# Sainte-Laguë method

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The Webster/Sainte-Laguë Method is a Highest averages method used for allocating seats proportionally for representative assemblies with party list voting systems. It works like D'Hondt method, except that you use divisors 1, 3, 5, 7, ... instead of 1, 2, 3, 4, ...

In the modified Sainte-Laguë method, the first divisor is modified to 1.4. The sequence of divisors is then 1.4, 3, 5, 7, ... The modified Sainte-Laguë method is used for elections to the Danish parliament.

## Allocation

After all the votes have been tallied, successive quotas 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.[2]

## Extensions of theory

Several cardinal PR methods reduce to Sainte-Laguë if certain divisors are used. Some of which are:

## Notes

Webster, unlike D'Hondt, doesn't guarantee that a majority of voters will get at least half of the seats.[3]

35-seat example
Party Votes Votes % 2nd-to-last round seats 2nd-to-last round divisors Final seats Final divisors Seats %
A 503 50.3% 16 15.2424 (503/33) 17 14.3714 (503/35) 48.57%
B 304 30.4% 10 14.4762 (304/21) 11 13.2174 (304/23) 31.43%
C 193 19.3% 6 14.8461 (193/15) 7 12.8666 (193/15) 20%
Total seats awarded 32 35

If D'Hondt had been used, the final divisor would've been 27.944, with (results calculated by rounding down to the nearest number) Party A getting 18 seats out of 35, a 51.42% majority (503/27.944), B 10 seats (304/27.944), and C 6 seats.

## References

1. 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. Miller, Nicholas R. (February 2013), "Election inversions under proportional representation", Annual Meeting of the Public Choice Society, New Orleans, March 8-10, 2013 (PDF).
3. Miller, Nicholas R. (2014-12-05). "Election Inversions under Proportional Representation" (PDF). Scandinavian Political Studies. Wiley. 38 (1): 4–25. doi:10.1111/1467-9477.12038. ISSN 0080-6757. Retrieved 2020-03-24.