Part of the Politics and Economics series 
Electoral systems 


The highest averages, divisor, or divideandround 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 seatstovotes ratio (or divisor).^{[3]}^{: 30 } Such methods divide the number of votes, by the number of votesperseat, then round the 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 moreproportional results by most metrics and are less susceptible to apportionment paradoxes.^{[4]}^{[5]}^{[6]} In particular, divisor methods satisfy voteratio monotonicity and participation, i.e. voting for a party can never cause it to lose seats, unlike in the largest remainders methods.^{[5]}
YouTube Encyclopedic

1/5Views:530 90212 353 8513 603 5562 011 5525 482 174

How to Find the Average  Math with Mr. J

The Hardest Math Test

How To Solve Math Percentage Word Problem?

Anyone Can Be a Math Person Once They Know the Best Learning Techniques  PoShen Loh  Big Think

Math Antics  Mean, Median and Mode
Transcription
History
Divisor methods were first invented by Thomas Jefferson to comply with the 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.^{[6]}^{: 20 }
Apportionment would become a major topic of debate in Congress, especially after the discovery of pathologies in many superficiallyreasonable rounding rules.^{[6]}^{: 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 adhoc procedure designed to favor Republicancontrolled states. Had each state's electoral vote total been exactly equal to its entitlement, or had Congress used Webster or Hamilton's method (as it had since 1840), the 1876 election would have gone to Tilden instead of Hayes.^{[8]}^{[6]}^{: 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.^{[9]}
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.^{[10]}
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 congressional district, or the number of voters represented by each legislator.^{[11]}
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.^{[11]} 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:^{[11]}
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.^{[12]}
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 votes per seat. This method proceeds until all seats are allocated.^{[11]}
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.^{[11]} 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.^{[11]}
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.^{[13]}
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  √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/SainteLaguë  k + 1⁄2  Arithmetic  0.50 1.50 2.50 3.50 
Power mean (e.g. p = 2) 
^{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 
Jefferson (D'Hondt) method
Thomas Jefferson proposed the first divisor method in 1792.^{[11]} It assigns the representative to the state that would be most underrepresented at the end of the round.^{[11]} It remains the mostcommon method for proportional representation to this day.^{[11]}
Jefferson's method uses the sequence , i.e. (1, 2, 3, ...),^{[14]} which means it will always round a party's apportionment down.^{[11]}
apportionment never falls below the lower end of the ideal frame, and it minimizes the worstcase overrepresentation in the legislature.^{[11]} However, Jefferson's method performs poorly when judged by most metrics of proportionality.^{[15]} The rule typically gives large parties an excessive number of seats, with their seat share generally exceeding the ideal share rounded up.^{[6]}^{: 81 }
This pathology led to widespread mockery of Jefferson's 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".^{[6]}^{: 34 }
Adams' (Cambridge) method
Adams' method was conceived of by John Quincy Adams after noticing Jefferson's method allocated too few seats to smaller states.^{[16]} 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 post(k) = k, which is equivalent to always rounding up.^{[15]}
Adams' apportionment never exceeds the upper end of the ideal frame, and minimizes the worstcase underrepresentation.^{[11]} However, violations of the lower seat quota are common.^{[17]} Like Jefferson, Adams' method performs poorly according to most metrics of proportionality.^{[15]}
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.^{[18]}
Webster's (SainteLaguë) method
Daniel Webster's method 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.^{[11]}^{[19]}
Webster's method produces more proportional apportionments than D'Hondt's by almost every metric of misrepresentation.^{[20]} 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).^{[21]} It is also notable for minimizing seat bias even when dealing with parties that win very small numbers of seats.^{[22]} Webster's method can theoretically violate the ideal share rule, although this is extremely rare for even moderatelylarge parliaments; it has never been observed to violate quota in any United States congressional apportionment.^{[21]}
In small districts with no threshold, parties can 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 implicit threshold.^{[23]}
Hill's (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.^{[11]}
Hill's method tends to produce very similar results to Webster's method; when first used for congressional apportionment, the two methods differed only in whether they assigned a single seat to Michigan or Arkansas.^{[6]}^{: 58 }
Comparison of properties
Zeroseat apportionments
HuntingtonHill, 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.^{[11]} This property can be desirable (as when apportioning seats to states) or undesirable, in which case the first divisor may be adjusted to create a natural threshold.^{[24]}
Bias
There are many metrics of seat bias. While Webster's method is sometimes described as "uniquely" unbiased,^{[21]} 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.^{[21]}
By this definition, Webster's method is the leastbiased apportionment method,^{[22]} while HuntingtonHill exhibits a mild bias towards smaller states.^{[21]} However, other researchers have noted that slightly different definitions of bias, generally based on percent errors, find the opposite result (Hill's method is unbiased, while Webster's method is slightly biased towards large states).^{[22]}^{[25]}
In practice, the difference between these definitions is small when handling parties or states with more than one seat.^{[22]} Thus, both HuntingtonHill and Webster's method can be considered unbiased or lowbias methods (unlike Jefferson or Adams' methods).^{[22]}^{[25]} A 1929 report to Congress by the National Academy of Sciences recommended Hill's method,^{[26]} while the Supreme Court has ruled the choice of bias metric to be a matter of opinion.^{[25]}
Comparison and examples
Example: Jefferson
The following example shows how Jefferson's method can differ substantially from lessbiased methods such as Webster's. 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. Webster's method shows none of these properties, with a maximum error of 22.6%.
Jefferson's method  Webster's method  

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
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  Webster's 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
The following shows a workedout example for all voting systems. Notice how HuntingtonHill and Adams' methods give every party one seat before assigning any more, unlike Webster's or Jefferson's.
Jefferson method  Webster 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 
Properties
Monotonicity
Divisor methods are generally preferred by mathematicians to largest remainder methods^{[27]} because they are less susceptible to apportionment paradoxes.^{[28]} In particular, divisor methods satisfy population monotonicity, i.e. voting for a party can never cause it to lose seats.^{[28]} Such population paradoxes occur by increasing the electoral quota, which can cause different states' remainders to respond erratically.^{[6]}^{: 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.^{[28]}^{[6]}^{: Cor.4.3.1 }
MinMax inequality
Every divisor method can be defined using the minmax inequality. Letting brackets denote array indexing, an allocation is valid ifandonlyif:^{[11]}^{: 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.^{[11]}^{: 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 post(k) = avg(k, k+1).^{[11]}
Stationary family
A divisor method is called stationary^{[29]}^{: 68 } if for some real number , its signposts are of the form . The Adams, Webster, and Jefferson methods are stationary, while Dean and HuntingtonHill are not. A stationary method corresponds to rounding numbers up if they exceed the weighted arithmetic mean of k and k+1.^{[11]} Smaller values of r are friendlier to smaller parties.^{[22]}
Danish elections allocate leveling seats at the province level usingmember constituencies. It divides the number of votes received by a party in a multimember 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.^{[30]}
Power mean family
The power mean family of divisor methods includes the Adams, HuntingtonHill, Webster, Dean, and Jefferson methods (either directly or as limits). For a given constant p, the power mean method has signpost function post(k) = ^{p}√k^{p} + (k+1)^{p}. The HuntingtonHill method corresponds to the limit as p tends to 0, while Adams and Jefferson represent the limits as p tends to negative or positive infinity.^{[11]}
The family also includes the lesscommon 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:^{[31]}^{: 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 HuntingtonHill 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 the HuntingtonHill technique:^{[32]} Huntington argued the choice of apportionment method should not depend on how the equation for equal representation is rearranged, and only relative errors (i.e. the HuntingtonHill technique) satisfy this property.^{[31]}^{: 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.^{[33]} 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.^{[34]}
Modifications
Thresholds
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.^{[23]} Other countries modify the first divisor to introduce a natural threshold; when using Webster's method, the first divisor is often set to 0.7 or 1.0 (the latter being called the fullseat modification).^{[23]}
Majoritypreservation clause
A majoritypreservation clause guarantees any party winning a majority of the vote will receive at least half the seats in a legislature.^{[23]} 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).^{[23]} This is typically accomplished by adding seats to the legislature until an apportionment that preserves the majority for a parliament is found.^{[23]}
Quotacapped divisor method
A quotacapped divisor method is an apportionment method where we begin by assigning every state its lower quota of seats. Then, we add seats onebyone to the state with the highest votesperseat average, so long as adding an additional seat does not result in the state exceeding its upper quota.^{[35]} However, quotacapped 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.^{[36]}^{: Tbl.A7.2 }
References
 ^ ^{a} ^{b} Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Divisor Methods of Apportionment: Divide and Round", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 71–93, doi:10.1007/9783319647074_4, ISBN 9783319647074, retrieved 20210901
 ^ ^{a} ^{b} Pukelsheim, Friedrich (2017), "From Reals to Integers: Rounding Functions, Rounding Rules", Proportional Representation: Apportionment Methods and Their Applications, Springer International Publishing, pp. 71–93, doi:10.1007/9783319647074_4, ISBN 9783319647074, retrieved 20210901
 ^ ^{a} ^{b} Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0300027249.
 ^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Quota Methods of Apportionment: Divide and Rank", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 95–105, doi:10.1007/9783319647074_5, ISBN 9783319647074, retrieved 20240510
 ^ ^{a} ^{b} Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Securing System Consistency: Coherence and Paradoxes", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 159–183, doi:10.1007/9783319647074_9, ISBN 9783319647074, retrieved 20240510
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0300027249.
 ^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Exposing Methods: The 2014 European Parliament Elections", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 1–40, doi:10.1007/9783319647074_1, ISBN 9783319647074, retrieved 20240703
 ^ 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 00255769.
 ^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Divisor Methods of Apportionment: Divide and Round", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 71–93, doi:10.1007/9783319647074_4, ISBN 9783319647074, retrieved 20210901
 ^ Pukelsheim, Friedrich (2017), "From Reals to Integers: Rounding Functions, Rounding Rules", Proportional Representation: Apportionment Methods and Their Applications, Springer International Publishing, pp. 71–93, doi:10.1007/9783319647074_4, ISBN 9783319647074, retrieved 20210901
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} ^{o} ^{p} ^{q} ^{r} ^{s} ^{t} Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Divisor Methods of Apportionment: Divide and Round", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 71–93, doi:10.1007/9783319647074_4, ISBN 9783319647074, retrieved 20210901
 ^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Targeting the House Size: Discrepancy Distribution", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 107–125, doi:10.1007/9783319647074_6, ISBN 9783319647074, retrieved 20240510
 ^ Pukelsheim, Friedrich (2017), "From Reals to Integers: Rounding Functions, Rounding Rules", Proportional Representation: Apportionment Methods and Their Applications, Springer International Publishing, pp. 71–93, doi:10.1007/9783319647074_4, ISBN 9783319647074, retrieved 20210901
 ^ Gallagher, Michael (1991). "Proportionality, disproportionality and electoral systems" (PDF). Electoral Studies. 10 (1): 33–51. doi:10.1016/02613794(91)90004C. Archived from the original (PDF) on 20160304.
 ^ ^{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 00071234. S2CID 153414497.
 ^ "Apportioning Representatives in the United States Congress  Adams' Method of Apportionment  Mathematical Association of America". www.maa.org. Retrieved 20201111.
 ^ Ichimori, Tetsuo (2010). "New apportionment methods and their quota property". JSIAM Letters. 2: 33–36. doi:10.14495/jsiaml.2.33. ISSN 18830617.
 ^ The allocation between the EU Member States of the seats in the European Parliament (PDF) (Report). European Parliament. 2011.
 ^ SainteLaguë, André. "La représentation proportionnelle et la méthode des moindres carrés." Annales scientifiques de l'école Normale Supérieure. Vol. 27. 1910.
 ^ Pennisi, Aline (March 1998). "Disproportionality indexes and robustness of proportional allocation methods". Electoral Studies. 17 (1): 3–19. doi:10.1016/S02613794(97)000528.
 ^ ^{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 00278424. PMC 348194. PMID 16592744.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Favoring Some at the Expense of Others: Seat Biases", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 127–147, doi:10.1007/9783319647074_7, ISBN 9783319647074, retrieved 20240510
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Tracing Peculiarities: Vote Thresholds and Majority Clauses", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 207–223, doi:10.1007/9783319647074_11, ISBN 9783319647074, retrieved 20240510
 ^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Truncating Seat Ranges: MinimumMaximum Restrictions", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 225–245, doi:10.1007/9783319647074_12, ISBN 9783319647074, retrieved 20240510
 ^ ^{a} ^{b} ^{c} Ernst, Lawrence R. (1994). "Apportionment Methods for the House of Representatives and the Court Challenges". Management Science. 40 (10): 1207–1227. doi:10.1287/mnsc.40.10.1207. ISSN 00251909. JSTOR 2661618.
 ^ Huntington, Edward V. (1929). "The Report of the National Academy of Sciences on Reapportionment". Science. 69 (1792): 471–473. ISSN 00368075.
 ^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Quota Methods of Apportionment: Divide and Rank", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 95–105, doi:10.1007/9783319647074_5, ISBN 9783319647074, retrieved 20240510
 ^ ^{a} ^{b} ^{c} Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Securing System Consistency: Coherence and Paradoxes", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 159–183, doi:10.1007/9783319647074_9, ISBN 9783319647074, retrieved 20240510
 ^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "From Reals to Integers: Rounding Functions and Rounding Rules", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 59–70, doi:10.1007/9783319647074_3, ISBN 9783319647074, retrieved 20210901
 ^ "The Parliamentary Electoral System in Denmark". Archived from the original on 20160828.
 ^ ^{a} ^{b} Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0300027249.
 ^ Lauwers, Luc; Van Puyenbroeck, Tom (2008). "Minimally Disproportional Representation: Generalized Entropy and Stolarsky MeanDivisor Methods of Apportionment". SSRN Electronic Journal. doi:10.2139/ssrn.1304628. ISSN 15565068. S2CID 124797897.
 ^ Wada, Junichiro (20120501). "A divisor apportionment method based on the Kolm–Atkinson social welfare function and generalized entropy". Mathematical Social Sciences. 63 (3): 243–247. doi:10.1016/j.mathsocsci.2012.02.002. ISSN 01654896.
 ^ Agnew, Robert A. (April 2008). "Optimal Congressional Apportionment". The American Mathematical Monthly. 115 (4): 297–303. doi:10.1080/00029890.2008.11920530. ISSN 00029890. S2CID 14596741.
 ^ Balinski, M. L.; Young, H. P. (19750801). "The Quota Method of Apportionment". The American Mathematical Monthly. 82 (7): 701–730. doi:10.1080/00029890.1975.11993911. ISSN 00029890.
 ^ Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0300027249.