The Wiki "Droop quota " article is flawed. suggestions for revision are ignored.
so here is what it should say:
(an even more refined better version is farther below)
In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff or Newland-Britton quota[a]) is the minimum number of votes needed for a candidate to be certain to be elected under STV systems used today. It is the preferred quota, being known to be less likely than the Hare quota, to give majority of seats to a minority party.[1] It is the smallest portion of votes that elects the correct number of members to fill the seats, but no more than that number.
Droop quota is the number obtained by dividing the total number of valid votes cast in a district by a number that is one more than the number of places to be filled (members to be elected) and increasing the result by a small amount. (Often it is rounded up to the next whole number).[2]
With most of the successful candidates having a vote tally equal to the quota, each party will receive its due share of seats, as much as the number of seats in the district can allow anyway. (Of course in STV elections, in odd exceptions one to three candidates might be be elected with more or less than quota.)[3]
The Droop quota generalizes the concept of a majority to multiple-winner elections: just as a majority (more than half of votes) guarantees a candidate can be declared the winner of a one-on-one election, having more than one Droop quota's worth of votes measures the number of votes a candidate needs to be guaranteed victory in a multiwinner election.
Swiss physicist Hagenbach-Bischoff also put his name to the Droop quota.[4]
Hagenbach-Bischoff was quite clear that his desired quota was one where no more could be elected by quota than the number of empty seats -- "Hagenbach-Bischoff was aware of the possibility and formulated the calculation of this quota in such a way it is always the smallest integer greater than V/(S+1)." (from Dancisin, Misinterpretation of the H-B quota https://www.unipo.sk/public/media/18214/09%20Dancisin.pdf or https://www.academia.edu/3877678/MISINTERPRETATION_OF_THE_HAGENBACH_BISCHOFF_QUOTA (I have added bold to the important word in that sentence)] The Hagenbach-Bischoff system is his application of this quota to election contests.
Besides establishing winners, the Droop quota is used to define the number of excess votes, votes not needed by a candidate who has been declared elected. In proportional quota-rule systems such as STV and CPO-STV, these excess votes are transferred to other candidates, preventing them from being wasted.
The Droop quota was first devised by the English lawyer and mathematician Henry Richmond Droop (1831–1884), as an alternative to the Hare quota.
Hagenbach-Bischoff also wrote on the quota in 1888, in his study entitled Die Frage der Einführung einer Proportionalvertretung statt des absoluten Mehres.
Both were clear that their quota was some number just larger than votes/seats plus 1, As Droop put it, "the whole number next greater than the quotient obtained by dividing mV, the number of votes, by n + 1, will be called the quota."[1][2]
(see Henry R. Droop, "On Methods of Electing Representatives," Journal of the Statistical Society of London, Vol. 44, No. 2. (Jun., 1881), pp. 141–202 (Reprinted in Voting matters, No. 24 (Oct., 2007), pp. 7-46)
Hagenbach-Bischoff also wrote on the quota in 1888, in his study entitled Die Frage der Einführung einer Proportionalvertretung statt des absoluten Mehres.
Dancisin:
the calculation of the electoral quota is defined verbally as follows: “Zu der Wahl eines Vertreters genügt eine bestimmte Zahl von Stimmen, die wir Wahlzahl nennen; dieselbe wird erhalten, indem man die Zahl der Wähler durch die um eins vermehrte Zahl der Vertreter dividirt und die auf den so erhaltenen Quotienten nächstfolgende ganze Zahl nimmt” (1888, s. 9).
This can be translated as follows:
the electoral quota can be calculated by dividing the number of valid votes by the number of seats plus one. The result of this calculation must subsequently be rounded up to the nearest integer, which represents the actual electoral number (quota).
E. Hagenbach-Bischoff also considered the possibility of the result calculated according to the formula Q = V/(S+1) being an integer. In the circumstances, the quota has to be increased by one vote (Hagenbach-Bischoff, 1905, p. 7).
This can be turned into a mathematical formula, namely Q = [V/(S+1)]+1.1 Hagenbach-Bischoff’s intention behind increasing the number of seats in the denominator by one was to ensure that the highest number of seats gets distributed among the individual parties concerned as soon as possible (in the first count)."
Today the Droop quota is used in almost all STV elections, including those in the Republic of Ireland, Northern Ireland, Malta, and Australia. It is also used in South Africa to allocate seats by the largest remainder method.
Standard Formula
The exact form of the Droop quota for a �-winner election is given by the formula:[b]
total votes�+1
plus a fraction, or plus 1, or rounded up to next whole number.
Sometimes, the Droop quota is written as a share (i.e. percentage) of the total votes, in which case it has value of more than 1⁄k+1.
Any candidate who attains quota or exceeds it is declared elected.
Derivation
The value of Droop quota can be seen by considering what would happen if k candidates (called "Droop winners") attain the Droop quota. The prove of its value is to see whether an outside candidate could defeat any of these candidates.
In this situation, each quota winner's share of the vote equals or exceeds 1⁄k+1, while all the unelected candidates' share of the vote, even if taken together, is less than Droop quota. Thus, even if there were only one unelected candidate who held all the remaining votes, they would not be able to defeat any of the Droop winners.
Example in STV
The following election has 3 seats to be filled by single transferable vote. There are 4 candidates: George Washington, Alexander Hamilton, Thomas Jefferson, and Aaron Burr. There are 102 voters, but two of the votes are spoiled.
The total number of valid votes is 100, and there are 3 seats. The Droop quota is therefore 1003+1=25, plus 1 = 26. These votes are as follows:
45 voters | 20 voters | 25 voters | 10 voters | |
1 | Washington | Burr | Jefferson | Hamilton |
2 | Hamilton | Jefferson | Burr | Washington |
3 | Jefferson | Washington | Washington | Jefferson |
First preferences for each candidate are tallied:
Washington: 45
Hamilton: 10
Burr: 20
Jefferson: 25
Only Washington has quota -- 26 votes. As a result, he is immediately elected. Washington has 19 excess votes that can be transferred to their second choice, Hamilton. The tallies therefore become:
Washington: 26
Hamilton: 29
Burr: 20
Jefferson: 25
Hamilton is elected, so his 3 excess votes are redistributed. Thanks to Hamilton's support, Jefferson receives 28 votes to Burr's 20 and is elected.
Sometimes there may be a tie between two candidates. The tiebreaking rules are discussed below.
Incorrect or nonstandard variants
[this is a topic in the Wiki article but actually is not important to me so have dropped it here]
Different versions of Droop
At least six different versions of the Droop quota appear in various legal codes or definitions of the quota. Some claim that, depending on which version is used, a failure of proportionality may arise in small elections. Common variants include:
Droop: ⌊votes/seats+1+1⌋
Hagenbach-Bischoff: ⌈votes/seats+1⌉ or sometimes: ⌊votes/seats+1⌋+1
Unusual: ⌊votes/seats+1⌋ ⌊votes/seats+1+12⌋
Accidental: votes+1/seats+1
Droop and Hagenbach-Bischoff derived new quota as a replacement for the Hare quota (votes/seats). Their quota was meant to produce more proportional results by having the quota as low as thought to be possible. Their quota was basically a number just larger than votes/seats plus 1.
This formula may yield a fraction, which was a problem as early STV systems did not use fractions. Droop went to votes/seats plus 1, plus 1, rounded down, as shown. Hagenbach-Bischoff went to votes/seats +1, rounded up or another formula, as shown. Hagenbach-Bischoff proposed a quota that is "the whole number next greater than the quotient obtained by dividing , the number of votes, by " (where n is the number of seats).
Due to the use of fractions in many STV systems today, rounded-off variants of the Droop and Hagenbach-Bischoff quota may not be needed.
The Britton or Newland-Britton quota, sometimes called the "exact Droop" quota, is not rounded off and is slightly smaller than the Droop quota as Droop originally proposed it. Its formula is vote/seats plus 1, with no rounding off or addition. Although it is smaller than Droop, which is billed as the lowest possible workable quota, N-B is said to be workable according to its originators, R.A. Newland and F.S. Britton.
It is un-necessary to ensure the quota is larger than vote/seats plus 1.
When using the exact Droop quota (votes/seats plus 1) or any variant where the quota is slightly less than votes/seats plus 1, such as in an unusual formula shown above (votes/seats plus 1, rounded down), it is possible for one more candidate to reach the quota than there are seats to fill. However, as Newland and Britton noted in 1974, this is not a problem: if the last two winners both receive a Droop quota of votes, it would mean a tie. Rules are in place to break a tie, and ties can occur regardless of which quota is used.
Even the Imperiali quota, a quota smaller than Droop, can work as long as rules indicate that relative plurality or some other method is to be used where more achieve quota than the number of empty seats. Auxiliary rules are needed where quota is below the level of H-B quota just in case more candidates garner quotas than can be given seats. This is especially true when quotas are used in list PR situations with large DM.
(The difference is that under Droop some candidates are declared elected at the end when they have vote totals fewer than quota, while under Imperiali some candidates are declared elected at the end when they have vote totals larger than quota, the amount that early winners received.
Under Droop some winners have less, under Imperiali some winners have more!)
Confusion with the Hare quota
The Droop quota is sometimes confused with the more intuitive Hare quota. This is discussed in Comparison of the Hare and Droop quotas.
The Droop quota is today the most popular quota for STV elections.
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see these books by Droop:
“On the Political and Social Effects of Different Methods of Electing Representatives,” published originally in 1869
(not easy to find today, unfortunately -- not available in Hathi Trust online or by google search)
Proportional representation as applied to the election of local governing bodies.
Published 1871 (Hathi trust online)
excerpts:
As every representative is elected to represent one of these two parties, the nation, as represented in the assembly, appears to consist only of these two parties, each bent on carrying out its own programme. But, in fact, a large proportion of the electors who vote for the candidates of the one party or the other really care much more about the country being honestly and wisely governed than about the particular points at issue between the two parties; and if this moderate non-partisan section of the electors had their separate representatives in the assembly, they would be able to mediate between the opposing parties and prevent the one party from pushing their advantage too far, and the other from prolonging a factious opposition.
With majority voting they can only intervene at general elections, and even then cannot punish one party for excessive partisanship, without giving a lease of uncontrolled power to their rivals.
...
It will, I believe, hardly be disputed, that the claim of a representative assembly to have the decisions of a majority of its members accepted as the decision of the whole country, depends upon the theory that these decisions do in general correspond to what the majority of the whole body of electors in the country would decide, if they had leisure sufficiently to investigate each of the questions to be decided, and an opportunity to vote upon it.
…a representative assembly in which all parties and sections of parties and all diversities of opinions are represented proportionally, will be much easier to deal with, than an assembly in which the particular differences of opinion upon which the division into two parties is founded, are represented to an exaggerated degree, while subordinate divisions of parties and the various opinions existing upon other questions are only represented by the chance opinions of individual members, and not by members authorised to speak upon these points in the name of their constituents.
...
…even if the number of politically neutral voters continued unaltered, but in fact with single voting applied to constituencies returning five or more members a-piece, each elector will have at least six different candidates with six different sets of opinions to choose between; and, therefore, if he has any political opinions at all, he will be able to find at least one candidate he cares to vote for…
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Here in even bolder style is what the Wikiepda article Droop quota should say:
In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff) is the minimum number of supporters a party or candidate needs to receive in a district to guarantee they will win at least one seat in a legislature.[1][2]
Just as any candidate with more than half of all votes is guaranteed to be declared the winner in single-seat election, any candidate who holds at least a Droop quota's worth of votes is guaranteed to win a seat in a multiwinner election. (Even if there is no official electoral threshold to that effect, there is no way for all the seats to be filled by others each having more votes than that.)[2]
Besides establishing winners, the Droop quota is used in many electoral systems to establish how many votes remain with a successful candidate and thus how many of their votes are surplus votes and available to be transferred to other candidates in order to prevent them from being wasted. Such transfers are done in STV, quota-based proportional Largest remainder method and expanding approvals systems.[2]
The Droop quota was first devised by the English lawyer and mathematician Henry Richmond Droop (1831–1884), as an alternative to the Hare quota.[3] As Droop put it, "the whole number next greater than the quotient obtained by dividing mV , the number of votes, by n + 1, will be called the quota."[4]
Later, Eduard Hagenbach-Bischoff (1833-1910) also wrote on the quota in his studies, Die Frage der Einführung einer Proportionalvertretung statt des absoluten Mehres (1888) and Die Verteilungsrechnung beim Basler Gesetz nach dem Grundsatz der Verhältniswahl (1905). Both Droop and Hagenbach-Bischoff gave their quota as some number just larger than votes/seats plus 1.[5]
Today, the Droop quota is used in almost all STV elections, including those in Australia,[6] the Republic of Ireland, Northern Ireland, and Malta.[7] It is also used in South Africa to allocate seats by the largest remainder method.[8][9]
Standard Formula
total vote/s𝑘+1 plus 1 or rounded up.
Sometimes, the Droop quota is written as a share of all votes, in which case it has value 1⁄k+1, plus 1.
A candidate who, at any point, holds at least a Droop quota's worth of votes is guaranteed to win a seat.[16]
Derivation
The Droop quota can be derived by considering what would happen if k candidates (who we call "Droop winners") have achieved the Droop quota. The goal is to identify whether an outside candidate could defeat any of these candidates. In this situation, if each quota winner's share of the vote equals 1⁄k+1 plus 1, while all unelected candidates' share of the vote, taken together, would be less than 1⁄k+1 votes. Thus, even if there were only one unelected candidate who held all the remaining votes, they would not be able to defeat any of the Droop winners.[2] Newland and Britton noted that while a tie for the last seat is possible, such a situation can occur no matter which quota is used.[10][17]
Example in STV
The following election has 3 seats to be filled by single transferable vote. There are 4 candidates: George Washington, Alexander Hamilton, Thomas Jefferson, and Aaron Burr. There are 104 voters, but two of the votes are spoiled.
The total number of valid votes is 102, and there are 3 seats. The Droop quota is therefore 1023+1=25.5. Rounded up, that is 26.[18] These votes are as follows:
preferences marked | 45 voters | 23 voters | 22 voters | 10 voters |
1 | Washington | Jefferson | Burr | Hamilton |
2 | Hamilton | Burr | Jefferson | Washington |
3 | Jefferson | Washington | Washington | Jefferson |
First preferences for each candidate are tallied:
Washington: 45
Jefferson: 23
Burr: 22
Hamilton: 10
Only Washington has at least 26 votes. As a result, he is declared elected. Washington has 19 excess votes that are now transferred to their second choice, Hamilton. The tallies therefore become:
Washington: 26
Jefferson: 23
Burr: 22
Hamilton: 29
Hamilton is elected. There is still one seat remaining to be filled so his excess votes are transferred. Thanks to the four vote transfer from Hamilton, Jefferson accumulates 27 votes to Burr's 22 and is declared elected. That fills the last empty seat.
If ties happen, pre-set rules deal with them, usually by reference to whom had the most first-preference votes.
Under plurality rules (such as block voting or SNTV), Burr would have been elected to a seat in the first round. But under STV he did not collect any transfers and Jefferson was proven to be the more generally supported candidate.
Burr, as a representative of a minority, would have been elected if his supporters numbered 26, but as they did not and as he did not receive any transfers from others, he was not elected and his voice was not heard in the chamber following the election.
Different versions of Droop
At least six different versions of the Droop quota appear in various legal codes or definitions of the quota.[19] Some claim that, depending on which version is used, a failure of proportionality may arise in small elections.[10][17]
Common variants include:
Droop:⌊votes/seats+1+1⌋
Hagenbach-Bischoff:⌈votes/seats+1⌉ or sometimes:⌊votes/seats+1⌋+1
Unusual:⌊votes/seats+1⌋ ⌊votes/seats+1 + 1/2⌋
Accidental:
votes+1seats+1
Droop and Hagenbach-Bischoff derived new quota as a replacement for the Hare quota (votes/seats). Their quota was meant to produce more proportional results by having the quota as low as thought to be possible. Their quota was basically a number just larger than votes/seats plus 1.
This formula may yield a fraction, which was a problem as early STV systems did not use fractions. Droop went to votes/seats plus 1, plus 1, rounded down, as shown.[2] Hagenbach-Bischoff went to votes/seats +1, rounded up or another formula, as shown. Hagenbach-Bischoff proposed a quota that is "the whole number next greater than the quotient obtained by dividing 𝑚𝑉, the number of votes, by 𝑛+1" (where n is the number of seats).[19]
Due to the use of fractions in many STV systems today, rounded-off variants of the Droop and Hagenbach-Bischoff quota may not be needed.
The Britton or Newland-Britton quota[10][a], sometimes called the "exact Droop" quota, is not rounded off and is slightly smaller than the Droop quota as Droop originally proposed it. Its formula is vote/seats plus 1, with no rounding off or addition. Although it is smaller than Droop, which is billed as the lowest possible workable quota, N-B is said to be workable according to its originators, R.A. Newland and F.S. Britton.[22]
It is un-necessary to ensure the quota is larger than vote/seats plus 1. When using the exact Droop quota (votes/seats plus 1) or any variant where the quota is slightly less than votes/seats plus 1, such as in an unusual formula shown above (votes/seats plus 1, rounded down), it is possible for one more candidate to reach the quota than there are seats to fill.[19] However, as Newland and Britton noted in 1974, this is not a problem: if the last two winners both receive a Droop quota of votes, it would mean a tie. Rules are in place to break a tie, and ties can occur regardless of which quota is used.[10][17] Even the Imperiali quota, a quota smaller than Droop, can work as long as rules indicate that relative plurality or some other method is to be used where more achieve quota than the number of empty seats. Auxiliary rules are needed where quota is below the level of H-B quota just in case more candidates garner quotas than can be given seats. This is especially true when quotas are used in list PR situations with large DM.[23] (Rose, International Encyclopedia of Elections, p. 233)
Spoiled ballots should not be included when calculating the Droop quota. Some jurisdictions fail to specify in their election administration laws that valid votes should be the base for determining quota.[citation needed]
Confusion with the Hare quota
The Droop quota is often confused with the Hare quota. While the Droop quota gives the number of voters needed to mathematically guarantee a candidate's election, the Hare quota gives the number of voters represented by each winner by exactly linear proportionality.
As a result, the Hare quota is said to give somewhat more proportional outcomes,[24] by having large parties waste more votes and thus promoting representation of smaller parties. But sometimes under Hare a majority group will be denied the majority of seats, thus denying the principle of majority rule in such settings as a city council elected at-large. By contrast, the Droop quota is more biased towards large parties than any other admissible quota.[24] The Droop quota sometimes allows a party representing less than half of the voters to take a majority of seats in a constituency.[24][2]
The Droop quota is today the most popular quota for STV elections.
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