# Sainte-Laguë method

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 identical or almost identical results. The D'Hondt method gives similar results too, but favours larger parties compared to the Sainte-Laguë method.

The Sainte-Laguë method is used in New Zealand, Norway, Sweden, Bosnia and Herzegovina, Latvia, Kosovo, Denmark (for the 40 supplementary seats in the national parliament), 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). It was also 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. It has been proposed by the Green Party in Ireland as a reform for use in Dáil Éireann elections,[1] and by the United Kingdom Conservative-Liberal Democrat coalition government in 2011 as the method for calculating the distribution of seats in elections to its upper house.[2]

## Description of the method

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

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

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. 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 $\left( quot = \frac{V}{s+1}\right)$.

## 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. The latter, invented for legislative apportionment rather than elections, uses a quota as in the largest remainder method but the quota (called a divisor) is adjusted as necessary so that the sum of allocated seats after being rounded gives the required total. One way to accomplish this, is to start with a large divisor, so that no seats are allocated after rounding. Then the divisor is successively decreased until one seat, two seats, three seats and finally the total number of seats are allocated. The values of the divisors corresponding to an increase of one are the number of votes of a party divided by the number of seats plus 0.5 allocated to that party so far. That means the seats are allocated according to the Sainte-Laguë method as described above.

## Modified Sainte-Laguë method

Some countries, e.g. Nepal, Norway and Sweden, replace the first divisor with 1.4. 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. If there is a restriction as to how small parties are allowed to earn seats, the modification does not have any effect when many seats are distributed, as every party will earn at least one seat anyway.

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, slightly increasing the odds of winning a seat for small national parties.[clarification needed][3]

## References

1. ^ Ireland's Green Party website;
2. ^ "House of Lords Reform Draft Bill" (PDF). Cabinet Office. May 2011. p. 16.
3. ^ Norway's Ministry of Local Government website; Stortinget; General Elections; The main features of the Norwegian electoral system; accessed 22 August 2009