A couple of weeks ago, I was playing with apportionment using the Hamilton method (which Congress no longer uses). Now, I've written a program to apportion seats using the Huntington-Hill method, which is the method that Congress uses today. Having run it several times, I've come to the following conclusion:
Congress got it wrong in 2000.
A short overview of the way that Huntington-Hill works:
- Give every state 1 seat to start.
- Assign priorities to the states, based on each state's population.
- Repeat the following steps until all seats are apportioned:
- Find the state with the highest priority.
- Assign that state another seat.
- Re-calculate its priority to reflect the additional seat.
This is the exact algorithm that my program used. When I ran it, my result matched 99.5% of the actual apportionment -- that is, out of 435 seats, there were only two differences. North Carolina lost one seat to Utah. This (the fact that NC and UT switch seats) is notable in itself, for reasons I will discuss later.
In attempting to explain this discrepancy, I looked at the ratio between each state's population and its highest priority. I call this ratio the "initial ratio".
Because of the way that the Huntington-Hill method is defined, that initial ratio should always equal the square root of 2, which is approximately 1.414214.
On average, the initial ratios that came from my program differed from the expected value by one part in ten million, an error that can be attributed to rounding. Specifically, the mean of the deviations was 0.000015%, with a standard deviation of 0.000019%. The expected deviation is 0, so the initial ratios I calculated agree with what Huntington-Hill should produce.
Congress's numbers, however, don't agree: The priorities that the Census Bureau published yield initial ratios that differ on average by a few parts in a thousand. Here, the mean of the deviations was 0.212%, with a standard deviation of 0.065%. In this case, the calculated mean deviation is 3.3 error bars away from the expected deviation. Congress's numbers do not agree with what Huntington-Hill should produce.
Why do they not agree? I have come up with three possibilities, in order of decreasing likelihood:
- Congress is not using published 2000 Census data to apportion seats
- Congress is not using the Huntington-Hill method to apportion seats
- Congress is not doing the math correctly
Now, a bit more about why the shift from NC to UT is relevant: When seats were apportioned after the 2000 Census, Utah missed getting the last seat by about 800 people. They went to the Supreme Court twice, arguing first that the Census did not count the thousands of Utah residents working abroad as missionaries, then that the Census had shafted Utah statistically (see Utah v. Evans).
According to my analysis, of course, the state of Utah was right -- but for a completely different reason.
How do we resolve this particular slight to the people of Utah? In the 110th Congress, the House passed the DC House Voting Rights Act of 2007, which would have expanded the House to 437, made DC's Delegate a full Representative, and given a seat to the state that was 'next in line' after the 2000 reapportionment -- which, of course, was Utah. The House passed it, but it didn't survive cloture in the Senate... although seven of the eight Senate seats we took this year were held by Republicans who voted against it.
One way or the other, Congress needs to know that something wasn't right with its apportionment in 2000. We need to make sure they fix the problem before 2011.