Huntington–Hill method
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The Huntington–Hill method is a way of allocating seats proportionally for representative assemblies such as 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 chamber of the United States Congress, with the Senate being the upper chamber. Together they ...
. The method assigns seats by finding a modified divisor ''D'' such that each constituency's priority quotient (its population divided by ''D''), using the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of the lower and upper quota for the divisor, yields the correct number of seats that minimizes the percentage differences in the size of subconstituencies. When envisioned as a proportional
electoral system 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 is effectively a highest averages method of
party-list proportional representation Party-list proportional representation (list-PR) is a subset of proportional representation electoral systems in which multiple candidates are elected (e.g., elections to parliament) through their position on an electoral list. They can also be u ...
in which the divisors are given by D=\sqrt, ''n'' being the number of seats a state or party is currently allocated in the apportionment process (the lower quota) and is the number of seats the state or party ''would'' have if it is assigned to the party list (the upper quota). Although no legislature uses this method of apportionment to assign seats to parties after an election, it was considered for
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elections under the ill-fated House of Lords Reform Bill. This method is how the United States House of Representatives assigns the number of representative seats to each state — the purpose for which it was devised — and the
Census Bureau The United States Census Bureau (USCB), officially the Bureau of the Census, is a principal agency of the Federal Statistical System of the United States, U.S. Federal Statistical System, responsible for producing data about the Americans, Ame ...
calls it the method of equal proportions. It is credited to
Edward Vermilye Huntington Edward Vermilye Huntington (April 26, 1874November 25, 1952) was an American mathematician. Biography Huntington was awarded the B.A. and the M.A. by Harvard University in 1895 and 1897, respectively. After two years' teaching at Williams College ...
and Joseph Adna Hill.


Allocation

In a legislative election under the Huntington–Hill method, after the votes have been tallied, the qualification value would be calculated. This step is necessary because in an election, unlike in a legislative apportionment, not all parties are always guaranteed at least one seat. If the legislature concerned has no exclusion threshold, the qualification value could be a predefined quota, such as the Hare, Droop, or
Imperiali quota The Imperiali quota is a formula used to calculate the minimum number, or quota, of votes required to capture a seat in some forms of single transferable vote or largest remainder method party-list proportional representation voting systems. It i ...
. In legislatures which use an exclusion threshold, the qualification value would be equipotent to the threshold, that is: :\text \left(\frac\right). where * ''total votes'' is the total valid poll; that is, the number of valid (unspoilt) votes cast in an election. * ''total seats'' is the total number of seats to be filled in the election. Every party polling votes equal to or greater than the qualification value would be given an initial number of seats, again varying if whether or not there is a threshold: In legislatures which do not use an exclusion threshold, the initial number would be 1, but in legislatures which do, the initial number of seats would be: :\text\left(\frac\right). with all fractional remainders being rounded up. In legislatures elected under a mixed-member proportional system, the initial number of seats would be further modified by adding the number of single-member district seats won by the party before any allocation. Determining the qualification value is not necessary when distributing seats in a legislature among states pursuant to census results, where all states are guaranteed a fixed number of seats, either one (as in the US) or a greater number, which may be uniform (as in
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) or vary between states (as in
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). It can also be skipped if the Huntington-Hill system is used in the nationwide stage of a
national remnant National remnant (Turkish ''Ulusal artık'' or ''milli bakiye'') is a method used to allocate seats in a party-list proportional representation electoral system that has electoral districts. The system uses a Largest remainder method to determine ...
system, because the only qualified parties are those which obtained seats at the subnational stage. After all qualified parties or states received their initial seats, 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, as in other Highest Averages methods, for each qualified party or state, and seats would be repeatedly allocated to the party or state having the highest quotient until there are no more seats to allocate. The formula of quotients calculated under the Huntington–Hill method is :A_n = \frac where: * ''V'' is the population of the state or the total number of votes that party received, and * ''s'' is the number of seats that the state or party has been allocated so far.


Example

Even though the Huntington–Hill system was designed to distribute seats in a legislature among states pursuant to census results, it can also be used, when putting parties in the place of states and votes in place of population, for the mathematically equivalent task of distributing seats among parties pursuant to an election results in a
party-list proportional representation Party-list proportional representation (list-PR) is a subset of proportional representation electoral systems in which multiple candidates are elected (e.g., elections to parliament) through their position on an electoral list. They can also be u ...
system. A party-list PR system requires large multi-member districts to function effectively. In this example, 230,000 voters decide the disposition of 8 seats among 4 parties. Unlike the D'Hondt and Sainte-Laguë systems, which allow the allocation of seats by calculating successive quotients right away, the Huntington–Hill system requires each party or state have at least one seat to avoid a
division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
error. In the U.S. House of Representatives, this is ensured by guaranteeing each state at least one seat; in a single-stage PR election under the Huntington–Hill system, however, the first stage would be to calculate which parties are eligible for an initial seat. This could be done by excluding any parties which polled less than a predefined quota, and giving every party which polled at least the quota one seat. In this case, the qualified parties stay the same regardless of quota. Each eligible party is assigned one seat. With all the initial seats assigned, the remaining five seats are distributed by a priority number calculated as follows. Each eligible party's (Parties A, B, and C) total votes is divided by (the square root of the product of 1, the number of seats currently assigned, and 2, the number of seats that would next be assigned), then by approximately 2.45, 3.46, 4.47, 5.48, 6.48, 7.48, and 8.49. The 5 highest entries, marked with asterisks, range from 70,711 down to 28,868. For each, the corresponding party gets another seat. For comparison, the "Proportionate seats" column shows the exact fractional numbers of seats due, calculated in proportion to the number of votes received (For example, ). If the "Total Seats" column is less than the "Proportionate seats" column (Parties C and D in this example) the party is under-represented. Conversely, if the "Total Seats" column is greater than the "Proportionate seats" column (Parties A and B in this example) the party is over-represented. If the number of seats was equal in size to the number of votes cast, this method would guarantee that the apportionments would equal the vote shares of each party. In this example, the results of the apportionment is identical to one under the D'Hondt system. However, as the
District magnitude An electoral district, also known as an election district, legislative district, voting district, constituency, riding, ward, division, or (election) precinct is a subdivision of a larger state (a country, administrative region, or other poli ...
increases, differences emerge: all 120 members of the
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,
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 ...
's unicameral legislature, are elected under the D'Hondt method. Had the Huntington–Hill method, rather than the D'Hondt method, been used to apportion seats following the elections to the
20th Knesset The members of the 20th Knesset were 2015 Israeli legislative election, elected on 17 March 2015 and sworn in on 31 March 2015. Members of the Knesset Replacements The twentieth Knesset had the highest number of replacements in Israeli histo ...
, held in 2015, the 120 seats in the 20th Knesset would have been apportioned as follows: Compared with the actual apportionment,
Kulanu Kulanu ( he, כולנו, lit. ''All of Us'') was a centrist political party in Israel founded by Moshe Kahlon that focused on economic and cost-of-living issues. History The party was established on 27 November 2014 following months of spec ...
would have lost one seat, while
The Jewish Home The Jewish Home ( he, הַבַּיִת הַיְהוּדִי, HaBayit HaYehudi) is an Orthodox Jewish and religious Zionist political party in Israel. It was originally formed by a merger of the National Religious Party, Moledet, and Tkuma in ...
would have gained one seat.


Footnotes


References

{{DEFAULTSORT:Huntington-Hill Method Apportionment methods