# The Pamphlet Collection of Sir Robert Stout: Volume 53

# Incomplete Voting Papers

## Incomplete Voting Papers.

There is a point of some practical importance to be considered in connection with the proposed method. If the number of candidates was large, some of the electors might not be able to make out a complete list of the candidates in order of preference. We have then to consider how voting papers, on which the names are not all marked in order of preference, are to be dealt with. Such a voting paper may be called incomplete. In order to examine this question, let us first suppose, for the sake of simplicity, that there are only three candidates A, B, C, and that the votes tendered are of one of the forms AB, BA, C, that is to say, that all the electors who put A first put B second, that all who put B first put A second, and that all who vote for C mark no second name. In accordance with the proposed method, for each paper of the form AB, two votes would be given to A and one to B; and for each paper of the form BA, two votes would be given to B and one to A. The question arises, however: is a paper of the form C, that is, a plumper for C, to be counted as one vote or as two votes for C? If it be counted as one vote only, it is clear that C might be defeated even if he had an absolute majority of first votes in his favour. For if we suppose AB=BA=*a*, and C=*c*, it is clear that the scores of A and B will each be equal to 3*a*, and that of C to *c*. Thus C will be defeated unless *c* > 3*a;* but if c > 2*a*, there is an absolute majority for C. Hence, then, we may be led into error if each plumper for C be counted as one vote only. If, on the other hand, a plumper be counted as two votes, it is clear that C might win even if there were an absolute majority against him. For the score of C will now be 2*c*, and C will win if 2*c* > 3*a*. But if *2c* < 4*a*, there is an absolute majority against C. Thus we should also be led into error if each plumper be counted as two votes. If, however, we agree to count a plumper as three halves of a vote, neither of these errors could occur. This course is readily seen to be the proper one in any case of three candidates, for it clearly amounts to assuming that the electors who plump for C are equally divided as to the merits of A and B. For if *a*^{1} *b*^{1}, *c*^{1} denote the numbers of plumpers for A, B, C respectively, and if we agree to consider all the electors who plump for A as being equally divided as to the merits of B and C, the effect of the *a*^{1}plumpers for A would be to give 2 *a*^{1} votes to A, and ½ *a*^{1} each
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to B and C. Now, as we are only concerned with the differences of the totals polled for each candidate, we see that the result of the first scrutiny will he the same if we take away ½ *a*^{1} votes from each candidate. Thus the result will come out the same if we give ¾ *a*^{1} votes to A, and none to B or C, so far as the plumpers are concerned. Similarly the result will not be altered if the *b*^{1} plumpers for B be counted, as 3/2 *b*^{1} votes for B and nothing for G and A, and so for C's plumpers. Thus the final result will be in accordance with the views of the electors, if each plumper be reckoned as three halves of a vote

The assumption that the electors who plump for A are equally divided as to the merits of B and C, appears to be perfectly legitimate, for the electors have an opportunity of stating their preference, if they have one, and as they have, in the case supposed, declined to express any, it may be fairly concluded that they have none.

At the final scrutiny (if held), all plumpers for the candidate who has been rejected will have no effect.

If there be more than three candidates, and incomplete papers are presented, we should have to make a similar assumption, viz., that in all cases where the preference is not fully expressed, the elector has no preference as regards the candidates whom he has omitted to mark on his voting paper. Thus, for example, if there be four candidates, A, B, C, D, a plumper for A ought to count as two votes for A and none for B, C, D. Again, a voting paper on which A is marked first and B second, and on which no other names are marked, ought to count as two and ahalf votes for A and three halves of a vote for B. If there be more than four candidates the varieties of incomplete papers would be more numerous, and the weights to be allotted to each would be given by more complicated rules. Practically it would be best to count one vote for each plumper in the case in which only one candidate is marked on a voting paper; one for the last, and two for the first, when two names only are marked on a voting paper; one for the last, two for the next, and three for the first, when three names only are marked on a voting paper, and so on, giving in all cases one vote to the candidate marked lowest on any paper, and as many votes to the candidate marked first as there are names marked on the paper. By this means the rules for computing the votes would be the same in all cases and at all scrutinies. We have seen, it is true, that page 37 this method may lead to error. The error has the effect of decreasing the votes for the candidates who are marked on any incomplete paper, and it arises solely in consequence of the papers being incomplete. Thus, if the electors do not fully express their preference, the effect is to injure the chances of their favourite candidates. If, then, we adopt the plan just described for incomplete papers, it will be sufficiently simple for practical purposes, and its use will tend to elicit from electors a full statement of their various preferences.