D'Hondt method

The D' Hondt method (after the Belgian lawyers Victor D' Hondt, also Divisor with rounding, in Anglo-Saxon: Jefferson procedures in Switzerland: Hagenbach -Bischoff method ) is a method of proportional representation ( seat allocation method ) as it is (see proportional representation ) is needed for example in elections with the principle of distribution proportional to convert votes into mandates deputies.

The method can be used in the form of five mathematically equivalent algorithms or variants always generate the same seat allocation result:

• As a two- step procedure
• As a highest averages method,
• As Rangmaßzahlverfahren,
• As Pairwise Comparison method or
• Described as a quasi- ratio method as the Swiss physicist Eduard Hagenbach - Bischoff.

In the USA, the future president Thomas Jefferson with rounding in 1792 made ​​based on the eponymous Divisorverfahrens a proposal for population- proportional distribution of seats in the U.S. House of Representatives to the states. The method was used until 1840, when it was replaced by the Hamiltonian method (Designation in the English-speaking for the Hare- Niemeyer method), less disadvantaged the smaller parties.

In Germany, the D' Hondt method was used up to and including 1983 for calculating the distribution of seats in elections to the German Bundestag in the 1987 election, however, by the Hare- Niemeyer method replaced (see above, the development in the U.S. 147 years ago ). The D' Hondt method had been replaced already in 1970 when calculating the Committee Instrumentation For the same reason, and by the same method.

In elections to some state parliaments, local councils, judicial selection committees or councils, the D' Hondt method is still used today, but - because of its distorting effect of proportional (systematic disadvantage smaller parties, see below) - also becoming increasingly rare.

In Austria, the D' Hondt method in the third investigation in elections to the National Council shall be applied (see NRWO ).

Example of calculation

If the election of a committee to multiple parties, the proportional seat share on the basis of the voting share ( Ideal claim ) is integer only in rare cases. Therefore, a method for calculating an integer number of seats is necessary that each party receives in the body.

When using the d' Hondt method of dividing the number of votes received by a party in succession by an increasing sequence of natural numbers (1, 2, 3, 4, 5, ..., n). The fractions obtained are referred to as maximum numbers. So here the original " number of votes " - - As the basis of this division (dividend ) is always the starting number is used here. The dividend is always the same in each column and is by the changing divisor ( here: 1, 2, 3, ...) shared.

The maximum figures are then sorted in descending order by size. The determined sequence specifies the order of award seats. There will be so many maximum numbers into account, as seats are to be awarded on the committee. In this example, 10 seats are allocated. The 10 largest maximum numbers (darker shading) are descending distributed by size of their associated parties. The last and smallest maximum number for which a party still receives a seat, specifies the value representation (also representing weight ) of their seats. The representative value is the ratio of votes and number of seats a party. Party A represents to each seat 104, Party B and Party C 84.5 123 voters. Not only in absolute terms but also in relation to their share of the vote party B is in the body much more strongly represented than party C.

When using the two-stage procedure, the number of votes of all parties by a suitable (not necessarily all) of the number are (divisor ) is divided and rounded the results. The number can be determined by trial and error. It is at most equal to that maximum number, which leads to a mandate as the last. This maximum number is always appropriate. Each number which leads to the correct total number of seats is suitable. In the example, the seat allocation also results in division by 84, that is, for each full 84 votes each party receives a seat

Properties

Error minimization ( minimax criterion)

D' Hondt maximizes the minimum ( lowest ) value representation ( votes per seat). That is, for a given election result there is no other seat allocation method in which the votes - seats relationship of the party is higher with the lowest votes - seat ratio than the votes - seats relationship of the party with the lowest votes - seat ratio by D' Hondt.

Conversely, the representative value determining the success value as the ratio of seats per vote for a party (the reciprocal of the representative value). Consequently, D' Hondt minimizes the maximum ( highest ) value of success (seats per voice ).

Majority condition

D' Hondt meets the majority requirement, but not the minority condition. That is, a party that unites at least 50 % of the vote will also receive at least 50 % of the seats. Conversely, a party that does not by at least 50 % of the vote, while 50 % of the seats obtained when all the other parties have a worse outcome votes.

The fulfillment of the majority condition is "bought" by the systematic preference for larger parties. If, however, to ensure that one party receives an absolute majority, ie at least one vote more than half of the votes, an absolute majority of seats, with the seat number must be odd.

That D' Hondt with even total number of seats the absolute majority condition is fulfilled not in principle, the following example shows: Number assigned unit seats: 10 Number of Votes of valid votes: 1000 Party A: . 505 votes, Party B 495 votes. As a result, both parties so do not get 5 seats and Party A, the absolute majority of (at least) 6 seats.

The problem would be eliminated by the party assigned by an absolute majority if it has not received an absolute majority of seats, an extra seat, and the total number of seats is made ​​so odd. However, if the total number of seats of the Board be an even number in all circumstances, a provision should be made, after which the largest party receives a basic seat and only the remaining seats are distributed to D' Hondt, which would create an additional proportional distortion.

Rate condition

As with all other Divisor, the rate condition can be violated ( see extreme example in the next section ), after which the number of seats a party only by less than 1 of their ideal claim or their ratio ( votes times divided by total number of votes number of seats ) should be changed to:

• According to the D' Hondt method can not only obtain a (large ) party to the next whole number rounded upwards seat claim, but even one or more seats beyond;

• However, the reverse is not possible because the method does not, but rather fulfills the quota condition after top to bottom; that is, no (small ) party may receive less seats than corresponds to their rounded quota.

Disadvantage of smaller parties

The seat allocation may differ significantly from proportionality ( proporzverzerrende effect in the form of systematic discrimination against smaller parties ). This effect is enhanced by large differences in party strength, a high number antretender parties and a low number to forgiving seats.

Extreme example: number assigned unit seats: 10 Number of Votes of valid votes: 1000th Party A wins 600 votes, 7 other parties win together 400 voices ( including no more than 59 ). As a result, party A receives a vote share of 60 % every 10 seats.

General rule, for n to be awarded seats, the largest party receives all n seats if their share of the vote more than n times larger than that of the second largest party. Thus, the strongest party can obtain all the seats in an arbitrarily small share of the vote if the party number is correspondingly large. If the vote share of the largest party exactly n times as large as the second strongest, both parties have the same right to the nth seat, which consequently needs to be raffled.

Comparison with the Hare - Niemeyer method and the Sainte- Laguë method

The example of the Schleswig-Holstein state election in 2005 can be illustrated that the D' Hondt method smaller parties at a disadvantage against larger, the Hare- Niemeyer method and the Sainte- Laguë method does not. In Schleswig-Holstein the D' Hondt method was used until 2009 in state elections; from 2012 is considered the Sainte- Laguë method.

After the official provisional result, the distribution of seats was by the two methods as follows:

The relative deviation from the ideal claim indicates the percentage by which deviates the representation of a party with seats in parliament of their won in the election vote share:

• Is the relative deviation from the ideal positive claim, the party obtained by the seat allocation method an advantage because it is more strongly represented in the Parliament, as it corresponds to their share of the vote;
• Is the relative deviation from the ideal negative claim, the party obtained by the seat allocation process at a disadvantage because it is weakly represented in the Parliament, as it corresponds to their share of the vote.

Multiple application

Particularly problematic is the application of the D' Hondt method, when the total electoral area divided into sub-regions and in each a fixed number of deputies is elected. The application of the D' Hondt method then leads to the number of sub-areas to a multiplication of the effect of disadvantage smaller parties.

The parliamentary elections in 1949 and 1953 were such cases. Each state was formed (1953 apart from the locking clause control) a self-contained, independent electoral area in which twice as many MP ( plus potential overhang mandates) were chosen as there were constituencies.