Quotas and Effective Votes as percentage of whole -- Hare, Droop or NB -- District Quota contains same or fewer votes than list PR electoral threshold

Tom Monto
Feb 18, 2021
21 min read

Updated: Mar 8

As seen in the table below, the number of votes that makes up quota hardly grows at all as a multi-member districts is given more seats. From a two-seat district to a ten-seat district, quota rises only 60 while the number of votes in the district goes up by almost 2000.

The portion of Effective Votes grows magnificently from a minimum of 2/3rds in a two-seat district to a minimum of 91 percent in a 10-seat district, assuming all successful candidates win with full quotas, the Droop quota is used and there are no exhausted ballots.

An exhausted ballot can only happen if the voter did not mark enough back-up preferences to cover all the candidates.

There are cases where candidates are elected without quota - they are the last remaining candidates as the field of candidates thins to a point where there are only as many candidates as the number of empty seats. This can only happen if votes are relatively evenly spread over the candidates.

The opposite case is where some candidates are so popular that they take quota in the first count and are declared elected, filling the seats before any candidates are eliminated and votes transferred. This never happened in STV elections in Canada but did happen in Alternative Voting elections where quota was 50 plus 1.

If such did happen in STV, the percentage of Effective Votes is exactly as presented in the table below.

But it is more likely that some candidates will be eliminated and some or most of the remaining ones will achieve quota and be declared elected to fill the seats. The next table applies here as well.

But it can happen that not all the seats are filled by the time the field of candidates thins to the number of remaining open seats. In that case, likely the number of votes still in play has also decreased by some votes becoming exhausted un-transferable due to not having any un-used marked back-up preferences.

In that case the number of Effective Votes may be lower than in the next table. However that is not to say that some of the votes who cast what turned out to be exhausted votes are disappointed by the election. Some of the exhausted ballots are marked for candidates already elected, so the voters in some cases sees one (or more) of his or her preferred candidates elected anyway. without his or her vote being needed.

Oddly, in cases where candidates are elected with partial quotas at the end, the proportion of Effective Votes may be larger than where candidates are elected with quota. They may be larger or not, but for sure there is only one candidate whose supporters did not elect anyone. The only candidate who is not elected and whose votes are not transferred is the final one eliminated (or declared defeated).

Whereas when candidates win with full quotas, there may be many voters whose votes were not used to elect anyone. In both cases the number of votes cannot be more than a quota, but in one there may be a noticeable number of candidates whose supporters did not elect anyone. The supporters of this group of middle-ranking candidates may feel disenfranchised as their votes were pretty much ignored.

Luckily, most of the historical Canada STV elections came to an end with candidates being elected with partial quotas after the last elimination takes place and only the votes of one candidate are ignored. At that stage, when the last elimination takes place, the only candidates that the votes could be transferred to are going to be elected anyway.

It could happen that the field of candidates is thinned to the point of equality with the number of remaining open seats by the election of a candidate. In that case, the only ignored votes are the surplus of that winner!

Effective Votes in a FPTP district can be as low as 18 percent of the votes cast and are not often as large as 63 percent of the votes.

Math Proof for these statements

This example assumes 240 votes for each seat

(actually districts do not have precisely the same ratio of pop. per seat, and the turnout can vary considerably from district to district)

(Districts of that same size are grouped to make multi-member districts)

Effective Vote = votes used to elect winners if successful candidates all have quota.

Droop quota is used in this example.

Votes in this table can mean either:

votes bearing first preferences marked for the candidate or

a combination of votes bearing first preferences and votes bearing back-up preferences marked for the candidate.

1 district (FPTP)

240 votes 33 p.c. (or fewer) is enough to be elected = 80 Min. Effective vote = 80

--------------------------------

(in 2-seat district Droop quota is 1/3 plus 1 but in table below, for shortness sake, 1/3 is used as Droop.)

MULTI-SEAT DISTRICT (where 240 votes cast per member)

2-seat district (STV) 480 votes 33 p.c. is quota = 160 Minimum Effective votes= 320 2/3

3-seat district (STV) 720 25 p.c. is quota = 180 Minimum Effective votes = 540

4-seat districts (STV) 960 20 p.c. is quota = 192 Minimum Effective votes = 768

5-seat districts (STV) 1200 17 p.c. is quota = 200 Minimum Effective votes = 1000

6-seat districts (STV) 1440 14 p.c. is quota = 206 Minimum Effective votes = 1236

7-seat districts (STV) 1680 12 p.c. is quota = 210 Minimum Effective votes = 1470

8-seat districts (STV) 1920 11 p.c. is quota = 213 Minimum Effective votes = 1707

9-seat districts (STV) 2160 10 p.c. is quota = 216 Minimum Effective votes = 1944 10-seat districts (STV) 2400 9 p.c. is quota = 218 Minimum Effective vote = 2182 (91 p.c.)

It is probably easier for a small party like the Greens to get a seat when only 10 percent of the vote in a district is required to win a seat than under FPTP where a third of the vote in the district (1/3rd of 240 in this example) is the basic minimum (more or less) and the amount needed might be as large as 50 percent plus one (when only two evenly-matched candidates compete)

Note: Minimum Effective Vote is the number or proportion of votes (approximately) that is expected to be used to elect the winners. Not all members are elected with full quota but also some may be elected with more than quota. Almost never does the elected candidate get exactly quota - those with full quota get more than quota, although the surplus votes in many (but not all) cases are transferred away.

But at the end, some elected candidates are recorded as having more than quota.

So Minimum Effective vote may be considered to be the normal expected outcome.

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It is surprising that some people get heated about which quota is “correct”. There are a half dozen in use and, except for the most bizarre, the choice makes little difference.

The number of seats effects the accuracy of proportional representation more than the quota rule does. Recall that 5-seat PR districts create fewer “wasted votes” than 3-seat districts. That means fewer votes go to parties (or people) who fail to win a seat, and each party's share of seats is closer to its share of votes.

Give the district enough seats so the quota is less than the voters in each significant faction.

[Or if you are like Prime Minister Trudeau and are worried bout extremist politicians being elected, then set quota at an amount just above the support level of the "extremist" politicians. That seems self-serving but is likely the process when a country changes its electoral threshold.

(It is even the cause of a country denying itself a fair election system as in Canada's case.)

Four percent is a high electoral threshold under a list PR system.

Threshold of one percent is obviously more open.

The People's Party peaked in 2021 at about 5 percent, it seems.

Liberals, Conservatives, NDP, Green, Bloc Quebecois, People's Party all got more than 2 percent.

The Communist Party, Mavericks, Christian Heritage, Rhinoceros and many more all got less than one percent, so if you want to prevent all but the big six from getting elected representation, then set electoral threshold at 1 percent.

STV does not use electoral threshold, but the quota used at the district is about same as the total votes/elected members, so set your chamber size at 100 or less and you are sure that the quota used in a multi-member district will be equal to about 1 percent of the votes cast overall.

Below I show why I believe that.]

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A Brief History of Quotas Toward Inclusive Democracy

This page focuses on optional quotas for electing ensemble councils [multi-member contests].

Nicolaus Tideman and Daniel Richardson, wrote an excellent history of all STV quotas which is available online Better Voting Methods Through Technology (retrieved May 2015).

Plurality rule can leave a large majority of the voters unrepresented.

Even where majority rules, close to half of voters can be unrepresented, as is common in close two-way races.

A Brief History of Quotas

Thomas Hare developed STV to end such failures by older rules.

Hare's simple quota of ballots needed to win a seat made every vote count:

Quota = Votes cast / Seats.

If there are 100 votes and 5 seats, the quota is 20 ,and all 100 ballots are needed to elect 5 reps. [assuming they all get full quotas, which is seldom or never the case].

This is called either the simple quota or the Hare quota.

Sometimes the last candidate eliminated is the political opposite of the last candidate elected. [and sometimes it isn't]

[If the vote count process continues past the last member being elected, which there is no reason for it to do so:]

Ballots are forced to transfer to a very low preference or, if that preference is not marked, the ballot cannot be transferred, is thrown out and the last rep is elected with less than a quota.

Critics found that simple quota can under-represent a majority as shown in case 1 below. [The Hare quota can be unfair to the largest party.]

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

Quota = (Voters / Seats + 1) + 1 vote. (The term "+ 1 vote" avoids a tie for the last seat.)

This quota has been refined further by Newland and Britton [the NB quota] and by Irwin Mann.

Most jurisdictions that use STV use one of these three quotas. [three: Hare, Droop, NB -- most use Droop]

Critics point out that, unlike Hare's simple quota, the Droop and NB are designed to leave 1 quota of voters with no rep. The failure of Hare quota to empower the majority can lead to more serious political turmoil than the failure of Droop quota to represent a minority. [actually, he means the failure of Droop to empower the supporters of less-popular candidates, who may or may not belong to small or medium parties, that make up the remaining quota]

But Hare's failure may be unlikely

[not always does one party take a majority of votes in a district]

and Droop's failure almost certain in a particular electorate [district]

[but even if almost inevitable, the democratic failure is much less, the candidates of the unelected quota are often additional members of parties already elected in other seats in the district, i.e. the candidates not represented include perhaps a third Liberal or a second Conservative, like that.]

The easy quota, (Votes+1) / (Seats+1), is easy to remember. It is between Droop and Hare. So it does not fail in quota-borderline cases. (Such cases can make Droop or Hare elect too many or too few winners for the majority.

[This fourth type of quota seem un-necessary to me]

Tideman, Dr. L. Bruce Anderson and others offer many examples to show how this happens.

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

from The_Single_Transferable_Vote.pdf

The Choice of a Quota

After Hare's idea of transferring votes of eliminated candidates [and surplus votes of winners], the next significant improvement in STV came in 1868 when another London barrister, H. R. Droop, proposed that the quota be reduced from the integer part of N/K to the integer part of [N/(K + 1)] + 1 (Hoag and Hallett, 1926, pp. 177, 378-80).

The Droop quota is the smallest integer quota such that no more than K candidates can have a quota of votes. [where quota is arrived at by votes/K+1 +1]

The value of the Droop quota can be seen by considering the following example.

[in small fields of candidates, early elimination of a candidate may prevent that party from getting its full share of seats. But this is seldom a problem.

And even if it is, it is not clear to me that in all cases Droop will save the day. see "[Droop not capable of always achieving majority rule]" below]

Suppose that two Democrats, D and E, and two Republicans, R and S, are competing for three positions. There are 100 votes, distributed as follows:

24 rank the candidates DERS,

23 EDSR,

32 RSDE, and

21 SRED.

Notice that the first two groups of voters rank both Democrats ahead of both Republicans, while the second two groups rank both Republicans first. Since a majority of the electorate ranks both Republicans ahead of both Democrats, one would expect the Republicans to be awarded two of the three positions.

But if the quota is set at 33 in accordance with Hare's proposal, then no candidate receives a quota of first-place votes, and the candidate with the fewest votes, namely S, is eliminated, so that Democrats are awarded two of the three seats even though only 47 percent of the electorate favor the Democrats.

[on other hand, under STV as in most eleciton systems, party popularity is usually measured by reference just to first preferences.

54 voters gave first preferences to Republican candidates, 47 to Democrat candidates.

so same thing!]

On the other hand, if Droop's suggestion is employed, then the quota is 26. R is elected with a surplus of 6, which is transferred to S, securing S's election, and the faction with a majority of the vote is awarded the majority of the seats.

In view of examples of this sort, most proponents of STV recommend that the number of voters be divided by K + 1 in computing the quota.

[see Proof footnote below]

But there is some sentiment against [Droop], dividing by K + 1, because this means that the vote-counting process creates K equal-sized groups that are represented and one group of the same size that is not represented.

However, dividing by K + 1 can be regarded as a generalization of the principle that just more than half the votes are needed to win when one person is elected. And there seems to be no other way to avoid embarrassing cases of a majority coalition receiving a minority of the positions.

[Droop not capable of always achieving majority rule]

While the introduction of the Droop quota improves the performance of STV, in some examples a majority coalition is still awarded a minority of the positions.

[Again I say in small fields of candidates, early elimination of a candidate may prevent that party from getting its full share of seats. But this is seldom a problem. Why? because never in Canada in elections is only a battle between two parties.

Third parties and Independents do run and do take voters and are eliminated (in most cases ) and whose transfers change balance between two main parties.]

Suppose for example that four Democrats, D, E, F, and G, [bold below] and four Republicans, R, S, T and U, are contesting an election in which there are 80 voters and seven positions to be filled.

The votes are distributed as follows:

11 STUREFGD

11 RSTUDEFG

11 TURSFGDE

10 DEFGRSTU

10 EFGDSTUR

10 FGDETURS

9 GDEFURST

8 URSTGDEF

This example has a pattern similar to the previous example, ...

Since 39 voters [marked first preference for Democrats] while 41 [marked first preference for] Republicans, one would expect a majority of the positions to go to the Republicans.

However, the Droop quota is [80/(7 + 1)] + 1 = 11. Therefore the three Republicans with 11 votes are elected, and then the fourth Republican is eliminated, resulting in the election of the four Democrats. [it is not common for most of a party's candidates to be elected before the first elimination of a candidate of that party.]

Newland-Britton Quota (NB quota)

The desired result [Republican majority] is achieved with a quota of N/(K + 1) [NB quota], along with a tie-breaking rule to deal with the possibility that [more than K] candidates receive quotas. Such a quota was proposed by Robert Newland and Frank Britton (1973) and will therefore be called an NB quota.

The difficulty with the NB quota is that it does not treat certain ties fairly.

[the author here extends tie to mean cases where two different voting blocks rank two candidates in first two ranks and he lumps them together to make a composite voting block and then says that that composite voting block is tied with two winners who got quota with individual support. He then says that one candidate of the composite voting block is eliminated and one of the composite voting block is in running for tie-breaker with the two individual winners.

But under STV ou would not know what voters' first two chocies are anyway.]

For example, suppose that four candidates are competing for two positions and there are 12 votes distributed as follows:

4 for DFGE,

4 EFGD,

3 FGED, and

1 GFDE.

The NB quota is 4, so candidates D and E are elected. But since there are four voters who rank F and G ahead of all other candidates, it seems unfair that both F and G are invariably eliminated.

[Mann's quota = reversion to Droop at least in part]

A mechanism for achieving a fairer result has been offered by Irwin Mann (1973). Mann's rule is that a candidate is not declared elected until he or she has more than an NB quota of votes, with all of the excess above the NB quota treated as surplus. [Thus Mann is saying use Droop. Having to get more than NB is same as achieving or surpassing Droop. 12/3 = 4 + 1 = 5]

Thus, for this example,

no one is elected upon the distribution of the first-place votes,

G is eliminated,

that vote is transferred to F, and

in view of the resulting tie, two of D, E, and F are chosen by lot to be elected.

[actually normal tie-breaker rule is to look at first preferences. FGED only had 3 first preferences, so in that case if that rule is applied, D and E would be declared winners anyway.]

...

The Refinement-Comprehensibility Trade-Off

Each refinement of STV answers an objection to an earlier version, but at some cost in computations or in making the rule less comprehensible. Are such refinements worth their costs?

Some members of the Electoral Reform Society who are concerned with spreading the acceptance of STV - and have considerable experience with trying to explain it - believe that any rules more sophisticated than those introduced in the early 1970s for elections in Northern Ireland (a variation on the Senatorial rules) would be unacceptable to a general electorate.

Others would stop at the Newland and Britton rules or the Meek rules.

However, whatever the tolerance of computational cost and complexity may be, the more sophisticated rules provide valuable insights into the cost of overcoming limitations of the simpler rules.

Also, experimenting with sophisticated rules can reveal the losses, if any, from using simpler rules. In experimenting with alternative rules for STV, I have found that, when more sophisticated rules make differences in real elections, they generally produce outcomes that the simpler rules would have produced with changes of very few votes. [that is, most simple STV elections produce same result as sophisticated-rule systems]

Still, the more sophisticated rules generally do yield outcomes that are more defensible when [it is estimated that a different result would be produced. Not that this all strictly hypothetical.] Thus it is sensible to use the most-sophisticated STV rules that engender no significant opposition for their sophistication, and rest assured that no great harm is done.

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

Proof footnote

2The proof that Hare's proposal satisfies the "proportionality for solid coalitions" condition described in the previous note is as follows:

Suppose that there is a solid coalition, with a size greater than or equal to J quotas, for the set of candidates C, and that there are at least J candidates in C. The number of voters not in the coalition is at most N(K - J)/K, which is enough to provide at most (K - J) quotas.

Therefore the number of candidates that can be elected without using votes from the coalition is at most K -J. The votes of voters in the coalition are assigned initially to candidates in C, and since the coalition ranks all of the candidates in C ahead of all other candidates, these votes remain assigned to candidates in C when candidates are eliminated, as long as there are candidates in C that remain unelected and uneliminated.

Since the coalition has enough members to elect J candidates, it is not possible for the last unelected candidate in C to be eliminated until J candidates from C have been elected. Since those outside the coalition can elect at most K - J candidates without votes from the coalition, the election cannot end before J candidates from C have been elected.

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

STV district quota contains same votes as valid votes/members in chamber (approx.)

Here I show why I believe that.]

Fairness of MMDs is shown by fact that where ranked votes and quota is used, quota in the MMD, no matter what DM, is not far off of basic valid votes/seats quotient, or is lower in many cases, often as low or lower than the electoral threshold used in some list PR systems.

90 members overall (approx. number of AB MLAs)

say an MMD of 9 seats

population in district is 1/10th of pop (if distrcits are equally sized per DM)

votes cast in district is roughly 1/10th of votes cast overall.

overall quota in district is 1/10th of that so roughly 1 percent of overall votes cast.

35 members (approx. number of PEI MLAs, AB MPs)

say an MMD of 7 seats

1/5th of population, 1/5th of votes cast overall,

quota is 1/8th of that so about 2.6 percent of votes cast overall.

That perhapss explains how Ireland's STV is calculated as having low GI, even though its max. DM in district is only about 5 or so.

looking at Ireland Dail Eirann elecction, 2020

quota in each district is about same as one/160th of the votes cast, Dail Eirann having 160 members.

say we look at Cork South Central. it elects 4. 57,000 votes cast there

quota is 11,429

votes cast across Ireland in 2020 == 2.2M

divided by 160 = 13,757 (1/160th of votes cast)

Cork South Central has 1/40th of country's members. and almost exactly 1/40th of votes cast - 2.6 percent.

Cork South Central quota is 1/5th of that == 11,429 (taken from Wiki "Cork South")

11,429 is a bit lower than 13,757, the natural threshold across country.

One difference is: to win a seat, a party must take 11,429 (or thereabouts - quota varies slightly from district to district) in just one district. A party whose suporters are thinly spread across Ireland will not take a seat unless it gets quota (around 11,429) in any one district.

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

Here I look at Denmark 2022 election and see that overall electoral threshold was more of barrier to small parties than use of list PR and distrricting, or than STV would have been.

Email subject line not quite right

I mean:

Droop quota in a district is less or equal to amount needed to take a seat under list PR,

in many cases anyway

Supposing that the Droop quota is used in a nine-seat district, the Droop quota is 10% (plus one vote); in a three-seat district, it would be 25% (plus one vote). This electoral threshold looks significantly higher than for most party-list PR systems, based on percentage points.

But Droop quota in a district covering just part of a jurisdiction may be set at as few votes as an list PR system's electoral threshold, set at a lower percentage but based on the votes cast across a whole jurisdiction. (and also set at less than it takes to win a seat under list PR.)

For instance, in the 2022 Danish general election, the main electoral threshold of

2 percent in use meant 71,000 of the 3.534M votes cast overall were required for a party to be eligible for leveling seats,

while in the 10-seat North Zealand Folketing constituency, Droop quota (set at 9 percent) would have been 26,500 (1/11th of 292,000 valid votes).

In the North Zealand constituency in the 2022 election, held using list PR (where theoretical threshold is ten percent of district votes) the Conservative People's party with just 22,000 votes filled a seat, ten percent of the seats in the district.

(75 percent of votes cast in district were used to fill seats,

the other 25 percent of votes had been placed on parties that did not have more than 7 percent of votes cast.

1/10th was not solid threshold, due to the wasted votes problem.)

Meanwhile when levelling seats were allocated, the Independent Greens with almost 32,000 votes overall were not allocated any seats, not even as top-up.

Independent Greens took less than the threshold but did have enough votes to take a seat if they had had just two-thirds of their support in North Zealand (or another district).

<ref>https://www.dst.dk/valg/Valg1968094/other/Fordelingen-af-mandater-ved-FV2022.pdf</ref>

Perhaps that is a rule of fair elections:

Fair voting allocates a seat to any party with votes about equal to

total votes/total seats, whether seats are allocated overall, or in multi-member districts, barring cases where electoral threshold is set at higher than total votes/total seats.

Where districts are used, effective electoral threshold is likely to be about or less than overall votes/seats, but party must have its voters somewhat concentrated to get district seats.

Votes in districts where they are not plentiful enough to be due at least one seat are disregarded.

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Montopedia

What is, was and could be

Quotas and Effective Votes as percentage of whole -- Hare, Droop or NB -- District Quota contains same or fewer votes than list PR electoral threshold

A Brief History of Quotas Toward Inclusive Democracy

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