An apportionment paradox exists when the rules for apportionment in a political system produce results which are unexpected or seem to violate common sense.
To apportion is to divide into parts according to some rule, the rule typically being one of proportion. 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. This results, at times, in unintuitive observations, or paradoxes.
Several paradoxes related to apportionment, also called fair division, have been identified. In some cases, simple post facto adjustments, if allowed, to an apportionment methodology can resolve observed paradoxes. However, as shown by examples relating to the United States House of Representatives, and subsequently proven by the Balinski–Young theorem, mathematics alone cannot always provide a single, fair resolution to the apportionment of remaining fractions into discrete equal whole-number parts, while complying fully with all the competing fairness elements.
History
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:
First, the fair share of each state is computed, i.e. the proportional share of seats that each state would get if fractional values were allowed. Second, each state receives as many seats as the whole number portion of its fair share. Third, any state whose fair share is less than one receives one seat, regardless of population, as required by the United States Constitution. Fourth, any remaining seats are distributed, one each, to the states whose fair shares have the highest fractional parts.
The Hamilton method replaced a rounding method proposed by Thomas Jefferson, and was itself replaced by the Huntington–Hill method in 1941. Under certain conditions, the Huntington-Hill method can also give paradoxical results.
Examples of paradoxes
Alabama paradox
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.
The following is a simplified example (following the largest remainder method) with three states and 10 seats and 11 seats.
Observe that state C's share decreases from 2 to 1 with the added seat.
In this example of a 10% increase in the number of seats, each state's share increases by 10%. However, increasing the number of seats by a fixed % increases the fair share more for larger numbers (i.e., large states more than small states). In particular, large A and B had their fair share increase faster than small C. Therefore, the fractional parts for A and B increased faster than those for C. In fact, they overtook C's fraction, causing C to lose its seat, since the Hamilton method allocates according to which states have the largest fractional remainder.
The Alabama paradox gave rise to the axiom known as house monotonicity, which says that, when the house size increases, the allocations of all states should weakly increase.