Hamilton Apportionment Method
Constitutional Convention
The Hamilton method was the first proposed method for apportionment to be used in the House of Representatives. His method was adopted by Congress but it was later vetoed by President Washington. It is believed that he did this becase another method, Jefferson apportionment method, would give an adddtial representative to his home state of Virginia. Although it this method was not used when it was first introduced, it was later adopted in 1852 and used until 1911.
Using Hamilton Apportionment Method
First we need to determine the standard divisor, s.
Now, find the lower quota, Q, for each state and round down to the nearest whole number.
That number will be the minimum number of seats each state receives.
The last step is to assign any remaining seats that were not distributed when we found the lower quota. We will do this by using the Hamilton apportionment method. This method uses the remainder (decimal) of each Q(s). The states that have the remainder closest to the next greater whole number will recieve the extra seats.
EXAMPLE 1
A country has 3 states. State X has a population of 10,119, state Y has a population of 19,029 and state Z has a population of 3,475. There are 65 seats. What will the distribution of seats look like using Hamilton apportionment method?
total population = State X + State Y + State Z = 10,119 + 19,029 + 3,475
total poplulation = 32,623
standard divisor (s) = total population / number of seats = 32,623 / 65
standard divisor (s) = 501.892
Q(s) = state population /s
State X = 10,119 / 501.892 = 20.162
State Y = 19,029 / 501.892 = 37.915
State Z = 3,475 / 501.892 = 6.924
Each state gets its respective whole number as the minimum number of seats.
State X -20
State Y - 37
State Z - 6
When you add up this distribution it adds up to 63, meaning we have 2 seats left!
Now we will use the Hamilton apportionment method to distribute the two remaining seats.
* If the lower quota is less than 1, the state still reciveve a minimum of 1 seat.
Hamilton Apportionment Method
With the Hamilton method we will distribute the leftover seats to the states with the largest remainders (decimals) until there are none left.
State X 20.162
State Y 37.915
State Z 6.924
State Z has the highest decimal place at .924:
According to the Hamilton method, State Z will now recieve one of the leftover seats.
State Y has the second highest decimal at .915 and will recieve the last seat.
Here is what the the final results should look like. State Y and state Z received one extra seat each because they have the highest remainders at .915 and .924 respectively.
EXAMPLE 2
At Green High School, a group of students decide to create a student council. They conclude that the number of representatives each class year recieves should be reflective of the amount of students there are in each class. There are 1,372 Freshman, 976 Sophomores, 1,008 Juniors and 898 Seniors. There are 35 representatives total. The students decide to use the Hamilton apportionment method to determine the distribution of representatives.
First, find the total number of students
1,156 + 976 + 1,008 + 898
total number of students = 4,038
Now find the standard divisor (s)
4, 038 / 35 = 115.371
Next find Q(s) for each class year
Freshman - 1,156 / 115.371 = 10.02 ≅ 10
Sophomores - 976 / 115.371 = 8.451 ≅ 8
Juniors - 1,008 / 115.371 = 8.737 ≅ 8
Seniors - 898 / 115.371 = 7.784 ≅ 7
This only adds up to 33 seats so we have 2 left over. Now we will use Hamilton apportionment method to distribute the last 2 seats.
Seniors have the highest remainder with .784 so their number of representatives will increase by 1.
Juniors have the second highest remainder with .737 so their number of representatives will increase by 1.
Here is what the final counts should look like for the distribution of student council representatives.
Problems with the Hamilton Apportionment Method
New States Paradox
When a new state is added, the number of seats increases. Although populations of the previous states don’t change the distribution of seats can change.
Alabama Paradox
When seats are added, this changes the distribution, despite no change in population.
Some states lose or gain seats, even though the overall number of seats increased.
Population Paradox
Sometimes states with the largest population increase do not gain seats while states with little to no increase gain seats. Other times states with the greatest population decrease will not lose seats while the states with low population decrease will lose seats.
Examples with the Paradoxes
EXAMPLE 3
a) The fictional country has three states. Annoyance, Boredom, and Complaints, with a population of 2,436; 4,012; and 1,928. The country has a branch of government with proportional representation by state population. There are a total of 25 seats. Find the distribution of seats using the Hamilton apportionment method.
b) A new state, Demand, joins the country. Demand has a large population of 4,921. To account for Demand's addition to the country, 7 more seats will be added bringing the total to 32 seats.
c) What happened to the distribution of seats between part a and part b?
EXAMPLE 3 - solution
EXAMPLE 4
a) A large company, Summertime, owns three smaller companies named Live, Laugh, and Love. The CEO of the Summertime decides to create a collaborative team with representatives from each of the smaller companies. She decides that this new team will have 12 representatives and that the number of representatives will reflect the size of the three smaller companies.
b) Summertime just moved locations and now has a much larger conference room. The CEO decided to increase the number of representatives to 15. None of the companies gained or lost employees. Use the Hamilton apportionment method to determine the distribution of seats.
c) What did you notice happen with the addition of seats?
EXAMPLE 4 - solution
EXAMPLE 5
a) A new charter high school, Ocean View, opens up in the same city as the Green High School in Example 2. Ocean View is trying a new pilot program and is only enrolling Freshman and Seniors. The charter school enrolls 400 Freshmen and 250 Seniors. All these students transferred from Green High School in Example 2. At the same time, 70 Juniors transferred into Green High School. There is no change in the amount of Sophomore students. Find the distribution of the student council at Green High School using Hamilton apportionment with the new student population.
b) What do you notice about the new distribution of seats?
EXAMPLE 5 - solution
Need more help?
Check out this video that explains Hamilton apportionment and the apportionment paradoxes.
Mariela Duran - Grace Jackson - Hannah Quiroz