Chapter 15 Section 4 - Slide 1

Transcription

1 AND Chapter 15 Section 4 - Slide 1

2 Chapter 15 Voting and Apportionment Chapter 15 Section 4 - Slide 2

3 WHAT YOU WILL LEARN Flaws of apportionment methods Chapter 15 Section 4 - Slide 3

4 Section 4 Flaws of the Apportionment Methods Chapter 15 Section 4 - Slide 4

5 Three Flaws of Hamilton s Method The three flaws of Hamilton s method are: the Alabama paradox, the population paradox, and the new-states paradox. 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. Chapter 15 Section 4 - Slide 5

6 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. Chapter 15 Section 4 - Slide 6

7 Example: Demonstrating the Alabama Paradox A large company, with branches in three cities, must distribute 30 cell phones to the three offices. The cell phones will be apportioned based on the number of employees in each office shown in the table below. Office Total Employees Chapter 15 Section 4 - Slide 7

8 Example: Demonstrating the Alabama Paradox (continued) Apportion the cell phones using Hamilton s method. Does the Alabama paradox occur using Hamilton s method if the number of new cell phones increased from 30 to 31? Explain. Chapter 15 Section 4 - Slide 8

9 Example: Demonstrating the Alabama Paradox (continued) Based on 30 cell phones, the table is as follows: (te: standard divisor = 900/30 = 30) Office Total Employees Standard Quota Lower Quota Hamilton s apportionment Chapter 15 Section 4 - Slide 9

10 Example: Demonstrating the Alabama Paradox (continued) Based on 31 cell phones, the table is as follows: (te: standard divisor = 900/ ) Office Total Employees Standard Quota Lower Quota Hamilton s apportionment Chapter 15 Section 4 - Slide 10

11 Example: Demonstrating the Alabama Paradox (continued) When the number of cell phones increased from 30 to 31, office one actually lost a cell phone, while the other two offices actually gained a cell phone under Hamilton s apportionment. Chapter 15 Section 4 - Slide 11

12 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. Chapter 15 Section 4 - Slide 12

13 Example: Demonstrating Population Paradox A school district with five elementary schools has funds for 54 scholarships. The student population for each school is shown in the table below. School A B C D E Total Population in Population in Chapter 15 Section 4 - Slide 13

14 Example: Demonstrating Population Paradox (continued) Apportion the scholarships using Hamilton s method. If the school wishes to give the same number of scholarships two years later, does a population paradox occur? Chapter 15 Section 4 - Slide 14

15 Solution Based on the population in 2003, the table is as follows: (te: standard divisor = 5400/54 = 100) School A B C D E Total Population in Standard Quota Lower Quota Hamilton s apportionment Chapter 15 Section 4 - Slide 15

16 Solution (continued) Based on the population in 2005, the table is as follows: (te: standard divisor = 5450/ ) School A B C D E Total Population in Standard Quota Lower Quota Hamilton s apportionment Chapter 15 Section 4 - Slide 16

17 Solution (continued) In the school district in 2005, school B actually gives one of its scholarships to school A, even though the population in school B actually grew by 1 student and the population in School A remained the same. Chapter 15 Section 4 - Slide 17

18 New-States Paradox The new-states paradox occurs when the addition of a new group reduces the apportionment of another group. Chapter 15 Section 4 - Slide 18

19 Summary May violate the quota rule May produce the Alabama paradox May produce the population paradox May produce the newstates paradox Appointment method favors Hamilton Yes Yes Yes Large states Apportionment Method Jefferson Yes Large states Adams Yes Small states Webster Yes Small states Chapter 15 Section 4 - Slide 19