Congressional representation and mathematics

Summary: Though the Constitution dictates proportional representation by state, there are multiple methods for attempting to achieve fair apportionment.

Apportionment is the process of distributing a fixed resource on a proportional basis, particularly associated with government. The legislative branch of the U.S. federal government—and most U.S. states—is bicameral, meaning that two separate bodies deliberate on laws. Reflecting a great political compromise of American government, these bodies are formulated on two distinct representative principles. The U.S. Senate has equal representation from each state to ensure that states have equal voices. For the House of Representatives, the U.S. Constitution requires that “Representatives…shall be apportioned among the several States which may be included within this Union, according to their respective Numbers.”

This requirement ensures that larger states have a voice that fairly represents their greater constituencies. The primary mathematical challenge in most systems of representation is that typically not all representatives will represent the same number of citizens, and calculations rarely result in integers. Deciding a fair system of rounding for representative numbers for fractional constituencies has proven surprisingly challenging, and Congressional apportionment has generated substantial controversy throughout the history of the United States.

Numerous serious apportionment methods have been proposed. Most have names associated with the people who proposed them, such as third U.S. president Thomas Jefferson, and are generally classified as “divisor methods” or “quota methods.” Many systems have been used in the United States, and mathematicians have long investigated fair apportionment. In 1948, at the request of the National Academy of Sciences, mathematicians Luther Eisenhart, Marston Morse, and John von Neumann recommended the Huntington–Hill method, proposed by mathematician Edward Huntington and statistician Joseph Hill. Apportionment is a prominent aspect of social choice theory, extensively studied by mathematicians such as Peyton Young and Michel Balinski. There have also been innovative links between apportionment and other areas of mathematics, like just-in-time sequencing and scheduling problems for manufacturing.

Apportionment Methods

A state’s proportion of the total population of a country can be found by dividing the state’s population by the total population. The state’s fair share of the total seats in the nation’s legislature, called its “standard quota,” is the product of this proportion and the total seats. Alternatively, the standard quota can be found by using the standard divisor, which measures the average number of people per seat on a national basis, and is found by dividing the total population by the number of available seats.

For example, suppose that a small country consists of four states (A, B, C, D), with populations given as

The total population of this country is 9200, and State A has 791/9200 or approximately 8.6% of the population. If there are 25 seats in the country’s legislature, then State A’s standard quota is

seats. State A’s population therefore warrants slightly more than two seats but less than three. The standard divisor in this case is 9200/25 = 368 people per seat, and State A’s standard quota can also be represented as 791/368. Similarly, the standard quotas for States B, C, and D are calculated as 2.424, 18.986, and 1.440, respectively.

The requirement that each state be assigned an integer number of representatives forces a country to impose a systematic method for rounding standard quotas to whole numbers. It is reasonable to expect that any reasonable method will assign each state either its lower quota or its upper quota—its standard quota rounded down or up, respectively. This requirement is known as the Quota Rule. One method that arises naturally is to round up those standard quotas that are closest to the next number of seats. Specifically, one may choose to initially apportion each state its lower quota, which will always yield leftover seats. These surplus seats are distributed to the states whose standard quotas have the largest fractional part. This method is known as Hamilton’s Method, Vinton’s Method, or the Method of Largest Remainders, named for the first Secretary of the Treasury Alexander Hamilton and Congressman Samuel Vinton. In the above example, after assigning each state its lower quota, only 23 of the 25 seats have been apportioned. The first surplus seat is assigned to State C, whose standard quota is very close to 19, while the other surplus seat is assigned to State D (See Table 1).

Some apportionment methods solve the problem of apportionment by using a specific rounding rule and modifying the standard divisor if necessary. In Jefferson’s Method, for example, all quotas are rounded down to the integer part of the quota. As with Hamilton’s Method, this yields unassigned seats in an initial apportionment. Rather than distributing those surplus seats as Hamilton’s Method does, Jefferson’s Method instead modifies the divisor by making it smaller. This method makes it easier for the states to obtain a seat and allows the states’ quotas to grow larger. In a successful Jefferson apportionment, a modified divisor is found so that when the modified quotas are rounded down, the total number of seats apportioned is the desired number.

Adams’ Method, named for sixth U.S. president John Quincy Adams, is similar to Jefferson’s Method. Rather than rounding down, however, quotas are instead rounded up to the next largest integer. In this case, the initial attempt at apportionment results in too many seats being distributed and a modified divisor must be chosen that is larger than the standard divisor, reflecting a need to make it more difficult to obtain a seat.

Other divisor methods differ from the methods of Jefferson and Adams only in how the rounding is conducted. In Webster’s Method, named for Senator and Secretary of State Daniel Webster, for example, all quotas are rounded conventionally—to the nearest whole number. If a state’s quota has a fractional part that is 0.5 or greater, the quota is rounded up. Otherwise, it is rounded down. In other words, one can think of the tipping point for rounding in Webster’s method as being the arithmetic mean of a state’s lower and upper quota. In the Method of Huntington–Hill, a state’s cutoff for rounding is the geometric mean of the state’s lower and upper quotas. For this method, if a state’s lower quota is L and its upper quota is U, then the cutoff for rounding is

If the quota is less than the cutoff, then it is rounded down, otherwise it is rounded up. In Dean’s Method, named for mathematician and astronomer James Dean, the cutoff for rounding is the harmonic mean of the lower and upper quota, expressed algebraically as

Applying any of these methods requires searching for a modified divisor so that when the modified quotas are calculated and rounded according to the given rule, the number of seats distributed is the correct total.

Applying these divisor methods to the sample situation given above results in the apportionments seen in Table 2. A number of important aspects of apportionment can be seen in the table. First, the apportionment method makes a difference; different methods can yield different apportionments. Jefferson’s Method has a substantial bias toward larger states. Adams’s Method, on the other hand, is biased toward smaller states and can cause lower quota violations. Quota rule violations can occur with Webster’s Method as well, though they are relatively rare. Webster’s Method demonstrates little bias overall. The Huntington-Hill Method and, to a greater degree, Dean’s Method, have biases toward smaller states.

History of U.S. Apportionment

The U.S. Constitution mandates a decennial census. Congressional representatives are reapportioned every 10 years based on the results. Though the Constitution provided for an initial distribution of U.S. Congressional representatives, it specified no particular apportioning method. Following Constitutional ratification in 1787 and the census of 1790, the first apportionment was carried out. In 1792, Congress passed a bill instituting Hamilton’s Method, which George Washington vetoed. Congress then approved Jefferson’s Method, which was used through 1832 when a Quota Rule violation was observed. New York State had a standard quota of 38.59 seats, so New York should have received either 38 or 39 seats. However, Jefferson’s Method assigned New York 40 seats. John Quincy Adams and Daniel Webster immediately put forth separate bills calling for the adoption of the apportionment methods that carry their names. Though both bills failed, this was the last apportionment in which Jefferson’s Method was used.

Webster’s Method was used for the apportionment of 1842, but in 1852, Hamilton’s Method was adopted as “permanent” by Congress. In 1872, Hamilton’s Method was not applied correctly. In 1882, additional difficulties arose with the method itself. While considering different sizes for the House of Representatives, observers noted that with a House size of 299, Alabama would receive eight seats under Hamilton’s Method, but with a House size of 300, Alabama would receive only seven seats. This flaw, whereby increasing the number of seats to apportion can, in and of itself, cause a state to lose a seat, became known as the “Alabama paradox.” Congress sidestepped this issue in 1882 by increasing the size of the House to 325 seats, but the flaw led to their discarding Hamilton’s Method in 1901. Webster’s Method was used in the apportionments of 1901, 1911, and 1931, though no apportionment was completed after the 1920 census. In 1941, Congress adopted the Huntington–Hill Method as “permanent,” with the House size of 435 seats, which is the method still in use at the start of the twenty-first century, though controversy continues.

Impossibility

Many mathematicians and others have asked whether there is an “ideal” apportionment method that solves the apportionment problem in a reasonable way and is free of flaws such as the Alabama paradox and quota rule violations. In the 1970s, Balinski and Young proved that no such method exists. Every apportionment method will either potentially violate the quota rule or cause either the Alabama paradox or another problematic paradox called the “Population paradox,” whereby one state whose population is growing at a faster rate can lose a seat to a state with a slower growth rate. The search for perfection in apportionment is an inherently impossible task, but mathematicians continue to study the problems and paradoxes and seek new approaches to reduce bias.

Bibliography

American Mathematical Society. “Apportionment: Introduction.” http://www.ams.org/samplings/feature-column/fcarc-apportion1.

American Mathematical Society. “Apportionment II: Apportionment Systems.” http://www.ams.org/samplings/feature-column/fcarc-apportionii1.

Balinski, Michel, and Peyton Young. Fair Representation: Meeting the Ideal of One Man, One Vote. 2nd ed. Washington, DC: Brookings Institution Press, 2001.

Frederick, Brian. Congressional Representation & Constituents: The Case for Increasing the U.S. House of Representatives. New York: Routledge, 2009.