Apportionment paradox
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.
History An example of the apportionment paradox known as "the Alabama paradox" was discovered in the context of United States congressional apportionment in 1880,
The method for apportionment used during this period, originally put forth by Alexander Hamilton, but vetoed by George Washington and not adopted until 1852,
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.
Population paradox 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. In 1900, Virginia lost a seat to Maine, even though Virginia's population was growing more rapidly. More precisely, their theorem states that there is no apportionment system that has the following properties for more than three parties
The division of seats in an election is a prominent cultural concern. In 1876, the United States presidential election turned on the method by which the remaining fraction was calculated. Rutherford Hayes received 185 electoral college votes, and Samuel Tilden received 184. Tilden won the popular vote. With a different rounding method the final electoral college tally would have reversed.