Balinski–Young theorem

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.). The theorem shows that any possible method used to allocate the remaining fraction will necessarily fail to always follow quota. More precisely, their theorem states that there is no apportionment system that has the following 3 properties (as the example we take the division of seats between parties in a system of proportional representation):
 * It avoids violations of the Quota Rule: Each of the parties gets one of the two numbers closest to its fair share of seats. For example, if a party's fair share is 7.34 seats, it must get either 7 or 8 seats to avoid a violation; any other number will violate the rule.
 * It has House monotonicity: If the total number of seats is increased, no party's number of seats decreases.
 * It has Population monotonicity: If party A gets more votes and party B gets fewer votes, no seat will be transferred from A to B.

Specific systems
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. However, these methods will necessarily fail the quota rule in some elections.
 * No method may always follow quota and be free of the population paradox.
 * Largest remainder methods obey Quota Rules but have neither of the other two criteria. The Hamilton method of apportionment is actually a largest-remainder method which is specifically defined as using the Hare Quota

History
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. However, many mathematically analogous situations arise in which quantities are to be divided into discrete equal chunks. The Balinski–Young theorem applies in these situations: it indicates that although very reasonable approximations can be made, there is no mathematically rigorous way in which to reconcile the small remaining fraction while complying with all the competing fairness elements.

Related

 * Population monotonicity
 * House monotonicity
 * Quota rule