the mathematics of apportionment

7/27/2019 the Mathematics of Apportionment

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TERRIN [email protected]

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Dept of Computational Science & Mathematics

FCSIT, UNIMAS

Excursions in ModernMathematics

Sixth Edition

Peter Tannenbaum

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FCSIT, UNIMAS

Chapter 4The Mathematics of

Apportionment Making the Rounds

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Outline/learning Objectives• To state the basic apportionment

problem.

• To implement the methods of Hamilton,Jefferson, Adams and Webster to solveapportionment problems.

• To state the quota rule and determinewhen it is satisfied.

• To identify paradoxes when they occur.

• To understand the significance of Balanski and Young’s impossibilitytheorem.

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FCSIT, UNIMAS

The Mathematics of

Apportionment

4.1 Apportionment Problems

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The Mathematics of Apportionment

• We are dividing and assigning things.• We are doing this on a proportional

basis and in a planned, organized

fashion.

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Apportion- two critical elements

in the definition of the word




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State A B C D E F Total

Populatio

n (P)

1,646,000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000

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 people per seat.




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State A B C D E F Total

Population 1,646,000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000

Standard quota 32.92 138.72 3.08 41.82 13.70 19.76 250

Table 4-4 Republic of Parador: Standard Quotas for Each State

(SD = 50,000)

For example, take state A. To find a state’s standard quota,

we divide the state’s population by the standard divisor:

Quota = population/SD = 1,646,000/50,000 = 32.92




• The “states.” This is the term we will use

to describe the players involved in theapportionment.

• The “seats.” This term describes the set of

M identical, indivisible objects that arebeing 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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• Upper quotas. The quota rounded up

and is denoted by U .• Lower quotas. The quota rounded

down and denoted by L.

In the unlikely event that the quota is

a whole number, the lower and upperquotas are the same.

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FCSIT, UNIMAS

The Mathematics of

Apportionment 4.2 Hamilton’s Method (1757 -1804) and

the Quota Rule

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Hamilton’s Method

• Step 1. Calculate each

state’s standard

quota.

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State Population Step1

Quota

A 1,646,000 32.92

B 6,936,000 138.72

C 154,000 3.08

D 2,091,000 41.82

E 685,000 13.70

F 988,000 19.76

Total 12,500,000 250.00




Hamilton’s Method

• Step 2. Give toeach state itslower quota.

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Quota

Step 2

Lower Quota

A 1,646,000 32.92 32

B 6,936,000 138.72 138

C 154,000 3.08 3

D 2,091,000 41.82 41

E 685,000 13.70 13

F 988,000 19.76 19

Total 12,500,000 250.00 246




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

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Quota

Step 2

Lower Quota

Fractional

parts

Step 3

Surplus

Hamilton

apportionment

A 1,646,000 32.92 32 0.92 First 33

B 6,936,000 138.72 138 0.72 Last 139

C 154,000 3.08 3 0.08 3

D 2,091,000 41.82 41 0.82 Second 42

E 685,000 13.70 13 0.70 13

F 988,000 19.76 19 0.76 Third 20

Total 12,500,000 250.00 246 4.00 4 250




The Quota Rule

No state should be apportioned anumber 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.)14




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FCSIT, UNIMAS

The Mathematics of

Apportionment

4.3 The Alabama and Other Paradoxes

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The most serious (in fact, the fatal)

flaw of Hamilton's method iscommonly 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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With M = 200 seats and SD = 100, the

apportionment under Hamilton’smethod

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State Population Step 1 Step 2 Step 3 Apportionment

Bama 940 9.4 9 1 10

Tecos 9,030 90.3 90 0 90

Ilnos 10,030 100.3 100 0 100

Total 20,000 200.0 199 1 200

STD Quota




With M = 201 seats and SD = 99.5, the

apportionment under Hamilton’smethod

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State Population Step 1 Step 2 Step 3 Apportionment

Bama 940 9.45 9 0 9

Tecos 9,030 90.75 90 1 91

Ilnos 10,030 100.80 100 1 101

Total 20,000 201.00 199 2 201




The Hamilton’s method can fall victimto two other paradoxes called

• the population paradox- when state Aloses a seat to state B even though thepopulation of A grew at a higher rate than the population of B.

• the new-states paradox- that theaddition 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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FCSIT, UNIMAS

The Mathematics of

Apportionment

4.4 Jefferson’s Method

(1743-1826)

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Jefferson’s Method

• Step 1. Find a “suitable” divisor D. [ A suitable ormodified 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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Pick a D.



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The Mathematics of ApportionmentJefferson’s Method

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

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State Population Standard Quota

(SD = 50,000)

Lower Quota

(round down)

Modified Quota

(D = 49,500)

Jefferson

apportionment

A 1,646,000 32.92 32 33.25 33

B 6,936,000 138.72 138 140.12 140

C 154,000 3.08 3 3.11 3

D 2,091,000 41.82 41 42.24 42

E 685,000 13.70 13 13.84 13

F 988,000 19.76 19 19.96 19

Total 12,500,000 250.00 246 250

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Bad News- Jefferson’s 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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FCSIT, UNIMAS

The Mathematics of

Apportionment

4.5 Adam’s Method (1767 -1848)

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Th M h i f A i




Adam’s 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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State Population Quota

(D = 50,500)

A 1,646,000 32.59

B 6,936,000 137.35

C 154,000 3.05

D 2,091,000 41.41

E 685,000 13.56

F 988,000 19.56

Total 12,500,000

Th M h i f A i



The Mathematics of ApportionmentAdam’s Method

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

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State Population Quota

(D = 50,500)

Upper Quota

(D = 50,500)

Quota

(D = 50,700)

Adam’s

apportionment

A 1,646,000 32.59 33 32.47 33

B 6,936,000 137.35 138 136.80 137

C 154,000 3.05 4 3.04 4

D 2,091,000 41.41 42 41.24 42

E 685,000 13.56 14 13.51 14

F 988,000 19.56 20 19.49 20

Total 12,500,000 251 250

Th M h i f A i




Bad News- Adam’s method can produce

lower-quota violations!

We can reasonably conclude that

Adam’s method is no better (orworse) than Jefferson’s method– just

different.

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FCSIT, UNIMAS

The Mathematics of

Apportionment

4.6 Webster’s Method

(1782-1852)

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Th M th ti f A ti t




Webster’s 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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Th M th ti f A ti t




Step 2. Find the apportionment of each state by roundingits quota the conventional way.

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State Population Standard Quota

(D = 50,000)

Nearest

Integer

Quota

(D = 50,100)

Webster’s

apportionment

A 1,646,000 32.92 33 32.85 33

B 6,936,000 138.72 139 138.44 138

C 154,000 3.08 3 3.07 3

D 2,091,000 41.82 42 41.74 42

E 685,000 13.70 14 13.67 14

F 988,000 19.76 20 19.72 20

Total 12,500,000 250.00 251 250



S



SummaryTypes of Methods

• Quota Methods

– Hamilton, Lowndes

• Divisor Methods

– Jefferson, Adams, Webster, Huntington-Hill, Dean

Observations

• Satisfy Quota Rule: Hamilton, Lowndes• May violate Quota Rule

– May violate Upper Quota: Jefferson, Webster, Huntington-Hill, Dean

– May violate Lower Quota: Adams, Webster, Huntington-Hill, Dean

• Favor large states: Hamilton, Jefferson

• Favor small states: Lowndes, Adams

• Neutral to state size: Webster, Huntington-Hill34

Conclusion (continued)



Conclusion (continued)

• Balinski and Young’s impossibility

theorem

An apportionment method that does

not violate the quota rule and does

not produce any paradoxes is a

mathematical impossibility .

the mathematics of apportionment

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