TheInfoList

The D'Hondt method or the Jefferson method is a highest averages method for allocating seats in an electoral system and thus a type of party-list proportional representation. The method described is named in the United States after , who introduced the method for proportional allocation of seats in the
United States House of Representatives The United States House of Representatives is the lower house of the United States Congress, with the United States Senate, Senate being the upper house. Together they compose the national Bicameralism, bicameral legislature of the United St ...
in 1792, as well as in Europe after Belgian mathematician Victor D'Hondt, who described the methodology in 1878. There are two forms:
closed list Closed list describes the variant of party-list proportional representation where voters can (effectively) only vote for political parties A political party is an organization that coordinates candidates to compete in a country's elections ...
(under which a party selects the order of election of their candidates) and open list (under which 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. 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 and coalitions over scattered small parties. In comparison, the Webster/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. 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 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 Albania ( ; sq, Shqipëri or Shqipëria), officially the Republic of Albania ( sq, Republika e Shqipërisë), is a country in Southeast Europe, Southeastern Europe. It is located on the Adriatic Sea, Adriatic and Ionian Sea within the Mediter ...
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Angola , national_anthem = " Angola Avante"() , image_map = , map_caption = , capital = Luanda , religion = , religion_year = 2015 , religion_ref = , coordinates = , largest_city = capital , ...
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Argentina Argentina (), officially the Argentine Republic ( es, link=no, República Argentina), is a country located mostly in the southern half of South America. Sharing the bulk of the Southern Cone with Chile to the west, the country is also bordered b ...
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Armenia Armenia (; hy, Հայաստան, translit=Hayastan, ), officially the Republic of Armenia,, is a landlocked country located in the Armenian Highlands of Western Asia.The UN]classification of world regions places Armenia in Western Asia; the ...
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Austria Austria (, ; german: Österreich ), officially the Republic of Austria (german: Republik Österreich, links=no, ), is a landlocked Eastern Alps, East Alpine country in the southern part of Central Europe. It is composed of nine States of ...
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Belgium Belgium ( nl, België ; french: Belgique ; german: Belgien ), officially the Kingdom of Belgium, is a country in Western Europe. It is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to the s ...
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Bolivia Bolivia ; ay, Wuliwya ; Quechuan languages, Quechua: ''Puliwya'' , officially the Plurinational State of Bolivia, is a landlocked country located in western-central South America. The constitutional capital is Sucre, while the seat of go ...
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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 8.5 million square kilometers (3.2 million square miles) and with over 211 milli ...
, Burundi, Cambodia, Cape Verde, Chile, Colombia, Croatia, Denmark, the Dominican Republic, East Timor, Ecuador, El Salvador, Estonia, Fiji, Finland, Greenland, Guatemala, Hungary, Iceland, Israel, Japan, Luxembourg, Moldova, Monaco, Montenegro, Mozambique, Netherlands, Nicaragua, North Macedonia, Paraguay, Peru, Poland, Portugal, Romania, San Marino, Serbia, Slovenia, Spain, Switzerland, Turkey, Uruguay, and Venezuela. The system has also been used for the "top-up" seats in the Scottish Parliament, the National Assembly for Wales, Welsh Parliament and the London Assembly; in some countries for elections to the European Parliament; and during the 1997 Constitution of Thailand, 1997 Constitution era to allocate party-list parliamentary seats in Thailand. 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. 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 and for the allocation of ministers in the Northern Ireland Assembly.

# Allocation

After all the votes have been tallied, successive quotients 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: * ''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 ''i''th row and ''j''th column is the number of votes won by the ''i''th 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 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. 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. The chart below shows an easy way to perform the calculation:

### 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 (mathematician), 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 $p$, $s_p \in \left[0,1\right],\;\sum_p s_p = 1$, :$v_p$ – the vote share of party $p$, $v_p \in \left[0,1\right],\;\sum_p v_p = 1$. 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 $p$ are ::$r_p = v_p - \left(1-\pi^*\right) s_p,\; r_p \in \left[0, v_p\right], \sum_p\,r_p=\pi^*$. 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 $1/1.15 = 0.87 = 1 - \pi^*$. 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.,$1 - 0.87 = 0.13$. The decomposition of the votes into represented and residual ones is shown in the table below.

# 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. Ten days after the veto, Congress passed a new method of apportionment, now known as Jefferson's Method. Statesman and future US President devised the method in 1792 for the United States congressional apportionment, U.S. congressional apportionment pursuant to the 1790 United States Census, First United States Census. It was used to achieve the proportional distribution of seats in the United States House of Representatives, House of Representatives among the states until 1842. Victor D'Hondt 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 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 election threshold, 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 Shqipëria), officially the Republic of Albania ( sq, Republika e Shqipërisë), is a country in Southeast Europe, Southeastern Europe. It is located on the Adriatic Sea, Adriatic and Ionian Sea within the Mediter ...
(3% for single parties, 5% for coalitions of two or more parties, no threshold is applied for independent individuals); Denmark (2%); East Timor, Spain, Serbia, and Montenegro (3%); Israel (3.25%); Slovenia (4%); Croatia, Fiji, Romania and Tanzania (5%); Russia (5%); Turkey (10%); Poland (5%, or 8% for coalitions; but does not apply for ethnic-minority parties), Hungary (5% for single party, 10% for two-party coalitions, 15% for coalitions of 3 or more parties) and
Belgium Belgium ( nl, België ; french: Belgique ; german: Belgien ), officially the Kingdom of Belgium, is a country in Western Europe. It is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to the s ...

# 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 $\textstyle\frac$. In some cases such as the Elections in the Czech Republic, 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 Cymru – Welsh Parliament, 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, Australian Capital Territory Legislative Assembly, 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. The process involves 8 stages of scrutiny. ABC elections analyst Antony Green 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 (electoral alliance), 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 ( nl, België ; french: Belgique ; german: Belgien ), officially the Kingdom of Belgium, is a country in Western Europe. It is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to the s ...
the Election threshold, 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 (Spain), People's Party gained an absolute majority in the Congress of Deputies (Spain), Congress of Deputies 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) gained 1 seat for 969,946 votes, whereas Convergence and Union (Catalonia) gained 10 seats for 779,425 votes.