1. Chapter 15: Apportionment Part 1: Introduction

2. Apportionment To "apportion" means to divide and assign in proportion according to some plan. An apportionment problem is essentially a problem of deciding how to round fractional parts so as to maintain the sum of the whole. For example, consider the sum 20.34 + 18.3 + 1.36 = 40 We seek to round the terms of the sum to integer values and maintain the same total.

3. Apportionment Suppose we are given the sum 20.34 + 18.3 + 1.36 = 40 The “apportionment problem” is essentially to round the terms of the sum to integers and maintain a constant sum. For example, we may conclude with the sum 20 + 18 + 2 = 40. In general, given any real numbers q i > 0 for i = 1,2,…,n we seek integers a i > 0 such that

4. Apportionment Apportionment can also be considered as a problem of fair-division when not all participants have equal claim to the whole. There are many interesting apportionment problems but perhaps the most famous example is the apportionment of the seats in the United States House of Representatives.

5. Apportionment In the House Congress is made of the House of Representatives and the Senate. Currently there are 435 seats in the House of Representatives and 100 members of the Senate. Each of the 50 states sends a certain number of representatives to the House according to the size of that state's population. The larger the population, the more seats the state gets. Not only does apportionment decide the number of representatives each State has in Congress, but the number of seats a state has in the House determines the number of votes it has in the Electoral College. The number of Electoral votes given to a particular state is equal to the number of seats it has in the House and in the Senate added together. Because every state has 2 Senate seats, the real determining factor in how many electoral votes a state has is the number of seats in the House.

6. Apportionment In the House The Electoral College was formed as a compromise between those who wanted the president elected by a popular vote and those who wanted Congress to elect the president. There are a total of 538 Electoral votes (538 = 435 seats in the House + 100 Senate seats + 3 votes for the District of Columbia.) For example, Florida currently has 25 Congressional districts and one representative for each in the U.S. House of Representations. Florida also has 2 Senators and thus Florida currently has 25+2 = 27 Electoral votes in the Electoral College. To win the presidency, a candidate must get at least 270 (more than half) of the Electoral votes. While it is not written in law, the standard procedure is that the members of the Electoral College will vote according to whoever wins the plurality of the popular vote in their state.

7. Apportionment In the House It is interesting to note that the 435 seats of the House of Representatives are divided up among the 50 States of the Union – and that does not include Washington, D.C. The District of Columbia is not a state – and therefore has no representatives in Congress. In the 1960s, a bill was passed to allow 3 electoral votes for Washington, D.C. Based on our Constitution, every 10 years the government conducts a national census. Based on the census count, the number of representatives for each state is reapportioned. The government completed the 2000 census in 2001 and reapportioned the House in 2003.

8. Methods of Apportionment We will study the following methods of apportionment –Hamilton’s Method –Jefferson’s Method –Webster’s Method –Adam’s Method –Huntington-Hill Method We also consider several paradoxes of apportionment –The Alabama Paradox –The Population Paradox –The New States Paradox Finally, we conclude with Balinsky & Young’s Impossibility Theorem.

9. Fuzzy Math? We consider methods of apportionment that produce some surprising results. In the 1880s, a Texas Congressman named Roger Mills expressed his frustration with the unexpected result of a reapportionment of the House of Representatives as follows: I thought that mathematics was a divine science. I thought that mathematics was the only science that spoke to inspiration and was infallible in its utterances. I have been taught always that it demonstrated the truth. I have been told... that mathematics, like the voice of Revelation, said when it spoke, thus saith the Lord. But here is a new system of mathematics that demonstrates the truth to be false." Perhaps he felt like someone who said in the 2000 Presidential Campaign: “This is fuzzy math! “