# Rank-index method

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Rank-index_method
> Markdown URL: https://mediated.wiki/source/Rank-index_method.md
> Source: https://en.wikipedia.org/wiki/Rank-index_method
> Source revision: 1307695320
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [apportionment theory](/source/Mathematics_of_apportionment), **rank-index methods**[1]: Sec.8 are a set of [apportionment methods](/source/Apportionment_method) that generalize the [divisor method](/source/Divisor_method). These have also been called **Huntington methods**,[2] since they generalize an idea by [Edward Vermilye Huntington](/source/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](/source/Federal_states) or [political parties](/source/Political_party).
- A vector of fractions (t_1,\ldots,t_n) with \sum_{i=1}^n t_i = 1, representing *entitlements* - t_i represents the [entitlement](/source/Entitlement_(fair_division)) 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](/source/Divisor_method) 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:[1]: 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*](/source/House_monotonicity). 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](/source/Balance_(apportionment))* must be a rank-index method.[1]: Thm.8.3
- Every apportionment method that is *uniform*, [*house-monotone*](/source/House_monotonicity) and *[balanced](/source/Balance_(apportionment))* must be a rank-index method.[2]

## 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.[3] However, quota-capped divisor methods violate the [participation criterion](/source/Participation_criterion) (also called [population monotonicity](/source/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](/source/House_monotonicity). Moreover, quota-capped divisor methods satisfy the [quota rule](/source/Quota_rule).[5]: Thm.7.1

However, quota-capped divisor methods violate the [participation criterion](/source/Participation_criterion) (also called [population monotonicity](/source/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:

1. Party *i* gets more votes.
1. 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.
1. 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.[6]

## References

1. Balinski, Michel L. & Young, H. Peyton (1982). [*Fair Representation: Meeting the Ideal of One Man, One Vote*](https://archive.org/details/fairrepresentati00bali). New Haven: Yale University Press. ISBN 0-300-02724-9.

1. Balinski, M. L. & Young, H. P. (1977-12-01). ["On Huntington Methods of Apportionment"](https://epubs.siam.org/doi/pdf/10.1137/0133043). *SIAM Journal on Applied Mathematics*. **33** (4): 607–618. [doi:10.1137/0133043](https://doi.org/10.1137/0133043). [ISSN 0036-1399](https://www.worldcat.org/issn/0036-1399)

1. Balinski, M. L. & Young, H. P. (1975-08-01). ["The Quota Method of Apportionment"](https://doi.org/10.1080/00029890.1975.11993911). *The American Mathematical Monthly*. **82** (7): 701–730. [doi:10.1080/00029890.1975.11993911](https://doi.org/10.1080/00029890.1975.11993911). [ISSN 0002-9890](https://www.worldcat.org/issn/0002-9890)

1. Balinski, Michel L. & Young, H. Peyton (1982). [*Fair Representation: Meeting the Ideal of One Man, One Vote*](https://archive.org/details/fairrepresentati00bali). New Haven: Yale University Press. ISBN 0-300-02724-9.

1. Balinski, Michel L. & Young, H. Peyton (1982). [*Fair Representation: Meeting the Ideal of One Man, One Vote*](https://archive.org/details/fairrepresentati00bali). New Haven: Yale University Press. ISBN 0-300-02724-9.

1. Spencer, Bruce D. (December 1985). ["Statistical Aspects of Equitable Apportionment"](http://www.tandfonline.com/doi/abs/10.1080/01621459.1985.10478188). *Journal of the American Statistical Association*. **80** (392): 815–822. [doi:10.1080/01621459.1985.10478188](https://doi.org/10.1080/01621459.1985.10478188). [ISSN 0162-1459](https://www.worldcat.org/issn/0162-1459)

---
Adapted from the Wikipedia article [Rank-index method](https://en.wikipedia.org/wiki/Rank-index_method) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rank-index_method?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.