The (almost really) Complete Works of Lewis Carroll

October 11, 1884

Now that the Government scheme for Redistribution has been made public, it seems a fitting time for calling attention to the general principles on which Redistribution ought to be conducted.

In the article on “Parliamentary Elections,” which you did me the honour to publish on July 5, I explained a method for conducting Elections which (given that the various Districts had had Members assigned to them in proper proportions) would give the most equitable result possible: in the present article I propose to explain a method of Redistribution, by which Members would be equitably assigned to the various Districts. If both parties could agree to accept some such general principles for Redistribution and the conduct of Elections, each would feel secure that, whether it comprised a majority or a minority of the Electors, it would be fairly represented in the House, and there would be no objection, on either side, to passing the Franchise Bill, with the proviso that it should not come into operation till the Redistribution Bill had also been passed.

Assuming for a moment, for the sake of simplicity, that all Electors, whatever the value of their rateable property, and whether they be in town or country, are to have equal political weight, it is plain that an ideally perfect House would be one where each Member represented the same number of Electors, where the proportion between parties in the House was exactly the same as in the whole body of Electors, and where every Elector was represented by the same fraction of a Member. Such a result is however unattainable. The nearest possible approach to it would be to make the whole Kingdom into one gigantic District, and let each Elector give one vote only: in this case, if the House consisted of 600 Members, and if means were used to prevent votes being wasted, each Member would represent 1-601th of the whole body of Electors, the proportion of parties in the House would be almost exactly the same as in the Kingdom, and only 1-601th of the whole body of Electors would be unrepresented: but it would be impossible to conduct an Election on so gigantic a scale. The other extreme would be the scheme of “equal electoral Districts, each returning one Member,” which would leave nearly half of the whole body of Electors unrepresented.

Before settling a formula by which, given the number of Electors in a District, it could be calculated how many Members should be assigned to it, it would be necessary to agree what weight, if any, should be given to value of rateable property, and whether any difference should be made between town and country voters. The simplest way of giving effect to whatever were agreed on would be to have formulæ for multiplying the actual number of Electors in a District, and to use the number so obtained, instead of the actual number, in assigning Members to it: e. g. suppose it were agreed that the unit of rateable property should be £5 a year, and that the multiplier employed should be the square-root of the average value, then, in a District where the average value was £20 (4 times the unit), the multiplier would be ‘2’; and, if it were further agreed that town Electors should have 10 p. c. more weight than country Electors, then, in a District containing 20,000 town Electors and 30,000 country Electors, with an average value of £20 a year of rateable property, we should first add 2,000 to the number of town Electors, and then multiply the whole 52,000 by 2: that is, we should consider the District to contain, for the purpose of assigning Members, 104,000 Electors.

The question, how many Districts to make, which is the same thing as to ask how many Members to assign (on an average) to each District, and the question how many votes to allow each Elector to give, will be best considered in connection with the following Tables, which are calculated on the assumption that 6-11ths of the House are ‘red’ and 5-11ths ‘blue.’ Table I. gives the percentage, of the whole body of Electors, represented by the ‘red’ Members, Table II. the percentage represented by the ‘blue,’ and Table III. the percentage unrepresented:—

By inspecting these Tables we see that, the fewer the Districts (i. e. the greater the number of Members assigned to each District) and the smaller the number of votes each Elector can give, the more equitable is the result. With less then 10 Members to a District, or with more than one vote to an Elector, it is always possible that the majority of the Electors should be ‘blue,’ although 6-11ths of the House are ‘red.’ For example, if 3 Members be assigned to each District, and each Elector have 2 votes, the ‘reds’ in the House represent 33 p. c. of the Electors, and the ‘blues’ 28 p. c., while 39 p. c. are unrepresented. It might easily happen that 25 of these 39 were ‘blue’ and 14 ‘red’: in which case 53 p. c. of the Electors would be ‘blue,’ and only 47 p. c. ‘red,’ and yet the ‘reds’ would have a large majority in the House!

These Tables have been calculated for equal Districts, but probably the results would be much the same for unequal Districts, so long as very small Districts were as far as possible avoided.

We have now to determine a formula by which, given the number of Electors in a district (that is the nominal number, making due allowance for rateable property &c.) it may be calculated how many Members ought to be assigned to it. This formula should be such as to make the “quota,” necessary to return a Member, as nearly as possible the same for every District. Now, if ‘Q’ be this uniform quota, and if ‘e’ be the (nominal) number of Electors in a District, and ‘m’ the number of Members assigned to it, and if it be agreed that each Elector is to have one vote only, we know that Q must be just greater then $\frac{e}{m+1}$. Hence we set $m=\frac{e}{Q}-1$, as the formula required. For example, if the quota were 5,000, a District containing (nominally) 30,000 Electors would have 5 Members assigned to it.

It remains to be seen how to find an equitable value for Q. Let $e_1$ be the (nominal) number of Electors in District No. 1, and $m_1$ the number of Members assigned to it, and so on: also let E be the total (nominal) number of Electors, M the total number of Members, and D the number of Districts: then we have $$\begin{array}{rl}(m_1+1).Q& =e_1\\ (m_2+1).Q& =e_2\\ \text{\&c.}& \\ \therefore (M+D).Q& =E\text{; i.\hspace{0.17em}e. }Q=\frac{E}{M+D}\text{.}\end{array}$$

Let me say in conclusion that the object aimed at, in the methods I have proposed for Redistribution and the conduct of Elections, is fully as much to secure that the Majority of the Electors shall be duly represented as that the Minority shall be so. The same is true of the method (a less perfect one, as I believe) proposed by “the Proportional Representation Society.” In speaking to friends about this Society, I have had such answers as this:—“I have no interest in their proposal. All I care for is that the Majority of the Electors shall be fully represented in the House: it matters little whether the Minority are represented or not.” Such people seem to think that the existing system secures what they desire. In this they are utterly mistaken: there is ample room, under the existing system, for a large Majority of the Electors to find themselves in a woeful minority in the House. The only change for the worse, that I can think of, would be the method of “equal electoral Districts each returning one Member.” Almost any other change would be a change for the better: but any system which, like the one I have here proposed, secures that the Majority of the Electors shall be duly represented, must necessarily do the same for the Minority.

October 22, 1884

To the Editor of the St. James’s Gazette

Sir,—Mr. G. A. Simcox, in discussing my proposals (of Oct. 11) for Redistribution, asks “how is the average elector to be saved from throwing away his vote on a candidate who is safe without it?” This question does not belong to the subject of “Redistribution,” but to that of “Parliamentary Elections,” under which heading I fully discussed it in my paper of July 5, and there proposed that such votes should be at the absolute disposal of the candidate for whom they were given. I also discussed, but only to reject them, the other methods suggested by Mr. Simcox.

I am embodying both papers in a small pamphlet, which I hope to publish in a few days, and of which I shall be happy to send Mr. Simcox a copy.—I am, Sir, your obedient servant,