# Huntington–Hill method

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The Huntington–Hill method of apportionment assigns seats by finding a modified divisor D such that each constituency's priority quotient (its population divided by D), using the geometric mean 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. [1] When envisioned as a proportional electoral system, it is effectively a highest averages method of party-list proportional representation in which the divisors are given by ${\displaystyle \scriptstyle D={\sqrt {n(n+1)}}}$, n being the number of seats a state or party is currently allocated in the apportionment process (the lower quota) and n+1 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 House of Lords elections under the ill-fated House of Lords Reform Bill. [2]

## Contents

The 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. It is credited to Edward Vermilye Huntington and Joseph Adna Hill. [3]

## 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 would be the Hare quota, or

${\displaystyle {\frac {{\mbox{total}}\;{\mbox{votes}}}{{\mbox{total}}\;{\mbox{seats}}}}}$,

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.

In legislatures which use an exclusion threshold, the qualification value would be:

${\displaystyle {\mbox{exclusion}}\;{\mbox{threshold}}\;{\mbox{[percentage]}}({\frac {{\mbox{total}}\;{\mbox{votes}}}{\mbox{100}}})}$.

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:

${\displaystyle {\mbox{exclusion}}\;{\mbox{threshold}}\;{\mbox{[percentage]}}({\frac {{\mbox{total}}\;{\mbox{seats}}}{\mbox{100}}})}$

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 Brazil) or vary between states (as in Canada).

It can also be skipped if the Huntington-Hill system is used in the nationwide stage of a national remnant 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 quotients 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

${\displaystyle A_{n}={\frac {V}{\sqrt {s(s+1)}}}}$

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 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 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 the Hare quota, and giving every party which polled at least the Hare quota one seat. The Hare quota is calculated by dividing the number of votes cast (230,000) by the number of seats (8), which in this case gives a qualification value of 28,750 votes.

DenominatorVotesIs the party eligible or disqualified?
Party A100,000Eligible
Party B80,000Eligible
Party C30,000Eligible
Threshold28,750
Party D20,000Disqualified

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 1.41 (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 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 a 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, 100,000/230,000 × 8 = 3.48) [4]

Denominator1.412.453.464.475.486.487.488.49Initial
seats
Seats
won (*)
Total
Seats
Proportionate
seats
Party A70,711*40,825*28,868*22,36118,25715,43013,36311,7851343.5
Party B56,569*32,660*23,09417,88914,60612,34410,6909,4281232.8
Party C21,21312,2478,6606,7085,4774,6294,0093,5361011.0
Party DDisqualified00.7

If the number of seats was equal in size to the number of votes case for the qualified parties, this method would guarantee that the appointments 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 increases, differences emerge: all 120 members of the Knesset, Israel'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, held in 2015, the 120 seats in the 20th Knesset would have been apportioned as follows:

PartyVotesSeats (actual results, D'Hondt)Seats (hypothetical results, Huntington–Hill)Upper quota (Huntington–Hill)Geometric mean (Huntington–Hill)+/–
Likud 985,40830303130.500
Zionist Union 786,31324242524.500
Joint List 446,58313131413.490
Yesh Atid 371,60211111211.490
Kulanu 315,360109109.49–1
The Jewish Home 283,91089109.49+1
Shas 241,6137787.480
Yisrael Beiteinu 214,9066676.480
United Torah Judaism 210,1436676.480
Meretz 165,5295565.480
Source: CEC

Compared with the actual apportionment, Kulanu would have lost one seat, while The Jewish Home would have gained one seat.

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## References

1. "Congressional Apportionment". NationalAtlas.gov. Archived from the original on 2009-02-28. Retrieved 2009-02-14.
2. Draft House of Lords Reform Bill: report session 2010-12, Vol. 2. Google Books. 23 April 2012. ISBN   9780108475801 . Retrieved 6 November 2017.
3. "The History of Apportionment in America". American Mathematical Society. Retrieved 2009-02-15.
4. Note the slight favouring of the largest party over the smallest (if we subtract Proportionate seats from Total Seats Party A - the largest party - gets the highest total (0.5) while Party B only gets 0.2, Party C gets 0 and Party D gets -0.7)