Apportionment paradox

An apportionment paradox is a situation where an apportionment—a rule for dividing discrete objects according to some proportional relationship—produces results that violate notions of common sense or fairness.

Certain quantities, like milk, can be divided in any proportion whatsoever; others, such as horses, cannot—only whole numbers will do. In the latter case, there is an inherent tension between the desire to obey the rule of proportion as closely as possible and the constraint restricting the size of each portion to discrete values.

Several paradoxes related to apportionment and fair division have been identified. In some cases, simple adjustments to an apportionment methodology can resolve observed paradoxes. However, as shown by the Balinski–Young theorem, it is not always possible to provide a perfectly fair resolution that satisfies all competing fairness criteria.

An example of the apportionment paradox known as "the Alabama paradox" was discovered in the context of United States congressional apportionment in 1880, when census calculations found that if the total number of seats in the House of Representatives were hypothetically increased, this would decrease Alabama's seats from 8 to 7. An actual impact was observed in 1900, when Virginia lost a seat to Maine, even though Virginia's population was growing more rapidly: this is an example of the population paradox. In 1907, when Oklahoma became a state, New York lost a seat to Maine, thus the name "the new state paradox".

The method for apportionment used during this period, originally put forth by Alexander Hamilton, but vetoed by George Washington and not adopted until 1852, was as follows:

The Hamilton method replaced a rounding method proposed by Thomas Jefferson, and was itself replaced by the Huntington–Hill method in 1941.

The Alabama paradox was the first of the apportionment paradoxes to be discovered. The US House of Representatives is constitutionally required to allocate seats based on population counts, which are required every 10 years. The size of the House is set by statute.

After the 1880 census, C. W. Seaton, chief clerk of the United States Census Bureau, computed apportionments for all House sizes between 275 and 350, and discovered that Alabama would get eight seats with a House size of 299 but only seven with a House size of 300. In general the term Alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares. A similar exercise by the Census Bureau after the 1900 census computed apportionments for all House sizes between 350 and 400: Colorado would have received three seats in all cases, except with a House size of 357, in which case it would have received two.