Section 15.4
Flaws of the
Apportionment
Methods
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INB Table of Contents
2.3-2
Date
Topic
July 9, 2014
Section 15.4 Examples
48
July 9, 2014
Section 15.4 Notes
49
July 9, 2014
Practice Test 4
50
July 9, 2014
Practice Test 4 Workspace
51
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Page #
What You Will Learn

The Alabama Paradox

Population Paradox

The New-States Paradox
15.4-3
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Three Flaws of Hamilton’s Method
The three flaws of Hamilton’s method
are:
the Alabama paradox
15.4-4
the
population paradox
the
new-states paradox.
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
15.4-5
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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.
15.4-6
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Example 1: Demonstrating the
Alabama Paradox
Consider Stanhope, a small country with a population of 16,500 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.
15.4-7
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Example 1: Demonstrating the
Alabama Paradox
Round standard divisors and standard
quotas to the nearest hundredth.
15.4-8
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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.
15.4-14
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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.
15.4-15
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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.
15.4-17
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New-States Paradox
The new-states paradox occurs
when the addition of a new group
reduces the apportionment of another
group.
15.4-24
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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.
15.4-25
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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 newstates paradox occurs when the laptops are reapportioned.
15.4-27
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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.
15.4-30
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Balinski and Young’s Impossibility
Theorem
There is no perfect apportionment
method that satisfies the quota rule
and avoids any paradoxes.
15.4-31
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Summary
Apportionment Method
Hamilton
May violate the
quota rule
May produce the
Alabama paradox
May produce the
population paradox
May produce the
new-states paradox
Appointment
method favors
15.4-32
Jefferson
Adams
Webster
No
Yes
Yes
Yes
Yes
No
No
No
Yes
No
No
No
Yes
No
No
No
Large
states
Large
states
Small
states
Small
states
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