Rank-index method

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results In apportionment theory, rank-index methods[1]: Sec.8  are a set of apportionment methods that generalize the divisor method.

These have also been called Huntington methods,[2] since they generalize an idea by Edward Vermilye Huntington.

Like all apportionment methods, the inputs of any rank-index method are: Its output is a vector of integers

is the number of items allocated to agent i.

and decreasing in the current allocation

The apportionment is computed iteratively as follows: Divisor methods are a special case of rank-index methods: a divisor method with divisor function

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

This immediately follows from the iterative procedure.

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

This immediately follows from the min-max inequality.

Moreover: 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.

[3] 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.

[4]: Tbl.A7.2 Every quota-capped divisor method satisfies house monotonicity.

Moreover, quota-capped divisor methods satisfy the quota rule.

[5]: 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.

[5]: Tbl.A7.2  This occurs when: 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.