Editing Highest averages method (section)

== Specific methods ==
While all divisor methods share the same general procedure, they differ in the choice of signpost sequence and therefore rounding rule. Note that for methods where the first signpost is zero, every party with at least one vote will receive a seat before any party receives a second seat; in practice, this typically means that every party must receive at least one seat, unless disqualified by some [[electoral threshold]].<ref name="Pukelsheim-2017-5" />
{| class="wikitable"
|+Divisor formulas
!Method
!Signposts
!Rounding<br />of Seats
!Approx. first values
|-
|Adams
|{{math|''k''}}
|[[Ceiling function|Up]]
|{{math|0.00 1.00 2.00 3.00}}
|-
|Dean
|{{math|2÷({{frac|1|''k''}} + {{frac|1|''k''+1}})}}
|[[Harmonic mean|Harmonic]]
|{{math|0.00 1.33 2.40 3.43}}
|-
|[[Huntington–Hill method|Huntington–Hill]]
|<math>\sqrt{k(k+1)}</math>
|[[Geometric mean|Geometric]]
|{{math|0.00 1.41 2.45 3.46}}
|-
|Stationary<br />(e.g. {{math|1=''r'' = {{frac|1|3}}}})
|{{math|''k'' + ''r''}}
|[[Weighted arithmetic mean|Weighted]]
|{{math|0.33 1.33 2.33 3.33}}
|-
|[[Sainte-Laguë method|Webster/Sainte-Laguë]]
|{{math|''k'' + {{frac|1|2}}}}
|[[Arithmetic mean|Arithmetic]]
|{{math|0.50 1.50 2.50 3.50}}
|-
|Power mean<br />(e.g. {{math|1=''p'' = 2}})
|<math display=inline>\sqrt[p]{(k^p + (k+1)^p)/2}</math>
|[[Power mean]]
|{{math|0.71 1.58 2.55 3.54}}
|-
|[[D'Hondt method|Jefferson/D'Hondt]]
|{{math|''k'' + 1}}
|[[Floor function|Down]]
|{{math|1.00 2.00 3.00 4.00}}
|}

===Jefferson (D'Hondt) method===
{{Main article|D'Hondt method}}

[[Thomas Jefferson]] was the first to propose a divisor method, in 1792;<ref name="Pukelsheim-2017-1" /> it was later independently developed by Belgian political scientist [[Victor d'Hondt]] in 1878. It assigns the representative to the list that would be most underrepresented at the end of the round.<ref name="Pukelsheim-2017-1" /> It remains the most-common method for [[proportional representation]] to this day.<ref name="Pukelsheim-2017-1" />

Jefferson's method uses the sequence <math>\operatorname{post}(k) = k+1</math>, i.e. (1, 2, 3, ...),<ref name="Gallagher-1991" /> which means it will always round a party's apportionment down.<ref name="Pukelsheim-2017-1" />

Jefferson's apportionment never falls below the lower end of the [[quota rule|ideal frame]], and it minimizes the worst-case overrepresentation in the legislature.<ref name="Pukelsheim-2017-1" /> However, it performs poorly when judged by most other metrics of proportionality.<ref name="Gallagher-1992" /> The rule typically gives large parties an excessive number of seats, with their seat share often exceeding their entitlement rounded up.<ref name="Balinski-1982" />{{Rp||page=81}}

This [[Pathological (mathematics)|pathology]] led to widespread mockery of Jefferson's method when it was learned Jefferson's method could "round" [[New York (state)|New York]]'s apportionment of 40.5 up to 42, with Senator [[Mahlon Dickerson]] saying the extra seat must come from the "[[Ghosts of departed quantities|ghosts of departed representatives]]".<ref name="Balinski-1982" />{{Rp||page=34}}

===Adams' method===
Adams' method was conceived of by [[John Quincy Adams]] after noticing Jefferson's method allocated too few seats to smaller states.<ref name="MAA-2020" /> 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 new seat is added. The divisor function is {{Math|1=post(''k'') = ''k''}}, which is equivalent to always rounding up.<ref name="Gallagher-1992" />

Adams' apportionment never exceeds the upper end of the [[quota rule|ideal frame]], and minimizes the worst-case underrepresentation.<ref name="Pukelsheim-2017-1" /> However, like Jefferson's method, Adams' method performs poorly according to most metrics of proportionality.<ref name="Gallagher-1992" /> It also often violates the [[Quota rule|lower seat quota]].<ref name="Ichimori-2010" />

Adams' method was suggested as part of the Cambridge compromise for apportionment of [[European Parliament|European parliament]] seats to member states, with the aim of satisfying [[degressive proportionality]].<ref name="EU-2011" />

===Webster (Sainte-Laguë) method===
{{Main article|Sainte-Laguë method}}

The Sainte-Laguë or Webster method, first described in 1832 by American statesman and senator [[Daniel Webster]] and later independently in 1910 by the French mathematician [[André Sainte-Laguë|André Sainte-Lague]], uses the fencepost sequence {{Math|1=post(''k'') = ''k''+.5}} (i.e. 0.5, 1.5, 2.5); this corresponds to the standard [[Rounding|rounding rule]]. Equivalently, the odd integers (1, 3, 5...) can be used to calculate the averages instead.<ref name="Pukelsheim-2017-1" /><ref name="Sainte-2024" />

The Webster method produces more proportional apportionments than Jefferson's by almost every metric of misrepresentation.<ref name="Pennisi-1998" /> As such, it is typically preferred to D'Hondt by political scientists and mathematicians, at least in situations where manipulation is difficult or unlikely (as in large parliaments).<ref name="Balinski-1980" /> It is also notable for minimizing [[seat bias]] even when dealing with parties that win very small numbers of seats.<ref name="Pukelsheim-2017-2" /> The Webster method can theoretically violate the [[quota rule|ideal frame]], although this is extremely rare for even moderately-large parliaments; it has never been observed to violate quota in any [[United States congressional apportionment]].<ref name="Balinski-1980" />

In small districts with no [[Electoral threshold|threshold]], parties can [[vote management|manipulate]] Webster by splitting into many lists, each of which wins a full seat with less than a [[Hare quota]]'s worth of votes. This is often addressed by modifying the first divisor to be slightly larger (often a value of 0.7 or 1), which creates an [[electoral threshold|implicit threshold]].<ref name="Pukelsheim-2017-8" />

=== Huntington–Hill method ===
{{Main article|Huntington–Hill method}}
In the [[Huntington–Hill method]], the signpost sequence is {{Math|1=post(''k'') = {{sqrt|''k'' (''k''+1)}}}}, the [[geometric mean]] of the neighboring numbers. Conceptually, this method rounds to the integer that has the smallest [[Relative change#Logarithmic change|relative (percent) difference]]. For example, the difference between 2.47 and 3 is about 19%, while the difference from 2 is about 21%, so 2.47 is rounded up. This method is used for allotting seats in the US House of Representatives among the states.<ref name="Pukelsheim-2017-1" />

The Huntington-Hill method tends to produce very similar results to the Webster method, except that it guarantees every state or party at least one seat (see {{Section link|2=Zero-seat apportionments}}). When first used to assign seats in the [[US House of Representatives|House]], the two methods produced identical results; in their second use, they differed only in assigning a single seat to [[Michigan]] or [[Arkansas]].<ref name="Balinski-1982" />{{Rp|58}}