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The D'Hondt method, also called the Jefferson method or the greatest divisors method, is a method for allocating seats in parliaments among
federal states A federation (also known as a federal state) is a political entity characterized by a union of partially self-governing provinces, states, or other regions under a central federal government (federalism). In a federation, the self-governi ...
, or in
party-list An electoral list is a grouping of candidates for election, usually found in proportional or mixed electoral systems, but also in some plurality electoral systems. An electoral list can be registered by a political party (a party list) or can ...
proportional representation Proportional representation (PR) refers to a type of electoral system under which subgroups of an electorate are reflected proportionately in the elected body. The concept applies mainly to geographical (e.g. states, regions) and political divis ...
systems. It belongs to the class of highest-averages methods. The method was first described in 1792 by future U.S. president
Thomas Jefferson Thomas Jefferson (April 13, 1743 – July 4, 1826) was an American statesman, diplomat, lawyer, architect, philosopher, and Founding Father who served as the third president of the United States from 1801 to 1809. He was previously the natio ...
. It was re-invented independently in 1878 by Belgian mathematician
Victor D'Hondt Victor Joseph Auguste D'Hondt (; 20 November 1841 – 30 May 1901) was a Belgian lawyer and jurist of civil law at Ghent University. He devised a procedure, the D'Hondt method, which he first described in 1878, for allocating seats to candidates ...
, which is the reason for its two different names.


Motivation

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. Although all of these methods approximate proportionality, they do so by minimizing different kinds of disproportionality. The D'Hondt method minimizes the number of votes that need to be left aside so that the remaining votes are represented exactly proportionally. Only the D'Hondt method (and methods equivalent to it) minimizes this disproportionality. Empirical studies based on other, more popular concepts of disproportionality show that the D'Hondt method is one of the least proportional among the proportional representation methods. The D'Hondt slightly favours large
parties A party is a gathering of people who have been invited by a host for the purposes of socializing, conversation, recreation, or as part of a festival or other commemoration or celebration of a special occasion. A party will often feature ...
and
coalitions A coalition is a group formed when two or more people or groups temporarily work together to achieve a common goal. The term is most frequently used to denote a formation of power in political or economical spaces. Formation According to ''A Gui ...
over scattered small parties. In comparison, the
Webster/Sainte-Laguë method The Webster method, also called the Sainte-Laguë method () or the major fractions method, is a method for allocating seats in a parliament among federal states, or among parties in a party-list proportional representation system. The metho ...
, a different 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. 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. A method is consistent if it treats parties that received tied votes 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.


Usage

Legislatures using this system include those of
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Angola , national_anthem = " Angola Avante"() , image_map = , map_caption = , capital = Luanda , religion = , religion_year = 2020 , religion_ref = , coordina ...
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Armenia Armenia (), , group=pron officially the Republic of Armenia,, is a landlocked country in the Armenian Highlands of Western Asia.The UNbr>classification of world regions places Armenia in Western Asia; the CIA World Factbook , , and ''Ox ...
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Austria Austria, , bar, Östareich officially the Republic of Austria, is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine states, one of which is the capital, Vienna, the most populous ...
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Belgium Belgium, ; french: Belgique ; german: Belgien officially the Kingdom of Belgium, is a country in Northwestern Europe. The country is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to th ...
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Bolivia , image_flag = Bandera de Bolivia (Estado).svg , flag_alt = Horizontal tricolor (red, yellow, and green from top to bottom) with the coat of arms of Bolivia in the center , flag_alt2 = 7 × 7 square p ...
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Brazil Brazil ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the largest country in both South America and Latin America. At and with over 217 million people, Brazil is the world's fifth-largest country by area ...
, Burundi,
Cambodia Cambodia (; also Kampuchea ; km, កម្ពុជា, UNGEGN: ), officially the Kingdom of Cambodia, is a country located in the southern portion of the Indochinese Peninsula in Southeast Asia, spanning an area of , bordered by Thailan ...
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Chile Chile, officially the Republic of Chile, is a country in the western part of South America. It is the southernmost country in the world, and the closest to Antarctica, occupying a long and narrow strip of land between the Andes to the east a ...
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Colombia Colombia (, ; ), officially the Republic of Colombia, is a country in South America with insular regions in North America—near Nicaragua's Caribbean coast—as well as in the Pacific Ocean. The Colombian mainland is bordered by the Car ...
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Croatia , image_flag = Flag of Croatia.svg , image_coat = Coat of arms of Croatia.svg , anthem = "Lijepa naša domovino"("Our Beautiful Homeland") , image_map = , map_caption = , capit ...
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Denmark ) , song = ( en, "King Christian stood by the lofty mast") , song_type = National and royal anthem , image_map = EU-Denmark.svg , map_caption = , subdivision_type = Sovereign state , subdivision_name = Danish Realm, Kingdom of Denmark ...
, the
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East Timor East Timor (), also known as Timor-Leste (), officially the Democratic Republic of Timor-Leste, is an island country in Southeast Asia. It comprises the eastern half of the island of Timor, the exclave of Oecusse on the island's north-weste ...
,
Ecuador Ecuador ( ; ; Quechua: ''Ikwayur''; Shuar: ''Ecuador'' or ''Ekuatur''), officially the Republic of Ecuador ( es, República del Ecuador, which literally translates as "Republic of the Equator"; Quechua: ''Ikwadur Ripuwlika''; Shuar: ''Eku ...
,
Estonia Estonia, formally the Republic of Estonia, is a country by the Baltic Sea in Northern Europe. It is bordered to the north by the Gulf of Finland across from Finland, to the west by the sea across from Sweden, to the south by Latvia, a ...
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Fiji Fiji ( , ,; fj, Viti, ; Fiji Hindi: फ़िजी, ''Fijī''), officially the Republic of Fiji, is an island country in Melanesia, part of Oceania in the South Pacific Ocean. It lies about north-northeast of New Zealand. Fiji consists ...
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Finland Finland ( fi, Suomi ; sv, Finland ), officially the Republic of Finland (; ), is a Nordic country in Northern Europe. It shares land borders with Sweden to the northwest, Norway to the north, and Russia to the east, with the Gulf of B ...
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Greenland Greenland ( kl, Kalaallit Nunaat, ; da, Grønland, ) is an island country in North America that is part of the Kingdom of Denmark. It is located between the Arctic and Atlantic oceans, east of the Canadian Arctic Archipelago. Greenland is t ...
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Guatemala Guatemala ( ; ), officially the Republic of Guatemala ( es, República de Guatemala, links=no), is a country in Central America. It is bordered to the north and west by Mexico; to the northeast by Belize and the Caribbean; to the east by H ...
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Hungary Hungary ( hu, Magyarország ) is a landlocked country in Central Europe. Spanning of the Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to the northeast, Romania to the east and southeast, Serbia to the south, Croatia a ...
(in a mixed system),
Iceland Iceland ( is, Ísland; ) is a Nordic island country in the North Atlantic Ocean and in the Arctic Ocean. Iceland is the most sparsely populated country in Europe. Iceland's capital and largest city is Reykjavík, which (along with its s ...
,
Israel Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated ...
,
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,
Luxembourg Luxembourg ( ; lb, Lëtzebuerg ; french: link=no, Luxembourg; german: link=no, Luxemburg), officially the Grand Duchy of Luxembourg, ; french: link=no, Grand-Duché de Luxembourg ; german: link=no, Großherzogtum Luxemburg is a small lan ...
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Mozambique Mozambique (), officially the Republic of Mozambique ( pt, Moçambique or , ; ny, Mozambiki; sw, Msumbiji; ts, Muzambhiki), is a country located in southeastern Africa bordered by the Indian Ocean to the east, Tanzania to the north, Malawi ...
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Nicaragua Nicaragua (; ), officially the Republic of Nicaragua (), is the largest country in Central America, bordered by Honduras to the north, the Caribbean to the east, Costa Rica to the south, and the Pacific Ocean to the west. Managua is the cou ...
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North Macedonia North Macedonia, ; sq, Maqedonia e Veriut, (Macedonia before February 2019), officially the Republic of North Macedonia,, is a country in Southeast Europe. It gained independence in 1991 as one of the successor states of Socialist Feder ...
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Paraguay Paraguay (; ), officially the Republic of Paraguay ( es, República del Paraguay, links=no; gn, Tavakuairetã Paraguái, links=si), is a landlocked country in South America. It is bordered by Argentina to the south and southwest, Brazil to th ...
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Peru , image_flag = Flag of Peru.svg , image_coat = Escudo nacional del Perú.svg , other_symbol = Great Seal of the State , other_symbol_type = Seal (emblem), National seal , national_motto = "Fi ...
,
Poland Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populous ...
,
Portugal Portugal, officially the Portuguese Republic ( pt, República Portuguesa, links=yes ), is a country whose mainland is located on the Iberian Peninsula of Southwestern Europe, and whose territory also includes the Atlantic archipelagos of ...
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Romania Romania ( ; ro, România ) is a country located at the crossroads of Central Europe, Central, Eastern Europe, Eastern, and Southeast Europe, Southeastern Europe. It borders Bulgaria to the south, Ukraine to the north, Hungary to the west, S ...
,
San Marino San Marino (, ), officially the Republic of San Marino ( it, Repubblica di San Marino; ), also known as the Most Serene Republic of San Marino ( it, Serenissima Repubblica di San Marino, links=no), is the fifth-smallest country in the world an ...
,
Serbia Serbia (, ; Serbian language, Serbian: , , ), officially the Republic of Serbia (Serbian language, Serbian: , , ), is a landlocked country in Southeast Europe, Southeastern and Central Europe, situated at the crossroads of the Pannonian Bas ...
,
Slovenia Slovenia ( ; sl, Slovenija ), officially the Republic of Slovenia (Slovene: , abbr.: ''RS''), is a country in Central Europe. It is bordered by Italy to the west, Austria to the north, Hungary to the northeast, Croatia to the southeast, an ...
,
Spain , image_flag = Bandera de España.svg , image_coat = Escudo de España (mazonado).svg , national_motto = ''Plus ultra'' (Latin)(English: "Further Beyond") , national_anthem = (English: "Royal March") , i ...
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Turkey Turkey ( tr, Türkiye ), officially the Republic of Türkiye ( tr, Türkiye Cumhuriyeti, links=no ), is a list of transcontinental countries, transcontinental country located mainly on the Anatolia, Anatolian Peninsula in Western Asia, with ...
,
Uruguay Uruguay (; ), officially the Oriental Republic of Uruguay ( es, República Oriental del Uruguay), is a country in South America. It shares borders with Argentina to its west and southwest and Brazil to its north and northeast; while bordering ...
and
Venezuela Venezuela (; ), officially the Bolivarian Republic of Venezuela ( es, link=no, República Bolivariana de Venezuela), is a country on the northern coast of South America, consisting of a continental landmass and many islands and islets in th ...
. The system is used for the "top-up" seats in the Scottish Parliament, the Senedd (Welsh Parliament) and the London Assembly; in some countries for elections to the
European Parliament The European Parliament (EP) is one of the legislative bodies of the European Union and one of its seven institutions. Together with the Council of the European Union (known as the Council and informally as the Council of Ministers), it adopts ...
; and was used during the 1997 Constitution era to allocate party-list parliamentary seats in
Thailand Thailand ( ), historically known as Siam () and officially the Kingdom of Thailand, is a country in Southeast Asia, located at the centre of the Indochinese Peninsula, spanning , with a population of almost 70 million. The country is bo ...
. A modified form was used for elections in the Australian Capital Territory Legislative Assembly, but this was abandoned in favour of the
Hare–Clark electoral system Hare-Clark is a type of single transferable vote electoral system of proportional representation used for elections in Tasmania and the Australian Capital Territory. The method for the distribution of preferences is similar to other voting systems ...
. 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 The European Parliament (EP) is one of the legislative bodies of the European Union and one of its seven institutions. Together with the Council of the European Union (known as the Council and informally as the Council of Ministers), it adopts ...
and for the allocation of ministers in the Northern Ireland Assembly. It is also used to calculate the results in German and Austrian works council elections.


Procedure

After all the votes have been tallied, successive
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
s are calculated for each party. The party with the largest quotient wins one seat, and its quotient is recalculated. This is repeated until the required number of seats is filled. The formula for the quotient is. See in particular the section "Sainte-Lague"
pp. 174–175
\text = \frac where: * is the total number of votes that party received, and * 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 parties and seats. Then a grid of numbers can be created, with rows and columns, where the entry in the th row and th column is the number of votes won by the th party, divided by . The winning entries are the 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 are divided by 1, then by 2, 3, and 4 (and then, if necessary, by 5, 6, 7, and so on). The 8 highest entries, marked with asterisks, range from 100,000 down to 25,000. For each, the corresponding party gets a seat. 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. The chart below shows an easy way to perform the calculation. 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 comparison, the "True proportion" 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.


Further examples

A worked-through example for non-experts relating to the 2019 elections in the UK for the European Parliament written by Christina Pagel is available as an online article with the institute UK in a Changing Europe. A more mathematically detailed example has been written by British mathematician Professor Helen Wilson.


Approximate proportionality under D'Hondt

The D'Hondt method approximates proportionality by minimizing the largest seats-to-votes ratio among all parties. This ratio is also known as the advantage ratio. For party p \in \, where P is the overall number of parties, the advantage ratio is a_p=\frac, where *s_p – the seat share of party p, s_p \in ,1\;\sum_p s_p = 1, *v_p – the vote share of party p, v_p \in ,1\;\sum_p v_p = 1. The largest advantage ratio, \delta = \max_p a_p, captures how over-represented is the most over-represented party. The D'Hondt method assigns seats so that this ratio attains its smallest possible value, \delta^* = \min_ \max_p a_p, where \mathbf=\ is a seat allocation from the set of all allowed seat allocations \mathcal. Thanks to this, as shown by Juraj Medzihorsky, the D'Hondt method splits the votes into exactly proportionally represented ones and residual ones, minimizing the overall amount of the residuals in the process. The overall fraction of residual votes is \pi^* = 1 - \frac. The residuals of party are r_p = v_p - (1-\pi^*) s_p,\; r_p \in , v_p \sum_p\,r_p=\pi^*. For illustration, continue with the above example of four parties. The advantage ratios of the four parties are 1.2 for A, 1.1 for B, 1 for C, and 0 for D. The reciprocal of the largest advantage ratio is . The residuals as shares of the total vote are 0% for A, 2.2% for B, 2.2% for C, and 8.7% for party D. Their sum is 13%, i.e., . The decomposition of the votes into represented and residual ones is shown in the table below.


Jefferson and D'Hondt

The method was first described in 1792 by
Thomas Jefferson Thomas Jefferson (April 13, 1743 – July 4, 1826) was an American statesman, diplomat, lawyer, architect, philosopher, and Founding Father who served as the third president of the United States from 1801 to 1809. He was previously the natio ...
, in a letter to
George Washington George Washington (February 22, 1732, 1799) was an American military officer, statesman, and Founding Father who served as the first president of the United States from 1789 to 1797. Appointed by the Continental Congress as commander of th ...
regarding the apportionment of seats in the
United States House of Representatives The United States House of Representatives, often referred to as the House of Representatives, the U.S. House, or simply the House, is the Lower house, lower chamber of the United States Congress, with the United States Senate, Senate being ...
: It was invented independently in 1878 in Europe, by Belgian mathematician
Victor D'Hondt Victor Joseph Auguste D'Hondt (; 20 November 1841 – 30 May 1901) was a Belgian lawyer and jurist of civil law at Ghent University. He devised a procedure, the D'Hondt method, which he first described in 1878, for allocating seats to candidates ...
, who wrote: 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 George Washington (February 22, 1732, 1799) was an American military officer, statesman, and Founding Father who served as the first president of the United States from 1789 to 1797. Appointed by the Continental Congress as commander of th ...
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. Ten days after the veto, Congress passed a new method of apportionment, now known as Jefferson's Method. Statesman and future US President
Thomas Jefferson Thomas Jefferson (April 13, 1743 – July 4, 1826) was an American statesman, diplomat, lawyer, architect, philosopher, and Founding Father who served as the third president of the United States from 1801 to 1809. He was previously the natio ...
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 House of Representatives is the name of legislative bodies in many countries and sub-national entitles. In many countries, the House of Representatives is the lower house of a bicameral legislature, with the corresponding upper house often c ...
among the states until 1842.
Victor D'Hondt Victor Joseph Auguste D'Hondt (; 20 November 1841 – 30 May 1901) was a Belgian lawyer and jurist of civil law at Ghent University. He devised a procedure, the D'Hondt method, which he first described in 1878, for allocating seats to candidates ...
presented his method in his publication , 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 Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated ...
as the Bader–Ofer system. Jefferson's method uses a quota (called a divisor), as in the
largest remainder method The largest remainder method (also known as Hare–Niemeyer method, Hamilton method or as Vinton's method) is one way of allocating seats proportionally for representative assemblies with party list voting systems. It contrasts with variou ...
. The divisor is chosen as necessary so that the resulting quotients, disregarding any fractional
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In algeb ...
s, 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
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s 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 Albania ( ; sq, Shqipëri or ), or , also or . officially the Republic of Albania ( sq, Republika e Shqipërisë), is a country in Southeastern Europe. It is located on the Adriatic and Ionian Seas within the Mediterranean Sea and shares ...
(3% for single parties, 5% for coalitions of two or more parties, 1% for independent individuals);
Denmark ) , song = ( en, "King Christian stood by the lofty mast") , song_type = National and royal anthem , image_map = EU-Denmark.svg , map_caption = , subdivision_type = Sovereign state , subdivision_name = Danish Realm, Kingdom of Denmark ...
(2%);
East Timor East Timor (), also known as Timor-Leste (), officially the Democratic Republic of Timor-Leste, is an island country in Southeast Asia. It comprises the eastern half of the island of Timor, the exclave of Oecusse on the island's north-weste ...
,
Spain , image_flag = Bandera de España.svg , image_coat = Escudo de España (mazonado).svg , national_motto = ''Plus ultra'' (Latin)(English: "Further Beyond") , national_anthem = (English: "Royal March") , i ...
,
Serbia Serbia (, ; Serbian language, Serbian: , , ), officially the Republic of Serbia (Serbian language, Serbian: , , ), is a landlocked country in Southeast Europe, Southeastern and Central Europe, situated at the crossroads of the Pannonian Bas ...
, and
Montenegro ) , image_map = Europe-Montenegro.svg , map_caption = , image_map2 = , capital = Podgorica , coordinates = , largest_city = capital , official_languages = M ...
(3%);
Israel Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated ...
(3.25%);
Slovenia Slovenia ( ; sl, Slovenija ), officially the Republic of Slovenia (Slovene: , abbr.: ''RS''), is a country in Central Europe. It is bordered by Italy to the west, Austria to the north, Hungary to the northeast, Croatia to the southeast, an ...
and
Bulgaria Bulgaria (; bg, България, Bǎlgariya), officially the Republic of Bulgaria,, ) is a country in Southeast Europe. It is situated on the eastern flank of the Balkans, and is bordered by Romania to the north, Serbia and North Macedo ...
(4%);
Croatia , image_flag = Flag of Croatia.svg , image_coat = Coat of arms of Croatia.svg , anthem = "Lijepa naša domovino"("Our Beautiful Homeland") , image_map = , map_caption = , capit ...
,
Fiji Fiji ( , ,; fj, Viti, ; Fiji Hindi: फ़िजी, ''Fijī''), officially the Republic of Fiji, is an island country in Melanesia, part of Oceania in the South Pacific Ocean. It lies about north-northeast of New Zealand. Fiji consists ...
,
Romania Romania ( ; ro, România ) is a country located at the crossroads of Central Europe, Central, Eastern Europe, Eastern, and Southeast Europe, Southeastern Europe. It borders Bulgaria to the south, Ukraine to the north, Hungary to the west, S ...
,
Russia Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the ...
and
Tanzania Tanzania (; ), officially the United Republic of Tanzania ( sw, Jamhuri ya Muungano wa Tanzania), is a country in East Africa within the African Great Lakes region. It borders Uganda to the north; Kenya to the northeast; Comoro Islands and ...
(5%);
Turkey Turkey ( tr, Türkiye ), officially the Republic of Türkiye ( tr, Türkiye Cumhuriyeti, links=no ), is a list of transcontinental countries, transcontinental country located mainly on the Anatolia, Anatolian Peninsula in Western Asia, with ...
(7%);
Poland Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populous ...
(5%, or 8% for coalitions; but does not apply for ethnic-minority parties),
Hungary Hungary ( hu, Magyarország ) is a landlocked country in Central Europe. Spanning of the Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to the northeast, Romania to the east and southeast, Serbia to the south, Croatia a ...
(5% for single party, 10% for two-party coalitions, 15% for coalitions of 3 or more parties) and
Belgium Belgium, ; french: Belgique ; german: Belgien officially the Kingdom of Belgium, is a country in Northwestern Europe. The country is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to th ...
(5%, on regional basis). In the
Netherlands ) , anthem = ( en, "William of Nassau") , image_map = , map_caption = , subdivision_type = Sovereign state , subdivision_name = Kingdom of the Netherlands , established_title = Before independence , established_date = Spanish Netherl ...
, 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 Estonia, formally the Republic of Estonia, is a country by the Baltic Sea in Northern Europe. It is bordered to the north by the Gulf of Finland across from Finland, to the west by the sea across from Sweden, to the south by Latvia, a ...
, 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. 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 , image_flag = Flag of Croatia.svg , image_coat = Coat of arms of Croatia.svg , anthem = "Lijepa naša domovino"("Our Beautiful Homeland") , image_map = , map_caption = , capit ...
, 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 do not receive 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 do not 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'') (until 2017, when they were abolished) could not 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 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 meth ...
, 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 The Legislative Assembly of the Macau Special Administrative Region (; ) is the organ of the legislative branch of Macau. It is a 33-member body comprising 14 directly elected members, 12 indirectly elected members representing functional co ...
, a modified D'Hondt method is used. The formula for the quotient in this system is \textstyle\frac. In some cases such as the Czech regional elections, the first divisor (when the party has no seats so far, which is normally 1) was raised to favour larger parties and eliminate small ones. In the Czech case, it is set to 1.42 (approximately \sqrt, termed the Koudelka coefficient after the politician who introduced it). 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 In a first-past-the-post electoral system (FPTP or FPP), formally called single-member plurality voting (SMP) when used in single-member districts or informally choose-one voting in contrast to ranked voting, or score voting, voters cast thei ...
, D'Hondt is applied for the allocation of list seats, taking into account for each party the number of constituency seats it has won. When the seats allocated by D'hondt to a party are greater than the constituency seats that party has won, the extra seats are taken from list seats. In 1989 and 1992,
ACT Legislative Assembly The Legislative Assembly for the Australian Capital Territory (known in short as the ACT Legislative Assembly) is the unicameral legislature of the Australian Capital Territory (ACT). It sits in the Legislative Assembly Building on Civic Sq ...
elections were conducted by the
Australian Electoral Commission The Australian Electoral Commission (AEC) is the independent federal agency in charge of organising, conducting and supervising federal Australian elections, by-elections and referendums. Responsibilities The AEC's main responsibility is to ...
using the "modified d'Hondt" electoral system. The electoral system consisted of the d'Hondt system, the
Australian Senate The Senate is the upper house of the Bicameralism, bicameral Parliament of Australia, the lower house being the House of Representatives (Australia), House of Representatives. The composition and powers of the Senate are established in Chapter ...
system of proportional representation, and various methods for preferential voting for candidates and parties, both within and across party lines. The process involves 8 stages of scrutiny. ABC elections analyst
Antony Green Antony John Green (born 2 March 1960) is an Australian psephologist and commentator. He is the Australian Broadcasting Corporation's chief election analyst. Early years and background Born in Warrington, Lancashire, in northern England, Gre ...
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". 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 Belgium, ; french: Belgique ; german: Belgien officially the Kingdom of Belgium, is a country in Northwestern Europe. The country is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to th ...
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 The Congress of Deputies ( es, link=no, Congreso de los Diputados, italic=unset) is the lower house of the Cortes Generales, Spain's legislative branch. The Congress meets in the Palace of the Parliament () in Madrid. It has 350 members elect ...
with only 44% of the national vote. It may also skew results for small parties with broad appeal at a national level compared to small parties with a local appeal (e.g. nationalist parties). For instance, in the 2008 Spanish general election,
United Left (Spain) United Left ( es, Izquierda Unida , IU) is a federative political movement in Spain that was first organized as a coalition in 1986, bringing together several left-wing political organizations, most notably the Communist Party of Spain. IU was ...
gained 1 seat for 969,946 votes, whereas
Convergence and Union Convergence and Union ( ca, Convergència i Unió, CiU; ) was a Catalan nationalist electoral alliance in Catalonia, Spain. It was a federation of two constituent parties, the larger Democratic Convergence of Catalonia (CDC) and its smaller coun ...
(Catalonia) gained 10 seats for 779,425 votes. However, this is not D'Hondt method's fault but rather the division between regions, at it happens with
first-past-the-post In a first-past-the-post electoral system (FPTP or FPP), formally called single-member plurality voting (SMP) when used in single-member districts or informally choose-one voting in contrast to ranked voting, or score voting, voters cast thei ...
. Even then, D'Hondt method is fairer than first-past-the-post.


Notes


References


External links


Simulator Election calculus simulator based on the modified D'Hondt system

Calculations using the pure d'Hondt method

PHP Implementation of D'Hondt system

Java D'Hondt, Saint-Lague and Hare-Niemeyer calculator


* ttp://blog.stevenkellow.com/excel-dhondt-proportional-representation-calculator/ Downloadable Excel calculator for the D'Hondt method {{DEFAULTSORT:D'Hondt Method Apportionment methods fr:Scrutin proportionnel plurinominal#Méthode d'Hondt