Concordance (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 concordant if ${\displaystyle t_{i}>t_{j}}$ implies ${\displaystyle a_{i}\geq a_{j}}$ for all i,j.

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

There is a fine point with some divisor methods in which the divisor sequence starts with 0. For example, in the Adams apportionment method, the quotient of each agent whose current allocation is 0, is ${\displaystyle t_{i}/0}$ which is infinite. Therefore, if there are fewer items than agents, then the Adams method is, theoretically, allowed to allocate the objects arbitrarily, even giving more items to agents with smaller entitlements, which contradicts concordance. In practice, this seldom happens, as the number of items is usually larger than the number of agents. Formally, Adams' method is usually defined such that it returns an empty set whenever the number of items is smaller than the number of agents.