Talk:Droop quota

Math discussion

what exactly does $\left({Votes \over (Seats+1)}\right)+1$ equal? Kingturtle 23:29 18 Jun 2003 (UTC)

The quota you need to reach in Proportional Representation using the Single Transferable Vote to get elected.

Please stop changing this to lower case. It is a proper noun. It is the formal name of a formal electoral quota. It is not a generic term but a specific name of a specific item. FearÉIREANN 00:01 19 Jun 2003 (UTC)

So if there are 40 seats, and you get 10 votes, then the quota is 1.243? Kingturtle 00:04 19 Jun 2003 (UTC)

except that PR.STV does not operate on the basis of such large constituencies. The largest in Ireland at present is 5 seats. I think the largest ever was 8 seats. I don't think Malta which also uses PR.STV has larger constituencies. For example, if you have 100,000 votes cast in a constituency that has 5 seats, that produces (100,000/6) + 1 = 16,667. So the quota each candidate needs to reach is 16,667. Using PR.STV, each candidate when elected has that proportion of votes they have over the quota redistributed through lower preferences. If no candidate is elected, the bottom candidates are eliminated. Eventually through the distribution of surpluses or eliminations, five candidates will reach the quota, with not enough votes left for any candidate to reach the quota a sixth time. (If they did that would cause a problem as there are only 5 seats. So the quota is constructed to ensure that it is mathematically impossible for anyone other than the five candidates elected to reach the quota. It works quite simply using PR.STV. FearÉIREANN 00:14 19 Jun 2003 (UTC)

That explanation is most helpful. It should be incorporated into the article. Kingturtle 00:18 19 Jun 2003 (UTC)

Don't complain, fix! ;-) Evercat 00:25 19 Jun 2003 (UTC)

It is a slight bit more complicated; eg how do they decide which votes are your surplus votes, do they always go for distributing a surplus or eliminating a low candidate, and also sometimes they eliminate a number en bloc. But it makes for riveting TV. Unfortunately Ireland is replacing manual counts which are slow and nail-biting (I covered the last Irish general election count in some Dublin constituencies for a Sunday newspaper until 2am!) with electronic voting, which can give counts in minutes. A lot of us election-anoraks will miss the fun, especially as there is a phenonemon known as a tallyman who is a professional vote watcher. They can watch ballot boxes being opened and incredibly predict how four of the five seats, sometimes five of the five, will go before the count even starts. So ireland would have three or four hours of tallymen's predictions from around the country, then 10-15 hours of nail-biting counts, often with demands of recounts and rechecks.

Here's an example of how it could work, using the above quota:

say I get 17,000 Number 1 votes. I would be declared elected. I would have a surplus of 333. Those 333 votes would be examined to see where there Number 2s went. If say 200 went to to Kingturtle, and you were previously at 16,660. In the second count, your total would increase by 200 to 16,860. You would be declared elected, having won the second seat. Your surplus (amount over the quota) would be 193. They would then be examined for Number 2s in the third count. If after their distribution no candidate was elected, the bottom candidate would have his/her total checked. Say mav had 500 votes. The returning officer would announce that no-one had been elected at the end of the third count, and he was now proceeding to the fourth count, the elimination of Mav and the distribution of his votes. This would go on until either five candidates had through surplus distribution or eliminations reached the quota or until the difference between the last three remaining candidates was such that even if the bottom candidate was eliminated and all their votes sent to the second placed candidate, that second placed candidate could not get ahead of the first placed candidate, in which case the last candidate would be declared elected "without having reached the quota".

Elections using PR.STV can be very exciting, with counts going on for hours, often until the ninth or tenth count. Its beauty is that most people's votes help get someone elected, whereas in plurality voting, often most people's votes have no impact whatsoever, as the candidate who wins wins with the biggest minority vote. If my first choice candidate is eliminated, they look at my ballot paper to see who was my second choice, if she is eliminated, they look at my third choice, etc etc until eventually my vote may help elect someone, even if it to get the eighth place candidate on my ballot elected if I prefer him to my nineth choice. It is one of the fairest electoral systems around, and also produces some of the most exciting electoral counts imaginable, often with the last seat going down to the wire with 50 votes or sometimes as little as 2 votes deciding who gets the last seat. FearÉIREANN 00:39 19 Jun 2003 (UTC)

===============

here's some truths: Actually what happened in historic practice of Droop quota is

votes/seats plus 1) rounded down then add 1

or votes/seats plus 1, rounded up.

A candidate whose vote tally equalled or exceeded that amount was elected.

there was no way that more could be elected than the number of open seats.

STV has been used to elect as many as 21 in NSW every election since 1991. and 25 in the 1925 Irish Senate vote. City of Winnipeg (Canada) elected ten in one contest from 1920 to 1948.

Soon even more --2025 - Western Australia scheduled to elect 37-member Senate (Legislative Council) in state-wide district.

Question above is off-base: "if there are 40 seats, and you get 10 votes, then the quota is 1.243?"

first STV means each voter has one vote.

secondly, if questions was -- if there are 4000 votes, and you are electing 10 seats, what is Droop? Droop quota is 364.

Droop quota is arrived at by rounding off to give whole number amount.

Of course it is possible to pass quota by a fraction if Gregory method is used to transfer surplus votes to another candidate. Tom Monto — Preceding unsigned comment added by 2604:3D09:8880:11E0:C8E0:3DDF:A94D:41CB (talk) 19:17, 24 April 2024 (UTC)[reply]

Next question from an inquiring mind. Droop? The name of the person who developed this system? Or does it refer to not allowing the quota to DROOP below a certain percentage? Do tell. :) Kingturtle 00:32 19 Jun 2003 (UTC)~

That I don't know. I think it may be the mathematician who first drew up the formula but that is just a rough guess. FearÉIREANN 00:39 19 Jun 2003 (UTC)

Looks to me like the capitalization is pretty equally distributed between "Droop Quota" and "Droop quota", according to a google search. One site even calls it "Droop's quota." Also:

Henry Richmond Droop designed a quota to avoid under representing a majority.

(from this site). So it would appear that the D should definitely be capitalized, but the Q need not be. -- Wapcaplet 14:14 24 Jun 2003 (UTC)

Not so.

As I have learned from experience, google searches are regularly garbage. For example, tens of thousands of entries say that the Prince of Wales's surname is Windsor. Hundreds say Mountbatten-Windsor. According to Buckingham Palace, it is MW. So tens of thousands of google hits are bullshit. I could fill this page up to 32K with 'facts' on google that are rubbish.
According to every academic textbook I have ever used (and I taught students about DQ for 8 years) it was capitalised. Some American english sources tend to decapitalise titles such as this. But such an approach is regarded as 'semi-literate'. As the US does not even use DQ or PR.STV, academics do not pay any heed to America's fixation with lower-case. It is treated as a proper noun and capitalised. FearÉIREANN 01:28 25 Jun 2003 (UTC)

We should get this right, because we're Number 1! Evercat 14:25 24 Jun 2003 (UTC)

I removed some of the parentheses from the formula. According to a note on Evercat's talk page, all of the parentheses must be there, but I would like some clarification as to why this is so. All of the parentheses seem redundant to me (as they would to anyone with mathematical experience). Also, Jtdirl, if you have taught the DQ for 8 years, how come you didn't know who it was named after? I found that out in 20 seconds, and I've never heard of the Droop quota before today.

Finally, if the parentheses must be there, then why are they not present in the example given later in the article?

(100,000/6) + 1 = 16,667

Shouldn't that be:

(100,000/(5+1)) + 1 = 16,667

or

(100,000/(6)) + 1 = 16,667

-- Wapcaplet 14:27 24 Jun 2003 (UTC)

Actually I wrote that bit... (based on JT's informal example) Evercat 14:29 24 Jun 2003 (UTC)

Ah. I guess the ultimate question is, how is:

\left({Votes \over {Seats+1}}\right)+1

ambiguous in any way? (unless we're talking about people who don't know that division has precedence over addition, or that the stuff on the bottom of the horizontal line has to be calculated before dividing the stuff on the top of the horizontal line by it. But then again, those people probably wouldn't know that parentheses take precedence over both, and/or wouldn't understand the remainder of the article.) How can this formula get a student failed? It expresses precisely the same quantity as the versions with extra parentheses. -- Wapcaplet 14:37 24 Jun 2003 (UTC)

I suppose some of the confusion may arise from the differences in how the TeX stuff is rendered (as PNG or HTML). Maybe we should format it without any math markup, to remove all doubt about how it's rendered and how it is to be calculated. Such as:

Votes / (Seats + 1) + 1

-- Wapcaplet 14:45 24 Jun 2003 (UTC)

BTW Wapcaplet, would a mathematician also be happy with this?

{Votes \over {Seats+1}}+1

That's really the version that was most contoversial... Evercat 15:24 24 Jun 2003 (UTC)

And it is totally unambiguous, in my opinion, to anyone who has ever been exposed to basic arithmetical notation. -- The Anome 16:59 24 Jun 2003 (UTC)

The formula is always written with the brackets and never without. The reason is because students constantly don't understand the formula without them; ie the order in which the maths are done. Remember the students using the formula aren't mathematicians but students of political science or history. Without the brackets, people not understanding it sometimes add the final +1 to the votes total, or the seats +1. As a result, the brackets are thought so important that students who write the formula without the brackets in many colleges are automatically failed unless their answer in an exam shows they do know the order in which the maths are done. Recently a book about an Irish election had its first print-run pulped because the typesetter left out the brackets. A second print run was ordered with the brackets in place. It isn't a case of the brackets being optional, a matter of opinion. If they aren't there the formula is dismissed as wrong and if wiki can't even get the Droop Quota right it would instantly be dismissed by political scientists as an amateurish sourcebook that their students should not use. FearÉIREANN 18:16 24 Jun 2003 (UTC)

Well, not always. My google searching turned up far more instances of the formula being written without brackets than it did of those with the brackets. And one does not need to be a mathematician to understand (and correctly interpret) the formula without the extra brackets; we're talking about elementary arithmetic here. If we insert extra parentheses into every formula on Wikipedia that students have had trouble understanding, it's gonna start looking like Lisp. I would be much obliged if you could provide an external source stating that the brackets are mandatory.

More often than not, google searches are useless and unreliable. See above. FearÉIREANN 01:30 25 Jun 2003 (UTC)

I didn't imply that they were useful or reliable; I just meant it as an example of some cases where the formula is written without brackets. Could you please provide a better source so I can confirm your claims? -- Wapcaplet 01:33 25 Jun 2003 (UTC)

An anonymous edit brings up another interesting question: The fraction may result in a decimal value; is this portion truncated, rounded off, or what? The formula does not make this clear, without the addition (as the anon contributor did) of floor or ceiling indicators. -- Wapcaplet 01:22 25 Jun 2003 (UTC)

Well, after a number of false starts at math markup, I think I got something everyone can be happy with. Might need a bit of rephrasing here and there, and I've guessed at the rounding-down thing until it can be confirmed (btw, the example seems to round up, which may be in contradiction of the floor notation). -- Wapcaplet 01:51 25 Jun 2003 (UTC)

The example is correct. You apparently missed the +1 part.

Ah, so it is. One month out of college and I already forgot how to add... -- Wapcaplet 17:20 25 Jun 2003 (UTC)

Re FearÉIREANN's "I taught students about DQ for 8 years" and yet not knowing the origin of the term, we see the hazards of waving academic credentials about. Wikipedia is just as open to input from the incompetent academic as from the qualified one, which is why we should be citing from the published works of accepted authorities rather than trying to claim personal authority. Any competent academic should be able to reel off the relevant chapters (if not page numbers and paragraphs!) of the authorities' works that are the basis of any assertion. Stan 04:27 25 Jun 2003 (UTC)

ow! i thought Henry Richmond Droop was a piss-take -- but no, apparently he's real. Tangerine

Well, he showed up in numerous different sites on a google search. Unless they're all misinformed, he's probably real. -- Wapcaplet 17:43 25 Jun 2003 (UTC)

I would be interested to find an authoritative source on how the formula is calculated. The formula given in the article predominates in the google searches I've done; I have found one site which states that it's:

total valid vote plus one divided by the number of seats plus one

(from a Tasmanian House of Assembly site, which is where I got some of the additions to the article). Anyhow, this explanation of the quota is highly ambiguous, since it can be interpreted as:

(total valid votes plus one) divided by (the number of seats plus one)

(total valid votes) plus (one divided by the number of seats) plus one

(total valid votes plus one) divided by (the number of seats) plus one

All of which are wrong, in comparison with the current formula; a fine example of the ambiguity of the English language :-) Anyway, this may cast into suspicion the other bits about Droop himself, so those may need editing by someone in the know.

Also, I found a Green Party of Canada site which states that the formula is:

the number obtained by dividing the total number of valid votes cast in a constituency by a number which is one more than the number of places to be filled (members to be elected) and increasing the result to the next whole number

Which is subtly different from the interpretation we've used of rounding down, then adding one. This statement is worded more precisely than the previous one, and the only interpretation I can get out of this is that the quota is:

\left\lceil {\frac {Votes}{Seats+1}}\right\rceil

Mathematically, this is very slightly different from the round-down-then-add-one version. Specifically, if the part inside the brackets/floor/ceiling comes out to be an integer, this formula will give a result one less than the formula(s) used in the article.

Edit: Unless by "increasing the result to the next whole number" they mean increasing it even if it's already a whole number, I just realized. So maybe it is correct. -- Wapcaplet

Once again, not having studied (or even heard of) this quota before yesterday, I would appreciate some pointers towards a more authoritative source. By the way, IANAM, but I have a great appreciation for mathematically unambiguous (and preferably correct) formulas, even if they are in an article on politics. No sense in confusing even the non-mathematicians. -- Wapcaplet 17:43 25 Jun 2003 (UTC)

This last version can't be correct. Imagine there are 100,000 votes and 4 seats this time. Under this version, that gives a quota of 20,000. But it would be possible for 5 candidates to meet that quota. Evercat 17:56 25 Jun 2003 (UTC)

No. It is perfectly straightforward and foolproof, once the parentheses are left in. You increase the number of seats by one, divide the total votes by that number, then add one to the final total. That means 100,000 divided by 5 = 20,000, +1 gives the quota of 20,001. So that means that when 4 candidates reach 20,001, there are 19,996 votes left, not enough for a another quota. It is that straight forward. There is no question of rounding up or rounding down. The Droop Quota is only used with PR.STV and that is only used in the Republic of Ireland and Malta and it has one straight-forward formula. The parentheses are included to avoid the very confusion that seems to be cropping up on this page. FearÉIREANN 18:12 25 Jun 2003 (UTC)

Most of my confusion, at least, stems from the problem of rounding. There is a question of rounding! Is it not conceivable that there are, hypothetically, 1000 votes and 6 seats? (1000 / (6+1)) + 1 = 143.85714... Should that be rounded up to 144, or truncated/rounded down to 143? No amount of parentheses will clear this up. -- Wapcaplet 18:46 25 Jun 2003 (UTC)

Yes, I mean Wapcaplet's formula above (without the +1) must be wrong. Evercat 18:13 25 Jun 2003 (UTC)

But I don't understand "there is no question of rounding up or rounding down". There must be, since division need not leave a whole number. If there are 99,999 votes and 4 seats, that leaves a quota of 20,000.8 so is that rounded up to 20,001 or down to 20,000? I'm presuming down, since in this case 20,000 is the lowest number that fulfils the requirement of not allowing more winners than seats. Evercat 18:42 25 Jun 2003 (UTC)

Having thought about it, I think the version that rounds down first then adds one is always the lowest number that doesn't allow more candidates to win than there are seats... Evercat 17:58 25 Jun 2003 (UTC)

Lipjhart has some interesting comments on the subject (LR stands for Largest Remainder):

Like LR systems, STV requires the choice of a quota, which in practice is always the Droop quota. However, it is defined in a slightly different way from the LR Droop quota: the quotient arrived at by dividing the total vote by the number of seats plus 1 is rounded up or, if the quotient is an integer, 1 is added. In the example of Table A.3, the LR Droop quota would be 25, but the STV Droop quota is 26.

This is the most precise explanation I've seen yet. It appears that it is different depending on whether it's being used in the context of STV or LR. The LR quota, according to this author, rounds down and stops before adding one. So perhaps such a distinction should be made in the article, as well. -- Wapcaplet 18:16 25 Jun 2003 (UTC)

Reverting my version was uncalled for. I think it's very clear on all the issues discussed here.

Your version is factually wrong, your use of capitalisation is incorrect, you use the wrong quota. Your interpretation is flawed. FearÉIREANN 22:24 25 Jun 2003 (UTC)

Your arrogance is misplaced. Not only is my version correct, it's unambiguous and if you think my quota is different than the current quota as interpreted by the example then I would guess that mathematics isn't your strongest area. The "Droop Quota vs Droop quota" is a minor issue. The latter just outranks the former in search results (and makes more sense as well). And this "parenthesis babysitting" is ridiculous; the division line speaks for itself. And if needed, the correct interpretation of the formula can even be deduced from the sentence following it in my version. So you can see that I've covered all the bases.

This isn't a maths page, it is a page describing a formula used in political science to produce a quota for use in PR.STV. The formula I used is the formula used by political scientists. No other form of formula is used. Your formula may be the same in formal mathematics but this isn't about mathematics it is about a formula used in political science. As to the capitalisation, Droop Quota is treated as a proper noun and is written as such and should no more be decapitalised that President of the United States should be written as President of the united states. And the 'parenthesis babysitting' is simply ensuring that the formula is written as the formula is written and used by political science. Nothing more and nothing less. The problem with your text was that it turned something that is simply (simply!) describing a formula and how it relates to electoral politics into a mini-treatise on the theoretical mathematics behind it. While that has its uses, its effect was to turn a page meant to simply and straight-forwardly explain what the formula means by the people who would be seeking the information (people interested in the practicalities of electoral science) into something so complicated that non-mathematicians would run a mile from it. A daughter article could very well be constructed to analyse in mathematical terms the workings of the formulæ. But what you did inadvertently obscured the simple question of what is the Droop Quota and how is it used, by going in depth into information that people using the formula would not concern the formula, given that it is not used in mathematics but is simply used as part of a process of election. That is why I made the changes, and BTW I think we were both caught in an edit war and that may be the reason that your version was lost. Wiki is going so incredibly slow and I resorted to a cut and paste to save an extra couple of paragraphs I had added in before I got timed out or the modem started disconnecting. (Wiki is really infuriating right now with its slowness).

I do think a detailed of the mathematical nature of DQ might be useful, but be careful that it doesn't turn an article that is not about maths into a largely maths article. That isn't what people using the page would be using it for. The page would mostly be visited by students of electoral processes, not mathematicians (just as turning some maths page into a long treatise on how maths shaped elections might not be of much interest to maths fans and would be unlikely to be visited by political students. A linked daughter article explaining the mathematical theory behind it might make more sense. FearÉIREANN 23:37 25 Jun 2003 (UTC)

May I suggest as a compromise that we go back to the version that had both the brackety version and the mathematically precise version? Evercat 23:28 25 Jun 2003 (UTC)

That is OK with me. As I said we appear to have been caught in an edit conflict when I made changes. FearÉIREANN 23:37 25 Jun 2003 (UTC)

Ok, I'm willing to give in regarding the parenthesis issue, but your formula still suffers from the fact that it's not stated anywhere to round down. And that's hardly a complicated operation, is it? And what exactly is wrong with my example (as I find yours with footnote to be too convulated)? I'm also not convinced with the following claim inside the article:

This quota is designed to avoid the possibility of under-representing the majority.

You are right above the above line. (I've just come to the page to remove the line and put an explanation here!) Basically, PR.STV is designed to ensure that party political support in a popularly elected house of parliament is relatively proportional to a party's percentage support base. Single party constituencies are thought likely to decrease the degree of relative proportionality, with the higher the number of seats per constituency, the greater the degree of proportionality achieved. The DQ is used as the means to achieve a quota which can only be achieved by the number of candidates identical to the number of seats available. For in a five seat constituency it is mathematically impossible for more than five candidates to achieve the quota and so be elected.

re the question - it isn't exactly stated where to round down - you don't. Ever. You never have to round up or down. By adding 1 to the final result when you divide the TVP by (seats +1), you get a number that can never be achieved by more candidates than the number of seats available. Rounding up or down never arises and never can arise.

If that is true, then perhaps it should be stated in the article. Anyone who is familiar with fractions is going to wonder what to do with that fractional part, in the situations where it occurs. If the fractional part is totally immaterial, and should be thrown away, then say that (i.e., round down). Rounding must occur, in some sense, to convert a fractional number to an integral one. Obviously you can't have a fractional part of a vote, or a fractional candidate. It's just that all of us math guys are wondering whether 0.67 of a vote should be counted as 1 vote, or as 0 votes :) -- Wapcaplet 03:09 26 Jun 2003 (UTC)

As to your example: it uses terms that are never used in describing the workings of either the PR.STV or the DQ, its reference to 'rounding' is completely wrong, as is its mention of 'votes with positive weights' , 'depleted', 'floor' , 'decaying'. "Since votes with positive weights may become depleted . . ." what does that mean? I am rewriting my paragraph to make it clearer. FearÉIREANN 03:05 26 Jun 2003 (UTC)

I've re-made the page into the compromise version, which I hope we can all live with. I think this page is an interesting example of a page which is related to both political science and mathematics, and this debate indicative of the different ways of doing things these 2 camps bring. :-) Evercat 02:47 26 Jun 2003 (UTC)

It doesn't really have to be math-like. That rounding bit is quite important though, since it can make the difference of whether or not a candidate can meet the quota in some cases. I'm not sure what the "under-representing the majority" bit was. I got that from another site, but you won't hurt my feelings by removing it :) I like the compromise version. Simple for those who need it, detailed for those who are interested. -- Wapcaplet 03:09 26 Jun 2003 (UTC)

I'm sure we can end the controversy over the rounding if we can just get an answer to this simple question:

Say the Droop Quota formula gives the value 5000.5 : does a candidate need to get 5000 or 5001 votes? Evercat 03:20 26 Jun 2003 (UTC)

If you mean the formula as given by Jtdirl, then the answer is 5000 as 1 is already supposed to be added to it to prevent seats+1 candidates from reaching the quota. So the example you give is a bit ackward and potentially confusing.

Wouldn't it be better if the part of the text starting from the example is integrated with the Single Transferable Vote article instead, as it don't pertain uniquely to the Droop Quota?

I would advise against it. Millions of people cannot understand even the concept of the Droop Quota because their only experience is of electoral systems were quotas don't matter as the winner is the person with the largest number, even if it is the largest minority vote. So it is useful to explain on this page why and what a quota is, that means explaining that some electoral systems require them to function. Wiki is great for links, but one thing I think can be a problem is if we don't give enough info to grasp the basics but rely on links. It is all too east to be puzzled by something, hit a link, read a text, have to hit another one and end up a couple of links away where you started trying to work your way back. It turns reading wiki into attempting to do a jigsaw. I think every page should carry enough basic info so that someone reading the one page knows what the thing is and has a rough idea of the context in which it works. We should aim for each page to have enough info to ensure that someone when they leave that one page could explain the gist of what something is to a friend, not say "em I'm not sure. I have to explore a couple of links more before I can really know what it is all about." We can go into a lot more detail elsewhere but this page should allow someone to be able to walk away without leaving more confused than when they came in. FearÉIREANN 03:59 26 Jun 2003 (UTC)

I think the fundamental problem here is that various references disagree with one another. Therefore, attempting to clear away the confusion by giving a single clear answer is impossible. Some sources say that the Droop quota is the same as the Hagenbach-Bischoff quota, others give different formulae for the two, some even give entirely different formulae for the Droop quota: see the apparently authoritative http://www.aec.gov.au/_content/What/voting/elec_sys/03.htm for an example.

I think NPOV, rather than appeals to authority or credentials, is going to have to be used here. If we can point to a piece of Irish electoral legislation, and quote it as "in the Republic of Ireland, the Droop Quota (capitalized as a proper noun) is defined by the ... Act of 19xx as ...", and "according to the Australian Electoral Commission, the Droop quota is defined as ...". And of course, "In a paper entitled ..., written in 18xx, Henry Richmond Droop originally defined the Droop quota as ...". Doing this will require someone to actually look up the primary references, rather than citing secondary sources.

-- Anon.

This is what I was going for with splitting the formula into two sections (the less formal, and the more formal). If we can find the original authorities, that would be great. Here's the title of Droop's original book:

Droop, Henry Richmond, On methods of electing representatives. London, Macmillan and co., 1868

As for the rounding question, I give up. Hopefully some of our external sources can answer that question. Perhaps Jtdirl has, or can provide us with, some good ones? -- Wapcaplet 14:42 26 Jun 2003 (UTC)

math proof of the "seats +1"

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)[reply]

The "exact Droop" is what is described in the article. This is not Droop at all. It perhaps is Britton quota or Britton-Newland quota.

although it is not Droop, it works just fine as long as there are rules to cope with any problematic tie.

I doubt that there are any elections where there is not one single exhausted vote, This is important because if there is even one exhausted vote, ther is no way that too many candidates can get quota to take seats, even if votes/seats+1 is used.

And even if there is no exhausted vote and "seats plus 1" candidates achieve quota, then just break the tie.

quota of votes/seats+1 works but it is not Droop. It is only one number different from real Droop so the diff is minuscule, but the formula votes/seats+1 is simpler than Droop -- votes/seats+1, plus 1. 68.150.205.46 (talk) 04:38, 22 August 2024 (UTC)[reply]

Quota

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)[reply]

Contested move request

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)[reply]

Droop Quota → Droop quota — "quota" is common in this context. —81.104.175.145 21:22, 27 June 2007 (UTC)[reply]

The discussion above suggests that this is not uncontroversial. Some people are adamant that it is a proper noun, and deserving of a capital D. --Stemonitis 10:04, 28 June 2007 (UTC)[reply]

Who said anything about the D? I'm not proposing a {{lowercase}} 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)[reply]

Sorry — my mistake. I meant Q, of course. --Stemonitis 05:47, 30 June 2007 (UTC)[reply]

In the absence of any other objection, can we now move as uncontested? 81.104.175.145 20:28, 1 July 2007 (UTC)[reply]

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)[reply]

Requested move

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)[reply]

This article has been renamed from Droop Quota to Droop quota as the result of a move request. --Stemonitis 16:24, 10 July 2007 (UTC)[reply]

Erroneus formula?

Currently, the two formulae present ( $\left({\frac {TotalValidPoll}{\left(Seats+1\right)}}\right)+1$ 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)[reply]

Big mistake?

I don't believe that writing like this

{\frac {100}{2+1}}+1=34{\frac {1}{3}}

could be correct. I would write:

{\frac {100}{2+1}}+1=34+{\frac {1}{3}}

Kar.ma 07:29, 15 September 2007 (UTC)[reply]

Compound fractions are one of my pet-peeves too, but their use is unfortunately too widespread to change now, and using + just looks awkward. Either way, I prefer 103/3. ⇌Elektron 16:56, 15 September 2007 (UTC)[reply]

actually it should 100/3 = 33 1/3 becomes 33. 33 plus 1 = 34 as final answer (not 34 1/3) 2604:3D09:887C:7B70:606B:35C9:AB25:F021 (talk) 19:25, 25 August 2023 (UTC)[reply]

actually it should 100/3 = 33 1/3 rounded down becomes 33, plus 1 = 34 as final answer

OR

actually it should 100/3 = 33 1/3. raised to next higher integer becomes 34 as final answer.

for Droop's own words, see below. (recent edits.)

Tom 68.150.205.46 (talk) 02:20, 17 May 2024 (UTC)[reply]

Droop is not better than Hagenbach-Bischoff!!!

Okay suppose we have an instant runoff election, where we have 50 votes with Party-A as first preference and Party-B as second preference. We also have 50 votes with Party-B as first preference and Party-A as second preference.

Using the Hagenbach-Bischoff quota of 50, both parties reach quota, and there is a tie.

Using the Droop Quota of 51, neither party reaches the quota, but as they have the same number of first preference votes, neither can be eliminated; and we have a tie anyway.

So using the Droop Quota does not eliminate ties. It is a much more ugly formula, and has the property that parties with majority support, can recieve a minority of seats.

Can anybody give me a good reason why the world is still using the Droop Quota!

Zfishwiki (talk) 06:31, 6 May 2008 (UTC)[reply]

With Hagenbach-Bischoff you could have too many people elected and need a tie-breaker to unelect one of them;with Droop you could have too few elected and need a tie-breaker to elect one. Some people feel that the former is unsatisfactory: once you have won (reached the quota or whatever) then you should be safe and happy. --Rumping (talk) 23:51, 25 April 2011 (UTC)[reply]

Mostly because of confusion and off-by-one errors. In reality, the Droop quota and Hagenbach-Bischoff quotas are the same, but there are a bajillion annoying variants that differ by a single vote.

I've edited the article to use the exact (correct form) Droop quota. The rounded Droop quotas are presented later on as simplifications. Closed Limelike Curves (talk) 03:17, 13 February 2024 (UTC)[reply]

The idea is that the smaller the quota, the more proportional the result, and in almost all elections there is at least one "exhausted" or non-transferable vote even as soon as the second count, so even with H-B quota, there is little or no chance of having too many pass quota. The example of two candidates each getting 50 votes and getting H-B quota is not an STV problem. it being a single winner contest. with a district electing two or more members and three or more candidates running, there is little chance that two of them will have tie and both pass quota in the first count, even with H-B. — Preceding unsigned comment added by 2604:3D09:887C:7B70:A220:A8BE:46DD:80C8 (talk) 22:07, 31 August 2023 (UTC)[reply]

Actually H-B quota and Droop, being both the same, and each putting quota at more than votes/(seats plus 1), there is no way that too many can make quota than needed to fill open seats.

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." from Droop. "On methods of electing representatives (1881)". Voting Matters (24): 29.reprint. http://www.mcdougall.org.uk/voting-matters/ISSUE24/I24P3.pdf

Dancisin, Misinterpretation of H-B Quota is clear on the mistake that many make when they think H-B (and Droop too) is only votes/(seats plus 1). see https://www.researchgate.net/profile/Vladimir-Dancisin/publication/266030518_MISINTERPRETATION_OF_THE_HAGENBACH-BISCHOFF_QUOTA/links/5423ef8b0cf238c6ea6e7bfc/MISINTERPRETATION-OF-THE-HAGENBACH-BISCHOFF-QUOTA.pdf 68.150.205.46 (talk) 02:41, 17 May 2024 (UTC)[reply]

Actually H-B quota and Droop, being both the same, and each putting quota at more than votes/(seats plus 1), there is no way that too many can make quota than needed to fill open seats

H-B quota and Droop, being both the same, and each putting quota at more than votes/(seats plus 1), there is no way that too many can make quota than needed to fill open seats.

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." from Droop. "On methods of electing representatives (1881)". Voting Matters (24): 29.reprint. http://www.mcdougall.org.uk/voting-matters/ISSUE24/I24P3.pdf

Dancisin, Misinterpretation of H-B Quota is clear on the mistake that many make when they think H-B (and Droop too) is only votes/(seats plus 1). see https://www.researchgate.net/profile/Vladimir-Dancisin/publication/266030518_MISINTERPRETATION_OF_THE_HAGENBACH-BISCHOFF_QUOTA/links/5423ef8b0cf238c6ea6e7bfc/MISINTERPRETATION-OF-THE-HAGENBACH-BISCHOFF-QUOTA.pdf 68.150.205.46 (talk) 02:42, 17 May 2024 (UTC)[reply]

Party list system

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)[reply]

Say what?

"The difference between the two quotas comes down to what the quota implies. Winners elected under a Hare system represent that proportion of the electorate; winners under a Droop system were elected by that proportion of the electorate."

This does not seem defensible to me. Please defend it.

—JLundell talk 00:36, 8 August 2010 (UTC)[reply]

This was an attempt to capture the key substantive difference between the two systems in a single phrase without getting mathematical.

The Droop quota is the minimum vote count required for candidate to be considered "elected", so in a single seat election 50%+1 votes is enough to be elected, in a two-seat election one third plus 1 is enough. Even under a Hare quota system, a candidate that receives at least the Droop quota will still always get elected.

However a winner in a single seat election is the representative for the entire electorate, 100%, which is the same as the Hare quota; and in a two-seat election each elected candidate in effect represents half the voters, again the same as the Hare quota. This is not a coincidence, it is what the Hare quota is.

Chalky (talk) 06:57, 11 August 2010 (UTC)[reply]

The quota (Droop or Hare, whatever is used) is not actually the minimum to take a seat. In many real-life STV elections you find winners elected with less than quota (at the end when the field of candidates is thinned down the number of remaining open seats) That is, in STV elections held using Optional preferential or Semi-optional preferential voting, as described in Wiki "Optional preferential voting" which are most of the STV elections today. Only in elections where each voter ranks all the candidates do you see all winners elected with quota, and this is more likely when Droop is used, but even when Droop is used, it happens often that winners are elected with less than quota. To say that they are declared elected because they would eventually accumulate quota is an assumption, an unnecessary assumption I think. Those who are elected with partial quota are elected at a point when the number of candidates is reduced to the number of remaining open seats - no further elimination can take place so votes can not be transferred. The winners at that point in the count are the most popular and thus are deemed to be most worthy of election. A concise way to describe STV is to say that STV elects the most popular, whether by attaining quota when others don't or by being the most popular when no further transfers can take place. The last part does not apply when every voter ranks every candidate (or comes close to it) but that is seldom the case. — Preceding unsigned comment added by 68.150.212.252 (talk) 07:16, 15 August 2023 (UTC)[reply]

"The last part does not apply when every voter ranks every candidate" may be the crux of the matter. Suppose a ballot that ranks all candidates but two is considered to be two ballots, each with half the usual weight, with the first ballot putting the two candidates last and in one order and the second ballot putting the two candidates last and in the other order. Similarly, a ballot that ranks all but

k

candidates could be interpreted as

k!

ballots, each with a weight of

1 / k!

, that covers all ways to order the

k

candidates missing from the original ballot as last. With that interpretation, even the incomplete ballots are effectively complete, and all candidates do eventually reach the quota, even the candidates who are chosen when the number of candidates is reduced to the number of open seats.

It is in this sense that all candidates do achieve the quota. Finding a way to make that intuitive and useful in the article is another issue. :-). —

Q

uantling (talk | contribs) 16:04, 15 August 2023 (UTC)[reply]

I don't see need to see incomplete ballots as being complete. I think common practice is a ballot is good until the marked preferences do not provide instructions about a transfer, if need to transfer comes around. (then it is declared rejected or is put in exhausted pile or left with winning candidate in case of surplus transfer). I don't see importance of the "complete" or "incomplete" label. Even if each voter marked only one choice, the results would be more proportional than under FPTP. (Just look at Vanuatu elections where SNTV is used.) And I consider a candidate getting quota when the candidate actually accumulates a number of votes that is equal to or more than quota. What could happen if transfers were extended past the last seat being filled or if ballots are weighed differently than they are, is based on unnecessary assumptions, and not part of STV as it is described in election law in any jurisdiction where it is used, as far as I know. It is a common misconception that the quota is the amount required to win a seat but looking at almost any STV election you will find a member or two members or more than that elected with less than quota. Common so no one blames someone for thinking so but still erroneous. Getting quota guarantees you a seat but it is possible to be elected with less, but not to be depended on. Quota - and transfers themselves - are not only thing that makes STV more proportional than FPTP - it is single voting in a multi-member district. And the use of quota and transfers merely polish up the rough fairness seen among the front runners even in the first count, most of whom will be elected in the end anyway. I think it is not bad thing for candidates to be elected with less than quota when transfers are ended and the most popular remaining candidates are declared elected. (anyway it is no worse than FPTP - every successful candidate under FPTP is elected that way) so there is really no need to try to pretend that being elected with less than quota never happens. It is the reality. There may be reasons for it as you logically maintain but that is beside the point - it happens. STV is very intuitive as you say and flexible - all seats might be filled with quota in first count, some successful candidates might never get quota, votes might transfer across party lines so Gallagher Index may not apply even though most voters (80 percent) are happy that their vote was used to actually elect someone - but the practical effect in all cases is - the most popular are elected - whether through quota or by having a relative lead (plurality) at the end. The election by partial quota is a good thing in that it is proof that there are no candidates neither elected nor eliminated (or declared defeated) so that is good thing as far as votes used effectively goes. Preceding unsigned comment added by 68.150.212.252 (talk) 07:16, 17 August 2023 (UTC)

Considering incomplete ballots as if they were multiple fractionally-weighted completed ballots is useful if one does not like the "fact" that some candidates are elected only once they achieve quota and others seemingly are elected despite not having achieved quota. It's a way of understanding that the appearance of fewer votes for some candidates is due only to the hidden information that incomplete ballots represent. Because fairness is an important part of voting, I think it is important to indicate that all candidates do have to achieve quota in this extended sense. I would rather see the article discuss how this fairness is achieved; rather than highlighting that some candidates apparently get away with fewer votes, which to the uninformed reader might appear to be evidence of unfairness. —

Q

uantling (talk | contribs) 20:34, 18 August 2023 (UTC)[reply]

there is no "seeming" about some candidates being elected with less than quota - many STV elections show that result.

I don't understand the term "multiple fractionally-weighted completed ballots" and don't really wish to - as it is not part of any electoral rules I have ever seen.

under STV, there is fairness --the most-popular candidates are elected - as proven by receiving quota or by being the most-popular when the field of candidates is thinned to the number of remaining open seats, at which time further transfers are un-necessary and generally the process of filling seats ends. 2604:3D09:887C:7B70:D426:1433:3C80:C6BE (talk) 23:37, 24 August 2023 (UTC)[reply]

I apologize for the mouthful that "multiple fractionally-weighted completed ballots" is. It is just saying that if an election is among 3 candidates,

A

,

B

, and

C

, then a single ballot that indicates

A

as first but does not rank

B

and

C

is effectively two ballots,

A > B > C

and

A > C > B

, each with half a vote.

This view of an incomplete ballot does not change the set of winners nor the order that they are discovered, etc. in any way. However, it does make all ballots effectively complete. —

Q

uantling (talk | contribs) 23:53, 24 August 2023 (UTC)[reply]

In particular, the article text says

... many voters may vote for only a small proportion of the candidates on the ballot .... Those votes are known as 'NTs', or 'non transferable votes', or "exhausted votes,", and their removal from the votes still in play before or during the vote count process may reduce the number of votes available to such an extent that there may not be enough votes still in play for the last candidates to reach the quota.

I don't object to this text because it too is a reasonable way to look at things, but I would like to also have text that indicates how things change when each incomplete ballot is modeled as representing all possible completions of itself. In particular, with this thinking, there will be enough votes still in play for the last candidates to reach the quota. —

Q

uantling (talk | contribs) 13:15, 25 August 2023 (UTC)[reply]

I have gone ahead boldly making an edit along these lines. If you find it substandard please try to fix it rather than reverting it.

I also made several other edits. Please judge those independently of the bold edit. Thank you —

Q

uantling (talk | contribs) 13:41, 25 August 2023 (UTC)[reply]

I can see we are striving for understanding

IMO, this statement in the article is fraught with problems:

"While in theory every STV election should see the right number of candidates elected through reaching the quota, in practice where the STV system allows it, many voters may vote for only a small proportion of the candidates on the ballot paper, such as only those candidates from one party, or even only one candidate."

to parse it,

"While in theory every STV election should see the right number of candidates elected through reaching the quota,"

with IRV and Hare (I know it is not STV but it seems we are discussing it anyway) with even one exhausted vote, there is no way for a candidate to get 100 percent of the vote unless you arbitrarily treat the ballot as bearing preference(s) that are not marked.

...

in practice [s/b in real-life elections] where the STV system allows it, many voters may vote [s/b mark preferences] for only one or just a [few] of the candidates on the ballot paper, such as only those candidates from one party."

to carry on/explain,

In these cases it may be impossible for the ballot to be used to elect someone or on the other hand it may be used to elect someone. Under STV, eighty percent or so of valid votes are generally used to actually elect someone. STV's fairness goes deeper than high rate of effective votes - Even if he vote itself is not used to elect someone, one or more of the preferences marked on the ballot may be elected without the help of the vote.

STV is flexible - some votes are used to elect someone and others (perhaps only 20 percent) are not, but the presence of multiple winners in a district means great proportion of satisfied voters. And further, the mixed party rep. produced by STV means that even if the preferences marked on ballot are not elected, likely someone running in the district for the same party or a similar party will get a seat. Every substantial party will get at least some representation in the district, and with votes being able to be transferred across party lines, each side of the equation (left or right, development versus environment, labour versus Capital) is represented if a candidate of the least-popular side accumulates quota or close to it, perhaps as few as 16 percent of the vote or less.

If the "incomplete" ballot, bearing preferences for less than all, is due to be transferred and has no usable preference, it is declared NT or exhausted. and that outcome is more likely (but never certain) if only one or a few preferences are marked, but it can also happen even if all but one or two preferences are marked.

A marked preference is not used if the vote is not transferred at all or if the marked preference has already been eliminated or elected when the ballot is due to be transferred and the preference comes to be used. if the preference has already been elected, the voter is likely satisfied even if the vote itself was not used to get that result.

to explain:

not every incomplete ballot will be "scheduled" to be transferred. a vote cast in the first count for a candidate who wins (who wins at any step in the process) will never be transferred, except as a surplus vote. in the whole-vote transfer method, not every vote is considered surplus, not even partially as happens to all votes received by winner under Gregory methods.

talking about single-winner contests is not necessary in this article as it is about STV, not just about all forms of ranked voting.

and talking about IRV and Hare confuses issue IMO, as it creates desire for all votes to be considered as used and perhaps even for all winners to be elected with quota. when most electoral rules are clear, The ballots are taken as is, with no assumption made as to preferences not marked, and votes are not split into halves,

Most electoral rules are clear (even if almost all explanations ae not) that seats are filled by quota or by relative plurality at the end. I doubt there is an election run according to a rule that says each candidate will get quota. most explanations say that, for simplicity or out of misunderstanding but the election itself always has go-around allowed at the end.

even under full-pref voting, it is possible for a vote to be exhausted, because even if every candidate is marked, mistakes happen and say the voter marks 10 twice, the vote will be used for the first nine preferences and then if it is due to be transferred again, it will be exhausted as having no usable preference, although "complete" in a manner of speaking.

that is part of why the distinction between complete and incomplete does not strike me as important. 2604:3D09:887C:7B70:606B:35C9:AB25:F021 (talk) 19:13, 25 August 2023 (UTC)[reply]

section "effect of incomplete ballot" should go at end of article just before "see also" as it is a complication (I think an un-necessary complication) and not as important as telling reader what Droop means, how it works, and how it is different from other quotas. 2604:3D09:887C:7B70:606B:35C9:AB25:F021 (talk) 19:29, 25 August 2023 (UTC)[reply]

For the individual sentences that you indicate could use changes, please either edit the article directly, or explicitly mention here what new wording you would use. The section move could be appropriate ... though discussion of incomplete ballots is probably applicable regardless of the quota chosen, so maybe we don't need that section at all for this article. What do you think? —

Q

uantling (talk | contribs) 18:16, 26 August 2023 (UTC)[reply]

yes, I think remove the section on incomplete ballots, as it is an issue not just about DQ.

I will try to get to the sentence-edit soon. thanks for support. 2604:3D09:887C:7B70:A220:A8BE:46DD:80C8 (talk) 22:10, 31 August 2023 (UTC)[reply]

How is the surplus distributed?

I understand how the quota is arrived at: that's not the problem. This is: 'Andrea has more than 34 votes. She therefore has reached the quota and is declared elected. She has 11 votes more than the quota so these votes are transferred to Carter.' First, this example does not explain why Carter, rather than Brad, gets those votes. Second, and more importantly, will somebody please explain, in simple English, how the "surplus" of an elected candidate is distributed. For instance, if a candidate is elected and has a surplus of 1000 votes, I surmise that the 2nd preferences are examined of that 1000 votes and his surplus is distributed accordingly? But are the last 1000 votes chosen, or is the 1000 surplus chosen at random? Or are the 2nd preferences of *all* his votes examined? For example, if 25% of his total votes have a 2nd preference named as Candidate B, then B will get 25% of that 1000 vote surplus? 86.42.16.3 (talk) 00:17, 23 February 2011 (UTC)[reply]

All 45 of Andrea's votes had Carter as second choice, so the surplus votes go to Carter not Brad. How is more complicated; see Counting Single Transferable Votes#Surplus re-allocation --Rumping (talk) 23:55, 25 April 2011 (UTC)[reply]

yes if next usable preference on the relevant ballots show 25 percent for A, then A will get 25 percent of the surplus.

relevant ballots can be all votes held by the winner or only those in the last parcel that was transferred to the winner, (depending on variant of STV used)

surplus transfer can be done by moving a set proportion (25 percent) of the relevant ballots to A (Malta-Ireland whole-vote system)

or moving all the relevant ballots at a set value (25 percent) to A. (Gregory method)

Tom Monto 2604:3D09:8880:11E0:C8E0:3DDF:A94D:41CB (talk) 19:57, 24 April 2024 (UTC)[reply]

Software recommendation needed

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)[reply]

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)[reply]

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Merging Proportionality for Solid Coalitions page

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)[reply]

Disagree. The Droop quota page is a discussion of what the Droop quota is. The Proportionality for Solid Coalitions (PSC) page is a discussion as to what criteria you should compare the various quotas (e.g., Hare Quota, Hagenbach-Bischoff quota). Think of it as the difference between the Ranked pairs page and the Monotonicity page. Brvhelios (talk) 22:50, 24 March 2019 (UTC)[reply]

Merge with Hagenbach-Bischoff quota

In both modern descriptions and the original description of the method given by Hagenbach-Bischoff, both quotas are treated as being the same. Some textbooks maintain a minor distinction, where the term "Droop quota" is used for the H-B quota after rounding up to the nearest whole number (including a fencepost error made by legislators). Both quotas have precisely the form:

$quota -> .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}votes⁄seats+1$

i.e. the correct procedure for applying the Droop/H-B quota is to calculate results by taking the limit as the quota approaches the lower bound given by the "true" H-B quota.

For convenience of administration (prior to modern electronic voting systems, when actual physical ballots had to be shuffled around), this procedure was described as $votes ⁄ seats +1$ , rounded up (equivalent to the presentation here). However, with modern electronic voting, there is no reason why the quota must be a whole number, and trying to maintain distinction just confuses readers. Even for those jurisdictions that do use the rounded variant, the difference comes down to less than a single vote, making the distinction practically irrelevant.

As such, I propose merging the Hagenbach-Bischoff quota into this article, with a section mentioning some authors maintain a minor distinction where the Droop quota is defined as the Hagenbach-Bischoff quota "after rounding up." Closed Limelike Curves (talk) 20:11, 28 December 2023 (UTC)[reply]

Completed. –Maximum Limelihood Estimator 20:19, 7 May 2024 (UTC)[reply]

I agree with merging of H-B and Droop but the arrticle should clearly say that both are something larger than votes/seats plus 1.

H-B wrote (as explained in Dancisin, Misinterpretation...:)

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 have 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, or Q = [V/(S+1)+1] (brackets [ ] denoting the floor function).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).

This makes it clear that H-B quota is always larger than votes/seat +1, just as Droop quota is also. 68.150.205.46 (talk) 05:34, 12 May 2024 (UTC)[reply]

recent edits

Why my recent edits April 24 2024?

article as it stands is mis-leading to readers.

Droop is not votes/(seats plus 1) Droop is larger than that, if you look at any real-life application of Droop and any historical book on STV. please show me one that defines Droop as you formulate it -- votes/(seats plus 1)

so in the example, we agree 26 is the amount needed to be elected. you say Washington is elected because he exceeds Quota (which is taken wrongly to be 25).

but then when his surplus votes are transferred, he is left with just 25, not enough to be elected! 25 is thought to be Droop (wrongly) but anyway we just said he needed 26 to be elected, to win (the correct 26 figure) so when surplus votes are transferred, he should be left with 26 and that 25/26 discrepancy changes the the candidates' tallies down the line. Burr tie is no longer a possibility so a simple note about resolving ties is all that applies.

fencepost-mistake discussion is un-necessary IMO there are problems with proportionality and Droop versus Hare but fencepost mistake is not it. IMO

under both Droop and Hare, the votes of a group of voters elect a member and that member represents them - there is no Hare versus Droop thing here.

Problem with article seems to be - mis-identification of Droop (leaving out the rounding up or adding 1) - confusion about need to exceed Droop or just equal it, to win. likely these arise from awkward and pretty much inaccurate way the article lumps Droop in with H-B quota. Droop is only 1 more than HB but that small difference is important philosophically.

but article is clear that Droop and H-B is identical so that is fine. names reflect the two inventors, one English, the other German. (se below)

article is good start but just needs some fixing IMO 2604:3D09:8880:11E0:79F0:3444:FE20:77DB (talk) 22:10, 24 April 2024 (UTC)[reply]

Hi, some points. First, the Hagenbach-Bischoff quota is not "confused with" the Droop quota. The writings of both H-B and Droop use the exact same form of the quota (i.e. the quota used in this article rounded up). As a result, both are equivalent to each other. See the source provided in this article.

Whether the "Droop quota" refers to the whole-number (rounded up) version differs depending on context. Historically, the answer is it refers to the rounded-up form. Mathematically, the answer is that rounding-up is an accident of history (because Droop assumed a whole number of ballots, as in Hare's original random-transfer proposal).

The article here follows the convention set by the Electoral Reform Society, which refers to (and continues to refer to) the exact Droop quota as the "Droop quota", but mentions the existence of more complex minor variants (discouraged by mathematicians). –Maximum Limelihood Estimator 21:03, 7 May 2024 (UTC)[reply]

actually Electoral Reform Society says greater than vote/seats plus 1

from ERS Hare vs Droop – Electoral Quotas

The two main electoral quotas are Thomas Hare’s original quota – which is “total votes / total seats” and Henry Droop’s quota – which is “(total votes / (total seats + 1)) + 1 2604:3D09:8880:11E0:0:0:0:7044 (talk) 20:15, 15 May 2024 (UTC)[reply]

Source, please. I see nothing in the 1976 edition of the rules describing Droop's quota the way you say; the 1974 edition used the +1, but they corrected this error in 1976, noting (correctly) that the inclusion of the +1 causes STV to fail every mathematical property typically attributed to it. This includes proportionality for solid coalitions, homogeneity, etc. –Sincerely, A Lime 04:10, 17 May 2024 (UTC)[reply]

Sorry, I don't know 1976 rules but I do know what is on ERS website as of now.

here is link to Droop quota where Droop is defined as seats/(seats plus 1) plus 1 --

https://www.electoral-reform.org.uk/finding-the-finish-line-how-to-set-the-quota-under-stv/

It is right there on the website today.

I can't see how change from "exact Droop" (votes/seats plus 1) is so different in results from votes/(seats plus 1) plus 1. Both would be proportional. so can't understand why ERS in 1976 would say one is so proportional and the other is so not.

not big difference proportionally but getting definition correct in Wiki is big deal IMO 68.150.205.46 (talk) 06:42, 21 June 2024 (UTC)[reply]

Further Response: Simple math shows us that votes/seats +1 means that literally more can get quota than there are seats. That is why many sources says that Droop is votes/seats +1, +1 or rounded up or just a fracton more. even H-B in the document I linked to in the version of the article I put forward before says a candidate to be certain of election must get more than votes/seats plus 1. 25 out of 100 votes cast in three-seat contest is not Droop because it potentially allows more to get quota than there are seats. it is not definition in Humphreys book PR (1911) - he says votes/seats + 1, +1. I would put his book as more authoritatitve than the Lundell and Hill essay that can easily be seen to be flawed mathematically. Droop is mathematical. Whether it is rounded up, 1 added or fraction added to votes/seats +1 is no big diff. but that Droop is more than votes/seats +1 is imprtant and because present article does not say that, it is wrong. for short-hand some do say votes/seats +1 because it is just one off and it is shoerter, but for encyclopedia, we should be exact. at least exact enough to say Droop i "more than votes/seats +1." and if that is quota, then the example where a candidate is declared elected but left with less than than that quota (less than "more than votes/seats +1") is wrong. 26 is quota in the example - we are not using fractons of votes so 26 is smallest number "larger than votes/seats +1". under STV, winner is left with quota, not something less than quota. Droop is no "accident of history" - it was carefully thought out by both Droop and H-B.

both Droop and H-B are clearly greater than votes/seats plus 1 this paragraph would be useful addition to the article: 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."

Old Formula was good

this is the gist of the Formula section as of Jan 2024.

It is better than the formula that is now (May 2024) in the article.

Formula

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)[reply]

This article is wrong and self-contradictory

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)[reply]

"exact Droop quota" is not Droop/H-B quota

As this article is about the "exact droop quota", it should be entitled that, not "Droop quota".

a footnote actually says the article is about "exact droop quota" not Droop.

As this is the case, the article should be renamed or should be rewritten so it is about the Droop quota.

the "exact Droop quota" is not the same proportion of the valid votes cast as Droop.

The "exact Droop quota" is not the quota that Droop and H-B envisioned, which is the quota that Droop and H-B each named after themselves.

so I guess prevous remark that the article is wrong is itself wrong, just the article is mis-identified as it is not about the Droop quota. 68.150.205.46 (talk) 07:06, 26 May 2024 (UTC)[reply]

But actually article is wrong becasue it says "exact Droop qjuota" is derived from majority 50 percent plus 1,

but "exact Droop quota" is not 50 percent plus 1. 68.150.205.46 (talk) 07:09, 26 May 2024 (UTC)[reply]

Neither Droop nor H-B named it after themselves (as that's an easy way to be laughed out of the scientific community). Droop's original work contained an off-by-one error, which has been identified and corrected at different points (most recently Newland and Britton).

The term "Droop quota" has been used to refer to both the forms with and without the off-by-one error, as noted in the article. –Sincerely, A Lime 01:17, 30 May 2024 (UTC)[reply]

what yo9u call "exact Droop" is votes/(seats plus 1), so is 50 out of 100.

that is not majority.but when electing just 1 person, Droop and H-B quota is meant to be majority. as article says they are.

so I believe it is correct to either say Droop is something larger than "exact Droop" i.e something larger than (votes/seats plus 1), and you must equal or exceed ti to be certain of election, or one should say you must exceed "exact Droop" to be certain to be elected.

Currently article is self-contradictory saying "exact droop" is a majority, when it isn't.

I never said Droop or H-B named it after themselves. But I do say their formulation of Droop (and H-B) is differnt from what you call "exact Droop". That is easy to see by looking at their writings.

It is a difference of only 1. But causes vagueness that is un-necessary. it seems logical to use the definition that each of them used, which is votes/(seats plus 1) plus 1, or in a system that uses fractions, then anything larger than votes/(seats plus 1).

and if you need to exceed quota to be certain to be elected, then when you say two win with "exact Droop" and thereby a tie but that is no problem as any system may produce tires, yo uare merely saying they have not won by exceeding "exact Droop".

in real life, there are few ties received by winners (prior to transfer of surplus votes). if exceeding "exact Droop" is needed to be certain of being elected, then the quota (to be equalled or exceeded) is something more than exact Droop. and if exceeding "exact Droop" is needed to be certain of election, then the only way people win with just "exact Droop" is because you don't always need quota to be elected, only to be certain of being elected.

fact is having more than 50 percent in one-winner situation is majority, achieves quota, and is Droop. equalling "Exact Droop" is not majority, surpassing 50 percent is.

Ireland and Malta, the only two countries where STV is currently used at the national level to elect all members, use Droop. This is now, in modern times so I think it is unfair to say the Droop (as traditonally defined by Droop and H-B) is archaic and that all modern uses use a variant of "exact Droop."

I say drop talk of "exact Droop" and "archaic Droop" and just present Droop and H-B as those people themselves formulated them. the portion (a majority in a one-winner situation) that is to be equalled or exceeded to be certain of election, This is the quota that is used in Malta and Ireland, and Cambridge today. 68.150.205.46 (talk) 06:20, 21 June 2024 (UTC)[reply]

"Common errors section

Below is what common error should say. as well the section head itself should be changed. Droop's own quota (or an almost-identical formulation) is recorded here as an error!

my version is based on my interpreting the math notation properly. if it is mistake as CLC said in oct 2024, he or she should clarify what the math notation actually means.

Here's my version: The first variant in the top-left, votes/k+1 rounded up, is close to Droop's original proposition. It was thought important that no more could achieve quota than the number of open seats. It arose from Droop's discussion of the quota in the context of Hare's original proposal for STV, which assumed a whole number of ballots would be transferred and fractional votes would not be used.^[1] In such a situation, a fractional quota would be physically impossible, leading Droop to describe the next-best value as "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).^[2] In such a situation, rounding the number of votes upwards introduces as little error as possible, while maintaining the admissibility of the quota.^[2]

The top-right variant shows that the quota is votes/k+1 plus 1, rounded down. This is in effect equivalent to votes/k+1 rounded down, then add 1. This is Droop's original proposition of the Droop quota. It is the quota used in most or all of the STV systems used today, many of which conduct transfers using whole votes. Some hold that it is still needed in the context of modern fractional transfer systems, such as Gregory Method STV systems used in Australia. They apprehend that when using the exact Droop quota (votes/ k+1), it is possible for one more candidate than there are winners to reach the quota.^[2] 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, rules can be applied to break the tie, and ties can occur regardless of which quota is used.^[3]Cite error: The <ref> tag has too many names (see the help page). Hagenbach-Bischoff ascribed to the next-best value (shown in the top right example) as "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).^[2] In such a situation, rounding the number of votes upwards introduces as little error as possible, while maintaining the admissibility of the quota.^[2]

Some hold the misconception that these rounded-off variants of the Droop and Hagenbach-Bischoff quota are still needed, despite the use of fractions in fractional STV systems, now common today. As well even the addition of 1 to (votes/seats plus 1) is un-necessary. When using the exact Droop quota (votes/seats plus 1), it is possible for one more candidate than there are winners to reach the quota.^[2] 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, rules can be applied to break the tie, and ties can occur regardless of which quota is used.^[3]^[4] 2604:3D09:8880:11E0:79FB:98D6:5E02:5E5A (talk) 19:52, 18 October 2024 (UTC)[reply]

confusion between hare and Droop section

This paragraph below should be in the "Confusion between Hare and Droop" section.

It fixes a couple misconceptions in the article as written in late Oct 2024: The article as written confuses district results with overall rep in the chamber. Minority rule only happens in the chamber; Droop only applies in district. As well, article says Droop give majority of seats to party with less than majority of seats, while more importantly Hare is to blame for denying a majority party a majority of seats.

here's my version: The Droop quota is often confused with the more intuitive 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 in an ideally-proportional system, i.e. one where every voter is treated equally. As a result, the Hare quota gives more proportional outcomes, although sometimes under Hare a majority group in a district will be denied the majority of seats in a district.[1] By contrast, the Droop quota is not biased against large parties, as the Hare quota is. While the Hare quota sometimes denies majority of seats to a party with majority of votes, Droop more often ensures the largest party has majority of seats if it has majority of votes. 2604:3D09:8880:11E0:2454:D454:D3F5:C7B7 (talk) 21:20, 17 October 2024 (UTC)[reply]

or this version makes sense (it is currently (Oct. 18, 2024) what is on article but as CLC is defensive of changes, I don't expect my changes to survive) Droop and Hagenbach-Bischoff derived new quota as a replacement for the Hare quota (votes/seats). Their quota was meant to produce more proportional result by having the quota as low as thought to be possible. Their quota was basically votes/seats plus 1, plus 1. Such a formula may yield a fraction, which was a problem as STV system did not use fraction. Droop went to votes/seats plus 1, plus 1, rounded down (variant in top right).

On other hand, Hagenbach-Bischoff went to the first variant in the top-left, votes/seats plus 1 rounded up.^[1] Hagenbach-Bischoff ascribed to the next-best value (shown in the top right example) as "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).^[2]

Some hold the misconception that these rounded-off variants of the Droop and Hagenbach-Bischoff quota are still needed, despite the use of fractions in fractional STV systems, now common today.

As well even the addition of 1 to (votes/seats plus 1) is un-necessary. When using the exact Droop quota (votes/seats plus 1), it is possible for one more candidate than there are winners to reach the quota.^[2] 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, rules can be applied to break the tie, and ties can occur regardless of which quota is used.^[3]^[4] — Preceding unsigned comment added by 2604:3D09:8880:11E0:79FB:98D6:5E02:5E5A (talk) 19:55, 18 October 2024 (UTC)[reply]

Or this version:

There are at least six different versions of the Droop quota to appear in various legal codes or definitions of the quota, all varying by one vote. Some claim that, depending on which version is used, a failure of proportionality in small elections may arise. Common variants include:

Droop and Hagenbach-Bischoff derived new quota as a replacement for the Hare quota (votes/seats). Their quota was meant to produce more proportional result by having the quota as low as thought to be possible. Their quota was basically votes/seats plus 1, plus 1, the formula on the left on the first row.

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 (the variant on top right). Hagenbach-Bischoff went to votes/seats +1, rounded up, the variant in the middle of the top row. 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).

Some hold the misconception that these rounded-off variants of the Droop and Hagenbach-Bischoff quota are still needed, despite the use of fractions in fractional STV systems, now common today.

As well, it is un-necessary to ensure the quota is larger than vote/seats plus 1, as in the historical examples, the variant on the second row, and the formula on the right on the bottom row. When using the exact Droop quota (votes/seats plus 1) or any variant where the quota is slightly less than votes/seats plus 1, as in votes/seats plus 1, rounded down (the left variant on the third row), it is possible for one more candidate than there are seats to reach the quota. 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, rules can be applied to break the tie, and ties can occur regardless of which quota is used.

Spoiled ballots should not be included when calculating the Droop quota. However, some jurisdictions fail to correctly specify this in their election administration laws

======== 68.150.205.46 (talk) 02:58, 23 October 2024 (UTC)[reply]

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