1. Excursions in Modern Mathematics Sixth Edition
Peter Tannenbaum

2. Chapter 4 The Mathematics of Apportionment
Making the Rounds

3. The Mathematics of Apportionment Outline/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 Young’s impossibility theorem.

4. The Mathematics of Apportionment
4.1 Apportionment Problems

5. 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.

6. The Mathematics of Apportionment
Table 4-3 Republic of Parador (Population by State)
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
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
250 seats available (M), 12,500,000 total population (P)
SD = 12,500,000/250 = 50,000
SD (50,000) tells us each seat corresponds to 50,000 people

7. The Mathematics of Apportionment
Table 4-4 Republic of Parador: Standard Quotas for Each State (SD = 50,000)
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
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

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

9. The Mathematics of Apportionment
Standard Divisor (SD). The ratio of population to seats. SD = P/M
Standard quota. The exact fractional amount of seats the state would receive. q1, q2, … qN

10. The Mathematics of Apportionment
Lower quotas. The quota rounded down and is denoted by L.
Upper 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.

11. The Mathematics of Apportionment
Traditional Rounding
Rounding up: = 251
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

12. The Mathematics of Apportionment
The main goal is to discover a “good” apportionment method that…
Will always result in all the seats being apportioned.
Will always be fair.

13. The Mathematics of Apportionment
4.2 Hamilton’s Method and the Quota Rule

14. Hamilton’s Method Hamilton’s Method
Step 1. Calculate each state’s standard quota.
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

15. Hamilton’s Method Hamilton’s Method
Step 2. Give to each state its lower quota.
State
Population
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
Total
12,500,000
250.00
246

16. Hamilton’s Method
Step 3. Give the surplus seats to the state with the largest fractional parts until there are no more surplus seats.
State
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
Second 42 E
685,000
13.70 13
0.70 F
988,000
19.76 19
0.76
Third 20
Total
12,500,000
250.00
246
4.00 4
250

17. Hamilton’s Method Is state E happy? State A B C D E F Total
Standard quota
32.92
138.72
3.08
41.82
13.70
19.76
250
Lower quota 32
138 3 41 13 19
246
Residue
0.92
0.72
0.08
0.82
0.70
0.76 4
Order of Surplus
First
Last
Second
Third
Apportionment 33
139 42 20
Is state E happy?

18. Hamilton’s Method
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.)

19. The Mathematics of Apportionment
4.3 The Alabama and Other Paradoxes

20. The Alabama and Other Paradoxes
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.

21. The Alabama and Other Paradoxes
With M = 200 seats and SD = 100, the apportionment under Hamilton’s method
State
Population
Step 1
Step 2
Step 3
Apportionment
Bama
940
9.4 9 1 10
Tecos
9,030
90.3 90
Ilnos
10,030
100.3
100
Total
20,000
200.0
199
200

22. The Alabama and Other Paradoxes
With M = 201 seats and SD = 99.5, the apportionment under Hamilton’s method
State
Population
Step 1
Step 2
Step 3
Apportionment
Bama
940
9.45 9
Tecos
9,030
90.75 90 1 91
Ilnos
10,030
100.80
100
101
Total
20,000
201.00
199 2
201

23. The Alabama and Other Paradoxes
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.

24. The Mathematics of Apportionment
4.4 Jefferson’s Method

25. Jefferson’s Method
Step 1: Find a “suitable” divisor D. (guess-and- check)
Step 2: Using D as the divisor, compute each state’s “modified” quota.
Step 3: Each state is apportioned its modified lower quota.

26. Jefferson’s Method 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.

27. Jefferson’s Method Jefferson’s Method
Step 2. Each state is apportioned its lower quota.
State
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 D
2,091,000
41.82 41
42.24 42 E
685,000
13.70 13
13.84 F
988,000
19.76 19
19.96
Total
12,500,000
250.00
246
250

28. Jefferson’s Method
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.

29. Jefferson’s Method M = 250 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
Lower quota 32
138 3 41 13 19
246
Modified Divisor
M = 250

30. The Mathematics of Apportionment
4.5 Adam’s Method

31. Adam’s Method Step 1: Find a “suitable” divisor D. (guess-and- check)
Step 2: Using D as the divisor, compute each state’s “modified” quota.
Step 3: Each state is apportioned its modified upper quota.

32. Adam’s Method 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
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

33. Adam’s Method Adam’s Method
Step 2. Each state is apportioned its upper quota.
State
Population
Quota
(D = 50,500)
Upper Quota
(D = 50,700)
Adam’s
apportionment A
1,646,000
32.59 33
32.47 B
6,936,000
137.35
138
136.80
137 C
154,000
3.05 4
3.04 D
2,091,000
41.41 42
41.24 E
685,000
13.56 14
13.51 F
988,000
19.56 20
19.49
Total
12,500,000
251
250

34. Adam’s Method
Bad News- Adam’s method can produce lower-quota violations! We can reasonably conclude that Adam’s method is no better (or worse) than Jefferson’s method– just different.

35. The Mathematics of Apportionment
4.6 Webster’s Method

36. Webster’s Method
Step 1: Find a “suitable” divisor D. (guess-and- check)
Step 2: Using D as the divisor, compute each state’s “modified” quota.
Step 3: Each state is apportioned its quota rounded to the nearest integer (.5 rule)

37. Webster’s Method 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.

38. Webster’s Method
Step 2. Find the apportionment of each state by rounding its quota the conventional way.
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 B
6,936,000
138.72
139
138.44
138 C
154,000
3.08 3
3.07 D
2,091,000
41.82 42
41.74 E
685,000
13.70 14
13.67 F
988,000
19.76 20
19.72
Total
12,500,000
250.00
251
250

39. The Mathematics of Apportionment Conclusion
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
Hamilton 33
139 3 42 13 20
Jefferson
140 19
Adams
137 4 14
Webster
138
Note: Standard Quota is shown not the Modified Quota

40. The Mathematics of Apportionment
Hill’s Method (mini-excursion 1)

41. Hill’s Method From 1949 to today:
Proposed by Joseph A. Hill (Chief Statistician of the Bureau of the Census) in 1911.
Desire to find the “correct mathematical formula.”
Uses the geometric mean instead of the arithmetic mean to do the rounding.

42. Hill’s Method Arithmetic Mean: a + b / 2 a + b + c / 3 Geometric Mean:
Why?

43. Hill’s Method Arithmetic Mean:
Relevant any time several quantities add together to produce a total.
Geometric Mean:
Relevant any time several quantities multiply together to produce a product.

44. Hill’s Method Hill’s Method is similar to Webster’s:
Step 1: Find a “suitable” Divisor (D).
Step 2: Using D as the divisor, compute each states modified quota (modified quota = state pop/D).
Step 3: Find the apportionments by rounding each modified quota using the geometric mean as a reference (instead of the .5 rule).

45. Hill’s Method Quota Lower Quota Upper Quota Arithmetic Mean
Webster’s Apportion
Geometric Mean
Hill’s Apportion
32.59 32 33
32.5
32.49
3.08 3 4
3.5
3.46
123.49
3.49
10.48
5.46
25.26

46. Hill’s Method 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
Hamilton 33
139 3 42 13 20
Jefferson
140 19
Adams
137 4 14
Webster
138
Hill’s
Note: Standard Quota is shown not the Modified Quota

47. 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

48. The Mathematics of Apportionment Conclusion
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.