Huntington–Hill 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 priority number. Thus, the 51st seat would go to the most populous state (currently California). The priority number is determined by the ratio of the state population to the geometric mean of the number of seats it currently holds in the assignment process, n (initially 1), and the number of seats it would hold if the seat were assigned to it, n+1. Symbolically, the priority number A_n is

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

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. An equivalent, recursive definition is

${\displaystyle A_{m+1}={\sqrt {\frac {m}{m+2}}}\ A_{m}}$

${\displaystyle A_{n}={\sqrt {\frac {n-1}{n+1}}}\ A_{n-1}}$

where n is still the number of seats the state has before allocation of the next (in other words, for the mth allocation, n = m-1, where m > 1), and for n = 1, the initial A₁ is explicitly defined by the non-recursive formula as

${\displaystyle A_{1}={\frac {P}{\sqrt {2}}}}$

Consider the reapportionment following the 2010 U.S. census: beginning with all states initially being allocated one seat, the largest value of A₁ corresponds to the largest state, California, which is allocated seat 51. After being allocated its 2nd seat, its priority value decreases to its A₂ value, which is reordered to a position back in line. The 52nd seat goes to Texas, the 2nd largest state, because its A₁ priority value is larger than the A_n of any other state. However, the 53rd seat goes back to California because its A₂ priority value is larger than the A_n of any other state. The 54th seat goes to New York because its A₁ priority value is larger than the A_n 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 and reordered among the states, whereupon another state normally rises to the top of the list.

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 ${\textstyle D={\sqrt {n(n+1)}}}$, 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).

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 could be a predefined quota, such as the Hare, Droop, or Imperiali quota.

In legislatures which use an exclusion threshold, the qualification value would be equipotent to the threshold, that is:

${\displaystyle {\text{exclusion threshold [percentage]}}\left({\frac {\text{total votes}}{100}}\right)}$

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.

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:

${\displaystyle {\text{exclusion threshold [percentage]}}\left({\frac {\text{total seats}}{100}}\right)}$

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

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

where:

• V is the population of the state or the total number of votes that party received, and
• n is the number of seats that the state or party has been allocated so far.

Because squaring does not change the relative order of positive values, one can avoid the square root by instead comparing the values

${\displaystyle (A_{n})^{2}={\frac {V^{2}}{n(n+1)}}\,.}$