The Sainte-Laguë method (), also called the Webster method or the Schepers method (), is a highest averages apportionment method for allocating seats in a parliament among federal states, or among parties in a party-list proportional representation system. The Sainte-Laguë method shows a more equal seats-to-votes ratio for different sized parties among apportionment methods.
The method was first described in 1832 by American statesman and senator Daniel Webster. In 1842, the method was adopted for proportional allocation of seats in United States congressional apportionment (Act of 25 June 1842, ch 46, 5 Stat. 491). The same method was independently invented in 1910 by the French mathematician André Sainte-Laguë, and in 1980 by German physicist .
It is used for party-list proportional representation in Ecuador, Germany, Indonesia, Latvia and Nepal, as well as in modified form in Norway and Sweden. It is also used as part of MMP in New Zealand.
Motivation Proportional electoral systems attempt to distribute seats in proportion to the votes for each political party, e.g. a party with 30% of votes would receive 30% of seats. Exact proportionality is not possible because only whole seats can be distributed. Different apportionment methods, of which the Sainte-Laguë method is one, exist to distribute the seats according to the votes. Different apportionment methods show different levels of proportionality, apportionment paradoxes and political fragmentation. The Sainte-Laguë method minimizes the average seats-to-votes ratio deviation and empirically shows the best proportionality behaviorPennisi, Aline. "Disproportionality indexes and robustness of proportional allocation methods." Electoral Studies 17.1 (1998): 3-19. and more equal seats-to-votes ratio for different sized parties among apportionment methods. Among other common methods, the D'Hondt method favours large parties and coalitions over small parties. While favoring large parties reduces political fragmentation, this can be achieved with electoral thresholds as well. The Sainte-Laguë method shows fewer apportionment paradoxes compared to largest remainder methods such as the Hare quota and other highest averages methods such as d'Hondt method.
Description After all the votes have been tallied, successive quotients are calculated for each party. The formula for the quotient is See in particular the section "Sainte-Lague", pp. 174–175.
where:
V is the total number of votes that party received, and s is the number of seats that have been allocated so far to that party, initially 0 for all parties.
Whichever party has the highest quotient gets the next seat allocated, and their quotient is recalculated. The process is repeated until all seats have been allocated.
The Webster/Sainte-Laguë method does not ensure that a party receiving more than half the votes will win at least half the seats, which might not happen if a party with just over half the vote gets "rounded down" to under half the seats.. It also does not ensure that a party with a minority of the vote will not win a majority of the seats, for roughly the same reason.
Often there is an electoral threshold; that is, in order to be allocated seats, a minimum percentage of votes must be gained.
Example In this example, 230,000 voters decide the disposition of 8 seats among 4 parties. Since 8 seats are to be allocated, each party's total votes are divided by 1, then by 3, and 5 (and then, if necessary, by 7, 9, 11, 13, and so on by using the formula above) every time the number of votes is the biggest for the current round of calculation.
For comparison, the "True proportion" column shows the exact fractional numbers of seats due, calculated in proportion to the number of votes received. (For example, 100,000/230,000 × 8 = 3.48.)
round (1 seat per round)1234567Seats won (bold)Party A (100,000 votes)100,000 0+133,333 133,333 1+120,000 220,000 220,000 2+114,286 33Party B (80,000 votes)80,000 080,000 0+126,667 126,667 126,667 1+116,000 216,000 2+13Party C (30,000 votes)30,000 030,000 030,000 030,000 0+110,000 110,000 110,000 11Party D (20,000 votes)20,000 020,000 020,000 020,000 020,000 020,000 0+16,667 11
The 8 highest entries (in the current round of calculation) are marked by asterisk: from 100,000 down to 16,000; for each, the corresponding party gets a seat.
The below chart shows an easy way to perform the calculation:
Denominator /0.5 /1.5 /2.5 Seatswon (*) True proportion Party A 100,000* 33,333* 20,000* 3 3.5 Party B 80,000* 26,667* 16,000* 3 2.8 Party C 30,000* 10,000 6,000 1 1.0 Party D 20,000* 6,667 4,000 1 0.7Total 8 8
PartyPopular voteParty-list PR — Sainte-Laguë methodNumber of seatsSeats %Party A43.5%337.5%Party B34.8%337.5%Party C13.0%112.5%Party D8.7%112.5%TOTAL100%8100%
In comparison, the D'Hondt method would allocate four seats to party A and no seats to party D, reflecting the D'Hondt method's overrepresentation of larger parties.
Modified Sainte-Laguë method To reduce political fragmentation, some countries, e.g. Nepal, Norway and Sweden, change the quotient formula for parties with no seats (s = 0). These countries changed the quotient from V/1 to V/1.4, though from the general 2018 elections onwards, Sweden has been using V/1.2.Holmberg, Kaj (2019), "A new method for optimal proportional representation" . Linköping, Sweden: Linköping University Department of Mathematics, p.8. That is, the modified method changes the sequence of divisors used in this method from (1, 3, 5, 7, ...) to (1.4, 3, 5, 7, ...). This makes it more difficult for parties to earn only one seat, compared to the unmodified Sainte-Laguë's method. With the modified method, such small parties do not get any seats; these seats are instead given to a larger party.
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 (former counties) depends on the population and area of the county; each inhabitant counts one point, while each km2 counts 1.8 points. Furthermore, one seat from each constituency is allocated according to the national distribution of votes.Norway's Ministry of Local Government website; Stortinget; General Elections; The main features of the Norwegian electoral system ; accessed 22 August 2009
Threshold for seats An election threshold can be set to reduce political fragmentation, and any list party which does not receive at least a specified percentage of list votes will not be allocated any seats, even if it received enough votes to have otherwise receive a seat. Examples of countries using the Sainte-Laguë method with a threshold are Germany and New Zealand (5%), although the threshold does not apply if a party wins at least one electorate seat in New Zealand or three electorate seats in Germany. Sweden uses a modified Sainte-Laguë method with a 4% threshold, and a 12% threshold in individual constituencies (i.e. a political party can gain representation with a minuscule representation on the national stage, if its vote share in at least one constituency exceeded 12%). Norway has a threshold of 4% to qualify for leveling seats that are allocated according to the national distribution of votes. This means that even though a party is below the threshold of 4% nationally, they can still get seats from constituencies in which they are particularly popular.
Usage by country The Webster/Sainte-Laguë method is currently used in Bosnia and Herzegovina, Ecuador, Indonesia, Iraq, Latvia, Nepal, New Zealand, Norway and Sweden.
In Germany it is used on the federal level for the Bundestag, and on the state level for the legislatures of Baden-Württemberg, Bavaria, Bremen, Hamburg, North Rhine-Westphalia, Rhineland-Palatinate, Saxony and Schleswig-Holstein. To correct for the deficiency where a party can win a majority of votes but not a majority of seats, in federal elections the law provides such a party will receive extra seats until it has a majority of one.
Some cantons in Switzerland use the Sainte-Laguë method for biproportional apportionment between electoral districts and for votes to seats allocation.Bericht 09.1775.02 der vorberatenden Spezialkommission CountryLegislative bodyType of bodyList typeVariation of open lists (if applicable)Apportionment methodElectoral thresholdConstituenciesGovernmental systemNotesBosnia and HerzegovinaHouse of RepresentativesLower house of national parliamentOpen listSainte-Laguë methodParliamentary directorial republicEcuadorNational CongressUnicameral national parliamentClosed list—Sainte-Laguë methodGermanyBundestagLower house of national parliamentLocalized listSeparate vote for candidatesOnly first place candidate may win seat (but not guaranteed to)5% or 3 constituencies, first place for independents (only in constituencies)Constituencies (single-member)Federal parliamentary republicThe system was recently modified to an essentially (non-mixed) closed list proportional system with a local constituency vote to eliminate the need for overhang seats. In the new system, the number of seats a party can win is capped, if they "won" more seats by plurality, not all of their winners will be elected.Closed list—Sainte-Laguë methodFederal states (Länder)IndonesiaHouse of RepresentativesLower house of national parliamentOpen listSainte-Laguë method4%3 to 10 members constituenciesPresidential systemIraqLatviaSaeimaUnicameral national parliamentOpen listSainte-Laguë method5%Parliamentary republicNorwayStortingUnicameral national parliamentClosed list—modified Sainte-Laguë methodNo de jure threshold19 multi-member constituenciesParliamentary systemFirst divisor is 1,4 instead of 1.4% for leveling seatsOne seat in each constituency is used for nationwide levelingSwedenRiksdagUnicameral national parliamentOpen listMore open (5% of the party vote to override the default party-list)Swedish Election Authority: Elections in Sweden: The way its done (page 16)Sainte-Laguë method (leveling seats)4% nationally or 12% in a given constituencyCounties of Sweden (some counties are further subdivided)Parliamentary system The Webster/Sainte-Laguë method was used in Bolivia in 1993, in Poland in 2001, and the Palestinian Legislative Council in 2006. The United Kingdom Electoral Commission has used the method from 2003 to 2013 to distribute British seats in the European Parliament to constituent countries of the United Kingdom and the English regions. The modified method was used by Denmark for distributing direct seats from 1953 to 2005
The method 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 the House of Lords, the country's upper house of parliament.
History Webster proposed the method in the United States Congress in 1832 for proportional allocation of seats in United States congressional apportionment. In 1842 the method was adopted (Act of 25 June 1842, ch 46, 5 Stat. 491). It was then replaced by Hamilton method and in 1911 the Webster method was reintroduced.
Webster and Sainte-Laguë methods should be treated as two methods with the same result, because the Webster method is used for allocating seats based on states' population, and the Sainte-Laguë based on parties' votes. Webster invented his method for legislative apportionment (allocating legislative seats to regions based on their share of the population) rather than elections (allocating legislative seats to parties based on their share of the votes) but this makes no difference to the calculations in the method.