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.
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.
[1]: 227–235 An example of the apportionment paradox known as "the Alabama paradox" was discovered in the context of United States congressional apportionment in 1880,[1]: 228–231 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.
[1]: 228–231 In general the term Alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares.
In 1983, two mathematicians, Michel Balinski and Peyton Young, proved that any method of apportionment that does not violate the quota rule will result in paradoxes whenever there are four or more parties (or states, regions, etc.).
[4][5] More precisely, their theorem states that there is no apportionment system that has the following properties for more than three states[1]: 233–234 (as the example we take the division of seats between parties in a system of proportional representation): It is of note that any method of apportionment free of the Population Paradox will always be free of Alabama Paradox.
[4][5] They show a proof of impossibility: apportionment methods may have a subset of these properties, but cannot have all of them: The division of seats in an election is a prominent cultural concern.
[1]: 233 The Balinski–Young theorem applies in these situations: it indicates that although very reasonable approximations can be made, there is no mathematically rigorous way to reconcile the small remaining fraction while complying with all the competing fairness elements.
[1]: 233 In general, the response from mathematicians has been to abandon the quota rule as the less-important property, accepting that apportionment errors may sometimes slightly exceed one seat.