# Webster/Sainte-Laguë method

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The Webster/Sainte-Laguë method, often simply Webster method or Sainte-Laguë method (French pronunciation: ​[sɛ̃t.la.ɡy]), is a highest quotient method for allocating seats in party-list proportional representation used in many voting systems. It is named in Europe after the French mathematician André Sainte-Laguë and in United States after statesman and senator Daniel Webster. The method is quite similar to the D'Hondt method, but uses different divisors. In most cases the largest remainder method with a Hare quota delivers almost identical results. The D'Hondt method gives similar results too, but favors larger parties compared to the Webster/Sainte-Laguë method;[1] the Webster/Sainte-Laguë method is generally seen as more proportional but risks an outcome where a party with more than half the votes can win fewer than half the seats.[2] Often there is an electoral threshold, that is a minimum percentage of votes required to be allocated seats.

Webster first proposed the method in 1832 and 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). It was then replaced by Hamilton method and in 1911 the Webster method was reintroduced.[3] The method was again replaced in 1940, this time by the Huntington–Hill method. In France, André Sainte-Laguë introduced the method in his 1910 article. It seems that French and European literature was unaware of Webster until after World War II.

The Webster/Sainte-Laguë method is used in Bosnia and Herzegovina, Iraq, Kosovo, Latvia, 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, Bremen, Hamburg, North Rhine-Westphalia, Rhineland-Palatinate, and Schleswig-Holstein. In Denmark it is used for 40 out of the 179 seats in the Folketing, supplementing the D'Hondt method.

The Webster/Sainte-Laguë method was used in Bolivia in 1993, in Poland in 2001, 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. The 2019 Indonesian legislative election also utilized the method.[4]

The method has been proposed by the Green Party in Ireland as a reform for use in Dáil Éireann elections,[5] 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 country's upper house of parliament.[6] The United Kingdom Electoral Commission has used the method in 2003, 2007, 2010 and 2013 to distribute British seats in the European Parliament to constituent countries of the United Kingdom and the English regions.[7][8] The European Parliament (Representation) Act 2003 stipulates each region must be allocated at least 3 seats and that the ratio of electors to seats is as nearly as possible the same for each, the Commission found the Sainte-Laguë method produced the smallest standard deviation when compared to the D'Hondt method and Hare quota.[9][10]

## Description of the method

After all the votes have been tallied, successive quotients are calculated for each party. The formula for the quotient is[1]

${\displaystyle {\text{quotient}}={\frac {V}{2s+1}}}$

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; nor does its modified form.[11]

### 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 is divided by 1, then by 3, and 5 (and then, if necessary, by 7, 9, 11, 13, and so on). The 8 highest entries, marked with asterisks, range from 100,000 down to 16,000. For each, the corresponding party gets a seat.

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)

1 2 3 4 5 6 7 Seats won

(bold)

Party A quotient

seats after round

100,000

1

33,333

1

33,333

2

20,000

2

20,000

2

20,000

3

14,286

3

3
Party B quotient

seats after round

80,000

0

80,000

1

26,667

1

26,667

1

26,667

2

16,000

2

16,000

3

3
Party C quotient

seats after round

30,000

0

30,000

0

30,000

0

30,000

1

10,000

1

10,000

1

10,000

1

1
Party D quotient

seats after round

20,000

0

20,000

0

20,000

0

20,000

0

20,000

0

20,000

1

6,667

1

1

The below chart is an easy way to perform the calculation:

Denominator /1 /3 /5 Seats
won (*)
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 0.7
Total 8 8

The d'Hondt method differs by the formula to calculate the quotients ${\displaystyle \left({\text{quotient}}={\frac {V}{s+1}}\right)}$; using this formula, A would be allocated four seats and D none, reflecting the method's favoring of larger parties.[1]

## Webster, Sainte-Laguë, and Schepers

Webster proposed the method in United States Congress in 1832 for proportional allocation of seats in United States congressional apportionment. In 1842 the method was adopted (Act of June 25, 1842, ch 46, 5 Stat. 491). It was then replaced by Hamilton method and in 1911 the Webster method was reintroduced.[3]

According to some observers the method should be treated as two methods with the same result, because Webster method is used for allocating seats based on states' population and Saint Lague based on parties' votes.[12] 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.

Webster's method is defined in terms of a quota as in the largest remainder method; in this method, the quota is called a "divisor". For a given value of the divisor, the population count for each region is divided by this divisor and then rounded to give the number of legislators to allocate to that region. In order to make the total number of legislators come out equal to the target number, the divisor is adjusted to make the sum of allocated seats after being rounded give the required total.

One way to determine the correct value of the divisor would be to start with a very large divisor, so that no seats are allocated after rounding. Then the divisor may be successively decreased until one seat, two seats, three seats and finally the total number of seats are allocated. The number of allocated seats for a given region increases from s to s + 1 exactly when the divisor equals the population of the region divided by s + 1/2, so at each step the next region to get a seat will be the one with the largest value of this quotient. That means that this successive adjustment method for implementing Webster's method allocates seats in the same order to the same regions as the Sainte-Laguë method would allocate them.

In 1980 the German physicist Hans Schepers, at the time Head of the Data Processing Group of the German Bundestag, suggested that the distribution of seats according to d'Hondt be modified to avoid putting smaller parties at a disadvantage.[13] German media started using the term Schepers Method and later German literature usually calls it Sainte-Laguë/Schepers.[13]

## Modified Sainte-Laguë method

Some countries, e.g. Nepal, Norway and Sweden, change the quotient formula for parties that have not yet been allocated any seats (s = 0). These countries changed the quotient from V to V/1.4, though from the general 2018 elections onwards, Sweden has been using V/1.2.[14] 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 gives slightly greater preference to the larger parties over parties that would earn, by a small margin, a single seat if the unmodified Sainte-Laguë's method were used. With the modified method, such small parties do not get any seats; these seats are instead given to a larger party.[1]

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 square kilometer counts 1.8 points. Furthermore, one seat from each constituency is allocated according to the national distribution of votes.[15]

## Threshold for seats

Often a threshold or barrage is set, 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.

## References

1. ^ a b c d Lijphart, Arend (2003), "Degrees of proportionality of proportional representation formulas", in Grofman, Bernard; Lijphart, Arend (eds.), Electoral Laws and Their Political Consequences, Agathon series on representation, 1, Algora Publishing, pp. 170–179, ISBN 9780875862675 See in particular the section "Sainte-Lague", pp. 174–175.
2. ^ For example with three seats, a 55-25-20 vote is seen to be more proportionally represented by an allocation of 1-1-1 seats than by 2-1-0.
3. ^ a b Balinski, Michel L.; Peyton, Young (1982). Fair Representation: Meeting the Ideal of One Man, One Vote.
4. ^ "New votes-to-seats system makes elections 'fairer'". The Jakarta Post. 28 May 2018. Retrieved 19 April 2019.
5. ^ Ireland's Green Party website
6. ^ "House of Lords Reform Draft Bill" (PDF). Cabinet Office. May 2011. p. 16.
7. ^ (PDF). Electoral Commission https://www.electoralcommission.org.uk/sites/default/files/pdf_file/Distribution-of-UK-MEPs-among-electoral-regions.pdf. Retrieved 21 December 2019. Missing or empty |title= (help)
8. ^
9. ^ http://www.europarl.europa.eu/sides/getDoc.do?pubRef=-//EP//NONSGML+IM-PRESS+20070604IPR07417+EN+DOC+PDF+V0//EN&language=EN. Missing or empty |title= (help)
10. ^ McLean, Iain (1 November 2008). "Don't let the lawyers do the math: Some problems of legislative districting in the UK and the USA". Mathematical and Computer Modelling. 48 (9): 1446–1454. doi:10.1016/j.mcm.2008.05.025. ISSN 0895-7177.
11. ^ Miller, Nicholas R. (February 2013), "Election inversions under proportional representation", Annual Meeting of the Public Choice Society, New Orleans, March 8-10, 2013 (PDF).
12. ^ Badie, Bertrand; Berg-Schlosser, Dirk; Morlino, Leonardo, eds. (2011), International Encyclopedia of Political Science, Volume 1, SAGE, p. 754, ISBN 9781412959636, Mathematically, divisor methods for allocating seats to parties on the basis of party vote shares are identical to divisor methods for allocating seats to geographic units on the basis of the unit's share of the total population. ... Similarly, the Sainte-Laguë method is identical to a method devised by the American legislator Daniel Webster.
13. ^ a b "Sainte-Laguë/Schepers". The Federal Returning Officer of Germany. Retrieved 14 January 2016.
14. ^ Holmberg, Kaj (2019), "A new method for optimal proportional representation". Linköping, Sweden: Linköping University Department of Mathematics, p.8.
15. ^ Norway's Ministry of Local Government website; Stortinget; General Elections; The main features of the Norwegian electoral system; accessed 22 August 2009