# Fair apportionment

## Discrepancy of states

One can easily calculate the number of representatives a state is entitled to by dividing the population of the state (Pi) by the total population of the United States (P), and multiplying that by the number of seats in the House of Representatives (H). This yields the quota for the ith state, Qi = H × Pi/P. However this will generally not be an integer, but representatives are discrete objects, so the assigned number of representatives (Ai) cannot, in general, equal Qi. The objective is to get the assigned numbers as close to the quotas as possibles. The problem is determining what is meant by "close"; once that has been determined there are algorithms to make the assignment. We will illustrate this with the populations of the first 13 states in 1790, and the numbers of representatives which were assigned to them in the Constitution (the House size was 65). The fractional entries under population reflect that slaves were counted as .6. If the reader is surprised at how far the assigned values for some states are from their quotas, the reader may bear in mind that the Constitution was written before the 1890 census was taken, hence the populations of the states were not known.

 State Population Quota Assigned NH 141821.8 2.59 3 MA 475327.0 8.67 8 RI 68445.8 1.25 1 CT 236840.4 4.32 5 NY 331590.4 6.05 6 NJ 179569.8 3.27 4 PA 432878.2 7.89 8 DE 55541.2 1.01 1 MD 278513.6 5.08 6 VA 699264.2 12.75 10 NC 387846.4 7.07 5 SC 206235.4 3.76 5 GA 70842.4 1.29 3

There are several ways to measure the discrepancy between the assigned values and quotas. I shall calculate two of these for Rhode Island and Connecticut.

• What is the discrepancy between the assigned number and the quota measured as a difference?
RI has an assigned number which is .25 too small, and CT has an assigned number which is .68 too large.
• What is the discrepancy between the assigned number and quota measured as a ratio?
RI has the ratio Ai/Qi = .8, which is .2 below 1, but in CT Ai/Qi = 1.16 which is only .16 above 1. Hence which state is further from the goal has been changed by changing the definition of closeness.
The discrepancy (difference) between Ai and Qi was used as the measure of fairness of apportionment from 1850 until 1910, but because voters vote in districts rather than in the state at-large, the ratio is a better measure of the fairness of representation. The quantities Ai/Pi (=(Ai/Qi)(H/P)) giving representatives per voter or Pi/Ai giving voters per representative were used before 1850 and have been used since 1910 to determine the fairest apportionment.

Exercise: For which state is Ai-Qi greatest? least? What does this represent?
For which state is Ai/Pi (or Ai/Qi) greatest? least? What does this represent?
For which state is Pi/Ai (or Qi/Ai) greatest? least? What does this represent?

Using Ai/Qi or Qi/Ai is nice because unfairness is measured from the ratio 1.

## discrepancy between states

In order to increase or decrease the number of representatives from one state, the number of representatives of another state must be decreased or increased since the House size is fixed. Therefore some measure of fairness entailing all the states must be employed.

The first apportionment method employed (1791-1850), called Greatest Divisors or rejectected fractions and propounded by Thomas Jefferson, maximized the minimum district size. Since the district size is Pi/Ai, the minimum district size for the original apportionment is 23614 in Georgia. Since a small district size means more power per voter, maximizing the minimum district size entails minimizing the extent to which anyone has too much power. Note that Pi/Ai = (Qi/Ai)(P/H), hence the minimum of the ratio Qi/Ai is maximized or the maximum of Ai/Qi is minimized (overrepresentation is minimized)

The method used from 1850-1910, called largest fractions or Hare quota and propounded by Alexander Hamilton (but vetoed by George Washington), minimizes the sum over all states of |Ai - Qi|. For the original apportionment this is readily calculated as 11.59.

The present method (1910-date), called equal proportions or method of the geometric mean, minimizes |log((Ai/Pi)/(Aj/Pj))| over all i,j. For the original apportionment the maximum of this ratio is equal to .52 for the contrast of Georgia and North Carolina. This method minimizes the the disparity of representatives per voter between the states, measured as a ratio.

Both the method of greatest divisors and method of the geometric mean allows a discrepancy greater than 1 in magnitude between the Ai and Qi. The metod of largest fractions allows the "Alabama paradox" by which increasing the size of the House can result in a state losing a representative.

Competency: For the apportionment NH-2, MA-9, RI-1, CT-4, NY-6, NJ-3, PA-8,DE-1, MD-5, VA-14, NC-7, SC-4, GA-1;: calculate Qi-Ai for each state, Qi/Ai foreach state, Pi/Ai for each state (compare the minimum to the value calculated with the original apportionment), the sum over all states of |Ai - Qi| (compare this to the value calculated above for the original apportionment), and the maximum of |log((Ai/Pi)/(Aj/Pj))| over all i,j (compare this to the value for the original apportionment). (This is the apportionment for the method of greatest divisors.

I am really more interested in the differences Ai-Qi and the ratios Ai/Qi because these directly measure over or under representation.

Reflection:

Challenge:

May 2003