Sainte Laguë Formula Explained

 

Electorate Seats

The MP for the electoral district is the candidate who wins more votes than any other candidate.  He or she does not need to win more than half the votes cast.  Under an MMP electoral system MPs for the electoral districts are elected in exactly the same way as they would be under the First-Past-The-Post (FPP) electoral system.

Party List Seats

The number of party votes won by each registered party which has submitted a Party List is used to decide how many seats overall each party will have in Parliament. 

If, for example, the party vote for the Grandstand Party entitled it to a total of 54 seats in Parliament and it won 40 electorate candidate seats, it would gain 14 further seats which would be drawn from the Party List of the Grandstand Party.  Candidates may stand for Parliament both in an electoral district and on their Party’s List.  As a result, the first 14 candidates on the Grandstand Party’s rank-ordered Party List who had not been elected to Parliament to represent an electoral district would be declared elected as Party List MPs.

A procedure, known as the Sainte Laguë formula (after its founder) is used to decide the order in which political parties are awarded seats in Parliament.

Allocating 2005 General Election Parliamentary Seats using the Sainte-Laguë Formula

To determine the precise order in which all the seats in Parliament are allocated to the various political parties, the Electoral Act 1993 prescribes that a mathematical formula, called the Sainte-Laguë formula, be applied.  The nationwide party vote of each of the parties which qualified for representation in Parliament is divided by successive odd numbers starting with 1 (ie divided the party votes by 1, 3, 5, 7, 9, 11, 13, etc).  The 120 highest numbers (which are called quotients) determine both the number of seats for each party and the order in which they are allocated.  The following explains how the process works:

Step 1:

The Chief Electoral Officer draws up a table showing the name of each party shown on the party side of the ballot paper, the number of party votes it won, the percentage of all party votes it won and the number of electorate seats it won.  For the purposes of this explanation minor parties are combined under the heading ‘OTHER’.


Registered Parties

LABOUR
PARTY

 NATIONAL  PARTY

NEW ZEALAND FIRST PARTY

ACT
NEW ZEALAND

GREEN PARTY

UNITED
FUTURE

JIM ANDERTON’S PROGRESSIVE

MAORI PARTY

OTHER

TOTAL

Party Votes

935,319

889,813

130,115

34,469

120,521

60,860

26,441

48,263

29,828

2,275,629

% of all party votes

41.10%

39.10%

5.72%

1.51%

5.30%

2.67%

1.16%

2.12%

1.31%

100%

Number of
electorate seats won

31

31

0

1

0

1

1

4

0

69

Step 2:

The Chief Electoral Officer then excludes parties that are not eligible for Party List seats by deleting any party that has not won at least 5% of the total number of party votes and has not won at least one electorate seat (commonly termed the threshold).  Although ACT New Zealand, United Future, Jim Anderton’s Progressive and the Maori Party each gained less than 5% of the party votes they did win electorate seats, so are included.

Registered Parties
that gained 5% of total party votes or won at least 1 electorate seat

LABOUR
PARTY

NATIONAL PARTY

NEW ZEALAND FIRST PARTY

ACT
NEW ZEALAND

GREEN PARTY

UNITED
FUTURE

JIM ANDERTON’S PROGRESSIVE

MAORI PARTY

TOTAL

Party Votes

935,319

889,813

130,115

34,469

120,521

60,860

26,441

48,263

2,245,801

% of all party votes eligible for list seats

41.65%

39.62%

5.79%

1.53%.

5.37%

2.71%

1.18%

2.15%

100.00%

Number of electorate seats won

31

31

0

1

0

1

1

4

69


Notes:

Because the parties not reaching the threshold have been disregarded the percentage share for each of the remaining parties has increased.

Step 3:

The Chief Electoral Officer then divides the total party votes for each eligible party by a sequence of odd numbers starting with 1 (1, 3, 5, 7, 9, 11, 13, 15, etc), until enough quotients had been found to allocate all 120 seats.  In the table on the following page the italic numbers beside the highest 120 quotients indicate their order from highest to lowest.

Step 4:

The Chief Electoral Officer then counts the number of quotients each party has in the highest 120.

Step 5:

The Chief Electoral Officer then determines how many electorate seats each party has won, and allocates enough Party List seats to each party to bring the total number of seats up to the number to which it is entitled.

Step 6:

The Chief Electoral Officer then examines the list of candidates each party submitted on its Party List before the election, and deletes the names of any candidate who has won an electorate seat.  He then allocates each Party's list seats to its list candidates, starting at the top of the list and working down until he has allocated all the list seats to which that party is entitled.  He then declares these candidates elected to Parliament and advises the Clerk of the House of Representatives of their names.

Notes:

There are four further points to note about the process.  

  1. If a party that appears on the party vote side of the ballot paper wins more electorate seats than it is entitled to based on its share of the party vote, then it does not receive any list seats.  It keeps the extra seats, and the size of Parliament is increased by that number of seats until the next general election.  The increase in the size of Parliament is known as an overhang.  The number of seats won by other parties is not affected.[1]
  2. If a party has not nominated enough list candidates to fill all the seats to which it is entitled on the basis of its share of the party vote, the seats remain unfilled and the size of Parliament is reduced by that number of seats until the next general election.  The number of seats won by other parties is not affected.
  3. If an electorate seat is won by a candidate not representing a party contesting the party vote, the Chief Electoral Officer subtracts that number of seats from 120, and works out the allocation of seats between registered parties based on that lower number.
  4. The list nominated by a party at a general election is used to replace a list MP from that party in the event of their seat being vacated (Electoral Act 1993, Section 137). The Chief Electoral Officer asks the remaining candidates on the list in turn if they are willing to become an MP until a replacement candidate is found.  If no such candidate can be found from the list, the seat remains unfilled until the next general election.

    Parliament may, by resolution supported by 75% of all MPs, avoid filling a vacant list seat if the vacancy occurs within six months of the date Parliament is due to expire or if the Prime Minister has announced that a general election is to be held within six months of the date the vacancy occurred  (Electoral Act 1993, Section 136).

 


[1] An overhang occurred at the 2005 general election with the Maori Party winning more electorate seats (4) than it was entitled to based on its share of the party vote (3).  Accordingly, the size of Parliament increased to 121 seats.

 
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