Dean method
The Dean method (also: divisor method with harmonic 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.
The American mathematician and astronomer James Dean proposed the process named after him in 1832 as a method for a population-proportional distribution of the seats in the House of Representatives among the states. However, it was never used for this purpose. The Dean procedure is one of the classic seat allocation procedures and plays an important role in the question of the optimal method of representation.
description
The calculation of a seat allocation according to the Dean 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 rounded off 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.
Harmonic rounding: For a harmonious rounding , the harmonic mean of the two numbers is calculated to which a number should be rounded up or down. The harmonic 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 harmonic mean is defined by the reciprocal arithmetic mean of the reciprocal feature values (numbers). The harmonic mean of 2 and 3 results from the reciprocal value (reciprocal value) of the arithmetic mean of 1/2 and 1/3 and is exactly 2.4. The number 2.41 rounded harmonically to a whole number results in 3. Since 2.41> 2.4 must be rounded up (2.41 is above the rounding limit 2.4).
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 non-whole numbers obtained are rounded to the nearest whole number:
Party A: 23; Party B: 18; Party C: 10; Party D: 1
Comment on party D: The harmonic mean of 0 and 1 results according to the above. Rule from the reciprocal of the arithmetic mean of the values 1/0 and 1 and amounts to 0. That is, every number> 0, no matter how small, is rounded up to 1 as an integer. As a result, according to the Dean procedure, each party receives a seat with just one vote - provided the number of parties with at least one vote is not greater than the total number of seats to be allocated.
Division by zero: Although division by the number zero is not mathematically possible, the quotient can still be regarded as a number of the size "infinite". The harmonic mean of 0 and 1 results from the reciprocal of the arithmetic mean of "infinite" and 1. The arithmetic mean results in "infinite". Its reciprocal is 0.
The sum of the harmonically rounded quotients is 52. The divisor of 20 is therefore 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 harmonious rounding gives the following seat allocation result:
Party A: 22 seats; Party B: 17 seats; Party C: 10 seats; Party D: 1 seat
Maximum payment method
Alternatively, as with any other divisor method, the seat allocation according to Dean can also be calculated on the basis of the corresponding maximum number 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 maximum number method according to Dean is:
0; 1 1/3; 2 2/5; 3 3/7; 4 4/9; 5 5/11; 6 6/13; 7 7/15; 8 8/17; 9 9/19 etc.
The divisors result from the harmonious mean of the successive seating claims. The divisor for the n-th seat is also the rounding limit between the n-th and (n + 1) -th seat according to the algorithm described above.
Division by zero: Although division by the number zero is not mathematically defined, the quotient can still be viewed as a number of the size "infinite". The first maximum number of each party with at least one vote is therefore "infinite", so that no party - no matter how large - is assigned a second seat before all others with at least one vote have received their first seat.
Comparison with the Hill Huntington's disease
The Dean method generates i. d. Usually the same distribution of seats as the Hill-Huntington method (divisor method with geometric rounding). This is due to the fact that the difference between the harmonic and geometric 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 Dean (first numerical value) and Hill-Huntington (second numerical value):
0/0; 1.3333 / 1.4142; 2.4000 / 2.4495; 3.4286 / 3.4641; 4.4444 / 4.4721; 5.4545 / 5.4772; 6.4615 / 6.4807; 7.4667 / 7.4833; 8.4706 / 8.4853; 9.4737 / 9.4868 etc.
In both procedures, the decimal places or rounding limits tend towards the decimal value 5, but will never quite reach it. The rounding limit according to Dean is always smaller than according to Hill-Huntington. The difference with the 10th divisor is only 0.0131.