Highest averages method

A divisor method is parametrized by a function ${\displaystyle d(k)}$, mapping each integer ${\displaystyle k}$ to a real number (usually in the range ${\displaystyle [k,k+1]}$) .

The number of seats allocated to party ${\displaystyle i}$ is denoted by ${\displaystyle a_{i}}$. Initially, ${\displaystyle a_{i}}$ is set to 0 for all parties. Then, at each iteration, the next seat is allocated to a party which maximizes the ratio ${\displaystyle {\frac {t_{i}}{d(a_{i})}}}$. The method proceeds for h iterations, until all seats are allocated.

An equivalent definition directly gives the outcome of the divisor method as follows.

For an election, a quotient is calculated, usually the total number of votes divided by the number of seats to be allocated (the Hare quota). Parties are then allocated seats by determining how many quotients they have won, by dividing their vote totals by the quotient. Where a party wins a fraction of a quotient, this can be rounded down or rounded to the nearest whole number. Rounding down is equivalent to using the D'Hondt method, while rounding to the nearest whole number is equivalent to the Sainte-Laguë method. Rounding up is equivalent to using Adams' method. However, because of the rounding, this will not necessarily result in the desired number of seats being filled. In that case, the quotient may be adjusted up or down until the number of seats after rounding is equal to the desired number.

The tables used in the D'Hondt or Sainte-Laguë methods can then be viewed as calculating the highest quotient possible to round off to a given number of seats. For example, the quotient which wins the first seat in a D'Hondt calculation is the highest quotient possible to have one party's vote, when rounded down, be greater than 1 quota and thus allocate 1 seat. The quotient for the second round is the highest divisor possible to have a total of 2 seats allocated, and so on.

Formally, given the vector of entitlements ${\displaystyle \mathbf {t} }$ and house-size ${\displaystyle h}$, a divisor method can be defined as:

${\displaystyle \{\mathbf {a} |a_{i}=\operatorname {round} (t_{i}\cdot H){\text{ and }}\sum _{i=1}^{n}a_{i}=h{\text{ for some real number }}H\}}$

where the method of rounding is defined by the divisor function d.

Every divisor method can be defined using a min-max inequality: a is an allocation for the divisor method with divisor d, if-and-only-if[2]^: 78–81

${\displaystyle \max _{i}t_{i}/d(a_{i}+1)\leq \min _{i:a_{i}>0}t_{i}/d(a_{i})}$.

Every number in the range ${\displaystyle [\max _{i}t_{i}/d(a_{i}+1)~,~\min _{i:a_{i}>0}t_{i}/d(a_{i})]}$ is a possible divisor. If the range is not a singleton (that is, the inequality is strict), then the solution is unique; otherwise (the inequality is an equality), there are multiple solutions.[2]^: 83

The Sainte-Laguë method divides the number of votes for each party by the odd numbers (1, 3, 5, 7 etc.), or equivalently by 0.5, 1.5, 2.5, 3.5 etc. It corresponds to a divisor function ${\displaystyle d(k)=k+0.5}$.

It is considered more proportional than D'Hondt in terms of a comparison between a party's share of the total vote and its share of the seat allocation,[3] though it can lead to a party with a majority of votes winning fewer than half the seats. This system is more favourable to smaller parties than D'Hondt's method and to reduce political fragmentation is typically combined with electoral thresholds.

The used divisor sequence is 1, 2, 3, 4, etc., corresponding to the divisor function ${\displaystyle d(k)=k+1}$. It is called the D'Hondt formula.[4] This system tends to give larger parties a larger portion of seats than their portion of the electorate,[5] and thus incentivises strategic voting. This apportionment method guarantees that a party with a majority of voters will get at least half of the seats.

In the Huntington–Hill method, the divisor function is ${\displaystyle d(k)={\sqrt {k(k+1)}}}$, which makes sense only if every party is guaranteed at least one seat: this effect can be achieved by disqualifying parties receiving fewer votes than a specified number of votes, be it a percentage threshold or a quota such as Hare, Droop, or Imperiali. This method is used for allotting seats in the US House of Representatives among the states. Because squaring does not change the order of the computed ratios, the square root involved in comparing ${\displaystyle {\frac {t_{i}}{\sqrt {a_{i}(a_{i}+1)}}}}$ values can be avoided by instead comparing ${\displaystyle {\frac {t_{i}^{2}}{a_{i}(a_{i}+1)}}}$ values.

The Danish method is used in Danish elections to allocate each party's compensatory seats (or levelling seats) at the electoral province level to individual multi-member constituencies. It divides the number of votes received by a party in a multi-member constituency by the divisors growing by step equal to 3 (1, 4, 7, 10, etc.). Alternatively, dividing the votes numbers by 0.33, 1.33, 2.33, 3.33 etc. yields the same result. The divisor function is ${\displaystyle d(k)=k+1/3}$. This system purposely attempts to allocate seats equally rather than proportionately.[6]

The Imperiali highest average method (not to be confused with the Imperiali quota which is a Largest remainder method) has divisors 1, 1.5, 2, 2.5, 3, 3.5 etc., or equivalently 2, 3, 4, 5, etc., corresponding to a divisor function ${\displaystyle d(k)=k+2}$. It is designed to disfavor the smallest parties, akin to a "cutoff", and is used only in Belgian municipal elections. This method (unlike other listed methods) is not strictly proportional: if a perfectly proportional allocation exists, it is not guaranteed to find it.

Adams' method was conceived by John Quincy Adams for apportioning seats of the House to states.[7] He perceived Jefferson's method to allocate too few seats to smaller states. It can be described as the inverse of Jefferson's method; it awards a seat to the party that has the most votes per seat before the seat is added. The divisor function is ${\displaystyle d(k)=k}$.[8]

Like the Huntington-Hill method, this results in a value of 0 for the first seats to be appointed for each party, resulting in an average of ∞. It can only violate the lower quota rule.[9] This occurs in the example below.

Without a threshold, all parties that have received at least one vote, also receive a seat, with the obvious exception of cases where there are more parties than seats. This property can be desirable, for example when apportioning seats to electoral districts. As long as there are at least as many seats as districts, all districts are represented. In a party-list proportional representation election, it may result in very small parties receiving seats. Furthermore, quota rule violations in the pure Adams' method are very common.[10] These problems may be solved by introducing an electoral threshold.[8]

Like all apportionment methods, the inputs of any rank-index method are:

• A positive integer ${\displaystyle h}$ representing the total number of items to allocate. It is also called the house size, since in many cases, the items to allocate are seats in a house of representatives.

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

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

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

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

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

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:[18]^: Thm.8.1

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

Every rank-index method is house-monotone. This means that, when ${\displaystyle h}$ 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 ${\displaystyle 1,\ldots ,k}$, and apply the same method to their combined allocation ${\displaystyle h_{k}=a_{1}+\cdots +a_{k}}$, then the result is exactly the vector ${\displaystyle (a_{1},\ldots ,a_{k})}$. 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.[18]^: Thm.8.3
• Every apportionment method that is uniform, house-monotone and balanced must be a rank-index method.[19]