# Sainte-Laguë method

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The 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 after the French mathematician André Sainte-Laguë. The Sainte-Laguë method is quite similar to the D'Hondt method, but uses different divisors. In most cases the largest remainder method delivers almost identical results. The D'Hondt method gives similar results too, but favors larger parties compared to the Sainte-Laguë method.[1]

The Sainte-Laguë method is used in Iraq, New Zealand, Norway, Sweden, Bosnia and Herzegovina, Latvia, Kosovo, and Germany (on federal level for the Bundestag, on state level for the legislatures of Baden-Württemberg, Hamburg, Bremen, North Rhine-Westphalia, Schleswig-Holstein and Rhineland-Palatinate). Often there is a minimum percentage of votes required to be allocated seats.

The Sainte-Laguë method was used in Bolivia in 1993, in Poland in 2001, and in the elections to 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 method has been proposed by the Green Party in Ireland as a reform for use in Dáil Éireann elections,[2] 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.[3]

## 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]

$quot = \frac{V}{2s+1}$

where:

• V is the total number of votes that party received, and
• s is the number of seats that party has been allocated so far, initially 0 for all parties.

Whichever party has the highest quotient gets the next seat allocated, and their quotient is recalculated given their new seat total. The process is repeated until all seats have been allocated.

The 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.[4] For example, with seven seats available and the votes split 53,000, 24,000 and 23,000, the allocation would be three, two and two seats respectively.

denominator /1 /3 /5 /7 /9 /11 /13 Seats won (*)
Party A 53,000* 17,666* 10,600* 7,571 5,888 4,818 4,076 3
Party B 24,000* 8,000* 4,800 3,428 2,666 2,181 1,846 2
Party C 23,000* 7,666* 4,600 3,285 2,555 2,090 1,769 2

The d'Hondt method differs by the formula to calculate the quotients $\left( quot = \frac{V}{s+1}\right)$.[1]

## Sainte-Laguë and Webster

The Sainte-Laguë method is equivalent to the Webster method (named after its proponent, the U.S. senator Daniel Webster) in that they always give the same results, but the method of calculating the apportionment seems to be quite different.[5] 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 Droop 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.

## 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) from V to V/1.4. 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 larger preference to the larger parties over parties that would earn, with small margin, only a single seat if unmodified Sainte-Laguë's method were used. With the modified method, such small parties do not get any seat; 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 (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.[6]

## 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.