The following paper has been written and printed in great haste, as it was only on the night of Friday the 12th that it occurred to me to investigate the subject, which proved to be much more complicated than I had expected. Still I hope that I have given sufficient thought to it to escape the commission of any serious mistake.
I commence by considering certain known Methods of Procedure, in the case where some candidate must be elected, proving that each Method is liable, under certain circumstances, to fail in giving the proper result.
I then consider the question of ‘Election or no Election?’ proving that the two ordinary Methods of deciding it are unsound.
And I conclude by describing a Method of Procedure (whether new or not I cannot say) which seems to me not liable to the same objections as have been proved to exist in other cases.
C. L. D.
Ch. Ch., Dec. 18, 1873.
Chapter I. On the failure of certain Methods of Procedure, in the case where an Election is necessary.
§ 1. The Method of a Simple Majority.
In this Method, each elector names the one candidate he prefers, and he who gets the greatest number of votes is taken as the winner. The extraordinary injustice of this Method may easily be demonstrated. Let us suppose that there are eleven electors, and four candidates, a, b, c, d; and that each elector has arranged in a column the names of the candidates, in the order of his preference; and that the eleven columns stand thus:—
| a | a | a | b | b | b | b | c | c | c | d |
| c | c | c | a | a | a | a | a | a | a | a |
| d | d | d | d | c | c | c | d | d | d | c |
| b | b | b | d | d | d | d | b | b | b | b |
Here a is considered best by three of the electors, and second by all the rest. It seems clear that he ought to be elected; and yet, by the above method, b would be the winner—a candidate who is considered worst by seven of the electors!
§ 2. The Method of an Absolute Majority.
In this Method, each elector names the one candidate he prefers; and if there be an absolute majority for any one candidate, he is taken as the winner.
| b | b | b | b | b | b | a | a | a | a | a |
| a | a | a | a | a | a | c | c | c | d | d |
| c | c | c | d | d | d | d | d | d | c | c |
| d | d | d | c | c | c | b | b | b | b | b |
Here a is considered best by nearly half the electors (one more vote would give him an absolute majority), and never put lower than second by any; while b is put last by five of the electors, and c and d by three each. There seems to be no doubt that a ought to be elected; and yet, by the above Method, b would win.
§ 3. The Method of Elimination, where the names are voted on by two at a time.
In this Method, two names are chosen at random and proposed for voting; the loser is struck out from further competition, and the winner taken along with some other candidate, and so on, till there is only one candidate left.
| a | a | a | a | a | b | b | c | d | d | d |
| c | c | c | c | d | a | a | b | b | b | b |
| b | d | d | d | c | c | c | a | a | a | a |
| d | b | b | b | b | d | d | d | c | c | c |
Here it seems clear that a ought to be the winner, as he is considered best by nearly half the electors, and never put lower than third; while b and d are each put last by four electors, and c by three. Nevertheless, by the above Method, if (a, b) were put up first for voting, a would be rejected, and ultimately c would be elected. Again, if (a, c) were put up first, c would be rejected, and if (a, b) were put up next, d would be elected—but if (a, d), b would be elected.
Such preposterous results, making the Election turn on the mere accident of which couple is put up first, seem to me to prove this Method to be entirely untrustworthy.
§ 4. The Method of Elimination, where the names are voted on all at once.
In this Method, each elector names the one candidate he prefers: the one who gets fewest votes is excluded from further competition, and the process is repeated.
| b | b | b | c | c | c | d | d | d | a | a |
| a | a | a | a | a | a | a | a | a | b | c |
| d | c | d | b | b | b | c | c | b | d | d |
| c | d | c | d | d | d | b | b | c | c | b |
Here, while b is put last by three of the electors, and c and d by four each, a is not put lower than second by any. There seems to be no doubt that a’s election would be the most generally acceptable: and yet, by the above rule, he would be excluded at once, and ultimately c would be elected.
§ 5. The Method of Marks.
In this Method, a certain number of marks is fixed, which each candidate shall have at his disposal; he may assign them all to one candidate, or divide them among several candidates, in proportion to their eligibility; and the candidate who gets the greatest total of marks is the winner.
This Method would, I think, be absolutely perfect, if only each elector wished to do all in his power to secure the election of that candidate who should be the most generally acceptable, even if that candidate should not be the one of his own choice: in this case he would be careful to make the marks exactly represent his estimate of the relative eligibility of all the candidates, even of those he least desired to see elected; and the desired result would be served.
But we are not sufficiently unselfish and public-spirited to give any hope of this result being attained. Each elector would feel that it was possible for each other elector to assign the entire number of marks to his favorite candidate, giving to all the other candidates zero: and he would conclude that, in order to give his own favorite candidate any chance of success, he must do the same for him.
This Method is therefore liable, in practice, to coincide with ‘the Method of a Simple Majority,’ which has been already discussed, and, as I think, provided to be unsound.
§ 6. The Method of Nomination.
In this Method, some one candidate is proposed, seconded, and the votes taken for and against. This Method is fair for those electors only who prefer that candidate to any other, or else any other to him. But any other elector might say ‘I do not know whether to vote for or against a till I know who would come in if he failed. If I were sure b would come in, I would vote against a: otherwise, I vote for a.’
If this Method leads to a majority of votes being obtained for the proposed candidate, it is identical with ‘the Method of an absolute Majority,’ which was discussed in § 2. If a minority only is obtained, it may be thus represented:—
| b | b | c | c | d | d | a | a | a | a | a |
| a | a | a | a | a | a | b | b | c | c | d |
| c | c | b | b | b | b | c | c | b | b | b |
| d | d | d | d | c | c | d | d | d | d | c |
Here there seems no doubt that a ought to be elected; and yet, by the above Method, he would be rejected at once, and, whichever candidate came in, nine of the electors would say ‘We would rather have had a.’
Chapter II. On the failure of certain Methods of Procedure, in the case where it is allowable to have ‘no Election.’
§ 1. The Method of commencing with a vote on the question ‘Election or no Election?’
This Method has the strong recommendation that if ‘no Election’ be carried, it saves all further trouble, and it might be a just method to adopt, provided the electors were of two kinds only—one, which prefers ‘no Election’ to any candidate, even the best; the other, which prefers any candidate, even the worst, to ‘no Election.’ But it would seldom happen that all the electors could be so classed: and any elector who preferred certain candidates to ‘no Election,’ but preferred ‘no Election’ to certain other candidates, would not be fairly treated by such a procedure. He might say ‘It is premature to ask me to vote on this question. If I knew that a or b would be elected, I would vote to have an election; but if neither a nor b can get in, I vote for having none.’
Let us, however, test this Method by a case—representing ‘no Election’ by the symbol ‘○.’
| a | a | b | b | c | c | ○ | ○ | ○ | ○ | ○ |
| ○ | ○ | ○ | ○ | ○ | ○ | a | a | b | b | c |
| c | c | a | a | b | b | d | d | c | c | b |
| d | d | d | d | d | d | b | c | a | a | a |
| b | b | c | c | a | a | c | b | d | d | d |
Here there seems no doubt that ‘no election’ would be the most satisfactory result: and yet, by the above Method, an Election would take place, and in all probability b would be elected—a candidate regarding whom nine of the electors would say ‘I would rather have had no Election.’
§ 2. The Method of concluding with a vote on the question ‘Shall x (the successful candidate) be elected, or shall there be no Election?’
Here again a voter who preferred certain candidates to ‘no Election,’ but preferred ‘no Election’ to certain other candidates, would not be fairly treated. He might say ‘If you had taken a or b, I would have been content, but as you have taken c, I vote for no Election,’ and his vote might decide the point: while the other electors might say ‘If we had only known how it would end, we would willingly have taken a instead of c.’
But let us test this Method also by a case.
| b | b | b | b | b | ○ | a | a | a | a | a |
| a | a | a | a | a | b | ○ | ○ | ○ | ○ | ○ |
| c | c | c | c | c | a | b | b | b | b | b |
| d | d | d | d | d | c | c | c | c | c | c |
| ○ | ○ | ○ | ○ | ○ | d | d | d | d | d | d |
Here there seems to be no doubt that the election of a would be much more satisfactory than having no Election: and yet, by the above Method, b would first be selected from all the candidates, and ultimately rejected on the question of ‘b or no Election?’ while ten of the electors would say ‘We would rather have taken a than have no Election at all.’
The conclusion I come to is that, where ‘no Election’ is allowable, the phrase should be treated exactly as if it were the name of a candidate.
Chapter III. On a proposed Method of Procedure.
The Method now to be proposed is, in principle, a modification of No. 5, viz. ‘The Method of Marks,’ since it assigns to each candidate a mark for every vote given to him, when taken in competition with any other candidate.
Suppose that, in the opinion of a certain elector, the candidates stand in the order a, b, c, d: then his votes may be represented by giving a the number 3, b 2, c 1, and d 0.
Hence all that is necessary is that each elector should make out a list of the candidates, arranging them in order of merit.
If ‘no Election’ is allowable, this phrase should be placed somewhere in the list.
If the elector cannot arrange all in seccession, but places two or more in a bracket, a question arises as to how the bracketed names should be marked. The tendency of many electors being, as explained in Chap. I. § 5, to give to the favorite candidate the maximum mark, and bracket all the rest, in order to reduce their chances as much as possible, it is proposed, in order to counteract this tendency, to give to each bracketed candidate the same mark that the highest would have if the bracket were removed. This plan will furnish a strong inducement to avoid brackets as far as possible.
In order to illustrate this process, let us apply it to the various ‘Cases’ already considered.
| α | β | γ | δ | ε | ζ | |
| a | 25 | 27 | 23 | 24 | 21 | 37 |
| b | 12 | 18 | 15 | 15 | 21 | 33 |
| c | 20 | 11 | 14 | 14 | 20 | 16 |
| d | 9 | 10 | 14 | 13 | 10 | 5 |
| ○ | 38 | 19 |
It will be seen that in each case the candidate, whose election is obviously most to be desired, obtains the greates number of marks.
Chapter IV. Summary of Rules.
1. Let each elector make out a list of the candidates, (treating ‘no Election’ as if it were the name of a candidate), arranging them as far as possible in the order of merit, and bracketing those whom he regards as equal.
2. Let the names on each list be marked with the numbers 0, 1, 2, &c., beginning at the last.
3. Whenever two or more names are bracketed, each must have the mark which would belong to the highest, if there were no bracket.
4. Add up the numbers assigned to each candidate.
The first Rule is all with which the electors need trouble themselves. Rules 2, 3, 4 can all be carried out by one person, as it is merely a matter of counting.