Another Method for Cases of Bracketing.

Another plan might also be adopted for dealing with cases of bracketing. It is as follows. For each candidate in the first place count one vote; for each candidate in the second place count m₁ + 1 votes; for each candidate in the third place count m₁ + m₂ + 1 votes; for each candidate in the fourth place count m₁ + m₂ + m₃ + 1 votes; and so on. The plan now under consideration comes to the same thing as counting for the successive places the numbers 0, m₁, m₁ + m₂, . . . m₁ + m₂ + . . . + m_r-1, &c. instead of the proper numbers (1). Thus the errors for the successive places are

page 42

Hence we see that

(i.)	If the same number of candidates be bracketed for each place, the plan is accurate.
(ii.)	If m1 be greater than each of the numbers m2, m3, &c, that is, if more candidates are bracketed for the first place than for any other place—then the errors will be all positive, and the effect will be to give the elector more negative votes than he is entitled to, and, consequently, to increase unduly the chances of the candidates bracketed for the first place.
(iii.)	If m1 be less than each of the numbers m2, m3, &c.—that is, if fewer candidates are bracketed for the first place than for any other place — then the errors will be all negative, and the effect will be to give the elector fewer negative votes than he is entitled to, and, consequently, to decrease unduly the chances of the candidates placed at the top end of the elector's list.
(iv.)	If m1 be equal to the mean of the numbers m2, m3, &c, the elector will have just as many votes as he ought to have, but he will give more negative votes to some candidates and less to others than they ought to have.
(v.)	If m1 be not equal to the mean, then the elector will have more or less votes than he is entitled to, according as m1 is Greater or less than the mean.

The results just given apply to each scrutiny; but the numbers m₁, m₂, m₃, &c, will generally be altered at each scrutiny. Thus it is in general impossible to tell at the commencement of an election what will be the effect of different modes of bracketing. Sometimes the elector will get too many votes, sometimes too few. At some scrutinies the candidates at the top end of his list will get too many votes, and at others those at the lower end will get too many votes.

If there be one candidate only in each place except the last, or, in other words, if the only bracket be for the last place, we have the case of incomplete papers discussed above. In this case the plan just described, and the method adopted above, agree; and the effect is, as has already been pointed out, to give the elector too few votes; and this would be the case at each scrutiny, until all but one of the candidates in the bracket are rejected.

If, however, an elector bracket a number of candidates for the first place and arrange all the rest in order of merit, he would get more votes than be is really entitled to, and page 43 this would be the case at each scrutiny until all but one of the candidates in the bracket are rejected. Electors would very soon find this out. Each elector would ask himself the question, How must I vote in order to get as much electoral power as possible; and the answer would very soon be seen to be—I must bracket all the candidates I don't object to for the first place, and I must arrange all the rest in numerical order. Thus, instead of encouraging the electors to arrange all the candidates in order of merit, this plan would lead to each elector trying all he could to defeat objectionable candidates without expressing any opinion as to the relative merits of those he does not object to.