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):
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.
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||Size||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.
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.
Consider a legislature with a fixed number of "seats" to be divided among some number of districts ("states") according to the populations of the states. The population paradox occurs when two states have populations increasing at different rates and the state with the greater growth rate loses a legislative seat to the state with the lower growth rate.
The paradox arises because of rounding in the procedure for dividing the seats. See the apportionment rules for the United States Congress for an example.