Harmonic Mean Quota for very highly Proportional Representation in Single Transferable Vote (STV) Elections.


I did write this section later but I'm putting it first, because it's about principles that underlie the arithmetic that follows on this page. And which will explain more fully the terms Hare quota and Droop quota, incidentally mentioned here.

I define an election quota as a ratio of representation of votes per seat, which rationalises from the single-member constituency case to Proportional Representation in multi-member constituencies. (It is impossible to have PR in single member constituencies, because a proportion is an equality of ratios between at least two representatives in a constituency, elected on an equal number of votes per seat.)

A single-member constituency cannot be won by all the votes (which is what the Hare quota requires) unless the voters are absolutely disciplined or shepherded to a single choice of candidate. In effect, the Hare quota, in the extreme single-member case, is a quota of complete social control.

The Hare quota if it were achieved in the single-member constituency would remind you of those old Soviet elections where the candidates were elected with virtually all the votes, barring accidental short-falls. Utopia!

The other extreme is the minimum requirement for winning, given by the Droop quota, where a candidate can win a single seat with just half the votes. (If there was another candidate also with half the votes, he would have to win a random choice decider as well, such as by drawing lots.)

The Droop quota for electing a candidate does not require a truly decisive victory but one that could be explained purely on grounds of chance, if the majority is only slightly more than half the votes.

A random choice deficit is the opposite kind of deficit to the deterministic deficit. The argument applies more tenuously to two member constituencies. The first elected candidate might be deemed more prefered than the others, owing to a transferable surplus vote. But between a second and third candidate, it may not be clear that one is significantly more prefered than the other, in gaining the second seat thru the Droop quota.

So, there are these two quotas, especially in the single-member constituency, which require the vote either to be totally determined or allow it to be completely random. Neither is a truly democratic result.
It might happen on odd occasions that some extremely popular candidate gets sensationally high proportions of the vote. But as a routine happening, it is more like the hall-mark of a totalitarian state.

A single-member election, on First Past The Post, effectively reduces choice to a two-party system. Half the voters may abstain thru indifference between the limited choice of candidates, and the other half of the voters themselves split nicely between just two candidates.

This cannot even claim to be maiorocacy, the tyranny of the majority, when one quarter of the voters have a statistically insignificant majority over the other quarter who also bothered to vote.

This is predictable as a binomial distribution of random choice, refered to below and on my page: Choice Voting America?
A greater choice, of candidates than two, predicts a progressively larger turn-out with increased choice.

The Alternative Vote allows a ranked choice of candidates, so votes are not split between candidates and a winner must have an over-all majority. John Curtice surveyed its use in by-elections in Scottish local government and found very slight change in effect from FPTP.

Australians, who have used the Alternative Vote for over eighty years, say all the Alternative Vote does is put the post in First Past The Post.

The more seats per constituency, the less importance, with the Droop quota, that the contenders, for the final seat, may only have a random difference in support.
Likewise for the Hare quota, the more seats per constituency, candidates need progressively smaller proportions of the vote to be elected, and the less important the expectation that the last seat will be taken by a candidate with all the remaining votes, tho this is still too controling and unrealistic.

Large multi-member constituencies greatly diminish the excesses of the Hare quota and Droop quota, as exposed in the single-member constituency. But in principle an intermediary quota is required between their respective determinism and randomness, to establish a level of genuine choice, not control or chaos.

This page will show how the intermediate quota arithmetic is satisfied by two quotas which I've called simple harmonic mean quota and harmonic mean quota (in the latter case taking into account natural variations in electorates across constituencies, with uniform multi-member constituency systems being considered).


1. The turnout-reducing minimal choice of the single-member system is undemocratic.
Single-member constituencies minimise choice which minimises turn-out. As the binomial theorem predicts, a First Past The Post two-party system turn-out may be only 50%.

2. The single-member majority is an undemocratic decider to an undemocratic turn-out.
The minimal choice forces a minimal (Droop) quota, that is an over-all majority, allowing election of no more than random significance. That is an approximate 50%, for one of two candidates, of the 50% turn-out. Or, about 25% of the voters may decide the winner.
If any more do, their vote is unnecessary or wasted, anyway. And that is the point of a transferable vote, from candidates already elected, to help elect next prefered candidates.

3. Whereas the maximal (Hare) quota requires a totally determined vote all for one candidate, in a single member constituency.

4. Between totally determined and completely random choice a new average quota is required for a choice that is meaningful but free.

5. Since the Hare quota and Droop quota generate a harmonic series, their average must be a harmonic mean quota. This is what I've called the simple harmonic mean (SHM) quota. [v/(s+ )]

6. A more radical possibility is to have a harmonic mean (HM) quota [(s+2)v/(s+1)^2], whose purpose may be understood in terms of compensating for unequal electorates e.g. in a uniform multi-member system of naturally varying community sizes.

Hare and Droop quotas.

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Proportional representation was introduced for use in formal elections, at almost the same time that the theory of natural selection was published. Both innovations were independently discovered.

The Danish liberal, Carl Andrae slightly antedated the English barrister, Thomas Hare, in their original system of consistently reforming both the vote and the count. Votes are proportionately redistributed according to an order of preference, given by the voters, to show how they wanted the votes to go from successful candidates to those still trying to get elected to seats in a multi-member constituency.

In the single-member constituency elections won by one candidate getting more votes than another, equal representation merely means equal constituencies. Many or most of their voters might not have voted for the single-member who monopolises the constituency representation.

But Hare system of equal representation is much more rigorous. Candidates are elected on gaining equal shares, or quotas, of votes. Virtually all the voters are represented by someone they actually voted for, usually a first preference or at least a high preference.

This proportioning of the vote was originally done by the so-called Hare quota. This is simply the number of votes, v, divided by the number of seats, s. Or: v/s.

Hare system including Hare quota came out in 1859. By 1914, it was still in the example HG Wells gives in his great essay, The Disease Of Parliaments.

The Hare quota does not work for single-member constituencies, because it would mean that the winning candidate would need to win all the votes in the constituency. Whereas a decisive result could be obtained by winning just over one half the votes. This is not the number of votes divided by the number of seats, but the number of votes divided by one more than the number of seats. Or: v/(s+1).

This slightly different quota also has a name, the Droop quota, again named after the person who thought of it. In modern elections, the Droop quota is the standard quota for use with the single transferable vote.

Hare and Mill first advocated proportional representation, in terms of a national constituency with hundreds of seats. In that case, it makes very little difference which quota, Hare or Droop, is employed in the count.

Even by 1914, reformers were still thinking in terms of very large constituencies, covering the regions, if not the whole nation.
Gradually over the decades, reformers proposals have further conceded smaller and smaller multi-member constituencies, to make this system less equitable and less competitive, in mostly failed attempts to appease place-holding opponents.

But in fairly small multi-member constituencies, the Droop quota is regarded by some as not proportional enough.
This is especially so of small party activists hoping to glean a seat in Parliament.
While former Irish premier, John Brunton submitted a document that advocated STV for the European Union, the claim that it is not proportional enough has been used on the Continent to block STV for general use in the Euro-elections.

In a two-member constituency, Droop quota requires two candidates to be elected on one third of the votes each, giving a proportional representation (PR) of two thirds of the voters.
In a three-member constituency, the Droop quota requires the election of three candidates on one quarter of the votes each, for a PR of three quarters of the voters. And so on, for four, five, six etc member constituencies.

A discussion on the STV-voting e-mail group, about the relative merits of the Droop quota and Hare quota, refered to a test example in Comparative Electoral Systems (1982) by Robert A Newland.

That example had the Hare quota elect less candidates from a larger party than the smaller party, rather than the other way round with the Droop quota. This was because the smaller quota freed up votes for transfer from the more popular candidates in the larger party to its less popular candidate.

This example is all right as far as it goes. But it is based on a strict partisan divide between the voters. It is rooted in traditional British politics, when a Tory agent boasted that if he put up a pint pot of beer he could get it elected.

The example is something like the truth where you have absolute partisanship, as in an economically-based class war or doctrinally-based religious war. This is the situation of an entrenched duopoly that divides and rules.

Hypocritically incited conflicts apart, the Newland example is not typical. Common sense critics of contrived refutations say: People don't vote like that. That is a valid rebuttal of what might be called theorem syndrome. There is more to election method than deductive logic, vital tho that is.

As JFS Ross said, in Elections and Electors (1955), the way people normally vote, forms a normal distribution of choices, where opinions gradually shade off. Instead of two blocks of brilliant red and brilliant blue, the two political colors of left and right merge, such that the dominating color, in center, is more or less purple, so to speak.

The Newland assumption that party choice always trumps personal choice is based on historical conditioning to dogmatic partisanship. The example, tho valid in itself, obscures the fact that the smaller (Droop) quota yields a somewhat smaller PR than the bigger (Hare) quota. And, because a larger quota is somewhat more difficult to achieve, it more probably indicates the personal popularity of the candidate.

The Newland partisan inference that the Hare quota can give a less democratic result than the Droop quota, as a general rule, cannot be substantiated. A party is just a coalition of particular policies out of an astronomical number of other possible combinations that might make up a manifesto.

With the single transferable vote, the voters free order of preferences for candidates, in effect, is their own personal manifesto. There are thousands of them. The purpose of the STV count is to add up those thousands of manifestos into one composite or communal manifesto, not merely to arbitrate between two particular party manifestos.

The fact, that one particular party might lose out by a candidate, who is personally unpopuar, to another slightly smaller particular party, does not amount to a democratic indictment of the Hare quota compared to the Droop quota.

What one can say with assurance is that any candidate with not enough votes to be elected by the Hare quota might have enough to be elected by the Droop quota. And if that candidate happens to belong to a party, perhaps a small party that cannot muster more votes, then the Droop quota favors small parties more than the Hare quota. This general conclusion is in flat contradiction to Newlands conclusion, that depends on the primacy of partisan over personal popularity.

The principle of freely transferable voting makes no such presumption.

The Hare and Droop quotas respectively favor candidates with larger and smaller quotas of support, and thereby might respectively favor larger and smaller parties.
To neutralise any imbalance either way, would require an average quota between the two. Because Hare and Droop quotas generate a harmonic series, the suitable average would be the harmonic mean. Hence the simple harmonic mean (SHM) quota, shown in the following section.
There is also the possibility of a more radical harmonic mean (HM) quota, shown in a further section.

The Droop quota undoubtedly offers the necessary decisive choice, especially in the single-member constituency. With it, a majority of one vote is enough to win.
The trouble is that majorities, as in US presidential elections between two main candidates, tend not to be convincing. Especially if you factor in the skewing of the result from one candidate being able to spend more campaign money than the other.

The word, campaign, and the result, spoken of as a victory, give away the game as one of a ritual battle, rather than a rational debate. Statisticly, such single-member majority wins may be no more significant than tossing a coin.

It is time to question the standard use of the Droop quota in all STV elections.

There are four logical possibilities (by the binomial theorem) to an out-come between two candidates. One quarter the voters will like each candidate equally, and not bother to vote; another quarter dislikes each equally and wont vote. The other two quarters, making half, prefer one or other candidate.
That is a remarkably accurate prediction of the voting patterns of US elections, where half the electorate dont vote, and the win is usually not by a statisticly significant margin.

The French Second Ballot allows more choice and has higher turn-outs. In 1906, H G Wells, on The Future In America, pointed out a need for the Second Ballot. This is itself now an antiquated system.

To repeat the point, the Droop quota, when just in a single member constituency offering an over-all majority, may be little or nothing more than a result of chance rather than choice. The Droop quota for single majorities is not just an elective minimum quota but also may be a minimally elective quota.

An executive election such as the US presidency would proably be better conducted for a five-member ruling council by STV/PR. This would ensure a much higher turn-out. The binomial theorem predicts over 90 per cent, even for just five candidates instead of two (on the basis of two to the power of five, instead of two to the power of two, preference combinations, where only 2/32, instead of 2/4 of the voters would completely like or completely dislike all five).

This high turn-out most likely would be sustained by the voters knowing that their five most prefered candidates would be elected, as distinct from a preference vote merely giving a more sporting chance to all five candidates contending for a single seat.

Of course, there are logistical or practical difficulties that lower theoreticly predicted turn-outs under conditions of perfect voter access to the polls.

In a five-member ruling council, the President would just be the first among equals, much more representative of a vast continental nation. A single leader, chief of the armed forces, as the US president is, is designed more for a situation of war emergency. It may even encourage leadership by chronic war emergency to divert from really addressing an intractable state of the nation.

The moral is to avoid single member elections either to the legislature or the executive.

As a statistics student, i remeember our lecturer saying that a random sample size should be at least thirty. And I thought to myself at the time, that's about a random or binomial distribution of thirty-two (two to the power of five) which is just becoming meaningfully differentiated as a distribution.

For a (effective) choice of five candidates, the predicted non-turnout of one-sixteenth or 6.25% just falls short of the 95% confidence limit conventionly used in statistical tests to decide that a result is statisticly significant. So, you wouldnt want less than a five-member constituency, where the voters can be assured that the five most prefered candidates would be elected. Otherwise, a drop in turn-out may significantly reflect on the level of democratic legitimacy of the result.

One possible reason, why the turn-out in Irish STV elections has fallen, may be that the average number of seats per constituency has been reduced to between three and four-member consttuencies.

HG Wells advocated electoral reform as proportional representation by the single transferable vote in large constituencies. He explained it was necessary to state this formula because politicians were always trying to mutilate the essential method for their party prospects.

He regarded an inter-war reform proposal of four seats minimum as much too small.
I think if we are to get away from political expediency towards a scientific or knowledgable basis for election method, that suggests a minimum of five seats per constituency (six perhaps better). Tho, local exceptions might have to be taken into account. This suggests that Ulster is about right with their six-member STV system.

The historical change from using the Hare quota to the Droop quota for STV has less to do with logic than the relentless political pressure to make elections less competitive and less democratic.

That is why, when reform is expedient, its movers still sabotage representative democracy with a Mixed Member Proportional system, a mix of two least democratic kinds of system: single-member constituencies and a party list vote.

Had HG Wells formula of PR by STV in large constituencies been upheld, there probably would never have been a change of quota from Hare to Droop.

That is not to say that one is better than the other. They are both limit quotas. The Hare quota is an upper limit quota and the Droop quota is a lower limit quota. This means by definition that neither is representative but situated at the extremes of a range.

The representative quota of that range, being in terms of a harmonic series, is what I've called the simple harmonic mean quota: v/(s+ ), the subject of the next section.

The simple Harmonic Mean quota: SHM quota.

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The simple Harmonic Mean quota.

Step one. (This is already generally accepted.)

The Droop quota is the elective minimum quota. When all the seats have been filled by the Droop quota, there is exactly one quota left, so that no remaining candidate can have more votes than those elected to a seat.

It is possible that in the last round two candidates might tie with the quota for the final seat. In that case, lots are drawn or some other random method of choice.
This is very unlikely with large-scale elections and many candidates.

Plainly if the quota were any smaller, there would be a possibility of candidates getting elected while enough votes remained for other candidates to be inequitably denied chance of winning a seat.

Step two.
Taking the Hare quota, giving maximum representation, as the maximum quota.

Step three.
Since the Hare quota and Droop quota both form a harmonic series, the harmonic mean is the correct average to represent this maximum-minimum range of quotas.

The harmonic mean calculation is as follows. Add inverted terms: {s/v + (s+1)/v}/2 = (2s+1)/2v. And invert the result.
The harmonic mean is thus: 2v/(2s+1) = v/(s+1/2).

For a single-member constituency, the HM is 2/3 the votes; as should be between 1/1 and 1/2 the votes.
For a two member constituency, the HM is two-fifth or 12/30 the votes; as should be between one half and one third or 15/30 and 10/30 the votes.

And so on, as summarised in table 1.

Table 1: Droop & Simple Harmonic Mean quotas (in elective proportions of votes) & comparitive proportional representations.
Number of seats. Droop quota: v/(s+1) Droop PR: sv/(s+1) Simple Harmonic Mean quota: v/(s+1/2) Harmonic Mean PR: sv/(s+1/2)
1 1/2 1/2 = 50% 2/3 2/3 = 67%
2 1/3 2/3 =67% 2/5 4/5 = 80%
3 1/4 3/4 = 75% 2/7 6/7 = 85.7%
4 1/5 4/5 = 80% 2/9 8/9 = 88.9%
5 1/6 5/6 = 83.3% 2/11 10/11 = 90.9%
6 1/7 6/7 = 85.7% 2/13 12/13 = 92.3%
7 1/8 7/8 = 87.5% 2/15 14/15 = 93.3%

The simple Harmonic Mean quota is a less dramatic improvement in proportionality than the Harmonic Mean quota. (See succeeding sections.) Yet it still does not replace the need to resort rather to the Droop quota for single member constituencies. Tho, of course, single member constituencies, and the smallest multi-member constituencies of two or even three seats, should be avoided, anyway, as they cannot share, or adequately share, power by PR of the main opinion groups.

The SHM quota also for more constituency system equality.

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My early efforts to study election method involved generalising a rule given by JFS Ross for single-member systems. In my solitude and inexperience, I was much muddled in the attempt, tho I eventually got the following result.

Ross asked what is the most equitable way to represent local communities, without breaking them up arbitrarily at exactly the required average number of constituents per member of Parliament.

Say, there is an electorate of 30 million voters and a parliament of 500 members. Then the required number of constituents per member is 60,000.But every local community will not be conveniently made up of exactly that number. So the question is: how much variation in local constituencies is allowed, without their becoming too inequitable?

JFS Ross, in Elections and Electors, gave the rule for this. He says a constituency must not go above 80,000, because it would be more equitable to split more than 80,000 into two constituencies of 40,000, or more, voters each.In other words, the 60,000 required average must not be varied by more than plus or minus 20,000 voters.

To put that another way, the allowed constituency system range is 40,000, which is the difference between a permitted maximum of 80,000 minus the minimum permitted of 40,000 constituents.

This system range of 40,000 is given by the formula, V/(S + ), where V is the required average votes per constituency and S is the seats per constituency.Thus 60,000, divided by one plus a half, gives 40,000.

This system range formula or quota works for multi-member constituency systems, as well as single-member constituency systems.

For example, take a two member constituency system, whose required average votes per constituency is 120,000. Substituting into this system range formula:V/(S + ) = 120,000/(2 + ) = 48,000.

For the minimum permitted constituency, multiply by the number of seats, S = 2, to get 96,000.For the maximum permitted constituency, multiply by the number of seats plus one, S + 1 = 2 + 1 = 3, to get 144,000.

If the local community is greater than 144,000, it is more equitable to transfer at least 48,000 voters to another constituency, leaving it with more than 96,000 constituents, being nearer the required average of 120,000, than some electorate in excess of 144,000.

Having read this far, it will be seen that the system range quota, a cross-constituencies quota, is the same as the simple harmonic mean quota, as a representative within-constituencies quota.

We can say more than this.
Take the maximum permitted constituency, (S+1)V/(S + ). If the Droop quota, V/(S+1), is characterised as the elective minimum, then the minimum quota is: (S+1)V/(S + )(S+1) = V/(S + ).

At the other end of the permitted constituency range, is the minimum constituency, S.V/(S + ). If the Hare quota, V/S, is characterised as the elective maximum, then the maximum quota is: S.V/(S + )S = V/(S + ).

Thus, at the maximum and minimum permitted limits of constituency size, both extremes can use the same elective quota, namely the simple harmonic mean quota, V/(S + ), more equitably than either the Hare quota or Droop quota.

Since equality of representation is a condition of democracy, this is a further reason for using the SHM quota in STV elections.
My original reason was the need for a more representative quota, within a constituency, than either the Hare quota or Droop quota.

Now there is another reason, in the need also for a more representative quota across constituencies, in a system allowing for natural variations in size of local communities.

From the point of view of equal representation, it is as important to look across constituencies, as within constituencies. And in each case, the lesson is the same. Larger multi-member constituencies are required to reduce inequalities of representation.

Most obviously, the single-member system is the most inequitable. Within the single-member constituency, only half the voters may be represented. This is Only Half A Democracy, as Robert Newland said. But taken with the effect across constituencies as well, it is only half of half a democracy or a quarter of a democracy.
That is because, allowing for natural community size variations, the smallest permitted constituency is only half the size of the largest, which is therefore only half as well represented.

Andrew Carstairs, in Short History of Electoral Systems in Western Europe, uses this fact to denigrate STV elections in particular. The basis for this is that Irish constituencies historically have been whittled down to only three or four members per constituency.

Taking into account cross-constituency variations, Carstairs says this reduces the proportional representation to 3/4 of three quarters, equals 9/16, or 4/5 of 4/5, equals 16/25.

However, this reasoning holds for any voting system using small constituencies. For instance, it holds for the Additional Member System AMS, also called Mixed Member Proportional (MMP) system, with regard to the part of the system made up of single-member constituencies.

In short, the formula for democratic elections is: "Proportional Representation by the Single Transferable Votes in large constituencies." This was the formula frequently used by HG Wells, so it may simply be called "the Wells formula."

While I am handing out titles, it is the discoverer's prerogative to name a discovery, so the Simple Harmonic Mean (SHM) quota, which is also the Constituency System Range (CSR) quota, V/(S + 1/2), may be called "the Ross quota," after JFS Ross.

Why? You may ask.

Well, because Ross gave the rule for single member systems. Without that rule, I wouldnt have thought to generalise the rule for multi-member constituencies.

It is true, I rediscovered the same formula, in a different way, used for a different purpose. But it would be convenient to call the quota, either in its guise as SHM or CSR, simply "the Ross quota."

Ross was a great pioneer, whose name deserves a memorial.

There is more to this story. As a new student already fed-up with writing set essays, I rebelled against looking-up texts on the subject of voting method. After all, what was there to know about that? I'd sat up all general election night, hadnt I, like most people, those days?

Half way thru my essay of a complacent fool, I reluctantly faced the reality that I would have to reference a text or two. And one of these happened to be Elections and Electors by JFS Ross (1955).

My ignorant eyes were opened. Eventually, I joined the Electoral Reform Society. The secretary, Major Frank Britton, when I asked if they had the book, kindly sent me a copy. (He didnt ask for a payment but I did pay for it.)

So I owe Ross a considerable debt, and also Frank Britton.

People used to have a calling for reform. Whereas, the Alternative Vote referendum was an expedient which Enid Lakeman and Frank Britton would never have entertained.

Electoral reform has become just another career stepping stone for qualified administrators.

PR has come to mean Public Relations rather than Proportional Representation.

This web page is rather a hotch-potch (for which Ive been criticised) because I have just put down ideas as they came to me. This section was the latest to be inserted. And I havent considered how it fits in with the treatment of harmonic mean quotas and constituency systems, in the following sections, which are more complicated, but which were actually my first approach to the subject of this page.

The notional working of an Upper limit quota.

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Representative democracy is not the only kind of representation. Elementary statistics teaches to think of representation, of a distribution, by an average. Different kinds of distribution are better represented by different kinds of average.

The distribution may be thought of as a range of items at differing densities or frequencies along the range. The range may be infinite or it might have finite limits. A representative sample of items in the distribution would serve to calculate an average representative of the whole distribution. If just the two end limits are known and representative, they might be used to calculate an average representative of the whole range.

Consequently, the Droop quota might be considered as the lower limit of a range of proportional representation (PR). The Droop quota gives a PR of the quota multiplied by the number of seats. Or: sv/(s+1).
The Hare quota gives a PR of: sv/s = (s+1)v/(s+1).

These two figures suggest an upper limit of PR, as: (s+2)v/(s+1), being the third term in a series. It follows that the Hare quota gives the arithmetic mean (AM) PR of the upper and lower limits of PR. The working is:

AM PR = {sv/(s+1) + (s+2)v/(s+1)}/2 = (s+s+2)v/2(s+1) =2(s+1)v/2(s+1) = sv/s = v, which is Hare quota PR.

It may be objected that the upper limit PR is not workable. In a way, it is no more unworkable than the lower limit PR given by the Droop quota.
The Droop quota leaves up to half the voters unrepresented in a single-member constituency; up to one third the voters unrepresented in a double member constituency; up to one quarter the voters unrepresented in a three member constituency; and so on.

In other words, the lower limit PR leaves up to a quota of voters without the seat they are worth.

As to the upper limit PR, dividing it by the number of seats, s, for its quota, (s+2)v/s(s+1), this is too large for the voters to elect one of the seats. So, it cannot elect any representative in a single-member constituency. In two member constituency, it can only elect one member; in a three member constituency, only two members are electable; and so on.

Yet, the upper limit quota in a two-member constituency is subtly different from a single member constituency. Tho, it cannot elect more than one candidate, it can still transfer votes towards the election of a second candidate. There is a short-fall, yes, but short-falls are a common-place of First Past The Post elections.

It is not uncommon for the last candidate to be elected in a conventional (Droop quota) STV multi-member contest, not on having reached a full quota for the last seat, but merely on the strength of being first past the post against other remaining unelected candidates.

I have been confronted by this as a criticism of (conventional) STV without the critic realising it is a far more damning criticism of his favored single member system with a candidate winning who is merely first past the post, in the first and only round.

To sum up, the lower limit (Droop) quota leaves voters with a seat too few; the upper limit quota leaves voters with a seat too many. Either quota is a seat out in its representation.

The Upper Limit (UL) quota might be adapted to elect the full number of seats in the constituency. For instance, take the three member constituency. This quota is 5/12 of the vote. That is sufficient to elect two candidates on a combined total of 10/12 the vote.

This only leaves 2/12 the vote available to help elect a candidate to the third seat.
(It will be found with this upper limit quota, that in every multi-member constituency case, there are always remainders of just two portions of the vote available to help elect a candidate to the final seat.)

This means that another three portions of the vote must be found to add to these two remainder portions, to make up an electable quota of five portions. These three portions can only be obtained equitably by transfering votes, in order of preference for remaining candidates, from the 10/12 of the vote for the two already elected candidates, at a weight or transfer value of two times 3/2 portions.

Table 2 shows how the Upper Limit quota might work for the first few numbers of seats per constituency.

Table 2: How the Upper Limit quota could make-up its one-seat election short-fall.
Seats Elected candidates x UL quota = (s-1) x (s+2)v/s(s+1).

Remainder of 2 portions of votes available towards the final seat.

0, 1, 2, 3...elected candidates votes transferable portions towards last seat quota (made-up in each case by adding 2 portions remainder). Final seat quota (in portions of votes, v) equals elected candidates transfered vote portions plus remaindered two portions.
1 0 x v.3/2 = 0 0 x 1/0 = 1 1/2 + 2/2 = 3/2
2 1 x v.4/6 = v.4/6 1 x 2/1 = 2 2/6 + 2/6 = 4/6
3 2 x v.5/12 = v.10/12 2 x 3/2 = 3 3/12 + 2/12 = 5/12
4 3 x v.6/20 = v.18/20 3 x 4/3 = 4 4/20 + 2/20 = 6/20
5 4 x v.7/30 = v.28/30 4 x 5/4 = 5 5/30 + 2/30 = 7/30

In table 2, the single-member case is the trickiest to understand but can be derived back from the arithmetic logic of the multi-member constituency cases. No single seat is electable with the upper limit quota, which demands the impossible proportion of 3/2 the vote.

However, there is as usual a two portions remainder, that is 2/2 of the quota. The other 1/2 part of the vote may be considered as transfered from the election of zero seats. It is an arithmetic formality, I hasten to add.
In effect, the Upper Limit quota for a single-member constituency is like the Hare quota, which only elects a candidate who can win all the votes.

While impractical for large-scale elections, the Hare quota is actually the norm for self-representation, where one only needs ones own vote to represent one.

The moral is we have to be careful about assuming some all-purpose quota, when different quotas may be more suitable for some situations than others.

The Upper Limit (UL) quota, as its name suggests, is the largest quota, offering the most proportional representation on the PR scale or range. Or so it would seem in theory. Tho, the most I would venture to say in its favor is that its largest quota makes election more difficult and therefore makes it correspondingly more probable that the popularity of those elected may be regarded with more confidence.

However, my working of the upper limit quota is only notional, and somewhat tedious at that, so I wouldn't expect it to catch on. Indeed, as only a quota on a range limit, by definition it should not be considered a general-purpose quota.

Further reflection on this maximum-minimum quota range with Hare quota arithmetic mean is that the range can be applied to a uniform multi-member system (and in principle no doubt to a varying number of seats per constituency but the uniform system is simpler and all that is needed to make the point).

In practise, single-member constituency boundary drawing is obscured by gerrymandering, which Presidents Reagan and Ford agreed was the main US constitutional abuse.
Ford also addressed piper-paying campaign finance, which President Carter coolly called a disgrace.
The US Congress is an incumbents House without effective competition. That's unhealthy for a democracy, said Ford. Sweet-heart gerrymandering is the agreed boundary-drawing by neighboring representatives, of opposing parties, to give each other safe seats.

Because they are competing for government, British single member boundary-drawing is a hypocritical tug o' war between Labour and Tory to maximise their ratio of seats for votes.

Such political passion for dishonest advantage obscures merely natural factors. Geographicly large rural constituencies naturally tend to have less voters per seat than urban constituencies.
It can be supposed that given an average number of votes per constituency, v, then v/(s+1), the lower limit quota, may be the rate of the least votes per seat found in the sparsest populated rural constituency.
And v(s+2)/s(s+1) may be the rate of the most votes per seat found in the densest populated urban constituency.
The (arithmetic mean) average votes per seat is then v/s, ideally required for equal representation.

In principle, to compensate for this cross-constituency inequality in distribution of voters per seat, one could require the constituency with the lowest quota, across constituencies, to have the highest quota for the election count within the constituency, and vice versa.

This justifies on a compensatory principle, the use, across and within constituencies, of the maximum-minimum quota range with Hare quota arithmetic mean.
This is a crucial point, because the simple harmonic mean (SHM) quota, discussed above, is simple because it is not a cross-constituencies compensatory quota.

Harmonic Mean quota: HM quota.

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When all is said and done, the main purpose of the Upper Limit quota, is to usher-in a more representative quota than itself or the lower limit (Droop) quota for proportional representation.

The UL quota and Droop quotas are at the upper and lower limits of some more representative average quota. And it is known that the Hare quota, as arithmetic mean quota, is not always a suitably representative average.

Other possible averages, to represent this range of PR, are the geometric mean and the harmonic mean. Both these averages give a required smaller quota than the arithmetic mean (Hare) quota, tho only slightly smaller.

It may be observed that vote quotas, like Hare and Droop, form an harmonic series, and therefore it is logical to suppose that the harmonic mean best averages a range in a harmonic series.

The harmonic mean of the upper and lower limit quotas is calculated as follows. Invert the upper and lower limit quotas; add them and divide by two; then invert the result.

The working is:

{s(s+1)/v(s+2) + (s+1)/v}/2 =

{s(s+1) + (s+1)(s+2)}/2v(s+2) =

(s+1)(2s+2)/2v(s+2) =


Finally, inverting this result gives:


This is the Harmonic Mean (HM) quota.

Table 3 shows the striking improvement in PR with the Harmonic Mean quota compared to the Droop quota.
Qualifications of this advantage are discussed after the table.

Table 3: Droop & Harmonic Mean quotas (in elective proportions of votes) & comparitive proportional representations.
Number of seats. Droop quota: v/(s+1) Droop PR: sv/(s+1) Harmonic Mean quota: (s+2)v/(s+1) Harmonic Mean PR: s(s+2)v/(s+1)
1 1/2 1/2 = 50% 3/4 3/4 = 75%
2 1/3 2/3 =67% 4/9 8/9 = 88.89%
3 1/4 3/4 = 75% 5/16 15/16 = 93.75%
4 1/5 4/5 = 80% 6/25 24/25 = 96%
5 1/6 5/6 = 83.3% 7/36 35/36 = 97.22%
6 1/7 6/7 = 85.7% 8/49 48/49 = 97.96%
7 1/8 7/8 = 87.5% 9/64 63/64 = 98.44%

Importance of, and reservations about, the Harmonic Mean (HM) quota PR advantage over the Droop quota.

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The first row of table 3 is most off-putting with regard to the Harmonic Mean (HM) quota, because it stands at three quarters of the votes to elect a single-member. That is obviously impractical, as candidates are typically elected with not much more than half the votes. (Or less, if there is vote splitting between several candidates on a first past the post system.)

We are not going to be abandoning the Droop quota for the Harmonic Mean quota in single-member constituencies, at any rate. But this does not necessarily invalidate the general superiority of the HM quota. The HM quota is the best average or representative of a harmonic series of quotas. But the meaning of an average is that it is still only a best fit in general, not necessarily for a special case, like single member constituencies.

Single-member elections are really a special case, because there are no more seats to transfer surplus votes to. A single-member system is not a transferable voting system.

But there could be a (Droop quota) majority decision in favor of an issue. And there could also be a 75% (Harmonic Mean quota) super-majority decision, which makes that issue binding.

In the Electoral Reform Society, recently, it so happened that a 75% majority was required to make commitment to the single transferable vote binding on its council. The STV Action web-site records that the membership vote easily surpassed that level at over 90%.

One point needs to be mentioned in passing. This quota is not an excuse for inequitable voting thresholds. The HM quota having a three-quarters single-member quota does not mean that between two opposed issues, one of them needs only one-quarter the votes to win. There is no one-quarter vote quota to the HM quota.

The Canadian election referendums threshold of 60% created two inequitable quotas or thresholds, the other being 40% for the alternative decision that the politicians wanted to win. This was a fabricated disproportionate representation and was wrong and illegitimate.

In table 3, for two-member constituencies, it will be observed that the Harmonic Mean (HM) quota of 4/9 is not all that much greater than the Droop quota of one third. Yet the increase in proportional representation is more than 20%, a huge increase.
Of course, two member constituencies don't offer much diversity of representatives. A lot of seats in a large constituency is needed to make that possible.

Regarding three-member constituencies, the HM quota at 5/16 isn't all that much more than the Droop quota at one quarter, yet over three seats it multiplies out at a much bigger proportional representation, an increase of nearly 19%.

Considering successively larger multi-member constituencies, the trend of the argument continues. The HM quota gets closer and closer to the Droop quota, so that the increases in proportional representation steadily decline but still remain very decisively greater thru-out.

A striking example is that the Droop quota, for nine seats in a constituency, gives a PR of 90%. The HM quota for nine seats gives 99% PR. It would take the Droop quota 99 seats, in one constituency, to equal this PR of 99%.

I believe the trend, as the number of seats increases in a multi-member constituency, so that the HM quota becomes a not much larger quota to achieve than the Droop quota but gives much more proportional representation, I believe this is a potential game-changer in improving the prospects of the single transferable vote as the system of choice for all official elections in countries that genuinely want to be democratic.

The HM quota demolishes the argument for "STV Plus." This is a system designed to top-up the proportionality of conventional STV with party lists to ensure representation of small parties warranted on a partisan share of votes for seats over a large or very large constituency.

In any case, ad hoc or privileged list counts for particular groups, like parties, is wrong from either a scientific or democratic point of view, as I have been at pains to explain on many of my web pages about elections.

Moreover, conventional STV, as shown in Ireland, elected candidates from the tiny Green party. They even have been in coalition. Tho the proportionality has been whittled down in ireland by reducing the numbers of seats to three or four per constituency, the transfer of votes can help small parties, who may not be many voters first preferences but are still suffciently prefered to gain the odd seat here and there.

With the size constituencies originally introduced with STV into Ireland, I doubt that there would be any case for small parties to complain about the representativeness of conventional STV.

The Harmonic Mean quota is not a complete replacement for the Droop quota, any more than it would make sense in statistics to say that the average is a complete replacement for one of the range limits on which that average estimate of the whole distribution is based.

The Harmonic Mean quota essentially depends on the voters ranking sufficient candidates for a fairly exhaustive count, to make possible the recording of such highly proportional representation, as the Harmonic Mean quota affords.

STV will always remain dependent on the Droop quota, as long as there is a question of voters recording insufficient preferences for the count to elect candidates to every seat.

It may well become customary for STV election counts first with the Harmonic Mean quota. And if this cannot fill all the seats, having the computer run a second count with the Droop quota.

It might be asked what if both quota counts are run with different results? In that case, preference would go to the Harmonic Mean quota, because of the higher number of votes it requires for the election of each candidate.

In Australia, there is mostly a compulsory recording of preferences for all candidates. In that case, there seems no question that the Harmonic Mean quota would fill all the seats and record a correspondingly higher proportional representation than the conventional Droop quota.

Also in Australia, the random, rather than alphabetic, listing of candidates neutralises the donkey vote. Thus, voters, who take the trouble to prefer candidates, with discernment as to their capabilities, characters and their policies, are likely to have some effect. That is even if most voters simply follow party lines.

I actually don't approve of compulsory voting and compulsory recording of all preferences. Nor do I believe, for reasons mentioned above, in just voting for a party and letting the party managers dictate which of their candidates get into Parliament.

Therefore, I don't believe in the easy way out, this affords, for ensuring that STV elections would work with the Harmonic Mean quota.

The computer counted STV elections by the Meek method is used in some New Zealand local elections and for their Health boards. This involves a more thoro-going and systematic transfer of preferences than is generally possible with a manual count.

Another innovation is that, towards the end of the count, with less votes to go around, the Droop quota may be re-set, for a lower total vote, to help decide who will take a remaining seat.

I came across this in a technical paper by Woodall, most of which I didn't understand. He didn't actually say why it was right, just insist that it was, despite the disagreement of the ERS Council.

I think there is subtle difference in value emphasis here. The returning officer, who devises the counting rules, is more concerned to fall back on any rational recourse that will decide the contest without a legal challenge from a losing candidates agent.

Whereas a reform society is more concerned to promote the system in high principle without resorting to lower principles, however justifiable, which water down the over-all effect.

Both contradictory sides are right, according to their priorities. Their dilemma is essentially similar to the one that makes me promote harmonic mean quota in preference to Droop quota.That doesn't mean to say that an STV count using harmonic mean quota might not have to fall back on the Droop quota. Similarly Meek method may have to resort to resetting the Droop quota.

In fact my proposal, in the first instance, of the simple harmonic mean quota is less radical than re-setting the Droop quota, to a lower total vote, in Meek method.

All I am suggesting is that, to increase the PR of STV, the SHM quota be used in the count, in a multi-member system preferably with a minimum of five seats.
If there arent enough preferences to go around, then the Droop quota might have to be resorted-to, instead.

In any case, the long-term problem is to nourish a political culture that makes voters want to prefer comprehensive personal cross-party lists.

I do believe in a civic culture to replace donkey voters with informed citizens, thru competent education and fair debate, for participatory exercise of the public intelligence with regard to policies.

Measures to increase number of preferences given by voters should include encouraging candidature. The 2012 UK Police Commissioners elections had a losable deposit of 5000. Besides the rich, this left only party-backed candidates, the usual suspects to stand. This amounted to a sinister party monopoly control of police forces.

This election was on a Supplementary Vote, or two crosses, serving as a first and second preference only. Only one person is elected. The previous boards, tho only indirectly elected, were capable of being more representative, so that one party could not control the police force.

The turn-out was dire. Not remotely democratic.


The rationale for proposed harmonic mean quotas is that their problems are out-weighed by other short-comings of elections that we take for granted. When you factor in those democratic disadvantages, then there is a useful place for the harmonic mean quota [in the first place, perhaps the less radical simple harmonic mean quota, v/(s+)].

Harmonic mean quotas don't work for single-member constituencies but single-member constituencies don't work for democracy.

The central point was that there is a democratic deficit both in the Hare quota and the Droop quota. The former gives too deterministic a result and the latter too random a result. However, these contrary faults may be neutralised by taking their (representative) average, which for harmonic series is the harmonic mean.

Harmonic mean quotas require largish multi-member constituencies and a cultural commitment to nationally minded preference voting. I see the need for greater voter expression of preferences as a cultural problem largely due to the parties taking a possessive partisan view of how many orders of preference the voters should cast.
The parties think their divisions are more important than the nations unity. From the point of view of the direction a nation has to take democraticly, that attitude has to change. Cross-party issues may be the most important to most people.

Foot-note on the geometric mean to average keep values in preference-unpreference Binomial STV.

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I proposed an STV method to counter the criticism of STV known as "premature exclusion" of candidates. When surplus votes run-out, in the STV count, recourse is had to excluding the candidate who happens to have the least votes at that stage, tho conceivably he might have gone on to pick up a remaining seat.

To avoid this criticism, I invented a new method. Tho not applicable to the STV quota (as this page explains), I applied the geometric mean to a situation where there is a geometric decline, namely in the size of surplus votes transfered, after the successive election of candidates during a transferable voting count.

I used the geometric mean to average each candidates keep values derived from controled re-counts according to the binomial theorem of preference voting and (its order in reverse) unpreference voting.
The preferences can be given prior importance to the unpreferences.

The keep value is the quota divided by the total transferable vote from one candidate to others according to voters next preferences for remaining candidates. The keep value plus the transfer value equals one. The keep value signifies what the candidate gets to keep.

If the keep value happens to be exactly one, with therefore zero transfer value, the candidate has just enough votes to be elected, the quota, and no more to pass on to help his voters next prefered candidates.

Also departing from conventional STV, these keep values were calculated not only where there were vote surpluses to candidates elective needs but also by how much candidates might be still in deficit of the quota.

Besides my two research innovations of the harmonic mean and geometric mean into STV elections, in general practise, the Hare quota, which I considered as an arithmetic mean quota, has its place in the proportioning of voters per constituency.

Appendix: Definition of "quota."

I apologise for this pedantic appendix but I have been criticised for using the term, quota, for the over-all majority in single member constituencies. Apparently, an STV count rule book uses it solely in the sense of proportion, whose mathematical definition is an equality of ratios. That is, in votes to seats. You can only have proportional representation where there's more than one seat in the constituency.

When I refer to a quota of one-half, I am merely refering to its place early in the harmonic series. That is, simply as a ratio, rather than a proportion, in the case of single member constituencies. In multi-member constituencies, the Droop quota is a proportion, as well as a ratio.

The purpose of the Droop quota is proportional but it originates in a ratio of one-half, which is not.

It's the rationalisation from one-half over-all majority, to two one-thirds majority etc, that led to the Droop quota. At least I dont know any other way it could have come about. In any event, it is unavoidable to retain that rational understanding of the emergence of the Droop quota in our explanations - just as it is essential to also mark the distinguishing features between counts in single as against multi-member constituencies, that allow surplus votes to be transfered from candidates already elected to a quota.

When I use the term, Droop quota, I refer to the whole of the series, including the one-half ratio, electing a single member, tho it is not also a proportional representation unlike in two, three and succeeding multi-member constituencies.

This is a generally accepted use of the word, quota, which can just mean literally "how many," as well as a share or proportion.

The purpose of the Droop quota is proportional but, in the first instance, it is just a ratio (of one-half) from which its rationalisation was derived, which is why it should be seen as a whole series.

Richard Lung.
8; 11; 15; 19; 26 February; 21 march; minor elaboration 4 april 2013; 11 june 2013.

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