# D'Hondt method

The D'Hondt method (after the Belgian lawyer Victor D'Hondt ; also divisor method with rounding , in the Anglo-Saxon area: Jefferson method , in Switzerland : Hagenbach-Bischoff method ) is a method of proportional representation ( seat allocation method ), such as she z. B. in elections with the principle of proportional representation (see proportional representation ) is required to convert votes into members of parliament.

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

• as a two-step process,
• as a maximum payment method,
• as rank measure method,
• as a paired comparison procedure or
• as a quasi-quota method as described by the Swiss physicist Eduard Hagenbach-Bischoff .

## history

In the US, later made President Thomas Jefferson , based on the eponymous Divisorverfahrens with rounding in 1792 a proposal for the population proportional distribution of seats in the US House of Representatives to the individual states . The process was replaced in 1840 by the Hamilton process (term in the Anglo-Saxon language for the Hare-Niemeyer process ), which puts smaller parties - or in this case states - less at a disadvantage.

In Germany, the D'Hondt procedure was used until 1985 to calculate the distribution of seats in elections to the German Bundestag . It was replaced by the Hare-Niemeyer procedure, which was replaced by the Sainte-Laguë procedure in 2008 .

The D'Hondt procedure is still used today in elections to some state parliaments , municipal councils , judges' committees or works councils . It is used in state elections in Lower Saxony, Saxony and Saarland. The D'Hondt procedure used to apply in almost all countries. North Rhine-Westphalia is the only West German state in which it was never used in state elections.

In Austria , the D'Hondt procedure is used in the third preliminary investigation in elections to the National Council (see NRWO ), in university elections and in works council elections .

In the elections to the European Parliament , the D'Hondt procedure is used in most of the countries to allocate the national parliamentary seats.

## Calculation example

Political party Number of
Percentage
Seats
proportionally
Sit after
d'Hondt
Party A 416 41.6% 4.16 4th
Party B 338 33.8% 3.38 4th
Party C 246 24.6% 2.46 2
1000 100.00% 10 10
in the election of a 10-member committee
divisor Party A Party B Party C
1 416 (1) 338 (2) 246 (3)
2 208 (4) 169 (5) 123 (7)
3 138.7 (6) 112.7 (8) 82
4th 104 (9) 84.5 (10) 61.5
5 83.2 67.6 49.2
6th 69.3 56.3 41
Determination of the maximum numbers (the values
in brackets correspond to the order of assignment)

If several parties stand for the election of a body , the proportional seat share based on the share of the vote ( ideal claim ) is only in rare cases in whole numbers. Therefore, there is a need for a method of calculating an integer number of seats that each party on the panel receives.

When using d'hondt's maximum number method , the number of votes received by a party is divided successively by an ascending sequence of natural numbers (1, 2, 3, 4, 5, ..., n). The fractional numbers obtained are called maximum numbers. The starting number - in this case the original "number of votes" - is always used as the basis for this division (dividend). The dividend always remains the same in each column and is divided by the changing divisor (here: 1, 2, 3, ...).

The maximum numbers are then sorted in descending order according to their size. The order determined in this way indicates the order in which the seats are allocated. As many maximum numbers are taken into account as there are seats to be allocated on the committee. In this example 10 seats are allocated. The 10 largest maximum numbers (shaded) are distributed to the parties assigned to them in descending order according to their size. The last or smallest maximum number for which a party still receives a seat indicates the representation value (also representation weight) of its seats. The representation value is the ratio of the number of votes and seats of a party. Party A represents 104 voters with each seat, party B 84.5 and party C 123 voters. Not only in absolute terms, but also in relation to their share of the vote, party B is represented much more strongly than party C.

When using the two-stage procedure , the number of votes from all parties is divided by a suitable (not necessarily whole) number (divisor) and the results rounded off. The number can be determined by trying. It is at most equal to the maximum number that is the last to lead to a mandate. This maximum number is always suitable. Any number that results in the correct total number of seats is appropriate. In the example, the allocation of seats is also obtained by dividing by 84, i.e. for every full 84 votes, each party receives one seat.

## properties

### Minimizing errors (minimax criterion)

D'Hondt maximizes the minimum (lowest) representation value (votes per seat). I.e. If the election results are given, there is no other seat allocation procedure in which the vote-seat ratio of the party with the lowest vote-seat ratio is higher than the vote-seat ratio of the party with the lowest vote-seat ratio according to D'Hondt .

Conversely to the representation value , the success value is determined as the ratio of seats per vote for a party ( reciprocal value of the representation value ). Consequently, D'Hondt minimizes the maximum (highest) success value (seats per vote).

The party's success score is defined as ${\ displaystyle p \ in \ {1, \ dots, P \}}$

${\ displaystyle e_ {p} = {\ frac {s_ {p}} {v_ {p}}},}$

Where

${\ displaystyle s_ {p}}$- the seat share of the party , ,${\ displaystyle p}$${\ displaystyle s_ {p} \ in [0,1], \; \ sum _ {p} s_ {p} = 1}$
${\ displaystyle v_ {p}}$- the share of the vote the party , .${\ displaystyle p}$${\ displaystyle v_ {p} \ in [0,1], \; \ sum _ {p} v_ {p} = 1}$

The highest success score is defined as

${\ displaystyle \ delta = \ max _ {p} e_ {p}.}$

D'Hondt assigns seats in such a way that the success value is as low as possible and reaches the value

${\ displaystyle \ delta ^ {*} = \ min _ {\ mathbf {s} \ in {\ mathcal {S}}} \ max _ {p} a_ {p}}$,

where is a distribution of seats to the parties and the set of all possible seat distributions. Thanks to this function, D'Hondt divides the votes into exactly proportioned votes and remaining votes, which minimizes the proportion of remaining votes. The total share of the remaining votes is ${\ displaystyle \ mathbf {s} = \ {s_ {1}, \ dots, s_ {P} \}}$${\ displaystyle {\ mathcal {S}}}$

${\ displaystyle \ pi ^ {*} = 1 - {\ frac {1} {\ delta ^ {*}}}}$.

The party's remaining share of the votes is calculated as follows ${\ displaystyle p}$

${\ displaystyle r_ {p} = v_ {p} - (1- \ pi ^ {*}) s_ {p}, \; r_ {p} \ in [0, v_ {p}], \ sum _ {p } \, r_ {p} = \ pi ^ {*}}$.

For example, the three parties with 416, 338 and 246 votes that received 4, 4 and 2 seats. Your success values ​​are 0.96, 1.18, 0.81. The highest success value is 1.18. As a result, the fraction of the remaining votes is 1 - 1 / 1.18 = 0.155 or 15.5%. The remaining votes of the parties are 7.8%, 0% and 7.7%. This is shown in the table below.

Political party Percentage
Percentage
of seats according to
D'Hondt
Success value Remaining voices
after D'Hondt
Represented
voices
after D'Hondt
Party A 41.6% 40% 0.96 7.8% 33.8%
Party B 33.8% 40% 1.18 0.0% 33.8%
Party C 24.6% 20% 0.81 7.7% 16.9%
100.0% 100.0% - 15.5% 84.5%
Distribution of votes in the election of a 10-member committee

### Majority condition

D'Hondt meets the majority condition , but not the minority condition . I.e. a party that receives at least 50% of the votes also receives at least 50% of the seats. Conversely, a party that does not get at least 50% of the votes can still get 50% of the seats if all other parties have a poorer voting result.

Fulfilling the majority condition is “bought” through the systematic preference given to larger parties. However, if it is to be ensured that a party with an absolute majority of votes, i.e. more than half of the votes, also receives the absolute majority of the seats, the total number of seats must be odd .

The following example shows that D'Hondt does not fundamentally meet the absolute majority requirement with an even total number of seats: Number of seats to be allocated: 10, number of valid votes cast: 1000. Party A: 505 votes, Party B 495 votes. As a result, both parties get 5 seats and party A does not get an absolute majority of (at least) 6 seats.

The problem could be eliminated by allocating an additional seat to the party with an absolute majority of votes, if it did not receive an absolute majority of the seats, thereby making the total number of seats odd. However, if the total number of seats in the body is to be an even number under all circumstances, a regulation would have to be made according to which the largest party receives a basic seat and only the remaining seats are allocated to D'Hondt, which would create an additional distortion of proportional representation.

### Quota condition

As with all other divisor procedures, the quota condition can be violated (see extreme example in the next section), according to which the number of seats of a party should only deviate by less than 1 from its ideal claim or its quota (number of votes times number of mandates divided by total number of votes):

• According to the D'Hondt procedure, a (large) party can not only receive the seat entitlement rounded up to the nearest whole number, but even one or more seats in addition;
• however, the reverse case is not possible, since the procedure does not meet the quota condition upwards , but downwards ; d. H. no (small) party can get fewer seats than its quota rounded down.

### Discrimination against smaller parties

The allocation of seats can deviate significantly from proportionality (proportionality distorting effect in the form of systematic disadvantage for smaller parties). This effect is promoted by large differences in party strengths, a high number of parties standing and a low number of seats to be allocated.

Extreme example: Number of seats to be allocated: 10, number of valid votes cast: 1000. Party A wins 600 votes, 7 other parties together win 400 votes (none of them more than 59). As a result, party A receives all 10 seats with a share of 60% of the votes and one seat represents 60 votes of party A.

In general: If there are n seats to be allocated, the strongest party receives all n seats if its share of the vote is more than n times greater than that of the second strongest party. This means that the strongest party can get all the seats with any number of votes, provided the number of parties is correspondingly large. If the share of the vote of the strongest party is exactly n times as large as that of the second strongest, both parties have the same right to the nth seat, which consequently has to be raffled.

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

Using the example of the Schleswig-Holstein state elections in 2005, it can be illustrated that the D'Hondt process puts smaller parties at a disadvantage compared to larger parties, but not the Hare-Niemeyer process and the Sainte-Laguë process . Depending on the point of view, one could also say that the Hare-Niemeyer process and the Sainte-Laguë process favors smaller parties, as one seat represents fewer votes for them. In Schleswig-Holstein, the D'Hondt procedure was used in state elections until 2009; the Sainte-Laguë procedure has been in force since 2012.

According to the preliminary official result , the allocation of seats according to the two procedures was as follows:

Political party Number of votes Distribution of seats Relative deviation from the ideal
Ideal claim D'Hondt Hare-Niemeyer Sainte-Laguë D'Hondt Hare-Niemeyer Sainte-Laguë
CDU 576.100 29,077 30th 29 29 +3.175% −0.265% −0.265%
SPD 554,844 28.004 29 28 28 + 3.556% −0.015% −0.015%
FDP 94,920 4.791 4th 5 5 −16.507% +4.367% +4.367%
Green 89,330 4,509 4th 4th 4th −11.282% −11.282% −11.282%
SSW 51.901 2,620 2 3 3 −23.651% +14.524% +14.524%
total 1,367,095 69 69 69 69

The relative deviation from the ideal indicates the percentage by which the representation of a party with MPs in parliament deviates from its share of the votes won in the election:

• if the relative deviation from the ideal is positive, the party gains an advantage through the seat allocation procedure, since it is more represented in parliament than it corresponds to its share of the vote;
• if the relative deviation from the ideal is negative, the party is at a disadvantage through the seat allocation procedure, since it is less represented in parliament than its share of the vote corresponds to.
D'Hondt Hare-Niemeyer Sainte-Laguë D'Hondt Hare-Niemeyer Sainte-Laguë
CDU 576.100 30.11 30th 29 29 19.203 19,866 19,866
SPD 554,844 29.00 29 28 28 19,133 19,816 19,816
FDP 94,920 4.96 4th 5 5 23,730 18,984 18,984
Green 89,330 4.67 4th 4th 4th 22,333 22,333 22,333
SSW 51.901 2.71 2 3 3 25,951 17,300 17,300
total 1,367,095 71.45 69 69 69 19,813 19,813 19,813

### Multiple use

The use of the D'Hondt procedure can lead to a significantly disproportionate distribution of seats if the entire electoral area is divided into sub-areas and a fixed number of members is elected there, especially if few seats are available. The application of the D'Hondt method then leads to an increase in the disadvantageous effect of smaller parties, depending on the number of sub-areas. Such an allocation procedure exists in Switzerland and in many other countries, including Spain, Portugal, Belgium, Poland and Finland. In some of these countries threshold clauses exist either nationally or only at the constituency level. In Spain, the distribution of seats in the House of Representatives is particularly disproportionate due to the largely small constituencies. In Switzerland, the option of list connections reduces the disadvantages for small parties.

In the federal elections in 1949 and 1953, each federal state (apart from the threshold rule in 1953 ) formed a self-contained, independent electoral area.