1. Apportionment Systems
The methods below go under a surprisingly large number of names, partly because the methods were independently discovered for a variety of reasons. Here is a table of equivalency for these names.
Hamilton's method (for AP) conceptually starts by relaxing the requirement that the number of seats assigned to each state be an integer and looking at what the exact quota that each state is entitled to would be. This exact quota qi for state i can be computed by either of two calculations, each of which gives one a slightly different perspective. In the first instance, we can think of state i's share as the percent of the population state i has times the number of seats available. In the second instance one computes the number of people per seat (P/h) (i.e. the size of an ideal district) and divides this into the population of state i, to see its share
qi = (Pi /P)h = pi /(P/h)
One can think of qi as consisting of an integer part plus a fractional part. This integer part is referred to as lower quota, since intuitively each state should get at least this number of seats, and one more than this integer part is referred to as upper quota. Hamilton's method for the AP problem works by giving each state its lower quota. If there are any seats that have not been distributed, these are given out in the order of largest remainder, that is, in order of the size of the fractional parts. Obviously, there is a need for tie-breaking rules in the case that states have equal population. However, both with Hamilton's method and other methods we will discuss later, ties can result from other circumstances than equality of population. (Of course, in the case of CAP, the large numbers involved reduce the chance of such ties. Also, for the CAP problem, the discussion above must be modified so that the requirement that each state get one or more seats be dealt with.) Hamilton's method has a very appealing property. Each state gets either its lower quota or upper quota, that is, the number of seats that a state gets in the House of Representatives is either the largest integer less than or equal to a states' quota qi or one more than this number. Yet, as mentioned in the historical section Hamilton's method can fail, using a fixed set of populations, to guarantee that as the house size goes up, a state will not lose(!) a seat. Thus, Hamilton's method violates the sensible requirement, for some AP problems, of house size monotonicity.
The table above shows the results of two consecutive censuses where there are three regions and 100 seats to distribute to the three states in a regional legislature. Note that the population of A has gone up and the population of B has gone up, while the population of C has gone down between the two censuses.
We will suppose that we have 10 seats to distribute and note that the total population of all the states is 1100.
In grade school you probably learned how to round decimal numbers to the nearest integer. This procedure required that if the fractional part were .5 or more, one rounded up to the next largest integer; if the fractional part was smaller than .5, then you rounded down. If we apply this approach to the numbers in the example above we would give 6 seats to A, 3 seats to B, and 1 seat to C. Since these numbers add to 10, we can use these values to apportion the 10 seats.
Again we will suppose that we have 10 seats to distribute and note that the total population of all the states is 1100. Since there are 1100 people and 10 seats to distribute, ideally we would like to be able to have one district for every 110 people (i.e. 1100/10). This number 110 is known as the ideal district size. Notice that it usually will not be an integer, but we allow this. Using this ideal district size we can compute each state's exact quota.
Using the grade school approach to rounding we would give 6 seats to A, 2 seats to B, and 1 seat to C, which adds up to only 9 seats, one short of the 10 we must distribute. The Webster method approach to handling this problem is that one can modify the ideal district size to obtain a modified district size (MDS). By dividing the state populations by the MDS one gets modified quotas which hopefully, when rounded in the usual way, will distribute h seats. Since in our example we distributed too few seats, we must use a smaller MDS, which will increase the size of the fractional parts, so when we round them we distribute more seats (but not too many). We will use a MDS of 107.2. (Can you figure out from the calculation below why that value was chosen?) Generally there will be an interval of numbers which if we use any number in this range as an MDS, will distribute the desired h seats.)
When we round we give 6 seats to A, 3 seats to B and 1 seat to C, which adds up to 10 seats, the number we need to distribute.
Rounding all the fractions down gives the assignment of 7 seats to A, 2 seats to B, and 1 seat to C, which gives the desired total of 10.
Rounding all the fractions down gives the assignment of 6 seats to A, 2 seats to B, and 2 seats to C, which gives the desired total of 10.
Note that in this example all three methods give each state either its lower or upper quota. However, it is important to realize that this is not true of all examples. Furthermore the method used to apportion the United States House of Representatives (described below) need not obey this fairness rule.
Example 3 (continued)
The table above indicates the priority numbers for the Adams method. In the first row we divided by 0, in the second row by 1, in the third row by 2, etc.
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