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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 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 would decrease Alabama's share from 8 to 7. There was more to come—in the 1900s, Virginia lost a seat to Maine as a result of its population growing faster than Maine's. When Oklahoma became a new 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 no more or no fewer than its fair share of seats and the total number of seats increased by that amount.

The method for apportionment used during this period, originally put forth by Alexander Hamilton but not adopted until 1852, was as follows. First, the fair share of each state, i.e. the proportional share of seats that each state would get if fractional values were allowed, is computed. Next, the fair shares are rounded down to whole numbers, resulting in unallocated "leftover" seats. These seats are allocated to the states whose fair share exceeds the rounded-down number by the highest amount.

One might expect that the abundance of paradoxes is perhaps due to some deficiency of Hamilton's method. Indeed, a number of schemes have been proposed and four different methods signed into law (five counting repetitions). Amusingly, this vacillation has had less to do with mathematical than political considerations, such as the total number of seats that each party would be allotted by a given method. No method, however, has been found perfectly satisfactory in practice. It should therefore come as no surprise that in 1982, two mathematicians (Balinski and Young) developed a theorem showing that any method of apportionment will result in paradoxes. 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 one party gets more votes, whereas the other parties retain the same number of votes, that party does not get fewer seats.

• Alabama paradox
• New states paradox
• Population paradox

• The Constitution and Paradoxes