Sainte-Laguë method
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The Sainte-Laguë method of the highest average (equivalent to Webster's method or divisor method with standard rounding) is one way of allocating seats proportionally for representative assemblies with party list voting systems. It is named after French mathematician André Sainte-Laguë. The Sainte-Laguë method is closely related to the D'Hondt method, although without the latter's favoritism for larger parties.
The Sainte-Laguë method is used in New Zealand, Norway, Sweden, Denmark, Bosnia and Herzegovina, Latvia, Kosovo, Hamburg and Bremen. It was also used in Bolivia in 1993, in Poland in 2001, and in the elections to the Palestinian Legislative Council in 2006.
See the article on highest averages method for a comparison with the D'Hondt method.
Allocation
The Sainte-Laguë method is a divisor method, like the d'Hondt method, but with a different divisor.
After all the votes have been tallied, successive quotients are calculated for each list. The formula for the quotient is , where:
- V is the total number of votes that list received, and
- s is the number of seats that party has been allocated so far, initially 0 for all parties.
(The d'Hondt method uses as the formula).
Whichever list 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.
Example
| Party A | Party B | Party C | Party D | Party E | |
| Votes | 340,000 | 280,000 | 160,000 | 60,000 | 15,000 |
| Seat 1 | 340,000 | 280,000 | 160,000 | 60,000 | 15,000 |
| Seat 2 | 113,333 | 280,000 | 160,000 | 60,000 | 15,000 |
| Seat 3 | 113,333 | 93,333 | 160,000 | 60,000 | 15,000 |
| Seat 4 | 113,333 | 93,333 | 53,333 | 60,000 | 15,000 |
| Seat 5 | 68,000 | 93,333 | 53,333 | 60,000 | 15,000 |
| Seat 6 | 68,000 | 56,000 | 53,333 | 60,000 | 15,000 |
| Seat 7 | 48,571 | 56,000 | 53,333 | 60,000 | 15,000 |
| Total Seats | 3 | 2 | 1 | 1 | 0 |
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 is 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 resulting quotients sum to the required total of seats after being rounded off. One of a range of quotas will accomplish this, and applied to the above example of party lists this extends as integers from 112,000 to 120,000, the highest number always being twice that of the last average to which the Sainte-Lague method awards a seat if it is used rather than the Webster method; the lowest number always being twice that of the average to which the Sainte-Lague method would award the following seat (in the example the 8th seat).
Modified Sainte-Laguë method
Some countries, e.g. 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.