Apportionment Paradox - Impossibility Result

Impossibility Result

In 1982 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.). 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 party A gets more votes and party B gets fewer votes, no seat will be transferred from A to B.