1. 1 Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum

2. 2 Chapter 4 The Mathematics of Apportionment Making the Rounds

3. 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. 4 The Mathematics of Apportionment 4.1 Apportionment Problems

5. 5 The Mathematics of Apportionment We are dividing and assigning things We are dividing and assigning things. We are doing this on a proportional basis and in a planned, organized fashion We are doing this on a proportional basis and in a planned, organized fashion. Apportion- two critical elements in the definition of the word

6. 6 The Mathematics of Apportionment State ABCDEFTotal Population 1,646,0006,936,000154,0002,091,000685,000988,00012,500,000 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

7. 7 The Mathematics of Apportionment State ABCDEFTotal Population 1,646,0006,936,000154,0002,091,000685,000988,00012,500,000 Standard quota32.92138.723.0841.8213.7019.76250 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

8. 8 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.

9. 9 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.

10. 10 The Mathematics of Apportionment 4.2 Hamilton’s Method and the Quota Rule

11. 11 The Mathematics of Apportionment Hamilton’s Method Step 1. Calculate each state’s standard quota. StatePopulationStep1 Quota A 1,646,000 32.92 B 6,936,000138.72 C 154,000 3.08 D 2,091,000 41.82 E 685,000 13.70 F 988,000 19.76 Total12,500,000250.00

12. 12 The Mathematics of Apportionment Hamilton’s Method Step 2. Give to each state its lower quota. StatePopulationStep1 Quota Step 2 Lower Quota A 1,646,000 32.92 32 B 6,936,000138.72138 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 Total12,500,000250.00246

13. 13 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. StatePopulationStep1 Quota Step 2 Lower Quota Fractional parts Step 3 Surplus Hamilton apportionment A 1,646,000 32.92 320.92First 33 B 6,936,000138.721380.72Last139 C 154,000 3.08 30.08 3 D 2,091,000 41.82 410.82Second 42 E 685,000 13.70 130.70 13 F 988,000 19.76 190.76Third 20 Total12,500,000250.002464.004250

14. 14 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.)

15. 15 The Mathematics of Apportionment 4.3 The Alabama and Other Paradoxes

16. 16 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.

17. 17 The Mathematics of Apportionment With M = 200 seats and SD = 100, the apportionment under Hamilton’s method StatePopulationStep 1Step 2Step 3Apportionment Bama 940 9.4 91 10 Tecos 9,030 90.3 900 Ilnos10,030100.31000 Total20,000200.01991 200

18. 18 The Mathematics of Apportionment With M = 201 seats and SD = 99.5, the apportionment under Hamilton’s method StatePopulationStep 1Step 2Step 3Apportionment Bama 940 9.45 90 9 Tecos 9,030 90.75 901 91 Ilnos10,030100.801001101 Total20,000201.001992 201

19. 19 The Mathematics of Apportionment 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.

20. 20 The Mathematics of Apportionment 4.4 Jefferson’s Method

21. 21 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.

22. 22 The Mathematics of Apportionment Jefferson’s Method Step 2. Each state is apportioned its lower quota. StatePopulationStandard Quota (SD = 50,000) Lower QuotaModified Quota (D = 49,500) Hamilton apportionment A 1,646,000 32.92 32 33.25 33 B 6,936,000 138.72 138140.12 140 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 Total12,500,000250.00246 250

23. 23 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.

24. 24 The Mathematics of Apportionment 4.5 Adam’s Method

25. 25 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. StatePopulationQuota (D = 50,500) A 1,646,000 32.59 B 6,936,000137.35 C 154,000 3.05 D 2,091,000 41.41 E 685,000 13.56 F 988,000 19.56 Total12,500,000

26. 26 The Mathematics of Apportionment Adam’s Method Step 2. Each state is apportioned its upper quota. StatePopulationQuota (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 6,936,000137.35138136.80137 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 Total12,500,000 251 250

27. 27 The Mathematics of Apportionment 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.

28. 28 The Mathematics of Apportionment 4.6 Webster’s Method

29. 29 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.

30. 30 The Mathematics of Apportionment Step 2. Find the apportionment of each state by rounding its quota the conventional way. StatePopulationStandard Quota (D = 50,000) Nearest Integer Quota (D = 50,100) Webster’s apportionment A 1,646,000 32.92 3332.8533 B 6,936,000138.72139138.44138 C 154,000 3.08 33.073 D 2,091,000 41.82 4241.7442 E 685,000 13.70 1413.6714 F 988,000 19.76 2019.7220 Total12,500,000250.00251 250

31. 31 The Mathematics of Apportionment Conclusion Covered different methods to solve apportionment problems Covered different methods to solve apportionment problems named after Alexander Hamilton, Thomas Jefferson, John Quincy Adams, and Daniel Webster. Examples of divisor methods Examples of divisor methods based on the notion divisors and quotas can be modified to work under different rounding methods

32. 32 The Mathematics of Apportionment Conclusion (continued) Balinski and Young’s impossibility theorem 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.