# Rank-index method

In apportionment theory, rank-index methods$^{:Sec.8}$ are a set of apportionment methods that generalize the divisor method. These have also been called Huntington methods, since they generalize an idea by Edward Vermilye Huntington.

## Input and output

Like all apportionment methods, the inputs of any rank-index method are:
* A positive integer $h$ representing the total number of items to allocate. It is also called the house size.
* A positive integer $n$ 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 $(t_{1},\ldots ,t_{n})$ with $\sum _{i=1}^{n}t_{i}=1$, representing entitlements - $t_{i}$ represents the entitlement of agent $i$, that is, the fraction of items to which $i$ is entitled (out of the total of $h$).
Its output is a vector of integers $a_{1},\ldots ,a_{n}$ with $\sum _{i=1}^{n}a_{i}=h$, called an apportionment of $h$, where $a_{i}$ is the number of items allocated to agent i.

## Iterative procedure

Every rank-index method is parametrized by a rank-index function $r(t,a)$, which is increasing in the entitlement $t$ and decreasing in the current allocation $a$. The apportionment is computed iteratively as follows:
* Initially, set $a_{i}$ to 0 for all parties.
* At each iteration, allocate one item to an agent for whom $r(t_{i},a_{i})$ is maximum (break ties arbitrarily).
* Stop after $h$ iterations.
Divisor methods are a special case of rank-index methods: a divisor method with divisor function $d(a)$ is equivalent to a rank-index method with rank-index function $r(t,a)=t/d(a)$.

## Min-max formulation

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:$^{:Thm.8.1}$
   $\min _{i:a_{i}>0}r(t_{i},a_{i}-1)\geq \max _{i}r(t_{i},a_{i})$.

## Properties

Every rank-index method is house-monotone. This means that, when $h$ 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 $1,\ldots ,k$, and apply the same method to their combined allocation, then the result is exactly the vector $(a_{1},\ldots ,a_{k})$. 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.$^{:Thm.8.3}$
* Every apportionment method that is uniform, house-monotone and balanced must be a rank-index method.

## Quota-capped divisor methods

A quota-capped divisor method is an apportionment method where we begin by assigning every state its lower quota of seats. Then, we add seats one-by-one to the state with the highest votes-per-seat average, so long as adding an additional seat does not result in the state exceeding its upper quota. However, quota-capped divisor methods violate the participation criterion (also called population monotonicity)—it is possible for a party to lose a seat as a result of winning more votes.$^{:Tbl.A7.2}$
Every quota-capped divisor method satisfies house monotonicity. Moreover, quota-capped divisor methods satisfy the quota rule.$^{:Thm.7.1}$
However, quota-capped divisor methods violate the participation criterion (also called population monotonicity)—it is possible for a party to lose a seat as a result of winning more votes.$^{:Tbl.A7.2}$ This occurs when:
* Party i gets more votes.
* Because of the greater divisor, the upper quota of some other party j decreases. Therefore, party j is not eligible to a seat in the current iteration, and some third party receives the seat instead.
* Then, at the next iteration, party j is again eligible to win a seat and it beats party i.
Moreover, quota-capped versions of other algorithms frequently violate the true quota in the presence of error (e.g. census miscounts). Jefferson's method frequently violates the true quota, even after being quota-capped, while Webster's method and Huntington-Hill perform well even without quota-caps.