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

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:

8 votes for A,
7 votes for B,
6 votes for C,

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, b  = 13, c  = 13,
 a' = 8, b'  = 8, c  = 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.