1. Chapter 4: The Mathematics of Apportionment

2. "Representatives...shall be apportioned among the several States, which may be included within this Union, according to their respective Numbers... Article 1, Section 2. U.S. Constitution

3. The "Numbers" mentioned in the previous slide is talking about the population of each state. Even though we know this information should be used for calculating representation, it doesn't actually say HOW to calculate it.

4. Section 1: The Apportionment Problem In Fair Division, everyone was entitled to the same amount. In Apportionment, the "players" are entitled to different shares based on some information. In the most popular case in this chapter, it will be representatives based on population. Other examples could be giving out homework to students based on the amount of work they do in class, giving candy for time spent cleaning bedroom, etc.

5. Chapter 4 Sect 2 We will give out seats in the House of Representatives based on each state's population. This is called the STANDARD DIVISOR. Standard divisor = total population (SD) Total number of seats The standard divisor tells us the number of people per seat in the legislature

6. Now we have to know how many seats that are available (standard divisor) each state is entitled to based on their population. This will be fractional answers, but that's okay. We will fix this using the different methods we will learn in this chapter. We call this number (the seats each state is entitled to) the standard quota. Standard quota= each state's population (Sq) Standard divisor

7. Standard Quota Since each quota will come out to be a decimal, we will use the lower quota and the upper quota in apportioning out the seats. The lower quota is each state's quota rounded down NO MATTER WHAT FRACTIONAL PART IT HAS. For example, if a quota is 35.8, the lower quota is going to be 35.

8. The upper quota for each state is its standard quota rounded up, NO MATTER WHAT THE FRACTIONAL PART IS. If a quota is 35.1, you still round up to 36! In the next few sections, you'll learn the methods used to apportion out goods using either the lower quota or the upper quota.

9. Chapter 4 Sect 3: Hamilton's Method Step 1: find the standard divisor Step 2: find each state's standard quota Step 3: give each state it's LOWER QUOTA Step 4: give any surplus seats out to the states with the largest fractional parts ONE AT A TIME until all seats are given out.

10. Hamilton's Method example: Intergalactic Federation (Population in billions) PlanetABCDETotalPop15078173204295900 50 seats to be given out

11. Step 1: find the standard divisor Sd = 900 billion = 18 billion. 50 Step 2: find the standard quota for each state by diving its individual population by the standard divisor A: 150/18 = 8.33 B: 78/18 = 4.33 C: 173/18 = 9.61 D: 204/18 = 11.33 E: 295/18 = 16.38

12. Step 3: Give each state it's lower quota StateSqLqA8.38 B4.34 C9.619 D11.311 E16.3816 Total: 48

13. Step 4: Since the lower quotas add up to be only 48 seats, we have extra seats left over. StateSqLqA8.38 B4.34 C9.619 D11.311 E16.3816 We have two extra seats to give out. Those go to the two states with the highest fractional parts. That will be state C and state E.

14. Final apportionment State A: 8 seats State B: 4 seats State C: 10 seats (9+1 of the extras) State D: 11 seats State E: 17 seats (16+1 of the extras)

15. In each method we will use in the future, we will be using a MODIFIED DIVISOR. Instead of giving out the leftovers, we will modify our standard divisor so that we round, we get exactly the total number of seats we need.

16. In Jefferson's Method, we will modify our divisor so that if we round the standard quotas DOWN, we will get exactly the number of seats we have to give out. Chapter 4 Sect 4: Jefferson's Method StateSqLqA8.38 B4.34 C9.619 D11.311 E16.3816 In our previous example, our divisor was 18 billion. When we rounded down, it only added up to 48 seats. We want that number to go up to 50, so our sd has to go down.

17. To still round down and total exactly 50 seats, we will make our sd lower. We will try 16 billion. State Sq using md of 16 LqA9.379 B4.8754 C10.812510 D12.7512 E18.437518 Using 16 gave us a total of 53, which is too many. 16 was too low to use.

18. State Sq using md of 17.2 Lq A8.728 B4.534 C10.0610 D11.8611 E17.1517 Using a modified divisor of 17.2 will give me exactly 50 seats when I use the lower quota. BUT, it took me three tries to find the right md. I first used 17 then 17.5 before trying 17.2