How to conduct an election by the Single Transferable Vote 3rd Edition

First Edition, December 1972

In this publication, we have sought to provide a comprehensive handbook for anyone conducting an election. At the same time, we have given an explanation of the purpose, operation and effect of the Single Transferable Vote, which we hope will be of value to those who may have to advise or decide on improvements in election procedures in any organisation, large or small.

We hope also that this account of how the Single Transferable Vote operates (with minor variations) in public and other elections in the British Isles and elsewhere will be of interest both to the political commentator and to the politically aware citizen.

We are grateful to Robin Clarke, Peter Crisell, Peter Dean, James Knight, Enid Lakeman and Donald Macdonald who were kind enough to read the draft and make helpful suggestions, but we are responsible for the final text.

ROBERT A NEWLAND
FRANK S BRITTON

Third Edition, June 1997

In the 20 years since the publication of the second edition the Society has sadly lost through death both the principal authors of this booklet. Widespread availability of calculators and computers has also greatly eased the work of Returning Officers. This edition is the fruit of work started by the Society’s Technical Committee, and finalised by Colin Rosenstiel and James Woodward-Nutt.

The opportunity has been taken to simplify the structure of this publication, to clarify its wording and to remove some duplication. The principal changes are:

1. Withdrawal of the optional provision to place votes in suspense, which was introduced in the second edition. This appears to have been little used. We believe that the increased use of computers makes it unnecessary
2. A further reduction in the quota when it is safe to do so
3. All quotas calculated to two decimal places.

COLIN ROSENSTIEL
JAMES WOODWARD-NUTT

Note on the text, March 2018

The rules and methodology contained herein are designed to facilitate the running of an STV election with hand counting.

In the modern age, STV elections can be conducted entirely online or with scanning and counting machines.

Many organisation’s constitutions refer to the 3rd edition 1997 STV rules, so the text is preserved here.

Objectives of an electoral system

The objectives of a valid method of election may be defined as: –

1. to ascertain the electors’ wishes and, as far as possible, to give effect to them;
2. to ensure that as many as possible of those who take part have an effect, and an equal effect, on the result;
3. to ensure that nearly every elector can identify among those elected representatives of their choice whom they helped to elect;
4. to obtain, as far as practicable, proportional representation (PR) of whatever views, opinions and judgements motivate electors when they vote.

Various methods of election satisfy some of these objectives. For example, most methods of election seek to achieve some form of proportional representation, by (generally) giving most seats to the largest party or opinion group, and fewer seats to the next largest party or group.

But different methods of election attain these objectives with varying degrees of success and reliability, and some methods such as the X-vote are notably inefficient and uncertain. (For a discussion of why X-voting fails, see Robert A Newland Only Half a Democracy, Electoral Reform Society, (2nd edition) 1975.)

The Single Transferable Vote (STV) is a logical method of election designed to attain these objectives with economy, efficiency and certainty.

The essential features of the Single Transferable Vote (STV) are:-

Specification of the Single Transferable Vote

1. a single vote whereby each elector can choose their prospective representative from a number of candidates;
2. several representatives elected together to enable different viewpoints and opinions to be reflected;
3. election by quota, being the minimum number of votes which, if attained by as many candidates as there are places to be filled, leaves at most a quota of votes unused. This is the Droop Quota, being the total valid vote, divided by one more than the number of places to be filled. Thus, if seven representatives are elected together, and if each of seven candidates obtains a Quota of one-eighth of the votes, then at most one-eighth of the votes are unused;
4. the single vote to be transferable according to preferences expressed by the elector to enable the surpluses of candidates who exceed the quota, and the votes of candidates with no possibility of election to be transferred to other, continuing, candidates until the required number of representatives is elected.

Operation of the Single Transferable Vote

The voting papers, each showing an elector’s order of preference for one or more of the candidates, and each representing a single vote, are sorted according to first preferences and counted.

The quota for election is determined. All the transferable papers of any candidate with a surplus above the quota are transferred to other, continuing, candidates in accordance with next available preferences expressed by the electors, the transfer value being determined by sharing the surplus equally between the transferable papers.

Candidates with fewest votes are then excluded in turn and their voting papers are transferred to continuing candidates in accordance with the next available preferences on those papers.

Transfers of surpluses and exclusions continue until the desired number of candidates is elected.

Effect of the Single Transferable Vote

The sorting of voting papers according to first preferences in effect arranges the electors into generally unequal groups, each group supporting a single candidate. The transfers of surpluses and exclusions reduce the groups in number according to the number of places to be filled, and make the initially unequal groups each approximately equal to a quota.

The electorate is thus arranged into the desired number of nearly equal opinion groups, each group with its own representative.

Nearly every vote is effective in helping to secure the election of a chosen candidate.

Nearly every elector has an equal effect on the result and is directly represented by someone whom that voter has helped to elect.

In voting, different electors may attach different weights to several criteria simultaneously. The Single Transferable Vote gives proportional representation of this opinion structure of the electorate with an accuracy dependent only on the number of representatives simultaneously elected.

The Single Transferable Vote gives freedom of choice to electors and ensures, as far as possible, that the choice is satisfied and not distorted or frustrated.

Size of constituency

The proportion of electors represented and the accuracy of the proportional representation of opinion obtained increases with the number of representatives elected together.

Constituency size and electors assured of a representative of their choice

For the councils or executive committees of many organisations, representatives to fill all vacancies can with advantage be elected together.

For the larger electorate and membership of a parliament, in order to avoid problems of communication, while retaining sufficiently large constituencies, it would be appropriate to divide a country into multi-member constituencies returning some three to five members in rural areas, and five to seven members in the conurbations.

For district councils, rural wards would return some five to seven members, and urban wards seven to nine members.

Voting paper

The voting paper must enable electors to exercise their single votes for their preferred candidates by expressing their first preferences. It must also permit them to indicate, if they desire, their subsequent orders of preference for any of the other candidates. The number of preferences which may be expressed bears no relation to the number of places to be filled. A voting paper is valid providing that a first preference is clearly expressed. Later preferences are contingency choices only, which may or may not be expressed, and, if expressed, may or may not be considered.

Two forms of voting paper are in use, the elector in one case numbering and in the other case listing the candidates in order of preference (see overleaf).
The first form of voting paper (A) (number in order of preference) is widely used in public elections. The names of the candidates are shown, generally alphabetically, and the elector numbers the candidates 1,2,3, . . . until indifferent to their order.

Voting Paper A

The second form of voting paper (B) (list in order of preference) is used by many organisations. The electors are provided with a separate schedule of candidates. The electors enter the names of their first preferences on the voting papers, followed by the names of other candidates in order of preference until indifferent to their order. The voting papers contain at most one fewer spaces than the number of candidates, since under no circumstances can a voting paper be transferred to the candidate whom the elector would place last of all the candidates.

Voting Paper B

This form of voting paper discourages any tendency to assign preferences alphabetically and facilitates the sorting and transfer of papers in the counting process.

In some cases, it may be thought desirable to assign to candidates codes which are used instead of their names.

Nominations

The Single Transferable Vote requires no special procedure with regard to nominations. In small elections using the second form of voting paper, the write-in of candidates without formal nomination may be permitted if desired, although it would be prudent to confirm the consent of such candidates before the commencement of the count.

Withdrawals

The withdrawal of candidates after nomination day up to the time of the commencement of the count does not necessitate a postponement of the election, since preferences for such candidates are merely passed over during the count without disadvantage to any elector.

Declaration of the result

The purpose of an election by the Single Transferable Vote is to elect a number of representatives of equal status to represent the electorate.

The declaration of the result comprises the total vote, the total valid vote, the quota for election, and the names of those elected. If an order is desired, this is provided by the order of election. In addition, the first preference vote for each candidate and the names of the candidates excluded may be given.

The votes credited to elected candidates at the end of the count have, of course, no special significance since they have been made equal, or nearly equal, by transfer. Further information is given, if desired, by the publication of the election result sheet giving details of all stages of the count.

Elections to fill individual posts

The Single Transferable Vote can be used to fill a single office, such as that of Prime Minister, Mayor, Chairman, Secretary or Treasurer of any organisation, and is then commonly known as the Alternative Vote.

The Quota for election is then one-half of the votes. The Alternative Vote avoids election on a minority of votes, but proportional representation is, of course, unattainable in filling a single vacancy.

When direct elections for officers and executive committee or council are held simultaneously, the counts are completed consecutively according to seniority of office. As in the case of a withdrawal, this enables preferences for any candidate already elected to a higher office to be passed over during a later count, without disadvantage to any elector.

Organising the hall

The arrangement of the counting hall should reflect the operations which have to be carried out at each stage of the count after the total vote has been determined at the first stage.

These operations are:-

1. Sorting the voting papers
2. Checking the sorting and counting the papers
3. Checking the counting
4. The assembly of the papers for each candidate, and the compilation of the vote record forms
5. the compilation of the result sheet, and the forms for the transfer of surpluses and exclusion of candidates

Various arrangements are possible, depending on the physical constraints, and on the size of the election. For a small election, one set of sorting trays would be adequate, but for a large election several sets should be provided.

The Count

4.1 The count is divided into a number of stages. At the first stage the voting papers are counted to determine the total vote. They are then sorted according to their first preferences, and any papers which are invalid are removed. The total number of valid votes is then found and the quota calculated. Any candidates who have at least a quota of first preference votes are deemed elected at this stage.

4.2 Each subsequent stage of the count is concerned either with the transfer of surplus votes of a candidate whose vote exceeds the quota, or with the exclusion of one or more candidates with the fewest votes.

4.3 This procedure continues until either sufficient candidates have reached the quota to fill all the seats, or there is the same number of candidates left as unfilled seats.

4.4 These rules refer to the various forms published by the Electoral Reform Society. The use of these forms is optional, but where they are used, the various options should be made easier, particularly for those not experienced in conducting STV counts.

4.5 In the rules below, words in bold type indicate that there is a definition in the glossary (section 6).

First stage

In a public election, it is necessary to include certain formalities, such as unsealing and opening the ballot boxes at the start, checking the number of papers in each and ascertaining that the candidates and their agents are content at the conclusion of each stage. For simplicity, these have been omitted from these instructions.

5.1 First stage

5.1.1 Count all the voting papers to determine the total number of votes cast.

5.1.2 Sort the voting papers into first preferences, setting aside any invalid papers. Count the number of invalid papers, and subtract this from the total vote to get the total valid vote.

5.1.3 Check the sorting, and count the papers for each candidate into bundles, inserting a counting slip (green) in each bundle marked with the name of the candidate, the number of papers, and ‘first stage’. For very small elections, the use of counting slips may be dispensed with.

5.1.4 Check the counting. Enter on each candidate’s vote record form (yellow) the total number of first preference votes.

5.1.5 Copy the candidates’ votes from the vote record forms onto a result sheet (white), and check that their total is the same as the total valid vote.

5.1.6 Calculate the quota by dividing the total valid vote by one more than the number of places to be filled. Take the division to two decimal places. If the result is exact that is the quota. Otherwise ignore the remainder, and add 0.01.

5.1.7 Considering each candidate in turn in descending order of their votes, deem elected any candidate whose vote equals or exceeds
(a) the quota, or
(b) (on very rare occasions, where this is less than the quota), the total active vote, divided by one more than the number of places not yet filled,
up to the number of places to be filled, subject to paragraph 5.6.2.

5.1.8 That completes the first stage of the count. Now proceed to section 5.2 below.

Subsequent stages

5.2.1 Each subsequent stage will involve either the distribution of a surplus, or, if there is no surplus to distribute, the exclusion of one or more candidates.

5.2.2 If one or more candidates have surpluses, the largest of these should now be transferred. However the transfer of a surplus or surpluses is deferred and reconsidered at the next stage, if the total of such surpluses does not exceed either:
(a) The difference between the votes of the two candidates who have the fewest votes, or
(b) The difference between the total of the votes of two or more candidates with the fewest votes who could be excluded under rule 5.2.5, and the vote of the candidate next above.

5.2.3 If one or more candidates have surpluses which have not been deferred, transfer the largest surplus. If the surpluses of two or more candidates are equal, and they have the largest surplus, transfer the surplus of the candidate who had the greatest vote at the first stage or at the earliest point in the count, after the transfer of a batch of papers, where they had unequal votes. If the votes of such candidates have been equal at all such points, the Returning Officer shall decide which surplus to transfer by lot.

5.2.4 The transfer of a surplus constitutes a stage in the count. Details of how to do this are in section 5.3. If, after completing the transfer, there are still any untransferred surpluses, and not all the places have been filled, proceed as in paragraph 5.2.2

5.2.5 If, after all surpluses have been transferred or deferred, one or more places remain to be filled, the candidate or candidates with the fewest votes must be excluded. Exclude as many candidates together as possible, provided that:
(a) Sufficient candidates remain to fill all the remaining places
(b) The total votes of these candidates, together with the total of any deferred surpluses, does not exceed the vote of the candidate next above.

If the votes of two or more candidates are equal, and those candidates have the fewest votes, exclude the candidate who had the fewest votes at the first stage or at the earliest point in the count, after the transfer of a batch of papers, where they had unequal votes. If the votes of such candidates have been equal at all such points the Returning Officer shall decide which candidate to exclude by lot.

5.2.6 Details of how to exclude a candidate are given in section 5.4.

5.2.7 Exclusion of one or more candidates constitutes a stage in the count. If, after completing this, there are any surpluses to transfer, and not all the places have been filled, proceed as in paragraph 5.2.2. Otherwise proceed to exclude further candidates as in paragraph 5.2.5.

Transfer of a surplus

5.3.1 If a surplus arises at the first stage, select for examination all the papers which the candidate has received.

5.3.2 If a surplus arises at a later stage, because of the transfer of another surplus or the exclusion of a candidate or candidates, select only the last received batch of papers, which gave rise to the surplus.

5.3.3 Examine the selected voting papers and sort them into their next available preferences for continuing candidates. Set aside as non-transferable papers any on which no next available preference is expressed.

5.3.4 Check the sorting, count and bundle the papers now being transferred to each candidate, also any non-transferable papers. Insert a counting slip in each bundle marked with the stage number, the name of the candidate to whom the papers are being transferred, and the number of papers in the bundle.

5.3.5 Count the number of transferable papers and enter the number for each candidate on the vote record forms.

5.3.6 Prepare a surplus form (pink). Copy the number of papers for each candidate from the vote record forms to the surplus form, and check the total.

5.3.7 Calculate the total value of the transferable papers. If this exceeds the surplus, determine the transfer value of each paper by dividing the surplus by the number of transferable papers, to two decimal places, ignoring any remainder. If the total value does not exceed the surplus, the transfer value of each paper is its present value.

5.3.8 Calculate the value to be credited to each candidate by multiplying the transfer value by the number of papers, check the totals, and enter these on the surplus form.

5.3.9 Copy the values to be credited, and the non-transferable difference arising from the neglected remainder, from the surplus form to the vote record forms and to the result sheet.

5.3.10 Add these values to the previous votes for each candidate, and add the non-transferable difference to the previous total of non-transferable votes, entering the figures onto the vote record forms and the result sheet.

5.3.11 Add up the new total number of votes on the result sheet, and check that this still equals the original total valid vote.

5.3.12 Complete the counting slips with the transfer value of each paper, and place the bundles of voting papers for each candidate with those previously received. In a small election, where counting slips are not being used, each ballot paper should be marked with its transfer value.

5.3.13 Considering each continuing candidate in turn in descending order of their votes, deem elected any candidate whose vote now equals or exceeds
(a) the quota, or
(b) the total active vote, divided by one more than the number of places not yet filled,
up to the number of places remaining to be filled, subject to paragraph 5.6.2.

Transfer of the votes of excluded candidates

Take together all the bundles of papers which are currently credited to the candidate or candidates to be excluded, and arrange them in batches in descending order of transfer value. Check that the number and total value of the papers in each batch agrees with the numbers on the vote record forms and the result sheet. Prepare an exclusion form (blue).

5.4.2 First, take the batch of papers with the highest transfer value. Sort them according to the next available preferences for continuing candidates, and set aside as non-transferable papers any on which no next available preference is expressed.

5.4.3 Check the sorting, count and bundle the papers for each candidate and any non-transferable papers. Insert a counting slip in each bundle stating the stage, the name of the candidate to whom the papers are being transferred, the number of papers, and the transfer value of each paper. If counting slips are not being used, the transfer value should be marked on each paper.

5.4.4 Check the counting and enter the number of papers for each candidate and the number of non-transferable papers on the vote record forms.

5.4.5 Copy the number of papers to be transferred to each candidate and the number of non-transferable papers, from the vote record forms onto a column of the exclusion form, and check the total.

5.4.6 Determine the total value of the papers for each candidate and that of the non-transferable papers and check the total.

5.4.7 Copy the total values for each candidate from the exclusion form to the vote record forms, and place the bundles of voting papers for each candidate with those previously received.

5.4.8 If any papers have become non-transferable before any candidate has been deemed elected, recalculate the quota as in paragraph 5.1.6, ignoring the non-transferable vote.

5.4.9 Considering each continuing candidate in turn in descending order of their votes, deem elected any candidate whose vote now equals or exceeds
(a) the quota, or
(b) the total active vote, divided by one more than the number of places not yet filled,
up to the number of places remaining to be filled, subject to paragraph 5.6.2.

5.4.10 Ensure that no further papers are given to candidates who are no longer continuing candidates because they have been deemed to be elected after transferring a batch of papers.

5.4.11 As in paragraph 5.4.2 and subsequently, sort and transfer each batch of papers in turn in descending order of transfer value, complete a column of the exclusion form for each batch, and deem candidates to be elected as appropriate.

5.4.12 After all the batches of papers have been transferred, the right hand (totals) column on the exclusion form should be completed and these totals checked against the vote record form(s) of the excluded candidate(s).

5.4.13 Copy the total values to be credited from the exclusion form to the vote record forms and to the result sheet, and add these to the previous totals for each candidate.

5.4.14 Copy the new vote for each candidate from the vote record forms onto the result sheet, and the new non-transferable vote from the exclusion forms onto the result sheet.

5.4.15 Add up the new total vote on the result sheet and check that this agrees with the original total valid vote.

Completion of the count

5.5.1 If a proposed exclusion of one or more candidates would leave only the same number of continuing candidates as there are places remaining unfilled, all such continuing candidates shall be deemed to be elected.

5.5.2 If, at any point in the count, the number of candidates deemed to be elected is equal to the number of places to be filled, no further transfers of papers are made, and the remaining continuing candidate(s) are formally excluded.

5.5.3 The count is now completed.

5.5.4 Declare elected all those candidates previously deemed to be elected.

Notes

5.6.1 Calculation of the total active vote may be simplified if the Count Control Form (beige) is used. This form enables the Returning Officer to keep a continuous check on the number of votes which are required for election of a candidate at any point in the count, by deducting the quotas (or actual votes if less) of the candidates deemed elected, and the total of non-transferable votes, from the total valid vote, to give the total active vote.

5.6.2 If, when candidates should be deemed elected under sections 5.1.7, 5.3.13 or 5.4.9, two or more have the same number of votes, and there are not sufficient places left for them all, then the one or more to be deemed elected shall be selected in descending order of votes at the first stage or at the earliest point in the count, after the transfer of a batch of papers, where they had unequal votes. If, however, their votes have been equal at all such points, then none of them shall be deemed elected at that stage.

5.6.3 If a re-count is conducted where a decision has been determined by lot, and the relevant votes are still equal in the recount, the earlier determination shall still hold.

5.6.4 These rules refer to the various forms published by the Electoral Reform Society. The use of these forms is optional, but where they are used, the various options should be made easier.

in alphabetical order

6.1 Batch: a bundle containing all the papers of one value in a transfer.

6.2 Candidate’s vote: the value of voting papers credited to a candidate at any point in the count.

6.3 Continuing candidate: a candidate not yet deemed elected or excluded.

6.4 Count Control form (beige): a form designed to be used to keep a continuous note of the total active vote, and hence the vote required for election of a candidate at any point in the count.

6.5 Counting slip (green): a slip inserted with a bundle of voting papers, showing the stage at which the papers are transferred, the name of the candidate to whom they are transferred, the number of papers in the bundle, and the transfer value of each paper.

6.6 Deemed elected: status of a candidate who is elected subject to formal confirmation.

6.7 Exclusion form (blue): a form showing the distribution of batches of papers in descending order of transfer value from one or more excluded candidates to continuing candidates.

6.8 First preference: this is shown by the figure “1” standing alone against only one candidate on a voting paper; or the name or code of a candidate entered on a voting paper as first preference.

6.9 Invalid paper: a voting paper on which no first or only preference is expressed, or on which any first preference is void for uncertainty.

6.10 Next available preference: the next subsequent preference in order, passing over earlier preferences for candidates already deemed elected or excluded. There is no next available preference where the next sequential preference for a continuing candidate is uncertain.

6.11 Non-transferable difference: the difference between the value of a surplus and the total new value of the papers transferred, which arises from ignoring the remainder when calculating the transfer values to two decimal places.

6.12 Non-transferable paper: a voting paper on which no next available preference for a continuing candidate is expressed, or on which any next available preference is void for uncertainty.

6.13 Non-transferable vote: the value credited as non-transferable at any point in the count.

6.14 Quota: the vote which, if attained by as many candidates as there are places to be filled, leaves at most a quota for all other candidates; the total valid vote divided by one more than the number of places to be filled, or a lesser value calculated as in paragraph 5.4.8.

6.15 Result sheet (white): a sheet showing the vote credited to each and every candidate, and the non-transferable vote at successive stages of the count.

6.16 Stage of the count: the determination of the first preference vote for each candidate (first stage)
or the transfer of a surplus
or the exclusion of a candidate, or two or more candidates at the same time, and the transfer of their votes.

6.17 Subsequent preferences: shown by the figures “2”, “3”, etc., standing alone against different candidates on a voting paper; or the names or codes of candidates entered in order on a voting paper as second, third, etc., preferences.

6.18 Surplus: the amount by which a candidate’s vote exceeds the quota.

6.19 Surplus form (pink): A form showing the calculation of the transfer value and the distribution of transferable papers from a candidate deemed elected to continuing candidates.

6.20 Total active vote: the sum of the votes credited to all continuing candidates, plus any votes awaiting transfer.

6.21 Total valid vote: the total number of valid voting papers.

6.22 Transfer value: the value, being unity or less, at which a voting paper is transferred from an elected or an excluded candidate to a continuing candidate. Where counting slips are not used, it is recommended that this value be marked on each paper at the time of transfer.

6.23 Transferable paper: a voting paper which, having been allocated to a candidate, bears a next available preference for a continuing candidate.

6.24 Valid voting paper: a voting paper on which a first or an only preference is unambiguously expressed.

6.25 Value: the value of a voting paper is unity, or a lower value at which it was last transferred.

6.26 Vote record form (yellow): a form showing the vote credited to any one candidate, or showing the non-transferable vote, at successive stages of the count.

by-election

No purpose is served by holding a by-election, since a representative so elected would represent the dominant opinion group in the particular multi-member constituency, and not necessarily the opinion group of the vacating member.

There are three possibilities:

The vacancy may be filled by recounting all the original voting papers for the constituency, passing over all preferences for the vacating representative, and for any other candidate who now withdraws. With the provision that no other previously elected representative should be excluded, the count proceeds until a stage when a new representative has been elected. This representative, together with the surviving representatives, will reflect the original wishes of the electorate. This method requires that the original voting papers should be retained under secure conditions.

The vacancy may be filled by co-option. A person could be co-opted who reflects, as far as possible, the opinion group of the vacating representative. The party, if any, of the vacating representative might be invited to nominate a candidate for the elected body to co-opt. Alternatively, the last formally excluded candidate could be co-opted.

The vacancy may be left unfilled. When a large number of representatives has been elected together by the Single Transferable Vote, it may be thought that the surviving representatives can adequately represent the electorate until the next election.

First Stage

The model election is devised to illustrate the main features of an STV count. There are eleven candidates, and six places to be filled.

First Stage

The total vote is 758. The voting papers are sorted according to first preferences for the eleven candidates. Three voting papers are marked with several crosses, and two have the figure ‘1’ against more than one candidate. These five invalid papers are set aside. The vote for each candidate is entered onto the vote record forms and on the election result sheet. The total valid vote is 753. The number of places to be filled is six. The Quota is determined by dividing the total valid vote by one more than the number of places to be filled, seven, continuing the calculation to two decimal places, and rounding up.

753/7 = 107.57142 Quota = 107.58

One candidate, Smith, with a vote of 134, exceeds the quota, and is deemed elected. The first line of the upper part of the Count Control form is completed.

Second Stage

Smith has a surplus of 134 – 107.58 = 26.42 This exceeds the difference between the votes of the last two candidates, (24 – 23 = 1). Since the transfer of the surplus could thus change the order of the last two candidates, and thereby determine the candidate next to be excluded, the surplus must be transferred.

Smith has 134 papers for review, of which 123 prove to be transferable. Since the total present value of these papers, 123, exceeds the surplus of 26.42, this surplus is shared between the transferable papers, to give, for each paper, a transfer value to two decimal places of:

26.42/123 = 0.21

Each of the candidates is credited on their vote record form with the value of the papers received, at the transfer value of 0.21, and these credits, together with the non-transferable difference arising from the neglected remainder

26.42 – (123 x 0.21) = 26.42 – 25.83 = 0.59

are recorded on the result sheet, which again shows the total valid vote at 753.

Third stage

Since there is now no outstanding surplus, one or more candidates must next be excluded. The total of the votes of the last two candidates exceeds the vote of the candidate next above. Therefore the last candidate only, Monk, is excluded.

Monk’s papers are arranged in descending order of transfer value,

23 @ 1 = 23
2 @ 0.21 = 0.42

23+.42 = 23.42

The 23 whole value papers are transferred first, passing over any preferences for Smith, who is already elected, and are entered on the vote record forms. A second candidate, Duke, now exceeds the Quota, and is deemed elected. The second line of the upper part of the Count Control form is completed.

The two papers at 0.42 are then transferred, passing over any preferences for Duke as well as Smith.

Each candidate is credited on their vote record form with the value of the papers received at the different transfer values, and these credits, together with the value of the non-transferable papers are recorded on the election result sheet, again giving the total of 753.

Fourth stage

Duke now has a surplus of 108.68 – 107.58 = 1.10. This is less than the difference between the votes of the two candidates now last,

32.25 – 30.51 = 1.74

and is also less than the difference between the total vote of the last two candidates and the candidate next above,

64.21 – (32.25 + 30.51) = 64.21 – 62.76 = 1.45

Thus, because the transfer of the surplus could not change the order of the last two candidates or of the last three candidates, on either ground the transfer of the surplus is deferred.

The last two candidates are now excluded together, because their total vote, together with the untransferred surplus, is

(32.25 + 30.51) + 1.10= 62.76 + 1.10 = 63.86

which is less than the vote of the candidate next above, at 64.21 while the total vote of the last three candidates, together with the untransferred surplus, is

(32.25 + 30.51 + 64.21) + 1.10 = 128.07

which is much greater than the vote of the candidate next above, at 64.84.

Their papers are arranged in descending order of transfer value,

51 @ 1 = 51
56 @ 0.21 = 11.76

51+11.76= 62.76

The 51 whole-value papers are transferred first, passing over any preferences for any candidate already elected or excluded, and are entered on the vote record forms. A third candidate, Carpenter, now exceeds the quota, and is deemed elected. The third line of the upper part of the Count Control form is completed.

The 56 papers at 0.21 are then transferred, passing over any preferences for Carpenter as well as for the other elected and excluded candidates.

The vote record forms and the election result sheet are completed as before.

There are 6.79 non-transferable votes and three candidates have been deemed elected. The reduced vote required for election is calculated on the first line of the lower part of the Count Control form, confirming that no-one else can be deemed elected yet.

Fifth stage

Carpenter and Duke now both have surpluses whose total value is

14.77 + 1.10 = 15.87

This exceeds the difference between the votes of the two candidates now last

68.26 – 67.10 = 1.16

Since the transfer of these surpluses could change the order of the last two candidates, the larger surplus is transferred.

Carpenter’s surplus has arisen through the exclusion of Glazier and Wright. The transfer of Carpenter’s surplus is thus in effect the completion of the exclusion of these two candidates. Hence the papers for review are the 34 papers last received, of which 8 prove to be transferable. Since the total present value of these papers, 8 does not exceed the surplus of 14.77, these papers are transferred at a transfer value equal to the value at which they were received.

Each candidate is credited on his vote record form with the value of the papers received, and these credits together with the non-transferable difference:

14.77 – 8 = 6.77

are entered as before on the result sheet.

A further line of the lower part of the Count Control form is completed. The reduced vote required for election is now 104.18. This is exceeded by Prince’s vote of 104.31. Prince is therefore deemed elected and a fourth line of the upper part of the Count Control form is now completed. A third line of the lower part of the Count Control form calculates the new reduced vote required for election, confirming that no-one else can be deemed elected yet.

Sixth stage

Duke still has a surplus of 1.10. This is less than the difference between the votes of the last two candidates,

70.26 – 68.10 = 2.16

Because the transfer of the surplus could not change the order of the last two candidates, the transfer of the surplus is again deferred. The candidate now last, Abbot, is excluded.

Abbot‘s papers are arranged in descending order of transfer value,

66 @ 1.00 = 66
10 @ 0.21 = 2.10
66+2.10 = 68.10

The 66 whole value papers are transferred first. Vicar now exceeds the quota and is the fifth candidate to be deemed elected. A fifth line of the upper part of the Count Control form is completed, followed by a fourth line in the lower part of that form. This calculates that the vote now required for election is 96.41.

This is exceeded by Freeman’s vote of 101.62. Freeman is therefore deemed elected to fill the last place.

No further transfers of papers are made. The ten papers value 0.21 each remain with Abbot, the vote record forms and the election result sheet are completed, and the remaining candidate, Baron, is formally excluded.

Smith, Duke, Carpenter, Prince, Vicar and Freeman are declared elected, and the count is concluded.

Vote Record Form

Count Control Form

* Quota or actual vote, whichever is less

Calculation of Vote Required for Election

Observations on model election

Proportional representation

The transfers at successive stages of the count reveal three groups or ‘parties’. The first preferences for candidates of these parties is shown at the first stage:

At the first stage, of the six leading candidates, two are artisans, three nobles and one independent. At the end of the count, however, consequent upon the successive transfers of votes, of the six elected candidates, two are artisans, two nobles, one cleric and one independent, which is proportional to the support for the ‘parties’. Thus proportional representation of opinion, as represented by the ‘parties’ has been achieved.

The term ‘parties’ is used here in the very general sense to indicate any criterion which significantly motivates electors when they vote. In some elections, parties may not be discernible.

Questions

The following exercises are designed to illustrate the decisions and calculations which have to be made during the course of a count in an election by the Single Transferable Vote.

9.1 Determine whether voting papers marked as below are valid or invalid, and if valid, up to what preference are they transferable:

(a) 1
(b) 3,4,1,2
(c) 2,3,4,1,3
(d) 2,1,3,1
(e) 3,5,2,1
(f) 1,4,3&5,2
(g) X,X

9.2 Calculate the quota for election in each of the following cases:

9.3 Calculate the surplus, the transfer value and the non-transferable difference in each of these cases arising at the first stage:

9.4 Calculate the surplus, the transfer value and the non-transferable difference in each of these cases arising at a later stage:

9.5 Candidates to be excluded received papers in the order shown. Re-arrange the papers in batches in transfer order:

(a) 137 @ 1, 71 @ 0.30, 36 @ 0.12, 8 @ 1, 4 @ 0.30
(b) 17 @ 1, 94 @ 0.18, 43 @ 0.07, 33 @ 0.18, 12 @ 0.07

9.6 The vote for each candidate at the first stage is as shown. There are five places to fill. Calculate the quota. Decide the candidates deemed to be elected, and the Returning Officer’s action in each case.

9.7 The vote for each candidate at a later stage is shown. There are five places to fill. Candidates A & B are already deemed to be elected. Decide which further candidate(s) are deemed to be elected, and the Returning Officer’s action in each case:

Answers

9.1

(a) valid, 1st
(b) valid, 4th
(c) valid, 2nd
(d) invalid
(e) valid, 3rd
(f) valid, 2nd
(g) invalid

9.2

(a) 762.00
(b) 87.25
(c) 4770.84
(d) 34.81

9.3

(a) 38.09, 0.30, 1.19
(b) 34, 0.04, 0
(c) 486, 0.29, 15.91
(d) 12, 0, 12

9.4

(a) 90.24, 1.00, 6.24
(b) 3.27, 0.40, 0.07
(c) 18.54, 0.21, 1.74
(d) 135.81, 0.91, 1.13

9.5

(a)

145 @ 1

75 @ 0.30

36 @ 0.12

(b)

17 @ 1

127 @ 0.18

55 @ 0.07

9.6

Quota 949.50

(a) A elected, transfer A’s surplus

(c) A elected; exclude H & J together(b) A elected; exclude J

(d) A & B elected; transfer A’s surplus

9.7

(a) C elected; transfer B’s surplus

(c) C elected; D elected; exclude G

(b) C & D elected, exclude G

(d) C elected; D elected; E elected; Formally exclude G & F together