Like all apportionment methods, the inputs of any rank-index method are:
- A positive integer representing the total number of items to allocate. It is also called the house size.
- 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 fractions with , representing entitlements - represents the entitlement of agent , that is, the fraction of items to which is entitled (out of the total of ).
Its output is a vector of integers with , called an apportionment of , where is the number of items allocated to agent i.
Every rank-index method can be defined using a min-max inequality: a is an allocation for the rank-index method with function r, if-and-only-if:[1]: Thm.8.1
.
Every rank-index method is house-monotone. This means that, when increases, the allocation of each agent weakly increases. This immediately follows from the iterative procedure.
Every rank-index method is uniform. This means that, we take some subset of the agents , and apply the same method to their combined allocation, then the result is exactly the vector . In other words: every part of a fair allocation is fair too. This immediately follows from the min-max inequality.
Moreover:
- Every apportionment method that is uniform, symmetric and balanced must be a rank-index method.[1]: Thm.8.3
- Every apportionment method that is uniform, house-monotone and balanced must be a rank-index method.[2]