1. Flaws of the Apportionment Methods
Section 14.4
Flaws of the Apportionment Methods

2. What You Will Learn
Upon completion of this section, you will be able to:
Determine if a given apportionment demonstrates the Alabama paradox.
Determine if a given apportionment demonstrates the population paradox.
Determine if a given apportionment demonstrates the new-states paradox.

3. Three Flaws of Hamilton’s Method
The three flaws of Hamilton’s method are:
the Alabama paradox
the population paradox
the new-states paradox.

4. Three Flaws of Hamilton’s Method
These flaws apply only to Hamilton’s method and do not apply to Jefferson’s method, Webster’s method, or Adam’s method.
In 1980 the Balinski and Young’s Impossibility Theorem stated that there is no perfect apportionment method that satisfies the quota rule and avoids any paradoxes.

5. Alabama Paradox
The Alabama paradox occurs when an increase in the total number of items to be apportioned results in a loss of an item for a group.

6. Example 1: Demonstrating the Alabama Paradox
Consider Stanhope, a small country with a population of 15,000 people and three states A, B, and C. There are 150 seats in the legislature that must be apportioned among the three states, according to their population. Show that the Alabama paradox occurs if the number of seats is increased to 151.

7. Example 1: Demonstrating the Alabama Paradox
Round standard divisors and standard quotas to the nearest hundredth.

8. Example 1: Demonstrating the Alabama Paradox
Solution
With 150 seats, the standard divisor is
15,000 ÷ 150 = 100
The standard quotas for each state and the apportionment for each state are shown in the following table.

9. Example 1: Demonstrating the Alabama Paradox
Solution

10. Example 1: Demonstrating the Alabama Paradox
Solution
With 151 seats, the standard divisor is
15,000 ÷ 151 ≈ 99.34
The standard quotas for each state and the apportionment for each state are shown in the following table.

11. Example 1: Demonstrating the Alabama Paradox
Solution

12. Example 1: Demonstrating the Alabama Paradox
Solution
When the number of seats increased from 150 to 151, state A’s apportionment actually decreased, from 8 to 7.
This example illustrates the Alabama paradox.

13. Population Paradox
The population paradox occurs when group A loses items to group B, even though group A’s population grew at a faster rate than group B’s.

14. Example 2: Demonstrating the Population Paradox
Consider Alexandria, a small country with a population of 100,000 and three states A, B, and C. There are 100 seats in the legislature that must be apportioned among the three states. Using Hamilton’s method, the apportionment is shown in the table.

15. Example 2: Demonstrating the Population Paradox

16. Example 2: Demonstrating the Population Paradox
Suppose that the population increases according to the table below and that the 100 seats are reapportioned. Show that the population paradox occurs when Hamilton’s method is used.

17. Example 2: Demonstrating the Population Paradox
Solution
Calculate the percent increases.
State A has an increase of 399 people. Therefore, state A has a percent increase of

18. Example 2: Demonstrating the Population Paradox
Solution
State B has an increase of 100 people. Therefore, state B has a percent increase of

19. Example 2: Demonstrating the Population Paradox
Solution
State C has an increase of 185 people. Therefore, state C has a percent increase of

20. Example 2: Demonstrating the Population Paradox
Solution
All three states had an increase in their population, but state B is increasing at a faster rate than state A and state C.
The standard divisor using the new population is

21. Example 2: Demonstrating the Population Paradox
Solution
The table shows the reapportionment, using Hamilton’s method using the standard divisor

22. Example 2: Demonstrating the Population Paradox
Solution
State B has lost a seat to state A even though state B’s population grew at a faster rate than state A’s.
As a result, we have an example of the population paradox.

23. New-States Paradox
The new-states paradox occurs when the addition of a new group, and additional
items to be apportioned, reduces the previous apportionment of another group.

24. Example 3: Demonstrating the New-States Paradox
The Oklahoma Public Library System has received a grant to purchase 100 laptop computers to be distributed between two libraries A and B. The 100 laptops will be apportioned based on the population served by each library. The apportionment using Hamilton’s method is shown in the table.

25. Example 3: Demonstrating the New-States Paradox
The standard divisor is

26. Example 3: Demonstrating the New-States Paradox
Suppose that an anonymous donor decides to donate money to purchase six more laptops provided that a third library, C, that serves a population of 625, is included in the apportionment. Show that the new-states paradox occurs when the laptops are reapportioned.

27. Example 3: Demonstrating the New-States Paradox
Solution
Total population is 10, = 10,625 and the total number of laptops to be apportioned is = 106.
The new standard divisor is

28. Example 3: Demonstrating the New-States Paradox
Solution
The new standard quota and Hamilton’s apportionment are shown here.

29. Example 3: Demonstrating the New-States Paradox
Solution
Before library C was added, library B would receive 79 laptops.
By adding a new library and increasing the total number of laptops to be apportioned, library B ended up losing a laptop to library A.
Thus, we have a case of the new-states paradox.

30. Balinski and Young’s Impossibility Theorem
There is no perfect apportionment method that satisfies the quota rule and avoids any paradoxes.

31. Summary Small states Large states Large states
Appointment method favors No
Yes
May produce the new-states paradox
May produce the population paradox
May produce the Alabama paradox
May violate the quota rule
Webster
Adams
Jefferson
Hamilton
Apportionment Method