US House Apportionment (The Huntington-Hill Method)

Like most pages on voting methods, this is a little dry.
For a more engaging explainer, check out CGP Grey’s video below. This is a supplement to his
National Popular Vote Interstate Compact video

History

This is the first debate over proportional representation and would be a framework for us to understand fairness in representation in multi-member bodies. The history of US apportionment features debate between Alexander Hamilton and Thomas Jefferson over the most appropriate way for the seats to be allocated to the states. The Apportionment Act of 1792 was a political concoction initially approved in a North v. South split, favoring the Northern states, and resulting in 120 seats. This allocation did not have the described method but was politically viable to pass the house and senate. George Washington issued a presidential veto, describing it as unfair and unconstitutional.

 

United States [Philadelphia] April 5 1792.

Gentlemen of the House of Representatives

I have maturely considered the Act passed by the two Houses, intitled, “An Act for an apportionment of Representatives among the several States according to the first enumeration,” and I return it to your House, wherein it originated, with the following objections.

First—The Constitution has prescribed that representatives shall be apportioned among the several States according to their respective numbers: and there is no one proportion or divisor which, applied to the respective numbers of the States will yield the number and allotment of representatives proposed by the Bill.

Second—The Constitution has also provided that the number of Representatives shall not exceed one for every thirty thousand; which restriction is, by the context, and by fair and obvious construction, to be applied to the seperate and respective numbers of the States: and the bill has allotted to eight of the States, more than one for thirty thousand.

George Washington.

The Jefferson/D’hondt Method

The first used apportionment method was contained in Art. I, § 2, cl. 3 of the Constitution.  Jefferson recommended apportioning the states sequentially so that the largest state got the first seat, but then had their population de-weighted by half, and then the sums recalculated, and the highest state again getting the next seat. After the first Census in 1790, Congress passed the Apportionment Act of 1792 and adopted the Jefferson method to apportion U.S. Representatives to the states based on population. The Jefferson method was used until the 1830 census.

The population of each state is calculated in the Census. The states are given a quotient equal to their population. The state with the largest quotient wins one seat, and its quotient is recalculated per the formula below. This is repeated until the required number of seats is filled.

quotient = V s + 1{\displaystyle {\text{quot}}={\frac {V}{s+1}}}

where:

  • V is the population of the state, and
  • s is the number of seats that state has been allocated so far, initially 0 for all states.

The population for each state is divided, first by 1, then by 2, then 3, up to the total number of seats to be allocated for the state.

StatesSeats (Vetoed)Seats (Enacted)
New Hampshire54
Massachusetts1614
Vermont32
Rhode Island22
Connecticut87
New York1110
New Jersey65
Pennsylvania1413
Delaware21
Maryland98
Virginia2119
Kentucky22
North Carolina1210
South Carolina76
Georgia22
Total Seats120105

The results of the initial and enacted Apportionment Act of 1792, utilizing the Jefferson Method

The Webster Method (1840-1850, 1910-1930)

The Webster method, proposed in 1832 by Daniel Webster and adopted for the 1840 Census, allocated an additional Representative to states with a fractional remainder greater than 0.5.

quotient = V 2s + 1

The Webster/Sainte-Laguë method divides the number of votes for each party by the odd numbers (1, 3, 5, 7, etc.) and is sometimes considered more proportional than D’Hondt in terms of a comparison between a party’s share of the total vote and its share of the seat allocation. This system can favor smaller parties over larger parties and so encourage splits. Dividing the votes numbers by 0.5, 1.5, 2.5, 3.5, etc. yields the same result.

The Webster/Sainte-Laguë method is sometimes modified by increasing the first divisor to e.g. 1.4, to discourage very small parties from gaining their first seat “too cheaply”.

The Hamilton Method (1850-1900)

The Hamilton (Yes, that Hamilton)/Vinton (largest remainder) method was used from 1850 until 1900. The Vinton or Hamilton method was shown to be susceptible to an apportionment paradox.The Apportionment Act of 1911, in addition to setting the number of U.S. Representatives at 435, returned to the Webster method, which was used following the 1910 and 1930 censuses (no reapportionment was done after the 1920 census).

total votes total seats {\displaystyle {\frac {\text{total votes}}{\text{total seats}}}} 

The Huntington-Hill Method (1941-Current)

The current method, known as the Huntington–Hill method or method of equal proportions, was adopted in 1941 for reapportionment based on the 1940 census and beyond. The revised method was necessary in the context of the cap on the number of Representatives set in the Reapportionment Act of 1929.

The Huntington–Hill method of apportionment assigns seats by finding a modified divisor D such that each constituency’s priority quotient (its population divided by D), 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 sub constituencies.[1] 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 \(D={\sqrt {n(n+1)}}\), 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).

A comparison of the Huntington-Hill method vs the Sainte-Lague/Webster and D’Hondt methods.