Math 132: Foundations of Mathematics Amy Lewis Math Specialist IU1 Center for STEM Education May 27, 2010 Math 132: Foundations of Mathematics

14.4 Flaws of Apportionment Methods Understand and illustrate the following: Alabama paradox Population paradox New-states paradox May 27, 2010 Math 132: Foundations of Mathematics

Math 132: Foundations of Mathematics Apportionment “The very mention of Florida outraged the Democrats. Florida’s contested electoral votes helped elect a Republican president who had lost the popular vote.” What election is this quote referring to? 1976: Rutherford B. Hays v. Samuel J. Tilden What happened? May 27, 2010 Math 132: Foundations of Mathematics

Fair Apportionment Method Although Hamilton’s method may appear to be a fair and reasonable apportionment method, it also creates some serious problems: Alabama paradox Population paradox New-states paradox May 27, 2010 Math 132: Foundations of Mathematics

Math 132: Foundations of Mathematics Hamilton’s Method Calculate each group’s standard quota. Round each standard quota down to the nearest whole number (the lower quota). Initially, give each group its lower quota. Give the surplus items, one at a time, to the groups with the largest decimal parts until there are no more surplus items. May 27, 2010 Math 132: Foundations of Mathematics

Math 132: Foundations of Mathematics Alabama Paradox An increase in the total number of items to be apportioned results in the loss of an item for a group. State A B C Total Population 5015 4515 470 10,000 What happens when the number of seat in congress is increased from 200 to 201? Start by finding the standard divisor for 200 seats. May 27, 2010 Math 132: Foundations of Mathematics

Math 132: Foundations of Mathematics Alabama Paradox State A B C Total Population 5015 4515 470 10,000 Standard Quota 100.3 90.3 9.4 200 Lower Quota 100 90 9 199 Surplus Seats 1 10 Now let’s see happens when the number of seat in congress is increased from 200 to 201? Calculate the new standard divisor and allocate seats. May 27, 2010 Math 132: Foundations of Mathematics

Math 132: Foundations of Mathematics Alabama Paradox State A B C Total Population 5015 4515 470 10,000 Standard Quota 100.8 90.75 9.45 201 Lower Quota 100 90 9 199 Surplus Seats 1 101 91 Is this fair? May 27, 2010 Math 132: Foundations of Mathematics

The Population Paradox Group A loses items to group B, even though the population of group A grew at a faster rate than group B. A small country has 100 seats in the congress, divided among the three states according to their respective populations. The table below shows their population before and after the country’s population increase. State A B C Total Original Population 19,110 39,090 141,800 200,000 New Population 19,302 39,480 200,582 May 27, 2010 Math 132: Foundations of Mathematics

The Population Paradox State A B C Total Original Population 19,110 39,090 141,800 200,000 New Population 19,302 39,480 200,582 Use Hamilton’s method to apportion the 100 congressional seats using the original problem. Find the percentage increase in the population of states A and B. Use Hamilton’s method to apportion the 100 congressional seats using the new population. May 27, 2010 Math 132: Foundations of Mathematics

The Population Paradox State A B C Total Original Population 19,110 39,090 141,800 200,000 Standard Quota 9.56 19.55 70.9 100 Lower Quota 9 19 70 98 Surplus Seats 1 Final Apportionment 10 71 May 27, 2010 Math 132: Foundations of Mathematics

The Population Paradox State A B C Total Original Population 19,110 39,090 141,800 200,000 New Population 19,302 39,480 200,582 Percent Increase State A: 1.004% State B: .9977% Who should benefit from the increased population? May 27, 2010 Math 132: Foundations of Mathematics

The Population Paradox State A B C Total New Population 19,302 39,480 141,800 200,582 Standard Quota 9.62 19.68 70.69 100 Lower Quota 9 19 70 98 Surplus Seats 1 Final Apportionment 20 71 What happened to state A’s apportionment? May 27, 2010 Math 132: Foundations of Mathematics

The New-States Paradox The addition of a new group changes the apportionments of other groups. When Oklahoma became a state, they decided that they would get 5 representatives, raising the number of seats from 386 to 391. This, however, changed the apportions for other states.

The New-States Paradox A school district has 2 HS, East High (2574 students) and West High (9426 students). The school district has a counseling staff of 100 counselors. Apportion the counselors to the two schools.

The New-States Paradox Standard Divisor: 120 State East High West High Total Number of Students 2,574 9,426 12,000 Standard Quota 21.45 78.55 100 Lower Quota 21 78 99 Surplus Seats 1 Final Apportionment 79

The New-States Paradox Suppose that North High School is added to the district with 750 students. The district hires 6 counselors for this new school. What is the new apportionment of counselors? State East High West High North High Total Number of Students 2,574 9,426 750 12,750 Standard Quota 21.40 78.37 6.24 106 Lower Quota 21 78 6 105 Surplus Seats 1 Final Apportionment 22

So what?

Math 132: Foundations of Mathematics Homework Bring your old tests and any clarifying questions that you may have about problems that were difficult for you. Final Class: Friday, May 28th!!! May 24, 2010 Math 132: Foundations of Mathematics