Math 101

Homework #3

Instructions. This is due on March 17 at the beginning of class. It is worth 20 points. Papers lose 3 points per day late. You may work in pairs, but there is to be no discussion between pairs.

Do not just answer the questions asked below. Write a report that explains the problem and answers the questions in an organized, readable manner. You will be graded on the correctness of your work and on the quality of your explanations.

Problems.

In this assignment, you will see examples of several of the unfairness problems that can plague any apportionment system. Hamilton and Jefferson are picked on here, partly because they’re the easiest methods to compute and partly because it is less common for Webster or Hill-Huntington to make a mistake.

Start with a small nation with three states. State A has population 10,030. State B has population 9030. State C has population 940. Use Hamilton’s method to find an apportionment with house size 200. Logically, what do you think should happen if one more seat is added to the house? Find out what Hamilton’s method does by reworking the apportionment with house size 201. Explain what is weird and unfair about this scenario. (This problem is called the Alabama paradox for historical reasons.)

The next example starts with two states. State A has population 8955. State B has population 1045. Find the Hamilton apportionment with house size 100. Now, suppose that state C joins the nation. If state C has population 525, use your work on states A and B to argue that the new state should have 5 representatives. Then find the Hamilton apportionment for the new house with three states and house size 105. Explain what is weird and unfair about this result (which is an example of the new states paradox).

Next, we have a nation with five states. State A has population 150, B has 78, C has 173, D has 204 and E has 295. Find the Hamilton apportionment with house size 50. Suppose that the population of state C increases to 181 and state E increases to 296. Would you expect the apportionment to change? If it does change, how do you expect it to change? Find out what Hamilton’s method does by reworking the apportionment with the new populations. Explain what is weird and unfair about this result (which is an example of the population paradox).

The last example is a nation of six states. A has population 5,737,037. B has 3,107,576. C has 1,124,660. D has 947,154. E has 920,610. F has 511,456. Compute the Jefferson apportionment with house size 30. Comparing your apportionment to the quotas for each state, explain what is weird and unfair about this result. This is an example of a violation of the quota condition.