01:640:103 Chap.5 Quotas, Divisors and Hamilton's Method
Quotas, Divisors and Hamilton's Method
The apportionment problem
The Constitution requires House seats to be apportioned among the states according to population, but leaves the method to Congress. The difficulty is that seats are indivisible while exact proportional shares are not whole numbers.
The same problem arises whenever whole items, such as buses, nurses or faculty positions, are shared in proportion to a measure such as riders, patients or enrolment, and the course keeps the words seats, states and population for all of them.
Divisors, quotas and the Quota Rule
The standard divisor is what one seat represents on average: add up the population and share it across the seats.
A state's standard quota is its population divided by the standard divisor, its exact proportional share. The lower quota rounds that down and the upper quota rounds it up. The Quota Rule says every state should receive either its lower or its upper quota.
Giving every state its lower quota always leaves seats over, and giving every state its upper quota always needs too many.
Hamilton's method and its paradox
Hamilton's method gives each state its lower quota and hands the leftover seats one at a time to the largest decimal parts, so it always satisfies the Quota Rule.
It passed Congress after the 1790 Census, drew the first presidential veto, and was later used from 1850 through 1911. Its weakness is the Alabama paradox: adding a seat, with nothing else changed, can take a seat away from a state.
In the guide's example, raising a school district's reading specialists from 20 to 21 costs one school its second specialist.
Who gains from Hamilton's ranking
After Hamilton's method, compare each state's population per seat with the standard divisor. States that received a surplus seat have fewer people per representative than average, and states that did not have more.
In the guide's example the district with the most students per specialist is Pine, at about 396, while Ridge, which won a surplus position, has 270. Residents of the more crowded districts have less influence on their representative, which is the resentment the text identifies. The divisor methods of the next chapter try to remove it by giving every state the same treatment.
What this chapter covers
- 01
The apportionment problem
- 02
Standard divisor and standard quotas
- 03
Lower quotas, upper quotas and the Quota Rule
- 04
Hamilton's method step by step
- 05
Population per seat and the Alabama paradox
Worked example · free
Twenty seats for three counties
- 1Standard divisor = 10,000 ÷ 20 = 500 people per seat.
- 2Standard quotas are 10.6, 6.2 and 3.2; the lower quotas 10, 6 and 3 total 19.
- 2The one surplus seat goes to the largest decimal part, Oak's 0.6.
- 1Check that 11 + 6 + 3 = 20.
Key terms
- Standard divisor
- The standard divisor is the population total over the seat count, the average population per seat.
- Standard quota
- A standard quota is a state's population divided by the standard divisor, its exact proportional share of seats.
- Quota Rule
- The Quota Rule allows each state just two seat counts: its standard quota rounded down or rounded up.
- Alabama paradox
- The Alabama paradox is a state losing a seat when the total number of seats increases and nothing else changes.
- Lower quota
- A lower quota is a state's standard quota rounded down to the nearest whole number.
Quotas, Divisors and Hamilton's Method FAQ
Why can't every state simply receive its standard quota?
Standard quotas are rarely whole numbers, and a seat cannot be split between two states. Every apportionment method is a rule for turning fractional shares into whole seats that still add to the total.
How is the Alabama paradox related to voting theory?
It resembles a monotonicity failure. A change that should help, or at least not hurt, one side ends up costing it, here a seat lost when the house grows.
Does Hamilton's method ever break the Quota Rule?
No. Every state starts at its lower quota and gains at most one surplus seat, so it always ends at its lower or upper quota. Its weaknesses are the Alabama paradox and resentment between states that do and do not receive surplus seats.
What units does the standard divisor have?
People per seat, or in non-political problems whatever the population measures per item, such as students per specialist or riders per bus. It is the size of an average district if every seat represented exactly the same number of people.
Exam move
Lay out every apportionment as a table with columns for population, quota, lower quota, decimal part, rank and seats, and finish by adding the seats column.
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