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The Pamphlet Collection of Sir Robert Stout: Volume 53

Condorcet's Practical Method

Condorcet's Practical Method.

This method was proposed in 1793 by Condorcet, and appears to have been used for some time at Geneva. It is described at pp. 36—41 of vol. xv. of Condorcet's collected works (edition of 1804), and may be used in the case of any number of candidates for any number of vacancies. We are at present concerned only with the case of a single vacancy; and for the sake of simplicity I describe Condorcet's method for the case in which there are only three candidates.

Two scrutinies may be necessary in order to ascertain the result of the election in this method. At the first scrutiny one vote is counted for each first place assigned to a candidate, and if any candidate obtains an absolute majority of the votes counted he is elected. But if no one obtain such an absolute majority a second scrutiny is held. At the second scrutiny one vote is counted for each first place, and one vote for each second place, exactly as in the first page 14 scrutiny on the Venetian method, and the candidate who obtains most votes is elected. At first sight we might suppose that this method could not lead to error. Comparing it with the Venetian method, described above, we see that Condorcet supplies a remedy for the obvious defect of the Venetian method—that is to say, the rejection of a candidate who has an absolute majority is now impossible. A little examination, however, will show, as seems to have been pointed out by Lhuilier,* that the method is not free from error. For, let us suppose that there are sixteen electors, of whom five put A first and B second, five put C first and B second, two put A first and C second, two put B first and A second, and two put C first and A second. Then the result of the first scrutiny will be, for A, B, C, seven, two, seven votes respectively. Thus, no one baving an absolute majority, a second scrutiny is necessary. The result of the second scrutiny will be—for A, B, C, eleven, twelve, and nine votes respectively. Thus B, having the largest number of votes, is elected. This result, however, is not in accordance with the views of the majority of the electors. For the proposition, "B is better than A," would be negatived by a majority of two votes, and the proposition, "B is better than C," would also be negatived by a majority of two votes, so that in the opinion of the electors B is worse than A and also worse than C, and, therefore, ought not to be elected.

Summing up the results we have arrived at, we see that each of the methods which have been described may result in the return of a candidate who is considered by a majority of the electors to be inferior to each of the other candidates. Some of the methods—viz., the double vote method, the method of Borda, and the Venetian method—may even result in the rejection of a candidate who has an absolute majority of votes in his favour as against all comers. It would, however, be quite impossible for such a result to occur on the single vote method, or the methods of Ware and Condorcet.

* See Montuela's Histoire des Mathématiques, vol. iii., p. 421.