Rank-index method

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:

• A positive integer ${\displaystyle h}${\displaystyle h} representing the total number of items to allocate. It is also called the house size.

• A positive integer ${\displaystyle n}${\displaystyle 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 ${\displaystyle (t_{1},\ldots ,t_{n})}${\displaystyle (t_{1},\ldots ,t_{n})} with ${\displaystyle \sum _{i=1}^{n}t_{i}=1}${\displaystyle \sum _{i=1}^{n}t_{i}=1}, representing entitlements - ${\displaystyle t_{i}}${\displaystyle t_{i}} represents the entitlement of agent ${\displaystyle i}${\displaystyle i}, that is, the fraction of items to which ${\displaystyle i}${\displaystyle i} is entitled (out of the total of ${\displaystyle h}${\displaystyle h}).

Its output is a vector of integers ${\displaystyle a_{1},\ldots ,a_{n}}${\displaystyle a_{1},\ldots ,a_{n}} with ${\displaystyle \sum _{i=1}^{n}a_{i}=h}${\displaystyle \sum _{i=1}^{n}a_{i}=h}, called an apportionment of ${\displaystyle h}${\displaystyle h}, where ${\displaystyle a_{i}}${\displaystyle a_{i}} is the number of items allocated to agent i.

Every rank-index method is parametrized by a rank-index function ${\displaystyle r(t,a)}${\displaystyle r(t,a)}, which is increasing in the entitlement ${\displaystyle t}${\displaystyle t} and decreasing in the current allocation ${\displaystyle a}${\displaystyle a}. The apportionment is computed iteratively as follows:

• Initially, set ${\displaystyle a_{i}}${\displaystyle a_{i}} to 0 for all parties.
• At each iteration, allocate one item to an agent for whom ${\displaystyle r(t_{i},a_{i})}${\displaystyle r(t_{i},a_{i})} is maximum (break ties arbitrarily).
• Stop after ${\displaystyle h}${\displaystyle h} iterations.

Divisor methods are a special case of rank-index methods: a divisor method with divisor function ${\displaystyle d(a)}${\displaystyle d(a)} is equivalent to a rank-index method with rank-index function ${\displaystyle r(t,a)=t/d(a)}${\displaystyle 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:^{[1] : Thm.8.1}

${\displaystyle \min _{i:a_{i}>0}r(t_{i},a_{i}-1)\geq \max _{i}r(t_{i},a_{i})}${\displaystyle \min _{i:a_{i}>0}r(t_{i},a_{i}-1)\geq \max _{i}r(t_{i},a_{i})}.

Every rank-index method is house-monotone. This means that, when ${\displaystyle h}${\displaystyle 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 ${\displaystyle 1,\ldots ,k}${\displaystyle 1,\ldots ,k}, and apply the same method to their combined allocation, then the result is exactly the vector ${\displaystyle (a_{1},\ldots ,a_{k})}${\displaystyle (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 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]

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]

• Mathematical theorems
• Apportionment methods

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