Excursions in Modern
Mathematics
Sixth Edition
Peter Tannenbaum
1
Chapter 4
The Mathematics of
Apportionment
Making the Rounds
2
The Mathematics of Apportionment
Outline/learning Objectives
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3
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
Young’s impossibility theorem.
The Mathematics of
Apportionment
4.1 Apportionment
Problems
4
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.
5
The Mathematics of Apportionment
Table 4-3 Republic of Parador (Population by State)
State
Population
A
1,646,000
B
6,936,000
C
154,000
D
2,091,000
E
685,000
F
988,000
Total
12,500,000
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
6
The Mathematics of Apportionment
Table 4-4 Republic of Parador: Standard Quotas for Each State
(SD = 50,000)
State
7
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
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 Mathematics of Apportionment
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8
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.
The Mathematics of Apportionment
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
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.
9
The Mathematics of
Apportionment
4.2 Hamilton’s Method
and the Quota Rule
10
The Mathematics of Apportionment
Hamilton’s Method
 Step 1. Calculate each
state’s standard quota.
11
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
12,500,000
250.00
Total
The Mathematics of Apportionment
Hamilton’s Method
 Step 2. Give to each
state its lower quota.
State
Step1
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
12,500,000
250.00
246
Total
12
Population
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.
State
13
Population
Step1
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
D
2,091,000
41.82
41
0.82
E
685,000
13.70
13
0.70
F
988,000
19.76
19
0.76
Third
12,500,000
250.00
246
4.00
4
Total
3
Second
42
13
20
250
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.)
14
The Mathematics of
Apportionment
4.3 The Alabama
and Other
Paradoxes
15
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.
16
The Mathematics of Apportionment
With M = 200 seats and SD = 100, the
apportionment under Hamilton’s method
State
17
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
The Mathematics of Apportionment
With M = 201 seats and SD = 99.5, the
apportionment under Hamilton’s method
State
18
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 Mathematics of Apportionment
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19
The Hamilton’s 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.
The Mathematics of
Apportionment
4.4 Jefferson’s
Method
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The Mathematics of Apportionment
Jefferson’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 down.
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The Mathematics of Apportionment
Jefferson’s Method
 Step 2. Each state is apportioned its lower quota.
State
22
Population
Standard Quota
(SD = 50,000)
Lower Quota
Modified Quota
(D = 49,500)
Hamilton
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
12,500,000
250.00
246
Total
250
The Mathematics of Apportionment
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.
23
The Mathematics of
Apportionment
4.5 Adam’s Method
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The Mathematics of Apportionment
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.
State
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
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Population
12,500,000
The Mathematics of Apportionment
Adam’s Method
 Step 2. Each state is apportioned its upper quota.
State
26
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
The Mathematics of Apportionment
Bad News- Adam’s method can produce lowerquota violations!
We can reasonably conclude that Adam’s
method is no better (or worse) than Jefferson’s
method– just different.
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The Mathematics of
Apportionment
4.6 Webster’s
Method
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The Mathematics of Apportionment
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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The Mathematics of Apportionment
Step 2. Find the apportionment of each state by
rounding its quota the conventional way.
State
30
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
12,500,000
250.00
251
Total
250
The Mathematics of Apportionment
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 Young’s impossibility
theorem
An apportionment method that does
not violate the quota rule and does not
produce any paradoxes is a
mathematical impossibility.
32