# Sainte-Laguë method

The Sainte-Laguë method [ sɛ̃tlaˈɡy ] (in the Anglo-Saxon area Webster method; other names: divisor method with standard rounding , method of half fractions, method of odd numbers ) is a method of proportional representation (a seat allocation method ), as it is e.g. B. in elections with the principle of proportional representation (see proportional representation ) is required to convert votes into members of parliament.

## history

In 1832, the American politician Daniel Webster promoted the process as part of an investigation into the allocation of the US state's mandate claims in the US House of Representatives , but it failed to prevail - until it was finally used from 1880 to 1940.

The French mathematician André Sainte-Laguë was the first at the beginning of the 20th century to justify the method with the optimal fulfillment of the equal success value of the votes.

Since the 9th  legislative period (beginning of 1980), the procedure has been used in Germany at the suggestion of the physicist and Bundestag administrative employee Hans Schepers for the allocation of the committee seats of the German Bundestag. After the flare-up of specialist discussions at the end of the nineties, the use of the procedure is also gaining ground in legislative elections : it was and is used so far in the mayor elections in Bremen (since 2003) and Hamburg (since 2008) as well as the state elections in North Rhine-Westphalia (since 2010), Rhineland-Palatinate (2011) , Baden-Württemberg (2011) and Schleswig-Holstein (2012) , in federal elections since 2009 and in local elections in Bavaria (2020). The seats to which Germany is entitled in the European Parliament have also been allocated to the lists of the parties since 2009 according to this rule. Experts expect the procedure to be included in further federal and state electoral laws .

In Switzerland, the Sainte-Laguë method was used as part of the introduction of the biproportional divisor method ( double Pukelsheim ) to appoint parliaments in three cantons: Zurich (since 2006), Aargau and Schaffhausen (both in 2008). In these cantons, the standard rounding also applies to municipal elections - be it with or without “Pukelsheim”. In 2011, the canton of Basel-Stadt introduced the pure Sainte-Laguë procedure for the election of its parliament (Grand Council).

## Calculation method

The Sainte-Laguë method is a divisor or maximum number method and its system is therefore comparable with the D'Hondt method, among other things . However, while the D'Hondt method generally rounds off the seat claims (divisor method with rounding off), the Sainte-Laguë method uses the standard rounding (divisor method with standard rounding ) .

When using the maximum number method, the number of votes is not replaced by the numbers 1; 2; 3; ... but by 0.5; 1.5; 2.5; ... (alternatively by 1; 3; 5; ...) and the seats are allocated in the order of the highest resulting maximum number. As a result, the distribution distortions in favor of large parties, which are inherent in the D'Hondt process, do not occur. The allocation of seats to Sainte-Laguë is neutral to the strength of the parties.

The result of the Sainte-Laguë procedure can also be determined in other ways, which for each election result lead to the same seat allocation as the maximum number procedure described:

Divisor method
The votes of the parties are divided by a suitable divisor (votes per seat) and rounded according to standard rounding. If too many seats are distributed as a result, the calculation has to be repeated with a larger divisor, in the opposite case with a smaller divisor.
Rank metric method
When determining the composition of committees in the Bundestag, the Sainte-Laguë method is not used as a maximum number, but as a ranking method. By calculating the reciprocal of the respective maximum numbers and then multiplying by the total number of votes, one obtains measures of rank. The seats are allocated in the order of the lowest ranking metrics.

Due to the consistency of the procedure - which is given in all divisor procedures - the possible leaps in the Hare-Niemeyer procedure according to the Alabama paradox and the voter growth paradox inherent in all quota procedures are excluded.

## Calculation example according to the maximum number method

A total of 15 seats are available in a parliament.
10,000 votes have been cast, of which 5200 are for party X, 1700 for party Y and 3100 for party Z.
Now the number of votes for each party is multiplied by 0.5; 1.5; 2.5; ... divided, the results are listed. (In the example: 5200 divided by 0.5 results in 10,400.) Then the allocation is made: The highest number gets place 1, the second highest place 2, and so on, until all (here 15) places in parliament have been allocated. This results in the following picture:

divisor       Party X       Party Y       Party Z
0.5 1   10,400.00 4     3,400.00 2     6,200.00
1.5 3     3,466.67 10   1,133.33 6     2,066.67
2.5 5     2,080.00 680.00 8     1,240.00
3.5 7     1,485.71 485.71 12       885.71
4.5 9     1,155.56 377.78 15       688.89
5.5 11       945.45 309.09 563.64
6.5 13       800.00 261.54 476.92
7.5 14       693.33 226.67 413.33
8.5 611.76 200.00 364.71

Party X receives seats 1, 3, 5, 7, 9, 11, 13 and 14. A total of 8 of the 15 seats.
Party Y receives seats 4 and 10. A total of 2 of the 15 seats.
Party Z receives seats 2, 6, 8, 12 and 15. A total of 5 of the 15 seats.

## Calculation example using the divisor method

With the same input data, i.e. 15 seats to be allocated, 5200 votes for party X, 1700 for party Y and 3100 for party Z, the result is the same seat distribution by looking for a suitable divisor, which is then divided by rounding. Such a divisor is, for example, 685, because it results in

• 5200: 685 = 7.59 ..., rounded 8 seats for party X,
• 1700: 685 = 2.48 ..., rounded 2 seats for party Y,
• 3100: 685 = 4.52…, rounded 5 seats for party Z.

One can easily check that the same distribution of seats results with each divisor in the range from 680 (exclusive) to 688 89 (inclusive). With smaller divisors, on the other hand, there are too many seats overall, with larger ones too few seats. These limits also appear in the table above for the maximum number procedure: 688 89 is for the last, the 15th seat distributed, and 680 is the next maximum number, so it would be used in the distribution of a 16th seat. In particular, with the obvious quotient of the number of votes and the number of seats as a divisor (in the example with 666 23 ) and commercial rounding, the desired total number of seats is not obtained.

## Modified Sainte-Laguë method ("balanced method")

In Sweden, a modified Sainte-Laguë method , also known as the balanced method, is used. The first divisor is not 1, but 1.2 and the series of divisors is 1.2; 3; 5; 7 etc. (alternatively 0.6; 1.5; 2.5; 3.5 ...). As a result, the hurdle for small parties to get a mandate is higher, but still lower than with D'Hondt . In Sweden, the first divisor 1.4 was used until the 2014 election, but the change in the party landscape made it necessary to adjust this in order to continue to ensure proportional representation. In Norway, 1.4 is used as the first divisor for the distribution of constituency seats.