The Principles of Parliamentary Representation/Chapter III

by Charles L. Dodgson

 

Chapter III.

 

Principles to be observed in conducting Elections.

 

§ 1. Number of Votes each Elector may give.

The two extreme cases are (1) to let each Elector give as many votes as there are Members to be returned by the District; (2) to let him give one vote only.

The effect of each of these methods, and of the intermediate methods which lie between them, will be best understood by considering the following Tables of percentages.

We will first find general formulæ for determining what number of Electors, in a given District, is necessary and sufficient to secure the return of one Candidate, of 2, of 3, &c.

Let e =	No. of	Electors in the District,
m =	. . . . .	Members assigned to it,
v =	. . . . .	votes each Elector can give,
s =	. . . . .	seats it is desired to fill,
x =	. . . . .	Electors required.

Also let it be assumed that an Elector may not give 2 votes to the same Candidate. (N.B. 'cumulative' voting is discussed at p. 27.)

Now, in order that x may be sufficient to fill s seats, it must be large enough to make it impossible for the other (e − x) Electors to fill (m + 1 − s) seats; since the two events are incompatible, so that, if the latter were possible, the former would be impossible. To effect this, each of the s Candidates must have more votes than it is possible to give to each of (m + 1 − s) rival Candidates.

In order that x may be necessary, it must be only just large enough for the purpose.

It will be necessary to consider the following 4 cases separately. Observe that > means 'greater than,' ≯ means 'not greater than,' and ∴ means 'therefore'.

Case (a)v is ≯ s, and also ≯ (m + 1 − s);
Case (b) . . . . . > s but ≯ (m + 1 − s);
Case (c) . . . . . ≯ s, but > (m + 1 − s);
Case (d) . . . . . > s and also > (m + 1 − s).

In case (a), the x Electors can give vx votes, which, divided among s Candidates, supply them with vx/s votes apiece. Similarly, the (e − x) Electors can give v.(e − x) votes, which, divided among (m + 1 − s) Candidates, supply them with v.(e − x)/m + 1 − s votes apiece. Hence we must have

vx/s >	v.(e − x)/m + 1 − s,
	where v divides out;
∴ x.(m + 1 − s) >	se − sx;
∴ x.(m + 1) >	se;
∴ x >	se/m + 1.

In case (b), each of the x Electors can only use s of his v votes, since he can only give one to each Candidate: hence the x Electors can only give sx votes, thus supplying s Candidates with x votes apiece. But the (e − x) Electors can, as in case (a), supply (m + 1 − s) Candidates with v.(e − x)/m + 1 − s votes apiece. Hence we must have

x >	v.(e − x)/m + 1 − s
∴ x.(m + 1 − s) >	ve − vx;
∴ x.(m + 1 − s + v) >	ve;
∴ x >	ve/(m + 1 − s + v).

In case (c), the x Electors can, as in case (a), supply s Candidates with vx/x votes apiece. But each of the (e − x) Electors can only use (m + 1 − s) of his votes: hence the (e − x) Electors can only give (m + 1 − s).(e − x) votes, thus supplying (m + 1 − s) Candidates with (e − x) votes apiece. Hence we must have

vx/s >	e − x;
∴ vx >	se − sx;
∴ x.(s + v) >	se;
∴ x >	se/s + v.

In case (d), the x Electors can, as in case (b), supply s Candidates with x votes apiece. And the (e − x) Electors can, as in case (c), supply (m + 1 − s) Candidates with (e − x) votes apiece. Hence we must have

x >	e − x;
∴ 2x >	e;
∴ x >	e/2.

Tabulating these results, we have the following formulæ.

	Data.	Formulæ.
(a)	v ≯ s ≯ m + 1 − s	x > se/m + 1
(b)	v > s ≯ m + 1 − s	x > ve/m + 1 − s + v
(c)	v ≯ s > m + 1 − s	x > se/s + v
(d)	v > s > m + 1 − s	x > e/2

By these formulæ the following Table is calculated. It shows, for a given District, what percentage of the Electors is necessary and sufficient to secure the return of one Candidate, of 2, of 3, &c.

The 2nd line in the 3d section represents the well-known "three-cornered constituency." Observe (by comparing it with the next line) that it makes it too hard for a minority to fill one seat, and too easy for a majority to fill all.

 

TABLE III.

No. of Members ret. by District.	No. of votes each Elector can give.	No. of Seats it is desired to fill.
		1	2	3	4	5	6
1	1	51
2	2 1	51 34	51 67
3	3 2 1	51 41 26	51 51 51	51 61 76
4	4 3 2 1	51 43 34 21	51 51 41 41	51 51 61 61	51 58 67 81
5	5 4 3 2 1	51 45 38 29 17	51 51 43 34 34	51 51 51 51 51	51 51 58 67 67	51 56 63 72 84
6	6 5 4 3 2 1	51 46 41 34 26 15	51 51 45 38 29 29	51 51 51 43 43 43	51 51 51 58 58 58	51 51 56 63 72 72	51 55 61 67 76 86

 

In examining this Table, we notice, first, the uniformity of the upper line in each section (i.e. the percentages required when each Elector can give as many votes as there are seats to fill). Here, in every case, more than half the Electors must agree, in order to fill one single seat: but, when once this number have mustered, they have it in their power to fill all the seats! 'C'est le premier pas qui coûte.'

This absurdity diminishes gradually, from line to line, as we look down each section; the lowest line (i.e. the percentages required when each Elector can give one vote only) being always the most reasonable. One of the most startling anomalies is the 4th line of the 6th section. Here we see that, out of 100 Electors, we must muster 34 in order to fill one seat: with four more Electors, we can fill the second seat: with five more, the third: but 'then comes the tug of war'; to win the fourth seat, we actually need fifteen more Electors!

Lastly, comparing together the lowest lines of the several sections, we notice that they gradually improve as we move down from section to section, requiring a smaller percentage to fill one seat, thus giving a minority a better chance of being represented, and a larger percentage to fill all, thus leaving a smaller number unrepresented. This last figure (the right-hand end of each lowest row) represents the percentage of the Electors in the Kingdom who would be represented in the House, supposing all the Districts similar to the one under consideration: and this percentage we find to rise, from 51 in the case of single-Member Districts, to 86 in the case of six-Member Districts.

The obvious conclusion is—let the Districts be as large as possible, and let each Elector give one vote only.

The effect, on the composition of the House, will be yet more clearly seen by considering the following three Tables, which are calculated on the assumption that, in any District, all proportions, between 'red' and 'blue,' are equally probable, and that 6-11ths of the House are 'red' and 5-11ths' blue.' Table IV. gives the percentage of the whole body of Electors represented by the 'red' Members, Table V. the percentage represented by the 'blue,' and Table VI. the percentage unrepresented:—

 

TABLE IV.

Number of Members assigned to each District.		Number of votes each Elector can give.
		6.	5.	4.	3.	2.	1.
	1.	....	....	....	....	....	28
	2.	....	....	....	....	28	37
	3.	....	....	....	28	36	42
	4.	....	....	28	35	40	44
	5.	....	28	33	39	43	46
	6.	28	32	36	40	44	48

TABLE V.

Number of Members assigned to each District.		Number of votes each Elector can give.
		6.	5.	4.	3.	2.	1.
	1.	....	....	....	....	....	23
	2.	....	....	....	....	23	31
	3.	....	....	....	23	30	34
	4.	....	....	23	29	34	37
	5.	....	23	28	32	36	38
	6.	23	27	30	34	37	38

 

TABLE VI.

Number of Members assigned to each District.		Number of votes each Elector can give.
		6.	5.	4.	3.	2.	1.
	1.	....	....	....	....	....	49
	2.	....	....	....	....	49	32
	3.	....	....	....	49	34	24
	4.	....	....	49	36	26	19
	5.	....	49	39	29	21	16
	6.	49	41	34	26	19	14

By inspecting these Tables, we see two things:—

First, that the fewer and larger the Districts, i.e. the greater the number of Members returned (on an average) by each District, the more equitable the result. This conclusion we have already arrived at, from general considerations. (See p. 6, line 1.) We observe, further, that the advantage, in fairness of result, increases rapidly at first and more slowly afterwards. For instance, in Table VI, if each Elector be allowed one vote only, the change from single-Member to two-Member Districts changes the percentage of unrepresented Electors from 49 to 32 (i.e. deducts about 1-3rd); whereas the change, from 5-Member to 6-Member Districts, only changes the percentage from 16 to 14 (i.e. deducts only 1-8th). The conclusion is that the important point is to have as few single-Member, and even as few 2-Member, Districts as possible; but that, when we have got as far as to Districts returning 4 or 5 Members each, it is hardly worth while to go further.

Secondly, we see that the fewer the number of votes (down to the least possible, viz. 'one') that each Elector is allowed to give, the more equitable the result. We observe, further, that the advantage, in fairness of result, increases slowly at first and more rapidly afterwards. For instance, in Table VI, if 6 Members be assigned to a District, the change from 6 votes to 5 only changes the percentage of unrepresented Electors from 49 to 41 (i.e. deducts less than 1-6th); whereas the change from 2 votes to one changes it from 19 to 14 (i.e. deducts more than 1-4th). We observe, further, that the system of allowing each Elector as many votes as there are seats to fill produces, in every case, the same result, (the most, inequitable that it is possible to produce by any variation in these data,) viz. that it leaves about 49 p. c. of the Electors unrepresented. The system (already discussed at p. 4) of "equal electoral Districts, each returning one Member" is only a particular instance of this general law.

The method of 'cumulative voting' (where an Elector can give two or more votes to the same Candidate) will usually have no other effect than to increase the 'specific gravity'—so to speak—of a vote. Let each Elector have 4 votes, with permission to 'lump' them if he chooses, and in the end you will find most of the votes given in lumps of 4, and the result much the same as if each Elector had had one vote only.

The conclusion is that the important point is to let each Elector give one vote only.

§ 2. Formula for determining, after the poll is closed, the quota of Votes needed to return a Member.

By a process, exactly similar to that employed at p. 9, we may prove that, if 'r' be the number of recorded votes, and 'm' the number of Members to be returned, the quota must be just greater than r/m + 1. For example, if 55,000 votes had been given, and the District had to return 6 Members, the quota needed to return one Member would be just greater than 7,857 and 1-7th: i.e., a Member, having 7,858 votes, would be returned. Similarly, anything just greater than 15,714 and 2-7ths would be enough (if the votes could be reckoned en masse) to return 2 Members: i.e., if 2 Members of the same party had 15,715 votes between them, both could be returned. We shall prove, further on, that such reckoning of votes is equitable and ought to be provided for.

This quota must be carefully distinguished from the one discussed at p. 9. If a District, returning one Member, contains 10,001 Electors, the quota needed, before the poll is closed, to make it certain that 'A' will be returned, is 5,001; but, if only 8,001 vote, the quota needed, after the poll is closed, to return him, is only 4,001. For the purpose of assigning Members to a District, it is fair to proceed as if all the Electors were sure to vote; but, for the purpose of returning Members, we can count only the votes that are actually recorded.

 

§ 3. Method for preventing waste of Votes.

Assuming it to be agreed that each District is to return 2 or more Members, and that each Elector is to give one vote only, we have now to consider what is to be done when 2 or more Candidates of the same party have got, among them, enough votes to be returned, but when some have got more than the quota, and others less. It is obviously not fair that the party should fail in bringing in their rightful number of Members, merely by an accidental disarrangement of votes; but how to make an equitable transfer of the superfluous votes is by no means so obvious.

Various methods have been proposed for this: of which I will consider two:—

(1) "The Proportional Representation Society" proposes to let each Elector hand in a list of Candidates, marked in the order of his preference; and that his vote, if not required for his No. 1, should be transferred to his No. 2, and, if not required for him, then to No. 3, and so on. One great objection to this method is the confusion it would cause in the mind of an ignorant Elector, who, though quite able to name his favourite Candidate, would be utterly puzzled if told to arrange 5 or 6 names in order of merit. But a much stronger objection is the difficulty, of deciding to which of the remaining Candidates the surplus votes shall go: e.g. if 8,000 be the quota needed to return a Member, and if 6,000 lists be headed 'A B,' and 4,000 'A C,' which 2,000 are to be transferred? Mr. J. Parker Smith, in a Pamphlet entitled "Preferential Voting," says (at p. 2), "The course which is exactly fair to B and C is that the votes which are transferred should be divided between them in the same proportion as that in which the opinions of the whole number of A's supporters is divided." (This would require, in the above instance, that 3-5ths of the 2,000, i.e. 1,200, should be taken from the 'A B' lists, and 2-5ths, i.e. 800, from the 'A C' lists.) He adds, "This principle avoids all uncertainty, and is indisputably fair." He then proceeds to show that if, instead of counting and arranging the surplus votes, they be taken "in a random order," the chances are very great that they will come out nearly in this proportion. And he further adds (at p. 4), that "the element of chance will not be of importance as between the different parties, but only as between different individual Candidates of the same party." Now all this rests on the assertion that this mode of dividing the surplus votes, whether effected by counting or left to chance, is "indisputably fair:" and this assertion I entirely deny. The following instance will serve the two purposes, of showing that this method may easily lead to gross injustice, and of showing that the difficulty may easily arise between candidates of opposite parties.

Take a town of 39,999 Electors, returning 3 Members, so that 10,000 votes will suffice to return a Member; let there be 4 'red' Candidates, A, B, C, D, and one 'blue,' 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, since there are more than two full quotas who put 'A B' first, and, over and above these, more than one quota who put 'A C' first. 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 certain event, if the lists are divided by rule, and the most probable event, if they are divided at random, 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 B is returned. Erasing "B" from the remaining lists, we now have 5,015 "D," 6,985 "C," and 7,999 "Z"; so that Z is returned with a majority of more than 1,000 over C. And the 'reds' must derive what consolation they can from the reflection that their rejected Candidate really had 2,161 more supporters than the successful 'blue'!

While fully agreeing, then, with the Proportional Representation Society as to the propriety of allowing only one vote to each Elector, I think I have sufficiently proved the fallacy of its method for disposing of surplus votes.

(2) A mechanical method of recording votes was suggested, in a letter signed "F. R. C.," in the St. James' Gazette for Aug. 1. Each Elector is to pass (unseen) through one of a set of turnstiles, (each Candidate having a separate turnstile), which will mechanically record his vote. The records are to be periodically examined, and the results placarded outside, in order that Electors, on seeing that a Candidate has already got votes enough to secure his return, may cease to vote for him. Several objections, each by itself fatal, may be made to this method. One is that, if the periods were short enough to prevent waste of votes, the inspection would destroy the secrecy of the ballot, as it would be known who had just voted, and the result of his voting would be at once placarded; whereas, if the periods were long enough to avoid this, time would be allowed for large waste of votes. Another is that, as the quota, necessary to return a Candidate, could not be fixed till the poll had closed, it would be impossible to know, during the Election, whether a Candidate had or had not received votes enough to secure his return. Another is that, if part of the machinery went wrong, so as (for instance) to record a total of votes greater than the number of Electors, the mistake could not (as it can with voting-papers) be rectified, but the Election would have to be held over again.

Having proved, then, that the method of arranged lists will not serve fairly to dispose of surplus votes, and yet that we cannot prevent such votes being given, we have now to find, if possible, a fair method for disposing of them. Clearly somebody must have authority to dispose of them: it cannot be the Elector (as we have proved); it will never do to refer it to a Committee. There remains the Candidate himself, for whom the votes have been given. This seems to solve the whole difficulty. The Elector must understand that, in giving his vote to A, he gives it him as his absolute property, to use for himself, or to transfer to other Candidates, or to leave unused. If he cannot trust the man, for whom he votes, so far as to believe that he will use the vote for the best, how comes it that he can trust him so far as to wish to return him as Member?

 

§ 4. Method for preventing the Electors in one District from being influenced by the results of Elections in other Districts.

That Electors are liable to such influences may be proved both a priori and a posteriori. On the one hand, it is a tendency of human nature, too well-known to need proving, to surrender one's own judgment in order to be on the winning side. In the words of the immortal Mr. Pickwick, "it's always best on these occasions to do what the mob do." "But suppose there are two mobs?" suggested Mr. Snodgrass. "Shout with the largest," replied Mr. Pickwick. On the other hand, no one, who has ever watched the progress of a General Election, can need to be reminded how obviously the local Elections of the later days have 'followed suit,' under the irresistible influence of those of the earlier days. "The secret of success," it has been well said, "is to succeed:" and there can be little doubt that the party, which fails in carrying a majority of the local Elections at first, is heavily handicapped during the rest of the contest.

Supposing it admitted that such an influence does exist in General Elections as now managed, and that it is an influence to be avoided, the remedy is not far to seek: let the local Elections be so arranged that all, or nearly all, the results may be announced at the same time.

This arrangement would no doubt be unwelcome to certain 'pluralists,' who are now able to vote in several different Districts. Possibly, in such exceptional cases, voting-papers might be allowed. But, even if no remedy could be found, the justice of allowing one Elector to vote as if he were, "like Cerberus, three gentlemen at once," seems so doubtful that the objection hardly deserves serious consideration.

§5. Conduct of Elections.

The practical working of the principles, which have now been demonstrated, would be as follows:—When the poll is closed, let the total number of votes recorded be divided by the number of Members to be returned increased by one, and let the returning-officer announce the whole number next greater than the quotient as the quota needed to return one Member. Similarly, the whole number next greater than twice the quotient will be the quota needed to return two, and so on.

Let him further announce the number of votes given for each Candidate, atid also announce as "returned" any Candidate who has received the quota needed to return one. If there are still Members to return, let him appoint a time and place for all the Candidates to appear before him; and any two or more Candidates may then formally signify that they wish their votes to be clubbed together, and may nominate so many of themselves as can be returned by the votes so clubbed. They must of course include in their nomination any of themselves who have been already declared to be returned. Let the returning-officer add together the votes of these Candidates, and, if the amount be not less than the necessary quota, let him declare to be duly returned the Candidates so nominated.

As an example, suppose that a District is to return 5 Members, and that there are 4 'red' Candidates, A, B, C, D, and 3 'blue,' X, Y, Z. Then the returning-officer might announce as follows:—

 

Votes given for
C	.	15,000
X	.	9,000
D	.	8,001
Z	.	8,000
B	.	7,500
A	.	6,500
Y	.	6,000
	6	60,001
		10,000	and 1-6th
Quota needed to return
1 Member .	.	10,001
2 Members	.	20,001
3 Members	.	30,001
4 Members	.	40,001
5 Members	.	50,001
I hereby declare C to be duly returned.
Four vacancies remain to be filled.
(Signed)

 

The Candidates might then appear before the returning-officer, and B, C, D might formally declare that they wished to club their votes; and, as the sum total of their votes is 30,501, they would be declared to be "returned": similarly, X, Y, Z might club their votes, naming X and Z as the Candidates to be returned; and, as the sum total of their votes is 23,000, X and Z would be declared to be "returned."

 

Such Candidates would have to sign some such paper as the following:—

 

We, the undersigned, for whom the recorded votes, as stated below, amount to    , which is not less than    , the quota announced as needed to return   Candidates, hereby declare that we desire the said votes to be clubbed together. And we nominate, as Candidates whom we desire to be returned by the said votes, in addition to        , who have been already declared to be duly returned,        .
	Names.	Votes.
Signed,
	Sum total of votes

 

This method would enable each of the parties in a District to return as many Members as it could muster the proper quota for, no matter how the votes were distributed. There would be no risk of a seat being left vacant through rivalry between two Candidates of the same party: an unwritten law would soon come to be recognised—that the one with fewest votes should give way. With Candidates of two opposite parties, such a difficulty could not arise at all: one or other of them could always be returned by the surplus votes of his own party. The only exception to this would be the occurrence (a very rare one) of an exact balance of votes. This might happen, even in the case of a single-Member constituency, if each of 2 Candidates got exactly half the votes. Of course, in such a case, somebody must give a casting-vote.