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The two names for these methods—highest averages and divisors—reflect two different ways of thinking about them, and their two independent inventions. However, both procedures are equivalent and give the same answer. [1] Divisor methods are based on rounding rules, defined using a signpost sequence post(k), where k ≤ post(k) ≤ k+1.
In general, a "fair" apportionment is one in which each allocation is as close as possible to the quota . An apportionment method may return a set of apportionment vectors (in other words: it is a multivalued function). This is required, since in some cases there is no fair way to distinguish between two possible solutions.
Huntington-Hill uses a continuity correction as a compromise, given by taking the geometric mean of both divisors, i.e.: [4] A n = P n ( n + 1 ) {\displaystyle A_{n}={\frac {P}{\sqrt {n(n+1)}}}} where P is the population of the state, and n is the number of seats it currently holds before the possible allocation of the next seat.
The D'Hondt method, [a] also called the Jefferson method or the greatest divisors method, is an apportionment method for allocating seats in parliaments among federal states, or in proportional representation among political parties. It belongs to the class of highest-averages methods.
When using the Hare quota, this rule is called Hamilton's method, and is the third-most common apportionment rule worldwide (after Jefferson's method and Webster's method). [1] Despite their intuitive definition, quota methods are generally disfavored by social choice theorists as a result of apportionment paradoxes.
The Hare quota is often used to set electoral thresholds and to calculate apportionments under party-list proportional representation when using the largest remainder method. In such cases, the Hare quota gives unbiased apportionments that do not favor either large or small parties. [ 1 ]
Different apportionment methods show different levels of proportionality, apportionment paradoxes and political fragmentation. The Sainte-Laguë method minimizes the average seats-to-votes ratio deviation [ 2 ] and empirically shows the best proportionality behavior [ 3 ] and more equal seats-to-votes ratio for different sized parties [ 1 ...
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 ]