Cases of Inconsistency.

We have now to consider what is the result of the proposed method in those cases in which there is not a majority for one candidate against each of the others. The methods which have been described have been shown to be erroneous by examining cases in which either one candidate has an absolute majority of the electors in his favour, or a candidate A is inferior to B and also to C, or a candidate A is superior to B and also to C. Now it is not necessary that any of these cases should occur. If a single person has to place three candidates in order of preference he can do so, and it would be quite impossible for any rational person to arrive at the conclusions

B is superior to C	(1)
C is superior to A	(2)
A is superior to B	(3)

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When, however, we have to deal with a body of men, this result may easily occur, and no one of the candidates can be elected without contradicting some one of the propositions stated above. If this result does occur, then, no matter what result any method of election may give, it cannot be demonstrated to be erroneous. We have examined several methods, and all but the one now proposed have been shown to lead to erroneous results in certain cases. It may fairly be urged, then, that that method which cannot be shown to be erroneous in any case has a greater claim to our consideration than any of the other methods which can be shown to erroneous. On this ground alone I think the method proposed ought to be adopted for all cases.

We can, however, give other reasons in favour of the method proposed. We have seen that it gives effect to the views of the majority in all cases except that in which the three results (1), (2), (3) are arrived at. In this case there is no real majority, and we cannot arrive at any result without abandoning some one of the three propositions (1), (2), (3). It seems most reasonable that that one should be abandoned which is affirmed by the smallest majority. Now, if this be conceded, it may be shown that the proposed method will give the correct result in all cases. For it is easily seen that the majorities in favour of the three propositions (1), (2), (3) are respectively 2a, 2b, 2c. Hence, then, in the case under consideration, a, b, c, must be all positive. Let us suppose that a is the smallest of the three. Then we abandon the proposition (1), and consequently C ought to be elected. Now let us see what the proposed method leads to in this case. B's score at the first scrutiny is 2N—c + a, and this is necessarily less than 2N, because c is greater than a, and each is positive. Again, C's score is 2N —a + b, and this is necessarily greater than 2N, because b is greater than a, and each is positive. Thus B is below the average, and C is above the average. Therefore, at the first scrutiny B goes out and C remains in. If A goes out also, C wins at the first scrutiny. But if A does not go out, C will beat A at the second scrutiny. Thus C wins in either case, and, therefore, the proposed method leads to the result which is obtained by abandoning that one of the propositions (1), (2), (3) which is affirmed by the smallest majority. We have already seen that in the case in which the numbers a, b, c are not all of the same sign, the proposed method leads to the correct result. Hence, then, if it be admitted that when page 19 we arrive at the three inconsistent propositions (1), (2), (3) we are to abandon the one which is affirmed by the smallest majority, it follows that the proposed method will give the correct result in all cases.

We have, then, arrived at two results. First, that if the electors affirm any two of the propositions (1), (2), (3) and affirm the contrary of the remaining one, and so affirm three consistent propositions, then the result of the method of election which is here proposed, will be that which is the logical consequence of these propositions, whilst the methods in ordinary use may easily give a different result. Second, that if the electors affirm the three propositions (1), (2), (3) which are inconsistent, then the result of the method proposed is that which is the logical consequence of abandoning that one of the three propositions which is affirmed by the smallest majority.