Huntington–Hill method

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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 the average should be calculated 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.:^[4]

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

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. Each time a state is assigned a seat, its n is incremented by 1, causing its priority value to be reduced.

Consider the reapportionment following the 2020 U.S. census. After every state is given one seat:

1. The 51st allocated seat goes to the state with the largest A₁ priority value, which in this instance is the largest-population state, California, with the value 39,576,757/√(1(1+1)) = 27,984,993 ≈ 28.0 M. Its A₂ priority value is then 39,576,757/√(2(2+1)) ≈ 16.2 M.
2. The 52nd seat goes to Texas, the 2nd largest state, because its A₁ ≈ 20.6 M is larger than the A_n of any other state.
3. The 53rd seat goes back to California, because its A₂ ≈ 16.2 M is again larger than the A_n of any other state; it now has three seats.
4. The 54th seat goes to Florida, the 3rd largest state, because its A₁ ≈ 15.3 M is the largest at this point.
5. The 55th seat is New York’s, the 4th largest state, because its A₁ ≈ 14.3 M is now the largest.
6. The 56th seat goes again to Texas for three seats, as its A₂ ≈ 11.9 M is the largest.
7. The 57th seat is then California’s for four seats, as its A₃ ≈ 11.4 M is the largest.
8. The 58th seat then goes to Pennsylvania, the 5th largest state, with its A₁ ≈ 9.2 M being the largest.

This process continues until all remaining seats are assigned. This process is illustrated in this table, where the maximum A value is highlighted, along with the increased number of seats assigned to that state:

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.^[4] In the U.S. House of Representatives, this is ensured by guaranteeing each state at least one seat;^[4] in party-list representation, small parties would likely be eliminated using some electoral threshold, or the first divisor can be modified.

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 √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.

• Edward Vermilye Huntington
• Highest averages method
• Joseph Adna Hill