## Schedule 1A New Zealand method of counting single transferable votes

Schedule 1A: inserted, on 1 January 2004, by regulation 42 of the Local Electoral Amendment Regulations 2003 (SR 2003/391).

#### Purpose of schedule

##### 1

[Revoked]Schedule 1A clause 1: revoked, on 7 July 2004, by section 19(7) of the Local Electoral Amendment Act 2004 (2004 No 62).

#### Interpretation

##### 2

In this schedule, unless the context otherwise requires,—

AAFD Method means the Ahead At First Difference Method set out in Part 5

exhausted, in relation to a voting document, means the point at which no further votes from that voting document can be redistributed by the operations in clause 10

non-transferable votes means the votes remaining untransferred when a voting document becomes exhausted

preference means a preference expressed for a candidate on a voting document as a rank, for example, first, second, or third

PRN means the pseudo-random number generated by the PRNG method

PRNG method means the Pseudo Random Number Generation Method set out in Part 5.

## Part 1 Multi-member vacancies

*Counting*

##### 3

In the case of multi-member vacancies, votes are counted under the New Zealand method of counting single transferable votes by recognising preferences as provided for in Part 3 and carrying out the steps in clauses 5 to 18 until counting is complete. Counting is complete when the number of successful candidates is equal to the number of vacancies or any of clauses 19 to 23 (which relate to exceptional cases) provide that counting is complete. Counting is also complete when the number of successful candidates and hopeful candidates is equal to the number of vacancies. In this case, the hopeful candidates become successful candidates. When counting is complete, successful candidates are elected.

*Interpretation*

##### 4

In this part, unless the context otherwise requires,—

hopeful candidates means all candidates before the commencement of step 1

vote includes non integer values.

*Step 1*

##### 5

Calculate a quota using the following formula:

q = (v − vnt) ÷ (n + 1) + 0.000 000 001

where—

- q
is the quota

- v
is the total number of valid voting documents

- vnt
is the number of non transferable votes

- n
is the number of vacancies

and q is truncated to 9 decimal digits after the point with no rounding.

##### 6

A candidate whose first preferences equal or exceed the quota becomes successful.

*Step 2*

##### 7

Give each successful candidate a keep value calculated using the following formula:

kc = q ÷ vc

where—

- kc
is the keep value for the successful candidate

- q
is the quota for this step

- vc
is the candidate’s votes

and kc is truncated to 9 decimal digits after the point and rounded up if not exact.

##### 8

Give hopeful candidates a keep value of 1.0.

##### 9

Give withdrawn candidates a keep value of 0.

##### 10

Allocate votes from each voting document according to the following scheme:

vm = (1 – k1) × (1 – k2) × (1 – km-1) × km

where—

- vm
is the votes to allocate to the mth preference candidate on the voting document

- k1
is the keep value for the first preference candidate on the voting document

- k2
is the keep value for the second preference candidate on the voting document

- km
is the keep value for the mth preference candidate on the voting document

and the product of each multiplication on the right hand side of the equation must be calculated to 9 decimal digits after the point and rounded up if not exact and the multiplications must be calculated from left to right.

##### 11

Calculate a new quota using the following formula:

q = (v − vnt) ÷ (n + 1) + 0.000 000 001

where—

- q
is the quota

- v
is the total number of valid voting documents

- vnt
is the number of non transferable votes

- n
is the number of vacancies

and q is truncated to 9 decimal digits after the point with no rounding.

##### 12

A hopeful candidate becomes successful if his or her votes equal or exceed the quota.

##### 13

Exclude the hopeful candidate with the least votes if the sum of his or her votes and the total surplus is less than the votes of any other hopeful candidate or if the total surplus is less than 0.0001.

##### 14

In clause 13, total surplus means the sum of all candidates’ surpluses. Surplus means the number of votes that a candidate has in excess of the quota.

*Step 3 and subsequent steps*

##### 15

Repeat the operations set out in the second step with the changes set out in clauses 16 and 17.

##### 16

Set the keep value of excluded candidates to 0.

##### 17

Use the following formula instead of the formula in clause 7:

kc = q × kc ÷ vc

where—

- kc
is the keep value for the successful candidate

- q
is the quota for this step

- vc
is the candidate’s votes

and the product of q and kc is truncated to 9 digits after the point and rounded up if not exact before being divided by vc and kc is also truncated to 9 digits after the point and rounded up if not exact.

*Exceptional cases*

##### 18

If 1 or more of clauses 19 to 23 apply, the operations in steps in clauses 5 to 17 must be modified by carrying out the operations set out in the clause or clauses that apply.

*Ties*

##### 19

This clause applies if a candidate with the lowest number of votes is to be excluded but 2 or more candidates share the lowest number of votes. If this clause applies, exclude the candidate identified by the AAFD method as the candidate to exclude. If the AAFD method does not identify a single candidate to exclude, exclude the candidate with the lowest PRN.

*Zero votes: Case 1*

##### 20

This clause applies to any candidate who, at the commencement of step 1, has no first preferences and no other preferences. A candidate to whom this clause applies, becomes withdrawn.

*Zero votes: Case 2*

##### 21

This clause applies if, at the commencement of step 1, there are fewer candidates with votes than positions to be filled. If this clause applies, the candidates who attracted votes are successful, the counting is complete, and, for the avoidance of doubt, the remaining vacancies are extraordinary vacancies.

*Candidate not able to be elected*

##### 22

A candidate becomes withdrawn if, at the commencement of step 1, any of the following provisions apply to him or her: section 71(2) of the Act (which relates to death, incapacity, or invalid or cancelled nomination), section 88(1) of the Act (which relates to a candidate being elected as both mayor and member), or section 88A of the Act (which relates to a candidate being elected as both member and community board member).

Schedule 1A clause 22: amended, on 29 June 2013, by section 46 of the Local Electoral Amendment Act 2013 (2013 No 40).

*Stable state in counting*

##### 23

This clause applies if the operations in clauses 5 to 22 result in a state where any number of iterations would not result in further candidates becoming excluded or successful. If this clause applies, the candidate with the lowest total votes becomes excluded. Recommence counting at the step following the step at which the stable state occurred.

## Part 2 Single member vacancies

*Interpretation*

##### 24

In this Part, unless the context otherwise requires,—

hopeful candidates means all candidates before the commencement of step 1

votes means whole number values.

##### 25

In the case of single member vacancies, before the commencement of step 1, votes are counted under the New Zealand method of counting single transferable votes by recognising preferences as provided for in Part 3 and by carrying out the steps in clauses 26 to 33 until a candidate is elected.

*Step 1*

##### 26

Determine the absolute majority of votes using the following formula:

m = (v − vnt) ÷ 2

where—

- m
is the absolute majority of votes

- v
is the number of voting documents

- vnt
is the total number of non-transferable votes

and if m is not a whole number, it is rounded up to the next whole number.

##### 27

A candidate is elected if his or her first preference votes equal or exceed the absolute majority of votes.

*Step 2*

##### 28

If no candidate is elected, the candidate with the fewest first preference votes becomes excluded.

##### 29

For each voting document on which a withdrawn candidate is the first preference, redistribute that vote to the candidate who is the next preference on that voting document.

##### 30

For each voting document on which the candidate excluded under clause 28 was the first preference, redistribute that vote to the candidate who is the next preference on that voting document.

##### 31

A candidate is elected if his or her votes equal or exceed the absolute majority of votes.

##### 32

If no candidate is elected, repeat the operations in clauses 26 to 31.

*Exceptional cases*

##### 33

If any of clauses 34 to 37 apply, the steps in clauses 26 to 32 must be modified by carrying out the operations set out in the clause or clauses that apply.

*Ties*

##### 34

This clause applies if—

##### (a)

a candidate with the lowest number of votes is to be excluded but 2 or more candidates share the lowest number of votes; or

##### (b)

a candidate whose votes equal or exceed the absolute majority of votes is to be elected but 2 candidates’ votes equal the absolute majority of votes.

If this clause applies, in the case of (a), exclude the candidate identified by the AAFD method as the candidate to exclude. If the AAFD method does not identify a candidate to exclude, exclude the candidate with the lowest PRN. In case of (b), the candidate who is not excluded is elected.

*Zero votes*

##### 35

This clause applies if, at the commencement of step 1, only 1 candidate has any votes. If this clause applies, that candidate is elected.

*Candidate not able to be elected*

##### 36

Clause 22 applies, with any necessary modifications, to single member vacancies.

## Part 3 Recognising preferences

##### 37

*Example 1: Not unique*

A voting document that expresses these preferences—

Candidate A | 1 | ||

Candidate B | 2 | ||

Candidate C | 3 | ||

Candidate D | 2 | Not unique | |

Candidate E | 4 |

generates the following preferences for the purposes of counting votes:

First preference | Second preference | Third preference | Fourth preference | Fifth preference |

A | None | None | None | None |

*Example 2: Not consecutive*

A voting document that expresses these preferences—

Candidate A | 1 | ||

Candidate B | 4 | Not consecutive | |

Candidate C | 5 | ||

Candidate D | 2 | ||

Candidate E | 6 |

generates the following preferences for the purposes of counting votes:

First preference | Second preference | Third preference | Fourth preference | Fifth preference |

A | D | None | None | None |

Schedule 1A clause 37(b): amended, on 7 July 2004, by section 19(8) of the Local Electoral Amendment Act 2004 (2004 No 62).

##### 38

Do not recognise a preference for a withdrawn candidate. In the case of voting documents that contain a preference for a withdrawn candidate, recognise preferences lower in rank on the same voting document as preferences of the rank stated.

*Example 3: Withdrawn candidate*

A voting document that expresses these preferences—

Candidate A | 1 | ||

Candidate B | 4 | Withdrawn candidate | |

Candidate C | 3 | ||

Candidate D | 2 | Withdrawn candidate | |

Candidate E | 5 |

generates the following preferences for the purposes of counting votes:

First preference | Second preference | Third preference | Fourth preference | Fifth preference |

A | None | C | None | E |

## Part 4 Polls

##### 39

In the case of polls, votes are counted under the single transferable voting system by applying Part 1 or Part 2, with any necessary modifications, as if every reference to a candidate or candidates is a reference to the matter or matters that are the subject of the poll, and references to withdrawn candidates are of no effect.

## Part 5 Methods relevant to tie breaking

*Ahead at first difference method (AAFD method)*

##### 40

To use the Ahead At First Difference Method determine which tied candidate, or candidates, did not have more votes than another tied candidate at the earliest step at which the candidates had different numbers of votes. If one candidate is identified, exclude him or her.

*PRNG method*

##### 41

Allocate a unique pseudo-random whole number (a PRN number) for each candidate at each stage of the counting.

##### 42

To generate PRNs, calculate x, y, and z using the following formulae:

x = c+5

y = n

z = (v + 1 000 (v rem 10)) rem 30 323

where—

- c
is the number of candidates

- n
is the number of vacancies

- v
is the total number of valid voting documents

- rem
is the remainder operator such that a rem b gives the remainder of dividing whole number a by whole number b.

##### 43

Generate a random whole number rc using the following formulae:

x = (171x) rem 30 269

y = (172y) rem 30 307

z = (170z) rem 30 323

rc = (10 000x) div 30 269 + (10 000y) div 30 307 + (10 000z) div 30 323

where—

- rc
is a pseudo-random number

- div
is the integer division operator such that a div b gives the whole number quotient of dividing whole number a by whole number b.

##### 44

Repeat the step in clause 43 four times, discarding the first 4 values of rc.

##### 45

Assign the current value of rc to the first candidate.

##### 46

Repeat the step in clause 43 until a pseudo-random number r results that is distinct from all previous pseudo random numbers assigned to candidates. Assign rc to the next candidate.

##### 47

Repeat the step in clause 43 until all candidates have been assigned a pseudo-random number.

##### 48

For the second and subsequent steps, replace the pseudo-random number for each candidate with the candidate’s PRN at the previous step subtracted from 10 000.