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Meek’s Method

NEW! Computer Journal article that describes Meek's method of counting STV votes:
'Algorithm 123 - Single Transferable Vote by Meek's Method' (166k)*
*This document is in Adobe Acrobat (.pdf) format. You need to have the Adobe Acrobat Reader installed on your computer. You can download a free version from the Adobe site.

The following paragraphs are extracts from Schedule 1 to the Local Electoral Act which explain how Meek's method has been adapted for use ïn New Zealand.

Every Single Transferable Voting system for elections has the following features:

voting by order of preference for the candidates:

a quota for election, calculated from the number of votes and the number of positions to be filled:

a first count of first preferences only, and the election of any candidate who equals or exceeds the quota (except in the special case of a multi-way tie):

redistribution of surplus votes (above the quota) for any candidate in accordance with the voter's further preferences, and election of any candidate who then reaches the quota:

when no further distribution of surpluses is possible, the exclusion of the candidate who then has the fewest votes, and redistribution of those votes:

further counting, election, redistribution of surpluses, and exclusion, as necessary, until all positions for election are filled.

The following points explain in what ways Meek’s method is different to other forms of STV:

Vote transfer
Votes are transferred to the next preference of the voter in the exact order indicated by the voter on the voting document unless the candidate has already been excluded.

Value of surpluses
The total value of a surplus or surpluses is shared in due proportion across both transferable and non-transferable voting documents.

Sharing of votes
If a candidate is elected later in the count, or an elected candidate receives further votes, the surplus to be transferred is shared across all voting documents credited to that candidate in due proportions, not just across the voting documents that gave immediate rise to the surplus.

Recalculation of quota
As votes become non-transferable (eg, because the number of preferences recorded in the voting document is exhausted), the quota is recalculated to reflect the smaller total of votes remaining active. The new quota then applies to already elected candidates as well as others, giving them further surpluses to redistribute.

Need for computer technology
Because the procedure required to conduct a count using Meek's method of counting votes requires a candidate to be assigned a scaling factor (a keep value) representing the proportion of each vote that will actually be credited to each candidate, the number of calculations involved requires the count to be conducted using computer technology rather than by means of a manual count.

General description of process used to conduct count at election under system of Single Transferable Voting using Meek's method of counting votes

Interpretation
In the following clauses, unless the context otherwise requires:

non-transferable vote means a voting document on which no next preference for a non-excluded candidate is indicated or can be identified by the electoral officer

surplus means the number of votes that a candidate obtains at any stage of the counting of votes in excess of the quota

total surplus means the sum of all candidates' surpluses at any stage of the counting of votes.

Election to fill mayoral or single member vacancy

Before commencing the official count, the electoral officer rejects as informal any voting document on which the voter's first preference is not able to be determined.

The number of valid voting documents is determined.

The absolute majority of votes is the number of votes sufficient to secure the election of any candidate. It is determined by dividing the total number of valid votes, excluding non-transferable votes, by 2. If the number is not a whole number, the quotient is rounded up to the next whole number.

The number of first preference votes recorded for each candidate is determined and recorded.

If any candidate obtains an absolute majority of votes, that candidate is treated as being elected and the count ceases.

If no candidate has an absolute majority of votes, the candidate who has the fewest votes is excluded and each voting document for that candidate, unless it is non-transferable, is counted to the candidate next in the order of the elector's preference.

This process is continued until 1 candidate has an absolute majority of votes.

Equal votes
If, on any count, 2 or more candidates have an equal number of votes and 1 of them has to be excluded, the electoral officer determines which candidate had the fewest votes the first time the candidates' totals were different and excludes the candidate with the lowest total votes.

Ties
If the candidates had an equal number of votes at all stages of the count, a random (or pseudo-random) process is used to choose which candidate is excluded.

Multi-member vacancies
Before commencing the official count, the electoral officer rejects as informal any voting document on which the voter's first preference is not able to be determined, or on which a unique first preference vote is recorded for a candidate who is no longer available for election.

The number of valid voting documents is determined.

Formulas

Quota
The quota is calculated (to 9 decimal digits after the point with any remainder being disregarded) in accordance with the following formula:

q =	v + 0.000000001
	----- n + 1

where:

q is the quota
v is the total number of valid votes, less the number of non-transferable
votes
n is the total number of members to be elected.

Keep value
When a candidate's votes exceed the quota, a new keep value is calculated in accordance with the following formula:

k =	ck x q
	------- cv

where:

k is the candidate's new keep value
ck is the candidate's current keep value
q is the current quota
cv is the candidate's current votes.

In calculating the keep value under subclause (1), both the multiplication and the division are taken to 9 decimal digits after the point and, in each case, rounded up if not exact.

General procedure

The number of valid voting documents is counted and the quota determined. The quota is adjusted downwards as voting documents become non-transferable.

At each stage in the count, each candidate has an associated keep value, which indicates the proportion of every vote, or part of a vote, received by that candidate that is kept by him or her.

Each candidate's keep value is initially set at 1.0, and that candidate retains the full value of all votes, or parts of votes, received. If any candidate withdraws, that candidate's keep value is set to 0.0.

If a candidate reaches the current quota for election, a new keep value is calculated, indicating the value of each vote or part of a vote retained by him or her.

A candidate is excluded at any stage of the count if the sum of his or her votes and the total surplus is less than the votes of any other non-excluded candidate.

If a candidate is excluded, his or her keep value is set to 0.0.

Any candidate whose votes equal or exceed the current quota is elected.

Counting of votes

The first iteration

The votes credited to each candidate are set to 0.0.

Non-transferable votes are set to 0.0.

The first preferences for the candidates are tallied and the initial quota is determined.

Any candidate whose votes equal or exceed the current quota is elected.

Second and subsequent iterations

A new value of the quota is calculated to allow for any increases in non-transferable votes.

A new keep value for each elected candidate is calculated.

The votes are recounted using the latest keep values, which automatically perform the required redistributions, and increase the non-transferable votes, if relevant.

Any candidate whose vote now equals or exceeds the current quota is elected.

If the sum of the lowest candidate's votes and the total surplus is less than the votes of any other non-excluded candidate, or if the total surplus is less than 0.0001, the lowest candidate is excluded and his or her keep value reset to 0.0.

Iterations continue until the required number of members are elected.

Equal votes
If, on any count, 2 or more candidates have an equal number of votes and 1 of them has to be excluded, the electoral officer determines which candidate had the fewest votes the first time the candidates' totals were different and excludes the candidate with the lowest total votes.

Ties
If the candidates had an equal number of votes at all stages of the count, a random (or pseudo-random) process is used to choose which candidate is excluded.

Algorithm used to calculate Meek’s method
The following paragraphs give some further detail on the algorithm used to calculate Meek’s method and how computer programs using the algorithm will be certified.

Algorithm and article
Meek's method of counting votes requires the use of Algorithm 123 in the form described in an article in The Computer Journal (UK), Vol 30, No 3, 1987, pp 277-81 (the article). (A discussion of the mathematical equations that prove the existence and uniqueness of the method is set out in the article).

Authority for modifications
Algorithm 123 (in the form described in the article) and the formulas and procedures proposed in the article may, for the purposes of designing a computer program to implement Meek's method of counting votes, incorporate the following modifications:

(a) the formula used for determining the quota for election quoted in paragraph 2.5 of the article, (ie, "total votes---total excess)/(number of seats plus 1)"), may be modified by adding the figure 0.000000001 to the product of that calculation (any remainder after the ninth decimal digit being disregarded). (The addition of one-billionth of a vote to the quota removes the unlikely possibility of a tie):

(b) any other modification authorised by regulations made under this Act.

Duties of programmers and certifiers

Duties of programmers
Every person responsible for the design of a computer program intended to implement Meek's method of counting votes must take all reasonable steps to ensure that the program produces outcomes that are consistent with Algorithm 123 in the form described in the article and any authorised modifications.

Duties of certifiers
A computer program may not be used at any election or poll under this Act for the purpose of implementing Meek's method of counting votes unless a certifier appointed for the purpose by the Secretary for Local Government has first certified that the program produces outcomes that are consistent with Algorithm 123 (in the form described in the article) and any authorized modifications.

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Last updated: 20/04/2006