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. He follows these steps:
Definition3.1.1Hamilton’s Method
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.
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:
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).
Assuming that the total from Step 3 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.
Example3.1.2
Figure3.1.3 Video solution by David Lippman
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):
County
Population
Kent
162,310
New Castle
538,479
Sussex
197,145
Total
897,934
1. First, we determine the divisor: 897,934/41 = 21,900.82927
2. Now we determine each county’s quota by dividing the county’s population by the divisor:
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:
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:
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
Example3.1.4
Figure3.1.5 Video solution by David Lippman
Use Hamilton’s method to apportion the 75 seats of Rhode Island’s House of Representatives among its five counties.
County
Population
Bristol
49,875
Kent
166,158
Newport
82,888
Providence
626,667
Washington
126,979
Total
1,052,567
1. The divisor is 1,052,567/75 = 14,034.22667
2. Determine each county’s quota by dividing its population by the divisor:
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:
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:
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.
Definition3.1.6Quota 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.