Editing Highest averages method (section)

== Definitions ==
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.<ref name="Pukelsheim-2017-1" />

Divisor methods are based on [[rounding]] rules, defined using a ''[[signpost sequence]]'' {{Math|post(''k'')}}'','' where {{Math|''k'' ≤ post(''k'') ≤ ''k''+1}}''.'' Each signpost marks the boundary between natural numbers, with numbers being rounded down if and only if they are less than the signpost.'''<ref name="Pukelsheim-2017-5" />'''

=== Divisor procedure ===
The divisor procedure apportions seats by searching for a ''divisor'' or ''[[electoral quota]]''. This divisor can be thought of as the number of votes a party needs to earn one additional seat in the legislature, the ideal population of a [[Constituency|congressional district]], or the number of voters represented by each legislator.<ref name="Pukelsheim-2017-1" />

If each legislator represented an equal number of voters, the number of seats for each state could be found by dividing the population by the divisor.<ref name="Pukelsheim-2017-1" /> However, seat allocations must be whole numbers, so to find the apportionment for a given state we must round (using the signpost sequence) after dividing. Thus, each party's apportionment is given by:<ref name="Pukelsheim-2017-1" />

<math>\text{seats} = \operatorname{round}\left(\frac{\text{votes}}{\text{divisor}}\right)</math>

Usually, the divisor is initially set to equal the [[Hare quota]]. However, this procedure may assign too many or too few seats. In this case the apportionments for each state will not add up to the total legislature size. A feasible divisor can be found by [[Root-finding algorithms|trial and error]].<ref name="Pukelsheim-2017-3" />

=== Highest averages procedure ===
With the highest averages algorithm, every party begins with 0 seats. Then, at each iteration, we allocate a seat to the party with the ''highest vote average,'' i.e. the party with the most [[Seats-to-votes ratio|votes per seat]]''.'' This method proceeds until all seats are allocated.<ref name="Pukelsheim-2017-1" />

However, it is unclear whether it is better to look at the vote average ''before'' assigning the seat, what the average will be ''after'' assigning the seat, or if we should compromise with a [[continuity correction]]. These approaches each give slightly different apportionments.<ref name="Pukelsheim-2017-1" /> In general, we can define the averages using the signpost sequence:

<math>\text{average} := \frac{\text{votes}}{\operatorname{post}(\text{seats})}</math>

With the highest averages procedure, every party begins with 0 seats. Then, at each iteration, we allocate a seat to the party with the ''highest vote average,'' i.e. the party with the most votes per seat''.'' This method proceeds until all seats are allocated.<ref name="Pukelsheim-2017-1" />