# D'Hondt method

The D'Hondt method[a] or the Jefferson method is a highest averages method for allocating seats, and is thus a type of party-list proportional representation. The method described is named in the United States after Thomas Jefferson, who introduced the method for proportional allocation of seats in the United States House of Representatives in 1791, and in Europe after Belgian mathematician Victor D'Hondt, who described it in 1878 for proportional allocation of parliamentary seats to the parties. 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 approximately in proportion to the number of votes received. For example, if a party wins one-third of the votes then it should gain about one-third 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 ensure that the parties' seat allocations, which are of whole numbers, are as proportional as possible.[1] In comparison, Sainte-Laguë method, a divisor method, reduces the reward to large parties, and it generally has benefited middle-size parties at the expense of both large and small parties.[2] Meanwhile, empirical studies show that the D’Hondt method is one of the least proportional among the proportional representation methods. The D'Hondt slightly favours large parties and coalitions over scattered small parties.[3][4][5][6] The axiomatic properties of the D’Hondt method were studied and they proved that the D’Hondt method is the unique consistent, monotone, stable, and balanced method that encourages coalitions.[7][8] A method is consistent if it treats parties which received tied vote equally. By monotonicity, the number of seats provided to any state or party will not decrease if the house size increases. A method is stable if two merged parties would neither gain nor lose more than one seat. By coalition encouragement of the D’Hondt method, any alliance cannot lose the seat.

Legislatures using this system include those of Albania, Angola, Argentina, Armenia, Aruba, Austria, Belgium, Bolivia, Brazil, Bulgaria, Burundi, Cambodia, Cape Verde, Chile, Colombia, Croatia, the Czech Republic, Denmark, the Dominican Republic, East Timor, Ecuador, El Salvador, Estonia, Fiji, Finland, Guatemala, Hungary, Iceland, Israel, Japan, Kosovo, Luxembourg, Macedonia, Moldova, Monaco, Montenegro, Mozambique, Netherlands, Nicaragua, Northern Ireland, Paraguay, Peru, Poland, Portugal, Romania, San Marino, Scotland, Serbia, Slovenia, Spain, Switzerland, Turkey, Uruguay, Venezuela and Wales.

The system has also been used for the "top-up" seats in the London Assembly; in some countries for elections to the European Parliament; and during the 1997 Constitution era to allocate party-list parliamentary seats in Thailand.[9] A modified form was used for elections in the Australian Capital Territory Legislative Assembly, but this was 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.[10]

## Allocation

After all the votes have been tallied, successive quotients are calculated for each party. The formula for the quotient is[11][1]

${\displaystyle {\text{quot}}={\frac {V}{s+1}}}$

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, up to the total number of seats to be allocated for the district/constituency. Say there are p parties and s seats. 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. 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, each party's total votes is divided 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 "Proportionate seats" column shows the exact 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 (*)
Proportionate
seats
Party A 100,000* 50,000* 33,333* 25,000* 20,000 16,666 14,286 12,500 4 3.5
Party B 80,000* 40,000* 26,666* 20,000 16,000 13,333 11,428 10,000 3 2.8
Party C 30,000* 15,000 10,000 7,500 6,000 5,000 4,286 3,750 1 1.0
Party D 20,000 10,000 6,666 5,000 4,000 3,333 2,857 2,500 0 0.7
Total 8 8

## Jefferson and D'Hondt

The Jefferson and the D'Hondt methods are equivalent. They always give the same results, but the methods of presenting the calculation are different. George Washington 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.[12] Ten days after the veto, Congress passed a new method of apportionment, now known as Jefferson’s Method. The U.S. statesman and future President Thomas 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 until 1842.[13]

Victor D'Hondt presented his method in his publication 'Système pratique et raisonné de représentation proportionnelle', published in Brussels in 1882.

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. The tasks are mathematically equivalent, putting states in the place of parties and population in place of votes. In some countries, the Jefferson system is known by the names of local politicians or experts who introduced them locally. For example, it is known in Israel as the Bader-Ofer system.

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 smallest number larger than the next number which would award a seat in the D'Hondt calculations.

Applied to the above example of party lists, this range extends as integers from 20,001 to 25,000. More precisely, any number n for which 20,000 < n ≤ 25,000 can be used.

## Threshold

In some cases, a threshold or barrage is set, and any list which does not achieve 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 coalitions of two or more parties, no threshold is applied for independent individuals); Denmark (2%); East Timor, Spain, and Montenegro (3%); Israel (3.25%); Slovenia (4%); Croatia, Fiji, Romania, Tanzania and Serbia (5%); Russia (5%); Turkey (10%); Poland (5%, or 8% for coalitions; but does not apply for ethnic-minority parties); Czech Republic (5% for single party, 10% for two-party coalitions, 15% for coalitions of 3 parties, 20% for coalitions of 4 or more parties),[14] Hungary (5% for single party, 10% for two-party coalitions, 15% for coalitions of 3 or more parties) and Belgium (5%, on regional basis). In the Netherlands, a party must win enough votes for one strictly proportional 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 vote threshold simplifies the process of seat allocation and discourages fringe parties (those that are likely to gain very few votes) from competing in the elections. Obviously, the higher the vote threshold, the fewer the parties that will be represented in parliament.[15]

The method can cause a hidden threshold.[16][17] 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%.[18] 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%. 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 such cartels. Smaller parties often form pre-election coalitions to make sure they get past the election threshold creating a coalition government. 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 are distributed proportionally (including to the list which has already received the majority bonus). In Italian local elections a similar system is used, where the party or coalition of parties linked to the elected mayor automatically receives 60% of seats; unlike the French model though the remainder of the seats are not distributed again to the largest party.

## 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 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 ${\displaystyle \textstyle {\frac {V}{2^{s}}}}$.

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.

In 1989 and 1992, ACT Legislative Assembly elections were conducted by the Australian Electoral Commission using the "modified d'Hondt" electoral system. The electoral system consisted of the d'Hondt system, the Australian Senate system of proportional representation, and various methods for preferential voting for candidates and parties, both within and across party lines.[19] The process involves 8 stages of scrutiny.

Some systems allow parties to associate their lists together into a single kartel in order to overcome the threshold, while some systems set a separate threshold for cartels. 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. Therefore some parties pool their voters in order to gain more (or any) seats.

### Regional D'Hondt

In most countries, seats for the national assembly are divided on a regional or even a provincial level. This means that seats are first divided between individual regions (or provinces) and are then allocated to the parties in each region separately (based on only the votes cast in the given region). The votes for parties that have not gained a seat at the regional level are thus discarded, so they do not aggregate at a national level. This means that parties which would have gained seats in a national distribution of seats may still end up with no seats as they did not gain enough votes in any region. This may also lead to skewed seat allocation at a national level, such as in Spain in 2011 where the People's Party gained an absolute majority in the Congress of Deputies with only 44% of the national vote.[1]

## Notes

1. ^ The name D'Hondt is sometimes spelt as "d'Hondt". For example it is customary in the Netherlands to write such surnames with a lower-case "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

1. ^ a b c Gallagher, Michael (1991). "Proportionality, disproportionality and electoral systems" (PDF). Electoral Studies. 10 (1). doi:10.1016/0261-3794(91)90004-C. Archived from the original (pdf) on November 16, 2013. Retrieved 30 January 2016.
2. ^ "Election - Plurality and majority systems". Encyclopedia Britannica. Retrieved 2018-04-30.
3. ^ Pukelsheim, Friedrich (2007). "Seat bias formulas in proportional representation systems" (PDF). 4th ECPR General Conference. Archived from the original (PDF) on 7 February 2009.
4. ^ Schuster, Karsten; Pukelsheim, Friedrich; Drton, Mathias; Draper, Norman R. (2003). "Seat biases of apportionment methods for proportional representation" (pdf). Electoral Studies. 22 (4). doi:10.1016/S0261-3794(02)00027-6.
5. ^ Benoit, Kenneth (2000). "Which Electoral Formula Is the Most Proportional? A New Look with New Evidence" (pdf). Political Analysis. 8 (4): 381–388. doi:10.1093/oxfordjournals.pan.a029822.
6. ^ Lijphart, Arend (1990). "The Political Consequences of Electoral Laws, 1945-85". The American Political Science Review. 84 (2): 481–496. doi:10.2307/1963530.
7. ^ Balinksi, Young, M.L., H.P. (1978). "The Jefferson method of Apportionment". SIAM Rev. 20 (2): 278–284.
8. ^ Balinski, Young, M.L., H.P. (1979). "Criteria for proportional representation". Oper. Res. 27: 80–95.
9. ^ Aurel Croissant and Daniel J. Pojar, Jr., Quo Vadis Thailand? Thai Politics after the 2005 Parliamentary Election Archived April 19, 2009, at the Wayback Machine., Strategic Insights, Volume IV, Issue 6 (June 2005)
10. ^ "D'Hondt system for picking NI ministers in Stormont". BBC News. 11 May 2011. Retrieved 7 July 2013.
11. ^ 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.
12. ^ "Washington exercises first presidential veto - Apr 05, 1792 - HISTORY.com". HISTORY.com. Retrieved 2018-05-02.
13. ^ Caulfield, Michael. "Apportioning Representatives in the United States Congress - Jefferson's Method of Apportionment". Mathematical Association of America. Retrieved 25 June 2017.
14. ^ Lebeda, Tomáš (2001), Hlavní proměnné proporčních volebních systémů (PDF), 37 (4), Sociologický časopis, p. 442, ISSN 0038-0288
15. ^ "Electoral_Systems". faculty.georgetown.edu. Retrieved 2018-05-05.
16. ^ Venice Commission (2008). Comparative report on thresholds and other features of electoral systems which bar parties from access to parliament (Report). Council of Europe. Retrieved February 14, 2016.
17. ^ Gallagher, Michael; Mitchell, Paul (2005). "Appendix C:Effective threshold and effective magnitude". The Politics of Electoral Systems (PDF). Oxford University Press. ISBN 9780199257560.
18. ^ Oikeusministeriö. Suhteellisuuden parantaminen eduskuntavaaleissa.
19. ^ ACT, PositionTitle=Electoral Commissioner; SectionName=ACT Electoral Commission; CorporateName=Elections (2015-01-06). "Modified d'Hondt Electoral System". www.elections.act.gov.au. Retrieved 2018-05-05.