1. The unit of representation is equal to the total number of valid votes cast at the election, divided by the number of seats.

2. Each party is entitled to one seat for every whole unit of representation contained in the aggregate votes polled by all its candidates, and the odd seat goes to the party which has the larger remainder.

The fact that the last seat has to be assigned to the party which has the larger remainder is sometimes advanced as an objection, but it is evidently the fairest possible division that the size of the electorate will permit. Of course, the larger the electorate the more accurately proportioned will be the representation. Hence the representation would be most accurate if the whole assembly were elected in one large electorate. But if, for the sake of convenience, the assembly be elected in a large number of electorates in which the relative proportions of two parties vary the gains which a party makes in some electorates will be balanced by losses in others, so that the final result would be almost as accurate as if the whole country were polled as one electorate. It must be remembered that the result in any electorate cannot be foreseen, and that it is a matter of chance which party gains the advantage. Now, if the limits of variation comprise even a single unit of representation, each party will stand an equal chance of gaining, and therefore the laws of chance will ensure that the gains balance the losses in the different electorates. Supposing a party which averages 40 per cent. in the whole country to vary between 30 per cent. and 50 per cent, in the different electorates (which may be taken as a fair assumption), the unit of representation should equal 20 per cent., or one-fifth. Under these conditions the laws of chance will ensure correct representation, so long as the electorates do not contain less than five seats.

The above facts furnish a complete answer to the arguments advanced by Mr. J.W. M'Cay, ex-M.L.A., in a series of articles in the Age against the application of proportional representation to the Federal Senate. While apparently recognizing that it is utterly impossible for the minority to secure a majority of the representation, he based his objection solely on the fact that a minority is able with electorates containing an even number of seats to secure one-half of the representation, and thus lead to what he terms "the minority block."

The force of the objection will entirely depend on the size of the minority which is able thus to thwart the will of the majority. The Federal Senate will consist of 36 senators, each of the original States contributing six. No reasonable man would complain if the minority, being only entitled to 17 senators, actually returned 18, but Mr. M'Cay points out that it is possible for a minority entitled to 15 senators to return 18. To bring about this result he makes the absurd assumption that in each of the six States the minority polls exactly two whole units of representation, and a bare majority of a third unit. It is safe to say that this would not happen once in a thousand years. If the relative proportions of the two parties vary in the slightest in the different States some must be under and some over the assumed proportion. It is most probable that it will be under it in three States and over it in the other three States; and, under these circumstances, the party will return 15 senators, the exact number to which it is entitled. It may happen to be under the assumed proportion in only two of the States and over in the other four, and that the party will get one more senator than it is entitled to; but it is extremely improbable that it will get two more, and virtually impossible that it will get three more senators than its just proportion. Mr. M'Cay's conclusion that proportional representation can only be used in electorates returning an odd number of representatives is shown to be entirely unwarranted. Equally fallacious is Professor Nanson's rebutting statement that "scientific proportionalists recommend odd electorates." While the number of States remains even, the mathematical chance of a minority securing one-half of the representation is precisely the same whether the States return an odd or an even number of senators. As a matter of fact, the danger of a minority securing one-half of the representation is much greater at the intermediate elections for the Senate, when each State returns three senators, the reason being the smaller field.

We have dwelt at some length on the preceding example, because it serves to refute another error into which some of the proportionalists have fallen. It is held that the unit of representation should be ascertained by dividing the total votes, not by the number of seats, but by the seats increased by one. This unit is generally known as the Droop quota, having been proposed in a work published by Mr. H.R. Droop in 1869. Since one vote more than one-half of the total votes is sufficient for election in a single-seat electorate, it is argued that one vote more than one-third suffices in a two-seat electorate, one vote more than one-fourth in a three-seat electorate, and so on. The unit in a six-seat electorate would be one-seventh of the votes instead of one-sixth, and it is pointed out that by this means the whole six seats would be filled by whole units, leaving an unrepresented residuum of one-seventh of the votes divided between the two parties.

The error lies precisely as before in concentrating attention on one of the electorates, and in neglecting the theory of probability. The Droop quota introduces the condition that each party must pay a certain minimum number of votes for each seat, and the real distinction is that, instead of the minority and the majority having an equal chance of securing any advantage, the chances are in the same proportion as their relative strengths. If the majority be twice as strong as the minority, it will have twice the chance of gaining the advantage. To prove this, consider the position of a one-third minority in a number of five-seat electorates. The Droop quota being one-sixth of the votes, the minority will secure two seats or 40 per cent. in those electorates where it is just over one-third, and one seat or 20 per cent. where it is just under. Since the mathematical chances are that it will be over in one half and under in the other half, it will, on the average, secure only 30 per cent., although entitled to 33 per cent. Again, if the 670 members of the House of Commons were elected in three to five-seat electorates, and the Droop quota used as proposed by Sir John Lubbock, and if the Ministerialists were twice as strong as the Oppositionists, they would, on the average, return 30 more members than the two-thirds to which they are entitled, and this would count 60 members on a division.

The following table illustrates the erroneous result obtained by applying the Droop quota when a number of grouped-electorates are concerned. It will be noticed that where parties are nearly equal it makes very little difference which unit is used:—

STRENGTH OF
PARTY		AVERAGE REPRESENTATION
Five-Seat ElectoratesTen-Seat Electorates
10 per cent.6 per cent.2 per cent.
20 per cent.14 per cent.17 per cent.
30 per cent.26 per cent.28 per cent.
40 per cent.38 per cent.39 per cent.
50 per cent.50 per cent.50 per cent.

The Droop quota, therefore, gives, not proportional, but disproportional representation.

Election by Each Party of its Most Popular Candidates.—Still keeping in mind the six-seat electorate for the Federal Senate, we may note that there are two rival systems in the field—the scrutin de liste or Block Vote, in which each elector votes for any six of the candidates, and the Hare system, which allows each elector an effective vote for one candidate only. The adoption of either of these systems would be unfortunate. To force each elector to vote for six candidates is probably to require him to vote for more than he is inclined to support, and certainly for more than his party is entitled to return; and, also, to put it in the power of the majority to return all six senators. To allow him to vote for one candidate only, on the other hand, is to break up both parties into factions by allowing the favourites of sections within the parties to be elected, instead of those most in general favour with all sections composing each party. An intermediate position is therefore best. No elector should be required to vote for more than three candidates, and no elector should be allowed to vote for less. Because in the first place it is evident that each party will, on the average, return three senators, and, secondly, it may be taken for granted that even the minority will nominate at least three candidates. Two alternative proposals may be submitted as fulfilling these conditions:—