Wikipedia

Highest averages method

The highest averages, divisor, or divide-and-round methods[1] are a family of apportionment algorithms that aim to fairly divide a legislature between several groups, such as political parties or states.[1][2] More generally, divisor methods can be used to round shares of a total, e.g. percentage points (which must add up to 100).[2]

The methods aim to treat voters equally by ensuring legislators represent an equal number of voters by ensuring every party has the same seats-to-votes ratio (or divisor).[3]: 30  Such methods divide the number of votes by the number of votes-per-seat then round this total to get the final apportionment. In doing so, the method approximately maintains proportional representation so that a party with e.g. twice as many votes as another should win twice as many seats.[3]: 30 

The divisor methods are generally preferred by social choice theorists to the largest remainder methods, as they produce more-proportional results by most metrics and are less susceptible to apportionment paradoxes.[4][5][3][6] In particular, divisor methods avoid the population paradox and spoiler effects, unlike the largest remainder methods.[5]

History

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Divisor methods were first invented by Thomas Jefferson to comply with the United States Constitution's 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 of pathologies in many superficially-reasonable rounding rules.[3]: 20  Similar debates would appear in Europe after the adoption of proportional representation, typically as a result of large parties attempting to introduce thresholds and other barriers to entry for small parties.[7] Such apportionments often have substantial consequences, as in the 1870 reapportionment, when Congress used an ad-hoc apportionment to favor Republican states.[8] Had each state's electoral vote total been exactly equal to its entitlement, or had Congress used Sainte-Laguë or a largest remainders method (as it had since 1840), the 1876 election would have gone to Tilden instead of Hayes.[8][9][3]: 3, 37 

Definitions

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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. Each signpost marks the boundary between natural numbers, with numbers being rounded down if and only if they are less than the signpost.[2]

Divisor procedure

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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 congressional 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]

 

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 trial and error.[10]

Highest averages procedure

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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 votes per seat. This method proceeds until all seats are allocated.[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.[1] In general, we can define the averages using the signpost sequence:

 

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.[1]

Specific methods

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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.[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   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) 		  Power mean 0.71 1.58 2.55 3.54
D'Hondt k 1 Down 1.00 2.00 3.00 4.00

D'Hondt (Jefferson) method

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Main article: D'Hondt method

Thomas Jefferson proposed the first divisor method in 1792;[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.[1] It remains the most-common method for proportional representation to this day.[1]

The d'Hondt method uses the sequence  , i.e. (1, 2, 3, ...),[11] which means it will always round a party's apportionment down.[1]

Apportionment never falls below the lower end of the ideal frame, and it minimizes the worst-case overrepresentation in the legislature.[1] However, the d'Hondt method performs poorly when judged by most metrics of proportionality.[12] The rule typically gives large parties an excessive number of seats, with their seat share generally exceeding the ideal share rounded up.[3]: 81 

This pathology led to widespread mockery of the d'Hondt method when it was realized it would "round" 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 representatives".[3]: 34 

Adams' method

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Adams' method was conceived of by John Quincy Adams after noticing the d'Hondt method allocated too few seats to smaller states.[13] It can be described as the inverse of the d'Hondt 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 post(k) = k, which is equivalent to always rounding up.[12]

Adams' apportionment never exceeds the upper end of the ideal frame, and minimizes the worst-case underrepresentation.[1] However, violations of the lower seat quota are common.[14] Like d'Hondt, Adams' method performs poorly according to most metrics of proportionality.[12]

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

Sainte-Laguë (Webster) method

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Main article: Sainte-Laguë method

The Webster or Sainte-Laguë method, first described in 1832 by American statesman and senator Daniel Webster and later independently invented in 1910 by the French mathematician André Sainte-Laguë, uses the fencepost sequence post(k) = k .5 (i.e. 0.5, 1.5, 2.5); this corresponds to the standard rounding rule. Equivalently, the odd integers (1, 3, 5...) can be used to calculate the averages instead.[1][16]

The Sainte-Laguë method produces more proportional apportionments than d'Hondt 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 minimizing seat bias even when dealing with parties that win very small numbers of seats.[19] The Sainte-Laguë method can theoretically violate the ideal share rule, although this is extremely rare for even moderately-large parliaments; it has never been observed to violate quota in any United States congressional apportionment.[18]

In small districts with no threshold, parties can manipulate Sainte-Laguë 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 implicit threshold.[20]

Huntington–Hill method

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Main article: Huntington–Hill method

In the Huntington–Hill method, the signpost sequence is post(k) = √k (k 1), the geometric mean of the neighboring numbers. Conceptually, this method rounds to the integer that has the smallest 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.[1]

The Huntington-Hill method tends to produce very similar results to the Sainte-Laguë method, except that it guarantees every state or party at least one seat (see Highest averages method § Zero-seat apportionments). When first used to assign seats in the House, the two methods produced identical results; in their second use, they differed only in assigning a single seat to Michigan or Arkansas.[3]: 58 

Comparison of properties

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Zero-seat apportionments

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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 to 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.[21]

Bias

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There are many metrics of seat bias. While the Sainte-Laguë method is sometimes described as "uniquely" unbiased,[18] this uniqueness property relies on a technical definition of bias as the expected difference between a state's number of seats and its ideal share. 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 ideal share.[18]

By this definition, the Sainte-Laguë 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 on percent errors, find the opposite result (The Huntington-Hill method is unbiased, while the Sainte-Laguë 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 Sainte-Laguë methods can be considered unbiased or low-bias methods (unlike d'Hondt or Adams' methods).[19][22] A 1929 report to Congress by the National Academy of Sciences recommended the Huntington-Hill method,[23] while the Supreme Court has ruled the choice to be a matter of opinion.[22]

Comparison and examples

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Example: d'Hondt

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The following example shows how the d'Hondt method can differ substantially from less-biased methods such as Sainte-Laguë. 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 Sainte-Laguë method shows none of these properties, with a maximum error of 22.6%.

d'Hondt Sainte-Laguë
Party Yellow White Red Green Purple Total Party Yellow White Red Green Purple Total
Votes 46,000 25,100 12,210 8,350 8,340 100,000 Votes 46,000 25,100 12,210 8,350 8,340 100,000
Seats 11 6 2 1 1 21 Seats 9 5 3 2 2 21
Ideal 9.660 5.271 2.564 1.754 1.751 21 Ideal 9.660 5.271 2.564 1.754 1.751 21
Votes/Seat 4182 4183 6105 8350 8340 4762 Votes/Seat 5111 5020 4070 4175 4170 4762
% Error 13.0% 13.0% -24.8% -56.2% -56.0% (100.%) (% Range) -7.1% -5.3% 15.7% 13.2% 13.3% (22.6%)
Seats Averages Signposts Seats Averages Signposts
1 46,000 25,100 12,210 8,350 8,340 1.00 1 92,001 50,201 24,420 16,700 16,680 0.50
2 23,000 12,550 6,105 4,175 4,170 2.00 2 30,667 16,734 8,140 5,567 5,560 1.50
3 15,333 8,367 4,070 2,783 2,780 3.00 3 18,400 10,040 4,884 3,340 3,336 2.50
4 11,500 6,275 3,053 2,088 2,085 4.00 4 13,143 7,172 3,489 2,386 2,383 3.50
5 9,200 5,020 2,442 1,670 1,668 5.00 5 10,222 5,578 2,713 1,856 1,853 4.50
6 7,667 4,183 2,035 1,392 1,390 6.00 6 8,364 4,564 2,220 1,518 1,516 5.50
7 6,571 3,586 1,744 1,193 1,191 7.00 7 7,077 3,862 1,878 1,285 1,283 6.50
8 5,750 3,138 1,526 1,044 1,043 8.00 8 6,133 3,347 1,628 1,113 1,112 7.50
9 5,111 2,789 1,357 928 927 9.00 9 5,412 2,953 1,436 982 981 8.50
10 4,600 2,510 1,221 835 834 10.00 10 4,842 2,642 1,285 879 878 9.50
11 4,182 2,282 1,110 759 758 11.00 11 4,381 2,391 1,163 795 794 10.50

Example: Adams

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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.

Adams' Method Sainte-Laguë Method
Party Yellow White Red Green Purple Total Party Yellow White Red Green Purple Total
Votes 55,000 17,290 16,600 5,560 5,550 100,000 Votes 55,000 17,290 16,600 5,560 5,550 100,000
Seats 10 4 3 2 2 21 Seats 11 4 4 1 1 21
Ideal 11.550 3.631 3.486 1.168 1.166 21 Ideal 11.550 3.631 3.486 1.168 1.166 21
Votes/Seat 5500 4323 5533 2780 2775 4762 Votes/Seat 4583 4323 5533 5560 5550 4762
% Error -14.4% 9.7% -15.0% 53.8% 54.0% (99.4%) (% Range) 3.8% 9.7% -15.0% -15.5% -15.3% (28.6%)
Seats Averages Signposts Seats Averages Signposts
1 ∞ ∞ ∞ ∞ ∞ 0.00 1 110,001 34,580 33,200 11,120 11,100 0.50
2 55,001 17,290 16,600 5,560 5,550 1.00 2 36,667 11,527 11,067 3,707 3,700 1.50
3 27,500 8,645 8,300 2,780 2,775 2.00 3 22,000 6,916 6,640 2,224 2,220 2.50
4 18,334 5,763 5,533 1,853 1,850 3.00 4 15,714 4,940 4,743 1,589 1,586 3.50
5 13,750 4,323 4,150 1,390 1,388 4.00 5 12,222 3,842 3,689 1,236 1,233 4.50
6 11,000 3,458 3,320 1,112 1,110 5.00 6 10,000 3,144 3,018 1,011 1,009 5.50
7 9,167 2,882 2,767 927 925 6.00 7 8,462 2,660 2,554 855 854 6.50
8 7,857 2,470 2,371 794 793 7.00 8 7,333 2,305 2,213 741 740 7.50
9 6,875 2,161 2,075 695 694 8.00 9 6,471 2,034 1,953 654 653 8.50
10 6,111 1,921 1,844 618 617 9.00 10 5,790 1,820 1,747 585 584 9.50
11 5,500 1,729 1,660 556 555 10.00 11 5,238 1,647 1,581 530 529 10.50
Seats 10 4 3 2 2 Seats 11 4 4 1 1

Example: All systems

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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 Sainte-Laguë or d'Hondt.

d'Hondt method Sainte-Laguë method Huntington–Hill method 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 seat allocation seat allocation seat allocation 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

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The following table allows the user to calculate the apportionment for any stationary signpost function. In other words, it rounds an apportionment if the value is above the selected bar.

Properties

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Monotonicity

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Divisor methods are generally preferred by mathematicians to largest remainder methods[24] because they are less susceptible to apportionment paradoxes.[5] In particular, divisor methods satisfy population monotonicity, i.e. voting for a party can never cause it to lose seats.[5] Such population paradoxes occur by increasing the electoral quota, which can cause different states' remainders to respond erratically.[3]: Tbl.A7.2  Divisor methods also satisfy resource or house 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 

Min-Max inequality

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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 

max ⁠votes[party]/ post(seats[party])⁠ ≤ min ⁠votes[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 

Method families

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The divisor methods described above can be generalized into families.

Generalized average

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In general, it is possible to construct an apportionment method from any generalized average function, by defining the signpost function as post(k) = avg(k, k 1).[1]

Stationary family

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A divisor method is called stationary[25]: 68  if for some real number  , its signposts are of the form  . The Adams, Sainte-Laguë, 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 k and k 1.[1] Smaller values of r are friendlier to smaller parties.[19]

Danish elections allocate leveling 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 by post(k) = k 1⁄3; this aims to allocate seats closer to equally, rather than exactly proportionally.[26]

Power mean family

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The power mean family of divisor methods includes the Adams, Huntington-Hill, Sainte-Laguë, Dean, and d'Hondt methods (either directly or as limits). For a given constant p, the power mean method has signpost function post(k) = p√kp (k 1)p. The Huntington-Hill method corresponds to the limit as p tends to 0, while Adams and d'Hondt represent the limits as p tends to negative or positive infinity.[1]

The family also includes the less-common Dean's method for 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:[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 to Edward 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 

Stolarsky mean family

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Similarly, the Stolarsky mean can be used to define a family of divisor methods that minimizes the generalized entropy index of misrepresentation.[28] 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.[29]

Modifications

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Thresholds

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Main article: 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.[20] Other countries modify the first divisor to introduce a natural threshold; when using the Sainte-Laguë method, the first divisor is often set to 0.7 or 1.0 (the latter being called the full-seat modification).[20]

Majority-preservation clause

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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]

Quota-capped divisor method

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Main article: 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.[30] 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.[3]: Tbl.A7.2 

References

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  1. ^ a b c d e f g h i j k l m n o p q r s t u v w Pukelsheim, Friedrich (2017). "Divisor Methods of Apportionment: Divide and Round". Proportional Representation: Apportionment Methods and Their Applications. Cham: Springer International Publishing. pp. 71–93. doi:10.1007/978-3-319-64707-4_4. ISBN 978-3-319-64707-4. Retrieved 2021-09-01.
  2. ^ a b c d Pukelsheim, Friedrich (2017). "From Reals to Integers: Rounding Functions, Rounding Rules". Proportional Representation: Apportionment Methods and Their Applications. Springer International Publishing. pp. 59–70. doi:10.1007/978-3-319-64707-4_3. ISBN 978-3-319-64707-4. Retrieved 2021-09-01.
  3. ^ a b c d e f g h i j k l m n Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9.
  4. ^ Ricca, Federica; Scozzari, Andrea; Serafini, Paola (2017). "A Guided Tour of the Mathematics of Seat Allocation and Political Districting". In Endriss, Ulle (ed.). Trends in Computational Social Choice. Lulu.com. pp. 49–68. ISBN 978-1-326-91209-3. Archived from the original on 2024-10-08. Retrieved 2024-10-08.
  5. ^ a b c d e Pukelsheim, Friedrich (2017). "Securing System Consistency: Coherence and Paradoxes". Proportional Representation: Apportionment Methods and Their Applications. Cham: Springer International Publishing. pp. 159–183. doi:10.1007/978-3-319-64707-4_9. ISBN 978-3-319-64707-4. Retrieved 2024-05-10.
  6. ^ Dančišin, Vladimír (2017-01-01). "No-show paradox in Slovak party-list proportional system". Human Affairs. 27 (1): 15–21. doi:10.1515/humaff-2017-0002. ISSN 1337-401X.
  7. ^ Pukelsheim, Friedrich (2017). "Exposing Methods: The 2014 European Parliament Elections". Proportional Representation: Apportionment Methods and Their Applications. Cham: Springer International Publishing. pp. 1–40. doi:10.1007/978-3-319-64707-4_1. ISBN 978-3-319-64707-4. Retrieved 2024-07-03.
  8. ^ a b Argersinger, Peter H., ed. (2012), ""Injustices and Inequalities": The Politics of Apportionment, 1870–1888", Representation and Inequality in Late Nineteenth-Century America: The Politics of Apportionment, Cambridge: Cambridge University Press, pp. 8–41, doi:10.1017/cbo9781139149402.002, ISBN 978-1-139-14940-2, archived from the original on 2018-06-07, retrieved 2024-08-04, Apportionment not only determined the power of different states in Congress but, because it allocated electors as well, directly affected the election of the president. Indeed, the peculiar apportionment of 1872, adopted in violation of the prevailing law mandating the method of allocating seats, was directly responsible for the 1876 election of Rutherford B. Hayes with a popular vote minority. Had the previous method been followed, even the Electoral Commission would have been unable to place Hayes in the White House.
  9. ^ Caulfield, Michael J. (2012). "What If? How Apportionment Methods Choose Our Presidents". The Mathematics Teacher. 106 (3): 178–183. doi:10.5951/mathteacher.106.3.0178. ISSN 0025-5769. JSTOR 10.5951/mathteacher.106.3.0178.
  10. ^ Pukelsheim, Friedrich (2017). "Targeting the House Size: Discrepancy Distribution". Proportional Representation: Apportionment Methods and Their Applications. Cham: Springer International Publishing. pp. 107–125. doi:10.1007/978-3-319-64707-4_6. ISBN 978-3-319-64707-4. Retrieved 2024-05-10.
  11. ^ Gallagher, Michael (1991). "Proportionality, disproportionality and electoral systems" (PDF). Electoral Studies. 10 (1): 33–51. doi:10.1016/0261-3794(91)95004-C. Archived from the original (PDF) on 2016-03-04.
  12. ^ a b c Gallagher, Michael (1992). "Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities" (PDF). British Journal of Political Science. 22 (4): 469–496. doi:10.1017/S0007123400006499. ISSN 0007-1234. S2CID 153414497.
  13. ^ "Apportioning Representatives in the United States Congress - Adams' Method of Apportionment | Mathematical Association of America". www.maa.org. Archived from the original on 9 June 2024.
  14. ^ Ichimori, Tetsuo (2010). "New apportionment methods and their quota property". JSIAM Letters. 2: 33–36. doi:10.14495/jsiaml.2.33. ISSN 1883-0617.
  15. ^ The allocation between the EU Member States of the seats in the European Parliament (PDF) (Report). European Parliament. 2011. Archived (PDF) from the original on 2024-05-12. Retrieved 2024-01-26.
  16. ^ Sainte-Laguë, André. "La représentation proportionnelle et la méthode des moindres carrés." Archived 2024-05-15 at the Wayback Machine Annales scientifiques de l'école Normale Supérieure. Vol. 27. 1910.
  17. ^ Pennisi, Aline (March 1998). "Disproportionality indexes and robustness of proportional allocation methods". Electoral Studies. 17 (1): 3–19. doi:10.1016/S0261-3794(97)00052-8. Archived from the original on 2024-04-24. Retrieved 2024-05-10.
  18. ^ a b c d e Balinski, M. L.; Young, H. P. (January 1980). "The Webster method of apportionment". Proceedings of the National Academy of Sciences. 77 (1): 1–4. Bibcode:1980PNAS...77....1B. doi:10.1073/pnas.77.1.1. ISSN 0027-8424. PMC 348194. PMID 16592744.
  19. ^ a b c d e f Pukelsheim, Friedrich (2017). "Favoring Some at the Expense of Others: Seat Biases". Proportional Representation: Apportionment Methods and Their Applications. Cham: Springer International Publishing. pp. 127–147. doi:10.1007/978-3-319-64707-4_7. ISBN 978-3-319-64707-4. Retrieved 2024-05-10.
  20. ^ a b c d e f Pukelsheim, Friedrich (2017). "Tracing Peculiarities: Vote Thresholds and Majority Clauses". Proportional Representation: Apportionment Methods and Their Applications. Cham: Springer International Publishing. pp. 207–223. doi:10.1007/978-3-319-64707-4_11. ISBN 978-3-319-64707-4. Retrieved 2024-05-10.
  21. ^ Pukelsheim, Friedrich (2017). "Truncating Seat Ranges: Minimum-Maximum Restrictions". Proportional Representation: Apportionment Methods and Their Applications. Cham: Springer International Publishing. pp. 225–245. doi:10.1007/978-3-319-64707-4_12. ISBN 978-3-319-64707-4. Retrieved 2024-05-10.
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  24. ^ Pukelsheim, Friedrich (2017). "Quota Methods of Apportionment: Divide and Rank". Proportional Representation: Apportionment Methods and Their Applications. Cham: Springer International Publishing. pp. 95–105. doi:10.1007/978-3-319-64707-4_5. ISBN 978-3-319-64707-4. Retrieved 2024-05-10.
  25. ^ Pukelsheim, Friedrich (2017). "From Reals to Integers: Rounding Functions and Rounding Rules". Proportional Representation: Apportionment Methods and Their Applications. Cham: Springer International Publishing. pp. 59–70. doi:10.1007/978-3-319-64707-4_3. ISBN 978-3-319-64707-4. Retrieved 2021-09-01.
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