D'Hondt method

# D'Hondt method

Discussion

Encyclopedia
The d'Hondt method is a highest averages method
Highest averages method
The highest averages method is the name for a variety of ways to allocate seats proportionally for representative assemblies with party list voting systems....

for allocating seats in party-list proportional representation
Party-list proportional representation
Party-list proportional representation systems are a family of voting systems emphasizing proportional representation in elections in which multiple candidates are elected...

. The method described is named after Belgian
Belgium
Belgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...

mathematician Victor D'Hondt
Victor D'Hondt
Victor D’Hondt was a Belgian lawyer, salesman, jurist of civil law at Ghent University, and mathematician. He devised a procedure, the D'Hondt method, which he first described in 1878, for allocating seats to candidates in party-list proportional representation elections...

who described it in 1878. In comparison with the Sainte-Laguë method
Sainte-Laguë method
The Sainte-Laguë method is one way of allocating seats approximately proportional to the number of votes of a party to a party list used in many voting systems. It is named after the French mathematician André Sainte-Laguë. The Sainte-Laguë method is quite similar to the D'Hondt method, but uses...

, d'Hondt slightly favours large parties
Political party
A political party is a political organization that typically seeks to influence government policy, usually by nominating their own candidates and trying to seat them in political office. Parties participate in electoral campaigns, educational outreach or protest actions...

and coalitions over scattered small parties.

Legislatures using this system include those of Albania
Albania
Albania , officially known as the Republic of Albania , is a country in Southeastern Europe, in the Balkans region. It is bordered by Montenegro to the northwest, Kosovo to the northeast, the Republic of Macedonia to the east and Greece to the south and southeast. It has a coast on the Adriatic Sea...

, Argentina
Argentina
Argentina , officially the Argentine Republic , is the second largest country in South America by land area, after Brazil. It is constituted as a federation of 23 provinces and an autonomous city, Buenos Aires...

, Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...

, Belgium
Belgium
Belgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...

, Brazil
Brazil
Brazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...

, Bulgaria
Bulgaria
Bulgaria , officially the Republic of Bulgaria , is a parliamentary democracy within a unitary constitutional republic in Southeast Europe. The country borders Romania to the north, Serbia and Macedonia to the west, Greece and Turkey to the south, as well as the Black Sea to the east...

, Cambodia
Cambodia
Cambodia , officially known as the Kingdom of Cambodia, is a country located in the southern portion of the Indochina Peninsula in Southeast Asia...

, Cape Verde
Cape Verde
The Republic of Cape Verde is an island country, spanning an archipelago of 10 islands located in the central Atlantic Ocean, 570 kilometres off the coast of Western Africa...

, Chile
Chile
Chile ,officially the Republic of Chile , is a country in South America occupying a long, narrow coastal strip between the Andes mountains to the east and the Pacific Ocean to the west. It borders Peru to the north, Bolivia to the northeast, Argentina to the east, and the Drake Passage in the far...

, Colombia
Colombia
Colombia, officially the Republic of Colombia , is a unitary constitutional republic comprising thirty-two departments. The country is located in northwestern South America, bordered to the east by Venezuela and Brazil; to the south by Ecuador and Peru; to the north by the Caribbean Sea; to the...

, Croatia
Croatia
Croatia , officially the Republic of Croatia , is a unitary democratic parliamentary republic in Europe at the crossroads of the Mitteleuropa, the Balkans, and the Mediterranean. Its capital and largest city is Zagreb. The country is divided into 20 counties and the city of Zagreb. Croatia covers ...

, Czech Republic
Czech Republic
The Czech Republic is a landlocked country in Central Europe. The country is bordered by Poland to the northeast, Slovakia to the east, Austria to the south, and Germany to the west and northwest....

, Denmark
Denmark
Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...

, East Timor
East Timor
The Democratic Republic of Timor-Leste, commonly known as East Timor , is a state in Southeast Asia. It comprises the eastern half of the island of Timor, the nearby islands of Atauro and Jaco, and Oecusse, an exclave on the northwestern side of the island, within Indonesian West Timor...

Ecuador , officially the Republic of Ecuador is a representative democratic republic in South America, bordered by Colombia on the north, Peru on the east and south, and by the Pacific Ocean to the west. It is one of only two countries in South America, along with Chile, that do not have a border...

, Estonia
Estonia
Estonia , officially the Republic of Estonia , is a state in the Baltic region of Northern Europe. It is bordered to the north by the Gulf of Finland, to the west by the Baltic Sea, to the south by Latvia , and to the east by Lake Peipsi and the Russian Federation . Across the Baltic Sea lies...

, Finland
Finland
Finland , officially the Republic of Finland, is a Nordic country situated in the Fennoscandian region of Northern Europe. It is bordered by Sweden in the west, Norway in the north and Russia in the east, while Estonia lies to its south across the Gulf of Finland.Around 5.4 million people reside...

, Guatemala
Guatemala
Guatemala is a country in Central America bordered by Mexico to the north and west, the Pacific Ocean to the southwest, Belize to the northeast, the Caribbean to the east, and Honduras and El Salvador to the southeast...

, Hungary
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...

, Iceland
Iceland
Iceland , described as the Republic of Iceland, is a Nordic and European island country in the North Atlantic Ocean, on the Mid-Atlantic Ridge. Iceland also refers to the main island of the country, which contains almost all the population and almost all the land area. The country has a population...

, Israel
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...

, Japan
Japan
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...

, Luxembourg
Luxembourg
Luxembourg , officially the Grand Duchy of Luxembourg , is a landlocked country in western Europe, bordered by Belgium, France, and Germany. It has two principal regions: the Oesling in the North as part of the Ardennes massif, and the Gutland in the south...

, Republic of Macedonia
Republic of Macedonia
Macedonia , officially the Republic of Macedonia , is a country located in the central Balkan peninsula in Southeast Europe. It is one of the successor states of the former Yugoslavia, from which it declared independence in 1991...

, Republic of Moldova, Montenegro
Montenegro
Montenegro Montenegrin: Crna Gora Црна Гора , meaning "Black Mountain") is a country located in Southeastern Europe. It has a coast on the Adriatic Sea to the south-west and is bordered by Croatia to the west, Bosnia and Herzegovina to the northwest, Serbia to the northeast and Albania to the...

, the Netherlands
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...

, Northern Ireland
Northern Ireland
Northern Ireland is one of the four countries of the United Kingdom. Situated in the north-east of the island of Ireland, it shares a border with the Republic of Ireland to the south and west...

, Paraguay
Paraguay
Paraguay , officially the Republic of Paraguay , is a landlocked country in South America. It is bordered by Argentina to the south and southwest, Brazil to the east and northeast, and Bolivia to the northwest. Paraguay lies on both banks of the Paraguay River, which runs through the center of the...

, Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...

, Portugal
Portugal
Portugal , officially the Portuguese Republic is a country situated in southwestern Europe on the Iberian Peninsula. Portugal is the westernmost country of Europe, and is bordered by the Atlantic Ocean to the West and South and by Spain to the North and East. The Atlantic archipelagos of the...

, Romania
Romania
Romania is a country located at the crossroads of Central and Southeastern Europe, on the Lower Danube, within and outside the Carpathian arch, bordering on the Black Sea...

, Scotland
Scotland
Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...

, Serbia
Serbia
Serbia , officially the Republic of Serbia , is a landlocked country located at the crossroads of Central and Southeast Europe, covering the southern part of the Carpathian basin and the central part of the Balkans...

, Slovenia
Slovenia
Slovenia , officially the Republic of Slovenia , is a country in Central and Southeastern Europe touching the Alps and bordering the Mediterranean. Slovenia borders Italy to the west, Croatia to the south and east, Hungary to the northeast, and Austria to the north, and also has a small portion of...

, Spain
Spain
Spain , officially the Kingdom of Spain languages]] under the European Charter for Regional or Minority Languages. In each of these, Spain's official name is as follows:;;;;;;), is a country and member state of the European Union located in southwestern Europe on the Iberian Peninsula...

, Turkey
Turkey
Turkey , known officially as the Republic of Turkey , is a Eurasian country located in Western Asia and in East Thrace in Southeastern Europe...

, Uruguay
Uruguay
Uruguay ,officially the Oriental Republic of Uruguay,sometimes the Eastern Republic of Uruguay; ) is a country in the southeastern part of South America. It is home to some 3.5 million people, of whom 1.8 million live in the capital Montevideo and its metropolitan area...

and Wales
Wales
Wales is a country that is part of the United Kingdom and the island of Great Britain, bordered by England to its east and the Atlantic Ocean and Irish Sea to its west. It has a population of three million, and a total area of 20,779 km²...

.

The system has also been used in Northern Ireland
Northern Ireland
Northern Ireland is one of the four countries of the United Kingdom. Situated in the north-east of the island of Ireland, it shares a border with the Republic of Ireland to the south and west...

to allocate the ministerial positions in the Northern Ireland Executive
Northern Ireland Executive
The Northern Ireland Executive is the executive arm of the Northern Ireland Assembly, the devolved legislature for Northern Ireland. It is answerable to the Assembly and was established according to the terms of the Northern Ireland Act 1998, which followed the Good Friday Agreement...

; for the 'top-up' seats in the London Assembly
London Assembly
The London Assembly is an elected body, part of the Greater London Authority, that scrutinises the activities of the Mayor of London and has the power, with a two-thirds majority, to amend the mayor's annual budget. The assembly was established in 2000 and is headquartered at City Hall on the south...

; in some countries during elections to the European Parliament
European Parliament election, 2004
Elections to the European Parliament were held from 10 June 2004 to 13 June 2004 in the 25 member states of the European Union, using varying election days according to local custom...

; and during the 1997 Constitution
1997 Constitution of Thailand
For legal and historical context, see the Constitutions of Thailand articlethumb|275px|Bangkok's [[Democracy Monument, Bangkok|Democracy Monument]]: a representation of the 1932 Constitution sits on top of two golden offering bowls above a turret....

-era for allocating party-list parliamentary seats in Thailand
Thailand
Thailand , officially the Kingdom of Thailand , formerly known as Siam , is a country located at the centre of the Indochina peninsula and Southeast Asia. It is bordered to the north by Burma and Laos, to the east by Laos and Cambodia, to the south by the Gulf of Thailand and Malaysia, and to the...

. A modified form was used for elections in the Australian Capital Territory Legislative Assembly
Australian Capital Territory Legislative Assembly
The Australian Capital Territory Legislative Assembly is the unicameral legislature of the Australian Capital Territory...

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
European Parliament
The European Parliament is the directly elected parliamentary institution of the European Union . Together with the Council of the European Union and the Commission, it exercises the legislative function of the EU and it has been described as one of the most powerful legislatures in the world...

.

## Allocation

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 parties and seats. Then create a grid of numbers, with rows and columns, where the entry in the -th row and -th column is the number of votes won by the party divided by . For the purposes of this explanation, call the highest entries in the entire grid 'special entries'; each party is given as many seats as there are special entries in its row. The entries are sometimes called 'distribution figures'.

Example: 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 distribution figures are highlighted in bold, ranging from 100,000 down to 25,000. For each distribution figure in bold, the corresponding party gets a seat.
 /1 /2 /3 /4 /5 /6 /7 /8 Seats won Party A 100,000 50,000 33,333 25,000 20,000 16,666 14,286 12,500 4 Party B 80,000 40,000 26,666 20,000 16,000 13,333 11,428 10,000 3 Party C 30,000 15,000 10,000 7,500 6,000 5,000 4,286 3,750 1 Party D 20,000 10,000 6,666 5,000 4,000 3,333 2,857 2,500 0

Government formation: This is repeated for each district/constituency in the country. Often the party with the highest number of seats will get to select the chief negotiators to form a government and is awarded the Prime Minister's seat. Its seats may be majority, minority, or a coalition, according to whether voters have spread their votes across few or many parties.

Proportionality: D'Hondt does not produce absolute/pure proportionality: in relation to their total vote, the two larger parties are slightly advantaged. Party C gets what it deserves, and Party D received too few votes for a seat. To dispute Party A's fourth seat (*25,000) Party D would have needed a minimum of 25,000 votes. So the seat allocation across these 4 parties is a fair, if not absolutely proportional, reflection of their vote.

Party system produced: The method tends to allow for 3 nationwide parties to be present in parliament, or sometimes four if voters spread their vote more evenly across the four leading parties. However, regionally-based parties with concentrated support in certain districts may well get enough votes there to win a handful of seats in parliament. In other words the D'Hondt method also allows for regional-nationalist parties to be represented.

District magnitude effects: With the D'Hondt seat allocation method, district magnitude or constituency size has quite an effect. If our model constituency, above, had only 5 seats, they would all go to Parties A (*100,000, *50,000, *33,333) and B (*80,000, *40,000) and Party C would not get the (*30,000) seat, being short of 3,334 votes to take it from Party A. And in the opposite case, if the district contained just one more seat, then Party D would have a chance of getting it (contending with A and B for the 20,000 distribution number, the next highest in the table). So the larger the number of seats in a district/constituency, the more likely that a fourth nationwide party will get a seat - as long as voters spread their votes.

From party seat to winning candidate: How the parties' allocate the seats they have won to their candidates: in 'Closed List' PR systems, parties tend to present a list of candidates for all available seats (as if they could win them all), ranked by order of priority. Then the top candidates receive the seats won in order of their rank. In an 'Open List' system, in addition to the party's order, voters can add their own ranking of candidates to show where they disagree with the party's ranking. Or the party may have listed their candidates without any preference, leaving it to the voters to rank them.

## D'Hondt and Jefferson

The d'Hondt method is equivalent to the Jefferson method (named after the U.S.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...

statesman Thomas Jefferson
Thomas Jefferson
Thomas Jefferson was the principal author of the United States Declaration of Independence and the Statute of Virginia for Religious Freedom , the third President of the United States and founder of the University of Virginia...

) in that they always give the same results, but the method of calculating the apportionment is different. Jefferson devised the method in 1792 for the U.S. congressional apportionment
United States congressional apportionment
United States congressional apportionment is the process by which seats in the United States House of Representatives are redistributed amongst the 50 states following each constitutionally mandated decennial census. Each state is apportioned a number of seats which approximately corresponds to its...

pursuant to the First United States Census, with the objective being the proportional distribution of seats in the House of Representatives
United States House of Representatives
The United States House of Representatives is one of the two Houses of the United States Congress, the bicameral legislature which also includes the Senate.The composition and powers of the House are established in Article One of the Constitution...

among the states
U.S. state
A U.S. state is any one of the 50 federated states of the United States of America that share sovereignty with the federal government. Because of this shared sovereignty, an American is a citizen both of the federal entity and of his or her state of domicile. Four states use the official title of...

rather than distributing seats in a legislature among parties pursuant to an election (the problems are functionally equivalent if one puts states in the place of parties and population in place of votes). Jefferson's method uses a quota, as in the largest remainder method
Largest remainder method
The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems...

, but the quota (called a divisor) is adjusted as necessary so that the resulting quotients, disregarding any fractional remainders, sum to the required total (so the two methods share the additional property of not using all numbers, whether of state populations or of party votes, in the apportioning of seats). One of a range of quotas will accomplish this, and applied to the above example of party lists this extends as integers from 85,001 to 93,333, 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.

## Variations

In some cases, a threshold
Election threshold
In party-list proportional representation systems, an election threshold is a clause that stipulates that a party must receive a minimum percentage of votes, either nationally or within a particular district, to obtain any seats in the parliament...

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%), Spain and Montenegro (3%), Slovenia (4%), Czech Republic, Hungary, Romania and Serbia (5%), Russia(7%), Turkey (10%), Poland (5%, or 8% for coalitions) and Belgium (5%, on regional basis). In the Netherlands
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...

, 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
Estonia
Estonia , officially the Republic of Estonia , is a state in the Baltic region of Northern Europe. It is bordered to the north by the Gulf of Finland, to the west by the Baltic Sea, to the south by Latvia , and to the east by Lake Peipsi and the Russian Federation . Across the Baltic Sea lies...

, 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. 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
Croatia
Croatia , officially the Republic of Croatia , is a unitary democratic parliamentary republic in Europe at the crossroads of the Mitteleuropa, the Balkans, and the Mediterranean. Its capital and largest city is Zagreb. The country is divided into 20 counties and the city of Zagreb. Croatia covers ...

, 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 "fallen lists" (lists that don't get at least 5%). If many votes are lost in this manner, a list that gets barely more than 5% will still get a seat, whereas if there is a small number of parties that all 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 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
Hagenbach-Bischoff system
The Hagenbach-Bischoff system is a variant of the D'Hondt method, used for allocating seats in party-list proportional representation. It usually uses the Hagenbach-Bischoff quota for allocating seats, and for any seats remaining the D'Hondt method is then applied so that the first and subsequent...

, 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
Macau
Macau , also spelled Macao , is, along with Hong Kong, one of the two special administrative regions of the People's Republic of China...

, a modified D'Hondt method is used. The formula for the quotient in this system is .