Other Apportionment

Algorithms and Paradoxes

NC Standard Course of Study

• Competency Goal 2: The learner will analyze data and apply probability concepts to solve problems.

• Objective 2.03: Model and solve problems involving fair outcomes:– Apportionment. – Election Theory. – Voting Power. – Fair Division.

Hamiltonian and Jeffersonian Methods

• The Hamilton and Jefferson method of apportionment have fallen out of favor because they produce numerous paradoxes in that they take seats from smaller populations.

• The Jefferson method especially favors large states.

Divisor Methods

• The Jefferson method is one of several divisor methods of apportionment.

• The term divisor is used because the methods determine quotas by dividing the population by an ideal ratio or an adjusted ratio.

• This ratio is the divisor.

Webster and Hill Methods

• These are the most used methods today.

• They differ in the way that the round the quotas.

Webster Method

• This method uses the rounding method that is most familiar to us.

• A quota above or equal to 11.5 receives 12 seats and a quota below 11.5 receives 11 seats.

Hill Method• The Hill method is a little more

complicated.• Instead of using the arithmetic mean as

does the Webster method, the Hill method uses the geometric mean of the integers directly above and below the quota.

• If the quota exceeds the geometric mean the Hill method rounds up and if it doesn’t exceed it is rounded down.

Geometric Mean

• The geometric mean of two numbers is the square root of their product.

• For example, a quota between 11 and 12 would have to exceed = 11.4891 to receive 12 seats.

11 12

Practice Problem

• Let’s determine the apportionment of the 20 student council seats using the Webster and Hill methods.Class Size

Quota Hamilton Jefferson Webster Hill

46410.3

1

240 5.33

196 4.36

Class Size

Quota Hamilton Jefferson Webster Hill

Practice Problems

1. a. Complete the following apportionment table for the 21-seat Central High student council described:

Class Size

Quota Hamilton Jefferson Webster Hill

464

240

196

Practice Problems (cont'd)

b. The Jefferson method distributes only 19 seats. Determine which classes are given an additional seat and give the final distribution of the 21 seats according to the Jefferson method.

Webster and Hill• Both the Webster and Hill methods

apportion 22 seats. In such cases the ideal ratio must be increased until one of the classes loses a seat.

• The sophomore class, for example will lose a seat under the Webster method if its quota drops below 10.5.

• This requires an adjusted ratio of 464 10.5= 44.1905.

Adjusting the Council

• For the sophomore class to lose a seat under the Hill method, its quota must drop below 464 10.4881=44.2407.

c. Complete the following table of adjusted ratios for the Webster and Hill methods:

Class Size Webster Hill

464 464 10.5 = 44.1905

240

196

464 10 11 44 2407 .

Practice Problems (cont'd)

d. List the adjusted ratios for the Webster method in increasing order. The ideal ratio must be increased until it passes only the first ratio in your list. This class will lose one seat. Give the final Webster apportionment.

Practice Problems (cont'd)

e. List the adjusted ratios for the Hill method in increasing order. The ideal ratio must be increased until it passes only the first ratio in your list. This class will lose one seat. Give the final Hill apportionment.

f. If the council has 21 seats, which method(s) would be favored by each class.

Practice Problems (cont'd)• Is it possible to measure the fairness of

an apportionment method?• One way to do this is to measure the

discrepancy between a quota and the actual apportionment.

• If a quota is 11.25 and 11 seats are apportioned, the unfairness is 0.25 seats.

• If 12 seats are apportioned, the unfairness is 0.75 seats.

Practice Problems (cont'd)• Use the apportionment for the 20 seat council

to measure the discrepancy for each class by means of each method. Record the results in the following table:

Class Size

Quota Hamilton Jefferson Webster Hill

464 10.31

240 5.33

196 4.36

Total discrepancy

Practice Problems (cont'd)

b. According to this measure, which method is fairest?

Problems with the Divisor methods• None of the divisor methods is plagued

by the paradoxes that caused the demise of the Hamilton method.

• Divisor methods, however, can cause problems.

3. Mountain High School has four classes. The student council has 30 members distributed among the four classes by means of the Webster method of apportionment.

Practice Problems (cont'd)

a. What is the ideal ratio?b. Complete the following Webster

apportionment table:

Class Size QuotaInitial Webster Apportionment

1,105 22.1 22

185

130

80

Practice Problems (cont'd)

c. Because the Webster method apportions too many seats, the ideal ratio must be decreased. Calculate the adjusted ratio necessary for each class to lose a seat, and complete the following table:

Class Size Adjusted Ratio

1,105

185

130

80

1105 215 513953 . .

Practice Problems (cont'd)

d. Increase the ideal ratio until the extra seat is removed, State the final apportionment.

e. Explain why the results would be considered unfair to the freshman class.

Practice Problems (cont'd)• This situation is known as a violation

of quota. Its occurs whenever a class (district, state) is given a number of seats that is not equal to the integer part of its quota or one more than that.

• It can occur with any divisor method and is considered a flaw of the divisor methods.

Adjusted Ratio Table• Notice

that the freshman quota drops much more rapidly than do the others.

Adjusted Ratio

Freshman

Sophomore

JuniorSenio

r

50.00 22.10 3.70 2.60 1.60

50.20 22.01 3.69 2.59 1.59

50.40 21.92 3.67 2.58 1.59

50.60 21.84 3.66 2.57 1.58

50.80 21.75 3.64 2.56 1.57

51.00 21.67 3.63 2.55 1.57

51.20 21.58 3.61 2.54 1.56

51.40 21.50 3.60 2.53 1.56

51.60 21.41 3.59 2.52 1.55

51.80 21.33 3.57 2.51 1.54

52.00 21.25 3.56 2.50 1.54