Chapter 15 Section 4 - Slide 1Copyright © 2009 Pearson Education, Inc.

AND

Copyright © 2009 Pearson Education, Inc. Chapter 15 Section 4 - Slide 2

Chapter 15

Voting and Apportionment

Chapter 15 Section 4 - Slide 3Copyright © 2009 Pearson Education, Inc.

WHAT YOU WILL LEARN

• Preference tables• Voting methods• Flaws of voting methods• Standard quotas and standard

divisors• Apportionment methods• Flaws of apportionment methods

Copyright © 2009 Pearson Education, Inc. Chapter 15 Section 4 - Slide 4

Section 4

Flaws of the Apportionment Methods

Chapter 15 Section 4 - Slide 5Copyright © 2009 Pearson Education, Inc.

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 6Copyright © 2009 Pearson Education, Inc.

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 7Copyright © 2009 Pearson Education, Inc.

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.

900489250161Employees

Total321Office

Chapter 15 Section 4 - Slide 8Copyright © 2009 Pearson Education, Inc.

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 9Copyright © 2009 Pearson Education, Inc.

Example: Demonstrating the Alabama Paradox (continued) Based on 30 cell phones, the table is as follows: (Note: standard divisor = 900/30 = 30)

900489250161Employees

291685Lower Quota

301686Hamilton’s

apportionment

16.38.335.37Standard Quota

Total321Office

Chapter 15 Section 4 - Slide 10Copyright © 2009 Pearson Education, Inc.

Example: Demonstrating the Alabama Paradox (continued) Based on 31 cell phones, the table is as follows: (Note: standard divisor = 900/31 ≈ 29.03)

900489250161Employees

291685Lower Quota

311795Hamilton’s

apportionment

16.848.615.55Standard Quota

Total321Office

Chapter 15 Section 4 - Slide 11Copyright © 2009 Pearson Education, Inc.

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 12Copyright © 2009 Pearson Education, Inc.

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 13Copyright © 2009 Pearson Education, Inc.

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.

5400106311339331538733Population

in 2003

5450111211339331539733Population

in 2005

D E TotalCBASchool

Chapter 15 Section 4 - Slide 14Copyright © 2009 Pearson Education, Inc.

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 15Copyright © 2009 Pearson Education, Inc.

Solution

Based on the population in 2003, the table is as follows:

(Note: standard divisor = 5400/54 = 100)

16

15

15.38

1538

B

9

9

9.33

933

C

11

11

11.33

1133

D

54

52

5400

Total

11

10

10.63

1063

E

733Population in

2003

7Lower Quota

7Hamilton’s

apportionment

7.33Standard

Quota

ASchool

Chapter 15 Section 4 - Slide 16Copyright © 2009 Pearson Education, Inc.

Solution (continued) Based on the population in 2005, the table is as

follows: (Note: standard divisor = 5450/54 ≈ 100.93)

15

15

15.25

1539

B

9

9

9.24

933

C

11

11

11.23

1133

D

54

53

5450

Total

11

11

11.02

1112

E

733Population in

2005

7Lower Quota

8Hamilton’s

apportionment

7.26Standard

Quota

ASchool

Chapter 15 Section 4 - Slide 17Copyright © 2009 Pearson Education, Inc.

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 18Copyright © 2009 Pearson Education, Inc.

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 19Copyright © 2009 Pearson Education, Inc.

Summary

Small states

Small states

Large states

Large states

Appointment method favors

NoNoNoYesMay produce the new-states paradox

NoNoNoYesMay produce the

population paradox

NoNoNoYesMay produce the Alabama paradox

YesYesYesNoMay violate the

quota rule

WebsterAdamsJeffersonHamilton

Apportionment Method