The Apportionment Problem Section 9.1

Objectives: 1. Understand and illustrate the Alabama paradox. 2. Understand and illustrate the population paradox. 3. Understand and illustrate the new-states paradox. 4. Understand Balinski and Young’s Impossibility Theorem.

Textbook: Page 508 – 509/Understanding Apportionment Read through to Table 9.1 In your own words, succinctly tell what this passage is about.

Example 1: Identify the integer and the fractional part

Example 2: Identify the integer and the fractional part

Using the Hamilton Method The Hamilton method uses fractional parts to apportion representatives. 1. Determine the exact number of board members: i. percent of stockholders X size of board 2. Assign the integer part i. If there are more members to be allocated, then go to step Assign additional members according to the fractional parts i. The 1 st additional member goes to the company having the largest fractional part ii. The 2 nd additional member, if any, goes to the company with the 2 nd largest fractional part. i. Continue in this manner until you have assigned all additional members.

Example 3: Using the Hamilton Apportionment Method 12 Member Board CompanyPopulation Determine Percentage Step 1: Board Members Deserved Step 2: Assign Integer Part Naxxon47 Aroco37 Eurobile16 Total100

Example 4: TB pg. 517/4 Jungle World Theme Park EmployeesPopulation Percent of Employees Step 1: Members Deserved Step 2: Assign Integer Part Performers Food Wrkrs Maintenance Wrkrs Total

Section 9.1 Assignment TB pg. 516/1 – 7 odd Remember to write problems and show ALL work.

Section 9.1 Part II Alabama Paradox and Truncating the Fractional Part of a Number

Key Terms: Alabama Paradox – an increase in the total number of items to be apportioned results in the loss of an item for a group. Apportion – to divide according to a plan; to allot. Truncate – to shorten by cutting off. Note: sometimes it is necessary to truncate a number to keep the percentage from adding up to more than 100%.

Alabama Paradox: Illustrating the Alabama Paradox: A small country with a population of 10,000 is composed of 3 states. According to the country’s constitution, the congress will have 200 seats, divided among the 3 states according to their respective populations. Illustrating the Alabama Paradox. State Populatio n % of Pop. Step 1:Step 2:Step 3: A5015 B4515 C470 Total10,000

Example 5: Using Hamilton’s Method determine deserved seats. Illustrating the Alabama Paradox. StatePopulation % of Pop. Step 1:Step 2:Step 3: A5015 B4515 C470 Total10,000

Example 6: What happens if the number of seats in congress increases to 201. Illustrating the Alabama Paradox. StatePopulation % of Pop. Step 1:Step 2:Step 3: A5015 B4515 C470 Total10,000

Section 9.1 Assignment Part 1 TB pg. 517/11 and 12 (worksheet online) Remember to write problems and show ALL work.

Example 7: Alabama Paradox Using Hamilton’s Method Assume that there are now 10 members on the board. Oil Consortium Board Company % Stockholders (in thousands) Step 1:Step 2:Step 3: Naxxon47 Aroco37 Eurobile16 Total100

Example 8: Truncate the number to: a. Hundredths b. Ten thousandths

Example 9: Truncate the number to: a. Tenths b. Thousandths

Example 10: Presenting Survey Results Adjusting a list of numbers. Exact Percentage (column A) Percentage rounded to tenths (column B) Original Date Truncated to Tenths Part That is Discarded Final Results Taxes Educati on Crime14.75 Health Care Total100

Example 11: Adjusting a list of numbers A group of consumers was asked how they expected their spending to change in the next six months. Adjust the percentages in the following table so that they are shown to the tenths place and their sum is % Exact % % Rounded to tenths Original Data Truncated Discarded Part Final Results Spending Increase Spending Decrease Spending Stay the Same Unsure Total100.00%

Historical Highlight TB pg. 513/Apportionment U. S. History

Section 9.1 Assignment Part 2 Class work: TB pg. 517/13 – 22 Remember you must write problems and show ALL work to receive credit for this assignment.

Section 9.1 Part III Average Constituency, Absolute Unfairness, and Relative Unfairness

Key Term: Average Constituency – the quotient: population of state number of representatives from state NOTE: Comparing the representatives of two states A and B, we saw that state A is more poorly represented than state B, if the average constituency of A is larger than the average constituency of B.

Example 12: Average Constituency Finding the average constituency a. If the 420-member electricians union has three representatives on the United Labor Council, what is the average constituency of this group?

Example 13: Average Constituency Determine which group is more poorly represented. a. If the 420-member electricians’ union has three representatives on the United Labor Council, what is the average constituency of this group? b. If the 440-member plumbers’ union has four representatives on the council, are the electricians or the plumbers more poorly represented?

Key Term: Absolute Unfairness (of a state) – the difference between the larger average constituency and the smaller one. If State A has the larger average constituency, then the absolute fairness is: (avg. constituency of A) – (avg. constituency of B) NOTE: if the two states have the same average constituency, then we say that the two states are equally well represented.

Example 14: Find Absolute Unfairness Assume that state X has a population of 974,116 with four representatives and state Y has a population of 730,779 with three representatives. Compute the absolute unfairness for this apportionment.

Example 15: Find Absolute Unfairness Suppose the Weaver’s Guild, with 1,672 members, has six delegates on the National Art Commission and the Artists’ Alliance, with 1,535 members, has five delegates. Calculate the absolute unfairness for this assignment of delegates.

Key Term: Relative Unfairness (for 2 states) – the quotient: the absolute unfairness of the apportionment the smaller average constituency of the two states

Example 16: Determine Relative Unfairness If state A has a population of 11,710 and five representatives and state B has a population of 16,457 and seven representatives, calculate the relative unfairness of the apportionment.

Example 17: Determine Relative Unfairness Suppose state A has a population of 935,000 and five representatives, whereas state B has a population of 2,343,000 and eleven representatives. Determine the relative unfairness of this apportionment.

Section 9.1 Assignment Part 3 TB pg. 518/ Remember to write problems and show ALL work.