Apportionment and Electing a US President

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

Electing the President of the United States is not based on the popular vote that each candidate gets in the presidential election. Rather, to be elected President a candidate must be endorsed by the Electoral College. The "founding fathers" did not fully trust the "people" (it took a long time for women to achieve an equal role with men as part of the electoral process). So the election of the President was "filtered" through the Electoral College. The Electoral College is in essence (the details are quite complex) a "weighted voting game" where the 51 players, the 50 states and District of Columbia, cast blocks of votes. How large a block of votes does a state get to cast? This number comes back to the number of seats the state has in the House of Representatives. Within mathematics, the mathematics behind how many seats a state gets in the House of Representatives is known as the apportionment problem.

We will get to the details of the way the number of seats assigned for each state in the House of Representatives is done but in the spirit of mathematical modeling we will consider a simpler version of the apportionment problem which still has a rich mathematical structure.

Example:

A college has the following enrollments in three courses usually taken by incoming freshmen.

C (College Algebra) 978
S (Statistics) 500
L (Liberal Arts) 322

The dean of the college has agreed to fund 30 tutors for the students in these classes to be assigned in a way that is proportional to the enrollments in the classes, but no tutor can be "shared" between the students of more than one type of course. What is a reasonable (fair) way of assigning the tutors to the courses?

Activity 1

Determine an apportionment of the 30 tutors to the three courses based on the enrollments of the courses.

Comment: In fact, many reasonable methods have been found to solve this problem. Sometimes these methods give the same answer but often they give different answers to questions of this kind.

Activity 2

Determine an apportionment of the 31 tutors to the three courses based on the enrollments of the courses.

Comment: The only change between Activity 1 and Activity 2 is that the number of tutors to be allocated to the three courses has increased from 30 to 31. In the language of the apportionment problem the number of tutors is often called the "house size" because this number plays the same role in this problem as the role of the number seats in a parliament such as the House of Representatives, where the states are assigned seats based on their populations.

Activity 3

Apportion first 10 seats and then 11 seats in a student government at a small college to three student parties which got A = 237, B = 333 votes, and C 430 votes respectively.


Question:

After completing Activities 1,2, and 3 are you happy with the method you used to carry out the apportionment in these two cases?