Huntington–Hill method

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 be assigned. 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.

Example

Consider an example to distribute 8 seats between three parties A, B, C having respectively 100,000, 80,000 and 30,000 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 (Parties A, B, and C) total votes is divided by ≈ 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.

References

References

1. "Congressional Apportionment". NationalAtlas.gov.
2. "U.S. Code Title 2, Section 2a: Reapportionment of Representatives.".
3. "Computing Apportionment". [[United States Census Bureau]].
4. Pukelsheim, Friedrich. (2017). "Divisor Methods of Apportionment: Divide and Round". Springer International Publishing.
5. Pukelsheim, Friedrich. (2017). "Divisor Methods of Apportionment: Divide and Round". Springer International Publishing.