# 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.

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