Chapter XXII: Appendix: To Schedule

EXAMPLE OF AN ELECTION CONDUCTED ON THE SYSTEM OF PROPORTIONAL REPRESENTATION SET OUT ABOVE

Let it be assumed that there are five members to be elected, and that there are ten candidates.

The valid papers are drawn from the general heap of ballot papers and arranged in separate parcels under the names of the candidates marked with the figure 1. (Rule 1.)

Each separate parcel is counted (Rule 1) and the total of all the valid votes is ascertained (Rule 2). It is found that the total of all the valid votes is 6000.

This total is divided by six (_i.e._ the number which exceeds by one the number of vacancies to be filled), and 1001 (_i.e._ the quotient 1000 increased by one) is the number of votes sufficient to elect a member, and is called the "quota" (Rule 2).

The result of the count may be supposed to be as follows:--

A 2,009 Elected B 952 C 939 D 746 E 493 F 341 G 157 H 152 I 118 K 93 ----- 6,000

A's votes exceed the quota and he is declared elected (Rule 3).

_First Transfer_.

It now becomes necessary to transfer A's surplus votes (Rule 4 (1)). A has in fact (2009 less 1001 or) 1008 surplus votes. All A's 2009 voting papers are examined and arranged in separate sub-parcels according to the second preferences indicated thereon (Rule 4 (1) (_a_)). A separate sub-parcel is also formed of those papers on which no second preference is shown, and which are therefore not capable of transfer. (Rule 4 (1) (_b_).) The result is found to be as follows. (Rule 4 (1) (_c_).)

A second preference is shown for G on 1,708 papers
" " " D " 257 "
" " " E " 11 "
" " " F " 28 "
-----
Total of votes capable of transfer 2,004 "
No second preference is shown on 5 "
-----
Total of A's votes 2,009

The total number of votes to be transferred is 1008, and it is necessary that they should be taken from the several sub-parcels in the proportions which the latter bear to all the votes capable of transfer; that is, there must be transferred, _e.g.,_ to G a number of votes bearing the same proportion to 1008, the total to be transferred, as 1708, the number of votes in G's sub-parcel, bears to 2004, the total of votes capable of transfer. In other words the number of the ballot papers on which each candidate is next preference must be multiplied by a fraction of which the surplus is the numerator and the total of votes capable of transfer the denominator, in order to ascertain the number of votes to be transferred to the candidate in question. In making the transfers fractions of votes are neglected (Rule 4 (1) ( e) and (f)).

The process is as follows:--

To G there are to be transferred 1,708 x 1,008 / 2,004 = 589 votes

" D " " " 257 x 1,008 / 2,004 = 129 "

" E " " " 11 x 1,008 / 2,004 = 5 "

" F " " " 28 x 1,008 / 2,004 = 14 "
-------
1,007

859, 129, 5 and 14 votes are now transferred to G, D, E, and F respectively, the particular voting papers taken being those last filed in their sub-parcels, and therefore at the top of the sub-parcels. These voting papers are added in separate sub-parcels to G, D, E, and E (Rule 4 (2)).

Their totals then become--

G . . . . . 157 + 859 = 1,016 D . . . . . 746 + 129 = 875 E . . . . . 493 + 5 = 498 F . . . . . 341 + 14 = 355

All the other voting papers in A's parcel (1002 in number) are set aside as finally dealt with (Rule 4 (3)), the figure 1002 being the quota 1001 with the addition of the one further vote of the surplus which, owing to the disregard of fractions, is not transferred. G having obtained more than the quota is now declared elected (Rule 5), and the poll stands as follows:--

A 1,002 Elected G 1,016 Elected B 952 C 939 D 875 E 498 F 355 H 152 I 118 K 93

_Second Transfer_

G has now more than the quota, and his surplus votes (1016 less 1001 or 15) would have to be transferred (Rule 6(1)) were it not for the provisions of Rule 6(7). But under that rule, the process of transferring a surplus is postponed in a case where the surplus is less than the difference between the two lowest candidates on the poll, and where, therefore, the transfer would produce no practical effect. In this case the difference between I and K, the two lowest candidates, is 118 - 93, or 25, and therefore it is not necessary to transfer G's surplus.

The returning officer proceeds to distribute the votes of the candidates with the smallest totals (Rules 7 and 8).

K's parcel is therefore examined and is found to contain 89 papers on which F is next preference, and 4 on which C is next preference.

Therefore 89 votes are transferred to F and 4 to C.

The poll now stands--

A 1,002 Elected G 1,016 Elected B 952 C 943 D 875 E 498 F 444 H 152 I 118

No further candidate has the quota.

_Third Transfer_

The difference between I and H exceeds G's surplus, which therefore is allowed to remain (Rule 6 (7)), and the votes of I as now lowest on the poll have now to be distributed in the same manner as K's (Rule 8). But as the combined votes of H and I, together with G's surplus (152 + 118 + 15 = 285), are less than 444, the total of F, the next highest candidate, the returning officer avails himself of Rule 7 (2), and distributes both H and I's votes at one operation.

I's parcel is found to contain 107 papers on which D and 11 on which B is next preference, and H's parcel is found to contain 108 papers on which B is next preference, and 44 on which there is no available preference marked. (In some cases, some or one of A, G, I, H, and K are marked as next in order of preference on the papers examined, but as all of them are already either elected or excluded they are left out of account.) Therefore, 107 votes are transferred to D, and 119 (108 + 11) to B, while 44 are set aside as finally dealt with (Rule 7 (1)). The result is to give B the quota, and he is declared elected.

The poll now stands--

A 1,002 Elected G 1,016 Elected B 1,071 Elected D 982 C 943 E 498 F 444

_Fourth Transfer_

B has now a surplus of 70 votes, and it is necessary to distribute this (Rules 7 (4), 6, and 4) as it exceeds the difference between E and F, which is 54 (Rule 6 (7)).

For this purpose only the 119 votes last transferred are taken into account (Rule 6 (2)).

These are examined and arranged in sub-parcels, in the same manner as A's votes were examined and arranged, with the following result: A next preference is shown for E on 84 papers. No further preference is shown on 35 papers. The total number of votes capable of transfer (84) is thus greater than the surplus (70), but, as there is only one possible transfer, the process is simple: 84 x 70/84 = 70; and so the 70 votes last filed in E's sub-parcel are transferred to E.

The poll now stands--

A 1,002 Elected G 1,016 Elected B 1,001 Elected D 982 C 943 E 568 F 444

_Fifth Transfer_

G's surplus is still not distributable (Rule 6(7)), but F is now lowest on the poll and his votes have to be distributed (Rule 8).

On examination it is found that of F's 444 papers, 353 show a next preference for C, and the remainder, 91, contain no further preference.

The 353 are transferred to C, who thus has more than the quota, and is declared elected, and the 91 are set aside as finally dealt with (Rule 7(1)).

The poll now stands--

A 1,002 Elected G 1,016 Elected B 1,001 Elected C 1,296 Elected D 982 E 568

This terminates the election; for, even if all C's surplus votes (295) and all G's surplus votes (15) were transferred to E, his poll would only amount to 878. But D's votes (982) exceed this total, D is therefore declared elected (Rule 9 (2)).

The final result is that A, G, B, C, and D are elected.

Public Notice of the Result of the Poll and of the Transfer of Votes

Number of valid votes ... 6,000 Number of members to be elected ... 5 Quota ... 1,001

[column names-- ] N: Names of Candidates V: Votes TA: Transfer of A's surplus RA: Result TK: Transfer of K's Votes RK: Result THI: Transfer of H and I's Votes RHI: Result TB: Transfer of B's surplus TB: Result TF: Transfer of F's Votes RF: Final Result

N: V: TA: RA: TK: RK: THI: RHI: TB: TB: TF: RF:

A 2,009 -1,007 1,002 -- 1,002 -- 1,002 -- 1,002 -- 1,002(E) B 952 -- 952 -- 952 +119 1,071 -70 1,001 -- 1,001(E) C 939 -- 939 + 4 943 -- 943 -- 943 +353 1,296(E) D 746 +129 875 -- 875 +107 982 -- 982 -- 982(E) E 493 + 5 498 -- 498 -- 498 +70 568 -- 568 F 341 + 14 355 +89 444 -- 444 -- 444 -444 -- G 157 +859 1,016 -- 1,016 -- 1,016 -- 1,016 -- 1,016(E) H 152 -- 152 -- 152 -152 -- -- -- -- -- I 118 -- 118 -- 118 -118 -- -- -- -- -- K 93 -- 93 -93 -- -- -- -- -- -- -- ___ ___ ___ ___ ___ ___ Effective votes 6,000 -- 6,000 -- 6,000 -- 5,956 -- 5,956 -- 5,865 Preferences exhausted -- -- -- +44 44 -- 44 +91 135 Total valid votes 6,000 -- 6,000 6,000 -- 6,000 -- 6,000 -- 6,000

[Candidates A, B, C, D, and G are elected.]

[Footnote 1: The rules contained in this schedule were examined and approved by the Select Committee of the House of Lords in 1907. They are substantially identical with those embodied in the Transvaal Municipal Act of 1909, and used in the municipal elections of Pretoria and Johannesburg in 1909, as well as in the model elections conducted by the Proportional Representation Society in 1906, 1908, and 1910.]