Rank-index method

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 $(t\_1,\backslash ldots,t\_n)$ with $\backslash sum\_\{i=1\}^n\; t\_i\; =\; 1$, 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 $a\_1,\backslash ldots,a\_n$ with $\backslash sum\_\{i=1\}^n\; a\_i\; =\; h$, called an apportionment of $h$, where $a\_i$ is the number of items allocated to agent i.

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 $r(t,a)\; =\; t/d(a)$.

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:{{Rp|Thm.8.1}}

$\backslash min\_\{i:\; a\_i\; >\; 0\}\; r(t\_i,\; a\_i-1)\; \backslash geq\; \backslash max\_\{i\}\; r(t\_i,\; a\_i)$.

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 $1,\backslash ldots,k$, and apply the same method to their combined allocation, then the result is exactly the vector $(a\_1,\backslash ldots,a\_k)$. 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.{{Rp|Thm.8.3}}
• Every apportionment method that is uniform, house-monotone and balanced must be a rank-index method.

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.{{Cite journal |last1=Balinski |first1=M. L. |last2=Young |first2=H. P. |date=1975-08-01 |title=The Quota Method of Apportionment |url=https://doi.org/10.1080/00029890.1975.11993911 |journal=The American Mathematical Monthly |volume=82 |issue=7 |pages=701–730 |doi=10.1080/00029890.1975.11993911 |issn=0002-9890|url-access=subscription }} 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.{{Cite book |last1=Balinski |first1=Michel L. |url=https://archive.org/details/fairrepresentati00bali |title=Fair Representation: Meeting the Ideal of One Man, One Vote |last2=Young |first2=H. Peyton |publisher=Yale University Press |year=1982 |isbn=0-300-02724-9 |location=New Haven |url-access=registration}}{{Rp|Tbl.A7.2}}

Every quota-capped divisor method satisfies house monotonicity. Moreover, quota-capped divisor methods satisfy the quota rule.{{Cite book |last1=Balinski |first1=Michel L. |url=https://archive.org/details/fairrepresentati00bali |title=Fair Representation: Meeting the Ideal of One Man, One Vote |last2=Young |first2=H. Peyton |publisher=Yale University Press |year=1982 |isbn=0-300-02724-9 |location=New Haven |url-access=registration}}{{Rp|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.{{Rp|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.{{Cite journal |last=Spencer |first=Bruce D. |date=December 1985 |title=Statistical Aspects of Equitable Apportionment |url=http://www.tandfonline.com/doi/abs/10.1080/01621459.1985.10478188 |journal=Journal of the American Statistical Association |language=en |volume=80 |issue=392 |pages=815–822 |doi=10.1080/01621459.1985.10478188 |issn=0162-1459|url-access=subscription }}

{{References list}}{{voting systems}}

Category:Mathematical theorems

Category:Apportionment methods