The Sainte-Laguë method (French pronunciation: [sɛ̃t.la.ɡy]), also called the Webster method or the Schepers method (German pronunciation: [ˈʃeːpɐs]), 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 Hans Schepers. 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 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
quotient = V 2 s + 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, 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 (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.)
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:
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. 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.
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.
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