Balance (apportionment)

There is a resource to allocate, denoted by ${\displaystyle h}$. For example, it can be an integer representing the number of seats in a house of representatives. The resource should be allocated between some ${\displaystyle n}$ agents, such as states or parties. The agents have different entitlements, denoted by a vector ${\displaystyle t_{1},\ldots ,t_{n}}$. For example, t_i can be the fraction of votes won by party i. An allocation is a vector ${\displaystyle a_{1},\ldots ,a_{n}}$ with ${\displaystyle \sum _{i=1}^{n}a_{i}=h}$. An allocation rule is a rule that, for any ${\displaystyle h}$ and entitlement vector ${\displaystyle t_{1},\ldots ,t_{n}}$, returns an allocation vector ${\displaystyle a_{1},\ldots ,a_{n}}$.

An allocation rule is called balanced if ${\displaystyle t_{i}=t_{j}}$ implies ${\displaystyle |a_{i}-a_{j}|\leq 1}$ for all i,j. Equivalently, ${\displaystyle t_{i}=t_{j}}$ implies ${\displaystyle a_{i}\geq a_{j}-1}$ for all i,j.

All known apportionment methods are balanced. In particular, both Highest averages methods and Largest remainder methods are balanced.

Every apportionment method that is anonymous, exact and coherent, is also balanced.[3][4]^{: Lem.2.1, p.7}[5]^: 112