1. Other Paradoxes and Apportionment Methods
Section 9.4
Other Paradoxes and Apportionment Methods

2. Objectives: Find the standard divisors and standard quotas.
Understand the Apportionment problem.
Use Hamilton’s method with quotas.
Understand the Population Paradox and New-states Paradox.
Understand the quota rule.
Use Jefferson’s method.
Use Adam’s method.
Use Webster’s method.

3. Terms:
Standard Divisor – found by dividing the total population under consideration by the number of items to be allocated.
Standard Quota – (for a particular group) found by dividing that group’s population by the standard divisor.
Lower Quota – standard quota rounded down to the nearest whole number.
Upper Quota – standard quota rounded up to the nearest whole number.

4. Calculating Standard Divisor
Standard Divisor = total population number of allocated items

5. Example 1: Calculate the Standard Divisor
According to the constitution, of Margaritaville, the congress will have 30 seats, divided among the 4 states.
Population of Margaritaville by State
State A B C D
TOTAL
Population (in thousands)
275
383
465
767
1890

6. Example 2: Calculate the Standard Divisor
According to the country’s constitution, the congress will have 200 seats, divided among the 5 states.
Population of Amador by State
State A B C D E
Total
Population (in thousands)
1112
1118
1320
1515
4935
10,000

7. Calculating Standard Quota
To calculate standard quota, you must first find the standard divisor.
Standard Quota = population of a particular group standard divisor

8. Example 3: Calculate the Standard Quota
Standard quotas are obtained by dividing each state’s population by the standard divisisor.
Population of Margaritaville by State
State A B C D
TOTAL
Population (in thousands)
275
383
465
767
1890
Standard Quota

9. Example 4: Calculate the Standard Quota Population of Amador by State
According to the country’s constitution, congress will have 200 seats.
Population of Amador by State
State A B C D E
TOTAL
Population (in thousands)
1112
1118
1320
1515
4935
10,000
Standard Quota

10. Some Good Advice:
Keep in mind that the standard divisor is a single number that we calculate once and then use for the entire apportionment process. However, we must compute the standard quota individually for each state.

11. Study Tip:
Due to rounding, the sum of the standard quotas can be slightly above or slightly below the total number of allocated items.

12. The Apportionment Problem:
The apportionment problem is to determine a method for rounding standard quotas into whole numbers so that the sum of the numbers is the total number of allocated items.

13. Example 5: Finding lower and upper quotas.
Population of Margaritaville by State
State A B C D
TOTAL
Population (in thousands)
275
383
465
767
1890
Standard Quota
4.3651
6.0794
7.3810
Lower Quotas
Upper Quotas

14. Section 9.4 Assignments Classwork:
TB pg. 547/1 – 12…find standard divisors and standard quotas only!
Must write problems and show ALL work.

15. 4 Methods
There are 4 different apportionment methods, which we will look at to solve the apportionment problem.
Hamilton’s Method (already talked about)
Jefferson’s Method
Adam’s Method
Webster’s Method`

16. Method 1 Hamilton’s Method Calculate each group’s standard quota
Round each standard quota down to the nearest whole number, thereby finding the lower quota. Initially, give to each group its lower quota.
Give the surplus items, one at a time, to the groups with the largest decimal parts in their standard quotas until there are no more surplus items.

17. Example 6:
A rapid transit service operates 130 buses along six routes A, B, C, D, E, and F. The number of buses assigned to each route is based on the average number of daily passengers per route, given in the table. Use Hamilton’s method to apportion the buses.
Rapid Transit Service
Route A B C D E F
Total
Avg Number of Passengers
4360
5130
7080
10,245
15,535
22,650
65,000

18. Example 6: Rapid Transit Service A B 5130 C 7080 D 10,245 E 15,535 F
Route
Passengers
Standard Quota
Lower Quota
Decimal Part
Surplus Buses
Final Apportionment A
4360 B
5130 C
7080 D
10,245 E
15,535 F
22,650
Total
65,000

19. The Quota Rule:
A group’s apportionment should be either its upper quota or its lower quota.
An apportionment method that guarantees that this will always occur is said to satisfy the quota rule, such as the Hamilton Method.

20. Hamilton’s Method:
This would be the best method for apportionment, if its only problem was the Alabama paradox, but there are other paradoxes that occur from this method.

21. Population Paradox:
The population paradox occurs when state A’s population is growing faster than state B’s population, yet A loses a representative to state B. (We are assuming that the total number of representatives in the legislature is not changing.)

22. Population Paradox and the Hamilton Method
The Graduate school at Great Eastern University used the Hamilton method to apportion 15 graduate assistantships among the colleges of education, liberal arts, and business based on their undergraduate enrollments.
Use Hamilton’s method to allocate the graduate assistantships to the three colleges
Assume that after the allocation was made in part a) that education gains 30 students, liberal arts gains 46, and the business enrollment stays the same. Reapportion the graduate assistantships again using the Hamilton method.
Explain how this illustrates the population paradox.

23. Assign 2 Additonal Assistantships
Example 7:
Apportion the 15 graduate assistantships before enrollments increase.
Graduate School – Great Eastern University
College
# of Students
Standard Quota
Integer Parts
Fractional Parts
Assign 2 Additonal Assistantships
Education
940
Liberal Arts
1470
Business
1600
Total
4010

24. Assign 2 Additonal Assistantships
Example 8:
Apportion the 15 graduate assistantships with the increased enrollment.
Graduate School – Great Eastern University
College
# of Students
Standard Quota
Integer Parts
Fractional Parts
Assign 2 Additonal Assistantships
Education
970
Liberal Arts
1516
Business
1600
Total
4086

25. New-states Paradox
The new-states paradox occurs when a new state is added, and its share of seats is added to the legislature causing a change in the allocation of seats previously given to another state.

26. Example 9: New-States Paradox
A small country, Namania, consists of three states A, B, and C with populations given in the following table. Namania’s legislature has 37 representatives that are to be apportioned to these states using the Hamilton method.
Apportion these representatives using the Hamilton method.
Assume that Namania annexes the country Darelia whose population is 3,000 (in thousands). Give Darelia its current share of representatives using the current standard divisor and add the number to the total number of representatives of Namania. Reapportion Namania again using the Hamilton method.
Explain how the new-states paradox occurred.

27. Example 9: (a) Country of Namania – 37 Seats A B 6040 C 3350 Total
State
Pop. (in thousands)
Standard Quota
Integer Parts
Fractional Parts
Assign Additonal Reps A
2750 B
6040 C
3350
Total
12,140

28. Example 9: (b) Country of Namania A B 6040 C 3350 D(arelia) 3000 Total
State
Pop. (in thousands)
Standard Quota
Integer Parts
Fractional Parts
Assign Additonal Reps A
2750 B
6040 C
3350
D(arelia)
3000
Total
15,140

29. Example 9: (c) Country of Namania – 37 Representatives
State
Pop. (in thousands)
Standard Quota
Integer Parts
Fractional Parts
Assign Additonal Reps A
2750
8.38 8
0.38 B
6040
18.41 18
0.41 19 C
3350
10.21 10
0.21
Total
12,140 36 37
Country of Namania – 46 Representatives
State
Pop. (in thousands)
Standard Quota
Integer Parts
Fractional Parts
Assign Additonal Reps A
2750
8.36 8
0.36 9 B
6040
18.35 18
0.35 C
3350
10.18 10
0.18
D(arelia)
3000
9.11
0.11
Total
15,140 45 46

30. Method 2: Jefferson’s Method
Use trial and error to find a modified divisor which is smaller than the standard divisor for the apportionment.
Calculate the modifed quota (state’s population/modified divisor) for each state and round it down.
Assign that number of representatives to each state.
(Keep varying the modifed divisor until the sum of these assignments is equal to the total number being apportioned.)
Note: This method was adopted in 1791 and used until the apportionment of 1832, when NY received 40 seats, with a standard quota of Due to this violation the Jefferson Method was never used again.
After this the Hamilton method was resurrected by Congress.

31. Example 10: Find modified divisor, should be lower than standard divisor.
Rapid Transit Service
Route
Passengers
Modified Quota
Modified Lower
Final Apportionment A
4360 B
5130 C
7080 D
10,245 E
15,535 F
22,650
Total
65,000

32. Note:
If the total numbers assigned is too small, then we need larger modified quotas.
In order to have larger modified quotas, you will need to find smaller modified divisors.

33. Example 11: Jefferson’ Method
Rapid Transit Service
Route
Passengers
Modified Quota
Modified Lower
Final Apportionment A
4360 B
5130 C
7080 D
10,245 E
15,535 F
22,650
Total
65,000

34. Method 3: Adam’s Method
Use trial and error to find a modified divisor which is larger than the standard divisor for the apportionment.
Calculate the modified quota (state’s population/modified divisor) for each state and round it up.
Assign that number of representatives to each state.
(Keep varying the modified divisor until the sum of these assignments is equal to the total number being apportioned.)

35. Example 12: Find Modified Divisor, should be larger than standard divisor
Rapid Transit Service
Route
Passengers
Modified Quota
Modified Upper
Final Apportionment A
4360 B
5130 C
7080 D
10,245 E
15,535 F
22,650
Total
65,000

36. Example 13: Find Modified Divisor, should be larger than standard divisor
Rapid Transit Service
Route
Passengers
Modified Quota
Modified Upper
Final Apportionment A
4360 B
5130 C
7080 D
10,245 E
15,535 F
22,650
Total
65,000

37. Background:
In 1832, Daniel Webster suggested an apportionment method that sounds like a compromise between Jefferson’s and Adams’. He suggested if the decimal part was greater than 0.5, then we round up to the next whole number, whereas if the fractional part is less than 0.5, then we round to the whole number.

38. Method 4: Webster’s Method
Use trial and error to find a modified divisor.
Calculate the modified quota for each state and round it in the usual way.
Assign that number of representatives to each state.
(Keep varying the modified divisor until the sum of these assignments is equal to the total number being apportioned.)

39. Example 14: Rapid Transit Service A B 5130 C 7080 D 10,245 E 15,535 F
Route
Passengers
Modified Quota
Modified Upper
Final Apportionment A
4360 B
5130 C
7080 D
10,245 E
15,535 F
22,650
Total
65,000

40. Section 9.4 Assignment Classwork: TB pg. 547/23 – 26, 39 – 42, and 54
Remember you must write problems and show ALL work to receive credit for this assignment.
Due Friday, Nov. 04, 2011
Reminder: If the assignment is not turned in by due date, then 10 points are minus for each day late, up to 3 days. On the 4th day it is an automatic 0.