Apportionment

How are the number of seats per state assigned by the

Constitution?

Terminology

• Ideal Ratio

• Quota

• Lower Quota

• Upper Quota

• Quota Rule

• Paradox

• Truncate

Key Philosophies

• Hamilton

• Jefferson

• Adams

• Webster

• Huntington-Hill

• Dean

Hamilton Method

• Seats are awarded by dividing the class size by the ideal ratio. The population of each state is divided by the ideal ratio.

• These quantities are truncated and those integer values assign seats.

• Remaining seats are assigned to the largest portion of the truncated decimal repeatedly until all unassigned seats have been assigned.

Hamilton Method – Example 1

• Suppose we are assigning 23 seats to the new North Paulding Committee on Excellence. We want each seat to fairly represent the freshmen, sophomores, juniors, and seniors. There are 210 freshmen, 180 sophomores, 150 juniors and 135 seniors. How would we fairly apportion these seats using the Hamilton method?

Hamilton Method – Example 1

• Ideal Ratio = 675/23 = 29.35

• Freshmen 210/29.35 = 7.16 7 seats

• Sophomore 180/29.35 = 6.13 6 seats

• Juniors 150/29.35 = 5.11 5 seats

• Seniors 135/29.35 = 4.60 4 seats

• This assigns 22 of the 23 seats. The Hamilton method gives the one unassigned seat to the Seniors.

Hamilton Method – Example 2

• Suppose we are assigning 29 seats to the new North Paulding Committee on Excellence. We want each seat to fairly represent the freshmen, sophomores, juniors, and seniors. There are 270 freshmen, 190 sophomores, 180 juniors and 165 seniors. How would we fairly apportion these seats using the Hamilton method?

Hamilton Method – Example 2

• Ideal Ratio = 805/29 = 27.76• Freshmen 270/27.76 = 9.72 9 seats• Sophomore 190/27.76 = 6.84 6 seats• Juniors 180/27.76 = 6.48 6 seats• Seniors 165/27.76 = 5.94 5 seats• This assigns of the 26 seats. The Hamilton

method gives the first unassigned seat to the Seniors, the second unassigned seat to the Sophomores, and the third unassigned seat to the Freshmen.

Hamilton Method

• First method approved by Congress for apportionment.• President Washington exercised the veto power for the

first time to veto the usage of the Hamilton Method.• He favored the Jefferson Method. It is believed he

favored this method because it would give Virginia an extra seat in the House when the Hamilton Method did not.

• This method will not violate the Quota Rule, no state should receive more than its upper quota and no less than its lower quota. However, this method will cause paradoxes to occur.

Alabama Paradox

• http://www.cut-the-knot.org/ctk/Democracy.shtml#Alabama

• The Alabama paradox occurs when an increase in the total number of seats causes a state to lose one of its seats.

New State Paradox

• The new-states paradox occurs when addition of a new state with a parallel increase in a fair amount of seats affects apportionment of other states.

Population Paradox

• The population paradox occurs when a state with a higher grows rate loses a seat to a state with a lower growth rate, when the apportionment is recalculated on the basis of new figures.

Jefferson Method

• Jefferson Adjusted Ratio– Population/(Upper Quota)

• Whoever gives the best representation with the extra seat should receive the extra seat.

• Favors large states usually.

Jefferson Method

• Suppose we are assigning 29 seats to the new North Paulding Committee on Excellence. We want each seat to fairly represent the freshmen, sophomores, juniors, and seniors. There are 270 freshmen, 190 sophomores, 180 juniors and 165 seniors. How would we fairly apportion these seats using the Jefferson method?

Jefferson Method

• Find Ideal Ratio like in the Hamilton Method and the Quota like in the Hamilton Method

• Find the adjusted Jefferson Ratio.

• Whichever state is giving the best representation with that seat should receive the seat.

Jefferson Method

• Ideal Ratio is still 27.76• Freshmen quota is 9.73 and there upper

quota is therefore 10…so if they got an extra seat there seats would be representing 27 people per seat

• Sophomore quota is still 6.84 and there upper quota is therefore 7…so if they got an extra seat each of their seats would be representing 27.14 people

Jefferson Method

• Junior quota is still 6.48…so there upper quota is 7 and if they got an extra seat each of their seats would represent 25.71 people.

• Senior quota is still 5.94…so there upper quota is 6 and if they got an extra seat each of their seats would represent 27.5 people.

Jefferson Method

• So who gets the extra seats….– Freshmen get 10 seats– Sophomores get 7 seats– Juniors get 6 seats – Seniors get 6 seats

Jefferson Method

• Whoever gives the best representation with the left over seats should get them.

Adams Method

• Find Ideal Ratio same as other two methods

• Find Quota same as other two methods

• Assign seats by the truncated quota

• Assign extra seats using best representation and Adams adjusted ratio

• Find an Adams adjusted ratio by dividing the class size by the lower quota.

Adams Method

• Suppose we are assigning 27 seats to the new North Paulding Committee on Excellence. We want each seat to fairly represent the freshmen, sophomores, juniors, and seniors. There are 280 freshmen, 210 sophomores, 175 juniors and 145 seniors. How would we fairly apportion these seats using the Adams method? Hamilton? Jefferson?

Webster Method

• Find Ideal Ratio same as other three methods.

• Find Quota same as other three methods.• Assign seats by the truncated quota.• Assign extra seats using best

representation and Webster adjusted ratio.• Find a Webster adjusted ratio by dividing

the class size by the average of the upper and lower quotas.

Webster Method

• Suppose we are assigning 31 seats to the new North Paulding Committee on Excellence. We want each seat to fairly represent the freshmen, sophomores, juniors, and seniors. There are 400 freshmen, 320 sophomores, 300 juniors and 280 seniors. How would we fairly apportion these seats using the Webster method? Adams? Hamilton? Jefferson?

Huntington-Hill Method

• Find Ideal Ratio same as other four methods.

• Find Quota same as other four methods.

• Assign seats by the truncated quota.

• Assign extra seats using best representation and Huntington-Hill adjusted ratio.

• Find a Huntington-Hill adjusted ratio by dividing the class size by the geometric mean of the upper and lower quotas.

eanGeometricMLowerQuotaUpperQuota *

Huntington-Hill Method

• Suppose we are assigning 33 seats to the new North Paulding Committee on Excellence. We want each seat to fairly represent the freshmen, sophomores, juniors, and seniors. There are 480 freshmen, 420 sophomores, 380 juniors and 360 seniors. How would we fairly apportion these seats using the Webster method? Adams? Hamilton? Jefferson? Huntington-Hill?

Dean Method

• Find Ideal Ratio same as other four methods.

• Find Quota same as other four methods.

• Assign seats by the truncated quota.

• Assign extra seats using best representation and Dean adjusted ratio.

• Find a Dean adjusted ratio by dividing the class size by the harmonic mean of the upper and lower quotas.

anharmonicme

lowerquotaupperquota

112

Dean Method

• Suppose we are assigning 35 seats to the new North Paulding Committee on Excellence. We want each seat to fairly represent the freshmen, sophomores, juniors, and seniors. There are 500 freshmen, 420 sophomores, 360 juniors and 340 seniors. How would we fairly apportion these seats using the Webster method? Adams? Hamilton? Jefferson? Huntington-Hill? Dean?