Method Proposed.

Having pointed out the defects of the methods in common use, it now remains to describe the method proposed for adoption, and to show that it is free from these defects. It page 15 consists merely in combining the principle of successive scrutinies with the method of Borda, and at the same time making use of the preferential voting paper, so that the proposed method belongs to the third class. I propose, first, to describe and discuss the method for the case of three candidates, and then to pass on to the general case in which there may be any number of candidates.

Let us suppose, then, that there are three candidates, A, B, C. Each elector writes on his voting paper the names of two candidates in order of preference, it being clearly unnecessary to write down a third name. If we prefer it, the three names may be printed on the voting paper, and the elector may be required to indicate his order of preference by writing the figure I opposite the name of the candidate of his first choice, and the figure 2 opposite the name of the candidate of his second choice, it being clearly unnecessary to mark the third name. In order to ascertain the result of the election two scrutinies may be necessary.

At the first scrutiny two votes are counted for each first place and one vote for each second place, as in the method of Borda, Then if the two candidates who have the smallest number of votes have each not more than one-third of the whole number of votes, the candidate who has most votes is elected, as in Borda's method. But if one only of the candidates has not more than one-third of the votes polled (and some candidate must have less), then that candidate is rejected, and a second scrutiny is held to decide between the two remaining candidates. At the second scrutiny each elector has one vote, which is given to that one of the remaining candidates who stands highest in the elector's order of preference. The candidate who obtains most votes at the second scrutiny is elected.

The method may be more briefly described as follows:—Proceed exactly as in Borda's method, but instead of electing the highest candidate, reject all who have not more than the average number of votes polled. If two be thus rejected, the election is finished; but if one only be rejected, hold a final election between the two remaining candidates on the usual plan.

In order to show that the proposed method is free from the defects above described, it is necessary and it is sufficient to show that if the electors consider any one candidate, A, say superior to each of the others, B and C, then A cannot be rejected at the first scrutiny. For if A be not rejected at page 16 the first scrutiny he cannot fail to win at the second scrutiny. Let therefore the whole number of electors be 2N, and let the number who prefer B to C be N + a, and consequently the number who prefer C to B be N—a; similarly, let the number who prefer C to A be N + b, and therefore the number who prefer A to C be N — b, and let the number who prefer A to B be N +c, and therefore the number who prefer B to A be N — c. Then it is easy to see that the numbers of votes polled by A, B, C at the first scrutiny will be

2N — b + c, 2N — c + a, 2N — a + b

respectively. For if the compound symbol A B be used to denote the number of electors who put A first and B second, and similarly for other cases, it is clear that A's score at the first scrutiny will be

2AB + 2AC + BA + CA.

Now this expression can be written in the form

(AB + AC + CA) + (AC + AB + BA),

and it is clear that the three terms in the first pair of brackets represent precisely the number of electors who prefer A to B, which number has already been denoted by N + c. In the same way the remaining three terms represent the number of electors who prefer A to C, which number has been denoted by N — b. Hence the score of A on the first scrutiny is 2N — b + c. In exactly the same way it may be shown that the scores of B. C are 2N — c + a and 2N — a + b respectively. The sum of these three numbers is 6N, as it ought to be. Thus 2N is the mean or average of these three numbers, and consequently the highest of the three candidates must have more than 2N votes, and the lowest must have less than 2N votes. Now, let us suppose that a majority of the electors prefer A to B, and likewise that a majority prefer A to C; then c must be positive, and b must be negative. Hence the score of A, which has been shown to be 2N — b + c is necessarily greater than 2N, for it exceeds 2N by the sum of the two positive quantities — b and c. Thus A has more than 2N votes, that is, more than one-third, or the average of the votes polled. He cannot, therefore, be rejected at the first scrutiny, so that B or C or both must be rejected at the first scrutiny. If either of the two, B and C, be not rejected, A must win at the second scrutiny, for there is a majority for A against B, and also against C. Hence, then, it has page 17 been demonstrated that if the opinions of the electors are such that there is a majority in favour of A as against B, and likewise a majority in favour of A as against C, the method of election which is proposed will certainly bring about the correct result; whereas it has been shown by the consideration of particular examples that the methods in ordinary use may easily bring about an erroneous result under these circumstances. Thus the proposed method cannot bring about a result which is contrary to the wishes of the majority, so that the proposed method satisfies the fundamental condition.

The method which is proposed has, I think, strong claims. It is not at all difficult to carry out. The result will, as often as not, be decided on the first scrutiny. We simply require each elector to put down the names of two of the three candidates in order of preference. Then for each first name two votes are counted, and for each second name one vote is counted. The number of votes for each candidate is then found. The third part of the sum total may be called the average; then all candidates who are not above the average are at once rejected. The lowest candidate must, of course, be below the average. The second is just as likely to be below as above the average. If he is below, the election is settled; but if he is above the average, a second scrutiny is necessary to decide between him and the highest candidate.