Mathematics of apportionment

Mathematics of apportionment describes mathematical principles and algorithms for fair allocation of identical items among parties with different entitlements. Such principles are used to apportion seats in parliaments among federal states or political parties. See apportionment (politics) for the more concrete principles and issues related to apportionment, and apportionment by country for practical methods used around the world. Mathematically, an apportionment method is just a method of rounding fractions to integers. As simple as it may sound, each and every method for rounding suffers from one or more paradoxes. The mathematical theory of apportionment aims to decide what paradoxes can be avoided, or in other words, what properties can be expected from an apportionment method. The mathematical theory of apportionment was studied as early as 1907 by the mathematician Agner Krarup Erlang. It was later developed to a great detail by the mathematician Michel Balinsky and the economist Peyton Young. Besides its application to political parties, it is also applicable to fair item allocation when agents have different entitlements. It is also relevant in manpower planning - where jobs should be allocated in proportion to characteristics of the labor pool, to statistics - where the reported rounded numbers of percentages should sum up to 100%, and to bankruptcy problems. The inputs to an apportionment method are: A positive integer representing the total number of items to allocate. It is also called the house size, since in many cases, the items to allocate are seats in a house of representatives. A positive integer representing the number of agents to which items should be allocated. For example, these can be federal states or political parties. A vector of numbers representing entitlements - represents the entitlement of agent , that is, the amount of items to which is entitled (out of the total of ). These entitlements are often normalized such that .