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.

Input and output

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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.

Iterative procedure

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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:

  • Initially, set   to 0 for all parties.
  • At each iteration, allocate one item to an agent for whom   is maximum (break ties arbitrarily).
  • Stop after   iterations.

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  .

Min-max formulation

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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 

 .

Properties

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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]

Quota-capped divisor methods

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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:

  1. Party i gets more votes.
  2. 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.
  3. 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

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  1. ^ a b c Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9.
  2. ^ a b Balinski, M. L.; Young, H. P. (1977-12-01). "On Huntington Methods of Apportionment". SIAM Journal on Applied Mathematics. 33 (4): 607–618. doi:10.1137/0133043. ISSN 0036-1399.
  3. ^ Balinski, M. L.; Young, H. P. (1975-08-01). "The Quota Method of Apportionment". The American Mathematical Monthly. 82 (7): 701–730. doi:10.1080/00029890.1975.11993911. ISSN 0002-9890.
  4. ^ Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9.
  5. ^ a b Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9.
  6. ^ Spencer, Bruce D. (December 1985). "Statistical Aspects of Equitable Apportionment". Journal of the American Statistical Association. 80 (392): 815–822. doi:10.1080/01621459.1985.10478188. ISSN 0162-1459.