Talk:Droop quota

message: Im no theoretical math person, and im not good at coming up with them, but can follow them well.
Part of the main page should explain WHY it is "SEATS+1" and not just "SEATS".  And for that matter why are we putting faith in the droop? Im sure that there are other mathematical ways of doing this stuff that have equally as goofy names.  I just want to see the straight proof. 
It is obviously not the simplest thing in the world, or it would not have taken a mathmatician in the 1800's to figure it out.
EVERYTHING that follows on this "droop page" is based upon this formula, therefore it should really be explained in the most most detail.  Sure the example makes sense if you accept the droop formula at face value, but it does not make sense as a whole without a section about the origins of the actual foundations. 
Thanks, thats my 2 cents
(posted by 216.232.197.30)
----
Is this the statement you want proven?
::This gives the Droop Quota the special property that it is the smallest integral quota (although not the smallest quota) which guarantees that the number of candidates able to reach this quota cannot exceed the number of seats.
That statement's not too hard to show. Suppose that the number of candidates that reached the quota did exceed the number of seats: if you added up the votes for those candidates, then, you'd get more votes than there were in the election, so that can't happen.
And suppose that you used a quota less than the Droop quota; then you can imagine an election in which n+1 candidates get votes/(seats+1) votes, and they would all get a seat for having more than that quota, but then you've assigned more seats than you have. So the Droop quota is the smallest quota that works this way.
But perhaps you want some intuition on why the Droop quota works like this.
Suppose you're having an election with only one seat and two candidates. (Yes, then there's no need to use STV at all, but bear with me). The \frac{votes}{seats+1} + 1 quota comes out to be \frac{votes}{2} + 1, or in other words, it takes 50% + 1 votes, a majority, to get the seat. This is how you'd expect the election to work.
If you used \frac{votes}{seats}+1 instead, then it would require 100% + 1 of the votes to get the seat, and that's impossible.
Why do we use 50% + 1 in majority rule? Because it's the smallest number where it's impossible for more than one candidate to get 50% + 1 of the votes (that would make 100% + 2 votes).
If you include more candidates, but keep the one seat, you get an Instant Runoff election, where one candidate is guaranteed to get 50% + 1 after all the transfers happen.
Now generalize this. If there are two seats available, then you should be able to get a seat with 33% + 1 of the votes, because it's impossible for 3 or more seats to be assigned that way. At most two seats will be assigned. Likewise, you can get one of 3 seats with 25% + 1. And so on.
So if there are n seats available, it should take \frac{votes}{n+1} + 1 votes to get a seat. It's not really theoretical math, it's just taking advantage of the fact that you can't get more than 100% of the votes.
And since you're using Single Transferable Vote, you can keep transferring votes until someone gets a Droop quota, so all the seats will be assigned.
RSpeer 01:37, May 3, 2005 (UTC)
__________________________________________________________________
That explanation was excellent. Somehow I just assumed one would need true theoretical math to explain it.  I vote that that explanation, or a summary of it be on the main page.  Thanks again.
----
In systems that transfer fractional votes, it's reasonable to use the exact quota, \frac{votes}{n+1} with no rounding, the rounding being primarily a convenience for manual tabulation of votes. When the exact fractional quota is used, two approaches are possible.
In one interpretation, the algorithm requires that, to win a seat, a candidate must achieve a vote count strictly greater than the fractional quota. In an election for four seats, for example, it's not possible for five candidates to each have more than 1/5 of the votes.
The other alternative is used by Meek (see Meek's method). Meek requires that a candidate merely meet the exact fractional Droop quota, and points out that if there's a five-way tie for four seats, it's a true tie, and the tie should be broken by lot.
The difference between the exact quota and the rounded quota tends to be inconsequential for elections with very large quotas, but in small elections it's more likely to make a difference, as it did this July when the Green Party of the United States elected four Steering Committee seats with 94 ballots. The count with the exact quota of 18.8 yielded a different result than with the rounded quota of 19.
An excellent reference for STV details is Voting matters.
Jlundell 01:13, Aug 17, 2005 (UTC)
I mad a change to the wording of the explanation of the (more math-like version of the) formula, it previously said ...largest integer less than.. and I changed it to ...smallest integer greater than...; the earlier version described the result of the formula prior to applying the + 1 at the end, but not the Droop Quota (and would, if used as a quota, allow more candidates to meet the quota than seats were available!)
--Cmdicely 03:26, 24 July 2006 (UTC)
thread_title:  math proof of the "seats +1" 

message: I have removed a link to quota as of the disambiguation page repair process.
If a link is needed for quota, please feel free to re-insert it to the proper page. I don't think it is necessary.
FirefoxRocks 02:09, 12 December 2006 (UTC)
thread_title:  Quota 

message: The following request to move a page has been added to Wikipedia:Requested moves as an uncontroversial move, but this has been contested by one or more people. Any discussion on the issue should continue here. If a full request is not lodged within five days, the request will be removed from WP:RM.  —Stemonitis 10:04, 28 June 2007 (UTC)
replies: text: Who said anything about the D?  I'm not proposing a  here.  In this case, the simple fact of the matter is that "quota" is a common noun (whether it is named for a person called Droop or something else, it is still only a quota method named for it).  81.104.175.145 12:14, 29 June 2007 (UTC)
replies: text: Sorry — my mistake. I meant Q, of course. --Stemonitis 05:47, 30 June 2007 (UTC)
replies: text: In the absence of any other objection, can we now move as uncontested?  81.104.175.145 20:28, 1 July 2007 (UTC)
replies: text: The most appropriate course of action would be to lodge a full move request. The previous discussions show that the move cannot be treated as uncontroversial, so wider discussion (or at least the opportunity for it) is needed. A lack of opposition in this section is not necessarily indicative of anything. --Stemonitis 07:25, 2 July 2007 (UTC)
thread_title: Contested move request

message: Droop Quota → Droop quota — Article about a quota formula (used in proportional representation) named for someone called Droop, as opposed to some entity called the "Droop Quota", hence "quota" is not proper but common in this context, and should not be capitalized per WP:MOSCL. —81.104.175.145 13:40, 5 July 2007 (UTC)
thread_title: Requested move

message: Currently, the two formulae present (\left( \frac{Total Valid Poll }{\left( Seats+1 \right)} \right) + 1 and \left\lceil \frac{Votes}{Seats+1} \right\rceil) do not give the same value, even if we assume an integer number of votes. For a counterexample, have two votes and one seat - the first formula gives 2 (a majority is required to win), while the second gives 1 (two people can meet quota). User:Evercat's formula above (\left\lfloor \frac{Votes}{Seats+1} \right\rfloor+1 \cong \left\lfloor \frac{Votes}{Seats+1} + 1 \right\rfloor) gives the correct result (2) if we only allow integer votes (note that this is the first formula, rounded down). If we allow fractional votes, then I think it's sufficient to exceed votes/(seats+1), i.e. to be elected, a candidate must get more than 1 vote (so 1.1 will suffice). Elektron 15:19, 22 August 2007 (UTC)
thread_title:  Erroneus formula? 

message: This article discusses only the method to apply the DQ to a Single Transferrable Vote system. I came here wanting to learn how it is used with a Closed Party List election as in South Africa. Roger (talk) 14:28, 24 April 2009 (UTC)
thread_title:  Party list system 

message: Can anyone recommend a droop quota calculator, either as an online tool or downloadable software?
One should be linked here in external links.  Blue Rasberry (talk) 15:22, 1 June 2016 (UTC)
replies: text: Any basic (four-function) calculator will work. You just divide the number of votes by (one plus the number of seats). Closed Limelike Curves (talk) 03:19, 13 February 2024 (UTC)
thread_title:  Software recommendation needed 

message: Hello fellow Wikipedians,
I have just modified one external link on Droop quota. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
*Added archive https://web.archive.org/web/20100624102008/http://polmeth.wustl.edu/analysis/vol/8/PA84-381-388.pdf to http://polmeth.wustl.edu/analysis/vol/8/PA84-381-388.pdf
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
Cheers.—InternetArchiveBot (Report bug) 05:11, 14 September 2017 (UTC)
thread_title:  External links modified 

message: It would seem that Proportionality for Solid Coalitions should be merged into this page into a "Droop proportionality criterion" section.  I discovered that page only via the Droop proportionality criterion redirect.  I don't know the subject well enough to make the merge, but I'm hoping someone takes up the cause. -- RobLa (talk) 04:02, 3 December 2018 (UTC)
thread_title:  Merging Proportionality for Solid Coalitions page 

message: Put simply, Droop quota is the number of valid votes divided by one more than the number of seats to be filled, rounded down, and then add 1.
Sources differ as to the exact formula for the Droop quota. The Republic of Ireland uses:
* = Total number of valid (unspoiled) votes cast in an election.
* = total number of seats to be filled in the election.
* refers to the floor or integer portion of the number, sometimes written as 
* It is important to use the Total Valid Poll, which is arrived at by subtracting the spoiled and invalid votes from the total poll.
The Droop quota is the smallest number of votes that guarantees that no more candidates can reach the quota than the number of seats available to be filled. In a single winner election, in which STV becomes the same as instant-runoff voting, the Droop quota becomes a simple integral majority quota–that is, it will be equal to a simple majority of votes. The formula follows from the requirement that the number of votes received by winning candidates (the Droop quota) must be greater than the remaining votes that might be received by an additional candidate or candidates (the Droop quota – 1). 68.150.205.46 (talk) 03:01, 17 May 2024 (UTC)
thread_title:  Formula 

message: article wrong and self-contradictory:
first line says Droop is minimum required ... but then later says candidate must get more than Droop to be elected. 
anyways it is possible to be elected with less than quota under STV-- it happens alot in Ireland and malta, in fact anywhere except where full-preferential ranking is required.. 
"Common errors" lists "votes/(seats plus 1)" but that is what article defines as Droop.
(the two columns are confusing)
the left-most "unworkable" is perfectly fine and workable - anyone who get at least quota is elected. it will be likely someone with at least 1/2 more than such a quota but having quota set as votes/(seats 1), plus 1/2 is just fine if unorthodox.
In "Example"  after Washington's election and transfer away of his surplus votes, Washington is left with 25 which is wrong. Although that is wrongly called Droop, even the article admits he needed at least 26 to be elected. He should be left with 26 if we use H-B and Droops' own words to set Droop. they both say droop/H-B is a number greater than votes/(seats plus 1)
actually article should say this:
In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff or Newland-Britton quota) 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. 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).
With each successful candidate 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 candidates will be elected with more or less than quota.)
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. 
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.
Today the Droop quota is used in almost all STV elections, including those in the Republic of Ireland, Northern Ireland, Malta, and Australia.[citation needed] It is also used in South Africa to allocate seats by the largest remainder method.[citation needed]
Standard Formula
The exact form of the Droop quota for a �-winner election is given by the formula:
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 a number greater 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]
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. 2604:3D09:8880:11E0:0:0:0:7044 (talk) 20:56, 17 May 2024 (UTC)
thread_title:  This article is wrong and self-contradictory