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Section 12.1 Hamilton's Method

Alexander Hamilton proposed the method that now bears his name. His method was approved by Congress in 1791, but was vetoed by President Washington. It was later adopted in 1852 and used through 1911. He begins by determining, to several decimal places, how many things each group should get. Since he was interested in the question of Congressional representation, we’ll use the language of states and representatives, so he determines how many representatives each state should get.

Definition 12.1.1. Hamilton’s Method.

Hamilton's method follows the steps:

  1. Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the divisor.

  2. Divide each state’s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the quota. Since we can only allocate whole representatives, Hamilton resolves the whole number problem, as follows:

  3. Cut off all the decimal parts of all the quotas (but don’t forget what the decimals were). These are called the lower quotas. Add up the remaining whole numbers. This answer will always be less than or equal to the total number of representatives (and the “or equal to” part happens only in very specific circumstances that are incredibly unlikely to turn up).

  4. Assuming that the total from the previous step was less than the total number of representatives, assign the remaining representatives, one each, to the states whose decimal parts of the quota were largest, until the desired total is reached.

Make sure that each state ends up with at least one representative!

Note on rounding: Today we have technological advantages that Hamilton (and the others) couldn’t even have imagined. Take advantage of them, and keep several decimal places.

Example 12.1.2.

The state of Delaware has three counties: Kent, New Castle, and Sussex. The Delaware state House of Representatives has 41 members. If Delaware wants to divide this representation along county lines (which is not required, but let’s pretend they do), let’s use Hamilton’s method to apportion them. The populations of the counties are as follows (from the 2010 Census):

Table 12.1.3.
County Population
Kent 162,310
New Castle 538,479
Sussex 197,145
Total 897,934
Solution.

  1. First, we determine the divisor: \(897,934\div 41=21,900.82927\)

  2. Now we determine each county’s quota by dividing the county’s population by the divisor:
    Table 12.1.4.
    County Population Quota
    Kent 162,310 7.4111
    New Castle 538,479 24.5872
    Sussex 197,145 9.0017
    Total 897,934
  3. Removing the decimal parts of the quotas gives:
    Table 12.1.5.
    County Population Quota Initial
    Kent 162,310 7.4111 7
    New Castle 538,479 24.5872 24
    Sussex 197,145 9.0017 9
    Total 897,934 40
  4. We need 41 representatives and this only gives 40. The remaining one goes to the county with the largest decimal part, which is New Castle:
    Table 12.1.6.
    County Population Quota Initial Final
    Kent 162,310 7.4111 7 7
    New Castle 538,479 24.5872 24 25
    Sussex 197,145 9.0017 9 9
    Total 897,934 40 41
Example 12.1.7.

Use Hamilton’s method to apportion the 75 seats of Rhode Island’s House of Representatives among its five counties.

Table 12.1.8.
County Population
Bristol 49,875
Kent 166,158
Newport 82,888
Providence 626,667
Washington 126,979
Total 1,052,567
Solution.

  1. The divisor is \(1,052,567\div 75=14,034.22667\)

  2. Determine each county’s quota by dividing its population by the divisor:
    Table 12.1.9.
    County Population Quota
    Bristol 49,875 3.5538
    Kent 166,158 11.8395
    Newport 82,888 5.9061
    Providence 626,667 44.6528
    Washington 126,979 9.0478
    Total 1,052,567
  3. Remove the decimal part of each quota:
    Table 12.1.10.
    County Population Quota Initial
    Bristol 49,875 3.5538 3
    Kent 166,158 11.8395 11
    Newport 82,888 5.9061 5
    Providence 626,667 44.6528 44
    Washington 126,979 9.0478 9
    Total 1,052,567 72
  4. We need 75 representatives and we only have 72, so we assign the remaining three, one each, to the three counties with the largest decimal parts, which are Newport, Kent, and Providence:
    Table 12.1.11.
    County Population Quota Initial Final
    Bristol 49,875 3.5538 3 3
    Kent 166,158 11.8395 11 12
    Newport 82,888 5.9061 5 6
    Providence 626,667 44.6528 44 45
    Washington 126,979 9.0478 9 9
    Total 1,052,567 72 75

Note that even though Bristol County’s decimal part is greater than .5, it isn’t big enough to get an additional representative, because three other counties have greater decimal parts.

Hamilton’s method obeys something called the Quota Rule. The Quota Rule isn’t a law of any sort, but just an idea that some people, including Hamilton, think is a good one.

Definition 12.1.12. Quota Rule.

The Quota Rule says that the final number of representatives a state gets should be within one of that state’s quota. Since we’re dealing with whole numbers for our final answers, that means that each state should either go up to the next whole number above its quota, or down to the next whole number below its quota.

After seeing Hamilton’s method, many people find that it makes sense, it’s not that difficult to use (or, at least, the difficulty comes from the numbers that are involved and the amount of computation that’s needed, not from the method), and they wonder why anyone would want another method. The problem is that Hamilton’s method is subject to several paradoxes. Three of them happened, on separate occasions, when Hamilton’s method was used to apportion the United States House of Representatives.

The Alabama Paradox is named for an incident that happened during the apportionment that took place after the 1880 census. (A similar incident happened ten years earlier involving the state of Rhode Island, but the paradox is named after Alabama.) The post-1880 apportionment had been completed, using Hamilton’s method and the new population numbers from the census. Then it was decided that because of the country’s growing population, the House of Representatives should be made larger. That meant that the apportionment would need to be done again, still using Hamilton’s method and the same 1880 census numbers, but with more representatives. The assumption was that some states would gain another representative and others would stay with the same number they already had (since there weren’t enough new representatives being added to give one more to every state). The paradox is that Alabama ended up losing a representative in the process, even though no populations were changed and the total number of representatives increased.

The New States Paradox happened when Oklahoma became a state in 1907. Oklahoma had enough population to qualify for five representatives in Congress. Those five representatives would need to come from somewhere, though, so five states, presumably, would lose one representative each. That happened, but another thing also happened: Maine gained a representative (from New York).

The Population Paradox happened between the apportionments after the census of 1900 and of 1910. In those ten years, Virginia’s population grew at an average annual rate of 1.07%, while Maine’s grew at an average annual rate of 0.67%. Virginia started with more people, grew at a faster rate, grew by more people, and ended up with more people than Maine. By itself, that doesn’t mean that Virginia should gain representatives or Maine shouldn’t, because there are lots of other states involved. But Virginia ended up losing a representative to Maine.