Editing Highest averages method

{{Short description|Rule for proportional allocation}}
{{good article}}

{{Use American English|date=December 2024}}

{{Electoral systems|expanded=Proportional representation}}
The '''highest averages''', '''divisor''', or '''divide-and-round methods'''<ref name="Pukelsheim-2017-1" /> are a family of [[Apportionment (politics)|apportionment]] rules, i.e. algorithms for [[fair division]] of seats in a legislature between several groups (like [[Political party|political parties]] or [[State (sub-national)|states]]).<ref name="Pukelsheim-2017-1" />'''<ref name="Pukelsheim-2017-5" />''' More generally, divisor methods are used to round shares of a total to a [[Ratio|fraction]] with a fixed [[denominator]] (e.g. percentage points, which must add up to 100).'''<ref name="Pukelsheim-2017-5" />'''

The methods aim to treat voters equally by ensuring legislators [[One man, one vote|represent an equal number of voters]] by ensuring every party has the same [[seats-to-votes ratio]] (or ''divisor'').<ref name="Balinski-1982" />{{Rp||page=30}} Such methods divide the number of votes by the number of votes needed to win a seat. The final apportionment. In doing so, the method approximately maintains [[proportional representation]], meaning that a party with e.g. twice as many votes will win about twice as many seats.<ref name="Balinski-1982" />{{Rp||page=30}}

The divisor methods are generally preferred by [[Social choice theory|social choice theorists]] and mathematicians to the [[largest remainder method]]s, as they produce more-proportional results by most metrics and are less susceptible to [[apportionment paradox]]es.<ref name="Balinski-1982" /><ref name="Ricca-2017" /><ref name="Pukelsheim-2017-7" /><ref name="Dancisin-2017" /> In particular, divisor methods avoid the [[vote-ratio monotonicity|population paradox]] and [[spoiler effect]]s, unlike the largest remainder methods.<ref name="Pukelsheim-2017-7" />

== History ==

Divisor methods were first invented by [[Thomas Jefferson]] to comply with a [[Constitution of the United States|constitutional]] requirement, that states have at most one representative per 30,000 people. His solution was to divide each state's population by 30,000 before rounding down.<ref name="Balinski-1982" />{{Rp||page=20}}

Apportionment would become a major topic of debate in Congress, especially after the discovery of [[Apportionment paradox|pathologies]] in many superficially-reasonable rounding rules.<ref name="Balinski-1982" />{{Rp||page=20}} Similar debates would appear in Europe after the adoption of [[proportional representation]], typically as a result of large parties attempting to introduce [[Electoral threshold|thresholds]] and other [[barriers to entry]] for small parties.<ref name="Pukelsheim-2017-0" /> Such apportionments often have substantial consequences, as in the [[1870 United States census|1870 reapportionment]], when Congress used an ad-hoc apportionment to favor [[Republican Party (United States)|Republican]] states.<ref name="Argersinger-2012" /> Had each state's electoral vote total been exactly equal to [[Entitlement (fair division)|its entitlement]], or had Congress used [[Webster method|Webster's method]] or a [[largest remainders method]] (as it had since 1840), the [[1876 United States presidential election|1876 election]] would have gone to [[Samuel J. Tilden|Tilden]] instead of [[Rutherford B. Hayes|Hayes]].<ref name="Argersinger-2012" /><ref name="Caulfield-2012" /><ref name="Balinski-1982" />{{Rp||page=3, 37}}

== 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" />

== 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}}

== Comparison of properties ==

=== Zero-seat apportionments ===
Huntington-Hill, Dean, and Adams' method all have a value of 0 for the first fencepost, giving an average of ∞. Thus, without a threshold, all parties that have received at least one vote will also receive at least one seat.<ref name="Pukelsheim-2017-1" /> This property can be desirable (as when apportioning seats to [[Federated state|states]]) or undesirable (as when apportioning seats to party lists in an election), in which case the first divisor may be adjusted to create a natural threshold.<ref name="Pukelsheim-2017-9" />

=== Bias ===
There are many metrics of [[seat bias]]. While the Webster method is sometimes described as "uniquely" unbiased,<ref name="Balinski-1980" /> this uniqueness property relies on a [[Bias of an estimator|technical definition of bias]], which is defined as the [[Expected value|average]] difference between a state's number of seats and its [[Entitlement (fair division)|seat entitlement]]. In other words, a method is called unbiased if the number of seats a state receives is, on average across many elections, equal to its seat entitlement.<ref name="Balinski-1980" />

By this definition, the Webster method is the least-biased apportionment method,<ref name="Pukelsheim-2017-2" /> while Huntington-Hill exhibits a mild bias towards smaller parties.<ref name="Balinski-1980" /> However, other researchers have noted that slightly different definitions of bias, generally based on [[Percent error|''percent'' errors]], find the opposite result (The Huntington-Hill method is unbiased, while the Webster method is slightly biased towards large parties).<ref name="Pukelsheim-2017-2" /><ref name="Ernst-1994" />

In practice, the difference between these definitions is small when handling parties or states with more than one seat.<ref name="Pukelsheim-2017-2" /> Thus, both the Huntington-Hill and Webster methods can be considered unbiased or low-bias methods (unlike the Jefferson or Adams methods).<ref name="Pukelsheim-2017-2" /><ref name="Ernst-1994" /> A 1929 report to Congress by the [[National Academy of Sciences]] recommended the Huntington-Hill method,<ref name="Huntington-1929" /> while the [[Supreme Court of the United States|Supreme Court]] has ruled the choice to be a matter of opinion.<ref name="Ernst-1994" />

==Comparison and examples==

=== Example: Jefferson ===<!-- Excel spreadsheet for calculations:
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The following example shows how Jefferson's method can differ substantially from less-biased methods such as Webster. In this election, the largest party wins 46% of the vote, but takes 52.5% of the seats, enough to win a majority outright against a coalition of all other parties (which together reach 54% of the vote). Moreover, it does this in violation of quota: the largest party is entitled only to 9.7 seats, but it wins 11 regardless. The largest congressional district is nearly twice the size of the smallest district. The Webster method shows none of these properties, with a maximum error of 22.6%.
{| class="wikitable mw-collapsible" style="text-align:center; color:#000000;"
|+
! colspan="7" style="vertical-align:bottom; background-color:#ffffff;" |Jefferson
! style="background-color:#ffffff; font-weight:normal; text-align:left;" |
! colspan="7" style="background-color:#FFF;" |Webster
|- style="font-weight:bold;"
! style="vertical-align:bottom;" |Party
! style="vertical-align:bottom;" |Yellow
! style="vertical-align:bottom;" |White
! style="vertical-align:bottom;" |Red
! style="vertical-align:bottom;" |Green
! style="vertical-align:bottom;" |Purple
! style="vertical-align:bottom;" |Total
| rowspan="6" style="background-color:#ffffff; font-weight:normal; text-align:left;" |
!Party
!Yellow
!White
!Red
!Green
!Purple
!Total
|-
! style="vertical-align:bottom; font-weight:bold;" |Votes
| style="vertical-align:bottom; text-align:left;" |46,000
| style="vertical-align:bottom;" |25,100
| style="vertical-align:bottom;" |12,210
| style="vertical-align:bottom;" |8,350
| style="vertical-align:bottom;" |8,340
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |100,000
! style="font-weight:bold;" |Votes
| style="text-align:left;" |46,000
|25,100
|12,210
|8,350
|8,340
| style="text-align:right; font-weight:bold;" |100,000
|- style="text-align:left;"
! style="text-align:center; vertical-align:bottom; font-weight:bold;" |Seats
| style="vertical-align:bottom;" |11
| style="vertical-align:bottom;" |6
| style="vertical-align:bottom;" |2
| style="vertical-align:bottom;" |1
| style="vertical-align:bottom;" |1
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |21
! style="text-align:center; font-weight:bold;" |Seats
|9
|5
|3
|2
|2
| style="text-align:right; font-weight:bold;" |21
|- style="text-align:left;"
! style="text-align:center; vertical-align:bottom; font-weight:bold;" |Ideal
| style="vertical-align:bottom;" |9.660
| style="vertical-align:bottom;" |5.271
| style="vertical-align:bottom;" |2.564
| style="vertical-align:bottom;" |1.754
| style="vertical-align:bottom;" |1.751
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |21
! style="text-align:center; font-weight:bold;" |Ideal
|9.660
|5.271
|2.564
|1.754
|1.751
| style="text-align:right; font-weight:bold;" |21
|- style="text-align:left;"
! style="text-align:center; vertical-align:bottom; font-weight:bold;" |Votes/Seat
| style="vertical-align:bottom;" |4182
| style="vertical-align:bottom;" |4183
| style="vertical-align:bottom;" |6105
| style="vertical-align:bottom;" |8350
| style="vertical-align:bottom;" |8340
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |4762
! style="text-align:center; font-weight:bold;" |Votes/Seat
|5111
|5020
|4070
|4175
|4170
| style="text-align:right; font-weight:bold;" |4762
|- style="text-align:left;"
! style="text-align:center; vertical-align:bottom; font-weight:bold;" |% Error
| style="vertical-align:bottom;" |13.0%
| style="vertical-align:bottom;" |13.0%
| style="vertical-align:bottom;" | -24.8%
| style="vertical-align:bottom;" | -56.2%
| style="vertical-align:bottom;" | -56.0%
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |(100.%)
! style="text-align:center; font-weight:bold;" |(% Range)
| -7.1%
| -5.3%
|15.7%
|13.2%
|13.3%
| style="text-align:right; font-weight:bold;" |(22.6%)
|- style="font-weight:bold;"
! style="vertical-align:bottom;" |Seats
! colspan="5" |Averages
! style="vertical-align:bottom;" |Signposts
| style="background-color:#ffffff; font-weight:normal; text-align:left;" |
!Seats
! colspan="5" |Averages
!Signposts
|-
! style="vertical-align:bottom; font-weight:bold;" |1
| style="vertical-align:bottom; background-color:#63BE7B;" |46,000
| style="vertical-align:bottom; background-color:#AADBB8;" |25,100
| style="vertical-align:bottom; background-color:#D6EDDE;" |12,210
| style="vertical-align:bottom; background-color:#E3F2E9;" |8,350
| style="vertical-align:bottom; background-color:#E3F2E9;" |8,340
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |1.00
| rowspan="11" style="background-color:#ffffff; text-align:left;" |
! style="font-weight:bold;" |1
| style="background-color:#63BE7B;" |92,001
| style="background-color:#AADBB8;" |50,201
| style="background-color:#D5ECDD;" |24,420
| style="background-color:#E2F2E8;" |16,700
| style="background-color:#E2F2E9;" |16,680
| style="text-align:right; font-weight:bold;" |0.50
|-
! style="vertical-align:bottom; font-weight:bold;" |2
| style="vertical-align:bottom; background-color:#B1DEBF;" |23,000
| style="vertical-align:bottom; background-color:#D5ECDD;" |12,550
| style="vertical-align:bottom; background-color:#EAF5F0;" |6,105
| style="vertical-align:bottom;" |4,175
| style="vertical-align:bottom;" |4,170
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |2.00
! style="font-weight:bold;" |2
| style="background-color:#CAE8D4;" |30,667
| style="background-color:#E2F2E8;" |16,734
| style="background-color:#F0F8F5;" |8,140
| style="background-color:#F4F9F9;" |5,567
| style="background-color:#F5F9F9;" |5,560
| style="text-align:right; font-weight:bold;" |1.50
|-
! style="vertical-align:bottom; font-weight:bold;" |3
| style="vertical-align:bottom; background-color:#CBE9D5;" |15,333
| style="vertical-align:bottom; background-color:#E3F2E9;" |8,367
| style="vertical-align:bottom;" |4,070
| style="vertical-align:bottom;" |2,783
| style="vertical-align:bottom;" |2,780
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |3.00
! style="font-weight:bold;" |3
| style="background-color:#DFF1E6;" |18,400
| style="background-color:#EDF6F2;" |10,040
| style="background-color:#F6FAFA;" |4,884
|3,340
|3,336
| style="text-align:right; font-weight:bold;" |2.50
|-
! style="vertical-align:bottom; font-weight:bold;" |4
| style="vertical-align:bottom; background-color:#D8EEE0;" |11,500
| style="vertical-align:bottom; background-color:#EAF5EF;" |6,275
| style="vertical-align:bottom;" |3,053
| style="vertical-align:bottom;" |2,088
| style="vertical-align:bottom;" |2,085
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |4.00
! style="font-weight:bold;" |4
| style="background-color:#E8F4EE;" |13,143
| style="background-color:#F2F8F6;" |7,172
|3,489
|2,386
|2,383
| style="text-align:right; font-weight:bold;" |3.50
|-
! style="vertical-align:bottom; font-weight:bold;" |5
| style="vertical-align:bottom; background-color:#E0F1E7;" |9,200
| style="vertical-align:bottom; background-color:#EEF7F3;" |5,020
| style="vertical-align:bottom;" |2,442
| style="vertical-align:bottom;" |1,670
| style="vertical-align:bottom;" |1,668
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |5.00
! style="font-weight:bold;" |5
| style="background-color:#EDF6F2;" |10,222
| style="background-color:#F4F9F9;" |5,578
|2,713
|1,856
|1,853
| style="text-align:right; font-weight:bold;" |4.50
|-
! style="vertical-align:bottom; font-weight:bold;" |6
| style="vertical-align:bottom; background-color:#E5F3EB;" |7,667
| style="vertical-align:bottom; background-color:#F1F8F6;" |4,183
| style="vertical-align:bottom;" |2,035
| style="vertical-align:bottom;" |1,392
| style="vertical-align:bottom;" |1,390
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |6.00
! style="font-weight:bold;" |6
| style="background-color:#F0F7F5;" |8,364
| style="background-color:#F6FAFA;" |4,564
|2,220
|1,518
|1,516
| style="text-align:right; font-weight:bold;" |5.50
|-
! style="vertical-align:bottom; font-weight:bold;" |7
| style="vertical-align:bottom; background-color:#E9F5EF;" |6,571
| style="vertical-align:bottom;" |3,586
| style="vertical-align:bottom;" |1,744
| style="vertical-align:bottom;" |1,193
| style="vertical-align:bottom;" |1,191
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |7.00
! style="font-weight:bold;" |7
| style="background-color:#F2F8F6;" |7,077
|3,862
|1,878
|1,285
|1,283
| style="text-align:right; font-weight:bold;" |6.50
|-
! style="vertical-align:bottom; font-weight:bold;" |8
| style="vertical-align:bottom; background-color:#ECF6F1;" |5,750
| style="vertical-align:bottom;" |3,138
| style="vertical-align:bottom;" |1,526
| style="vertical-align:bottom;" |1,044
| style="vertical-align:bottom;" |1,043
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |8.00
! style="font-weight:bold;" |8
| style="background-color:#F4F9F8;" |6,133
|3,347
|1,628
|1,113
|1,112
| style="text-align:right; font-weight:bold;" |7.50
|-
! style="vertical-align:bottom; font-weight:bold;" |9
| style="vertical-align:bottom; background-color:#EEF7F3;" |5,111
| style="vertical-align:bottom;" |2,789
| style="vertical-align:bottom;" |1,357
| style="vertical-align:bottom;" |928
| style="vertical-align:bottom;" |927
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |9.00
! style="font-weight:bold;" |9
| style="background-color:#F5F9F9;" |5,412
|2,953
|1,436
|982
|981
| style="text-align:right; font-weight:bold;" |8.50
|-
! style="vertical-align:bottom; font-weight:bold;" |10
| style="vertical-align:bottom; background-color:#F0F7F4;" |4,600
| style="vertical-align:bottom;" |2,510
| style="vertical-align:bottom;" |1,221
| style="vertical-align:bottom;" |835
| style="vertical-align:bottom;" |834
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |10.00
! style="font-weight:bold;" |10
| style="background-color:#F6FAFA;" |4,842
|2,642
|1,285
|879
|878
| style="text-align:right; font-weight:bold;" |9.50
|-
! style="vertical-align:bottom; font-weight:bold;" |11
| style="vertical-align:bottom; background-color:#F1F8F6;" |4,182
| style="vertical-align:bottom;" |2,282
| style="vertical-align:bottom;" |1,110
| style="vertical-align:bottom;" |759
| style="vertical-align:bottom;" |758
| style="text-align:right; vertical-align:bottom; font-weight:bold;" |11.00
! style="font-weight:bold;" |11
| style="background-color:#F6FAFA;" |4,381
|2,391
|1,163
|795
|794
| style="text-align:right; font-weight:bold;" |10.50
|}

=== Example: Adams ===
The following example shows a case where Adams' method fails to give a majority to a party winning 55% of the vote, again in violation of their quota entitlement.

{| class="wikitable mw-collapsible" style="text-align:center;"
|+
|- style="font-weight:bold; vertical-align:bottom; background-color:#FFF;"
! colspan="7" style="background-color:#ffffff;" | Adams' Method
! style="vertical-align:middle; background-color:#ffffff; font-weight:normal; text-align:left;" |
! colspan="7" style="background-color:#ffffff;" | Webster Method
|- style="font-weight:bold; vertical-align:bottom;"
! Party
! Yellow
! White
! Red
! Green
! Purple
! Total
| rowspan="6" style="vertical-align:middle; background-color:#ffffff; font-weight:normal;" |
! Party
! Yellow
! White
! Red
! Green
! Purple
! Total
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | Votes
| 55,000
| 17,290
| 16,600
| 5,560
| 5,550
| style="text-align:right; font-weight:bold;" | 100,000
! style="font-weight:bold;" | Votes
| style="text-align:left;" | 55,000
| 17,290
| 16,600
| 5,560
| 5,550
| style="text-align:right; font-weight:bold;" | 100,000
|- style="text-align:left;"
! style="text-align:center; font-weight:bold;" | Seats
| 10
| 4
| 3
| 2
| 2
| style="text-align:right; font-weight:bold;" | 21
! style="text-align:center; font-weight:bold;" | Seats
| 11
| 4
| 4
| 1
| 1
| style="text-align:right; font-weight:bold;" | 21
|- style="text-align:left;"
! style="text-align:center; vertical-align:bottom; font-weight:bold;" | Ideal
| style="vertical-align:bottom;" | 11.550
| style="vertical-align:bottom;" | 3.631
| style="vertical-align:bottom;" | 3.486
| style="vertical-align:bottom;" | 1.168
| style="vertical-align:bottom;" | 1.166
| style="text-align:right; vertical-align:bottom; font-weight:bold;" | 21
! style="text-align:center; vertical-align:bottom; font-weight:bold;" | Ideal
| 11.550
| 3.631
| 3.486
| 1.168
| 1.166
| style="text-align:right; vertical-align:bottom; font-weight:bold;" | 21
|- style="text-align:left;"
! style="text-align:center; vertical-align:bottom; font-weight:bold;" | Votes/Seat
| style="vertical-align:bottom;" | 5500
| style="vertical-align:bottom;" | 4323
| style="vertical-align:bottom;" | 5533
| style="vertical-align:bottom;" | 2780
| style="vertical-align:bottom;" | 2775
| style="text-align:right; vertical-align:bottom; font-weight:bold;" | 4762
! style="text-align:center; font-weight:bold;" | Votes/Seat
| 4583
| 4323
| 5533
| 5560
| 5550
| style="text-align:right; font-weight:bold;" | 4762
|- style="text-align:left;"
! style="text-align:center; vertical-align:bottom; font-weight:bold;" | % Error
| style="vertical-align:bottom;" | -14.4%
| style="vertical-align:bottom;" | 9.7%
| style="vertical-align:bottom;" | -15.0%
| style="vertical-align:bottom;" | 53.8%
| style="vertical-align:bottom;" | 54.0%
| style="text-align:right; vertical-align:bottom; font-weight:bold;" | (99.4%)
! style="text-align:center; font-weight:bold;" | (% Range)
| style="vertical-align:bottom;" | 3.8%
| style="vertical-align:bottom;" | 9.7%
| style="vertical-align:bottom;" | -15.0%
| style="vertical-align:bottom;" | -15.5%
| style="vertical-align:bottom;" | -15.3%
| style="text-align:right; vertical-align:bottom; font-weight:bold;" | (28.6%)
|- style="font-weight:bold; vertical-align:bottom;"
! Seats
! colspan="5" | Averages
! Signposts
| style="vertical-align:middle; background-color:#ffffff; font-weight:normal;" |
! Seats
! colspan="5" | Averages
! style="text-align:right;" | Signposts
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 1
| style="background-color:#63BE7B;" | ∞
| style="background-color:#CDE9D6;" | ∞
| style="background-color:#CEEAD8;" | ∞
| style="background-color:#EDF6F2;" | ∞
| style="background-color:#EDF6F2;" | ∞
| style="text-align:right; font-weight:bold;" | 0.00
| rowspan="12" style="vertical-align:middle; background-color:#ffffff; text-align:left;" |
! style="font-weight:bold;" | 1
| style="background-color:#63BE7B;" | 110,001
| style="background-color:#CDE9D6;" | 34,580
| style="background-color:#CFEAD8;" | 33,200
| style="background-color:#EEF7F3;" | 11,120
| style="background-color:#EEF7F3;" | 11,100
| style="text-align:right; font-weight:bold;" | 0.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 2
| style="background-color:#EDF6F2;" | 55,001
| style="background-color:#F8FBFB;" | 17,290
| style="background-color:#F8FBFC;" | 16,600
| style="background-color:#FBFCFE;" | 5,560
| style="background-color:#FBFCFE;" | 5,550
| style="text-align:right; font-weight:bold;" | 1.00
! style="font-weight:bold;" | 2
| style="background-color:#CAE8D4;" | 36,667
| style="background-color:#EDF6F2;" | 11,527
| style="background-color:#EEF7F3;" | 11,067
| style="background-color:#EEF7F3;" | 3,707
| style="background-color:#EEF7F3;" | 3,700
| style="text-align:right; font-weight:bold;" | 1.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 3
| style="background-color:#F5F9F9;" | 27,500
| style="background-color:#FAFCFE;" | 8,645
| style="background-color:#FAFCFE;" | 8,300
| 2,780
| 2,775
| style="text-align:right; font-weight:bold;" | 2.00
! style="font-weight:bold;" | 3
| style="background-color:#DEF0E6;" | 22,000
| style="background-color:#F4F9F8;" | 6,916
| style="background-color:#F4F9F8;" | 6,640
| 2,224
| 2,220
| style="text-align:right; font-weight:bold;" | 2.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 4
| style="background-color:#F8FAFB;" | 18,334
| style="background-color:#FBFCFE;" | 5,763
| style="background-color:#FBFCFE;" | 5,533
| 1,853
| 1,850
| style="text-align:right; font-weight:bold;" | 3.00
! style="font-weight:bold;" | 4
| style="background-color:#E7F4ED;" | 15,714
| style="background-color:#F4F9F8;" | 4,940
| style="background-color:#F4F9F8;" | 4,743
| 1,589
| 1,586
| style="text-align:right; font-weight:bold;" | 3.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 5
| style="background-color:#F9FBFC;" | 13,750
| 4,323
| 4,150
| 1,390
| 1,388
| style="text-align:right; font-weight:bold;" | 4.00
! style="font-weight:bold;" | 5
| style="background-color:#ECF6F1;" | 12,222
| style="background-color:#F4F9F9;" | 3,842
| 3,689
| 1,236
| 1,233
| style="text-align:right; font-weight:bold;" | 4.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 6
| style="background-color:#FAFBFD;" | 11,000
| 3,458
| 3,320
| 1,112
| 1,110
| style="text-align:right; font-weight:bold;" | 5.00
! style="font-weight:bold;" | 6
| style="background-color:#EFF7F4;" | 10,000
| style="background-color:#F6FAFA;" | 3,144
| 3,018
| 1,011
| 1,009
| style="text-align:right; font-weight:bold;" | 5.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 7
| style="background-color:#FAFCFD;" | 9,167
| 2,882
| 2,767
| 927
| 925
| style="text-align:right; font-weight:bold;" | 6.00
! style="font-weight:bold;" | 7
| style="background-color:#F1F8F6;" | 8,462
| 2,660
| 2,554
| 855
| 854
| style="text-align:right; font-weight:bold;" | 6.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 8
| style="background-color:#FAFCFE;" | 7,857
| 2,470
| 2,371
| 794
| 793
| style="text-align:right; font-weight:bold;" | 7.00
! style="font-weight:bold;" | 8
| style="background-color:#F3F9F7;" | 7,333
| 2,305
| 2,213
| 741
| 740
| style="text-align:right; font-weight:bold;" | 7.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 9
| style="background-color:#FBFCFE;" | 6,875
| 2,161
| 2,075
| 695
| 694
| style="text-align:right; font-weight:bold;" | 8.00
! style="font-weight:bold;" | 9
| style="background-color:#F4F9F8;" | 6,471
| 2,034
| 1,953
| 654
| 653
| style="text-align:right; font-weight:bold;" | 8.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 10
| style="background-color:#FBFCFE;" | 6,111
| 1,921
| 1,844
| 618
| 617
| style="text-align:right; font-weight:bold;" | 9.00
! style="font-weight:bold;" | 10
| style="background-color:#F5FAF9;" | 5,790
| 1,820
| 1,747
| 585
| 584
| style="text-align:right; font-weight:bold;" | 9.50
|- style="vertical-align:bottom;"
! style="font-weight:bold;" | 11
| style="background-color:#FBFCFE;" | 5,500
| 1,729
| 1,660
| 556
| 555
| style="text-align:right; font-weight:bold;" | 10.00
! style="font-weight:bold;" | 11
| style="background-color:#F6FAFA;" | 5,238
| 1,647
| 1,581
| 530
| 529
| style="text-align:right; font-weight:bold;" | 10.50
|- style="font-weight:bold; text-align:left;"
! style="text-align:center;" | Seats
| 10
| 4
| 3
| 2
| 2
| style="font-weight:normal;" |
! style="text-align:center; vertical-align:bottom;" | Seats
| style="vertical-align:bottom;" | 11
| style="vertical-align:bottom;" | 4
| style="vertical-align:bottom;" | 4
| style="vertical-align:bottom;" | 1
| style="vertical-align:bottom;" | 1
| style="vertical-align:bottom;" |
|}

=== Example: All systems ===
The following shows a worked-out example for all voting systems. Notice how Huntington-Hill and Adams' methods give every party one seat before assigning any more, unlike Webster or Jefferson.
{| class="wikitable mw-collapsible" style="text-align:center"
|+
| style="background:#fff; border-color:#fff #aaa #aaa #fff" |
! colspan="6" |Jefferson method
! rowspan="16" |
! colspan="6" |Webster method
! rowspan="16" |
! colspan="6" |Huntington–Hill method
! rowspan="16" |
! colspan="6" |Adams method
|-
!party
|'''Yellow'''
|'''White'''
|'''Red'''
|'''Green'''
|'''Blue'''
|'''Pink'''
|'''Yellow'''
|'''White'''
|'''Red'''
|'''Green'''
|'''Blue'''
|'''Pink'''
|'''Yellow'''
|'''White'''
|'''Red'''
|'''Green'''
|'''Blue'''
|'''Pink'''
|'''Yellow'''
|'''White'''
|'''Red'''
|'''Green'''
|'''Blue'''
|'''Pink'''
|-
!votes
|47,000
|16,000
|15,900
|12,000
|6,000
|3,100
|47,000
|16,000
|15,900
|12,000
|6,000
|3,100
|47,000
|16,000
|15,900
|12,000
|6,000
|3,100
|47,000
|16,000
|15,900
|12,000
|6,000
|3,100
|-
!seats
|5
|2
|2
|1
|0
|0
|4
|2
|2
|1
|1
|0
|4
|2
|1
|1
|1
|1
|3
|2
|2
|1
|1
|1
|-
!votes/seat
|9,400
|8,000
|7,950
|12,000
|∞
|∞
|11,750
|8,000
|7,950
|12,000
|6,000
|∞
|11,750
|8,000
|15,900
|12,000
|6,000
|3,100
|15,667
|8,000
|7,950
|12,000
|6,000
|3,100
|-
|seat
! colspan="6" |seat allocation
! colspan="6" |seat allocation
! colspan="6" |seat allocation
! colspan="6" |seat allocation
|-
!1
|47,000
|
|
|
|
|
|47,000
|
|
|
|
|
|∞
|
|
|
|
|
|∞
|
|
|
|
|
|-
!2
|23,500
|
|
|
|
|
|
|16,000
|
|
|
|
|
|∞
|
|
|
|
|
|∞
|
|
|
|
|-
!3
|
|16,000
|
|
|
|
|
|
|15,900
|
|
|
|
|
|∞
|
|
|
|
|
|∞
|
|
|
|-
!4
|
|
|15,900
|
|
|
|15,667
|
|
|
|
|
|
|
|
|∞
|
|
|
|
|
|∞
|
|
|-
!5
|15,667
|
|
|
|
|
|
|
|
|12,000
|
|
|
|
|
|
|∞
|
|
|
|
|
|∞
|
|-
!6
|
|
|
|12,000
|
|
|9,400
|
|
|
|
|
|
|
|
|
|
|∞
|
|
|
|
|
|∞
|-
!7
|11,750
|
|
|
|
|
|6,714
|
|
|
|
|
|33,234
|
|
|
|
|
|47,000
|
|
|
|
|
|-
!8
|9,400
|
|
|
|
|
|
|
|
|
|6,000
|
|19,187
|
|
|
|
|
|23,500
|
|
|
|
|
|-
!9
|
|8,000
|
|
|
|
|
|5,333
|
|
|
|
|13,567
|
|
|
|
|
|
|16,000
|
|
|
|
|-
!10
|
|
|7,950
|
|
|
|
|
|5,300
|
|
|
|
|11,314
|
|
|
|
|
|
|15,900
|
|
|
|}

=== Stationary calculator ===
The following table calculates the apportionment for any stationary signpost function. In other words, it rounds an apportionment if the vote average is above the selected bar.
<div class="calculatorgadget-enabled calculator-container" style="display:none" data-calculator-refresh-on-load="true">
{| class="wikitable mw-collapsible" style="text-align:right;"
|-
!Party
!Yellow
!White
!Red
!Green
!Blue
!Pink
! style="text-align:left; border-left: groove 2px black; border-right: none;" | <u>Total</u>
! style="border-left: none; border-right: none;" |
|-
!Votes
|{{calculator|id=yellow|default=4600|size=4|min=0|step=50}}
|{{calculator|id=white |default=1600|size=4|min=0|step=50}}
|{{calculator|id=red   |default=1550|size=4|min=0|step=50}}
|{{calculator|id=green |default=1200|size=4|min=0|step=50}}
|{{calculator|id=blue  |default=600 |size=4|min=0|step=50}}
|{{calculator|id=pink  |default=450 |size=4|min=0|step=50}}
| style="text-align:left; border-left: groove 2px black; border-right: none;" |<u>{{calculator|id=total |type=plain  |size=4|formula=pink+yellow+white+red+green+blue}}</u>
| style="border-left: none; border-right: none;" |
|-
!Vote share
|{{calculator|id=percent_yellow|type=plain|size=3|formula=100*yellow/total|decimals=1}}%
|{{calculator|id=percent_white |type=plain|size=3|formula=100*white /total|decimals=1}}%
|{{calculator|id=percent_red   |type=plain|size=3|formula=100*red   /total|decimals=1}}%
|{{calculator|id=percent_green |type=plain|size=3|formula=100*green /total|decimals=1}}%
|{{calculator|id=percent_blue  |type=plain|size=3|formula=100*blue  /total|decimals=1}}%
|{{calculator|id=percent_pink  |type=plain|size=3|formula=100*pink  /total|decimals=1}}%
| style="text-align:left; border-left: groove 2px black; border-right: none;" |<u>100%</u>
| style="border-left: none; border-right: none;" |
|-
!Seats
|{{calculator|id=seats_yellow|type=plain|size=3|formula=trunc(1-signpost+(yellow/divisor))}}
|{{calculator|id=seats_white |type=plain|size=3|formula=trunc(1-signpost+(white /divisor))}}
|{{calculator|id=seats_red   |type=plain|size=3|formula=trunc(1-signpost+(red   /divisor))}}
|{{calculator|id=seats_green |type=plain|size=3|formula=trunc(1-signpost+(green /divisor))}}
|{{calculator|id=seats_blue  |type=plain|size=3|formula=trunc(1-signpost+(blue  /divisor))}}
|{{calculator|id=seats_pink  |type=plain|size=3|formula=trunc(1-signpost+(pink  /divisor))}}
| style="text-align:left; border-left: groove 2px black; border-right: none;" |<u>{{calculator|id=seats_total |type=plain|size=3|formula=seats_yellow+seats_white +seats_red   +seats_green +seats_blue  +seats_pink  }}</u>
| style="border-left: none; border-right: none;" |
|-
!Entitlement
|{{calculator|id=ent_yellow|type=plain|size=3|decimals=2|formula=(yellow/total)*seats_total}}
|{{calculator|id=ent_white |type=plain|size=3|decimals=2|formula=(white /total)*seats_total}}
|{{calculator|id=ent_red   |type=plain|size=3|decimals=2|formula=(red   /total)*seats_total}}
|{{calculator|id=ent_green |type=plain|size=3|decimals=2|formula=(green /total)*seats_total}}
|{{calculator|id=ent_blue  |type=plain|size=3|decimals=2|formula=(blue  /total)*seats_total}}
|{{calculator|id=ent_pink  |type=plain|size=3|decimals=2|formula=(pink  /total)*seats_total}}
| style="text-align:left; border-left: groove 2px black; border-right: none;" |<u>{{calculator label|for=signpost|'''Round x >''' }}</u>
| style="border-left: none; border-right: none;" |{{calculator|id=signpost|default=0.5|min=0|max=1|step=0.1|size=4}}
|-
! {{frac|votes|seat}}
|{{calculator|id=vs_yellow|type=plain|size=4|formula=yellow/seats_yellow|decimals=0}}
|{{calculator|id=vs_white |type=plain|size=4|formula=white /seats_white |decimals=0}}
|{{calculator|id=vs_red   |type=plain|size=4|formula=red   /seats_red   |decimals=0}}
|{{calculator|id=vs_green |type=plain|size=4|formula=green /seats_green |decimals=0}}
|{{calculator|id=vs_blue  |type=plain|size=4|formula=blue  /seats_blue  |decimals=0}}
|{{calculator|id=vs_pink  |type=plain|size=4|formula=pink  /seats_pink  |decimals=0}}
| style="text-align:left; border-left: groove 2px black; border-right: none;" |<u>{{calculator label|'''Divisor: '''|for=divisor}}</u>
| style="border-left: none; border-right: none;" |{{calculator|id=divisor|default=600|size=4|step=50}}
|}
</div>

== Properties ==

=== Monotonicity ===
Divisor methods are generally preferred by mathematicians to [[largest remainder method]]s<ref name="Pukelsheim-2017-6" /> because they are less susceptible to [[apportionment paradox]]es.<ref name="Pukelsheim-2017-7" /> In particular, divisor methods satisfy [[population monotonicity]], i.e. voting ''for'' a party can never cause it to ''lose'' seats.<ref name="Pukelsheim-2017-7" /> Such [[population paradox]]es occur by increasing the [[electoral quota]], which can cause different states' remainders to respond erratically.<ref name="Balinski-1982" />{{Rp|Tbl.A7.2}} Divisor methods also satisfy [[resource monotonicity|resource]] or [[house monotonicity]], which says that increasing the number of seats in a legislature should not cause a state to lose a seat.<ref name="Pukelsheim-2017-7" /><ref name="Balinski-1982" />{{Rp|Cor.4.3.1}}

=== Min-Max inequality ===
Every divisor method can be defined using the min-max inequality. Letting brackets denote array indexing, an allocation is valid if-and-only-if:'''<ref name="Pukelsheim-2017-1" />'''{{Rp|78–81}}<blockquote>{{Math|max {{sfrac|votes[party]  | post(seats[party])}} ≤ min {{sfrac|votes[party] | post(seats[party]+1)}}}}</blockquote>In other words, it is impossible to lower the highest vote average by reassigning a seat from one party to another. Every number in this range is a possible divisor. If the inequality is strict, the solution is unique; otherwise, there is an exactly tied vote in the final apportionment stage.'''<ref name="Pukelsheim-2017-1" />{{Rp|83}}'''

== Method families ==
The divisor methods described above can be generalized into families.

=== Generalized average ===
In general, it is possible to construct an apportionment method from any generalized [[average]] function, by defining the signpost function as {{Math|1=post(''k'') = avg(''k'', ''k''+1)}}.<ref name="Pukelsheim-2017-1" />

==== {{Anchor|Imperiali method|Danish method}}Stationary family ====
A divisor method is called '''stationary'''<ref name="Pukelsheim-2017-4" />{{Rp|68}} if for some real number <math>r\in[0,1]</math>, its signposts are of the form <math>d(k) = k+r</math>. The Adams, Webster, and d'Hondt methods are stationary, while Dean and Huntington-Hill are not. A stationary method corresponds to rounding numbers up if they exceed the [[weighted arithmetic mean]] of {{Math|''k''}} and {{Math|''k''+1}}.<ref name="Pukelsheim-2017-1" /> Smaller values of {{Math|''r''}} are friendlier to smaller parties.<ref name="Pukelsheim-2017-2" />

[[Elections in Denmark|Danish elections]] allocate [[leveling seat]]s at the province level using-member constituencies. It divides the number of votes received by a party in a multi-member constituency by 0.33, 1.33, 2.33, 3.33 etc. The fencepost sequence is given by {{Math|1=post(''k'') = ''k''+{{frac|1|3}}}}; this aims to allocate seats closer to equally, rather than exactly proportionally.<ref name="Denmark-2016" />

==== {{Anchor|Dean's method}}Power mean family ====
The '''[[power mean]] family''' of divisor methods includes the Adams, Huntington-Hill, Webster, Dean, and Jefferson methods (either directly or as limits). For a given constant {{Math|''p''}}, the power mean method has signpost function {{Math|1=post(''k'') = {{radic|''k''{{sup|''p''}} + (''k''+1){{sup|''p''}}|{{sub|''p''}}}}}}. The Huntington-Hill method corresponds to the limit as {{Math|''p''}} tends to 0, while Adams and Jefferson represent the limits as {{Math|''p''}} tends to negative or positive infinity.<ref name="Pukelsheim-2017-1" />

The family also includes the less-common '''Dean's method''' for {{Math|1=''p''=-1}}, which corresponds to the [[harmonic mean]]. Dean's method is equivalent to ''rounding to the nearest average''—every state has its seat count rounded in a way that minimizes the difference between the average district size and the ideal district size. For example:<ref name="Balinski-1982" />{{Rp||page=29}}<blockquote>The 1830 representative population of Massachusetts was 610,408: if it received 12 seats its average constituency size would be 50,867; if it received 13 it would be 46,954. So, if the divisor were 47,700 as Polk proposed, Massachusetts should receive 13 seats because 46,954 is closer to 47,700 than is 50,867.</blockquote>Rounding to the vote average with the smallest relative error once again yields the Huntington-Hill method because {{Math|1={{abs|log({{frac|x|y}})}} = {{abs|log({{frac|y|x}})}}}}, i.e. relative differences are reversible. This fact was central to [[Edward V. Huntington]]'s use of relative (instead of absolute) errors in measuring misrepresentation, and to his advocacy for Hill's rule:<ref name="Lauwers-2008" /> Huntington argued the choice of apportionment method should not depend on how the equation for equal representation is rearranged, and only the relative error (minimized by Hill's rule) satisfies this property.<ref name="Balinski-1982" />{{Rp||page=53}}

==== Stolarsky mean family ====
Similarly, the [[Stolarsky mean]] can be used to define a family of divisor methods that minimizes the [[generalized entropy index]] of misrepresentation.<ref name="Wada-2012" /> This family includes the [[logarithmic mean]], the [[geometric mean]], the [[identric mean]] and the [[arithmetic mean]]. The Stolarsky means can be justified as minimizing these misrepresentation metrics, which are of major importance in the study of [[information theory]].<ref name="Agnew-2008" />

== Modifications ==
=== Thresholds ===
{{Main|Electoral threshold}}
Many countries have electoral thresholds for representation, where parties must win a specified fraction of the vote in order to be represented; parties with fewer votes than the threshold requires for representation are eliminated.<ref name="Pukelsheim-2017-8" /> Other countries modify the first divisor to introduce a ''natural threshold''; when using the Webster method, the first divisor is often set to 0.7 or 1.0 (the latter being called the ''full-seat modification'').<ref name="Pukelsheim-2017-8" />

=== Majority-preservation clause ===
A majority-preservation clause guarantees any party winning a majority of the vote will receive at least half the seats in a legislature.<ref name="Pukelsheim-2017-8" /> Without such a clause, it is possible for a party with slightly more than half the vote to receive just barely less than half the seats (if using a method other than D'Hondt).<ref name="Pukelsheim-2017-8" /> This is typically accomplished by adding seats to the legislature until an apportionment that preserves the majority for a parliament is found.<ref name="Pukelsheim-2017-8" />

=== Quota-capped divisor method ===
{{Main|Rank-index method#Quota-capped divisor method}}

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.<ref name="Balinski-1975" /> 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.<ref name="Balinski-1982" />{{Rp|Tbl.A7.2}}

==References==
{{reflist|1=30em|refs=
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<ref name="Pukelsheim-2017-5">{{cite book|last=Pukelsheim |first=Friedrich |chapter=From Reals to Integers: Rounding Functions, Rounding Rules |date=2017 |title=Proportional Representation: Apportionment Methods and Their Applications |pages=59–70 |chapter-url=https://doi.org/10.1007/978-3-319-64707-4_3 |access-date=2021-09-01 |place= |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_3 |isbn=978-3-319-64707-4}}</ref>
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<ref name="Lauwers-2008">{{Cite journal |last1=Lauwers |first1=Luc |last2=Van Puyenbroeck |first2=Tom |date=2008 |title=Minimally Disproportional Representation: Generalized Entropy and Stolarsky Mean-Divisor Methods of Apportionment |url=http://dx.doi.org/10.2139/ssrn.1304628 |journal=SSRN Electronic Journal |doi=10.2139/ssrn.1304628 |s2cid=124797897 |issn=1556-5068}}</ref>
<ref name="Wada-2012">{{Cite journal |last=Wada |first=Junichiro |date=2012-05-01 |title=A divisor apportionment method based on the Kolm–Atkinson social welfare function and generalized entropy |url=https://www.sciencedirect.com/science/article/pii/S0165489612000194 |journal=Mathematical Social Sciences |volume=63 |issue=3 |pages=243–247 |doi=10.1016/j.mathsocsci.2012.02.002 |issn=0165-4896}}</ref>
<ref name="Agnew-2008">{{Cite journal |last=Agnew |first=Robert A. |date=April 2008 |title=Optimal Congressional Apportionment |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2008.11920530 |journal=The American Mathematical Monthly |language=en |volume=115 |issue=4 |pages=297–303 |doi=10.1080/00029890.2008.11920530 |s2cid=14596741 |issn=0002-9890}}</ref>
}}

{{voting systems}}

[[Category:Party-list proportional representation]]
[[Category:Apportionment methods]]