Huntington–Hill method

The Huntington–Hill method (sometimes method of equal proportions) is a highest averages method for assigning seats in a legislature to political parties or states. Since 1941, this method has been used to apportion the 435 seats in the United States House of Representatives following the completion of each decennial census.

The method minimizes the relative difference in the number of constituents represented by each legislator. In other words, the method selects the algorithm such that no transfer of a seat from one state to another can reduce the percent error in representation for both states.

Apportionment method
In this method, as a first step, each of the 50 states is given its one guaranteed seat in the House of Representatives, leaving 385 seats to assign. The remaining seats are allocated one at a time, to the state with the highest average district population, to bring its district population down. However, it is not clear if we should calculate the average before or after allocating an additional seat, and the two procedures give different results. Huntington-Hill uses a continuity correction as a compromise, given by taking the geometric mean of both divisors, i.e.:


 * $$ A_{n} = \frac{P}{\sqrt{n(n+1)}} $$

where P is the population of the state, and n is the number of seats it currently holds before the possible allocation of the next seat.

Consider the reapportionment following the 2010 U.S. census: after every state is given one seat:


 * 1) The largest value of A1 corresponds to the largest state, California, which is allocated seat 51.
 * 2) The 52nd seat goes to Texas, the 2nd largest state, because its A1 priority value is larger than the An of any other state.
 * 3) The 53rd seat goes back to California because its A2 priority value is larger than the An of any other state.
 * 4) The 54th seat goes to New York because its A1 priority value is larger than the An of any other state at this point.

This process continues until all remaining seats are assigned. Each time a state is assigned a seat, n is incremented by 1, causing its priority value to be reduced.

Division by zero
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 party-list representation, small parties would likely be eliminated using some electoral threshold, or the first divisor can be modified.

Examples
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 $√2 1 ≈ 1.41$, 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.

Knesset example
The Knesset (Israel's unicameral legislature), are elected by party-list representation with apportionment by 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.