The Mathematics of Apportionment

1. The Mathematics of Apportionment Notes 17 – Section 4.1

2. Essential Learnings • Students will understand the basic concepts and terminology of apportionment.

3. Apportionment • Without a doubt, the most important and heated debate at the Constitutional Convention concerned the makeup of the legislature. • The small states wanted all states to have the same number of representatives; the larger states wanted some form of proportional representation.

4. Apportionment • The Constitutional compromise, known as the Connecticut Plan, was a Senate, in which every state has two senators, and a House of Representatives, in which each state has a number of representatives that is a function of its population.

5. Apportionment • While the Constitution makes clear that seats in the House of Representatives are to be allocated to the states based on their populations (“According to their respective Numbers”), it does not prescribe a specific method for the calculations.

6. Apportionment • The Founding Fathers did not realize that Article I, Section 2, set the Constitution of the United States into a collision course with a mathematical iceberg known as the apportionment problem.

7. Apportionment • There are two critical elements in the dictionary definition of the word apportion : • (1) We are dividing and assigning things, and • (2) we are doing this on a proportional basis and in a planned, organized fashion.

8. Basic Concepts and Terminology • The basic elements of every apportionment problem are as follows: • The “states” - the term used to describe the parties having a stake in the apportionment. • Unless they have specific names (Azucar, Bahia, etc.), we will use A1, A2,…, AN, to denote the states.

9. Basic Concepts and Terminology • The “seats” - the term describes the set of Midentical, indivisible objects that are being divided among the N states. • For convenience, we will assume that there are more seats than there are states, thus ensuring that every state can potentially get a seat. (This assumption does not imply that every state must get a seat!)

10. Basic Concepts and Terminology • The “populations” - the set of N positive numbers (for simplicity we will assume that they are whole numbers) that are used as the basis for the apportionment of the seats to the states. • We will use p1, p2,…, pN, to denote the state’s respective populations and P to denote the total population P = p1 + p2 +…+ pN).

11. Basic Concepts and Terminology • The standard divisor(SD) - The ratio of population to seats. It gives us a unit of measurement (SD people = 1 seat) for our apportionment calculations.

12. Basic Concepts and Terminology • The standard quotas - the standard quota of a state is the exact fractional number of seats that the state would get if fractional seats were allowed. • We will use the notation q1, q2,…, qN to denote the standard quotas of the respective states.

13. Basic Concepts and Terminology • To find a state’s standard quota, we divide the state’s population by the standard divisor. • In general, the standard quotas can be expressed as fractions or decimals–round them to two or three decimal places.

14. Basic Concepts and Terminology • The lower quota is the standard quota rounded down and the upper quota is the standard quota rounded up. • In the unlikely event that the standard quota is a whole number, the lower and upper quotas are the same.

15. Basic Concepts and Terminology • We will use L’s to denote lower quotas and U’s to denote upper quotas. For example, the standard quota q1 = 32.92 has lower quota L1 = 32 and upper quota U1 = 33.

16. Apportionment • Our main goal in this chapter is to discover a “good” apportionment method–a reliable procedure that: • (1) will always produce a valid apportionment (exactly M seats are apportioned) and • (2) will always produce a “fair” apportionment.

17. Example – The Kitchen Capitalism • Mom has a total of 50 identical pieces of candy (let’s say caramels), which she is planning to divide among her five children (this is the division part). Like any good mom, she is intent on doing this fairly. Of course, the easiest thing to do would be to give each child 10 caramels–by most standards, that would be fair. Mom, however, is thinking of the long-term picture–she wants to teach her children about the value of work and about the relationship between work and reward.

18. Example – The Kitchen Capitalism • This leads her to the following idea: She announces to the kids that the candy is going to be divided at the end of the week in proportion to the amount of time each of them spends helping with the weekly kitchen chores–if you worked twice as long as your brother you get twice as much candy, and so on (this is the due and proper proportion part). Unwittingly, mom has turned this division problem into an apportionment problem.

19. Example – The Kitchen Capitalism • At the end of the week, the numbers are in. Table 4-1 shows the amount of work done by each child during the week. (Yes, mom did keep up-to-the-minute records!)

20. Example – The Kitchen Capitalism • According to the ground rules, Alan, who worked 150 out of a total of 900 minutes, should get 8 1/3 pieces. • Here comes the problem: Since the pieces of candy are indivisible, it is impossible for Alan to get his pieces–he can get 8 pieces (and get shorted) or he can get 9 pieces (and someone else will get shorted).

21. Example – The Kitchen Capitalism • A similar problem occurs with each of the other children. Betty’s exact fair share should be 4 1/3 pieces; Connie’s should be 9 11/18 pieces; Doug’s, 11 1/3 pieces; and Ellie’s, 16 7/18 pieces. • Because none of these shares can be realized, an absolutely fair apportionment of the candy is going to be impossible.

22. Example – The Congress of Parador • Parador is a small republic located in Central America and consists of six states: Azucar, Bahia, Cafe, Diamante, Esmeralda, and Felicidad (A, B, C, D, E, and F for short). • There are 250 seats in the Congress, which, according to the laws of Parador, are to be apportioned among the six states in proportion to their respective populations.

23. Example – The Congress of Parador • What is the “correct” apportionment? Table4-3 shows the population figures for the six states according to the most recent census.

24. Example – The Congress of Parador • The first step we will take to tackle this apportionment problem is to compute the population to seats ratio, called the standard divisor (SD). • In the case of Parador, the standard divisor is 12,500,000/250 = 50,000. • The standard divisor tells us that in Parador, each seat in the Congress corresponds to 50,000 people.

25. Example – The Congress of Parador • We can now use this yardstick to find the number of seats that each state should get by the proportionality criterion– all we have to do is divide the state’s population by 50,000. • For state A, if we divide the population of A by the standard divisor, we get 1,646,000/50,000 = 32.92. • This number is called the standard quota of state A.

26. Example – The Congress of Parador • Using the standard divisor SD = 50,000, we can quickly find the standard quotas of each of the other states. These are shown in Table 4-4 (rounded to two decimal places). • The sum of the standard quotas equals 250, the number of seats being apportioned.

27. Example – The Congress of Parador • Once we have computed the standard quotas, we get to the heart of the apportionment problem: • How should we round these quotas into whole numbers? • We all learned in school how to round decimals to whole numbers–round down if the fractional part is less than 0.5, round up otherwise. This kind of rounding is called rounding to the nearest integer, or simply conventional rounding.

28. Example – The Congress of Parador • Unfortunately, conventional rounding will not work in this example, and Table 4-5 shows why not–we would end up giving out 251 seats in the Congress, and there are only 250 seats to give out!

29. Example – The Congress of Parador • Conventional rounding of the standard quotas is not going to give us the solution to the apportionment of Parador’s Congress, what do we try next? • Our search for a good solution to this problem will be the theme of our journey through the rest of this chapter.

30. Assignment p. 144: 1, 3, 5, 6, 10