Compared to ideal proportional representation, the D'Hondt method reduces somewhat the political fragmentation for smaller electoral district sizes,[1] where it favors larger political parties over small parties.
It was re-invented independently in 1878 by Belgian mathematician Victor D'Hondt, which is the reason for its two different names.
In general, exact proportionality is not possible because these divisions produce fractional numbers of seats.
[1][8] A method is consistent if it treats parties that received tied votes equally.
Monotonicity means that the number of seats provided to any state or party will not decrease if the house size increases.
Then a grid of numbers can be created, with p rows and s columns, where the entry in the ith row and jth column is the number of votes won by the ith party, divided by j.
Note that in Round 1, the quotient shown in the table, as derived from the formula, is precisely the number of votes returned in the ballot.
Each party's vote is divided by 1, 2, 3, or 4 in consecutive columns, then the 8 highest values resulting are selected.
The quantity of highest values in each row is the number of seats won.
(For example, 100,000/230,000 × 8 = 3.48) The slight favouring of the largest party over the smallest is apparent.
[10] A more mathematically detailed example has been written by British mathematician Professor Helen Wilson.
[11] The D'Hondt method approximates proportionality by minimizing the largest seats-to-votes ratio among all parties.
The D'Hondt method assigns seats so that this ratio attains its smallest possible value,
Thanks to this, as shown by Juraj Medzihorsky,[4] the D'Hondt method splits the votes into exactly proportionally represented ones and residual ones.
The method was first described in 1792 by Statesman and future US President Thomas Jefferson, in a letter to George Washington regarding the apportionment of seats in the United States House of Representatives pursuant to the First United States Census:[1] For representatives there can be no such common ratio, or divisor which ... will divide them exactly without a remainder or fraction.
I answer then ... that representatives [must be divided] as nearly as the nearest ratio will admit; and the fractions must be neglected.Washington had exercised his first veto power on a bill that introduced a new plan for dividing seats in the House of Representatives that would have increased the number of seats for northern states.
It was used to achieve the proportional distribution of seats in the House of Representatives among the states until 1842.
[14] It was also invented independently in 1878 in Europe, by Belgian mathematician Victor D'Hondt, who wrote in his publication Système pratique et raisonné de représentation proportionnelle, published in Brussels in 1882[citation needed]: To allocate discrete entities proportionally among several numbers, it is necessary to divide these numbers by a common divisor, producing quotients whose sum is equal to the number of entities to be allocated.The system can be used both for distributing seats in a legislature among states pursuant to populations or among parties pursuant to an election result.
Applied to the above example of party lists, this range extends as integers from 20,001 to 25,000.
The D'Hondt method reduces political fragmentation by allocating more seats to larger parties.
In Estonia, candidates receiving the simple quota in their electoral districts are considered elected, but in the second (district level) and third round of counting (nationwide, modified D'Hondt method) mandates are awarded only to candidate lists receiving more than the threshold of 5% of the votes nationally.
Obviously, the higher the vote threshold, the fewer the parties that will be represented in parliament.
In French municipal and regional elections, the D'Hondt method is used to attribute a number of council seats; however, a fixed proportion of them (50% for municipal elections, 25% for regional elections) is automatically given to the list with the greatest number of votes, to ensure that it has a working majority: this is called the "majority bonus" (prime à la majorité), and only the remainder of the seats are distributed proportionally (including to the list which has already received the majority bonus).
The term "modified D'Hondt" has also been given to the use of the D'Hondt method in the additional member system used for the Scottish Parliament, Senedd (Welsh Parliament), and London Assembly, in which after constituency seats have been allocated to parties by first-past-the-post, D'Hondt is applied for the allocation of list seats, taking into account for each party the number of constituency seats it has won (that is, if a party has won 3 constituency seats, the divisor for that party in the first round will be 4, rather than 1).
ABC elections analyst Antony Green has described the modified d'Hondt system used in the ACT as a "monster ... that few understood, even electoral officials who had to wrestle with its intricacies while spending several weeks counting the votes".
In a system of proportional representation in which the country is divided in multiple electoral districts, such as Belgium the threshold to obtain one seat can be very high (5% of votes since 2003), which also favors larger parties.
In most countries, seats for the national assembly are divided on a regional or even a provincial level.
The D'Hondt method is used to elect the legislatures in Åland, Albania, Angola, Argentina, Armenia, Aruba, Austria, Belgium, Bolivia, Brazil, Burundi, Cambodia, Cape Verde, Chile, Colombia, Croatia, the Dominican Republic, East Timor, Estonia, Fiji, Finland, Greenland, Guatemala, Hungary (in a mixed system), Iceland, Israel, Italy (in a mixed system), Japan, Luxembourg, Moldova, Monaco, Montenegro, Mozambique, Netherlands, Nicaragua, North Macedonia, Paraguay, Peru, Poland, Portugal, Romania, San Marino, Serbia, Slovenia, Spain, Switzerland, Turkey, Uruguay and Venezuela.
[25] The system is also used in practice for the allocation between political groups of numerous posts (vice presidents, committee chairmen and vice-chairmen, delegation chairmen and vice-chairmen) in the European Parliament and for the allocation of ministers in the Northern Ireland Assembly.
[26] It is also used to calculate the results in German and Austrian works council elections.