Talk:Droop quota

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)

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)