Borda count

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MEMOIR

ON ELECTIONS BY BALLOT

By M. de Borda

It is a generally accepted opinion, and one against which I do not recall anyone ever having raised objections, that in an election by ballot, the plurality of votes always indicates the will of the electors—that is to say, that the candidate who obtains this plurality is necessarily the one whom the electors prefer to his competitors. But I am going to show that this opinion, which is true in the case where the election is between only two candidates, can lead to error in all other cases.

Let us suppose, for example, that the election is among three candidates A, B, C, and that the electors number 21. Let us further suppose that of these 21 electors, there are 13 who prefer candidate B to candidate A, and that only 8 prefer candidate A to candidate B; that these same 13 electors also give preference to C over A, while the other 8 give preference to A over C. It is clear that candidate A will have, in the collective opinion of the electors, a very marked inferiority, both with respect to B and with respect to C, since each of these latter, compared to candidate A, has 13 votes, whereas candidate A has only 8; from which it evidently follows that the will of the electors would exclude candidate A. Nevertheless, it could happen that in conducting the election in the ordinary manner, this candidate would have the plurality of votes. In effect, it is only necessary to suppose that among the 13 electors who favor candidates B & C, and who give one preference over the other, there are 7 who place B

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above C, and 6 who place C above B; then, in collecting the votes, we would have the following result:

8 votes for A,

7 votes for B,

6 votes for C.

Thus candidate A would have the plurality of votes, although, by hypothesis, the opinion of the electors was contrary to him.

Reflecting on the example reported, we see that candidate A has the advantage in the election result only because the two candidates B & C, who are superior to him, divided the votes of the 13 electors almost equally. One could compare them quite exactly to two Athletes, who, after having exhausted their strength against each other, would then be vanquished by a third weaker than each of them.

It follows from what we have just said that the ordinary manner of conducting elections is very defective, and the defect comes from the fact that in this form of election the electors cannot make known their opinion on the different candidates presented in a sufficiently complete manner. In effect, when among several candidates A, B, C, D, &c., one of the electors gives his vote to B, and another gives his to C, the first pronounces only on the superiority of B, relatively to all the competitors, and does not say what place he assigns to C among those he does not name. Similarly, the second, who accords C preference over all, does not say by this alone what place he assigns to B; however, this cannot be regarded as indifferent, because the one of the two who obtains a more distinguished place among those one does not name has, all things being equal otherwise, a claim of preference over the other, and in general the claim of each candidate to nomination by the electors is the result of the different places he occupies in the opinion of each elector; from which it follows that for a form of election to be good, it must give the electors the means to pronounce on the merit of each candidate, compared successively to the merits

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of each of his competitors. Now, there are for this two forms of election that one can equally adopt; in the first, each elector would assign places to the candidates presented, according to the degree of merit that he would recognize in each of them; in the second, one would make as many particular elections as there would be combinations among the candidates taken two by two, and thereby one would compare successively each candidate to all the others. It is easy to see that this latter form necessarily derives from the first, and that the one and the other would explain, as completely as is possible, the opinion of the electors for all the candidates presented; but the question is how one would conclude the result of the votes in these two kinds of election; and this is what I am going to examine in the remainder of this Memoir.

I shall begin with the first kind of election that I shall call election by order of merit. Let us first suppose that there are only three candidates presented, and that each elector has inscribed their three names on a ballot, ranking them according to the degree of merit that he attributes to each of them, and let

$$\begin{array}{lllll} 1. & A, & A, & B, & C, \\\ 2. & B, & C, & A, & B, \\\ 3. & C, & B, & C, & A, \end{array}$$

, &c. be these ballots; I shall first consider one of these ballots, for example, the first in which an elector has given first place to $A$, second to $B$, and third to $C$, and I say that the degree of superiority that this elector has accorded to A over B must be considered the same as the degree of superiority that he has accorded to B over C; in effect, as the second candidate B is equally susceptible to all degrees of merit comprised between the merits of the two other candidates A & C, one has no reason to say that the elector who has regulated the ranks among the three candidates wished to place him more or less near to A than to C, or, which is the same thing, that he attributed more superiority of the first over the second than of the second over the third. I say further that, because of the supposed equality among all the electors, each place assigned by one of the electors must be considered of the same value, and to suppose the same degree of merit that the place

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corresponding assigned to another candidate, or to the same by another elector whatsoever.

It follows from this that if one wishes to represent by $a$ the merit that each elector attributes to the last place, and by $a + b$ that which he attributes to the second, one must represent by $a + 2b$ the merit that belongs to the first, and it will be the same for places given by other electors, of which each last place will be equally represented by $a$, each second place by $a + b$, and each first place by $a + 2b$.

Let us now suppose that there are four candidates presented. One will prove by the same reasoning that the superiority of the first place over the second, that of the second over the third, and that of the third over the fourth, must be considered equal; and that the corresponding places given by different electors suppose the same degree of merit; from which one will conclude that the merits attributed by the electors to the fourth, third, second, and first places will be able to be represented by

$a, a + b, a + 2b, \text{ \& } a + 3b$.

It will be the same for a greater number of candidates presented.

This being established, it will be easy in any election whatsoever to compare the value of the votes accorded to different candidates. For this, one will multiply by $a$ the number of last-place votes given to each candidate; by $a + b$ the number of next-to-last votes; by $a + 2b$ the number of preceding votes, and so on; one will arrange all these different products for each candidate, and the sums of these products will represent the value of the votes accorded.

It is easy to see that in the question at hand, the quantities $a$ and $b$ can be whatever one wishes; one can therefore suppose $a = 1$ and $b = 1$, and then the value of the votes for each candidate will be represented by multiplying the

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number of last-place votes by 1, that of next-to-last votes by 2, that of preceding votes by 3, and so on up to the number of first-place votes, which will be multiplied by the very number of candidates.

Let us give an example of an election of this kind; let us again suppose 21 electors and three candidates presented $A, B, C$, and let

$$\begin{array}{ccccccccccccccccccccccc} A & A & A & A & A & A & A & A & B & B & B & B & B & B & B & C & C & C & C & C & C \\\ B & C & C & C & C & C & C & C & C & C & C & C & C & C & C & B & B & B & B & B & B \\\ C & B & B & B & B & B & B & B & A & A & A & A & A & A & A & A & A & A & A & A & A \end{array}$$

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

Votes for A:

8 first-place votes, multiplied by 3 = 24
13 third-place votes, multiplied by 1 = 13
Total: 37

Votes for B:

7 first-place votes, multiplied by 3 = 21
7 second-place votes, multiplied by 2 = 14
7 third-place votes, multiplied by 1 = 7
Total: 42

Votes for C:

6 first-place votes, multiplied by 3 = 18
14 second-place votes, multiplied by 2 = 28
1 third-place vote, multiplied by 1 = 1
Total: 47

from which one sees that the superiority of votes would be in favor of candidate C, that second place would be given to candidate B, and last place to candidate A.

It is to be remarked that if one had conducted the election in the ordinary manner, one would have had the following result:

8 votes for A,

7 votes for B,

6 votes for C,

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that is to say, that the plurality would have been for candidate $A$, who is last in the opinion of the electors, and that candidate $C$, who is truly first, would have had fewer votes than each of the other two.

Let us now suppose that one wishes to employ the method of particular elections [pairwise comparisons], and that there are equally three candidates presented $A, B, C$; as one can combine these three candidates taken two by two in three different ways, it will be necessary to hold three particular elections. Let the results of these elections be as follows:

1st election between $A$ & $B$ ...

$$ \begin{cases} a \text{ votes for } A \\ b \text{ votes for } B \end{cases} $$

2nd election between $A$ & $C$ ...

$$ \begin{cases} a' \text{ votes for } A \\ c \text{ votes for } C \end{cases} $$

3rd election between $B$ & $C$ ...

$$ \begin{cases} b' \text{ votes for } B \\ c' \text{ votes for } C \end{cases} $$

The question is to find the comparative value of the suffrages accorded to the three candidates. For this, we shall suppose that these elections are the result of an election by order of merit, which is always possible, because knowing the rank that each candidate occupies in the opinion of each elector, one can always determine the number of votes he ought to have in an election held between him and any other candidate.

This being established, let $y$ be the number of first-place votes that candidate $A$ would have obtained in this election by order of merit; $x$, the number of second-place votes; and $z$, the number of third-place votes. It is clear that then the value of the suffrages for candidate $A$ would be represented by $3y + 2x + z$; but $y + x + z =$ the total number of electors; let therefore this number $= E$; we will have, by eliminating $z$, the value of the suffrages for $A$ represented by $2y + x + E$, or simply by $2y + x$, because $E$ is common to all suffrages. Now, I remark that, for each first-place vote that candidate $A$ would have in the election

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by order of merit, he must have two votes in the particular elections; namely, one in the election between $A$ & $B$, and another in the election between $A$ & $C$; that for each second-place vote he would have in the election by order of merit, he will have only one in the particular elections; and that for third-place votes, he will have none.

From this one concludes that the number of votes he will have in all the particular elections, namely, $a + a'$, will be equal to $2y + x$; but we have just seen that this quantity $2y + x$ represents the value of the suffrages in the election by order of merit; therefore the quantity $a + a'$ will also represent it in the particular elections, that is to say, that the value of the suffrages accorded to a candidate will be represented by the sum of the votes he will have obtained in all the particular elections that concern him; which applies evidently to elections held among a greater number of candidates presented.

If one determines the values of $a, a', b, b', c, c'$, according to the supposition that the particular elections are the result of the election by order of merit reported above, one will find:

$$a = 8, \quad b = 13, \quad c = 13,$$

$$a' = 8, \quad b' = \cancel{13} \text{ [8]}, \quad c' = 13;$$

and consequently, one will have:

the suffrages of $A$ or $a + a' = 16,$ the suffrages of $B$ or $b + b' = \cancel{12} \text{ [21]},$ the suffrages of $C$ or $c + c' = 26;$

which gives between the three suffrages the same differences that were found by the first kind of election.

Moreover, we shall remark here that the second form of election which we have just spoken of would be embarrassing in practice, when a large number of candidates present themselves, because then the number of particular elections that would have to be held would be very great. According to this, one

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must prefer the form of election by order of merit, which is much more expeditious.

I shall terminate this Memoir by the examination of a particular question relative to the ordinary manner of holding elections. I have shown that in these elections, the plurality of votes is not always a certain indication of the will of the electors; but this plurality can be so great that it is not possible that the will of the electors is for another than for the one who has obtained this plurality. To determine in what cases this takes place, let $M$ be the number of candidates presented; $E$, the number of electors; $A$, the candidate who has the plurality; $B$, the one who, after candidate $A$, has the greatest number of votes; finally $y$, the votes for candidate $A$; and $z$, those for candidate $B$.

Let us suppose next that an election by order of merit is held among all the candidates; it is clear that then candidate $A$ will have a number of first-place votes $= y$, and that candidate $B$ will have a number $= z$. Now, all that could happen most unfavorable to candidate $A$ is that the electors who did not give him first place put him last, and that those who did not give first place to $B$ give him second place. In this case, as the value of first places is represented by $m$, that of second places by $m - 1$, and that of last places by $1$, one will have the value of the suffrages of $A = my + E - y$; and that of the suffrages of $B = mz + (m - 1)(E - z)$; it will therefore be necessary for the result of the election to be necessarily in favor of $A$, that one has:

$$ my + E - y > mz + (m - 1)(E - z), $$

$$ y > \frac{z + (m - 2)E}{m - 1}. $$

Let $m = 2$, one will have $y > z$, that is to say, that in the case where the election is held between only two candidates, the candidate who has the plurality is legitimately elected; thus in

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this case, but in that one only, the ordinary form of elections gives an exact result.

Let us suppose that candidate $B$ has all the votes that candidate $A$ does not have; then one will have $z = E - y$; putting this value in the expression above, one will have $y > E \cdot \frac{m - 1}{m}$.

If, in this latter expression, one makes $m = 3$, one will have $y = \frac{2}{3} E$, that is to say, that when there are three candidates presented, it is necessary, for one of the candidates to be assured of having the will of the electors, that he have more than two-thirds of the votes.

One will find similarly that when there are four candidates presented, $y$ must be greater than $\frac{3}{4}$ of $E$, and so on.

Finally, let the number of candidates be equal to the number of electors or greater than this number, the expression above $y > \frac{(m - 2)E + z}{m - 1}$ will become this one: $y > E - 1$, that is to say, that then the election cannot be rigorously decided except by unanimity, a result quite extraordinary which would justify the usage followed by a nation of the North in the election of its Kings.

It remains for me to observe, in finishing this Memoir, that all that we have said about elections applies equally to deliberations made by Bodies or Companies; these deliberations are in effect only kinds of elections between different opinions proposed, they are therefore subject to the same rules.

[Decorative floral ornament]

Mém. 1781. P p p p

endolith/Borda 1781.md

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