Download 2.4: More Apportionment Algorithms and Paradoxes

Discrete Mathematics
2.4: More Apportionment Algorithms and Paradoxes
Recall the example from lesson 2.3: CHS has 464 sophomores, 240 juniors, and 196 seniors. The 20
seats on the student council are divided among classes according to population. We want to determine
the number of seats awarded to each class. Ideal Ratio is 45.
Webster Method – do the initial apportionment by rounding quotas the familiar way (the way you should
always round decimals in a math class. The extra seat is awarded by determining the Webster Adjusted
Ratio. Award seats to the class with the largest adjusted ratio, take seats away from the class with the
smallest adjusted ratio.
Webster Adjusted Ratio 
Class Size
Arithmetic Mean
*Arithmetic mean is the average of two numbers.
Hill Method – Still used to apportion seats in the US House of Representatives. To do the initial
apportionment, round quotas using the geometric mean instead of the arithmetic mean. Extra seat is
awarded by determining the Hill Adjusted Ratio. Award seats to the class with the largest adjusted ratio,
take seats away from the class with the smallest adjusted ratio.
*Geometric mean of two numbers is the square root of their product. If a quota is below this
number, round the quota down, if a quota is above this number, round the quota up.
Hill Adjusted Ratio 
Class Size
Geometric Mean
Example: The Teenage Mutant Ninga Turtles fan club is getting so large that in order to make decisions
for the group, a council must be chosen to represent everyone. There are 15 seats on the council and
each turtle must be represented fairly. Use the table below to determine how many seats each turtle will
have representing him using the Webster and Hill Methods.
Ideal Ratio: 111.67
Webster Method
Turtle
Class
Size
Quota
Donatello
425
3.81
Raphael
375
3.36
Leonardo
225
2.01
Michelangelo
650
5.82
Initial Webster
Apportionment
Arithmetic
Mean for
Class
Webster
Adjusted
Ratio
Webster Final
Apportionment
Hill Method
Turtle
Class
Size
Quota
Donatello
425
3.81
Raphael
375
3.36
Leonardo
225
2.01
Michelangelo
650
5.82
Initial Hill
Apportionment
Geometric
Mean for
Class
Hill
Adjusted
Ratio
Hill Final
Apportionment