The (almost really) Complete Works of Lewis Carroll

May 15, 1884

To the Editor of the St. James’s Gazette

Sir,—The system of taking votes advocated by “The Proportionate Representation Society” labours under a very serious defect in its application, to which the attention of all interested in the question should be directed. An instance will make this clear.

The system proposed is that each voter shall have one vote; that he shall hand in a list of candidates numbered 1, 2, 3, etc.; and that it shall be counted as a vote for his No. 1, unless that candidate has already received enough votes to secure his return, in which case it shall be counted for his No. 2. The difficulty is that it will often depend on which lists are thus transferred whether the one or the other of two candidates shall be returned. If the lists in which A stands as No. 1 are of two kinds, some having B as No. 2 and some C, and if there are more than enough to return A, it may easily happen that, if the transferred lists are of the first kind, B will be returned, but if of the other kind, C.

Take a town containing 11,999 voters, and returning two members: so that 4,000 votes are enough to return a member. Let there be three candidates, A, B, and C; and let A have 8,000 supporters, 5,000 of whom take B as their No. 2, and the other 3,000 take C. Let B have 1,400 supporters and C 2,599. It does not signify whom these voters put as their No. 2, since A’s return is obvious, so that the only transferred lists are those on which he is No. 1.

Now A is returned with 4,000 votes to spare. Hence, if any teller has the opportunity of seeing the lists, he can easily arrange for the 4,000 transferred lists to contain 2,600 favourable to B, thus securing B’s return; or else to contain 1,401 favourable to C, thus securing C’s return.

But let us suppose all such cheating provided against, and that it is a matter of pure chance which 4,000, of the 8,000 lists headed “A,” are transferred. It is mathematically certain that the most probable event is that they will be divided between A and B in the same proportion—5 to 3—as the whole 8,000; i. e., that they will contain 2,500 lists headed “A B,” and 1,500 headed “A C.” Hence B will get 3,900 votes, and C 4,099; and C will be elected by a majority of nearly 200 over B. But there are 6,400 voters who prefer B to C, and only 5,599 who prefer C to B: so that, as a matter of fact, the unsuccessful candidate B has a majority of 801 over the successful candidate C:—I am, Sir, your obedient servant,

May 19, 1884

To the Editor of the St. James’s Gazette

Sir,—Mr. Cohen may rest assured that I should never think of applying the term “absurd” to any method of voting which had his support, though I do think that its practical application may prove in some cases “unfair.” The principle of the society is, in my belief, entirely right. Where the voters are divided into two parties, and where several members are to be returned, I hold that these should be divided, as nearly as possible, in the same proportion as the voters—e. g., if, in a town returning four members, rather more than five-eighths of the voters were Liberals and the rest Conservatives, the Liberals ought to return three of the members; if rather less than five-eighths, two. But this principle is not applicable where only two issues are possible; for then one party or the other must carry the day, and a compromise is no longer possible. And the case I proposed may easily be reduced to one of this sort.

Permit me to re-state the data, adding a new but not inconsistent hypothesis. A town containing 11,999 voters is to return two members, so that 4,000 votes are enough to return a member. There are three candidates—A, B, and C: 5,000 voters say “A B,” 3,000 “A C,” 1,400 “B A,” and 2,599 “C A.” The most probable result, if the Society’s method is adopted, is (as Mr. Cohen admits) to elect “A C.”

But this is a case where, as it appears to me, the society’s method is not fairly applicable; for there are, in fact, only two possible issues: the 6,400 voters, though differing as to the order in which they name the candidates, agree in wishing that A and B should be the two members returned; while the other 5,599 similarly wish to return A and C. Surely in this case, where no compromise is possible, the majority ought to have their wish, rather than the minority.

That the theory “A C ought to be elected” should commend itself with exceptional force to Mr. Arthur Cohen is not to be wondered at; but when he proposes to tell the unfortunate 6,400 voters that the reason the minority are allowed to carry all before them, and to return both their nominees, is that this is “the fair and proper way of giving effect to the desire of the voters,” and when Mr. Sidgwick adds his assurance that this method only claims “to give the minority their share (!),” I think the disappointed majority may be excused if they show some little coyness in accepting such doubtful consolation.—I am, Sir, your obedient servant,

P.S.—Mr. Sidgwick must surly have been reading the American Naturalist? “The snakes in this country may be divided into one species—the venemous.” Or else he is inspired by the poet who sang:

May 27, 1884

To the Editor of the St. James’s Gazette

Sir,—Having put before your readers, on the 15th and 19th of May, an instance where Sir John Lubbock’s method of taking votes fails to do justice, I propose now to state the additional rules needed to guard against such a result. Mr. Sidgwick has misunderstood me when he thinks it is my wish that the majority of a constituency shall return all the members, and that I object to giving the minority their share; and I think your readers are likely to misunderstand him when he says of me, “he demonstrates that this method will cause candidates to be elected who are not desired by the majority of the electors”: what I had demonstrated was something very different, namely, that it might cause one of two issues to triumph over the other, where (no compromise being possible) it had fewer supporters than the other issue; and that result I am sure neither Mr. Sidgwick, nor any other supporter of the method, would desire.

It may sound a paradox to say that this method enables the minority to return a fair share of members, and yet always gives the preference, when the question is whether one of two members shall be returned, or one of two issues (no compromise being possible) shall be adopted, to that side which has a majority of votes. Yet so it is. An instance will, I hope, make this clear. Take a town containing 29,999 voters, and returning five members. And let the voters be divided into two parties (call them “red” and “blue”). According to the method, 5,000 votes are enough to return a member. The reason is that each voter has only one vote, and that the five candidates who get the greatest numbers of votes are returned. Hence, a “red” candidate with 5,000 votes must be among the first five, for there cannot be more than four “blue” candidates who have as many votes as he: five such would require 25,000 votes, and there are only 24,999 to be had. Again, if the “reds” can muster 10,000 votes, they can return two members, by giving them 5,000 each; for there cannot be more than three “blue” candidates who have as many: four such would require 20,000 votes and there are only 19,999 to be had. Observe that, though the “reds,” being the minority, return two members, yet each of these has a majority of votes, compared with any rejected “blue” candidate.

The rules, required to complete the method, are as follows:—

(1). Divide the number of voters by the number of members to be returned, increased by one: and let n the lowest whole number greater than the quotient.

(2). If there are n lists in favour of A, A is returned: and so for B, or any other candidate.

(3). If there are n lists which suffice to return A, and which, erasing A, are in favour of no other than B; and other n lists which, erasing A, are in favour of B; B is returned: and so for A and C, or any other 2 candidates.

(4). If there are $2n$ lists, which suffice to return A and B and which, erasing A and B, are in favour of no other than C; and other n lists which, erasing A and B, are in favour of C; C is returned: and so for A, B, and D, or any other 3 candidates. Similarly for $3n$ lists, $4n$ lists, etc.

The words “no other than” are used in Rule 3, in order to meet the case of lists being handed in which contain A only. And there can be no doubt that, in thus disposing of the surplus lists, headed A, no such injustice can be done as I showed to be possible—and indeed probable—in the case examined in my former letters: for the lists, used to return A, could not, by being transferred, help any one but B; consequently the surplus lists may fairly be used in his interests.

Each rule must be applied as far as possible before taking the next. When all are exhausted, if there are still members to be returned, some other principle must be introduced. In the example, given in my former letters, these rules serve to return A, and then cease to be applicable.—I am, Sir, your obedient servant,

June 5, 1884

To the Editor of the St. James’s Gazette

Sir,—In reply to the charge which I brought (on the 15th of May) against the method of voting proposed by the “Society for Proportionate Representation,” that it is liable to bring in the wrong man, two pleas have been put forward: one, that the most probable result is also the most equitable; the other, that it can never be a matter of chance which of two candidates shall get in, unless they are of the same party.

The following instance will, I think, give the coup de grâce to both these pleas.

Take a town of 39,999 electors, returning three members, so that 10,000 votes will suffice to return a member; let there be four Liberal candidates, A, B, C, D, and one Conservative, Z; and let there be 21,840 lists “A B D,” 10,160 “A C B,” and 7,999 “Z.” There can be no shadow of doubt that, as a matter of justice, A, B, C, ought to be returned. Let us see what, under the society’s present rules, would be the most probable result.

The 32,000 lists headed “A” are of two kinds, bearing to each other the ratios of the numbers 273,127. Hence the most probable event is that the 10,000 lists, used in returning A, will contain 6,825 “A B D” and 3,175 “A C B.” Erasing “A” from the remaining lists, we have now in hand 15,015 “B D,” 6,985 “C B,” and 7,999 “Z”; so that Z is returned with a majority of more than 1,000 over C. And the Liberals must derive what consolation they can from the reflection that their rejected candidate really had 2,161 more supporters than the successful Conservative!—I am, Sir, your obedient servant,