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

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.