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 our 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 adjustments to an apportionment methodology can resolve observed paradoxes. Others, such as those relating to the United States House of Representatives, call into question notions that mathematics alone can provide a single, fair resolution.
The Alabama paradox was discovered in 1880, when it was found that increasing the total number of seats in the House of Representatives would decrease Alabama's seats from 8 to 7. There was more to come: when Oklahoma became a state in 1907, a recomputation of apportionment showed that the number of seats due to other states would be affected even though Oklahoma would be given a fair share of seats and the total number of seats increased by that number. Increasing the house from 386 to 391 members made New York lose a seat while Maine gained one, thus the name "the new state paradox". 
- 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.
In 1983 two mathematicians, Michel Balinski and Peyton Young, proved that any method of apportionment will result in paradoxes whenever there are three or more parties (or states, regions, etc.). More precisely, their theorem states that there is no apportionment system that has the following properties (as the example we take the division of seats between parties in a system of proportional representation):
- It follows the quota rule: Each of the parties gets one of the two numbers closest to its fair share of seats (if the party's fair share is 7.34 seats, it gets either 7 or 8).
- It does not have the Alabama paradox: If the total number of seats is increased, no party's number of seats decreases.
- It does not have the population paradox: If party A gets more votes and party B gets fewer votes, no seat will be transferred from A to B.
Methods may have a subset of these properties, but can't have all of them:
- A method may follow quota and be free of the Alabama paradox. Balinski and Young constructed a method that does so, although it is not in common political use.
- A method may be free of both the Alabama paradox and the population paradox. These methods are divisor methods, and Huntington-Hill, the method currently used to apportion House of Representatives seats, is one of them. By the impossibility result, these methods will necessarily fail to always follow quota.
- No method may always follow quota and be free of the population paradox.
Examples of paradoxes
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 8 seats with a House size of 299 but only 7 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.
|With 10 seats||With 11 seats|
|State||Population||Fair share||Seats||Fair share||Seats|
Observe that state C's share decreases from 2 to 1 with the added seat.
This occurs because increasing the number of seats increases the fair share faster for the large states than for the 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 examines which states have the largest fraction.
New states paradox
Given a fixed number of total representatives (as determined by the United States House of Representatives), adding a new state would in theory reduce the number of representatives for existing states, as under the United States Constitution each state is entitled to at least one representative regardless of its population. However, because of how the particular apportionment rules deal with rounding methods, it is possible for an existing state to get more representatives than if the new state were not added.
The population paradox is a counterintuitive result of some procedures for apportionment. When two states have populations increasing at different rates, a small state with rapid growth can lose a legislative seat to a big state with slower growth.
Some of the earlier Congressional apportionment methods, such as Hamilton, could exhibit the population paradox. However, divisor methods such as the current method do not.
- Apportioning Representatives in the United States Congress - Paradoxes of Apportionment, Michael J. Caulfield
- Balinski, Michel; H. Peyton Young (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. Yale Univ Pr. ISBN 0-300-02724-9.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- Balinski, Michel; H. Peyton Young (2001). Fair Representation: Meeting the Ideal of One Man, One Vote (2nd ed.). Brookings Institution Press. ISBN 0-8157-0111-X.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
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- Smith, Warren D. "Apportionment and rounding schemes".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- Cut-the-knot: The Constitution and Paradoxes
- The Constitution and Paradoxes
- Apportionment: Balinski and Young's Contribution