Quota rule

In mathematics and political science, the quota rule describes a desired property of a proportional apportionment or election method. It states that the number of seats that should be allocated to a given party should be between the upper or lower roundings (called upper and lower quotas) of its fractional proportional share (called natural quota). As an example, if a party deserves 10.56 seats out of 15, the quota rule states that when the seats are allotted, the party may get 10 or 11 seats, but not lower or higher. Many common election methods, such as all highest averages methods, violate the quota rule. If is the population of the party, is the total population, and is the number of available seats, then the natural quota for that party (the number of seats the party would ideally get) is The lower quota is then the natural quota rounded down to the nearest integer while the upper quota is the natural quota rounded up. The quota rule states that the only two allocations that a party can receive should be either the lower or upper quota. If at any time an allocation gives a party a greater or lesser number of seats than the upper or lower quota, that allocation (and by extension, the method used to allocate it) is said to be in violation of the quota rule. Another way to state this is to say that a given method only satisfies the quota rule if each party's allocation differs from its natural quota by less than one, where each party's allocation is an integer value. If there are 5 available seats in the council of a club with 300 members, and party A has 106 members, then the natural quota for party A is . The lower quota for party A is 1, because 1.8 rounded down equal 1. The upper quota, 1.8 rounded up, is 2. Therefore, the quota rule states that the only two allocations allowed for party A are 1 or 2 seats on the council. If there is a second party, B, that has 137 members, then the quota rule states that party B gets , rounded up and down equals either 2 or 3 seats.