Sainte-Laguë method

The Sainte-Laguë method, also called the Webster 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. The same method was independently invented in 1910 by the French mathematician André Sainte-Laguë.
It is used for party-list proportional representation in Ecuador, Germany, Indonesia, Kosovo, Latvia and Nepal, as well as in modified form in Denmark, 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 behavior 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
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 can happen when 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 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.

round '	1	2	3	4	5	6	7	Seats won
Party A	100,000 0+1	33,333 1	33,333 1+1	20,000 2	20,000 2	20,000 2+1	14,286 3	3'''
Party B	80,000 0	80,000 0+1	26,667 1	26,667 1	26,667 1+1	16,000 2	16,000 2+1	3
Party C	30,000 0	30,000 0	30,000 0	30,000 0+1	10,000 1	10,000 1	10,000 1	1
Party D	20,000 0	20,000 0	20,000 0	20,000 0	20,000 0	20,000 0+1	6,667 1	1

The 8 highest entries 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:
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. These countries changed the quotient from V/0.5 to V/0.7, though from the general 2018 elections onwards, Sweden has been using V/0.6. That is, the modified method changes the sequence of divisors used in this method from to. 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 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.

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, 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. 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, Kosovo, 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.
In Denmark it is used for leveling seats in the Folketing, correcting the disproportionality of the D'Hondt method for the other seats.
Some cantons in Switzerland use the Sainte-Laguë method for biproportional apportionment between electoral districts and for votes to seats allocation.

Country	Legislative body	Type of body	List type	Variation of open lists	Apportionment method	Electoral threshold	Constituencies	Governmental system	Notes
Bosnia and Herzegovina	House of Representatives	Lower house of national parliament	Open list		Sainte-Laguë method			Parliamentary directorial republic
Denmark	Folketing	Unicameral national parliament	Open list		D'Hondt method	2%
Denmark	Folketing	Unicameral national parliament	?		modified Sainte-Laguë method	2%
Ecuador	National Congress	Unicameral national parliament	Closed list	—	Sainte-Laguë method
Germany	Bundestag	Lower house of national parliament	Localized list	Separate vote for candidates	Only first place candidate may win seat	5% or 3 constituencies, first place for independents	Constituencies	Federal parliamentary republic	The system was recently modified to an essentially 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.
Germany	Bundestag	Lower house of national parliament	Closed list	—	Sainte-Laguë method	5% or 3 constituencies, first place for independents	Federal states	Federal parliamentary republic	The system was recently modified to an essentially 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.
Indonesia	House of Representatives	Lower house of national parliament	Open list		Sainte-Laguë method	4%	3 to 10 members constituencies	Presidential system
Iraq
Kosovo	Kuvendi	Unicameral national parliament	Open list		Sainte-Laguë method
Latvia	Saeima	Unicameral national parliament	Open list		Sainte-Laguë method	5%		Parliamentary republic
Norway	Storting	Unicameral national parliament	Closed list	—	modified Sainte-Laguë method	No de jure threshold	19 multi-member constituencies	Parliamentary system	First divisor is 1,4 instead of 1.
Norway	Storting	Unicameral national parliament	Closed list	4% for leveling seats	modified Sainte-Laguë method	One seat in each constituency is used for nationwide leveling	-	Parliamentary system	First divisor is 1,4 instead of 1.
Sweden	Riksdag	Unicameral national parliament	Open list	More open	Sainte-Laguë method	4% nationally or 12% in a given constituency	Counties of Sweden	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 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.