The Pamphlet Collection of Sir Robert Stout: Volume 53
Borda's Method
Borda's Method.
This method was proposed by Borda in 1770, but the first published description of it is in the volume for 1781 of the Memoirs of the Royal Academy of Sciences. For some remarks on the method see Todhunter's History of Probability, p. 433, where the method is described. In the case of three candidates, it is as follows. Each elector has three votes, two of which must be given to one candidate, and the third vote to another candidate. The candidate who obtains the greatest number of votes is elected.
In order to show that this method may lead to an erroneous result, suppose that there are twelve electors, of whom five prefer A to B and B to C, whilst two prefer A to C and C to B, and five prefer B to C and C to A. Then the votes polled will be, for A, fourteen; for B, fifteen; for C, seven. Thus B is elected. It is clear, however, that this result is wrong, because seven out of the whole twelve electors prefer A to B and C, so that, in fact, A has an absolute majority of the electors in his favour. Hence, then, Borda's method does not satisfy the fundamental condition, for it may lead to the rejection of a candidate who has an absolute majority of the electors in his favour.
It may be observed that the result of the poll on Borda's method may be obtained, in the case of three candidates, by adding together the corresponding results in the polls on the methods already described.
If there be n candidates, each elector is required to arrange them in order of merit; then for each highest place n—1 votes are counted; for each second place, n—2 votes, and so on; n— r votes being counted for each r th place, and no votes for the last place. The candidate who obtains the greatest number of votes is elected.
Borda does not give any satisfactory reason for adopting the method. Nevertheless he had great faith in it, and made use of it to test the accuracy of the ordinary or single vote method, and arrived at the extraordinary conclusion that in any case in which the number of candidates is equal to or exceeds the number of electors, the result cannot be depended upon unless the electors are perfectly unanimous. This in itself is sufficient to show that Borda's method must be capable of bringing about a result which is contrary to the wishes of the majority.
There is, however, another objection which is of great importance. Borda's method holds out great inducements to the electors to vote otherwise than according to their real views. For if an elector strongly desires the return of a particular candidate, he not only gives his two votes to that candidate, but he also takes care to give his remaining vote to the least formidable of the other candidates. The effect of this is to give a great advantage to secondrate candidates. Thus not only does Borda's method fail to interpret the true wishes of the electors, supposing that they vote honestly, but it holds out great inducements to them to vote otherwise than according to their real views.
Laplace discussed the question of the best mode of electing one out of several candidates, and by an analytical investigation was led to Borda's method. * He states distinctly that this method is the one indicated by the theory of probabilities. He then proceeds to point out the objection just stated, and expresses the opinion that the method would, without doubt, be the best if each elector would write the names of the candidates in what he thinks the order of merit. We have seen, however, that this is far from being the case.
* Journal de "Ecole Polytechnique, cahiers vii. and viii., pp. 169, 170; Théorie Analytique des Probabilités, pp. 101, 299; Todhunter's History of Probability, pp. 547, 548.