D'Hondt method
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The D'Hondt method^{[loweralpha 1]} (also known as Jefferson's method and BaderOfer method) is a highest averages method for allocating seats in partylist proportional representation. The method described is named after Belgian mathematician Victor D'Hondt, who described it in 1878. There are two forms: closed list (a party selects the order of election of their candidates) and an open list (voters' choices determine the order).
Proportional representation systems aim to allocate seats to parties in proportion to the number of votes received. For example, if a party wins onethird of the votes then it should gain onethird of the seats. In general, exact proportionality is not possible because these divisions produce fractional numbers of seats. As a result, several methods, of which the D'Hondt method is one, have been devised which guarantee that the parties' seat allocations are whole numbers that sum to the correct total, while aiming to preserve proportionality as far as is possible.^{[1]} In comparison with the SainteLaguë method, D'Hondt slightly favours large parties and coalitions over scattered small parties.^{[2]}
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, Estonia, Finland, Guatemala, Hungary, Iceland, Israel, Japan, Kosovo, Luxembourg, Macedonia, Moldova, Montenegro, Netherlands, Paraguay, Peru, Poland, Portugal, Romania, Scotland, Serbia, Slovenia, Spain, Turkey, Uruguay, and Wales.
The system has also been used for the 'topup' seats in the London Assembly; in some countries during elections to the European Parliament; and during the 1997 Constitutionera for allocating partylist parliamentary seats in Thailand.^{[3]} A modified form was used for elections in the Australian Capital Territory Legislative Assembly but abandoned in favour of the HareClark 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 vicechairmen, delegation chairmen and vicechairmen) in the European Parliament and for the allocation of ministers in the Northern Ireland Assembly.^{[4]}
Contents
Allocation
After all the votes have been tallied, successive quotients are calculated for each party. The formula for the quotient is^{[5]}
where:
 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.
Example
In this example, 230,000 voters decide the disposition of 8 seats among 4 parties. Since 8 seats are to be allocated, divide each party's total votes by 1, then by 2, 3, 4, 5, 6, 7, and 8. The 8 highest entries, marked with asterisks, range from 100,000 down to 25,000. For each, the corresponding party gets a seat.
For comparison, the "True proportion" column shows the fractional numbers of seats due, calculated in proportion to the number of votes received. (For example, 100,000/230,000 × 8 = 3.48) The slight favouring of the largest party over the smallest is apparent.
denominator  /1  /2  /3  /4  /5  /6  /7  /8  Seats won (*)  True proportion 

Party A  100,000*  50,000*  33,333*  25,000*  20,000  16,666  14,286  12,500  4  3.48 
Party B  80,000*  40,000*  26,666*  20,000  16,000  13,333  11,428  10,000  3  2.78 
Party C  30,000*  15,000  10,000  7,500  6,000  5,000  4,286  3,750  1  1.04 
Party D  20,000  10,000  6,666  5,000  4,000  3,333  2,857  2,500  0  0.70 
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, pick a number so that there is no need to examine the remainders.
Any number in one range of quotas will accomplish this, with the highest number in the range always being the same as the lowest number used by the D'Hondt method to award a seat if it is used rather than the Jefferson method, and the lowest number in the range being the next highest number in the D'Hondt calculations plus one.
Applied to the above example of party lists, this range extends as integers from 20,001 to 25,000.
Threshold
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 the D'Hondt method with a threshold are Albania (3% for single parties, 5% for two or more party coalitions, no threshold is applied for independent individuals); Denmark (2%); East Timor, Spain, and Montenegro (3%); Israel (3.25%); Slovenia (4%); Czech Republic, Croatia, Romania, and Serbia (5%); Russia (7%); Turkey (10%); Poland (5%, or 8% for coalitions; but does not apply for ethnicminority parties), 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%.^{[6]} 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 preelection 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).
Variations
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 HagenbachBischoff 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 firstpastthepost, d'Hondt is applied for the allocation of list seats taking into account for each party the number of constituency seats it has won.
Regional d'Hont
In some countries such as Spain or Czech Republic, seats for the National assembly are divided on a regional or even provincial level. This means the d'Hont method is applied to the total number of votes in each province which then divides up the limited number of seats. The votes for parties that have not gained one of the limited number of seats are thus discarded meaning they do not aggregate at a national level. This means that parties which have gained a 5% of the vote nationally may still end up with no seats as they did not gain enough votes in each provincial jurisdiction. This may also lead to skewed seat allocation at a national level such as in Spain in 2011 where the PP party gained an absolute majority in the National Assembly with only 44% of the national vote.^{[7]}
Notes
 ↑ The name D'Hondt is sometimes spelt as "d'Hondt". For example it is customary in the Netherlands to write such surnames with a lowercase "d" when preceded by the forename: thus Victor d'Hondt (with a small d), while the surname all by itself would be D'Hondt (with a capital D). However, in Belgium it is always capitalized, hence: Victor D'hondt.
References
 ↑ Gallagher, Michael. Proportionality, Disproportionality and Electoral Systems. Trinity College, Dublin.
 ↑ Pukelsheim, Friedrich (2007). "Seat bias formulas in proportional representation systems" (PDF). 4th ECPR General Conference.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ 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.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. See in particular the section "SainteLague", pp. 174–175.
 ↑ Oikeusministeriö. Suhteellisuuden parantaminen eduskuntavaaleissa.
 ↑ https://www.tcd.ie/Political_Science/staff/michael_gallagher/ElectoralStudies1991.pdf