![]() | This article may be too technical for most readers to understand.(January 2024) |
Part of the Politics and Economics series |
Electoral systems |
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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 with , called an apportionment of , where is the number of items allocated to agent i.
Every rank-index method is parametrized by a rank-index function , which is increasing in the entitlement 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 is equivalent to a rank-index method with rank-index 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
.
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:
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. [6]
![]() | This article may be too technical for most readers to understand.(January 2024) |
Part of the Politics and Economics series |
Electoral systems |
---|
![]() |
![]() ![]() |
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 with , called an apportionment of , where is the number of items allocated to agent i.
Every rank-index method is parametrized by a rank-index function , which is increasing in the entitlement 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 is equivalent to a rank-index method with rank-index 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
.
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:
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. [6]