# The Pamphlet Collection of Sir Robert Stout: Volume 53

# Condorcet's Theoretical Method

## Condorcet's Theoretical Method.

"There exists but one rigorous method of ascertaining the wish of the majority in an election. It consists in taking a vote on the respective merits of all the candidates compared two and two. This can be deduced from the lists upon which each elector has written their names in order of merit."

"But, in the first place, this method is very long. If there are only twenty candidates, in order to compare them two and two we must examine the votes given upon one hundred and ninety propositions, and upon seven hundred and eighty propositions if there are forty candidates. Often, indeed, the result will not be as satisfactory as we could wish, for it may happen that no candidate may be declared by the majority to be superior to all the others; and then we are obliged to prefer the one who is alone judged superior to a larger number; and amongst those who are judged superior to an equal number of candidates, the one who is either judged superior by a greater majority or inferior by a smaller. But cases present themselves where this preference is difficult to determine. The general rules are complicated and embarrassing in application." *(Œuvres de Condorcet*, vol. xv., pp. 28, 29.)

By this method Condorcet showed that the single vote method and the methods of Ware and Borda are erroneous. I do not think however, that any one has hitherto noticed
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that Borda's method may lead to the rejection of a candidate who has an absolute majority of the electors in his favour as against all comers. It has also been shown above by the help of this theoretical method that Condorcet's practical method is erroneous. Thus it will be seen that the theoretical method is of use in testing the accuracy of other methods. From the description which has been given above, however, it is not clear what the result of the theoretical method is, even in the simplest cases, when discordant propositions are affirmed, for if there be three candidates only, and with the notation already used, we have *a* = 1, *b* = 2, *c* = 3, each candidate is superior to one other candidate, and A is superior by most, whilst C is inferior by least. Thus, according to the above description, it is not certain which of the two, A or C, wins. In another passage, however,^{*} Condorcet explains how he deals with any case of three candidates, and the process he adopts in the case of inconsistent propositions is to reject the one affirmed by the smallest majority. This is exactly the process which has been described above, and which was shown to be in accordance with the method proposed. Thus it is clear that in the case of three candidates the result of the proposed method will always be the same as that of Condorcet's theoretical method.

The general rules for the case of any number of candidates as given by Condorcet^{†} are stated so briefly as to be hardly intelligible. Moreover, it is not easy to reconcile these rules with the statements made in the passage quoted above, and as no examples are given it is quite hopeless to find out what Condorcet meant.

^{*} *Œuvres*, vol. xiii., p. 259.

^{†} *Essai su*r *l'application de l' analyse a la probabilité des decisions rendues a la pluralité des voix*, pp. 125, 126.