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
An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections m ...
. It contrasts with various
highest averages methods (also known as divisor methods).
Method
The ''largest remainder method'' requires the numbers of votes for each party to be divided by a quota representing the number of votes ''required'' for a seat (i.e. usually the total number of votes cast divided by the number of seats, or some similar formula). The result for each party will usually consist of an
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 languag ...
part plus a
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 ...
. Each party is first allocated a number of seats equal to their integer. This will generally leave some
remainder seats unallocated: the parties are then ranked on the basis of the fractional remainders, and the parties with the largest remainders are each allocated one additional seat until all the seats have been allocated. This gives the method its name.
Quotas
There are several possibilities for the quota. The most common are:
the
Hare quota and the
Droop quota. The use of a particular quota with the largest remainders method is often abbreviated as "LR-
uota name, such as "LR-Droop".
The Hare (or simple) quota is defined as follows
:
It is used for legislative elections in
Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eig ...
(with a 5% exclusion threshold since 2016),
Ukraine
Ukraine ( uk, Україна, Ukraïna, ) is a country in Eastern Europe. It is the second-largest European country after Russia, which it borders to the east and northeast. Ukraine covers approximately . Prior to the ongoing Russian inv ...
(5% threshold),
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% threshold),
Lithuania (5% threshold for party and 7% threshold for coalition),
Tunisia,
Taiwan
Taiwan, officially the Republic of China (ROC), is a country in East Asia, at the junction of the East and South China Seas in the northwestern Pacific Ocean, with the People's Republic of China (PRC) to the northwest, Japan to the nort ...
(5% threshold),
Namibia
Namibia (, ), officially the Republic of Namibia, is a country in Southern Africa. Its western border is the Atlantic Ocean. It shares land borders with Zambia and Angola to the north, Botswana to the east and South Africa to the south and ea ...
and
Hong Kong
Hong Kong ( (US) or (UK); , ), officially the Hong Kong Special Administrative Region of the People's Republic of China (abbr. Hong Kong SAR or HKSAR), is a city and special administrative region of China on the eastern Pearl River Delta i ...
. The Hamilton method of apportionment is actually a largest-remainder method which uses the Hare Quota. It is named after
Alexander Hamilton, who invented the largest-remainder method in 1792. It was first adopted to
apportion the U.S. House of Representatives every ten years between 1852 and 1900.
The
Droop quota is the integer part of
:
and is applied in elections in
South Africa
South Africa, officially the Republic of South Africa (RSA), is the Southern Africa, southernmost country in Africa. It is bounded to the south by of coastline that stretch along the Atlantic Ocean, South Atlantic and Indian Oceans; to the ...
. The
Hagenbach-Bischoff quota is virtually identical, being
:
either used as a fraction or rounded up.
The Hare quota tends to be slightly more generous to less popular parties and the Droop quota to more popular parties. This means that Hare can arguably be considered more proportional than Droop quota. However,
an example shows that the Hare quota can fail to guarantee that a party with a majority of votes will earn at least half of the seats (though even the Droop quota can
very rarely do so).
The
Imperiali quota
:
is rarely used since it suffers from the defect that it might result in more seats being allocated than there are available (this can also occur with the
Hagenbach-Bischoff quota but it is very unlikely, and it is impossible with the Hare and Droop quotas). This will certainly happen if there are only two parties. In such a case, it is usual to increase the quota until the number of candidates elected is equal to the number of seats available, in effect changing the voting system to the Jefferson apportionment formula (see
D'Hondt method
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, or in party-list proportional representation systems. It belongs to the class of highest ...
).
Examples
These examples take an election to allocate 10 seats where there are 100,000 votes.
Hare quota
Droop quota
Pros and cons
It is relatively easy for a voter to understand how the largest remainder method allocates seats. The Hare quota gives an advantage to smaller parties while the Droop quota favours larger parties. However, whether a list gets an extra seat or not may well depend on how the remaining votes are distributed among other parties: it is quite possible for a party to make a slight percentage gain yet lose a seat if the votes for other parties also change. A related feature is that increasing the number of seats may cause a party to lose a seat (the so-called
Alabama paradox). The
highest averages methods avoid this latter paradox; but since no apportionment method is entirely free from paradox, they introduce others like quota violation (see
Quota rule).
Technical evaluation and paradoxes
The largest remainder method satisfies the
quota rule (each party's seats amount to its ideal share of seats, either rounded up or rounded down) and was designed to satisfy that criterion. However, that comes at the cost of
paradoxical behaviour. The
Alabama paradox is exhibited when an increase in seats apportioned leads to a decrease in the number of seats allocated to a certain party. In the example below, when the number of seats to be allocated is increased from 25 to 26 (with the number of votes held constant), parties D and E counterintuitively end up with fewer seats.
With 25 seats, the results are:
With 26 seats, the results are:
References
External links
Hamilton method experimentation appletat
cut-the-knot
{{voting systems
Party-list proportional representation
Apportionment methods