before a date of some importance in the history of democracy.

It was on this “Bastille Day” date in 1784 that Condorcet, then Permanent Secretary of the Academy, announced his intention to publish a book on the application of probability to the analysis of electoral decisions. To do so under the auspices of the Imprimerie Royale, both a positive report by two referees and a vote of approval by the members of the Academy was required. The two referees were chosen by Condorcet, and the required report was read to a meeting of the Academy just three days later—such an astonishingly short turnaround time given the length and complexity of Condorcet’s tome that historians generally believe the report was written by Condorcet himself.[15] Regardless of who actually wrote that report, it sufficed to secure the vote of approval necessary to go forward with publication. A possible delay then arose when Borda read his own paper on the arithmetic of voting to the Academy on July 21, 1784, apparently as a means to establish priority. In response to Borda’s paper, Condorcet took quick action to avoid a potential stumbling block to his own publication plans: using his authority as Permanent Secretary, he arranged for Borda’s paper to be included in the Academy’s proceedings that were then in preparation, as those would appear in print prior to the intended publication date for his own book. To quote Brian [2008, p. 3] once more:

This is why a paper, supposedly read in 1770, actually presented in 1784, was published in an official academic volume for the year 1781—the one under press in July 1784.

Because Condorcet’s Essai—actually written, supposedly reviewed by referees and already approved for publication by July 1784—was not officially published until 1785, Borda thus technically published on voting theory a year before Condorcet. To see why Condorcet’s work is nevertheless the more celebrated of the two as a pioneering contribution to voting theory, we now turn to a brief overview of the technical contents of both works, beginning with that of Borda.

[14] Brian does note that the topic of elections was under discussion in the Academy in 1770, and that it is likely that Borda had shared the idea behind his voting method at that time, but not in a published paper [Brian 2008, p. 3].

[15] Even over a century after its publication, Condorcet’s text has been described as “one of the most frequently cited, least-read and poorly-understood works in voting theory” [Urken 2008, p. 1].

# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Voting Theory in Borda’s “Memoire”

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

One of the standard voting methods studied in an undergraduate treatment of voting theory is named in honor of Borda’s 9-page paper [Borda 1784]. Under the Borda Count Method of Voting, candidates are assigned points based on their rankings across all ballots; for instance, 1 point for each last place vote, 2 points for each second-to-last place vote, and so on. The candidate with the highest total number of points is then selected as the winner of the election. Borda himself called this voting method an “election by order of merit,” and presented the following as an example.[16]

Suppose again 21 voters and three presented subjects A, B, C and let

A  A  A  A  A   A  A  A  B  B   B  B  B  B  B  C  C  C C  C  C
B  C  C  C  C  C  C  C  C  C  C  C  C  C C  B  B  B  B  B  B
C  B  B  B  B   B  B  B  A  A   A  A  A  A  A  A  A  A  A  A  A

be the 21 election ballots. One will have by what has been already said, the comparative value of votes by multiplying the first-place votes by 3, the second-place votes by 2, and the third by 1, which will give the following results.

Votes for A $$\left\{\begin{array}{c}8 \mbox{ first-place votes, multiplied by } 3 = 24\\13 \mbox{ third-place votes, multiplied by } 1 = 8\end{array}\right \}$$ 37.

Votes for B $$\left\{\begin{array}{c}7 \mbox{ first-place votes, multiplied by } 3 = 21,\\7 \mbox{ second-place votes, multiplied by } 2 = 14\\7 \mbox{ third-place votes, multiplied by } 1 = 7\end{array}\right \}$$ 42.

Votes for C $$\left\{\begin{array}{c}6 \mbox{ first-place votes, multiplied by } 3 = 6\\14 \mbox{ second-place votes, multiplied by } 2 = 28\\1 \mbox{ third-place votes, multiplied by } 1 = 1 \end{array}\right \}$$ 47.

From which one can see the superiority of votes will be in favor of subject C, that the second place will be given to subject B & the last to subject A.[17]

Borda continued:

It may be remarked that if one conducted the election in the usual manner, one would have had the following result:

That is to say that the plurality would have been for subject A, who is last in the opinions of the voters, & that the subject C, who is really the first, would have had fewer of the votes than each of the other two.[18]

Here, we hear Borda’s primary complaint about the Plurality Method of Voting: if only the first-place votes are considered in deciding the results of an election, then the wrong candidate could end up winning the election. He first expressed this complaint (and illustrated it by way of this same example) at the very start of his paper in order to illustrate how the Plurality Method of Voting could “induce an error” in elections involving more than two candidates.

It’s clear then that the subject A will have, in the collective opinion of the voters, a marked inferiority, as much with respect to B as with respect to C, because each of these latter [two], compared to the subject A has 13 votes, while the subject A only has 8; from this it clearly follows that the voice of the electors would exclude subject A.[19]

Having noted that a candidate who loses to every other candidate in head-to-head comparisons (in this example, candidate A) could end up winning the overall election under the Plurality Method of Voting,[20] Borda provided the following analogy to describe how this situation could come about:

In reflecting on the reported example, one sees that the subject A only has the advantage in the results of the [plurality] election, because the two subjects B & C, who are superior to him, almost equally divided between them the votes of the 13 electors [who did not place A in first place]. One could compare them fairly exactly to two athletes, who, after having spent their forces one against the other, were then defeated by a third [athlete] more feeble than either of them.[21]

Having provided an example to illustrate how election by order of merit (or the Borda Count Method of Voting) could be used to avoid such an error in at least some elections, Borda next considered a second method of voting which assigns point totals in a different fashion.

Let us suppose now that one wishes to use the method of special elections,[22] & that there are once more three subjects A, B, C presented; as one can combine these three subjects taken two by two in three different manners, it would be necessary to have three special elections. Let the results of these elections be as follows.

1.st election between A & B . . . $$\left\{\begin{array}{l}a \mbox{ votes for }A,\\b \mbox{ votes for } B,\end{array}\right .$$

2.nd election between A & C . . .  $$\left\{\begin{array}{l}a' \mbox{ votes for }A,\\c \mbox{ votes for } C,\end{array}\right .$$

3.rd election between B & C . . . $$\left\{\begin{array}{l}b' \mbox{ votes for }A,\\c' \mbox{ votes for } C,\end{array}\right .$$  . . . . [23]

Borda next discussed in general how the values of a,a',b,b',c,c' relate to the point totals of the candidates in an “election by order of merit,” showing that candidates A,B,C will have Borda count totals of a +a' + E, b + b' + E, c + c'+ E points respectively, where E is the number of voters casting ballots in the election. He then illustrated this idea by applying the “method of special elections” to his earlier example:

If one determines the values of a,a',b,b',c,c' under the supposition that the special elections are the result of the election by order of merit that reported above, one would find[24]

a = 8, = 13, = 13,
a' = 8, b'  = 8, = 13,

& consequently, one will have

the votes for A where a + a' = 16,
the votes for B where b + b' = 21,
the votes for C where c + c'; = 26,

which gives between the three votes the same differences that would have been found by the first type of election.[25]

In short, Borda algebraically proved, and then illustrated via an example, that the two methods of voting he presented are equivalent: not only do they lead to the same election results, but the point totals assigned to the candidates by each method are readily converted to those assigned by the other method.[26] Having established their equivalence, Borda further commented:

Besides [the fact that these two methods produce the same results], we will remark here that the second form of election of which we have spoken, would be awkward in practice, when a large number of candidates are presented, because the number of particular elections that it would be necessary to complete, will be very large. For this reason one should prefer the form of election by order of merit, which is much more expedient.[27]

In the next section of this article, we consider how Condorcet made use of pairwise comparisons, despite their “awkwardness,” in a considerably different fashion than Borda proposed.

[16] All French-to-English translations in this paper are due to its author. An English translation of Borda’s paper also appears in [de Grazia 1953].

[17] Borda 1784, p. 661.

[18] Borda 1784, p. 661.

[19] Borda 1784, pp. 657–658.

[20] A candidate who loses to all others in head-to-head comparisons is today called a Condorcet Loser. In general, the notion of a Condorcet Loser is not part of the standard undergraduate treatment of voting theory. As we remark in a later section of this article, the notion of a Condorcet Candidate—a candidate who wins against all others in head-to-head comparisons—is included in such a treatment. Although candidate C In Borda’s example does defeat both A and B in head-to-head comparisons (and is thus a Condorcet Candidate), Borda himself did not comment on this fact.

[21] Borda 1784, pp. 657–658.

[22] In French: elections particulières.

[23] Borda 1784, p. 662. Regrettably, Borda had earlier employed the variables a,b,c for a different purpose, which renders the paper somewhat cumbersome for direct use with students in its entirety.

[24] In our translation, we have corrected two typographical errors (both related to an incorrect value for b') that appeared in the original published version of Borda’s paper.

[25] Borda 1784, p. 663. We recall here that the Borda count totals were 37 = 16 + 21, 42 = 21 + 21, and 47 = 26 + 21 for A,B and C respectively, where there were E = 21 votes in this particular election.

[26] Borda did not explicitly say why he offered this second method, despite the fact that he personally understood that it was an equivalent to the first. One reason for his decision to describe both methods may have been to address potential resistance to his own preferred method (i.e., election by order of merit); after all, two-candidate elections are unproblematic (clearly, the candidate with the most votes wins), and  his “special elections” method is at least superficially based on a series of two-candidate elections..

[27] Borda 1784, pp. 663–664.

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

Author(s):
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].

# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Condorcet’s Use of Pairwise Comparisons and the Condorcet Paradox

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

Having given a particular example to exhibit the inherent flaw that can result in election decisions based on the Plurality Method of Voting, Condorcet next noted that there is one circumstance in which the Plurality Method of Voting could suffice to select the correct choice among candidates, namely when there is a candidate with a majority of the first place votes:

One sees therefore already that one should reject the form of election generally adopted: if one wishes to conserve it, it will be possible [to do so] only in the case where one is not required to elect on the spot, & where one could require that only someone who would have gathered more than half of the votes would be looked upon as elected. In this same case, this form [of voting] would again have the disadvantage of agreeing to look upon as non-elected the one who would have in reality has a very great plurality.[40]

Today, a candidate with a majority of the first-place votes is known as a Majority Candidate, and it is easy to see that a Majority Candidate is also necessarily a Condorcet Candidate. Similarly, it is straightforward to show that a Majority Candidate (if one exists) will necessarily win an election under both the Plurality and Borda Count Methods of Voting. This latter voting method was not, however, the one endorsed by Condorcet as an alternative to the former. Instead, he declared:

So one should in general substitute for this [plurality] form [of voting] that in which each Voter, [by] expressing the order according to which he places the Candidates, would pronounce at the same time on the preference that he would accord respectively to each.[41]

In other words, election decisions should be based on the outcomes of the head-to-head comparisons of every pair of candidates, which in turn can readily be determined from voters’ ranked orderings of the candidates. Importantly, Condorcet’s use of head-to-head comparisons differed from the way such comparisons featured in Borda’s “special elections” method. In that method, recall that Borda summed the numbers of votes cast for each candidate in all of the various pairs in a way that rendered that method equivalent to the basic Borda Count Method of Voting. Given this equivalency together with the dramatic increase that occurs in the number of pairs as the number of candidates increases, Borda ultimately rejected using head-to-head comparisons as an avoidable inconvenience. In contrast, Condorcet’s plan tallies the actual number of votes cast for each candidate in the pair only to determine which of the two candidates in the pair wins that particular comparison: did the voters find that A is better than B, or that B is better than A? Once this is decided, the vote totals themselves are discarded.

Unlike Borda, Condorcet also seemed undaunted by the number of head-to-head comparisons that may need to be tallied.[42] Nor was he daunted by a feature of his recommended voting method that is today known as Condorcet Paradox: the final results of the head-to-head comparisons can be non-transitive. Here is Condorcet’s first example of this situation.

Let us suppose indeed that in the example already chosen, where one has 23 votes for A, 19 for B, 18 for C, the 23 votes for A are for the proposition B is better than C; this proposition [B is better than C] will have a plurality of 42 votes against 18.

Let us next suppose that of the 19 votes in favor of B, there are 17 for B is better than C, & 2 for the contradictory proposition; this proposition C is better than A will have a plurality of 35 votes against 25. We suppose finally that of the 18 votes for C, 10 are for the proposition A is better than B, & 8 for the contradictory proposition, we will have a plurality of 33 votes against 27 in favor of the proposition A is better than B. The system that obtains the plurality will therefore be composed of three propositions,

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

This system is . . . one of the two [possible systems] that implies contradiction.[43]

For many students (and others), the fact that individual transitive preferences (an assumption implicit in the use of rank orderings) can lead to a intransitive group decision (with A > B, B > C, but C > A) is indeed a paradoxical aspect of the mathematics of social choice. How can the group outcome be so irrational when each individual within the group has voted rationally? As noted earlier, however, Condorcet himself was not fazed by this fact—indeed, he did not even view it as an impediment to the viability of the Pairwise Comparison Method of Voting. Instead, he devoted several pages of his Essai to a discussion of how to proceed with voting in order to produce the winner with the highest probability of being the correct choice in the event that election results do result in one of these “contradictory systems” [Condorcet 1785, pp. clxx–clxxvii; pp. 118–136]. While his full description of this somewhat complicated procedure in Part 2 of his Essai was not completely clear,[44] Condorcet’s concluding remarks about contradictory systems in his Preface provide a glimpse of the basic idea behind his procedure, both in its technical aspects and its societal implications:

Finally, let us observe that these contradictory systems cannot arise without indicating uncertainty in [the] opinions [of the voters], and they will not take place, neither if the votes being taken as usual, one of the Candidates has more than half of the votes, nor if one requires in order to admit the propositions which form the ballot, a plurality of one third.

There results from all the reflections that we have just made, this general rule, that whenever we are forced to elect, it is necessary to take successively all the propositions which have the plurality, beginning with those which have the largest, and pronounce according to the result formed by these first propositions, as soon as they form one, without having regard to the propositions less likely to follow them.

If by this means we disagree with the result least susceptible to error, or a result whose probability is greater than 1/2, and formed from two propositions which are more probable than their contradictories, we will at least have the one which does not oblige [us] to adopt the least probable proposals, and from which it results a lesser injustice between the Candidates, considered two by two.[45]

Today’s standard undergraduate treatment of voting theory features a simplified version of Condorcet’s voting procedure that drops his provision for handling cases in which Condorcet Paradox occurs. Known as the Pairwise Comparison Method of Voting, this method simply uses the number of comparisons won by each candidate to determine the overall election winner.

[40] Condorcet 1785, p. lviii.

[41] Condorcet 1785, p. lviii.

[42] Condorcet was, however, fully aware of the combinatorial challenges involved in using pairwise comparisons as part of one’s voting method, and remarked immediately following the preceding excerpt that:

One would draw from this order [in which candidates are ranked by each voter] the three propositions which should form each opinion, if there are three Candidates; the six propositions that should form each opinion, if there are four Candidates; the ten, if there are five, &c. in comparing the votes in favor of each of these propositions or their contraries [Condorcet 1785, p. lx].

Indeed, because of his emphasis on systems of propositions, the combinatorial challenges involved in Condorcet’s analysis went even further, as he himself noted:

One will have by this means the system of propositions, which will be formed . . . among the 8 systems possible for three candidates, the 64 systems possible for four Candidates, the 1024 systems possible for five Candidates & if one considers only those that do not imply a contradiction, there will be only 8 possible for three Candidates, 24 for four, 120 for five, and so on [Condorcet 1785, p. lx].

In the full analysis of the problems of voting that he provided in Part 2 of the Essai, he also computed these numbers for the general case if n candidates; see footnote 44 below for a statement of his results.

[43] Condorcet 1785, p. lxj. The actual sentence is: “This system is the third, & one of the two that implies contradiction.” The “third” here refers back to a slightly earlier section of his Preface [pp.lvj–lviij] in which Condorcet analyzed the eight different three-proposition systems that can result from an election involving three candidates, finding that two of the eight combinations favored A, two favored B, two favored C, and the final two involved non-transitive results. In Part 2 of his Essai¸ he also provided a full analysis of possible aggregate opinions in the case of n candidates, including a computation of how many of these opinions were contradictory. See also footnote 44.

[44] Condorcet’s statement of his proposed voting procedure appeared in Part 2 of his Essai [Condorcet 1785, pp. 125–126]:

1.° All possible opinions that do not imply a contradiction reduce to an indication of the order of merit that one judges to exist among the candidates. . . . Therefore for nCandidates, one will have $$n(n-1) \ldots 2$$ possible opinions. . . .

2.° Each Voter having thus given his opinion by indicating the candidates’ order of worth, if one compares them two by two, one will have in each opinion $$\frac{n(n-1)}{2}$$ propositions to consider separately. Taking the number of times that each is contained in the opinion of one of the q voters, one will have the number of voices who are for each proposition.

3.° One forms an opinion from those $$\frac{n(n-1)}{2}$$ propositions that agree with the most voices. If this opinion is among the $$n(n-1) \ldots 2$$ possible opinions, one regards as elected the Subject to whom this opinion accords the preference. If this opinion is among the $$2^{\frac{n(n-1)}{2}}-n(n-1)\ldots2$$ impossible opinions, then one successively deletes from that impossible opinion the propositions that have the least plurality, and one adopts the opinion from those that remain.

The instruction in 3.° that directs us to “successively deletes from that impossible opinion the propositions that have the least plurality” is the step in Condorcet’s algorithm that scholars have (rightfully) found to be vague. Unfortunately, Condorcet neglected to provide examples in his Essai or elsewhere that would clarify exactly how he meant this step to work. In his study of this particular feature of Condorcet’s voting theory, the economist and game theorist H. P. Young has cited the critical opinions of several Condorcet scholars, including that of the mathematician Isaac Todhunter (1820–1884): “The obscurity and self-contradiction are without any parallel, so far as our experience of mathematical works extends . . . no amount of examples can convey an adequate impression of the evils” (attributed in [Young 1988, p. 1234] to [Todhunter 1949, 352]).Young himself went on to propose a plausible hypothesis about Condorcet’s intended meaning, and described this interpreted version of the algorithm as “a novel and statistically correct rule for finding the most likely ranking of the alternatives” and Condorcet’s development of it as “one of the earliest applications of what today would be called ‘statistical hypothesis testing’” [Young 1988,, pp. 1231, 1235].

[45] Condorcet 1785, pp. lxix–lxx. True to the non-mathematical goals that Condorcet held for his Essai, he went on to write [Condorcet 1785, p. lxx]:

Moreover, the Voters must be enlightened, and all the more enlightened, as the questions which they decide are more complicated; otherwise we will find a form of decision which will preserve the fear of a false decision, but which at the same time making any decision almost impossible, will only be a means of perpetuating abuses and bad laws. Thus the form of the assemblies which decide the strength of men, is much less important for their happiness than the enlightenment of those who compose them: & the progress of reason will contribute more to the good of the Peoples than the forms of political constitutions.

# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Condorcet’s View of the Borda Count Method of Voting

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

As Condorcet noted, the Pairwise Comparison Method of Voting satisfies both the Majority and Condorcet Fairness Criteria. Of course, the Borda Count Method of Voting (Borda’s proposed alternative to the Plurality Method of Voting) also selects the Condorcet and  Majority Candidates as the overall winner in at least some cases (e.g., the two examples that we have seen thus far), and, because it is based on integer point totals (which always fall in transitive order), it avoids both the non-transitivity that can occur with Condorcet’s method, as well as the potentially tedious computations of pairwise comparison. Condorcet, having arranged for the publication of his paper, was well aware of Borda’s suggested alternative, which he described as follows:

A celebrated Geometer, who has observed before us the drawback of ordinary elections, has proposed a method, which consists of having each Voter give the order in which he ranks the candidates; of then giving to each first-place vote, the value of unity, for example; to each second-place vote a value less than unity; a value still smaller for each third-place vote, & so on, & of then choosing the candidate for whom the sum of these values, taken across all the voters, will be the largest.

This method has the advantage of being very simple, & one can without doubt, by setting the law of decreases for these values,[47] avoid many of the difficulties that the ordinary method has, of giving as the decision of the plurality one that is really contrary to it: but this method is not strictly protected from that drawback.

Although Condorcet did not say anything further about how “setting the law of decreases for these values” could lead to different election outcomes for the same set of preference ballots, his suggestion that the point values we assign for each place could change the election winner is one that can (and should) be explored by students. Also highly suitable for exploration by students is the election example that Condorcet gave in which the Condorcet Candidate loses the election under the Borda Count Method of Voting.  [48]

Having thus noted that Borda’s proposed method shares certain defects of the Plurality Method of Voting (e.g., violation of the Condorcet Fairness Criterion[49]), in addition to certain drawbacks of its own (e.g., susceptibility to manipulation of election outcomes by varying point assignments), Condorcet concluded his discussion on Borda’s method with remarks that might be taken as politely-veiled insults—never actually naming Borda, but making it clear to their fellow Academicians who this “celebrated Geometer” was! Any response that Borda may have made in return (assuming he was even aware of Condorcet’s remarks, hidden away as they were in his voluminous text) was either never recorded or has not yet been found in written records. In the next section, we consider the turn of events in France that almost certainly distracted both men from further discussion of the mathematics of voting theory.

[46] Condorcet 1785, p. clxxvij.

[47] Condorcet’s comments about the assignment of point values seem to have been targeted at a shortcoming in Borda’s presentation of his preferred method of voting. In particular, Borda argued that the assigned point must form an arithmetic sequence (i.,e., $$a, a+b, a+2b, \ldots$$ points, starting with last place on up) because “there is no reason to say that an elector who has settled the ranks between three subjects wanted to attribute more superiority to his first [choice] over his second [choice] than he attributed to his second [choice] over his third [choice]” [Borda 1784, p. 659]. Borda went on to remark that “because of the supposed equality between all voters, each place assigned by one of the electors, should be the same value” [Borda 1784, p. 659]. While this latter assumption by Borda (that the same point structure should be applied to the ballot of every voter) is necessary to make the method mathematically viable, his former assumption (that the point structure should form an arithmetic sequence) is not considered an essential feature of today’s Borda Count Method of Voting.

[48] In summary, the example provided by Condorcet (showing that the Borda Count Method of Voting violates the Condorcet Fairness Criterion) was a three-candidate election with 81 voters, 30 of whom ranked the candidates in the order A,B,C; 1 in the order A,C,B; 10 in the order C,A,B; 29 in the order B,A,C; 10 in the order B,C,A; and 1 in the order C,B,A. In this election, A is a Condorcet Candidate, whereas B would win the election under the Borda Count Method of Voting, regardless of how the point values for first and second place are assigned. Verifying this latter fact is an especially nice exploration for students.

[49] The Borda Count Method of Voting also violates the Majority Fairness Criterion, although Condorcet did not give an example to illustrate this fact.

# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Borda and Condorcet in Later Years

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

A landmark event that is often hailed as the official start of the French Revolution took place exactly five years after Condorcet announced his intention to publish on voting theory to the Academy, when the Bastille prison was destroyed by a group of Parisians on July 14, 1789. The events of the next decade significantly shaped the lives and works of all French citizens, including Borda and Condorcet. In this section, we complete our biographical sketch of their professional lives during that decade, by the end of which both had died. We again begin with some general historical context.

While the causes and events behind the French Revolution are numerous and complex, these certainly included the citizenry’s discontent with both the French monarchy in general and the fiscal crisis in which the monarchy found itself in particular. As a result of that crisis, the government of Louis XIV proposed the enactment of new tax measures in May 1789, as a means to stave off bankruptcy. Circumstances related to that proposal in turn led to the formation of a National Assembly[50] (later renamed the National Constitutional Assembly) by a group of individuals who demanded governmental restructuring, and, further, that the new structure be guaranteed by a written constitution. Grudgingly acknowledged by Louis XVI in June 1789, the National Assembly began to function as a governing body on July 9, 1789. Meanwhile, popular insurgencies began to arise throughout France, including the Paris uprising that culminated in the storming of the Bastille. On August 26, 1789, the Assembly approved the Declaration of the Rights of Man and of the Citizen,[51] premised on the belief that “­the ignorance, neglect, or contempt of the rights of man are the sole cause of public calamities and of the corruption of governments.” The “natural, unalienable and sacred rights of man” set forth in the Declaration’s seventeen articles[52] served as a set of guiding principles for the aspirations of the revolutionaries. The Declaration later became the preamble of the constitution[53] that was ratified by the Assembly on September 3, 1791, thereby establishing France as a constitutional monarchy with sovereignty residing effectively in the Assembly.

Figure 5: One of two remaining provisional meter sticks
(located in the Place Vendome and at 36 rue de Vaugirard respectively)
of the sixteen meter sticks that were originally mounted throughout Paris in 1796–1797
to provide the public with a means of learning the new measurement system. Photo by the author.

Borda’s contributions to the development of the metric system essentially constituted the full extent of his participation in revolutionary activities, as one might expect given his conservative political leanings. In contrast, Condorcet’s ever-increasing involvement in the political events of the time led to the (non-mathematical) works and the writings for which he is best remembered by the world-at-large today. In light of his Enlightenment values, Condorcet’s enthusiastic support of the ideals set forth in the Declaration of the Rights of Man and of the Citizen comes as no surprise. Echoes of his belief in the universality of equal rights can be heard in his assertion (as quoted in [Alder 2002]):

A good law ought to be good for all men,
as a good proposition [in geometry] is good for all men.[61]

In fact, Condorcet embraced this ideal beyond a literal interpretation of the phrase “all men.” He was ahead of his time, for instance, in supporting the abolition of slavery, as well as equal political rights for Jews, Protestants and women.[62] And, his commitment to social equality was much more than academic. When the National Constitutional Assembly was disbanded in September 1791 (its work in drafting a new constitution having been completed) and the Legislative Assembly took its place, Condorcet was elected as one of the delegates representing Paris on the new Assembly, where he quickly established himself as a torch bearer for the revolution. He served, for instance, as the secretary of the Assembly and led the efforts of the Assembly’s Committee on Education by drafting a plan for state-supported education[63] based on the principles of meritocracy and equality, in keeping with his belief that social progress required an enlightened citizenry.

It was also Condorcet who composed the Legislative Assembly’s declaration that justified the later suspension of the monarchy. Despite Louis XVI’s reluctant acceptance of France’s new constitutional monarchy, growing fears of counter-revolutionaries and mistrust of the King’s intentions led to the full arrest of the Royal family in 1792.[64] In September of that year, the Legislative Assembly was replaced by the National Convention and the (First) Republic of France was declared. Condorcet was again elected as a delegate to the Convention, and served as both its secretary and vice-president. He also chaired the committee that was charged with drafting the new republic’s constitution. Although Condorcet promptly began drafting a version of that constitution, its completion was delayed by his scholarly perfectionism and his extensive involvement in other political activities. In the interim, Louis XVI was condemned to death by the Convention and executed by guillotine in Paris in January 1793.[65] By the time Condorcet presented his careful and thorough draft constitution to the Convention, the relatively moderate political group with whom he was associated (the Girondists) was beginning to lose power. For this and other reasons,[66] Condorcet’s proposed constitution was rejected in June 1793, in favor of a hastily-drafted “Constitution of Year I” that was put forward by a far more radical group (the Jacobins). With the passage of the Jacobins’ constitution, a 9-member Committee of Public Safety[67] gained unlimited executive power and the infamous period of the revolution known as the Reign of Terror began soon thereafter. Between September 1793 and July 1794, Queen Marie-Antoinette and thousands of other French citizens from all social classes and professions were accused of counterrevolutionary actions and met their fate, as had Louis XVI, at the blade of the guillotine.[68

Although Condorcet himself escaped the guillotine, he did not outlive the Reign of Terror. After openly and passionately accusing the Committee of Public Safety of using fear tactics to gain passage of their version of the constitution, a warrant for his arrest was issued by the Convention in July 1793. Condorcet became a fugitive, and hid in the Parisian home of Madame Vernet[69] for the next eight months. During this time, Condorcet wrote his most famous philosophical work, Esquisse d'un tableau historique des progrès de l'esprit humain (Sketch of a historical picture of the progress of the human mind),[70] a declaration of his enlightenment belief in the power of rationality and perfectibility of human society. Concerned about the peril in which he was placing his protector, Condorcet fled Paris on foot on March 25, 1794, but was denounced to local authorities by an innkeeper in the nearby village of Clamart-de-Vignoble. Placed under arrest, he was transferred to a guarded house[71] located in the village of Bourg-la-Reine[72] on March 29, 1794. The following day, Condorcet was found dead in his makeshift cell; whether he took his own life, died of natural causes, or was murdered is unknown to this day.[73]

Figure 6: The gaoler opening the door of the prison cell to find Condorcet
lying dead on his bed. Stipple engraving by G. Aliprandi after J.H. Fragonard.
Wellcome Collection. Attribution 4.0 International (CC BY 4.0).

As for Borda, after retiring to his family estate during the Reign of Terror, he returned to Paris and resumed his work on the metric system. He died[74] after a long illness on February 20, 1799, at age 65, just months before the mètre des Archives, a platinum bar selected as the first prototype of the meter based on the results of the meridional survey effort that Borda coordinated, was placed in the National Archive.

The final years of Borda’s life, from 1796 to 1799, also witnessed the military rise of Napoléon Bonaparte (1769–1821), which in turn signaled the end of the revolutionary period in France. In the next section, we return to the history of the Academy of Sciences, where Napoléon exerted his influence as well.

[50] The formation of the National Assembly was instigated by delegates of the Third Estate who were summoned to Versailles by Louis XVI for an emergency meeting of the General Estates, a parliamentary entity that was convened (for the first time since 1614) with the express purpose of securing the necessary approval to enact his government’s tax proposals. Traditionally, each of the three Estates voted as a single group, despite the fact that the Third Estate represented a far greater proportion of the population than either the First or Second Estates, which represented the clergy and the aristocracy respectively. Although members of the Third Estate are generally referred to as “commoners,” its representatives at the Estates General predominantly belonged to the bourgeoisie, particularly the legal professions. After their demand that each deputy instead be given a vote was overruled by a declaration from the First Estate, the representatives of the Third Estate constituted itself as a National Assembly on June 17, 1789; they were later joined in their commitment to create a constitutional monarchy by members of the other two estates.

[51] Like the Declaration of Independence of the United States of America, written some 13 years earlier, primarily by Thomas Jefferson, the Declaration of the Rights of Man and of the Citizen clearly embodies Enlightenment ideals. It begins with a preamble that describes the Assembly’s goal in setting forth a set of “simple and incontestable principles” as an effort to combat the “ignorance, forgetfulness or contempt of the rights of man [that are] the only causes of public misfortunes and the corruption of Governments.” By serving as a unceasing reminder to “all members of the body politic . . . of their rights and their duties,” its authors asserted the desire that “the acts of the legislative power and those of the executive power . . . may . . . be the more respected” and that “the demands of the citizens . . . may always be directed toward the maintenance of the Constitution and the happiness of all.” Jefferson, who was then living in Paris as the US ambassador to France, served as a consultant to those authors, Emmanuel-Joseph Sieyès (1748–1836) and the Marquis de La Fayette (1737–1834), the latter of whom had served as a military consultant to the US during its war of independence from Britain. Unlike the US document, which served in a sense as a rationale for that country’s declaration of war aimed at ousting an oppressor, the French document was initially intended by its authors to serve as a rationale for a transition from an absolute to a constitutional monarchy, rather than an attempt to overthrow the monarchy per se.

[52] The seventeen articles of the Declaration were individually considered and approved by the National Assembly between August 20 and August 26, 1789.

[53] The preamble of the current French constitution still asserts the constitutional status of the Declaration of the Rights of Man and of the Citizen, by declaring that “The French people solemnly proclaim their attachment to the Rights of Man and the principles of national sovereignty as defined by the Declaration of 1789 . . .” [Constitution of the Fifth Republic of France, adopted October 4, 1958].

[54] These complaints were quite literally registered in the Cahiers de Doléances (Complaints Books) that were filed with King Louis XVI (at his invitation) by individuals, trade guilds and legislative units across France in preparation for the 1789 meeting of the General Estates. For example, of the 50 topics found among the complaints submitted by the Third Estate, weights and measures ranked fourteenth [Shapiro et al. 1998, p. 381]. Other frequently mentioned topics of complaint included personal liberties, taxes and the judiciary system.

[55] Condorcet was influential in convincing the National Assembly to approve funding for the Academy in connection with metric reform, although the proposal itself was put before the Assembly by Bishop Charles Maurice de Talleyrand, a former member of the First Estate who aligned himself with the revolutionary cause early on.

[56]According to both [Mascart 1919, p. 497] and [Heilbron 1990, p. 224], the other three members of the Commission on Weights and Measures were initially Jean-Louis Lagrange (1736–1813), the chemist Antoine Lavoisier (1743–1794) and the botanist Mathieu Tillet (1714–1791). After rendering an initial report to the Assembly on October 27, 1790, in which a decimal system was recommended, Lavoisier and Tillet were replaced by Pierre-Simon Laplace (1749–1827) and Gaspard Monge (1746–1818).

[57] The assigned tasks and original working group members were: triangulations and latitude determinations (Cassini IV, Méchain, Legendre), baselines (Meusnier, Monge), pendulum of Paris (Borda, Coulomb), weight of water (Lavoisier, Haüy), and comparison of old and new measures (Tillet, Brisson, Vandermonde). Responsibility for overall supervision of the endeavor was assigned to Borda, Condorcet, Lagrange, and Lavoisier. Two of these four (Condorcet and Lavoisier) died during the Reign of Terror that took place from July 1793 to September 1794. Several members of the original working groups also left the project due to “death and disinclination” [Heilbron 1990, p. 224]. See also [Mascart 1919, pp. 497–500].

[58] Heilbron [1989, p. 991] reports that estimates of the actual amount spent by the Academy to complete the survey range from 300,000 to millions of livres. An additional 500,000 livres in funding was also granted to the Committee of Public Instruction by the National Assembly in order “to defray all expenses relative to the establishment of the new measures, as well as the advances which are indispensable for the success of such work,” as stipulated in Article 21 of a National Assembly decree approved on April 7, 1795.

[59] Some have suggested that the Commission’s decision not to instead define the meter as the length of a pendulum with a half-period of one second (called a seconds pendulum) was due to an interest within the Academy in seeing a demonstration of the accuracy of Borda’s repeating circle. See, for instance, [Hahn 1971, p. 164].

[60] The full text of the legislative acts related to the official adoption of the metric system in France can be found in a report on the question of adopting the metric system in the United Kingdom that was later commissioned by Queen Victoria [Airy et al. 1869].The French decree of August 1, 1793, that legally adopted the metric system as that nation’s official system of weights and measures also mandated its obligatory use beginning July 1, 1794. In a new decree approved on April 7, 1795, the latter provision was rescinded indefinitely, due in part to the reluctance of the public to abandon its use of the existing units of measurements. (That same decree also indefinitely suspended the 1793 provision requiring the use of the decimal division of the day and the parts thereof.) Historian of science Kenneth Alder has offered a convincing explanation for why there was such resistance to actually using the metric system, despite the fact that it was complaints from the French people themselves that helped bring the new system into existence [Alder 1995; Alder 2002]. His book, The Measure of all Things: The Seven Year Odyssey and Hidden Error that Transformed the World [Alder 2002], also provides a riveting account of the surveying project behind the definition of the meter.

 Figure 7: Woodcut dated 1800 illustrating the new decimal units which became the legal norm in France on November 4, 1800, five years after the metrical system was first introduced. In the captions, each one of these six new units are followed by the old French unit in brackets.These were for length (metre), area (are = 100 m2), solid volume (stère = 1 m3), liquid volume (litre = 1 dm3), mass (gramme = the mass of 1 cm3 of water) and currency (franc).

[61] In the original French: “Une bonne loi doit être bonne pour tous les hommes, comme une proposition vrai est vrai pour tous.” Condorcet wrote his “Observations de Condorcet sur le vingt-neuvième livre de "L'esprit des lois" (”Observations on the twenty ninth book of “Spirit of Laws”) in 1780 as a commentary on a highly influential political theory treatise by Charles de Secondat, Baron de Montesquieu (1689–1755), De l'esprit des loix (The Spirit of the Laws), originally published anonymously in 1748. The publication details of Condorcet’s essay are not fully clear. It was not included in the 1804 edition of Condorcet’s collected works, but does appear in Volume 1 of the 1847 edition [Condorcet 1847, pp. 363–388]. In the intervening years, Condorcet’s essay appeared in its entirety in Thomas Jefferson’s 1811 English translation of de Tracy's Commentaire sur l'Esprit des Lois de Montesquieu par M. le comte de Destutt de Tracy (A Commentary and Review of Montesquieu's Spirit of Laws) [de Tracy 1811], and in the later French editions of that same work [de Tracy 1817; de Tracy 1819]. Antoine Louis Claude Destutt, comte de Tracy (1754–1836) was a French philosopher who is credited with coining the term "ideology" while being held in prison during the Reign of Terror.

[62] Condorcet wrote several essays advocating the abolition of slavery, the first of which, “Reflexions sur l’esclavage des nègres” (“Reflections on the enslavement of negros”), was published well before the revolution, in 1781. He was also a founding member, along with the military commander La Fayette, of the Société des Amis des Noirs (Society of the Friends of Negros), established in 1788. His 1790 essay, “Sur l’admission des femme au droit de Cité(“On the admission of women to the right of Citizenship”), further argued the ability to reason, a human attribute shared by men and women of all races, justifies granting equal rights to all:

The rights of men stem exclusively from the fact that they are sentient beings, capable of acquiring moral ideas and of reasoning upon them. Since women have the same qualities, they necessarily also have the same rights. Either no member of the human race has any true rights, or else they all have the same ones; and anyone who votes against the rights of another, whatever his religion, colour or sex, automatically forfeits his own.

[63] The plan for state-supported education that Condorcet helped to draft formed the basis of the national educational system that was eventually adopted, which in turn led to the creation of the École normale supérieure and the École centrale des travaux publics (later called the École polytechnique).

[64] The Royal family had been essentially under house arrest at the Palace of the Tuileries in Paris since the so-called Women's March on Versailles took place on October 6, 1789, in part as a reaction to the chronic hunger and increasing frustration brought about by the costs and scarcity of bread in the city. Concerns about Louis XVI’s intentions became more pronounced following his family’s aborted attempt to flee the country during the night of June 20–21, 1791. Nevertheless, the National Constitutional Assembly was willing to allow Louis to retain the throne under a constitutional monarchy at that time. According to the nineteenth-century historian François Auguste Marie Mignet (1796–1884), after Louis XVI examined the Assembly’s 1791 constitution, he sent a letter to the Assembly in which he wrote: “I accept the constitution. I engage to maintain it at home, to defend it from all attacks from abroad; and to cause execution by all the means it places at my disposal. I declare, that being informed of the attachment of the great majority of the people to the constitution, I renounce my claim to assist in the work, and that being responsible to the nation alone, no other person, now that I have made this renunciation, has a right to complain” [Mignet 1846, p. 110]. The Assembly is then reported to have subsequently invited Louis XVI to address their final session on September 29, 1789, where the King pronounced: “Tell [your fellow citizens] that the king will always be their first and most faithful friend; that he needs their love; that he can only be happy with them and by their means; the hope of contributing to their happiness will sustain my courage, as the satisfaction of having succeeded will be my sweetest recompence” [Mignet 1846, pp. 110–111].

[65] Although opposed to the death penalty on principle, Condorcet did not vote against the death penalty for Louis XVI, but instead chose to abstain. The abolition of the death penalty was, however, one of the 370 articles in Condorcet’s proposed version of the new constitution. In his 1791 “Opinion sur le jugement de Louis XVI” (“Opinion on the trial of Louis XIV”), a copy of which was owned by Thomas Jeffereson, Condorcet also presented his probabilistic view of voting to argue against executing the King.

[66] While the politics of the French revolution are too complex to recount here, the description of Condorcet’s abilities (or lack thereof) as a politician offered by historian John Herivel suggest another possible factor in the failure of Condorcet’s constitution [Herivel 1975, p. 247]:

Condorcet was no politician. His uncompromising directness of manner and inability to suffer illogical windbags in silence made him many enemies and few friends. His weak voice, lack of oratorical powers, and tendency to bore the Convention by the excessive height of his arguments was one of the tragedies of the Revolution.

[67] The Committee of Public Safety was actually created in April 1793, to replace the previous Committee of General Defense, an administrative body that was charged with protecting the new republic from internal and external attacks. In July 1793, Robespierre (1758–1794) became a member of the restructured Committee of Public Safety and served as its de facto leader until his own arrest and subsequent death by guillotine a year later essentially brought the Reign of Terror to a close.

Interestingly, the mathematician and military engineer Lazare Carnot (1753–1823) served as a member of the Committee of Public Safety from September 1793 until it was disbanded in March 1795. As a relatively conservative member of the committee, Carnot concentrated primarily on issues related to education and military operations. From 1795 to 1797, he continued to provide military leadership for the country as an elected member of the Directory, the five-member executive branch of the re-organized government that ruled France immediately following the dissolution of the Committee of Public Safety. As the Director in charge of military affairs, it was Carnot who appointed Napoléon Bonaparte (1769–1821) as general-in-chief of the Army of Italy. When Napoleon declared himself emperor of France in 1804, the republican-minded Carnot retired from public life. He lived out his final days in exile from France, devoting his energies to writing works on engineering and geometry.

[68] Estimates of the number of individuals that were executed in France during the Reign of Terror by the guillotine or other means range upwards of 17,000, excluding those who died at the hands of vigilantes or in prison while awaiting their trial. Lynn, who sets the number of official executions at 40,000, further estimates that the number of individuals who were arrested accounted for about 1 in every 50 French citizens, or approximately 300,000 individuals in all [Lynn n.d.]. Members of the clergy and aristocrats were disproportionately represented both among those arrested and those executed by the state.

[69]Located on Paris’ left bank on the Rue des Fossoyeurs (now 15 Rue Servandoni), Condorcet’s refuge for these months is situated just behind the beautiful Église Saint Étienne de Mont, where Blaise Pascal (1623–1662) is buried. Coincidentally, Olympe de Gouges (1745–1793), a playwright and author of the Declaration des Droites de Femmes et Citoyennes (Declaration of the Rights of Women and Citizenesses), lived on this same street, within half a block of Condorcet’s hiding place. Although it is unlikely that they associated with each other during this time, the two were almost certainly acquainted, given De Gouges’s attendance at the salon of Condorcet’s wife Sophie de Condorcet, née de Grouchy, at the Hôtel des Monnaies. Condorcet only met his protector Madame Vernet, the widow of the sculptor Louis-François Vernet (1744–1784), when mutual friends brought him to her home seeking refuge.

[70] Condorcet’s Esquisse was published posthumously by his wife, Sophie de Condorcet, née de Grouchy, in 1795. In 1799, she also published his Éloges des academicians (Eulogies for the Academicians). Although de Grouchy had filed for divorce while Condorcet was in hiding, she did so with his secret consent as a move to prevent the seizure of her property by the revolutionary government due to her marital connection to the fugitive Condorcet. This strategy failed when he died before the divorce was finalized. De Grouchy thereafter earned a living for herself and their daughter Eliza by painting miniature portraits. She also continued to edit Condorcet’s collected works, and remained active in France’s intellectual culture through her re-established salon. Although de Grouchy wrote many articles that were published anonymously, both before and after the revolution, the only one of her works to appear in her own name was the book Huit Lettres sur la Sympathie (The Letters on Sympathy), which was published alongside her translation of Adam Smith’s Theory of Moral Sentiment and Origins of Language in 1798.

[71] Located at 49 Grande Rue (now 81 Ave Le Clerc), the lower floor of the house in which Condorcet spent his last night was being used as a hair salon in 1999 when the author took the following photo (Figure 8).

[72] At the time, the town was called “Bourg-de-l’Égalité.” Bourg-la-Reine is also familiar to students of the history of mathematics as the birthplace of Évariste Galois (1811–1832), known both for his contributions to algebra and as a staunch supporter of the French republicans during the post-Napoléonic era.

[73] Although the location of his remains is also unknown to this day, memorials of Condorcet’s life and work include plaques at his final “residences” in Paris and Bourg-la-Reine, a statue at 15 Quai de Conti in Paris (between the Musée de Monnaie and the Institut du France), and the symbolic interment in the Pantheon in Paris of ashes taken (in 1989) from the cemetery in Bourg-la-Reine.

[74] Memorials of Borda’s life and work include a statue in the Place de la Halle in his birthplace of Dax, and a series of French naval vessels harboured in Brest (and nicknamed The Borda) that have successively served as the site of French Naval School since its founding in 1830. The current collection of scientific instruments and other objects in the Borda Museum in his hometown of Dax was founded on Borda’s own collection of curiosities. Borda’s name (but not Condorcet’s) is also listed, together with 71 other scientists, on the first floor of the Eiffel Tower. Both men have Paris streets named after them (Borda in the 3rd Arrondisement and Condorcet in the 9th).

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

Author(s):
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.

# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Borda and Condorcet in Today's Classroom

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

In this article, we have attempted to show that the temporarily “lost” texts written by Borda and Condorcet in the 1780s merit the attachment of their names to certain ideas that are part of today’s mathematical treatment of voting theory. Certainly, the technical content of their works shows that the two men did indeed study and care deeply about the concepts that are now attributed to them. The historical and institutional contexts in which they wrote further help to explain why the other pre-Arrow discoveries of voting theory (by Lull and Cusa in the medieval period and Dodgson in the nineteenth century) remain relatively unknown even today. In the brief historical remarks added to the 1963 second edition of Social Choice and Individual Values, Arrow himself confessed to a “certain want of diligence in tracking down the historical origins of social choice” [Arrow 1963, p. 93]. He then applauded the “excellent history” of the topic given in The Theory of Committee and Elections [Black 1958], “beginning with the work of Borda, and including that of Condorcet, Laplace, Nanson, Galton and, most especially C.L. Dodgson (Lewis Carroll)” [Arrow 1963, p. 94]. But Black was really the first to take note of Dogdson’s work, and Lull’s and Cusa’s ideas were re-discovered even later. For all three of these authors, readership of their works at the time they were written was quite limited. In contrast, the ongoing and well-documented discussions of election methods within the French Academy of Sciences were bound to be noticed once Arrow’s Impossibility Theorem was announced in 1951, as indeed they were.

Our second objective in this article has been to share the story of Borda’s and Condorcet’s treatment of issues related to elections involving more than three candidates with instructors who teach this topic, but are unfamiliar with why their names came to be associated with them. We have included details of the non-mathematical background to their works because of the rich social and cultural connections that could be made by bringing the broader historical context into the classroom. The lives and ambitions of our two authors further offer students a wonderful window through which to glimpse the human side of mathematics. Especially within a high school setting, the voting theory works of Borda and Condorcet could also serve as a unifying theme for an interdisciplinary unit on the French Enlightenment and Revolutionary period.

As we hope has been made clear, the nature and content of Borda’s and Condorcet’s original writings further offer an excellent vehicle for teaching students the technical aspects of voting theory themselves. The student project “The French Connection: Borda, Condorcet and the Mathematics of Voting Theory” (pdf)  interweaves excerpts from these two primary sources with exercises intended to engage students with the mathematical ideas contained in those excerpts. Designed to be completed via a combination of individual advance reading, small-group work and whole-class discussion over the course of approximately two weeks, this project contains all the content from a standard textbook treatment of this topic, including the Plurality, Plurality with Elimination, Borda Count and Pairwise Comparison Methods of Voting; the Majority, Condorcet, Independence of Irrelevant Alternatives and Monotonicity Fairness Criteria[82]; and the use of a Preference Schedule as a means to organize voter ballots. By drawing on Condorcet’s rich discussion of his own motivations for studying the problem of collective decision making, the project also goes beyond a standard textbook treatment in terms of its investigation of why Arrow’s Impossibility Theorem, and voting more generally, matters to their own lives.

We close by quoting the final paragraph from the student project, inspired by the writing of Condorcet, and offered as advice to students and instructors alike:

Voting is a privilege, one that is enjoyed by too few people in today’s world, and which can be too easily lost. Those of us who enjoy this privilege also bear the responsibility of helping to fulfill Condorcet’s vision of enlightened voters committed to the common good of society-at-large. Be sure to do your bit by becoming educated about the issues, identifying and setting aside your prejudices (we all have them, after all), and, most importantly, getting out there to vote.

[82] These latter two criteria, Independence of Irrelevant Alternatives and Monotonicity, are described in a non-historical fashion in a later section of the project, since neither Borda nor Condorcet discussed them. All other concepts in the project are introduced through some connection to excerpts from the primary sources.

# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – References

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

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# The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – About the Author

Author(s):
Janet Heine Barnett (Colorado State University – Pueblo)

Janet Heine Barnett joined the faculty at Colorado State University – Pueblo in 1990, and now holds the rank of Professor Emerita of Mathematics. A 1995–1996 fellow at the MAA Institute for History of Mathematics and Its Use in Teaching (funded by the NSF), her scholarly interests have long included the history of mathematics and its use in promoting mathematical understanding and as a vehicle for promoting teacher reflection on pedagogical issues. She currently serves as a PI on the NSF-funded project TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS), and has now written twenty Primary Source Projects through that grant and its parent and grandparent grants. She also currently serves as an editor of Convergence, with Amy Ackerberg-Hastings.

An active member of the MAA Rocky Mountain Section, Janet received the Section Certificate of Meritorious Service Award in 2007. She has also been recognized at the University level, regionally and nationally for her excellence in teaching, including the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics in 2016. In 2019, she was awarded the President’s Medallion for Distinguished Service to Education by Colorado State University – Pueblo.

Janet shares her passions for mathematics and history, as well as dance and yoga, with her husband and traveling companion George Heine, whom she met while serving as a Peace Corps volunteer in the Central African Republic (1982–1984).