# Hare-Niemeyer method

The Hare-Niemeyer procedure [ hɛəˈniːmajɐ ] (in Austria only "Hare'sches procedure", in the Anglo-Saxon area "Hamilton procedure"; also "quota procedure with residual settlement according to largest fractions") is a seat allocation procedure . It is used, for example, in elections with the principle of proportional representation (see proportional representation ) to convert votes into members of parliament .

## History and application

The method was propagated by the US politician Alexander Hamilton before the first US census in 1790, in order to distribute the seats in the US House of Representatives proportionally to the population among the individual states . However, it could not prevail against the use of D'Hondt's method . After the census in 1840, the Hamilton method was finally used and it was used for the last time in the census in 1890.

The procedure has been used to calculate the distribution of seats in the German Bundestag from the 1987 election until the 2005 election . The name used in Germany is derived from the London lawyer Thomas Hare and the German mathematician Horst F. Niemeyer , who had proposed the procedure in a letter to the Bundestag Presidium in October 1970 for the allocation of seats in the Bundestag committees .

In 2008, the Hare-Niemeyer process was replaced by the Sainte-Laguë process for federal elections .

The Hare-Niemeyer procedure is used in state elections in Bavaria, Berlin, Brandenburg, Hesse, Mecklenburg-Western Pomerania, Saxony-Anhalt and Thuringia.

Outside Germany, the Hare-Niemeyer process is used in Denmark, Italy, Greece and the Ukraine, among others. It is not used in Austria, in Switzerland only in the cantons of Vaud and Ticino . The Hare-Niemeyer procedure is usually used in Austria and Switzerland to distribute seats to constituencies proportionally to their population.

## Calculation method

The quota (the ideal seat entitlement) of each party is calculated according to the rule of three , and all quotas are rounded to whole numbers without adding the sum .

${\ displaystyle {\ frac {{\ text {Total number of seats}} \ cdot {\ text {Number of party votes}}} {\ text {Total number of votes}}} = {\ text {Quota}}}$

or, in other words:

${\ displaystyle {\ frac {\ text {number of party votes}} {\ text {total number of votes}}} = {\ frac {\ text {quota}} {\ text {total number of seats}}}}$

The second form makes it clear that the proportion of votes corresponds to the proportion of seats, with the number of seats rounded up or down according to the largest remainder. In practice it looks like this:

Each party is initially allocated seats equal to their quota rounded down. The remaining seats are allocated in the order of the highest divisional remainder of the quotas. If the remaining division is the same, the election supervisor will decide the lot. If applicable, only the votes of the parties that do not fall under a threshold clause are taken into account .

Political party be right Quota Seats
without a threshold clause
A. 216 22.34 22nd +1
B. 310 32.07 32
C. 22nd 2.28 2
D. 32 3.31 3
total 580 60.00 59 +1
with threshold clause
A. 216 23.22 23
B. 310 33.33 33
C.
D. 32 3.45 3 +1
total 558 60.00 59 +1

An example:

60 seats are to be allocated, which are to be distributed among four parties (A, B, C and D). A total of 580 votes were cast, which are distributed as shown in the table. This results in the following distribution of seats: In the first round, party A receives 22, party B 32, party C 2 and party D 3 seats. the remaining seat is assigned to party A, which has the highest decimal place.

If party C is not entitled to any seats due to a threshold clause, for example because of a 5% hurdle, the total number of votes to be taken into account is reduced by 22 to 558, and the respective quotas of the other parties A, B and D would be given by dividing by 558 to calculate. Party A would get 23, party B 33, party C 0 and party D 3 seats. In this case the remaining seat would go to party D, which now has the highest decimal place.

## properties

The Hare-Niemeyer procedure behaves neutrally with regard to the size of the parties, since the voting share (percentage of own votes from the total number of votes) is equal to the seat percentage (percentage of own seats from the total number of seats). In this way, it ensures compliance with the principle of equal choice. In contrast, other proceedings favor larger parties and disadvantage smaller ones (in particular the D'Hondt proceedings , in Switzerland Hagenbach-Bischoff proceedings ) or vice versa (in particular the Adams proceedings ).

The Hare-Niemeyer process is - like all quota processes - characterized by the inviolability of the quota condition : According to this, no party can receive more seats than its quota rounded up to the nearest whole number. At the same time, no party can receive fewer seats than its quota rounded down to the nearest whole number. This advantage does not exist with the Sainte-Laguë procedure . In the D'Hondt method , the quota condition is only met at the bottom, i.e. only the second of the above conditions.

The disadvantage of the procedure is the inconsistency resulting from the quota condition; the following paradoxes can arise:

## Individual evidence

1. Ilka Agricola , Friedrich Pukelsheim : Horst F. Niemeyer and the proportional method , Mathematical Semester Reports , Volume 64, 2017, pp. 129–146, doi : 10.1007 / s00591-017-0201-8 , online (freely accessible)