Excursions in Modern Mathematics Sixth Edition

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Excursions in Modern MathematicsSixth Edition

• Peter Tannenbaum

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Chapter 4The Mathematics of Apportionment

• Making the Rounds

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The Mathematics of ApportionmentOutline/learning
Objectives

• To state the basic apportionment problem.
• To implement the methods of Hamilton, Jefferson,
  Adams and Webster to solve apportionment
  problems.
• To state the quota rule and determine when it is
  satisfied.
• To identify paradoxes when they occur.
• To understand the significance of Balanski and
  Youngs impossibility theorem.

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The Mathematics of Apportionment

• 4.1 Apportionment Problems

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The Mathematics of Apportionment
Apportion- two critical elements in the
definition of the word

• We are dividing and assigning things.
• We are doing this on a proportional basis and in
  a planned, organized fashion.

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The Mathematics of Apportionment
Table 4-3 Republic of Parador (Population by
State)
 
The first step is to find a good unit of
measurement. The most natural unit of
measurement is the ratio of population to
seats. We call this ratio the standard divisor SD
P/M SD 12,500,000/250 50,000
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The Mathematics of Apportionment
Table 4-4 Republic of Parador Standard Quotas
for Each State (SD 50,000)
 
For example, take state A. To find a states
standard quota, we divide the states population
by the standard divisor Quota population/SD
1,646,000/50,000 32.92
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The Mathematics of Apportionment

• The states. This is the term we will use to
  describe the players involved in the
  apportionment.
• The seats. This term describes the set of M
  identical, indivisible objects that are being
  divided among the N states.
• The populations. This is a set of N positive
  numbers which are used as the basis for the
  apportionment of the seats to the states.

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The Mathematics of Apportionment

• Upper quotas. The quota rounded down and is
  denoted by L.
• Lower quotas. The quota rounded up and denoted
  by U.
• In the unlikely event that the quota is a whole
  number, the lower and upper quotas are the same.

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The Mathematics of Apportionment

• 4.2 Hamiltons Method and the Quota Rule

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The Mathematics of Apportionment

• Hamiltons Method
• Step 1. Calculate each states standard quota.

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The Mathematics of Apportionment

• Hamiltons Method
• Step 2. Give to each state its lower quota.

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The Mathematics of Apportionment

• Step 3. Give the surplus seats to the state with
  the largest fractional parts until there are no
  more surplus seats.

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The Mathematics of Apportionment

• The Quota Rule
• No state should be apportioned a number of seats
  smaller than its lower quota or larger than its
  upper quota. (When a state is apportioned a
  number smaller than its lower quota, we call it a
  lower-quota violation when a state is
  apportioned a number larger than its upper quota,
  we call it an upper-quota violation.)

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The Mathematics of Apportionment

• 4.3 The Alabama and Other Paradoxes

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The Mathematics of Apportionment

• The most serious (in fact, the fatal) flaw of
  Hamilton's method is commonly know as the Alabama
  paradox. In essence, the paradox occurs when an
  increase in the total number of seats being
  apportioned, in and of itself, forces a state to
  lose one of its seats.

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The Mathematics of Apportionment

• With M 200 seats and SD 100, the
  apportionment under Hamiltons method

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The Mathematics of Apportionment

• With M 201 seats and SD 99.5, the
  apportionment under Hamiltons method

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The Mathematics of Apportionment

• The Hamiltons method can fall victim to two
  other paradoxes called
• the population paradox- when state A loses a seat
  to state B even though the population of A grew
  at a higher rate than the population of B.
• the new-states paradox- that the addition of a
  new state with its fair share of seats can, in
  and of itself, affect the apportionments of other
  states.

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The Mathematics of Apportionment

• 4.4 Jeffersons Method

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• Jeffersons Method
• Step 1. Find a suitable divisor D. A suitable
  or modified divisor is a divisor that produces
  and apportionment of exactly M seats when the
  quotas (populations divided by D) are rounded
  down.

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The Mathematics of Apportionment

• Jeffersons Method
• Step 2. Each state is apportioned its lower quota.

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The Mathematics of Apportionment

• Bad News- Jeffersons method can produce
  upper-quota violations!
• To make matters worse, the upper-quota
  violations tend to consistently favor the larger
  states.

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The Mathematics of Apportionment

• 4.5 Adams Method

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The Mathematics of Apportionment

• Adams Method
• Step 1. Find a suitable divisor D. A suitable
  or modified divisor is a divisor that produces
  and apportionment of exactly M seats when the
  quotas (populations divided by D) are rounded up.

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The Mathematics of Apportionment

• Adams Method
• Step 2. Each state is apportioned its upper quota.

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The Mathematics of Apportionment

• Bad News- Adams method can produce lower-quota
  violations!
• We can reasonably conclude that Adams method is
  no better (or worse) than Jeffersons method
  just different.

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• 4.6 Websters Method

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• Websters Method
• Step 1. Find a suitable divisor D. Here a
  suitable divisor means a divisor that produces an
  apportionment of exactly M seats when the quotas
  (populations divided by D) are rounded the
  conventional way.

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• Step 2. Find the apportionment of each state by
  rounding its quota the conventional way.

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Conclusion

• Covered different methods to solve apportionment
  problems
• named after Alexander Hamilton, Thomas
  Jefferson, John Quincy Adams, and Daniel Webster.
• Examples of divisor methods
• based on the notion divisors and quotas can be
  modified to work under different rounding methods

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The Mathematics of Apportionment
Conclusion (continued)

• Balinski and Youngs impossibility theorem
• An apportionment method that does not violate
  the quota rule and does not produce any paradoxes
  is a mathematical impossibility.