Huntington–Hill method


The Huntington–Hill method of apportionment assigns seats by finding a modified divisor D such that each constituency's priority quotient, 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. 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, n being the number of seats a state or party is currently allocated in the apportionment process and n+1 is the number of seats the state or party would have if it is assigned to the party list. 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.
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.

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
where
In legislatures which use an exclusion threshold, the qualification value would be:
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:
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 or a greater number, which may be uniform or vary between states.
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
where:
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 by the number of seats, which in this case gives a qualification value of 28,750 votes.
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 total votes is divided by 1.41, 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. The slight favouring of the largest party over the smallest is apparent.
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:
Compared with the actual apportionment, Kulanu would have lost one seat, while The Jewish Home would have gained one seat.