Cases of Bracketing.

Under the head of "Incomplete Voting Papers" we have considered a case in which an elector does not fully express his preference. There is, however, another way in which an elector may fail to fully express his preference. An elector may have no difficulty in putting a number of candidates at the bottom of his list, and yet he may have considerable difficulty in deciding as to the precise order in which to place the candidates at the top end of his list. In such a case an elector might wish to put two or more candidates equal for the first, second, or some other place on his list. This may be called a case of bracketing. It is now to be shown that this system of bracketing can be permitted without causing any difficulty in the practical working of the system. Let us suppose that an elector brackets m₁ candidates for the first place, m₂ for the second place, and so on; so that m₁ + m₂ + m₃ + ... = n, the case in which one candidate only is put in the r^th place being provided for by supposing m_r = 1. Then in the poll-book already described enter the number one for each of the m₁ candidates in the first bracket, the number two for each of the m₂ candidates in the second bracket, the number three for each of the m₃ candidates in third bracket, and so on. Suppose, for example, that there are seven candidates, A, B, C, D, E, P, G, and that an elector wishes to bracket B, E for the first place and A, D, F for the second place, and that he does net care to say anything about C, G. Then he would mark his paper as shown in the margin. As nothing is said about C, G, we should consider them as bracketed for the third or last place. Now in order to record this vote in the poll-book it is merely necessary, as before, to copy the column of numbers on the page 40 voting paper into a column of the poll-book, taking care to write in two 3's in the two blank spaces opposite the names C, G. After copying the numbers from each ballot-paper into the poll-book and filling up all the vacant spaces, we should add up the different rows and proceed exactly as before to ascertain the result of the election. Thus it is clear that the method of dealing with the papers is exactly the same no matter how many or how few names be marked, nor how many are bracketed in the various brackets, and that there is very little risk of error in the process.

If this system of bracketing be permitted we at once get rid of the objection that the proposed method could only be used in a highly educated constituency, because it is only highly educated electors who can possibly arrange the candidates in order of merit. The method can easily be used by the most ill-informed electors. In fact, an elector, if he so pleased, could vote in exactly the same manner as in elections under the common "majority" system of voting in cases where there are several candidates—that is, the elector may simply cross out the names of all the candidates he objects to and leave uncancelled as many names as he pleases. In such a case the uncancelled names would all be considered bracketed for the first place, and the cancelled ones as bracketed for the second or last place.

Exactly as in the case of incomplete papers previously discussed, it is easy to see that the method just given is not strictly accurate, that the strictly accurate method would be too complicated for practical purposes, and that the error has the effect of decreasing the chances of success of the favourite candidates of the elector who resorts to bracketing. In fact it may be shown that the numbers which ought strictly to be entered in the poll-book for the candidates in the successive brackets are

Now the plan just described comes to the same thing in the end as entering instead of these the numbers

and as no one of the numbers m₁, m₂, m₃, &c, can be less than unity, it is easy to see that no one of the numbers (2) page 41 can be greater than the corresponding one of the numbers (1) that when no bracketing occurs the two sets (1), (2), are the same, and that the two sets agree until the first bracket is reached. Now observe that the numbers entered in the poll-book are in reality negative votes, and we see at once that the moment an elector begins to bracket, he diminishes the influence of his own vote on the result of the election, and also decreases the chances of success of all candidates who on his own list are placed higher than the bracket. Each additional bracket will have precisely the same effects. Thus it is clear that the effect of the proposed method will be to discourage the practice of bracketing. If we do not wish to discourage this practice we must resort to the accurate method, and use the numbers (1) instead of (2). This is not very difficult to do, but as it introduces a new method for the bracketed votes, it would give considerable extra trouble to the officers who make up the poll-books. The most convenient way of stating the accurate method would be as follows:—For each first place count one negative vote, for each second place count in addition ½ (m₁+ m₂) negative votes, for each third place count in addition to the last ½ (m₂ + m₃) negative votes, for each fourth place count in addition to the last ½ (m₃ + m₄) negative votes, and so on. As before remarked, the numbers for the successive places would be the natural numbers 1, 2, 3, 4, &c, until a bracket was arrived at. When brackets do occur we shall in general have to deal with half-votes, but no smaller fraction could occur.