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Tom Monto

PR Quota -- Hare versus Droop

Updated: Jul 18

Here's an article from the Electoral Reform Society newsletter August 2021

on the quota under STV, with my added content in square brackets attached.

[more on the quota at the end as well]


Finding the Finish Line: How to Set the Quota under STV

This is a guest post from Dylan Difford who has recently completed an MA in Politics at the University of Essex, focussing on party and voting systems in Britain and Europe.


The first meetings of the Proportional Representation Society quickly attracted many leading lights of the Victorian age – including Lewis Carroll, CP Scott (editor of what is now The Guardian) and Thomas Hare (the inventor of STV). The group quickly settled on Hare’s system as the best option for electoral reform.


The Proportional Representation Society may have since become the Electoral Reform Society, but it still promotes the Single Transferable Vote – though with a few tweaks to Hare’s original ideas, many of them picked up from real-life implementations of the system.


With STV, each constituency elects a small group of MPs and voters rank candidates according to their preferences. To work out who gets elected, you first need to work out the quota. Candidates that exceed the quota are elected, with any surplus votes (total votes minus the quota) transferred to each voter’s next choice as indicated on the ballot paper.


Once any candidates who beat the quota are elected, there is another round of counting to see if any other candidates have reached the quota, now the surplus votes have been transferred. If no candidate meets the quota in a particular round of counting, the candidate with the least votes is eliminated and their votes are redistributed to their voters’ next extant preference. This continues until all the seats have been filled.

But how do you work out where the quota – the winning line – is?


This is one of the things that has been tweaked since Hare’s original designs were laid out.


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”.


In a constituency electing three MPs where 960 votes have been cast, the Hare quota would be 320 and the Droop quota would be 241.


While nearly all STV elections today use the Droop quota and it is the preferred option of the Electoral Reform Society, some still advocate for Hare-STV – pointing to it typically producing slightly more proportional results and it being more favourable to smaller parties.

[Actually in some cases Droop works better for small parties - a party may have just enough votes to take a seat under Droop but not enough to take a seat under Hare!


As well, Droop means more votes of a winner are surplus so it actually helps larger parties that are due two or more seats.]


But Hare-STV does have a number of ‘quirks’…


Firstly, there is an inequality issue. Under Droop-STV, all elected candidates beat the quota and are thus elected on the same terms.

[Actually all successful candidates do not always win with quota when Droop is used -- we see that in STV elections conducted in Alberta.]

[Our different reading of the evidence likely arises from different data sets.

Dylan likely is looking at elections where voters must rank all candidates.

Alberta and Manitoba used optional-preferential voting (where there are more exhausted votes) versus full-preferential voting, historically used in many Australian STV applications (but recently some jurisdictions in Australia have switched to Optional-Preference Voting) In full-preferential voting, there are almost none exhausted votes and all or almost all successful candidates do get quota lto be elected.)

Although Droop quota was used in Canadian elections, in each election one, two or three candidates in each district won without quota, just by surviving until the field of candidates thinned to where there were only as many candidates as there were seats remaining to be filled.

And each time when there was only one more candidate than seats to be filled, the elimination of the least-popular candidate meant that there were only as many candidates as seats so that eliminated candidate's votes were not even transferred as those remaining after the elimination were declared elected.]


[Because Droop and Hare both elect some with less than quota, the inequality issue is not addressed by Hare any more than the use of Droop solves it, but I would say it is not important

- under FPTP we never look at whether a MP is elected with 4,000 votes or 44,000 votes but simply say "okay, he (or she) was elected".

While under STV, successful candidates' vote tallies are quota or a few percentiles above or below quota, so much more fair even if in most cases not every member is elected with quota.]

[see notes at bottom for list of members elected without quota under Droop]


But under Hare-STV it is practically impossible for all candidates to meet the quota. [Yes, more are elected with partial quota under Hare than under Droop but my point is it happen under Droop as well.]

[That is because even using full-preferential voting, in some systems the votes that are not filled out properly are used up to the point when the mistake annuls the vote (hence the controversy of the Langer vote) and thus there are not enough votes to fill available seats with quota each time at the end.]


As such, the fight for the final seat is awarded to the candidate with the most remaining votes, regardless of how short they have fallen of the quota.


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

My comments:

Some say a person can be said to be elected with quota - even when they aren't - if you think that they "would've gotten quota eventually."


But I consider a candidate getting quota when the candidate actually accumulates a number of votes that is equal to or amore 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 or even more than that elected with less.

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. 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.

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 in 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 - such as assumption that the candidate would take quota later - but that is beside the point - it happens.


STV is very 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 --


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.

==


Possible to be elected with less than quota?


The quota (Droop or Hare, whatever is used) is not actually the minimum needed 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 (having plurality) 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.


Marking full preferences is not actually important to produce proportionality or at least to produce more fair results than under FPTP -- 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.)


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


Then there is the majority rule problem. [see my notes at bottom]

In certain circumstances under the Hare quota, it is possible for a party that received slightly more than half of votes cast to win less than half of seats in a constituency. [For example this could be due to many votes remaining with a 1st Count winner, votes that would be transferred away to aid other candidates of the same party under Droop.]


Suppose that an STV election takes place between the Mountain Party and the Sea Party, with each running two candidates for three seats and 960 voters.

  • 510 voters give their first preference to a Mountain Party candidate – 340 for the first candidate [Candidate A] and 170 for the second candidate [Candidate B].

  • 450 voters choose the Sea Party, but with their voters more evenly splitting between the two candidates – 226 for the first [Candidate C] and 224 for the second [Candidate D].

All voters rate both candidates from their preferred party ahead of the candidates for the opposing party, with only half of voters afterwards ranking an opposing candidate.


Under a Hare-STV election, where the quota would be set at 320 (960 votes / 3 seats), the first Mountain candidate would be instantly elected, and their 20 surplus votes transferred to the other Mountain candidate – who now has 190 votes.

As this is fewer than either of the two Sea candidates, the remaining Mountain candidate is [declared defeated.] [W]ith only two candidates left for two vacant seats, the two Sea Party candidates are declared elected


Table: Hare-STV Election

[original presentation amended to fit STV vote counting as I know it]

Count 1 Count 2 Count 3

Mountain A 340 (elected) -20 320 ------------------

Sea C 226 0 226 (declared elected)

Sea D 224 0 224 (declared elected)

Mountain B 170 20 190 (declared defeated)

Quota 320


Not only are the two Sea Party candidates elected despite falling far short of the quota, but the Sea Party has also managed to take a majority of seats even though the majority of voters prefer the Mountain Party to the Sea Party.

Such a result would not be possible under Droop-STV, where a party that is preferred by at least half of voters will always take at least half of seats.


Indeed, if we repeat the election under Droop-STV, where the quota would be 241 (960 voters/(3 seats + 1)+1), the larger surplus of the Mountain A candidate would, when transferred, elect the Mountain B candidate in the 2nd count.

The final seat would go to the Sea Party.


Table: Droop-STV Election

[original presentation amended to fit STV vote counting as I know it]

[three seats to fill quota 320]

Count 1 Count 2 Count 3

Mountain A 340 (elected) -99 241 ------------

Sea C 226 0 226 8 234 declared elected

Sea D 224 0 224 6 230

Mountain B 170 99 269 (elected) -28 241 ------------------

Exhausted votes 14 14

Quota 241

Thus the Mountain Party elects two and the Sea Party elects one.


[Conversely, Droop is also better for small parties than Hare. The Droop quota being smaller than the Hare quota, a party that cannot win a seat under Hare may win a seat under Droop. So Droop is fairer tham Hare to large parties and to small parties too (sometimes).


But with the surplus votes of almost half of the successful candidates not being transferred (as discussed in other blogs) and in this case with the DM being only 3 and only two parties running candidates (an artificial case)- there needs to be only one elimination and the election counting is wrapped up. So in such a case the quota used may have little effect in practical terms.]


Of course, the relative downsides of Hare-STV are nothing like the major failures of the current First Past The Post system. Though examples like the above can happen, they are both a rarity and can be avoided by choosing the Droop quota instead.


FPTP, however, regularly leads to dozens of seats across the country where the majority of voters would have preferred an MP from another party and has left roughly half of voters with an MP they had no hand in electing.


The move from Hare to Droop also shows the adaptability of STV and how it has been improved over the years,* while FPTP remains the same failed voting system it was when the PRS was founded in late 1800s – still making the same mistakes, still failing to adequately represent British voters.


[Whether Hare or Droop quota is used, under STV each district in each election elects a mixed bag of representation, at least in any district where the number of reps is larger than three. (Even where the number of seats filled in a district contest is only two, still a mixed representation may be elected.). And in each election a large majority of the votes are used to elect someone, even if not the first choice candidate but at least someone preferred over someone else.


As the writer of the article mentioned, these are attributes held by STV that are not held by FPTP.]


[* Other variations in STV include

- varying the number of seats in a district, (maximum District magnitude is rising - an STV election contest electing 37 members - an all-time record-large DM for STV -- is set to happen in a year or two)

- whether to allow plumping - optional-preferential voting or full-preferential voting, Optional-Preference Voting puts little extra stress on voters compared to FPTP.)

- whether or not to accept X marked for the first choice or if it has to be a figure 1]



STV can even be combined with policies that guarantee minority or local representation.

For example, in Edmonton STV elections, two seats were guaranteed for the southside, achieved though the usual STV election by simple stipulation that if all but two seats have been filled and no southside candidates elected yet, the remaining seats are awarded to southside candidates. That is, when the number of remaining open seats was reduced to two and only two southside candidates remain, transferring of votes stops and the southside candidates are elected.


Or when there are only two empty seats remaining and there are more than two southside candidates still in the running, the remaining seats have to go to southside candidates, and the ones declared elected are determined by eliminating all the remaining northside candidates and transferring their votes, if possible, to the next usable back-up preference marked for a remaining southside candidate, and the two leading southside candidates would be declared elected.


That same kind of guarantee could be used under STV for women or any other group thought worthy of guaranteed representation.]


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


More on Quota


There is often confusion about quota in STV.


Even some official election sites say it is the actual requirement to be elected,

but in actuality it is the amount that guarantees election, and it is possible (customary for one or two or more) to be elected without it.


The issue of Journal of Electoral Administration (No. 18, 1996), previously referred to, has this to say on the point:

The "threshold of exclusion" is the lowest percentage of support that ensures a candidate will win no matter what other voters do.

With Cumulative Voting, this threshold is equal to (1/(seats +1)) plus 1. [This is the Droop quota and it works for STV as well.]


It goes on to say

In practice, a cohesive block of voters can elect a candidate with less than the threshold of exclusion as majority community voters are unlikely to distribute their vote evenly among the same number of candidates as the number of seats up for election (Cumulative Voting) or will mark at least some preferences to candidates of the minority slate ahead of majority-slate candidates (in STV). [or some votes will be exhausted, opening the door to success with less than quota.]


Amazingly, the article (of fairly recent date - 1997 vintage) seems to indicate that where MMDs are allowing majority-white voters to take all the seats, black groups are asking for and in legal cases securing an SMD to get at least one seat.


it is not the DM that is the problem. T


he method of voting - Block voting - is not mentioned but obviously is the problem. (that is, the districting is not the problem but the method used to fill the seats is the problem)


The MMD could be maintained and Cumulative Voting or STV or even Single Non-Transferable Voting would address the unfairness more or less elegantly without need for re-districting and use of inherently dis-proportional SMDs.


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


Mathmatical comparison of Hare and Droop


As Thomas Hare envisioned STV, he thought all U.K. MPs would be elected in one massivel contest. At that scale, there is none or only the slightest of difference between Hare and Droop. (The conducting of such a contest with such a massive DM without computers, or all kinds of time, is only possible with the random transfer of surplus votes. But as thta ws part of Hare's vioisn there was no problem that way. The random transfer might produce exactly the same result asthe more scientific "Exact method" (ala Ireland today) or WIGM (ala Scottish local elections today), or it might not -

but for sure, any form of STV would be more fair and proportional than the use of FPTP.



1859 UK election approx. 566,000 voters cast

(at that time, only men could vote and not all men even)

6454 MPs to be elected

Hare is 566,000 divided by 654  = 865

Droop is 566,000 divided by 655, plus 1 = 865


using actual number of valid votes -- 565,500

Hare   565,500/654 = 865

Droop  565,500/ 655, plus 1 = 864


For lesser vote totals than 565,500: there is a difference between Hare and Droop.


100,000 votes   10 seats

Hare =  10,000

Droop =   9091


200,000 votes 20 seats

Hare = 10,000

Droop = 9524


400,000 votes 20 seats

Hare   = 20,000

Droop = 19,048


so at vote totals of 100,000 to 400,000, no great difference between Hare and Droop really.


DM is more indicator of the difference than votes involved

With anything more than 4 seats, DM difference between Hare and Droop is less than 20 percent. (That is, Droop is more than 80 percent of Hare.)


Say a three-seat district of the two Sherwood Park districts and Fort Saskatchewan

using votes as cast in 2023 AB election

valid votes

Sherwood 12,965 UCP 11,000 NDP 13,000

Strathcona-Sherwood 26,125 UCP 14000 NDP 12,000

Ft. Sask. 24,326 UCP 14,000 NDP 9000

Total 63,416 UCP 39,000 NDP 34,000

Hare 21,139 UCP 1.84 NDP 1.61

Droop 15,855 UCP 2.46 NDP 2.14

So no difference in results between Hare and Droop.


But one can see that under Hare, NDP, the less-popular party, is closer to getting a second seat under Hare than it is under Droop. (The UCP, being more popular, are advantaged by Droop.)

But with UCP being more popular, a result where the NDP take more seats than the UCP in that district would be disproportional anyway.


Here's a better example

where UCP took a disproportional three-seat sweep in 2023:

a three-seat district of Drayton Valley, Lac Ste. Anne and Lacombe-Ponoka

In last AB election, UCP took all three seats.

valid votes

Drayton Valley, 22,000 UCP 17,000 NDP 5000

Lac Ste. Anne 22,000 UCP 15,000 NDP 6000

Lacombe-Ponoka 21,000 UCP 14,000 NDP 5000

Total 65,000 UCP 46,000 NDP 16,000


number of votes and quotas each party took

Quota Party total Party total

Hare 21,667 UCP 2.12 NDP .74

Droop 16,251 UCP 2.83 NDP .98

Thus under both Hare and Droop, UCP would take 2 seats and NDP would take 1.


This is more fair result than the Conservative sweep of all three seats that happened under FPTP.

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


Percentage difference between Hare and Droop

(one seat per 20,000 votes)

Total valid votes Hare Droop Diff (percentile) Droop from Hare

65,000 (3 seats) 21,667 16,251 25

65,000 (5 seats) 13,000 10,834 17

65,000 (6 seats) 10833 9286 14

65,000 (10 seats) 6500 5910 9

100,000 (3 seats) 33,333 25,001 25

100,000 (10 seats) 10,000 9091 9

200,000 (20 seats) 10,000 9524 5

400,000 (20 seats) 20,000 19,048 4.76


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

Notes:

Number of members elected with less than full quotas


it is said:

under Hare-STV it is practically impossible for all candidates to meet the quota. This is true if even one vote is found to be nontransferable .


Because in two-seat district Hare quota is 50 percent of the votes. 

for both to get Hare, you would need no non-transferable votes, and actually the vote transfers would go on past when it was clear who the two winners were. Moving surplus votes from one to the other just to bring the lower up to Hare, for no reason other than to prove a point that both winning candidates have Hare


it is said

Under Droop-STV, all elected candidates beat the quota and are thus elected on the same terms. 

but actually that is seldom the case-


Droop quota was used in Canadian STV elections in Alberta and Manitoba 1920s to 1950s

in the eight elections when Edmonton MLAs were elected by STV, every election saw one or more elected with partial quota.

in the eight elections when Calgary MLAs were elected by STV, every election saw one or more elected with partial quota.


in the nine elections when Winnipeg MLAs were elected by STV, only two STV contests saw all elected with full quota.


These both coincidentally were South Winnipeg, 1949, 1953.

at the end, a candidate was declared defeated and his votes were not transferred. that candidate had less than quota.

in fact the total of exhausted votes and that candidates votes did not add up to quota. leaving enough votes for four candidates to each get quota themselves.


Exhausted votes in Winn. South were about a thousand less than in the other two Winn. districts (likely voters in Winn. South marked more back-up preferences than in the other districts) and that led to relatively few exhausted votes and all four winning candidates getting quota in Winn. South in 1949 and 1953.


Hare allows a party with a majority of votes to take only a minority of seats.


say three parties Mountain Sea and Sky

960 votes in total Hare quota is 320

Mountain 340    170

Sea 160 70

Sky 140 80

1st count Mountain  A elected 

2nd Count  Mountain B gets 190 votes

3rd Count Sea B elim. votes go to Sea A 230

4th Count Sky B elim. Sky A 220

5th Count Mountain B declared defeated. 1 Mountain, 1 Sea, 1 Sky elected 

Mountain, with more than half the votes, got only a third of the seats.

This is case where Mountain was denied second seat because simply did not have two candidates at the end.

The three-party Hare quota example below shows where Mountin is denied second seat due to Hare being large enough to prevent Hare getting benefit from its majority of votes.


Droop

say four parties Mountain Sea Sky Cloud

960 votes in total Droop quota is 241

Mountain 340    170

Sea 130 40

Sky 110 50

Cloud  75  45

1st count Mountain  A elected 

2nd Count  Mountain B gets 269 votes

3rd Count Cloud B elim. Cloud A gets 120

4th Count  Sea B elim. votes go to Sea A 170

5th Count Sky B elim. Sky A 160

6th Count Cloud A elim. votes split between Sea A (230) and Sky A (220)

Sky A declared defeated. 2 Mountain, 1 Sea elected 

Mountain, with more than half the votes, gets a majority of seats, two-thirds of the seats.


Three-party Hare quota example

Three parties: Mountain Sea Sky Mountain has majority of votes

960 votes in total        Hare quota is 320

Mountain 340    170

Sea 205 40

Sky 155 50

1st count Mountain  A elected 

2nd Count  Mountain B gets 190 votes

3rd Count Sea B elim. votes go to Sea A 245

4th Count  Sky B elim. Sky A 205

5th Count Mountain B declared defeated. 

1 Mountain, 1 Sea 1 Sky elected 

Mountain, with more than half the votes, gets a minority of seats, one-third of the seats.


three-party Droop quota -- Mountain party does get majority of seats

three parties Mountain Sea Sky

960 votes in total        Droop quota is 241

Mountain 340    170

Sea 205 40

Sky 155 50

1st count Mountain  A elected 

2nd Count  Mountain B gets 269 votes

3rd Count Sea B elim. votes go to Sea A 245

4th Count  Sky B elim. Sky A 205

Three candidaes remaining, one seat taken,

Only two of the remaining candidates can win.

5th Count: Sky A is least-popular so is declared defeated.

Mountain B and Sky A declared elected

2 Mountain, 1 Sea elected 

Mountain, with more than half the votes, gets a majority of seats, two-thirds of the seats.


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

All PR systems have this in common:


Rather than electing one person per area, the areas are bigger and elects a group of MPs that closely reflect the way the area voted. The areas could be the size of a town, city county or large part of a country (such as an Australian state).

Netherleands and Israel actually use the whole country as a single at-large district.


Cities often do not have wards but use city-wide at-large election contests to elect city councillors.


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


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