Notes on Modeling: Session VII: Apportionment

Prepared by:

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


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The apportionment model finds application in both trying to assign seats in European democracies based on the votes that are given to different parties and in the United States in assigning seats in the House of the Representatives to the 50 states based on their populations. The US situation has the extra intriguing proviso that that all states must get at least one seat. However, the apportionment model can be used in many other settings where the objects to be distributed to claimants are not easily subdivided the way one can subdivide of water or money.

For example, suppose one has a college that has a required writing course aimed at students in different groups: humanities (Writing 401), sciences (Writing 402), and professional programs (Writing 403). Although in principle students are allowed to take any of the courses, the Provost of the college decides how many sections of Writing 401, 402, 403 to offer each semester. If she decides to offer 17 sections all together, what might be a fair system to for choosing how many Writing 401, 402, and 403 sections to offer?

It is a nice exercise to compare and contrast how different real world situations that have an "apportionment problem flavor." Such problems typically have similarities and differences. For example, for the writing course problem above it does not seem reasonable not to offer at least one section of each the writing courses. If the college has students who attend in an evening session only, it might seem reasonable to have sections of each kind in both the day and evening sessions.

In Europe and America two somewhat independent traditions of solving apportionment problems emerged. Many of the most natural methods and ideas were developed in parallel in both America and in Europe. Thus, D'Hondt and Jefferson came up with the same ideas basically. In the US the divisor method based on always rounding down is called Jefferson. However, it is not the case the Jefferson wrote a "paper" in which the details of the method are described. Rather, he endorsed an approach to computing apportionments which can be formalized today in the way we do. The names often used for these methods are in essence due to Michel Balinski and H. Peyton Young because their book Legislative Fairness became so central to the discussion of apportionment. At one time Balinski and Young both worked for the Graduate Mathematics Program of CUNY but now are in Paris and Oxford.

Although it seems very desirable and reasonable that an apportionment method assign a state whose exact quota is not an integer the integer value above or below its exact quota, none of the divisor methods are guaranteed to do so. Historically, an important phenomenon that we will discuss at our next meeting is the problem of house monotonicity. Since there were times during which the number of representatives in the House of Representatives was growing it seemed natural that one would want a state not to be assigned fewer seats in a larger sized House of Representatives. However, in 1880, in what has come to be known as the "Alabama Paradox"

it was observed that when a larger sized House of Representatives was apportioned using Hamilton's Method, as was the law at the time, Alabama got fewer seats in a larger House of Representatives. Since the divisor methods can be viewed as giving out the seats to be distributed one at a time using what I have been calling the "table method" (but sometimes is called the rank function method) it is clear that divisor methods are house monotone.

Somewhat later, the issue of population monotonicity was studied. Could a method assign fewer seats to a state that gained population and more seats to a state that lost population? This can actually happen! We will explore examples of this as we continue our discussions. The method we currently use to apportion the House of Representatives is known as the Huntington-Hill Method, though it sometimes is also referred to as the Method of Equal Proportions. However, this name does not give it a better claim to being a "good" method than any other name!

What also has to be developed is a way of comparing two different apportionments from some point of view. How can we tell whether one apportionment is better or worse than another (different) apportionment? We will devote attention to relative as well as absolute differences. We can try to measure how far apart apportionments are using concepts such as representatives per person or people per representative. It may seem that it would not make any difference whether a measure or its reciprocal were being used but it turns out that it does make an important difference. Another issue is whether or not one is measuring how good an apportionment is from some "global" point of view or from the point of view of how an interchange of seats between two states (parties) would affect matters.

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