Sainte-Laguë method
Encyclopedia
The Sainte-Laguë method is one way of allocating seats approximately proportional to the number of votes of a party to a party list
used in many voting systems. It is named after the French mathematician André Sainte-Laguë
. The Sainte-Laguë method is quite similar to the D'Hondt method
, but uses different divisors. In most cases the largest remainder method
delivers identical or almost identical results. The D'Hondt method gives similar results too, but favours larger parties compared to the Sainte-Laguë method.
The Sainte-Laguë method is used in New Zealand
, Norway
, Sweden
, Bosnia and Herzegovina
, Latvia
, Kosovo
, Denmark
(for the 40 supplementary seats in the national parliament), and Germany
(on federal level for the Bundestag
, on state level for the legislature
s of Baden-Württemberg
, Hamburg
, Bremen
, North Rhine-Westphalia
and Rhineland-Palatinate). It was also used in Bolivia
in 1993, in Poland
in 2001, and in the elections to the Palestinian Legislative Council
in 2006. A variant of this method, the modified Sainte-Laguë method, was used to allocate the proportional representation
(PR) seats in the Constituent Assembly poll of Nepal
in 2008. It has been proposed by the Green Party
in Ireland
as a reform for use in Dáil Éireann
elections, and by the United Kingdom
Conservative-Liberal Democrat coalition government
in 2011 as the method for calculating the distribution of seats in elections to its upper house
.
s are calculated for each party. The formula for the quotient is
where:
Whichever party has the highest quotient gets the next seat allocated, and their quotient is recalculated given their new seat total. The process is repeated until all seats have been allocated.
The d'Hondt method only differs by the formula to calculate the quotients
.
) in that they always give the same results, but the method of calculating the apportionment seems to be quite different. The latter, invented for legislative apportionment rather than elections, uses a quota
as in the largest remainder method
but the quota (called a divisor) is adjusted as necessary so that the sum of allocated seats after being rounded gives the required total. One way to accomplish this, is to start with a large divisor, so that no seats are allocated after rounding. Then the divisor is successively decreased until one seat, two seats, three seats and finally the total number of seats are allocated. The values of the divisors corresponding to an increase of one are the number of votes of a party divided by the number of seats plus 0.5 allocated to that party so far. That means the seats are allocated according to the Sainte-Laguë method as described above.
, Norway
and Sweden
, replace the first divisor with 1.4. This gives slightly larger preference to the larger parties over parties that would earn, with small margin, only a single seat if unmodified Sainte-Laguë's method were used. With the modified method, such small parties do not get any seat; these seats are instead given to a larger party. If there is a restriction as to how small parties are allowed to earn seats, the modification does not have any effect when many seats are distributed, as every party will earn at least one seat anyway.
Norway further amends this system by utilizing a two-tier proportionality. The number of members to be returned from each of Norway's 19 constituencies (counties) depends on the population and area of the county: each inhabitant counts one point, while each square kilometer counts 1.8 points. Furthermore, one seat from each constitutency is allocated according to the national distribution of votes, slightly increasing the odds of winning a seat for small national parties.
Party-list proportional representation
Party-list proportional representation systems are a family of voting systems emphasizing proportional representation in elections in which multiple candidates are elected...
used in many voting systems. It is named after the French mathematician André Sainte-Laguë
André Sainte-Laguë
André Sainte-Laguë was a French mathematician who was a pioneer in the area of graph theory.His research on seat allocation methods led to one being named after him, the Sainte-Laguë method...
. The Sainte-Laguë method is quite similar to the D'Hondt method
D'Hondt method
The d'Hondt method is a highest averages method for allocating seats in party-list proportional representation. The method described is named after Belgian mathematician Victor D'Hondt who described it in 1878...
, but uses different divisors. In most cases the largest remainder method
Largest remainder method
The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems...
delivers identical or almost identical results. The D'Hondt method gives similar results too, but favours larger parties compared to the Sainte-Laguë method.
The Sainte-Laguë method is used in New Zealand
New Zealand
New Zealand is an island country in the south-western Pacific Ocean comprising two main landmasses and numerous smaller islands. The country is situated some east of Australia across the Tasman Sea, and roughly south of the Pacific island nations of New Caledonia, Fiji, and Tonga...
, Norway
Norway
Norway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...
, Sweden
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
, Bosnia and Herzegovina
Bosnia and Herzegovina
Bosnia and Herzegovina , sometimes called Bosnia-Herzegovina or simply Bosnia, is a country in Southern Europe, on the Balkan Peninsula. Bordered by Croatia to the north, west and south, Serbia to the east, and Montenegro to the southeast, Bosnia and Herzegovina is almost landlocked, except for the...
, Latvia
Latvia
Latvia , officially the Republic of Latvia , is a country in the Baltic region of Northern Europe. It is bordered to the north by Estonia , to the south by Lithuania , to the east by the Russian Federation , to the southeast by Belarus and shares maritime borders to the west with Sweden...
, Kosovo
Kosovo
Kosovo is a region in southeastern Europe. Part of the Ottoman Empire for more than five centuries, later the Autonomous Province of Kosovo and Metohija within Serbia...
, Denmark
Denmark
Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...
(for the 40 supplementary seats in the national parliament), and Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
(on federal level for the Bundestag
Bundestag
The Bundestag is a federal legislative body in Germany. In practice Germany is governed by a bicameral legislature, of which the Bundestag serves as the lower house and the Bundesrat the upper house. The Bundestag is established by the German Basic Law of 1949, as the successor to the earlier...
, on state level for the legislature
Legislature
A legislature is a kind of deliberative assembly with the power to pass, amend, and repeal laws. The law created by a legislature is called legislation or statutory law. In addition to enacting laws, legislatures usually have exclusive authority to raise or lower taxes and adopt the budget and...
s of Baden-Württemberg
Landtag of Baden-Württemberg
The Landtag of Baden-Württemberg is the state diet of the German federal state of Baden-Württemberg. It convenes in Stuttgart and currently consists of 139 members of four Parties...
, Hamburg
Elections in Hamburg
The number of elections in Hamburg varies. Hamburg has a state election every four years, the elections for the state parliament. There are also elections to the federal diet of Germany, the local elections of the diet of the boroughs and every five years to the European Parliament...
, Bremen
Bürgerschaft of Bremen
The Bremische Bürgerschaft is the state diet of the Free Hanseatic City of Bremen. It currently consists of 83 members from five parties. The current majority is a coalition of the Social Democratic Party and the Alliance '90/The Greens, supporting Mayor and Senate president Jens Böhrnsen...
, North Rhine-Westphalia
Landtag of North Rhine-Westphalia
The Landtag of North Rhine-Westphalia is the state diet of the German federal state of North Rhine-Westphalia. It convenes in Düsseldorf and currently consists of 181 members of five Parties...
and Rhineland-Palatinate). It was also used in Bolivia
Bolivia
Bolivia officially known as Plurinational State of Bolivia , is a landlocked country in central South America. It is the poorest country in South America...
in 1993, in Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
in 2001, and in the elections to the Palestinian Legislative Council
Palestinian Legislative Council
The Palestinian Legislative Council, the legislature of the Palestinian Authority, is a unicameral body with 132 members, elected from 16 electoral districts in the West Bank and Gaza...
in 2006. A variant of this method, the modified Sainte-Laguë method, was used to allocate the proportional representation
Proportional representation
Proportional representation is a concept in voting systems used to elect an assembly or council. PR means that the number of seats won by a party or group of candidates is proportionate to the number of votes received. For example, under a PR voting system if 30% of voters support a particular...
(PR) seats in the Constituent Assembly poll of Nepal
Nepalese Constituent Assembly election, 2008
A general election for the Constituent Assembly was held in Nepal on 10 April 2008 after having been postponed from earlier dates of 7 June 2007 and 22 November 2007. The Nepalese Constituent Assembly will draft a new constitution; it will therefore decide, amongst other things, on the issue of...
in 2008. It has been proposed by the Green Party
Green Party (Ireland)
The Green Party is a green political party in Ireland. It was founded as the Ecology Party of Ireland in 1981 by Dublin teacher Christopher Fettes. The party became the Green Alliance in 1983 and in 1987 was renamed to its current title in English...
in Ireland
Ireland
Ireland is an island to the northwest of continental Europe. It is the third-largest island in Europe and the twentieth-largest island on Earth...
as a reform for use in Dáil Éireann
Dáil Éireann
Dáil Éireann is the lower house, but principal chamber, of the Oireachtas , which also includes the President of Ireland and Seanad Éireann . It is directly elected at least once in every five years under the system of proportional representation by means of the single transferable vote...
elections, and by the United Kingdom
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
Conservative-Liberal Democrat coalition government
United Kingdom coalition government (2010–present)
The ConservativeLiberal Democrat coalition is the present Government of the United Kingdom, formed after the 2010 general election. The Conservative Party and the Liberal Democrats entered into discussions which culminated in the 2010 coalition agreement, setting out a programme for government...
in 2011 as the method for calculating the distribution of seats in elections to its upper house
House of Lords
The House of Lords is the upper house of the Parliament of the United Kingdom. Like the House of Commons, it meets in the Palace of Westminster....
.
Description of the method
After all the votes have been tallied, successive quotientQuotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
s are calculated for each party. The formula for the quotient is
where:
- V is the total number of votes that party received, and
- s is the number of seats that party has been allocated so far, initially 0 for all parties.
Whichever party has the highest quotient gets the next seat allocated, and their quotient is recalculated given their new seat total. The process is repeated until all seats have been allocated.
The d'Hondt method only differs by the formula to calculate the quotients
.Sainte-Laguë and Webster
The Sainte-Laguë method is equivalent to the Webster method (named after its proponent, the U.S. senator Daniel WebsterDaniel Webster
Daniel Webster was a leading American statesman and senator from Massachusetts during the period leading up to the Civil War. He first rose to regional prominence through his defense of New England shipping interests...
) in that they always give the same results, but the method of calculating the apportionment seems to be quite different. The latter, invented for legislative apportionment rather than elections, uses a quota
Droop Quota
The Droop quota is the quota most commonly used in elections held under the Single Transferable Vote system. It is also sometimes used in elections held under the largest remainder method of party-list proportional representation . In an STV election the quota is the minimum number of votes a...
as in the largest remainder method
Largest remainder method
The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems...
but the quota (called a divisor) is adjusted as necessary so that the sum of allocated seats after being rounded gives the required total. One way to accomplish this, is to start with a large divisor, so that no seats are allocated after rounding. Then the divisor is successively decreased until one seat, two seats, three seats and finally the total number of seats are allocated. The values of the divisors corresponding to an increase of one are the number of votes of a party divided by the number of seats plus 0.5 allocated to that party so far. That means the seats are allocated according to the Sainte-Laguë method as described above.
Modified Sainte-Laguë method
Some countries, e.g. NepalNepal
Nepal , officially the Federal Democratic Republic of Nepal, is a landlocked sovereign state located in South Asia. It is located in the Himalayas and bordered to the north by the People's Republic of China, and to the south, east, and west by the Republic of India...
, Norway
Norway
Norway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...
and Sweden
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
, replace the first divisor with 1.4. This gives slightly larger preference to the larger parties over parties that would earn, with small margin, only a single seat if unmodified Sainte-Laguë's method were used. With the modified method, such small parties do not get any seat; these seats are instead given to a larger party. If there is a restriction as to how small parties are allowed to earn seats, the modification does not have any effect when many seats are distributed, as every party will earn at least one seat anyway.
Norway further amends this system by utilizing a two-tier proportionality. The number of members to be returned from each of Norway's 19 constituencies (counties) depends on the population and area of the county: each inhabitant counts one point, while each square kilometer counts 1.8 points. Furthermore, one seat from each constitutency is allocated according to the national distribution of votes, slightly increasing the odds of winning a seat for small national parties.
External links
- Java implementation of Webster's method at cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- Elections New Zealand explanation of Sainte-Laguë
- 2002 New Zealand election using Sainte-Laguë