Hill Huntington's disease
The Hill-Huntington method (also: divisor method with geometric rounding) is a method of proportional representation ( 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
The chief statistician of the US Bureau of the Census, Joseph A. Hill, proposed the procedure in a letter to the House of Representatives in 1911 for the population-based distribution of his seats in the states . The American mathematician and physicist Edward Vermilye Huntington supported Hill's proposal in 1921. Since 1941 the procedure named after both persons has been applicable to the above. Purpose required by law. Until then, the Webster method (in Germany Sainte-Laguë / Schepers method ) was used. Until 1900, the size of the House of Representatives was chosen in such a way that seat allocation was based on the Webster method and the Hamilton method (in GermanyHare-Niemeyer method ) was identical. Until 1880 only the Hamilton process was used, until 1840 the Jefferson process (in Germany D'Hondt process ).
Since 1911, the House of Representatives has consisted of 435 seats, the distribution of which across the states is redefined in all years divisible by 10 on the basis of a nationwide census. When a new state joins, additional seats are temporarily assigned to it until the next census.
In the first step, the ratios of the population of the states to the nationwide population are determined and multiplied by the 435 seats to be distributed. This results in the ideal seating requirements of the states. Due to the lack of integers, the ideal claims must be rounded to whole numbers that ultimately represent the actual seat claims. The Hill-Huntington method has been used to calculate the rounding since the 1950 census.
description
The calculation of a seat allocation according to the Hill-Huntington method is explained below using the example of a proportional representation.
Seats to be allocated: 50
Valid votes cast: 1000
Party A: 450 votes, Party B: 350 votes, Party C: 199 votes, Party D: 1 vote
The votes of the parties are divided by a suitable divisor . This must be determined empirically (by trial and error). The quotient of votes cast and seats to be allocated can be used as a benchmark for the divisor , in the example 20. The quotients resulting from the division are geometrically rounded to whole numbers. If these integers add up to 50, then the seat allocation calculation is correct. Each party receives seats equal to the whole number calculated for them. If the sum of these whole numbers is more or less than 50, the divisor is unsuitable and must be increased or decreased - until exactly 50 seats are distributed.
Geometric rounding: For a geometric rounding , the geometric mean of the two numbers to which a number should be rounded up or down is calculated. The geometric mean forms the rounding limit: If the number to be rounded is above the rounding limit, it is rounded up, otherwise it is rounded down. The geometric mean of two numbers is the square root of the product of the two numbers. The geometric mean of 2 and 3 is therefore the square root of 2 × 3 with a value of around 2.4495. The number 2.45 rounded geometrically results in 3 because 2.45 is greater than the rounding limit of 2.4495.
Division of the number of votes of the parties by the divisor 20:
Party A: 22.5; Party B: 17.5; Party C: 9.95; Party D: 0.05
The numbers obtained are geometrically rounded to whole numbers:
Party A: 23; Party B: 18; Party C: 10; Party D: 1; Total: 52
Comment on party D: The geometric mean of 0 and 1 results according to the above. Rule from the square root of 0 × 1 and is 0. Therefore, every positive number, no matter how small, is rounded up to 1 and the Hill-Huntington method gives each party a seat with just one vote - provided the number of parties indicates at least one vote is not greater than the total number of seats to be allocated.
The sum of the geometrically rounded quotients is 52, so the divisor of 20 is too small. A suitable divisor is 20.5. The following quotients result:
Party A: 21.95; Party B: 17.07; Party C: 9.71; Party D: 0.05
The geometric rounding gives the following seat allocation result:
Party A: 22 seats; Party B: 17 seats; Party C: 10 seats; Party D: 1 seat; Total: 50
Maximum payment method
Alternatively, the seat allocation according to Hill-Huntington can also be calculated on the basis of the corresponding maximum number method, as with any other divisor method. The number of votes of the parties is divided by a series of divisors . The resulting quotients are called maximum numbers . The seats are distributed to the parties in the order of the highest maximum number. This algorithm is more complex than the one described above. The advantage is that in the event of an enlargement or reduction of the committee to be elected by z. B. 1 seat can see at first glance which party would receive an additional seat or would have to do without a seat.
The divisor series for the Hill-Huntington maximum number method is:
0; 1.4142; 2.4495; 3.4641; 4.4721; 5.4772; 6.4807; 7.4833; 8,4853; 9.4868 etc.
The divisors result from the geometric mean of the successive seating claims. The divisor for the n th seat is at the same time the rounding limit between the n th and the ( n + 1) th seat according to the algorithm described above.
Division by zero: Although a division by 0 is generally not mathematically defined, in this specific case the quotient can be regarded as infinite. The first maximum number of each party with at least one vote is therefore infinite, so that no party - no matter how large - gets a second seat until all the others with at least one vote have received their first seat.
Comparison with the Dean method
The Hill-Huntington's disease generates i. d. Usually the same distribution of seats as the Dean method (divisor method with harmonic rounding). This is because the difference between the geometric and harmonic mean of two consecutive whole numbers is very small and tends towards zero as the size of the number pairs increases. Accordingly, the rounding limits and the divisor series for the maximum number method are very close to each other in both methods.
To illustrate a comparison of the divisor series according to Hill-Huntington (first numerical value) and Dean (second numerical value):
0/0; 1.4142 / 1.3333; 2.4495 / 2.4000; 3.4641 / 3.4286; 4.4721 / 4.4444; 5.4772 / 5.4545; 6.4807 / 6.4615; 7.4833 / 7.4667; 8.4853 / 8.4706; 9.4868 / 9.4737 etc.
In both procedures, the decimal places or rounding limits tend towards the decimal value 5, but will never quite reach it. The Hill-Huntington rounding limit is always greater than that according to Dean. The difference with the 10th divisor is only 0.0131.