The (almost really) Complete Works of Lewis Carroll

§ 1. Proposed Rules for Conducting an Election.

I.

Each elector shall write down the issue he desires (‘no Election’ being reckoned as an issue) and hand in the paper folded, with his name written outside: and the Chairman, or some one appointed by him, having before him a list of the electors, shall enter these issues against their names.

II.

If the Chairman find any issue having an absolute majority of votes, he shall communicate the list to the meeting. This issue shall then be formally moved, and, if none object, the Chairman shall declare it carried.

III.

If the Chairman shall find no issue having an absolute majority of votes, he shall communicate to the meeting the list of issues only, without stating who vote for each, and shall return the papers, that each elector may add the other issues, arranged in his order of preference. The Chairman shall enter these on his list, and then communicate the whole to the meeting.

IV.

If an issue be found which has a majority over every other taken separately, it shall be formally moved as in Rule II: but if none be found, the majorities being ‘cyclical’, opportunity shall be given for further debate. In ascertaining which of any pair of issues is preferred to the other, any elector whose paper contains one only of the two shall be reckoned as preferring that one, and any whose paper contains neither shall be considered as not voting.

V.

If the issues cannot be all arranged in one cycle, but form a cycle and a set of issues each of which is separately beaten by each of the cycle, it shall be formally moved that this cycle be retained and all other issues struck out, and, if none object, this shall be done.

VI.

If, a formal motion having been made that a certain issue be adopted, or that a certain cycle be retained and all other issues struck out, any one object, he may move as an amendment that a division be taken between the issue he desires and the issue so to be adopted, or any one of the cycle so to be retained. If every such amendment be lost on a division, the Chairman shall declare the original motion carried: but, if any such amendment be carried, by some voting contrary to their written papers, they shall be required to amend their papers, and the process shall begin again.

VII.

When the issues to be further debated consist of, or have been reduced to, a single cycle, the Chairman shall inform the meeting how many alternations of votes each issue requires to give it a majority over every other separately.

VIII.

If, when the majorities are found to be cyclical, any elector wish to alter his paper, he may do so: and if the cyclical majorities be thereby done away with, the voting shall proceed by former Rules: but if, when none will make any further alteration, the majorities continue cyclical, there shall be no election.

§ 2. The Legal Conditions.

In any election, when there are only two issues to vote on—for instance (there being only one candidate), ‘shall A be elected or not?’ or again (there being only two candidates, and it being understood that there is to be an election) ‘shall A or B be elected?’—and when the Chairman is able to give a casting vote, it is clear that there must be a majority for one or other issue, and in this case open voting is the obvious course.

But wherever there are three or more issues to vote on, any one of the following three cases may exist in the minds of the electors:—

(α) There may be one issue desired by an absolute majority of the electors.

(β) There may be one issue which, when paired against every other issue separately, is preferred by a majority of electors.

(γ) The majorities may be ‘cyclical,’ e. g. there may be a majority for A over B, for B over C, and for C over A.

The words of the Ordinance are “That Candidate for whom the greatest number of votes shall have been given shall be deemed elected.”

It seems to me that this may be complied with by either of two modes of election:—

In case (α) If a candidate be declared elected who, when all are voted on at once, has an absolute majority of votes.

In case (β) If a candidate be declared elected who, when paired with every other separately, is preferred by the majority of those voting.

But that is not complied with by the following mode:—

In case (γ) If a candidate be declared elected, though it is known that there is another who, when paired with him, is preferred by the majority of those voting.

Mode (α) needs no discussion. Failing this, it seems clear that mode (β) would be a satisfactory result, as any one who preferred some other candidate might be allowed to take a division between the two.

If modes (α) and (β) both fail, it shows that the majorities on the separate pairs are ‘cyclical,’ and if, after all possible discussion, this continues to be so, any election that may be arrived at must introduce mode (γ). My own opinion is that, under these circumstances, there ought to be ‘no Election’: two other courses might be suggested, which I will now consider.

§ 3. Courses that have been suggested for the case of ‘Cyclical Majorities.’

(1) That all candidates should be voted on at once, and the one who has the greatest number of votes should be elected.

This might be though to fulfil the letter of the law, if after the words ‘shall have been given’ we supply the words ‘in the final voting.’

Let us suppose that there are 11 electors, and 4 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 11 columns stand thus:—

Here the majorities are cyclical, in the order a d c b a, each beating the one next following.

Moreover, if we make a table of majorities in the separate pairs, in which the numerator of each fraction represents the number voting for the issue which stands at the top of that column and the denominator the number voting for the issue which stands at the end of that row, and in which every division, where the issue at the top of the column is beaten, is distinguished by placing the fraction in a parenthesis, we have

Here a and d each need 4 changes of votes to win, but b and c each need one only: for instance, the interchange of the two issues which are marked * would make b win. It seems clear that a has much less claim to be elected than either b or c (observe that he is put last by nearly half the electors, and only needs one interchange of votes to cause him to be beaten by every other candidate separately), and yet by the above course he would win.

Again, let there be 13 electors and 4 candidates.

Here the majorities are cyclical, in the order a b c d a; the table of majorities being:—

Here a, c, d each need 4 changes of votes to win, while b needs only one, for instance, the interchange of the two issues marked *. Yet by the above course a would win—a candidate whom this single interchange would cause to be beaten by every other candidate separately.

(2) That all candidates should be voted on at once, and the one who has the smallest number of votes should be struck out, and the process repeated till only two are left.

Here the majorities are cyclical, in the order a b c a. Moreover, a beates b (6 to 5), b beats c (6 to 5), but c beats a (7 to 4).

If any one is to be elected, it would seem that c has the strongest claim; but by the above method a would win—a candidate who is put last by nearly half the electors.

Again, let there be 15 electors and 4 candidates:—

Here there is a cyclical majority, in the order a b c d a; therefore by above Rule d is excluded: we now have—

Here there is again a cyclical majority, in the order a b c a; therefore c is excluded.

The candidates are now reduced to a and b, and a wins by a majority of 8 to 7.

But if we tabulate the majorities thus—

we see that a needs 6 changes of votes to win, b 5, c 2, and d only 1. It seems clear that d ought to win; yet he is the very first to be excluded by the above course.

Lastly, let us take a case in which these two courses bring in different candidates, neither of them being the one that ought to win.

Let there be 23 electors and 4 candidates.

Here the majorities are cyclical in the order a d c b a. The table of majorities is:—

Now, by course (1) a wins.

By course (2) d is excluded; but we still have a cyclical majority a c b a; we then exclude a, and c wins.

But, if we reckon how many changes of votes each needs to win, we find that a needs 5, c needs 6, and d needs 8; whereas b needs only 1—a single interchange, such as the two marked *, would give him a clear victory.

Note also that this single interchange would cause c (who is brought in winner by course (2)) to be beaten by every other candidate separately.

The instances I have taken seem to show that neither of these courses can be relied on to give a satisfactory result. But there is a stronger, and as I think a fatal, objection to both; namely, that any elector, who had not consented to this course being adopted, would have a very strong ground of appeal against the election if he were able to say “A was declared elected, and yet he had not ‘the greatest number of votes’ given for him, since he was beaten when paired against B.”

The conclusion I come to is that, in the case of persistent cyclical majorities, there ought to be ‘no Election.’

I am quite prepared to be told, with regard to the cases I have here proposed, as I have already been told with regard to others, ‘Oh, that is an extreme case: it could never really happen!’ Now I have observed that this answer is always given instantly, with perfect confidence, and without any examination of the details of the proposed case. It must therefore rest on some general principle: the mental process being probably something like this—‘I have formed a theory. This case contradicts my theory. Therefore this is an extreme case, and would never occur in practice.’

§ 4. Reasons for beginning with a vote on all issues at once.

One reason for this is that it may show an absolute majority for some one issue, and so save all further trouble. But another, and a stronger, reason is that, when a division is taken first of all between a certain pair of issues, there will very often be some of the electors who will not know which way to vote. I am not speaking of electors who are willing to vote contrary to their real opinion, but of electors generally.

An example or two will make this clear.

Suppose there are two vacancies, but that it is not necessary to fill both: and that a division is taken first of all on the question ‘Shall both vacancies be filled, or only one?’ An elector might reasonably say ‘I wish to elect A alone. If I were sure he would come in, I would vote for electing one only: but if B is preferred, then, rather than lose A, I would vote for electing two.’ And another might say ‘I wish to elect A and B, but I strongly object to C. If I were sure A and B would come in, I would vote for electing two: but if that would result in A and C coming in, then I vote for one only.’ How much simpler to allow the one to write down ‘A alone,’ and the other ‘A and B.’

Again, suppose it settled that two are to be elected, and a division to be taken between B and D. An elector might reasonably say ‘I wish to elect A at any rate: the other to be B or C. I do not care which: but I object to D. I would vote for B, if I were sure that A would be elected as the other. But if I knew that C would beat A on a division, I should wish to get C and A elected, and this might be effected by voting for D. I happen to know that C and A can each beat D, so that he has no real chance. My voting for him would not mean that I wish to bring him in, but that I with to keep B out, and so to get C and A elected, instead of C and B.’ How much simpler to allow him to write ‘A, and then B or C.’

§ 5. Reasons for allowing ‘no Election’ to be reckoned among the other issues.

Evidently an elector who desires ‘no Election’ ought to have some opportunity of voting on the question. And if it be not reckoned as an issue, it must be voted on, as a separate question, at the beginning or the end of the proceedings.

(1) The method of beginning 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.’

(2) The method of ending with a vote on the question ‘Shall X 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.’

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.

§ 6. Reasons for having a preliminary voting on paper and not open voting.

Suppose A to be the candidate whom I whish to elect, and that a division is taken between B and C; am I bound in honour to vote for the one whom I should really prefer, if A were not in the field, or may I vote in whatever way I think most favourable to A’s chances? Some say ‘the former,’ some ‘the latter.’ I proceed to show that, whenever case α fails to occur, and there are among the electors a certain number who hold the latter course to be allowable, the result must be a case of cyclical majorities.

Let there be 3 candidates, A, B, C, each preferred by about one-third of the electors; and suppose that, when a division is taken between A and B, A wins. A division is now taken between A and C, which of course depends on the votes of the B-party; perhaps a majority of them really prefer A, and if they voted accordingly A would win under case β; it might need only two or three to vote contrary to their real opinion to turn the division in favour of C. We have now got ‘A beats B, C beats A,’ and of course a division must be taken between B and C; this depends on the votes of the A-party, and, as before, it may only need two or three to vote contrary to their real opinion to prevent C winning the election. Thus we get ‘A beats B, C beats A, B beats C.’

This principle of voting makes an election more of a game of skill than a real test of the wishes of the electors, and as my own opinion is that it is better for elections to be decided according to the wish of the majority than of those who happen to have most skill in the game, I think it desirable that all should know the rule by which this game may be won. It is simply this:—‘In any division taken on a pair of issues neither of which you desire, vote against the most popular. There may be some one issue which, if all voted according to their real opinion, would beat every other issue when paired against it separately: but, by following this rule, you may succeed in getting it beaten once, and so prevent its having a clear victory, by introducing a cyclical majority. And this will give, to the issue you desire, a chance it would not otherwise have had.’

Now, it is impossible to prevent such votes being given: and even if a preliminary voting on paper should seem to lead to case α or β, it is impossible, when it comes to the final formal vote, to prevent votes being given contradictory to previous votes.

The advantages of having the preliminary voting taken on paper and not openly are, first, that each elector, not knowing exactly how the others are voting, has less inducement to vote contrary to his real opinion, so that a more trustworthy estimate is arrived at of the real opinion of the body of electors, and cyclical majorities are less likely to occur, than with open voting; and secondly, that if cyclical majorities do not occur in this process, they cannot occur in the formal voting except by some one or more of the electors giving votes inconsistent with their written opinions, and I think it desirable that in such a case the body of electors should know who they are that have so voted—a result which this method would secure.

I do not suppose that any one would be so unwilling to have it known that he has so voted that this publicity would prevent an artificial cyclical majority—for I am sure that those who do so believe it to be an honourable course to take, and so have no motive for desiring concealment—but I think it would increase the sense of the responsibility incurred by those who thus exercise their right of voting, and so make its occurrence less likely.

These written lists will also be, in many cases, a great saving of time. An example will best show this. Suppose there are 2 vacancies to be filled and 3 candidates, all recommended on various grounds by the examiners, and that the electors are divided among the following 6 issues, ‘A B’, ‘B A’, ‘A C’, ‘C A’, ‘B C’, ‘A alone.’ These, taken two and two, give 15 pairs: that is, it might require 15 divisions to be taken to get the information which the written lists furnish at once.