# Talk:Droop quota

## Math discussion

what exactly does ${\displaystyle \left({Votes \over (Seats+1)}\right)+1}$ 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:

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)

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.
1. 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.
2. 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:

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

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?

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

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:

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

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 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)

• Henry Richmond Droop. On the Political and Social Effects of Different Methods of Electing Representatives. London, 1869.
• Henry Richmond Droop. On methods of electing representatives. Journal of the Statistical Society of London 44 (1881) 141-196 [Discussion, 197-202].

--- Anon.

Whee, my library seems to have that last Journal - I'll try and grab it, tomorrow maybe... Evercat 14:56 26 Jun 2003 (UTC)

I honestly don't see what the point of confusing is! If you want the Droop Quota to have the special property of being the smallest possible quota such that no seats+1 candidates can reach it, you need the smallest integer larger than the ratio votes/(seats+1), i.e. if the ratio is an whole number you add 1 to it and if it's not, you round up. Rounding down and adding 1 to the ratio gives the correct result in both cases.

I'd agree, but not knowing anything about Droop Quota prior to this, I'd like to see this confirmed... Evercat 15:22 26 Jun 2003 (UTC)
If this is of any help: votes/(seats+1)+1 rounded down is equal to (votes+1)/(seats+1) rounded up. This may account for some of the sources (including the page about the Tasmanian House of Assembly).

I removed "(or Hagenbach-Bischoff Quota)" from the article as this seemed to be Droop but without the +1 at the end - although admittedly this was down to a Google search :-) and a page that compared the two... still, someone else must agree, since there's already a link here to Hagenbach-Bischoff Quota as a seperate article.... Evercat 22:30 26 Jun 2003 (UTC)

OK, sorry it took so long, but I finally got around to getting a hard copy of the relevant part of On methods of electing representatives. I admit that the use of terms can change, but what he describes is so completely in agreement with the Droop Quota article that I think it's reasonable to see Droop's paper as a definitive text.

Some points:

Droop gives the quota as being, and I quote:

the next whole number greater than ${\displaystyle {mV \over n+1}}$

where mV is the number of votes and n is the number of seats. This is, of course, mathematically equivalent to (votes / (seats + 1)) + 1, if this latter formula is always rounded down.

He also uses ${\displaystyle {mV \over n+1}+i}$ to refer to this value. Note that's an i, not a 1. i seems to refer to the value needed to take the quota up to the next whole number.

He does not use brackets. :-)

Evercat 17:08 7 Jul 2003 (UTC)

Thank you, Evercat! -- The Anome 17:17 7 Jul 2003 (UTC)

Nice work Evercat! We should definitely avoid the one with "i" in it, since that's used in math to refer to the imaginary number sqrt(-1). "the next whole number greater than..." is a good way to phrase it; it eliminates all ambiguity and is visually simpler than our existing versions. I like it. -- Wapcaplet 17:28 7 Jul 2003 (UTC)

OK, but for the avoidance of (my) doubt, the current "mathy" formula,

${\displaystyle \left\lfloor {\frac {Votes}{Seats+1}}\right\rfloor +1}$

is the same as what I just described. Right? Evercat 17:41 7 Jul 2003 (UTC)

To the best of my knowledge, yes. -- Wapcaplet 17:49 7 Jul 2003 (UTC)

Just to make things more complicated, some STV elections use fractional transfers (I once looked at a set of results where someone was eliminated with the glorious total of 0.01 of a vote - a candidate from a predecessor of the Official Monster Raving Loony Party). In that case, the quota only has to be the smallest number (including decimals to the precision being used) strictly larger than votes/(candidates plus one). --Henrygb 01:35, 25 Feb 2005 (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 ${\displaystyle {\frac {votes}{seats+1}}+1}$ quota comes out to be ${\displaystyle {\frac {votes}{2}}+1}$, 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 ${\displaystyle {\frac {votes}{seats}}+1}$ 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 ${\displaystyle {\frac {votes}{n+1}}+1}$ 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, ${\displaystyle {\frac {votes}{n+1}}}$ 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)

## 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)

## 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)

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)

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)
Sorry — my mistake. I meant Q, of course. --Stemonitis 05:47, 30 June 2007 (UTC)
In the absence of any other objection, can we now move as uncontested? 81.104.175.145 20:28, 1 July 2007 (UTC)
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)

## Requested move

Droop QuotaDroop 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)

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)

## Erroneus formula?

Currently, the two formulae present (${\displaystyle \left({\frac {TotalValidPoll}{\left(Seats+1\right)}}\right)+1}$ and ${\displaystyle \left\lceil {\frac {Votes}{Seats+1}}\right\rceil }$) 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 (${\displaystyle \left\lfloor {\frac {Votes}{Seats+1}}\right\rfloor +1\cong \left\lfloor {\frac {Votes}{Seats+1}}+1\right\rfloor }$) 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)

## Big mistake?

I don't believe that writing like this

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

could be correct. I would write:

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

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

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)

## 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)

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)

## 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)

## 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)

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)