| A jointPolitics andEconomics series |
| Social choice andelectoral systems |
|---|
 |
By results of combination By mechanism of combination By ballot type |
 Mathematics portal |
TheWebster method, also called theSainte-Laguë method (French pronunciation:[sɛ̃t.la.ɡy]), is ahighest averagesapportionment method for allocating seats in a parliament amongfederal states, or among parties in aparty-list proportional representation system. The Sainte-Laguë method shows a more equalseats-to-votes ratio for different sized parties[1] among apportionment methods.
The method was first described in 1832 by American statesman and senatorDaniel Webster. In 1842, the method was adopted for proportional allocation of seats inUnited States congressional apportionment (Act of 25 June 1842, ch 46, 5 Stat. 491). The same method was independently invented in 1910 by the French mathematicianAndré Sainte-Laguë.
Proportionalelectoral 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. Differentapportionment 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 andpolitical fragmentation. The Sainte-Laguë method minimizes the average seats-to-votes ratio deviation[2] and empirically shows the best proportionality behavior[3] and more equalseats-to-votes ratio for different sized parties[1] among apportionment methods. Among other common methods, the D'Hondt method favours largeparties andcoalitions over small parties.[1][4][5][6] While favoring large parties reducespolitical fragmentation, this can be achieved withelectoral thresholds as well. The Sainte-Laguë method shows fewerapportionment paradoxes compared tolargest remainder methods[7] such as theHare quota and otherhighest averages methods such asd'Hondt method.[8]
After all the votes have been tallied, successivequotients are calculated for each party. The formula for the quotient is[9]
where:
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.[10] 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 anelectoral threshold; that is, in order to be allocated seats, a minimum percentage of votes must be gained.
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 0.5, then by 1.5, and 2.5 (and then, if necessary, by 3.5, 4.5, 5.5, 6.5, 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) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Seats won (bold) |
|---|---|---|---|---|---|---|---|---|
| Party A (100,000 votes) | 200,000 0+1 | 66,666.67 1 | 66,666.67 1+1 | 40,000 2 | 40,000 2 | 40,000 2+1 | 28571.42 3 | 3 |
| Party B (80,000 votes) | 160,000 0 | 160,000 0+1 | 53,333.33 1 | 53,333.33 1 | 53,333.33 1+1 | 32,000 2 | 32,000 2+1 | 3 |
| Party C (30,000 votes) | 60,000 0 | 60,000 0 | 60,000 0 | 60,000 0+1 | 20,000 1 | 20,000 1 | 20,000 1 | 1 |
| Party D (20,000 votes) | 40,000 0 | 40,000 0 | 40,000 0 | 40,000 0 | 40,000 0 | 40,000 0+1 | 13,333.33 1 | 1 |
The 8 highest entries (in the current round of calculation) are marked by asterisk: from200,000 down to40,000; 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 | Seats won (*) | True proportion |
|---|---|---|---|---|---|
| Party A | 200,000* | 66,666.67* | 40,000* | 3 | 3.5 |
| Party B | 160,000* | 53,333.33* | 32,000* | 3 | 2.8 |
| Party C | 60,000* | 20,000 | 12,000 | 1 | 1.0 |
| Party D | 40,000* | 13,333.33 | 8,000 | 1 | 0.7 |
| Total | 8 | 8 | |||
| Party | Popular vote | Party-list PR — Sainte-Laguë method | |||
|---|---|---|---|---|---|
| Number of seats | Seats % |  | |||
| Party A | 43.5% | 3 | 37.5% | ||
| Party B | 34.8% | 3 | 37.5% | ||
| Party C | 13.0% | 1 | 12.5% | ||
| Party D | 8.7% | 1 | 12.5% | ||
| TOTAL | 100% | 8 | 100% | ||
In comparison, theD'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.[9]
When apportioning seats inproportional representation, it is particularly important to avoid bias between large parties and small parties to avoidstrategic voting.André Sainte-Laguë showed theoretically that the Sainte-Laguë method shows the lowest average bias inapportionment,[2] confirmed by different theoretical and empirical ways.[3][11]: Sec.5 TheEuropean 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 smalleststandard deviation when compared to theD'Hondt method and Hare quota.[12][13]
Theseats-to-votes ratio for a political party is the ratio between the fraction of seats and the fraction of votes for that party:
The Sainte-Laguë method approximates proportionality by optimizing the seats-to-votes ratio among all parties with theleast squares approach. First, the difference between the seats-to-votes ratio for a party and the ideal seats-to-votes ratio is calculated and squared to obtain the error for the party. To achieve equal representation of each voter, the ideal ratio of seats share to votes share is.
Second, the error for each party is weighted according to the vote share of each party to represent each voter equally. In the last step, the errors for each party are summed up. This error is identical to theSainte-Laguë Index.
It was shown[14] that this error is minimized by the Sainte-Laguë method.
To reducepolitical fragmentation, some countries, e.g.Nepal,Norway andSweden, change the quotient formula for parties with no seats (s = 0). These countries changed the quotient fromV/0.5 toV/0.7, though from the general 2018 elections onwards, Sweden has been usingV/0.6.[15] That is, the modified method changes the sequence of divisors used in this method from (0.5, 1.5, 2.5, 3.5, ...) to (0.7, 1.5, 2.5, 3.5, ...). 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.[9]
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.[16]
Webster proposed the method in the United States Congress in 1832 for proportional allocation of seats inUnited States congressional apportionment. In 1842 the method was adopted (Act of June 25, 1842, ch 46, 5 Stat. 491). It was then replaced byHamilton method and in 1911 the Webster method was reintroduced.[11]
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.[17] 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 thelargest 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 froms tos + 1 exactly when the divisor equals the population of the region divided bys + 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.[18] German media started using the term Schepers Method and later German literature usually calls it Sainte-Laguë/Schepers.[18]
Anelection threshold can be set to reducepolitical 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 forleveling 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.
The Webster/Sainte-Laguë method is currently used inBosnia and Herzegovina,Ecuador,Indonesia,[19]Iraq,[20][21]Kosovo,Latvia,Nepal,[22]New Zealand,Norway andSweden.
InGermany it is used on the federal level for theBundestag, and on the state level for thelegislatures ofBaden-Württemberg,Bavaria,Bremen,Hamburg,North Rhine-Westphalia,Rhineland-Palatinate,Saxony andSchleswig-Holstein.[23] 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.[24]
InDenmark it is used for leveling seats in theFolketing, correcting the disproportionality of the D'Hondt method for the other seats.[25]
Some cantons in Switzerland use the Sainte-Laguë method forbiproportional apportionment between electoral districts and for votes to seats allocation.[26]
The Webster/Sainte-Laguë method was used inBolivia in 1993, inPoland in2001, and thePalestinian Legislative Council in 2006. The United KingdomElectoral Commission has used the method from 2003 to 2013 to distribute British seats in theEuropean Parliament to constituent countries of the United Kingdom and the English regions.[27][28]
The method has been proposed by theGreen Party inIreland as a reform for use inDáil Éireann elections,[29] and by theUnited KingdomConservative–Liberal Democrat coalition government in 2011 as the method for calculating the distribution of seats in elections to theHouse of Lords, the country's upper house of parliament.[30]
The method belongs to the class ofhighest-averages methods. It is similar to theJefferson/D'Hondt method, but uses different divisors. The Jefferson/D'Hondt method favors larger parties while the Webster/Sainte-Laguë method doesn't.[9] 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.[31]
When there are two parties, the Webster method is the unique divisor method which is identical to theHamilton method.[32]: Sub.9.10
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.
With the reimplementation of the Sainte-Lague proportional representation system, candidates on lists ... will have a distinct advantage.
14th and 19thUS Secretary of State (1841-1843, 1850–1852) •U.S. Senator from Massachusetts (1827–1841, 1845–1850) | |||||
| Political career | |||||
| Electoral history |
| ||||
| Legacy | |||||
| Family |
| ||||
