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Apportionment: Introduction

Feature Column Archive

1. Introduction

Who would have thought that the United States Constitution would be the source of so much work for mathematicians! The reason stems from two strands of thought in the Constitution and at times these two separate threads of interest for mathematicians have coalesced.

The first strand of mathematics grows out of Article 1, Section 2 ( now amended) where it states that:

Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other persons.

Somewhat further in Section 2 it states:

The Number of Representatives shall not exceed one for every thirty thousand, but each State shall have at least one Representative;

This leads to a fascinating mathematical question which has come to be known as the apportionment problem (AP), and the related specific problem raised by the constitution which I will refer to as the constitutional apportionment problem (CAP).

We can formulate the AP mathematically as follows:

Given states s1, ..., sn with populations P1, ..., Pn and a positive integer h (think of h as the number of seats in the legislature), determine non-negative integers a1, ..., an where a1 + ... + an = h. (It is customary to think of the value h as given in advance and fixed, since currently the size of the House of Representatives is fixed; however, for some applications one might have the freedom to vary h as part of solving the problem.)

The CAP problem differs from the one above in requiring that each ai be greater than or equal to 1, or more generally (mathematicians like to generalize!) greater than or equal to bi where bi is some positive integer. The Constitution does not specify the h which started at 65 in 1790 and has grown to the now permanent value of 435, though when Alaska and Hawaii were admitted to the Union the value of h rose temporarily to 437.

At first glance the AP problem does not seem hard. If a state has 10 percent of the population and there are 37 items (seats in the parliament, computer systems, libraries, etc.) to apportion, then .10 (37) equals 3.7. In a parliament interpretation, the problem is we can not send 3.7 people to the legislature (though some feel they do not get full representation from whole bodies); 3.7 is not an integer! What should be done with those nuisance fractions? The quota principle (fairness rule) would say, in this example, that 3 or 4 representatives be assigned. With 3 representatives a state would be underrepresented, with 4 it would be over represented, but the method we currently use to apportion the House of Representatives could assign fewer than 3 or more than 4 representatives!

An additional mathematical strand grows out of Article 1, Section 2 of the Constitution (and picks up exactly where the first italicized section above ends), which states:

The actual enumeration shall be within three years after the first meeting of the Congress of the United States, and within every subsequent term of ten years, in such a manner as they shall by law direct.

The first census was made in 1790 and has occurred regularly at 10-year intervals, the most recent having been carried out in 2000. As the population of the United States has grown, as more Americans live for extended periods of time in other countries, and as people travel within the United States away from home for long periods, the problem of picking one day as the day to determine the official population of the United States and successfully counting all Americans on that day has become very complex. (Generally speaking federal employees who are either civilians or in the military, and their accompanying dependents are counted for apportionment purposes and assigned to a home state but employees of private firms stationed abroad are not counted for the purposes of apportionment.) It is widely agreed that it is especially hard to avoid undercounting the urban poor and migrant workers, for example, but there has been both political controversy and scholarly debate about what is to be done to deal with the situation. Whereas statistical sampling and adjustment techniques are in widespread use as tools for related situations, the highly charged political atmosphere associated with the census has resulted in a variety of court challenges and legislative restrictions to the way the Census Bureau has attempted to carry out its mandate to count all residents in America whether or not they are citizens). There is even debate about, for example, about how to define accuracy for the census since there are different reasonable approaches which vary in political consequences! Many mathematicians are employed by the Census Bureau to assist directly with the Census and related issues in applying the census data to such problems as apportionment, both at the federal and the state level.

You may find the apportionment population data for the last census, a graphic showing changes in the number members of the House of Representatives for each state based on the 2000 census, the United States overseas population, and US population growth data of interest.

There is another mathematical consequence of the CAP problem. When each state is assigned its number of the seats in the House of Representatives for the next 10 years, sometimes this number of seats is different from the assignment in the previous 10 years. This means that the congressional districts must be redrawn to either increase or decrease the number of representatives. (Even when the number of seats assigned to a state has not changed, population movement within a state might call for redistricting.) Building on a practice which began in England a long time earlier, many state legislatures, when redistricting, practice gerrymandering. This refers to the drawing of district lines in an exotic way to achieve some political goal, typically maximizing the number of seats which will be won by the same party as the one which controls the state legislature. In recent years, the Supreme Court has had to deal with many cases which were gerrymandered to achieve a particular racial goal, such as guaranteeing the election of at least one minority representative from a particular state with high minority population. Many states have turned to mathematicians to design districts which obey compactness criteria or other equity or political goals. Mathematical experts can also play a role in assisting with redistricting, whatever the goals.

The two strands mentioned above come together because the data from the census is used, among other things, as input to the procedure that has been developed to decide how many seats each state gets in the House of Representatives. It is also important to notice that apportionment ideas come into play in a wide variety of applied problems, many of which grow out of operations research. These problems arise whenever some collection of indivisible objects are to be distributed (or taken away) on the basis of data about the entities which are to share the objects. Thus, one might be distributing or cutting faculty lines in the different schools of a large university, assigning secretaries to the divisions of a new company, etc.

We will continue with a discussion of some of the rich and fascinating history of the apportionment problem, followed by a discussion of some of the mathematical questions to which it has given rise. We will also describe some of the mathematical issues that have arisen in conjunction with the census.

Joseph Malkevitch
York College (CUNY)


  1. Introduction
  2. The History of Apportionment in America
  3. Apportionment in Europe (and Other Democracies)
  4. References