Proportional Representation: A Study in Methods of Election

by John H. Humphreys

The difficulty can be made clear by a numerical instance. Take the case of an election for several seats, where the necessary quota is 6000, and where a favourite candidate, whom we will call A, has received the first votes of 10,000 voters. Though all those voters have agreed in putting the same candidate first, they are divided as to who may wish to be returned next. Six thousand of them put B as their second choice, and the other 4000 C. If the 6000 votes which A requires are drawn wholly from the AB votes, the result of the transfer will be that C is credited with 4000 votes and B with none. This would be clearly unfair, for, in reality, B has received among A's voters much more support than C. To use up the 4000 AC votes and only 2000 AB votes, and to transfer 4000 votes to B and none to C would be equally unfair to C. The course which is exactly fair to both 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. That is to say, strict justice will be done if every 1000 votes which are used or transferred are made up of 600 AB votes and 400 AC votes. Accordingly, A's quota of 6000 must be made up of 3600 AB votes and 2400 AC votes, and the 4000 papers left to be transferred will consequently consist of 2400 votes for B and 1600 votes for C.

This principle avoids all uncertainty, and is indisputably fair. It remains to consider how to carry it into effect. In most cases there would, in reality, be many more classes of votes than in the instance taken above. Even in such cases it is practicable, as will presently be shown, to divide the votes proportionately by an actual process of counting and separation. A certain amount of complication is, of course, introduced, but the extra labour involved does not seem impossible. The question whether this extra labour is necessary must be answered by examining the magnitude of the evil which it is sought to remedy.

If the votes are counted in a random order, it is clear there is a probability that the order in which they are drawn will correspond to the total numbers of each class in the ballot-box. It is reasonable to expect that when there are 10,000 ballot papers in an urn the composition of the first thousand drawn out will nearly be the same as that of any other thousand, or of the whole 10,000. The amount of this probability may be determined mathematically, and is very great.

This fact was clearly seen by Mr. Andrae, the statesman by whom the method of preferential voting was introduced into Denmark in 1855, and a mathematician of undisputed eminence. In answer to an objection of the kind now under discussion, he replied: "If this law of mine had already been in operation over the whole of Europe (including Turkey), for a period of 10,000 years, and if the elections in every part of Europe to which the law was applied were to take place, not every one, or three, or seven years, but every week in regular repetition, these elections throughout Europe, at the rate of a general European election per week, would still have to go on for more than a thousand times the period of years already stated; that is to say, for more than a thousand times ten thousand years, before the chances would be equal that the voting papers should come out of the urn in the order required to form the basis of this problem. Although, therefore, the supposed combination is, mathematically speaking, only an enormous improbability, yet, practically speaking, it is absolutely impossible."[2]

To state the matter more exactly, and as the result of an independent mathematical investigation, it appears that in the case we have stated, if 4000 voting papers were drawn out of A's heap at random, instead of the papers being carefully sorted and proportionately divided, the probability is that neither B nor C would gain or lose more than 11 votes. In other words, it is just even betting that the number of AB votes in the 4000 drawn would lie between 2411 and 2389 (inclusive), and consequently that the number of BC votes will lie between 1589 and 1611. The odds are more than 3 to 1 neither B nor C would gain or lose more than 20 votes, i.e. that the number of AB votes drawn will lie between 2420 and 2380; more than 10 to 1 that neither would gain or lose more than 30 votes; just 50 to 1 that neither would gain or lose more than 40 votes; and about 2000 to 1 that neither would gain or lose more than 60 votes. If the number of classes were larger or the number of votes to be drawn smaller, the effect would be much less. It will thus be seen that it is only in the case of very closely contested elections that the element of chance can affect the result. It will also be observed 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, since in almost all cases the electors who are agreed upon the candidate they most desire will also put for their second choice candidates of the same party.

In closely contested elections it must, of course, be admitted that as a result of this method, chance might decide which of two candidates of the same party should be elected. But in closely contested elections in large constituencies so many elements of chance are always and necessarily involved, that the introduction of a fresh one does not, in reality, make the result more arbitrary. Putting aside all the slight influences which at the last moment decide a score or two of featherweight votes, and assuming that every voter is profoundly convinced of the truth of his opinions, there remains the question of boundaries. A slight change in the line of the boundaries of the constituency might easily make a difference of fifty votes—a larger difference than what we are concerned with. To carry the dividing lines from North to South instead of from East to West, would, in many localities, completely alter the character of the representation.

These are, in reality, matters of chance, and more arbitrary in their nature than the order in which voting papers are drawn from an urn.

(2) Method of Eliminating the Chance Element

If, however, special precautions are still thought necessary, the following method of counting the votes appears to reduce, as far as practicable, the element of chance involved in the transfer of superfluous votes:—

The whole set of voting papers of the constituency being mixed, the papers, not yet unfolded, are drawn out one by one. Each is stamped, as it is drawn, with a corresponding number, 1, 2, … in order. It is then unfolded, and sorted according to the names of the candidates marked first and second upon it. Suppose there are six candidates, A, B, C, X, Y, Z; the votes of any candidate, A, will be sorted into six heaps, viz., A votes (i.e. votes where A only is voted for), AB, AC, AX, AY, and AZ votes. If A is found to have received more votes than he requires, the order in which the votes will be counted to him will be as follows: Use first the A votes, then use up those heaps where the second name also is that of a candidate who has received more than the necessary minimum. If these heaps give A more than he requires, take the same proportion out of each of such heaps, taking out of each heap the last drawn votes first. If, however, these heaps are used up without giving A as many votes as he requires, take an equal proportion of the votes of each of the remaining heaps—taking out of each heap the last drawn votes first.