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The D'Hondt method (mathematically but not operationally equivalent to Jefferson's method) is a highest averages method for allocating seats in party-list proportional representation. The method described is named after Belgian mathematician Victor D'Hondt, who described it in 1878. In comparison with the Sainte-Laguë method, D'Hondt slightly favours large parties and coalitions over scattered small parties. There are two forms: closed list and open list, which differ in whether each party selects the order of election of the party's candidates, or if the voter's choice can affect the order.
Legislatures using this system include those of Albania, Argentina, Austria, Belgium, Brazil, Bulgaria, Cambodia, Cape Verde, Chile, Colombia, Croatia, Czech Republic, Denmark, East Timor, Ecuador, England, Estonia, Finland, Guatemala, Hungary, Iceland, Israel, Japan, Luxembourg, Republic of Macedonia, Republic of Moldova, Montenegro, the Netherlands, Paraguay, Peru, Poland, Portugal, Romania, Scotland, Serbia, Slovenia, Spain, Turkey, Uruguay and Wales.
The system has also been used for the 'top-up' seats in the London Assembly; in some countries during elections to the European Parliament; and during the 1997 Constitution-era for allocating party-list parliamentary seats in Thailand. A modified form was used for elections in the Australian Capital Territory Legislative Assembly but abandoned in favour of the Hare-Clark system. The system is also used in practice for the allocation between political groups of a large number of 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.
- V is the total number of votes that party received, and
- s is the number of seats that party has been allocated so far, initially 0 for all parties.
The total votes cast for each party in the electoral district is divided, first by 1, then by 2, then 3, right up to the total number of seats to be allocated for the district/constituency. Say there are p parties and s seats. Then create a grid of numbers, 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. The s winning entries are the s highest numbers in the whole grid; each party is given as many seats as there are winning entries in its row.
If 8 seats are to be allocated, divide each party's total votes by 1, then by 2, 3, 4, 5, 6, 7, and 8. An example is given in the grid below. The 8 highest entries are highlighted in bold, ranging from 100,000 down to 25,000. For each bold figure, the corresponding party gets a seat.
|denominator||/1||/2||/3||/4||/5||/6||/7||/8||Seats won (*)|
D'Hondt and Jefferson
The d'Hondt method is equivalent to the Jefferson method (named after the U.S. statesman Thomas Jefferson). They always give the same results, but the methods of presenting the calculation are different. Jefferson devised the method in 1792 for the U.S. congressional apportionment pursuant to the First United States Census; it was used to achieve the proportional distribution of seats in the House of Representatives among the states, rather than distributing seats in a legislature among parties pursuant to an election; but the tasks are mathematically equivalent, putting states in the place of parties and population in place of votes.
Jefferson's method uses a quota (called a divisor), as in the largest remainder method. The divisor is chosen as necessary so that the resulting quotients, disregarding any fractional remainders, sum to the required total; in other words, so that there is no need to examine the remainders. One of a range of quotas will accomplish this. Applied to the above example of party lists, this extends as integers from 20,001 to 25,000, the highest number always being the same as the last average to which the d'Hondt method awards a seat if it is used rather than the Jefferson method, and the lowest number being the next average plus one.
In some cases, a threshold or barrage is set, and any list which does not receive that threshold will not have any seats allocated to it, even if it received enough votes to have otherwise been rewarded with a seat. Examples of countries using this threshold are Denmark and Israel (2%); East Timor, Spain, and Montenegro (3%); Slovenia (4%); Czech Republic, Croatia, Romania, and Serbia (5%); Russia (7%); Turkey (10%); Poland (5%, or 8% for coalitions), Hungary (5% for single party, 10% for two party coalition, 15% for 3 party coalition or more) and Belgium (5%, on regional basis). In the Netherlands, a party must win enough votes for one full seat (note that this is not necessary in plain d'Hondt), which with 150 seats in the lower chamber gives an effective threshold of 0.67%. 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.
The method can cause a hidden threshold. It depends on the number of seats that are allocated with the D'Hondt method. In Finland's parliamentary elections, there is no official threshold, but the effective threshold is gaining one seat. The country is divided into districts with different numbers of representatives, so there is a hidden threshold, different in each district. The largest district, Uusimaa with 33 representatives, has a hidden threshold of 3%, while the smallest district, South Savo with 6 representatives, has a hidden threshold of 14%. This favors large parties in the small districts. In Croatia, the official threshold is 5% for parties and coalitions. However, since the country is divided into 10 voting districts with 14 elected representatives each, sometimes the threshold can be higher, depending on the number of votes of "fallen lists" (lists that don't get at least 5%). If many votes are lost in this manner, a list that gets 5% will still get a seat, whereas if there is a small number votes for parties that don't pass the threshold, the actual ("natural") threshold is close to 7.15%. One fourteenth of the votes (7.15%) guarantees at least one representative. But the "actual" threshold depends on how many votes "larger" parties got. If the total of votes won by parties who got the seats is less than 70%, then the effective threshold is 5%. But if the total number of votes is more than 70%, then the threshold is higher (1/14 of percentage won by elected parties), approaching the theoretical 7.15%.
Some systems allow parties to associate their lists together into a single cartel in order to overcome the threshold, while some systems set a separate threshold for cartels. Smaller parties often form pre-election coalitions to make sure they get past the election threshold. In the Netherlands, cartels (lijstverbindingen) cannot be used to overcome the threshold, but they do influence the distribution of remainder seats; thus, smaller parties can use them to get a chance which is more like that of the big parties.
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 is distributed proportionally (including to the list which has already received the majority bonus).
The d'Hondt method can also be used in conjunction with a quota formula to allocate most seats, applying the d'Hondt method to allocate any remaining seats to get a result identical to that achieved by the standard d'Hondt formula. This variation is known as the Hagenbach-Bischoff System, and is the formula frequently used when a country's electoral system is referred to simply as 'd'Hondt'.
In the election of Legislative Assembly of Macau, a modified d'Hondt method is used. The formula for the quotient in this system is . 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, National Assembly for Wales, 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.
- Pukelsheim, Friedrich (2007). "Seat bias formulas in proportional representation systems". 4th ecpr General Conference.
- Aurel Croissant and Daniel J. Pojar, Jr., Quo Vadis Thailand? Thai Politics after the 2005 Parliamentary Election, Strategic Insights, Volume IV, Issue 6 (June 2005)
- "D'Hondt system for picking NI ministers in Stormont". BBC. 11 May 2011. Retrieved 7 July 2013.
- Lijphart, Arend (2003), "Degrees of proportionality of proportional representation formulas", in Grofman, Bernard; Lijphart, Arend, Electoral Laws and Their Political Consequences, Agathon series on representation 1, Algora Publishing, pp. 170–179, ISBN 9780875862675. See in particular the section "Sainte-Lague", pp. 174–175.
- Oikeusministeriö. Suhteellisuuden parantaminen eduskuntavaaleissa. http://www.om.fi/uploads/p0yt86h0difo.pdf