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Thehighest averages,divisor, ordivide-and-round methods^[1] are a family ofapportionment rules, i.e. algorithms forfair division of seats in a legislature between several groups (likepolitical parties orstates).^[1][2] More generally, divisor methods are used to round shares of a total to afraction with a fixeddenominator (e.g. percentage points, which must add up to 100).^[2]

The methods aim to treat voters equally by ensuring legislatorsrepresent an equal number of voters by ensuring every party has the sameseats-to-votes ratio (ordivisor).^[3]: 30 Such methods divide the number of votes by the number of votes per seat to get the final apportionment. By doing so, the method maintainsproportional representation, as a party with e.g. twice as many votes will win about twice as many seats.^[3]: 30

The divisor methods are generally preferred bysocial choice theorists and mathematicians to thelargest remainder methods, as they produce more-proportional results by most metrics and are less susceptible toapportionment paradoxes.^[3][4][5][6] In particular, divisor methods avoid thepopulation paradox andspoiler effects, unlike the largest remainder methods.^[5]

Divisor methods were first invented byThomas Jefferson to comply with aconstitutional 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.^[3]: 20

Apportionment would become a major topic of debate in Congress, especially after the discovery ofpathologies in many superficially-reasonable rounding rules.^[3]: 20 Similar debates would appear in Europe after the adoption ofproportional representation, typically as a result of large parties attempting to introducethresholds and otherbarriers to entry for small parties.^[7] Such apportionments often have substantial consequences, as in the1870 reapportionment, when Congress used an ad-hoc apportionment to favorRepublican states.^[8] Had each state's electoral vote total been exactly equal toits entitlement, or had Congress usedWebster's method or alargest remainders method (as it had since 1840), the1876 election would have gone toTilden instead ofHayes.^{[8][9][3]: 3, 37}

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 onrounding rules, defined using asignpost sequencepost(k), wherek ≤ 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.^[2]

The divisor procedure apportions seats by searching for adivisor orelectoral 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 acongressional district, or the number of voters represented by each legislator.^[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.^[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:^[1]

$${\displaystyle \text{seats}=\mathrm{round}\left(\frac{\text{votes}}{\text{divisor}}\right)}$$

Usually, the divisor is initially set to equal theHare 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 bytrial and error.^[10]

With the highest averages algorithm, every party begins with 0 seats. Then, at each iteration, we allocate a seat to the party with thehighest vote average, i.e. the party with the mostvotes per seat. This method proceeds until all seats are allocated.^[1]

However, it is unclear whether it is better to look at the vote averagebefore assigning the seat, what the average will beafter assigning the seat, or if we should compromise with acontinuity correction. These approaches each give slightly different apportionments.^[1] In general, we can define the averages using the signpost sequence:

$${\displaystyle \text{average}:=\frac{\text{votes}}{\mathrm{post}(\text{seats})}}$$

With the highest averages procedure, every party begins with 0 seats. Then, at each iteration, we allocate a seat to the party with thehighest vote average, i.e. the party with the most votes per seat. This method proceeds until all seats are allocated.^[1]

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 someelectoral threshold.^[2]

Divisor formulas

Method	Signposts	Rounding of Seats	Approx. first values
Adams	k	Up	0.00 1.00 2.00 3.00
Dean	2÷(1⁄k +1⁄k+1)	Harmonic	0.00 1.33 2.40 3.43
Huntington–Hill	${\displaystyle \sqrt{k(k+1)}}$	Geometric	0.00 1.41 2.45 3.46
Stationary (e.g.r =1⁄3)	k +r	Weighted	0.33 1.33 2.33 3.33
Webster/Sainte-Laguë	k +1⁄2	Arithmetic	0.50 1.50 2.50 3.50
Power mean (e.g.p = 2)	$\sqrt[p]{(k^p+(k+1{)}^p)/2}$	Power mean	0.71 1.58 2.55 3.54
Jefferson/D'Hondt	k + 1	Down	1.00 2.00 3.00 4.00

Thomas Jefferson was the first to propose a divisor method, in 1792;^[1] it was later independently developed by Belgian political scientistVictor d'Hondt in 1878. It assigns the representative to the list that would be most underrepresented at the end of the round.^[1] It remains the most-common method forproportional representation to this day.^[1]

Jefferson's method uses the sequence${\displaystyle \mathrm{post}(k)=k+1}$, i.e. (1, 2, 3, ...),^[11] which means it will always round a party's apportionment down.^[1]

Jefferson's apportionment never falls below the lower end of theideal frame, and it minimizes the worst-case overrepresentation in the legislature.^[1] However, it performs poorly when judged by most other metrics of proportionality.^[12] The rule typically gives large parties an excessive number of seats, with their seat share often exceeding their entitlement rounded up.^[3]: 81

Thispathology led to widespread mockery of Jefferson's method when it was learned Jefferson's method could "round"New York's apportionment of 40.5 up to 42, with SenatorMahlon Dickerson saying the extra seat must come from the "ghosts of departed representatives".^[3]: 34

Adams' method was conceived of byJohn Quincy Adams after noticing Jefferson's method allocated too few seats to smaller states.^[13] It can be described as the inverse of Jefferson's method; it awards a seat to the party that has the most votes per seatbefore the new seat is added. The divisor function ispost(k) =k, which is equivalent to always rounding up.^[12]

Adams' apportionment never exceeds the upper end of theideal frame, and minimizes the worst-case underrepresentation.^[1] However, like Jefferson's method, Adams' method performs poorly according to most metrics of proportionality.^[12] It also often violates thelower seat quota.^[14]

Adams' method was suggested as part of the Cambridge compromise for apportionment ofEuropean parliament seats to member states, with the aim of satisfyingdegressive proportionality.^[15]

The Sainte-Laguë or Webster method, first described in 1832 by American statesman and senatorDaniel Webster and later independently in 1910 by the French mathematicianAndré Sainte-Lague, uses the fencepost sequencepost(k) =k+.5 (i.e. 0.5, 1.5, 2.5); this corresponds to the standardrounding rule. Equivalently, the odd integers (1, 3, 5...) can be used to calculate the averages instead.^[1][16]

The Webster method produces more proportional apportionments than Jefferson's by almost every metric of misrepresentation.^[17] 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).^[18] It is also notable for minimizingseat bias even when dealing with parties that win very small numbers of seats.^[19] The Webster method can theoretically violate theideal frame, although this is extremely rare for even moderately-large parliaments; it has never been observed to violate quota in anyUnited States congressional apportionment.^[18]

In small districts with nothreshold, parties canmanipulate Webster by splitting into many lists, each of which wins a full seat with less than aHare 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 animplicit threshold.^[20]

In theHuntington–Hill method, the signpost sequence ispost(k) =√k (k+1), thegeometric mean of the neighboring numbers. Conceptually, this method rounds to the integer that has the smallestrelative (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.^[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 (seeHighest averages method § Zero-seat apportionments). When first used to assign seats in theHouse, the two methods produced identical results; in their second use, they differed only in assigning a single seat toMichigan orArkansas.^[3]: 58

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.^[1] This property can be desirable (as when apportioning seats tostates) 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.^[21]

There are many metrics ofseat bias. While the Webster method is sometimes described as "uniquely" unbiased,^[18] this uniqueness property relies on atechnical definition of bias, which is defined as theaverage difference between a state's number of seats and itsseat 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.^[18]

By this definition, the Webster method is the least-biased apportionment method,^[19] while Huntington-Hill exhibits a mild bias towards smaller parties.^[18] However, other researchers have noted that slightly different definitions of bias, generally based onpercent errors, find the opposite result (The Huntington-Hill method is unbiased, while the Webster method is slightly biased towards large parties).^[19][22]

In practice, the difference between these definitions is small when handling parties or states with more than one seat.^[19] Thus, both the Huntington-Hill and Webster methods can be considered unbiased or low-bias methods (unlike the Jefferson or Adams methods).^[19][22] A 1929 report to Congress by theNational Academy of Sciences recommended the Huntington-Hill method,^[23] while theSupreme Court has ruled the choice to be a matter of opinion.^[22]

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%.

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.

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.

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.

Divisor methods are generally preferred by mathematicians tolargest remainder methods^[24] because they are less susceptible toapportionment paradoxes.^[5] In particular, divisor methods satisfypopulation monotonicity, i.e. votingfor a party can never cause it tolose seats.^[5] Suchpopulation paradoxes occur by increasing theelectoral quota, which can cause different states' remainders to respond erratically.^{[3]: Tbl.A7.2} Divisor methods also satisfyresource orhouse monotonicity, which says that increasing the number of seats in a legislature should not cause a state to lose a seat.^{[5][3]: Cor.4.3.1}

Every divisor method can be defined using the min-max inequality. Letting brackets denote array indexing, an allocation is valid if-and-only-if:^{[1]: 78–81}

max⁠votes[party]/ post(seats[party])⁠ ≤ min⁠votes[party]/ post(seats[party]+1)⁠

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.^[1]: 83

The divisor methods described above can be generalized into families.

In general, it is possible to construct an apportionment method from any generalizedaverage function, by defining the signpost function aspost(k) = avg(k,k+1).^[1]

A divisor method is calledstationary^[25]: 68 if for some real number${\displaystyle r\in [0,1]}$, its signposts are of the form${\displaystyle d(k)=k+r}$. 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 theweighted arithmetic mean ofk andk+1.^[1] Smaller values ofr are friendlier to smaller parties.^[19]

Danish elections allocateleveling seats 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 bypost(k) =k+1⁄3; this aims to allocate seats closer to equally, rather than exactly proportionally.^[26]

Thepower mean family of divisor methods includes the Adams, Huntington-Hill, Webster, Dean, and Jefferson methods (either directly or as limits). For a given constantp, the power mean method has signpost functionpost(k) =^_p√k^p + (k+1)^p. The Huntington-Hill method corresponds to the limit asp tends to 0, while Adams and Jefferson represent the limits asp tends to negative or positive infinity.^[1]

The family also includes the less-commonDean's method forp=-1, which corresponds to theharmonic mean. Dean's method is equivalent torounding 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:^[3]: 29

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.

Rounding to the vote average with the smallest relative error once again yields the Huntington-Hill method because|log(x⁄y)| = |log(y⁄x)|, i.e. relative differences are reversible. This fact was central toEdward V. Huntington's use of relative (instead of absolute) errors in measuring misrepresentation, and to his advocacy for Hill's rule:^[27] 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.^[3]: 53

Similarly, theStolarsky mean can be used to define a family of divisor methods that minimizes thegeneralized entropy index of misrepresentation.^[28] This family includes thelogarithmic mean, thegeometric mean, theidentric mean and thearithmetic mean. The Stolarsky means can be justified as minimizing these misrepresentation metrics, which are of major importance in the study ofinformation theory.^[29]

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.^[20] Other countries modify the first divisor to introduce anatural threshold; when using the Webster method, the first divisor is often set to 0.7 or 1.0 (the latter being called thefull-seat modification).^[20]

A majority-preservation clause guarantees any party winning a majority of the vote will receive at least half the seats in a legislature.^[20] 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).^[20] This is typically accomplished by adding seats to the legislature until an apportionment that preserves the majority for a parliament is found.^[20]

Some electoral systems, such asSwitzerland,Israel, andEuropean Parliament elections inDenmark, permit surplus vote agreements, whereby two or more parties are regarded as a single bloc for the purposes of seat allocation; within this bloc, a second seat allocation takes place between the parties to the agreement.

Aquota-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.^[30] However, quota-capped divisor methods violate theparticipation criterion (also calledpopulation monotonicity)—it is possible for a party tolose a seat as a result of winningmore votes.^{[3]: Tbl.A7.2}

• Party-list proportional representation
• Apportionment methods

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