Apportionment is a task that may never have everyone agree on the proper method of implementation. The idea is to fairly assign a specific number of seats to various states or regions based on population. Obviously states with larger populations should have more seats awarded, but the exact method of division is far from agreed upon. The purpose of this assignment is to study the basics of two well-known methods of apportionment and discuss their fairness.

The first method is the Hamilton method. For this fictitious situation there are ten states with a total population of 532188 among which 100 seats are to be awarded. The first step is to find the divisor:

D = 532188 / 100 = 5321.88.

The next step is to find the standard and lower quotas. This will be done for the first state with the understanding that the procedure is repeated for all remaining states. The standard quota is found by dividing the state’s population by the divisor:

Standard Quota (1) = 15475 / 5321.88 = 2.9078

Lower Quota (1) = Integer (2.9079) = 2.

The last step before beginning apportionment is to also list the fractional part. This is merely what is left after subtracting the lower quota from the standard quota:

Fractional Part (1) = 2.9078 – 2 = 0.9078.

To begin, assign the lower quota to each state. The only exception is for a state whose lower quota is zero (like state 5 in this example). For this state assign one seat. In this example once this was done there were 96 seats assigned and 4 remaining. The distribution of assigned seats (in numerical order by state) is 2, 6, 18, 16, 1, 16, 8, 15, 10, and 4. In order to assign the remaining four seats the fractional parts are examined. Excluding state 5 (since it already has its upper quota of seats) examine the fractional parts and find the four largest. For this example those are states 1, 2, 8, and 10. Finally the four seats that remain to be awarded are assigned to the states with the larges fractional parts resulting in an apportionment of 3, 7, 18, 16, 1, 16, 8, 16, 10, and 5.

The average constituency for each state is found by dividing the population of the state by the number of seats awarded. In a perfect situation each state would have a constituency that is equal to the divisor. Obviously that is not the case, but states should have constituencies that are near the divisor. The constituency for state 1 is:

15475 / 3 = 5158.333.

The absolute unfairness of the apportionment is the difference between the largest constituency and the smallest constituency. The relative unfairness is ratio of the absolute unfairness to the smallest constituency. For this example these values are:

Absolute unfairness = 5521.625 – 369 = 5152.625

Relative unfairness = 5152.625 / 369 = 13.964.

The most obvious way that population change could affect this example is if 800 residents of state 2 either moved to or were reincorporated into state 4. This would reduce the population of state 2 to 34844 and increase the population of state 4 t o89146. The reason for picking these two states is that state 2 got the last additional seat due to the fractional parts and state 4 was next in line. The new fractional parts after this change would be

34844 / 5321.88 = 6.5473 0.5473

89145 / 5321.88 = 16.7508 0.7508.

This relatively small change in population would result in state 2 losing one of their seats and state 4 gaining that seat.

An Alabama paradox occurs when an increase in the total number of seats to be apportioned causes a state to lose one of its seats. The reason this happens is because states with larger populations have their fractional part increase at a greater rate. It is possible for enough states that are slightly lower than the victim state will have their fractional part surpass that of the victim state causing the loss of a seat. The Huntington-Hill method helps to avoid this paradox by assigning the additional seats using a geometric mean rather than a simple larger fractional part. Unfortunately it also has the possibility of violating the quota rule, where no state can be given less than its lower quota or more than its upper quota. As a possible check the apportionments were calculated (and results listed on the Excel sheet) for total seats of 101 through 106; no occurrence of the Alabama paradox was noted.

In the first application of the Huntington-Hill method the total number of seats awarded did not equal the number of seats available. Thus a new divisor needed to be attempted; an easy way to choose this was to divide the population by 99 instead of 100. This divisor produced a working solution, and is outlined here. The divisor and standard quotas were determined in the same manner as they were for the Hamilton method. After that the geometric mean of the upper quota and the lower quota for each state was found. For state 1 this resulted in:

Standard quota: 15475 / 5375.636 = 2.879

Geometric mean: (2 * 3)^{1/2} = 2.449.

Since the fractional part of the standard quota is larger than the fractional part of the geometric mean, state 1 is assigned a number of seats equal to its upper quota (3). Compare this with the result for state 3:

Standard quota: 98756 / 5375.636 = 18.371

Geometric mea: (18 * 19)1/2 = 18.493.

Here the fractional part of the standard quota is less than the fractional part of the geometric mean, so state 4 is assigned a number of seats equal to its lower quota (18). The final apportionment using this system is 3, 7, 18, 16, 1, 16, 8, 16, 10, and 5. Notice that this is the same apportionment achieved using the Hamilton method.

Since both apportionment methods result in the same division I feel that, at least in this case, apportionment is a fair way of dividing the seats. The only thing that possibly seems unfair is the constituency of state 5. The only way to rectify this that I see is to absorb state 5 into the other states leaving a total of 9 states. This is not a simple task, and would most certainly depend on which states bordered state 5. This was a very instructive project. It was complex enough to afford some rigor in determining the apportionments. It was also instructive in that it demonstrated both how apportionment methods can be fair and unfair.