Binomial STV and the Harmonic Mean quota.


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Preface to this survey of STV research by the author.

Binomial STV in practise.

Astentions-inclusive count.

Best number of seats for intelligent choice?

Binomial STV in theory.

Binomial STV as a data mining election.

Inter-active STV.

The Harmonic Mean quota.

References.



Preface to this survey of STV research by the author.

This web-site shows counts of my invention of Binomial STV, or Keep-value Averaged Binomial STV: (KAB) STV.

However, this page is a birds-eye view of Binomial STV. My research pages indeed merit the term web “site” because they were work in progress. This makes it hard for readers to follow, because they are following my own uncertain development of a new election system.

I was criticised for this failing in clarity, with regard to my page on the Harmonic Mean Quota, where I initially developed harmonic mean quotas in a constituency-system context, unlike that described here, which may be called in full, the simple harmonic mean quota, but which I usually just refer to as the harmonic mean quota (HM quota).


Binomial STV in practise.

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The single transferable vote transfers surplus votes, over an elective quota, to next prefered candidates, to help them also make the elective quota. When there are no more surplus votes, the count resorts to excluding the candidate with the least preferences, so that his next preferences may go towards electing the required number of representatives in a multi-member constituency.

Critics of this exclusion procedure have pointed out that it is arbitrary, because which candidate happens to have the least preferences, when the surplus votes run out, is a matter of chance.
This objection is to “premature exclusion.”

This author developed an STV count that avoids premature exclusion of candidates. I called it Binomial STV because it is a binary count of preference votes and un-preference votes. The only difference between the two counts is that, for the un-preference count, the voters preferences are counted in reverse, to determine the most un-prefered candidates.

The count of preferences is an election count, whereas the count of un-preferences is an exclusion count.

Both the strength of preference for candidates and the strength of un-preference for those candidates is determined in terms of each candidates “keep value.” (Keep values come from Brian Meek method of computer counting STV.)

In my new STV system, each candidate has both an election keep value and an exclusion keep value. Inverting all the candidates exclusion keep values turns this order of the most unprefered candidates into an order of the least unprefered candidates, thus approximating another set of election keep values. It is then possible to more broadly base an election on the average of the election keep values and the inverted exclusion keep values, for the candidates over-all keep values, to determine which candidates pass the elective quota.

In an STV election, every voter has one vote and each vote is of an equal value of one unit. Because this voting system is a transferable vote, typicly each vote will go partly towards electing a prefered candidate, with the rest of that unit vote being transferable, as a fractional surplus to a next prefered candidate, and so on.

Consequently, a candidate who has just enuf votes to achieve an elective quota, with no surplus votes transferable to next preferences, has a keep value of one, and a transfer value of zero, making a total of one. For, the keep value plus the surplus value always equals one. This ensures one person, one vote.

Candidates, with more votes than they need to be elected, have a keep value of less than one and a transfer value that makes up the difference from one.

One of my innovations was to allow the use, in my newly invented count, of keep values of more than unity, meaning that the surplus values are negative, which means that the candidate is not in surplus, but in deficit, of a quota and has not yet been elected, at some given stage in the count.

The use of deficit transfer values, as well as surplus transfer values, enables the relative strengths of support for the candidates to be measured thru-out every count. There is no premature exclusion of candidates. This invention has the advantage of making more use of preferential information than traditional STV count systems.

Inverting all the keep values in the exclusion count effectively makes that count another measure of the voters elective preference. It is possible to get an average measure of voters preference, by taking the appropriate average, of the election count and the inverted exclusion count, which is the geometric mean.

Thus, multiply each candidates election keep value by their inverted exclusion keep value, and take its square root.

(If you try to use the arithmetic mean, instead of the geometric mean, to arrive at an average keep value for each candidate, you will soon find that it does not add-up properly.)

In small scale elections, with just a few voters, a candidate might not get any votes. Using Binomial STV, it would be necessary to count each candidate as having at least his own one vote.

A simple example illustrates this. Suppose the quota is 100 votes. A candidate gets 150 votes, and is elected with a transferable surplus of 50 votes. His keep value is 100/150 = 2/3. Therefore the transfer value is the difference from each voters entitlement to one vote, which comes to 1 – 2/3 = 1/3 of a vote.

Now consider my innovation that the keep values, of candidates in deficit of a quota, are also counted. Suppose, at any given stage in the count, that another candidate only has 50 votes. His keep value is: 100/50 = 2. That implies a negative transfer value of 1 – 2 = -1.

Especially in small-scale elections, it is advisable that a candidate has at least his own vote. Because, with zero votes, his keep value becomes, in the above example: 100/0 = infinity.

Also, if the count is an exclusion count, where the keep value is inverted, then the keep value is zero.

The returning officer, averaging each candidates election and exclusion keep values, cannot give a meaningful result for a candidates over-all keep value, because the geometric mean does not work, for multiplying by zero or infinity.

Further-more, such small scale elections may yield way-out keep values, for candidates with just the odd vote, which are not really meaningful.

My examples for Binomial STV were necessarily very small, because of the complexity of the count, which normally would require a computer program. A very small number of votes renders doubtful the value of way-out keep-values.

Even for more numerous voters, say over a hundred, levels of statistical significance have to be agreed by contestants, before the election, as to when the more dispersed keep values, from the norm, be ruled-out as useful information in deciding, among remaining candidates, for election. This is a quandary but, as far as I know, it usually will be confined to the tail end of an election, to the last seat, or so, on a multi-member committee.

Traditional STV might be called uninomial STV because the counts are only of preferences. My system of Binomial STV also makes use of the counts of un-preference. The averaging of an election count and an inverted exclusion count, which I described above, might be called a first order STV count.

Traditional or uninomial STV is a zero order STV count.

A second order STV count is determined by employing the binomial theorem, with p for preference and u for un-preference, as the two terms in the theorem. Symbolically this may be written as: (p+u). And this factor symbolises the first order count.
The factor is actually taken to the power of one but the index number one is usually not written because the expansion of the theorem is the same as the factor itself.

Thus the second order count of my binomial STV is symbolised by the familiar quadratic equation: (p+u)^2 = p^2 + up + pu + u^2.

In this form of the equation, pu does not equal up, so they cannot be written as 2up = 2pu. To use the maths jargon, the two terms, pu and up, do not commute. They are a non-commutative algebra.

In symbolic logic, this expansion could be written as a four row truth table: pp, up, pu, uu.
This stands for a second order table of four logical possibilities: prefered preference; unpreferred preference; prefered un-preference; unprefered un-preference.

These four terms systematically determine the four different counts to be averaged for a second order STV count. The first order count has already provided a preference count, p, and an un-preference count, u. The second order count consists of two qualified preference counts, pp and up, and two qualified un-preference counts, uu and pu.

The prefered preference count (pp) is obtained by excluding the most prefered candidate and re-distributing his votes among next prefered candidates, in a so-qualified preference count.

The other qualified preference count, the un-prefered preference count excludes the most unprefered candidate, to redistribute or transfer to next prefered candidates.

Similarly, the unprefered un-preference count (uu) re-transfers the most unprefered candidates votes in an exclusion count of the most unprefered candidates. The prefered unpreference count (pu) operates by excluding the most prefered candidate for the re-transfer of his votes in reverse order of preference, in a further exclusion count of the most unprefered candidates.

(Thus, “pu” and “up” do not commute [as 2up or 2pu] because they are different operations symbolising different modes of count.)

Multiplying each candidates two keep values, from the two preference counts, (pp) and (up), and taking their square roots derives their geometric mean keep values. That is the average election count, which may be symbolised as:
P = {(pp)(up)}^1/2.

The average exclusion count, U, is similarly derived from (pu) and (uu).
Or: U = {(pu)(uu)}^1/2.

Multiply the average election count, P, by the inverse of the average exclusion count, 1/U, and take their square root to derive the over-all second-order STV count (which might be symbolised as B).
In symbols: B = (P/U)^1/2.

Both these examples of first-order and second order STV have used the power of rational measurement to determine the degree of exclusion of candidates, without “premature exclusion” of candidates found in traditional STV.

No candidates are deprived from a chance of winning a seat, at all, before all the stages of the count have been completed for the over-all result, and the relative standing of all the candidates is available, to decide which are the most representative to take the seats.

As the term, Binomial STV, implies, the qualifying of election and exclusion counts can be continued indefinitely in principle. A third order STV count would involve two to the power of three, which equals eight logical possibilities, in systematic recounts to determine the most probably prefered candidates.

These third order recounts would build on the second order recounts, just as the second order re-counts built on the first-order re-counts.

Higher orders of Binomial STV mine deeper and deeper into the preferential data. The more candidates there are, the more data to be mined. This possibility for thoro analysis suggests that Binomial STV is an information-rich procedure, offering some guarantee of its theoretical validity.

However, for most practical purposes, the mining will not have to be deep.

At present, traditional (zero-order) STV is incomparably better than all the unsatisfactory non-transferable voting systems that plague politics.

The foregoing discussion assumes that the voters expressed equally preferences and un-preferences. Suppose, however, that voters were twice as inclined to put their more prefered candidates, on the ballot paper, than their relatively un-prefered candidates.

This might be expressed symbolically by: (2p + u).

And so, in theory, election counts might be given twice as much weight as exclusion counts, where the balance of former to latter preferences justified it.

In preparing this explanation of Binomial STV, or (KAB) STV, I had to face further problems not adequately tackled, by the above possibility of weighting the importance of preference votes compared to un-preference votes. I thought that if there were twice as many votes just prefering half the candidates, rather than extending preferences to all the candidates, one might simply give twice the weight to the preference counts compared to the un-preference counts.

I gave up trying to come up with a weighting procedure that was not arbitrary or assumed. The next section tries a new tack.

Astentions-inclusive count.

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I decided that somehow the contours of the voters preferences had to be followed exactly. By this I mean, an accurate taking into account the fact that some voters preferences stop sooner than others. The structure of the count should reflect or represent typical imbalances, by which most voters only express a few higher preferences, with quite a small proportion of voters filling in all or nearly all preferences.

This putting in context of the count, the degree of imbalance between high and low preferences, would require every preference position to be counted, whether or not any given voters had expressed a preference in them.

This would even include a voter who expressed no preferences at all. If some voter folded a blank ballot paper and put it into the ballot box, that would count as a voter who prefered none of the candidates.

Blank balloting is equivalent to the “none of the above” option.

It would be wrong to allow this explicit option on a ballot paper, because it is giving a privileged position to just one possible voting permutation. Indeed, there are astronomical numbers of possible permutations for more than a few candidates.

This consideration also explains why group voting, or party lists options on the ballot paper, should be unconstitutional, because, like “none of the above,” giving privileged positions to just one ranked choice, for each party, of a huge number of possible permutations of preference for candidates.

So, a preference abstentions-inclusive count, the counting of all preference positions, whether or not filled by a voters choice of candidate, already clarifies two out-standing issues in election procedure. Altho this is a departure from existing STV counting practice, it would appear to be a principled one with useful implications.

(Just to clarify: by abstentions, I mean all the preferences that voters abstain from recording on their ballot papers. I don't mean formal refusals to cast a vote at all.)

How would the abstentions-inclusive count or all-preference positions count work? This could be done most simply by numbering every blank preference position according to its rank. Suppose an election of 12 candidates. And someone balloted a blank paper, equivalent to “none of the above.” This blank paper could be registered in the count by numbering blank preference positions a1 to a12 (meaning a for abstention one to abstention twelve) not for candidates, rather for 12 non-candidates.

This amounts to a twelve-fold expression of preference for non-candidates, or candidates who are not there. As they are absent and unknown, so we cannot say anything about them or their respective order, which is immaterial. That order merely is the instruction to transfer the preference for abstention, towards a quota for a mandatory vacancy or non-candidate. (Abstractly speaking, any unfilled seat goes to the most prefered non-candidate.)

If a voter only expresses one preference, leaving the rest of the ballot paper blank, the remaining 11 blank spaces may be registered for preference position abstentions a2 to a12.

This is not to be confused with the theoretical possibility, however improbable, that a voter might only wish to express dislike for one or more of the candidates. To merely vote against a candidate, the disliking voter could put the number 12 against that most disliked of candidates, because binomial STV includes independent counts of un-preference.

The 12 abstentions are un-differentiated and united expression of voters disinclination to prefer the 12 actual candidates. Therefore, the vote abstentions, considered as “ghosts” or non-candidates, are transferable, as a communal expression of the extent that the voters want none of the existing candidates. Nothing more definite can be asserted about the problem of an election that systematicly leaves one or more of its seats empty.

If the non-candidate preferences achieve a quota of abstention, then the voters have expressed sufficient objection to the candidates, that one of the seats, on the committee or in the multi-member constituency, is barred to the contenders in the current election. This would appear to be an effective expression of voters (frequently expressed) dissatisfaction with candidates.

This is a satisfactory conclusion for me, because a KAB STV introduction of an all-preference positions count or preference abstentions-inclusive count has enabled an election quality control of candidature.

The abstentions-inclusive principle displays some theoretical consistency. It is not just an ad hoc device to save the use of my Binomial STV for elections in which voters typically leave some of their preferences blank.

Besides candidature quality control, counting abstentions helps clear away the privileged status of the "none of the above" preference and hence, the party list preference orders or any other would-be privileged ranking.

Thus a fuller descriptive title for Binomial STV might be: Abstentions-inclusive Keep-value Averaged Binomial STV.


Best number of seats for intelligent choice?

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This new abstentions-inclusive procedure does raise the question of how best to obtain the maximum expression of preferences by voters.

The Irish constitutional convention voted to increase their constitutional minimum of 3 member constituencies to 5 member constituencies.

The Irish have had a centurys experience of STV, so this is as good an endorsement of minimum choice, as we are likely to find.
Northern Ireland has a six-member STV system to include the various political points of view.

This leaves the question of any desirable maximum number of seats, that the voters are likely to use all their preferences for.

Some relevant research was cited by Robert Ardrey, in his popular series on ethology. What size of group did people work best in? It was believed that mankind probably foraged in groups of an optimum size for the bounty of the terrain. Such ecological limits for survival might establish evolutionary precedent for how well people got on, say, in committees of a certain size.

Anyway, the studies found that some groups were too large and some too small for most productive effect. The number, the researchers settled on, was 11.
I remember this, because Ardrey quoted some wag as saying that Jesus had one too many disciples.

Therefore, I tentatively suggest that elections to, say, the Australian Federal Parliament, might be limited to 11 member constituencies. This is bearing in mind the overwhelming number of voters who simply take the easy way out by voting for a party ticket.
What ordinary members of the public can get to know all those stranger candidates? Human beings have chameleon personalities. Yet you have to figure some way of grading their qualities, or at least their policy commitments.

I’m not suggesting that 11 should always be the maximum. At least in certain circumstances, many more seats, in a constituency, might work splendidly. For many years, Cork elected all 21 councillors from one city constituency, on STV with optional preference voting.

It is a pity that trial and error in elections has been usually for partisan advantage, instead of disinterested democratic tests to optimise voter expression of choice.

Fundamental, to maximising choice, is the need to integrate democracy into community life. (This is briefly discussed, in the page on Scotlands electoral future.)


Binomial STV in theory.

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Meek method is a more systematic computer count of transferable surpluses, than was possible with hand-counted STV. The hand count transfers an elected candidates surplus votes to next preferences, but, at later stages of the count, simply passes over any more preferences for an already elected candidate.
A computer can re-calculate the already elected candidates keep value, thru adding the value of those further preferences to his total vote (keep value equals quota divided by candidates total vote) and hence the changed weight of his transfer value (which is: one minus the keep value) for next prefered candidates.

Like Meek method of computer-counted STV, Binomial STV is based on counting keep values, and would also have to be computer counted. Binomial STV is compatible with either hand or computer counts. Its election count can be that of any standard STV count, or any such standard count conducted in reverse, as an exclusion count.

Binomial STV can build on any standard STV count by progressing from a first order count (of a straight-forward election count and this count reversed for un-preference counting) to a second order count, that introduces such preference and unpreference counts, preferentially or unpreferentially qualified by transfering votes from the most prefered or unprefered candidate in a previous lower-order count.

Never the less, any official implementation of Binomial STV, as a computer-counted system, only would make sense in combination with the most accurate existing computer counted STV, namely Meek method, because maximising preferential information is the purpose of both methods and they complement each other to that end.


It should be emphasised that hand-count or Meek STV (tho both are just preferential, that is “uninomial,” and not also un-preferential) are incomparably better than any other election system in use, and generally fit for purpose.
But Binomial STV combats certain mote-in-your-eye critics of STV.

Unlike both traditional hand-counted STV and Meek STV, Binomial STV avoids the criticism of “premature exclusion,” employing a rational exclusion count of the preference votes in reverse, as well as the usual election count of preferences.

I was not the first person to think of a preference and reversed preference count. The Electoral Reform Society offered an article to one of its members, who thought of the idea, but he declined. I contacted him but he showed no desire to resume the idea.

As far as I know, I was the first person to publish the idea of counting reversed preferences and give practical examples. However, this required further new ideas.

Like Meek method, Binomial STV is based on keep values (which is where the concept comes from) but unlike Meek, I calculate keep values with deficit transfer values, as well as surplus transfer values. This means that all the candidates remain in the count til its completion, when, novely, they are all given a final relative popularity, according to their average keep values, of a series of systematic (keep-value) re-counts, based on a statistical logic of the binomial theorem.

An extravagant claim, associated with social choice theory, can easily be refuted. This is that voting methods have their advantages and disadvantages and generally one is not better than another. This was the Ontario government pretension that choice of voting methods is a sort of fashion statement.

John Allen Paulos (Beyond Numeracy) cited a ranked choice election, for a single vacancy, which contrived to derive five different results for five different voting methods. He claimed that social choice theory thus showed there was no clear choice of democratic voting method.

I examined this example and found that the difference in results was a function merely of the different extents to which the preferential information was used by each system.

The systems were first past the post, supplementary vote, alternative vote, Borda method and Condorcet pairing.

I modified the Condorcet pairing count by weighting the results to show the extent the possible pairs of candidates fared against each other. (This is a standard statistical technique. I didn’t know that such a method of weighted Condorcet pairing had been applied already by J G Kemeny.) This extra information was sufficient to make the Condorcet method result, once weighted, agree with Borda method, the two most information-gleaning methods used.

(The example is on my web-page: Keep-value Averaged STV.)

Condorcet pairing, weighted or not, still has its limitations because each pairing is confined to the relative preferences between any two candidates. Each dual neglects the context of relative preference between all the candidates, which Borda method estimates (but only estimates) by weighting the count more, the higher the preference or rank. Weighting the count with the harmonic series, a first choice would get one vote; a second choice gets half a vote; a third choice gets one-third of a vote, and so on.

Making most use of the preferential information is why, for example, STV is better than Borda method (as explained in my: Scientific method of elections: How to do it). More information efficiency is also justification for theoretical progress to Binomial STV.

It may be safely said that the anarchy of voting methods, in actual use, is not, as many politicians and academics would have us believe, any indication that there is no right and wrong in voting methods, and nothing need change. Information-poor voting methods are merely an all too extensive form of censorship of popular opinion and the common interest.

Social choice theory used Arrow Impossibility theorem, to demonstrate supposed limitations to the possibility of a democratic system.
(For further discussion, see my: Liberating democracy from Impossibility theorem.)

Arrow theorem was primarily demonstrated on the Alternative Vote, which is a rather undemocratic system to demonstrate the undemocratic nature of elections. The real elections paradox of social choice theory is that it has to acknowledge the most democratic system to show how undemocratic is democracy.

However, social choice theory has contented itself with the legend that STV is paradoxical or “non-monotonic,” in the maths jargon, by contriving a mini-election, in which a candidate loses by gaining votes. These are actually very marginal considerations, that have not been shown to affect electoral practise.

I have to mention this mote-in-your-eye criticism, because Ive seen it featured as a “disadvantage” of STV. A hypothetical case, never demonstrated in actual political elections, should not weigh against most of the votes wasted in First Past The Post. Nor should it warrant leaving the preference vote to one person, a dictator deciding the candidates order of election on a party list, and denying universal suffrage of preference voting.

Riker, quoted triumphantly in Labour party Plant report contrives a small scale example of STV in which no candidate achieves a surplus over the quota in the first or succeeding rounds. He considers a situation in which improving candidate Xs preferences at the expense of candidate Y, perversely loses X the seat.

But Binomial STV faithfully reflects the change. (The working is on my web-page: Keep-value averaging Binomial and Condorcet counts of Transferable Voting. [You don’t need to consider the subsequent Condorcet counts.])

The rationale of Binomial STV is just to maximise the preferential information. Any juggling the preferences, as Riker does, is reflected in the changed keep values, surplus or deficit, for every candidate.

I seem to remember, on an e-mail group, that someone, primed in social choice theory, asserted the “later no-harm” requirement for a voting system, that later preferences must not harm former preferences. I believe he was refering to my system, Binomial STV, but he gave no explanation, nor showed any understanding or indication that he had looked at the procedure.

In Borda method, which is for one seat, later preferences harm former preferences. How much harm depends on the mathematical series (arithmetic, harmonic or geometric series) used to weight a voters successive preferences.
As long as the single transferable vote in multi-member constituencies is transfering surplus votes to next preferences, from candidates already elected on a quota, then the next preferences cannot harm the former preferences.

Transfering surpluses may be expressed in terms of (surplus) transfer values, which do no later harm. On the same logical basis, Binomial STV extends this practise to calculating deficit transfer values, so that the weight of preference for all candidates is consistently measured in the count. Calculating keep values for candidates, whether from surplus or deficit transfer values, does not confer any harm from the candidates voters later preferences. It merely gives a comprehensive summary of preferential information.


As soon as it is seen that a once-and-for-all (uninomial) election count is only a special case of election system, and Binomial STV introduces systematic re-counts, then it becomes apparent that elections cannot be regarded as axiomatic deductions of incontrovertible winners.
Therefore, elections, as determinist counts, are explicitly replaced by elections as statistical approximations to the most likely preferences of the voters, derived by mining the preferential information on the ballot papers, with the required degree of thoroness.


Binomial STV as a data mining election.

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After I wrote the last sentence in the previous section, I found out, there was a computer specialty in "data mining" or "knowledge discovery." The usual specialist jargon made their exact procedure hard to follow. A problem is to classify masses of data in a searchable fashion. There is too much information to do this manually, so automatic methods using computer algorithms are necessary.

Transferable voting method has under-gone a comparable process. The hand count of STV, even with the help of a calculator, is less than completely thoro.
The computer algorithm of Meek STV does not pass over further surplus votes for an already elected candidate, without readjusting his keep value for a corresponding change in the transfer value, which will alter the weighting given to subsequent preferences.

This usually should make little or no difference to elections. It is a purist attitude of theoretical consistency, that if you are going to use a computer count, you might as well do it properly.

Yet, Meek method also incorporates the practise of reducing the Droop quota, or elective proportion of the votes, with every reduction in the number of voting papers that show no more preferences.
Woodall, the mathematician responsible, just asserted it was right, while noting the unanimous opposition of the Electoral Reform Society Council.

From the point of view of the returning officer, it is no doubt expedient to keep lowering the stile, to help-over the last few runners, in the marathon STV count, to fill all available seats.

For theoretical consistency, the council are right to uphold strict proportionality by not dropping the quota. Failure to do so, means that transfer values are not being calculated on an equal basis.
This can only be justified, if the existing Meek method is designed primarily to elect the most prefered candidates up to the number of seats, regardless of strictly equitable rules for their election.

Meek method, like manual STV, still excludes trailing candidates, when the transferable surplus votes have run out.

Binomial STV doesnt exclude candidates. It runs an exclusion count, as well as the election count. Abstentions are also counted, in order to preserve the structure of the preferential information, with regard to the balance of greater and lesser preferences.

So, in Binomial STV there is no question of lowering the quota, in step with the number of votes with no more available preferences. Instead, all the abstentions or unavailable preferences are taken into account, in each election or exclusion count.
(By abstentions, I mean all the preferences that voters abstain from recording on their ballot papers. I don't mean formal resfusals to cast a vote at all.)

The consequence is that Binomial STV may count enuf abstentions to reach a quota, leaving a seat in the multi-member constituency vacant.
(Further research can be done into maximising voters expression of preferences, to reduce the likelihood of unfilled seats.)

Of course, unfilled seats is what the preferences-terminating quota reduction rule, in Meek method, is meant to avoid.
Never the less, for theoretical consistency, as a data-mining algorithm, I would recommend my invention of Binomial STV to be used on Meek method but without quota reduction.

Binomial STV didnt seem a good name for my new form of STV, because higher orders of binomially qualified re-counts would hardly ever be needed for elections, in practise.
For purposes of data mining, the name "Binomial STV" is apposite, because higher orders, of Binomial STV, offer indefinitely greater extractions of preferential information.

(I would recommend, for theoretical reasons, the use, in large constituencies of my innovation of the Harmonic Mean quota, as average of the Hare and Droop quotas.)

Inter-active STV.

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Another of my web pages, on Real-time STV, considers the invention, in 1821, of the transferable voting principle, by Thomas Wright Hill. His school boys elected a committee by queueing behind the candidates. Boys transfered to other queues, for their second or next preferences, when their first choice either had more supporters, than he needed, or too few to hope to gather enough supporters, to win one of several seats at the committee table.

TW Hill STV is inter-active STV where the voters are changing their choices in response to each others decisions, in real time. This is much more flexible than the secret ballot, which has to be maintained to free voters from fear of retaliation for who they support or from dictation of who to support.

In freedom from fear, inter-active STV could be magnified to the scale of public elections, by displaying real time computations of voters choices, which they could amend in response to a display of the progress of the count for the various candidates.

An inter-active STV election resembles a settling down of oscillations or swings of support between surpluses and deficits of votes about an equilibrium level of support in the elective quota. In maths, this pattern is described as a damped vibration.

The point is that study of inter-active STV patterns could be expected to reveal new patterns of voter behavior. Such new patterns might be formulated and introduced as new rules into the standard rule book for a (non-interactive) STV election.

An inter-active STV reveals the prospect of voters trying to get the better of each other, as they watch each others voting behavior in real time. Gaming would become a natural pastime, both co-operative and competitive, which merely is to describe evolution.

There would be nothing wrong with such gaming, because it would be done on equal terms of universal instant access to both preferential voting and quota counting information. This would be a legitimate state-of-the art computer inter-active STV gaming.
It would be essentially a large-scale version of Thomas Wright Hill transferable voting between school boys lining up behind their favored candidates.

The real objection of Arrow Impossibility theorem, is not that all elections may be gamed in various ways but that they may be gamed inequitably information-wise and organisation-wise.

STV is the evolving voting method that best frees the voters from inequitable gaming. STV is the best voting method.

My characterisation of an inter-active STV election, as patterned like a damped vibration, is mathematical justification that inter-active STV should be introduced so that voters are free to express a pattern of behavior, of recognised scientific interest, denied to them by a formal (non-interactive) STV count.

More-over, this example is common sense explanation for Godel incompleteness theorem, with regard to scientific theories. A deductive theory starts from a few key first principles, suitably chosen to derive many consequences, testable as to how widely and well they fit observations.

But reality cannot be so rigidly pinned down. Theories are incomplete because reality is incomplete. The Creation is still being created and we can accept the same of theories, which not only relate reality but may partake of creating it.

The same reasoning applies to the social choice counterpart, of Godel theorem, in Arrow theorem. Election methods are not an undecidable contest between various systems as rigid sets of rules, with their good and bad points.

The reality was that an election started with a one-stage majority or simple plurality count. Then came a multi-stage count to guarantee a majority. But the exhaustive count or alternative vote was still regarded as one election, tho previously the one stage was the election.

So, election method is not merely a chaos of conflicting systems but has evolved and certainly has potential for further evolution, as this chapter shows. Tho, it is easy to disguise this evolution under a chaos of opportunist aberrations in election rules.

Theories themselves evolve but theorems, like Godel and Arrow, find incompleteness in theory, that is bound to be incomplete, because it is no more than a snapshot in time of how thinking evolves or develops in accord with consequences derived from its recognised limitations as a theory.

Incompleteness theorems will have to develop their own conceptions of incompleteness, beyond considering theory in terms of axiomatic determinism.

Theorem makers may be handicapped by a classical conception of maths as made up of once and for all proofs, the science of certainty.
Yet mathematicians also evolve their theorems, just as Arrow theorem has evolved, and needs to do so, considerably more, if it is to be relevant to the reality of a residue of indeterminacy in elections in general.

Elections may not be social choice theorists presumption of what goes to make right and wrong results. Arrow axioms may pre-determine some impossible election system.
But the point of the electoral exercise is to approach objectively, the subjective determinations of the voters as a whole, with reasonable degrees of confidence.

The few worked examples, on my Democracy Science web-site, gave stable results, even for the very small number of voters, inimical to decisive out-comes. Of course, many trials in Binomial STV would be needed to shake out the bugs.

After well over a century, the bugs are still being combed out of traditional STV, such as the need to replace the “inclusive Gregory method” with the “weighted inclusive Gregory method.” (Anthony Green: Elections Blog. Transfer Values in Northern Victoria Region.)


Nothing could be further from the truth, that you can just design your own electoral system, as the Jenkins Commission tried to do, with Alternative Vote Top-up. The Commission even ignored their in-house experts, who picked out some design flaws in their sad recommendation.
No recorded commission submission prepared a case for AV+ or its practical details. (And I would be surprised if any of the much greater mass of unrecorded submissions did either.) There was no more than the odd favorable mention buried in the mass of advice.

The report cited organisations and dignitaries but the great mass of members of the public were dismissed as too many to mention. That virtually no-one wanted AV Top-up would have been high-lighted had the Commission even bothered to add up the numbers of people who supported various systems. The unhidden public view would have told the world that the Commission wilfully ignored everybody.

(Except, that is, the Prime Minister, who privately wouldn’t give STV to Jenkins, as my account of the Ashdown diaries, relates.)

Likewise, the Ontario government, suitably cueing their hired academics, put the “leading question” (as the lawyers say for a misleading question) to their Citizens Assembly, that their business was to design an electoral system to suit Ontario. They even showed a picture of a choice of models of car.

Democracy was abolished in the alleged impossibility of right and wrong elections.


The Harmonic Mean quota.

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D R Woodall looked into Meek method, testing for whether the system was well-behaved, producing convergent results, and other maths I didn’t understand. But which should give pause to innuendo (as from Riker) that STV is “chaotic.”

As mentioned above, Meek method, used in New Zealand for District Health Boards and some local elections, re-calculates the (Droop) elective quota downwards, everytime voters drop out of the count by not expressing any more preferences on their ballot papers.

The Electoral Reform Society disagreed with quota reduction.
Who then was right, Woodall or the council?

Well, both are right according to their respective priorities. The priority of an election system designer is to ensure that a decisive result is achieved every election. There is no doubt that reducing the quota in step with reduced preferences at each count stage is an effective expedient for ensuring that all the seats are taken.

If you want half a dozen qualifiers for the high jump, you may have to lower the hurdle, more than once, before you get the six best high-jumpers.

The council also have a point. How can a system be fair that lowers the hurdle for some candidates, after others have had to scale a higher hurdle, leaving a relatively smaller surplus value of transferable votes for their voters next preferences?

As stated above, a (preference) abstentions-inclusive count with Binomial STV means there is no point in reducing the quota, because it is accepted that the voters may not express sufficient preferences to fill all the seats.

However, Droop quota reduction, during the STV count, is a convenient introduction to a more basic problem with the Droop quota, itself as an elective quota, even if you maintain it as a level hurdle for all candidates.

The Droop quota (votes divided by one more than the number of seats) itself is too low a hurdle, by which it takes only half the voters to elect a candidate in a single-member constituency. If two candidates both get half the votes, they have to draw lots for the winner.
If one of the candidates gets slightly more than half the votes, this counts as a decisive win by the Droop quota.But statisticly, the winer may not be significantly more popular than the loser, in elections of any size.

Originally, STV or the Hare system used the Hare quota (votes divided by seats). In a national constituency or even a regional constituency, there were so many seats that there is not much difference in the size of the Hare quota from the Droop quota.
But when you get down to a single seat constituency, the Hare quota requires a candidate to be elected with all the votes. Plainly, that is too high a hurdle, as Droop is too low a hurdle. In fact, they are the maximum and minimum quotas respectively.

The Hare quota could normally only elect a candidate, if all the voters defer to some authority, who chooses a candidate for them. That does not go beyond the herd instinct of follow my leader. List systems, that only allow voters to follow the party line, are institutionalised suppression of the diversity of popular choice. That is not democratic.

It should be apparent that the Droop quota is also less than democratic in its way. This quota count always leaves unrepresented a fraction of the voters, potentially large enough to take a seat, but for slight differences (or even no differences) in support attributable solely to chance.

What quota then will navigate between this Scilla and Charbydis of less than democratic quotas?

As the Hare and Droop quotas form a range between maximum and minimum limits, their average may be taken. As both limits form harmonic series, the suitable average, to take, is the harmonic mean.

To find their harmonic mean, invert both quotas, and add the inversions, dividing by two, then re-invert for the result.

Thus, Hare quota, V/S becomes S/V. Droop quota, V/(S+1) becomes (S+1)/V.

Adding and dividing by two gives: (S+S+1)/2V.

Re-inverting for the harmonic mean gives: 2V/(2S+1) = V/(S+ ).

The harmonic mean quota is the votes divided by one-half more than the number of seats.

How can you have half a seat? The same way a nation can have a birth rate of about two and a half children. It’s an average.

The HM quota does have a direct bearing on the progress of democratic electoral reform by STV elections. I believe a major stumbling block is the complaint of some election reformers, in the European Union and in Canada, that STV, using the Droop quota, isn’t proportional enuf.

The low hurdle of the Droop quota also facilitates elections in smaller multi-member constituencies than are desirable, from a democratic point of view. Only main-stream opinion can be represented, in three or four member constituencies. And this may discourage alternate voices from participating.

Larger constituencies would make the HM quota more achievable. The Droop quota could still be used as back-up, analogous to a tie-break when two candidates tie for a vacancy. Two candidates having a statisticly insignificant difference in votes between them are in the position of requiring a statistical tie-break, which might be rendered by resorting to the Droop quota, after finding the HM quota hurdle too high.

Some reformers ask, then why resort to a higher quota than Droop, if it more usually provides a decisive result, anyway?

Because, an election is also an exercise in providing the best level of democracy, not allowing chance to determine the out-come but as fully or proportionly representative as possible, consistent with the voters freedom of choice.


Richard Lung.
16 May 2015.


Further references:

Choosing Electoral Systems in Local Government in New Zealand by the STV task force, convened by the Department of Internal Affairs, describes Meek method.

My web-site page:
Science is ethics or "electics."
A new metafysics and model of reality synthesising the deterministic & statistical world-views.

(This page uses the HM quota as a simple arithmetic definition of how choice may operate within limits of determinism and chance, with philosophic implications for nature having an ethical reality, rather than just being determined or chance-driven.)


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