Talk:Droop quota

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 — Preceding unsigned comment added by 216.232.197.30 (talk • contribs) 01:37, 3 May 2005 (UTC)

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.— Preceding unsigned comment added by 216.232.197.30 (talk • contribs) 01:37, 3 May 2005 (UTC)

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)

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)

This complaint about the article has now ben addressed.

the article no longer presents exact droop as the Droop quota. 2604:3D09:8880:11E0:7C65:273D:FAF8:CE83 (talk) 19:36, 11 January 2025 (UTC)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

likely it is done as a largest remainder system. 2604:3D09:8880:11E0:7C65:273D:FAF8:CE83 (talk) 19:38, 11 January 2025 (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)

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)

"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.  :-). —Quantling (talk | contribs) 16:04, 15 August 2023 (UTC)

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. —Quantling (talk | contribs) 20:34, 18 August 2023 (UTC)

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)

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. —Quantling (talk | contribs) 23:53, 24 August 2023 (UTC)

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. —Quantling (talk | contribs) 13:15, 25 August 2023 (UTC)

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 —Quantling (talk | contribs) 13:41, 25 August 2023 (UTC)

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)

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)

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? —Quantling (talk | contribs) 18:16, 26 August 2023 (UTC)

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)

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)

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)

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)

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)

• 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)

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 -> 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)

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

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)

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)

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)

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)

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)

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)

This complaint about the article confusing exact Droop with Droop quota has now been addressed.

the article no longer presents exact droop as the Droop quota. 2604:3D09:8880:11E0:7C65:273D:FAF8:CE83 (talk) 19:49, 11 January 2025 (UTC)

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." — Preceding unsigned comment added by 2604:3D09:8880:11E0:0:0:0:7044 (talk) 20:16, 15 May 2024 (UTC)

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.

This complaint about the article has now been addressed.

the article no longer presents exact Droop as the Droop quota.

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)

This article is wrong and self-contradictory

article wrong and self-contradictory: [Jan. 2025: This complaint about the article has now been addressed. the article no longer presents exact Droop as the Droop quota.]

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)

Jan. 2025: This complaint about the article has now been addressed.

the article no longer presents exact Droop as the Droop quota. 2604:3D09:8880:11E0:7C65:273D:FAF8:CE83 (talk) 19:51, 11 January 2025 (UTC)

"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)

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)

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)

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)

Jan. 2025: This complaint about the article has now been addressed.

the article no longer presents exact Droop as the Droop quota. 2604:3D09:8880:11E0:7C65:273D:FAF8:CE83 (talk) 19:51, 11 January 2025 (UTC)

"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. 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). In such a situation, rounding the number of votes upwards introduces as little error as possible, while maintaining the admissibility of the quota.

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. 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. 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). In such a situation, rounding the number of votes upwards introduces as little error as possible, while maintaining the admissibility of the quota.

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. 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. 2604:3D09:8880:11E0:79FB:98D6:5E02:5E5A (talk) 19:52, 18 October 2024 (UTC)

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)

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. 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).

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. 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. — Preceding unsigned comment added by 2604:3D09:8880:11E0:79FB:98D6:5E02:5E5A (talk) 19:55, 18 October 2024 (UTC)

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)