Keep-value averaging Binomial and Condorcet counts of Transferable Voting.

Conventional STV, criticisms and reform.

By the single transferable vote ( STV or 'choice voting' ), candidates are each elected on winning a quota of the votes in a multi-member constituency. Some candidates win more votes than they need. These surplus votes are transferable to the elected candidate's voters' next preferences, by a method called the Senatorial Rules. All that elected candidate's voters' next preferences are taken into account but they can only be valued or weighted in proportion to the size of the surplus. This fractional weighting or value, at which surplus votes are transfered, is called the 'transfer value'.

The elected candidate only keeps the number of votes needed to be elected, that is the quota. The ratio, of the quota to all this candidate's votes, is the fraction of his votes that the candidate keeps. It is called the 'keep value.'

The elected candidate's keep value and transfer value are fractions that always add up to one. This is because all the candidate's voters must always have a vote, whose value adds up to no more and no less than one. This follows the democratic principle of one person one vote. All that varies is the proportion of the vote that goes to a first or second preference, and possibly further preferences.

However, at some stage in the STV count, there may be no more surplus votes to transfer, before all the seats in the multi-member constituency have been filled, by sufficient candidates achieving a quota. Then, the least likely candidate, to be elected, has to be excluded, to provide more votes to re-distribute to that candidate's voters' next preferences.

At this stage, traditional STV excludes the candidate with the least votes. Meek's method does likewise. The difference between the traditional manual count of STV and Meek's computer count of STV is that Meek can be much more systematic in using the senatorial rules to count the value of every next preference. The traditional count typicly has to take short-cuts to make a manual count managable. Hopefully, these short-cuts dont often make much or any difference to the more strictly accurate count usually only practical with a computer.

Thus, manual count and Meek's method differ in election of candidates. But they do not differ in the exclusion of candidates. Both essentially use the reverse of a First Past The Post, or 'simple majority' system to elect candidates. That is a "Last Past The Post" or simple minority system to exclude a candidate, who happens to have the least votes, once the election surpluses have run out.

In other words, a remnant of the old way of thinking has got left in the exclusion part of the count, known as proportional representation by the single transferable vote. This is ironic, because much of the academic criticism of PR by STV has come from censure of the effects of its last-past-the-post exclusion method.
For instance, Riker's example below, to expose the perversity ( 'non-monotonicity' ) of STV, supposed an STV election with no surplus of first preferences to transfer. This means an STV election that has to start with an exclusion. Such critics have discredited Last Past The Post exclusion, to try to discredit STV, as an election, in favor of a First Past The Post election system ( on its own or as part of a mixed system with party lists ).

The critics of STV have not put the blame for perverse results where it belongs, in the simple plurality system ( whether simple majority election or simple minority exclusion ). They have just focused on faults, apparent to a minor extent in STV's exclusion procedure, to excuse the same faults, apparent to a major extent in First Past The Post elections.

In the middle of the nineteenth century, within six months of the appearance of Hare's system ( the original STV ), John Stuart Mill noted this habit of critics to project or throw the faults of the simple majority system onto proportional representation.

Using proportional representation in the exclusion count as well as the election count, makes STV consistently a system of PR. That is to say when PR by STV can take the count no further in the election of candidates, then PR is used in the exclusion of candidates.

The exclusion count takes every order of preference, chosen by voters on their ballot papers, in reverse. Starting with the last preferences, a candidate's votes are added till they reach a quota, as a measure of exclusion. The exclusion count is conducted in the same way as the standard election count. When one realises that a count consists of a method of election and a method of exclusion, and that a method of exclusion is the reverse application of a method of election, it should no longer seem strange or controversial that one use the best available method for both election and exclusion.

The reason for using transferable voting to elect candidates also justifies its use to exclude candidates. STV is pre-eminent as a voting system because it uniquely follows the four scales of measurement, analysed by S S Stevens, and widely recognised in the sciences, in relation to problems of research. ( Measurement scales and voting methods are discussed on my web page about scientific method of elections. )

An exclusion quota brings the STV exclusion count up to the most powerful scale of measurement, the ratio scale. Likewise, the rational use of keep values for candidates in deficit as well as in surplus of a quota allows systematic re-counts, such as based on the binomial theorem or on Condorcet pairing, to derive average keep values, which may be more representative of the voters' choice than a single count.

AV and STV methods of excluding candidates.

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The simplest of elections can also be regarded as an exclusion. When two candidates contest a single office, the candidate with more votes is elected. One can also say that the candidate with less votes is excluded. I have described the single transferable vote as a generalisation of the simplest method of election from a single preference vote for a single majority count to a many-preference vote for a many-majority count.

But the count of STV for election of candidates and for exclusion of candidates has not been developed in a balanced or complementary manner. The transferable vote is of surpluses from candidates elected with more than enough of the required proportion or quota of votes in the constituency.

No such rule has applied for the exclusion, as distinct from election, of candidates in STV elections. Instead, STV has used the same method of exclusion used with the Alternative Vote ( AV ).

This matters, because far-out critics have argued against STV ever being used, simply for this employment of the AV exclusion or elimination count. This is not balanced criticism as I tried to explain on my web page against the Plant report. This Labour party committee didnt even consider STV as an option in its final report. This was all the more absurd in that the report recommended the Supplementary Vote, which is a limited form of the Alternative Vote.

The Alternative Vote is meant to elect the candidate with an over-all majority of votes in a single member constituency. If no one candidate has half or more of the first choice votes, then the candidate with the least first preferences is excluded. The excluded candidate's voters then have their votes re-distributed according to their second preferences. This process of elimination and redistribution is repeated till a candidate wins a majority.

This is what the STV count has resorted-to when there are no more surplus votes to transfer to help elect more candidates to the quota. However, STV is in multi-member constituencies. There is more than one seat to be won. The re-distributed votes of some excluded candidates may be more than enough to elect a further candidate. In that case, another elected candidate's surplus vote becomes available to help elect next prefered candidates.

The Supplementary Vote is a truncated version of the Alternative Vote. The two candidates with the most first preferences are the only ones not excluded in the first round. Voters for the excluded candidates then have their supplementary vote ( a second X-vote ) counted, if it is for one of the two remaining candidates.
In short, the Plant report chose a voting method that used a method they considered enough to disqualify STV for only partially using!

Sequential STV.

Sequential STV was devised to prevent the premature exclusion of candidates thru its last past the post exclusion method, by adapting the method of Condorcet.

The Condorcet method elects the candidate who comes off best in paired contests between all the candidates. J F S Ross did not recommend Condorcet pairings, because they neglect that higher preferences are more important than lower ones. Whereas the Senatorial Rules of transferable voting use a method called ( in simple statistics ) "weighting in arithmetic proportion" that gives due weight to the rank or order of choice.

However, Sequential STV systematically re-runs a conventional STV election with the winners pitted in turn against just one of the candidates not elected. The re-runs all involve just one more candidate than seat, so there is no need, in each of these sub-contests, to exclude a candidate before all the seats can be filled.

If the winners are the same as in the original contest, the election is over. But a new winner, replacing one of the old in the sub-contests, features thru-out another systematic series or sequences of sub-contests. Basicly, this goes on, till the situation either stabilises into consistent winners of the given number of seats, or a cycle appears, like a Condorcet cycle, which announces a dead-lock.

The sequence of sub-contests, that may throw up other winners than by simple STV, are rational in themselves. But their results compared to each other need not represent victories of the equal weight, that they are implicitly given. So, essentially, Ross's objection, about proper weighting, applies against Sequential STV

The moral of all this is to make the most rational measures when possible rather than just measures of more or less, which are less accurate for comparisons of results. The more approximate the scale of measurement the more room for errors. Identifiable patterns of error then become enshrined as paradoxes. Sequential STV conceivably can cause a voter's prior choice to lose an election, if the voter adds a second preference.

In other words, as the devisor of Sequential STV admits, it is still what the social choice theorists call "non-monotonic".

From a scientific point of view, Sequential STV has the merit that its process of elimination resembles a controlled experiment. But it seems to have the theoretical weakness of being ad hoc. The system has been devised solely for the purpose of preventing premature exclusion, without being inherent in simple STV.

Non-monotonic exclusion of candidates.

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It is worth looking at the exact criticism cited by second interim Plant report, appendix one. ( Afterwards, their example may be used to show how the innovation of keep-value averaged STV can over-come the objection described. )

The report quotes Riker's Liberalism Against Populism that STV is not monotonic:

By monotonicity is meant that if one or more voters change preferences in a direction favourable to x then the resulting change, if any, should be an improvement for x.

Riker gives an example of how STV can break the monotonic rule. Suppose four candidates compete for 26 votes to win two seats. The votes each winner needs is given by 26/(2+1), which is 9 rounded-up to the nearest whole number.

A first version of the voters' preferences is given in situation one. A second version differs only in that two voters have swapped their first and second preferences from first Y, second X, to first X, second Y. But this improved support for X, in the second situation, loses X the seat X wins in the first situation.

The perversity of this changed result is an example of a 'non-monotonic' counting procedure. ( See tables situation 1 and situation 2. )

Situation 1
Number of voters 1st choice 2nd choice 3rd choice 4th choice
9 W Z X Y
6 X Y Z W
2 Y X Z W
4 Y Z X W
5 Z X Y W

Situation 2
Number of voters 1st choice 2nd choice 3rd choice 4th choice
9 W Z X Y
6 X Y Z W
2 X Y Z W
4 Y Z X W
5 Z X Y W

As I mentioned at the time of the Plant report, this example of the transferable vote does not include its distinctive feature of transferable voting. Riker has given an example in which no candidates win more votes than they need on first preferences alone. So, the example has no surplus votes to transfer.

The tables of preferences could be turned. The fourth choices could be first, the third choices could be second, etc. In that case, the two seats would be won in the first round. W would win with a large surplus. There would not even be any need to transfer this surplus to next choices, because Y, too, would be elected on just enough votes, the quota of 9.

This turn-about may not seem relevant to Riker's example. But in fact it is, if transferable voting is used not only to elect candidates but to exclude them. When no candidate reaches a quota with only first preferences, then some other means must be used.

My first attempt at a proportional exclusion as well as election count of STV was "Reversible STV" as described in the appendix. Next described is my keep-valued transferable voting with Binomial counts and Condorcet pairing.

Reform demonstrations based on method critics' example.

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First order Binomial STV Count of Situation 1.

Starting with the first order count of Situation 1, we see whether Riker's contrived example can be given a clear result with this new version of STV and whether it is non-monotonic, like conventional STV.

Table 1: 1st order binomial STV count of situation 1.
(0) Candidates. (1) 1st preferences. (2) Preference keep values. (3) Last preferences. W, Y excluded. (4) Transfer W's surplus @ 9/17. (5) Unpreference keep values. (6) Inverse of (5). (7) Over-all keep values.
W 9 9/9 = 1 17 9 9/17 17/9 1.889
X 6 9/6 = 3/2 0 9x9/17 = 4.765 9/4.765 .529 .794
Y 6 9/6 = 3/2 9 9 9/14 1.5556 2.333
Z 5 9/5 0 8x9/17 = 4.235 9/4.235 .4706 .847
Total votes: 26 26

W has just enough votes to achieve the quota for election. Conventional STV would deem W to be irrevocably elected. But Binomial STV allows a small statistical margin of doubt, which makes W unelectable with its bad exclusion vote. The number of voters, 26, is almost large enough a sample on which to base such conclusions.

We need a simple statistical test of how significantly the candidates' achieved votes deviate from the required quota of election ( as discussed on my page, "Further example and development of First and Second order Binomial STV counts" ). That is to ask what is the deviation from a keep value of one. The lower and upper permitted deviations from a geometric mean keep value of one is calculated as the square of its upper and lower bounds. For two-seat elections, these bounds are given by Geometric Mean equals one equals the square root of the multiplied differences of the GM from lower and upper bounds: { 1 - (1/(2+1)}{ 1 + 1/2 }.

That is the lower bound is 1/3. Its deviation is one-third squared or one-ninth. One deviation below the GM is one minus one-ninth: 1 - 1/9 = 8/9. It is a matter of probability how many deviations a candidate's keep value can be below unity before his election is deemed beyond doubt significant. The bulk of the probability falls in the first deviation and most of the rest within the second. At the utmost, one wouldnt want to go beyond three deviations.

Candidate W has an elective keep value of exactly one. So any statistical margin of doubt, at all below one, raises the question of whether W might be unelected, if W has surpassed the exclusion quota and has an unpreference keep value significantly less than one - as is the case.

The upper bound is 1 + 1/2. Square its difference from the GM, to get ( 1/2 ) = 1/4. One-quarter is one above-bound deviation. Say, we allow two deviations above bound, then a candidate should not have a keep value of more than 1 + 2(1/4) = 1 + 1/2.

As it happens, both X and Y have preference keep values of 3/2. They are just close enough to an elective quota to ask whether they are at least not unpopular. Table 1 column 6 shows that this is the case for X but not for Y. We move on to the second order count to see if this will provide a more decisive result.

Second order binomial STV count of Situation 1.

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The second order count consists of four counts: the most prefered elected candidate has his votes re-distributed in a re-run of the preference count. ( The preference count is columns 1 and 2 of table 1. The unpreference count is columns 3, 4, 5 of table 1. Normally I would do separate tables but for so few candidates it was easier to run all the first order count tables into one table. ) So, table 2 is called a preference-qualified preference ( pp ) count.

Table 2: preference-qualified preference ( pp ) count of situation 1.
(0) Candidates. (1) 1st preferences. Z elected. (2) Z's surplus transfers @ 5/14. (3) X's 2 surplus votes transfer to Y. (4) pp keep values.
W - - - 9/9 = 1
X 6 11 9 9/11
Y 6 6 8 9/8
Z 14 9 9 9/14
Total votes: 26 26 26

A second of the required re-counts is that of unpreference-qualified preference ( up ). This is another qualified preference count. This time the first preferences of the most unprefered excluded candidate are re-distributed. But we find that the most prefered elected candidate and the most unprefered excluded candidate are one and the same candidate W. So, the pp count is the same as the up count which does not need to be repeated.

Likewise for finding the candidates' geometric mean preference keep values. Normally, one would multiply the pp keep values ( shown in table 2 col. 4 ) by the up keep values, then take the square root of each multiple. But in this special case, multiplying a number by itself and then finding its square root brings you back to the same number, the already calculated pp keep values, which are therefore all we need, here, for the second order preference keep values.

We find that this special case also applies here to calculating the second order unpreference keep values. In table 3, W, the most prefered candidate's unpreferences are re-distributed for a second order unpreference count.

Table 3: preference-qualified unpreference ( pu ) count of situation 1.
(0) Candidates. (1) W's last preferences redistributed. Y excluded. (2) Transfer Y's surplus @ 5/14. X excluded. (3) pu keep values.
W - - 9/17
X 4 9 9/9 = 1
Y 14 9 9/9 = 1
Z 8 8 9/8
Total votes: 26 26

The fourth required re-count is the redistribution of the most unprefered excluded candidate's last preferences for a qualified unpreference count. But this is the same as the count in table 3: the most prefered elected candidate's unpreferences re-distributed for the unpreference count. Once again, the most prefered elected and most unprefered excluded candidate is W.

The same logic follows as for the preference count in this special case. The pu keep values, here identical to the uu keep values, are all we need for the second order count of unpreference.

With regard to table 2 of the preference count, when W's votes were re-distributed, they all went to candidate Z, who in turn had a surplus. Some of that surplus might have gone back to W. A surplus value to W would have improved W's preference keep value, possibly putting W beyond the statistical bounds of doubt as an elected candidate.

In normal elections, we might expect some shared preference. Since voters for W next prefered Z, we would expect at least some voters for Z to next prefer W. People dont usually vote like Riker's contrived example to find logical fault with conventional STV and its admittedly crude last-past-the-post exclusion rule.

The preference and unpreference keep values from tables 2 and 3 are put together in table 4 and their over-all keep values shown in column 3.

Table 4: 2nd order count keep values for Situation 1.
(0) Candidates. (1) Preference keep values (from table 2, col. 4). (2) Inverse unpreference keep values (from inverse of table 3 col. 3). (3) Geometric mean keep value: square root of cols. (1) x (2).
W 1 17/9 1.374
X 9/11 1 .905
Y 9/8 1 1.061
Z 9/14 8/9 .756

The second order preference count brings Z and X ahead of W with no further high preferences. The inverse exclusion or "not unpopular" count confirms the preferences for Z and X, who are deemed elected to the two vacancies.

Situation 2.

Now to Riker's Situation 2, which modifies Situation 1 so that X gains 2 more first preferences from Y but, as a result, loses a conventional STV election using Last past the post exclusion. This is what is known in the jargon as a non-monotonic result. It is, in a small way, a perverse or illogical result and has been used by not over-scrupulous critics to discredit STV and eagerly seized-on by politicians.

Even my early STV reform, called Reversible STV passed the test of this example. And it's easy to show that Binomial STV does so, too. Here is the revised table 1a of the first order binomial count for Situation 2. Only the preference count is shown. The unpreference count is the same as for Situation 1.

Table 1a: 1st order preference count of Situation 2.
(0) Candidates. 1st preferences. Preference keep values.
W 9 1
X 8 9/8
Y 4 9/4
Z 5 9/5
Total votes: 26

In Situation 2, X's keep value has closed on the required unity, improving to one and one-eighth, from one and a half. Remembering the permissible deviations, worked out above, that puts X within one deviation from the required elective keep value of one. Given that X is also the least unpopular candidate, this makes X a likely winner of one of the two seats after the first order count, tho the outcome still requires a second order count to be decisive.

The second order count of Situation 2, which gave 2 first preference votes from Y to X, improves X's keep value from 9/11 to 9/13 and worsens Y's keep value from 9/8 to 9/6.

The point is that Binomial STV, whatever the faults this innovation may prove to have, does not have the drawback associated with conventional STV's last past the post exclusion.

Binomial STV with Keep valued Condorcet pairing.

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We use Situation 1 ( the less decisive of the two situations ) to try out Binomial STV supplemented by Condorcet pairing. Firstly we pair off all the possible combinations of candidates W, X, Y, Z. There are 6 pairs as is normal for Condorcet pairing. But we dont simply see which candidate wins the most contests. We see by how much each candidate wins or loses a binary contest, in terms of their complementary keep values. Then we take the geometric mean keep values of each candidate to see which is best.

Before we do this, we should bear in mind that Condorcet pairing loses the information of the voters' over-all orders of the candidates, measured by Binomial STV. And indeed we shall find, in this example, need to reconcile the two sets of measures, Binomial STV and Keep valued Condorcet pairing, to get a more representative over-all result.

Table 5: Keep valued Condorcet pairs of Situation 1: Quota = 26/(1+1) = 13. Divide all votes into 13 for the 4 candidates' 4 keep values each.
1st prefs. 2nd pref. W 2nd pref. X 2nd pref. Y 2nd pref. Z GM keep values: 4th root of multiplied four vote scores, per candidate, divided into the quota, 13.
W 9 9 9 13/9 = 1.444
X 17 20 8 .9313
Y 17 6 12 1.215
Z 17 18 14 .8004

Table 5 shows that Z has the lowest geometric mean keep value and is therefore the keep valued Condorcet winner. Admittedly, X also has a GM keep value below unity and was the other candidate to take a seat in a two-seat Binomial STV contest. However, this was a one vacancy contest, so let's see how Binomial STV handles one vacancy.

The first order count need not detain us. With the single seat vacancy requiring a quota of 13, no candidate comes close to winning on first preferences. My first reform, Reversible STV, reversed the preferences and excluded a candidate reaching a quota of unpreference. This wouldnt work either because W, the only candidate unpopular enough to be excluded, is the only candidate also to be elected.

Coming to second order Binomial STV, the preference-qualified preference ( pp ) count cannot be used either, because it only applies to a candidate prefered enough to reach an elective quota. As we've already shown for the two seat example, the unpreference-qualified preference ( up ) count is the same as the pp count. But here's the trick. Tho they are the same count, and the pp count is not allowed, the up count is allowed.

The latter is allowed, because W, as the most unprefered candidate, does achieve a quota, a quota of exclusion. So, we can conduct, in table 6, a second order preference count here, just so long as we label it the unpreference-qualified ( up ), and not the preference-qualified ( pp ), preference count, tho they are the same count.

In table 6, the weighted geometric mean, of the 2nd order Binomial STV preference keep values and the Condorcet preference keep-values, is explained near the end of the previous page on the Seven seat example supplemented by Condorcet group-pairing.
Here, the respective keep values, in columns (4) and (5) are given geometric weights of two to one ( out of three ), because the Binomial STV count is of twice as many candidates, four candidates, that the votes have to go round, instead of two for each Condorcet pairing contest.

Table 6: Weighted geometric mean keep values of 2nd order Binomial counts and Condorcet pairing counts.
(0) Candidates. (1) 1st prefs. Z elected. Transfer Z's 1 surplus vote. (2) up keep values. (3) Condorcet keep values. (4) up keep values to power of 2/3. (5) Condorcet keep values to power of 1/3. (6) Geometric mean keep values: (4) x (5).
W - 13/9 = 1.4444 1.4444 (1.444)^2/3 (1.444)^1/3 1.444
X 6+1 13/7 = 1.8571 .9313 1.51086 .9765 1.475
Y 6 13/6 = 2.1667 1.215 1.6744 1.06707 1.787
Z 14-1 13/14 = .9286 .8004 .9518 .92847 .884
Total votes: 26

From table 6, the keep-valued combination of 2nd order Binomial count and Condorcet pairing now shows only one candidate, Z, as having an elective keep value, in column (6). This is what we require of an election for only one vacancy.

Appendix: the former proposal of Reversible STV and criticisms.

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When no more votes can be transfered, as surpluses, to elect next prefered candidates, the count might be reversed for an exactly similar transferable vote to exclude least prefered candidates. ( Examples of former proposals, "Reversible STV" and the "Re-transferable Vote" are given on an archive page. )

Thus, the quota would not only be an elective quota. With a reversible STV, the same quota might also be an exclusive quota. Using Riker's example, W is the least widely liked candidate with a total of 17 fourth choices. W would be the first candidate to be excluded but for the fact that W also has 9 first choices, which are just enough to ensure W's election to the first of two seats.

W is representative enough ( in conventional STV thinking ) to be granted a seat, no matter how little the rest of the voters may care for him. An electorate of 26 voters is too small a number for such vagaries, as voters' difference in opinion over W, to be much wondered at.

Having to pass over W, for exclusion from the contest, still leaves Y with 9 fourth preferences, just enough to constitute a quota of exclusion. Y's first preferences are re-distributed. In situation 1, two of Y's votes go to X, already with six votes, for a sum of 8 votes. But four of Y's votes go to Z, already with 5, so that Z now has 9 votes and reaches the quota to be elected to the second seat.

In situation 2, Y only has four first choices to be re-distributed. But these are the four that ensure Z's election as in situation 1.

Thus Reversible STV elects W and Z in both situations and avoids Riker's objection, in his own example, to the traditional STV count as non-monotonic. The rule of rSTV refers candidates for exclusion on quotas of least choices. The traditional STV count depends for exclusion on the chance which candidate has the least votes, when the votes for surplus transfer happen to run out.

The traditional STV last past the post exclusion count has some degree of order about it, as to lesser popularity. But reversible STV's exclusion count, by quotas of least preference, employs a rational scale of measurement, which generally means a more powerful measurement for more accurate information.

However, there is criticism of reversible STV systems. Social choice theory is very rule-governed. And a reversible STV reverses the "Later-no-help" rule. However, this may be an advantage. Voters might state least preferences because they dont want certain candidates to be elected. But least wanted candidates' early exclusion may also help to prevent the premature exclusion of the voters' higher preferences, if they happen to be last past the post when the elective surpluses run out.

It is argued that, unlike STV, a reversible STV violates the "later-no-harm" rule: When voters' least preferences succeed in excluding a candidate, that candidates' voters' next elective preferences are transferable to candidates, who may be rivals to the excluding voters' highest preferences. Later harm, of a sort, may be done. But on balance, conventional STV really leaves the voters worse off. There might be less later harm from letting last-past-the-post fatefully exclude one of one's more prefered candidates, but that is surely worse than having reversible voting protect the most prefered candidates with least preferences to be excluded first. In other words, simple STV's last past the post exclusion rule invites a "sooner harm". Any average loss to "later harm" by reversible STV is more than offset by a gain against "sooner harm".

The later-no-harm rule has been so extended that it has become nothing more than the law of unintended consequences. Its original and proper application was to the fact that transferable surpluses offer no later harm to the more prefered candidates that the surpluses came from, because they were already elected. And this applies both to an election count and an exclusion count with preference or unpreference surpluses transferable.

Having said all that, the previous two paragraphs were written as a defense of my early reform called Reversible STV, which was not keep-value weighted and averaged over systematicly qualified re-counts as in Binomial STV and its supplemental Condorcet pairing. These statistical counts dont actually exclude ( or elect ) a candidate during the course of the election, except in a formal way for purposes of running controled re-counts.

Closing remarks on keep-value averaged transferable voting.

A requirement of good theory is that, in being implemented for a particular purpose, it also explains or helps with other problems. Averaged STV is more proportional as an election and is also an exclusion, no longer powerless against last past the post's premature exclusion. And it is possible to give a rational value not only to the highest preferences, but also the lowest preferences, thru Binomial STV, and the more middling preferences, thru keep-valued Condorcet pairing.

Voters may have new power of excluding candidates they perceive to be undesirable office-holders, as well as the power to elect desirables. There is no doubt that the public does have strong aversions to many politicians , which perhaps they would wish to express.

Moreover, the proportional representation is made higher than the minimum PR given by the Droop quota, which is S/(S+1), leaving up to 1/(S+1) of the voters unrepresented. S/(S+1) corresponds to an average keep value that is more than elective by one minus S/(S+1) equals 1/(S+1). But with a system of re-counts producing an averaged keep value, it is possible to consider corresponding unelective keep values of one plus 1/(S+1) equals (S+2)/(S+1).

Thus a typical candidate might get both the elective and unelective keep values in two counts, ( more or less ) so that his average keep value is their geometric mean. That is square root of(S(S+2)/(S+1). This keep value is less than one and still elective but more proportional than the Droop quota PR on its own, which is recognized as the lower bound of keep values, with (S+2)/(S+1) as a corresponding upper bound, which doesnt have to be elective in itself, because it is the average keep value that counts in a statistical re-count system.

Compared to (S+1)/S, the upper bound I used, the upper of the bounds, here, (S+2)/(S+1) would be slightly less generous as a basis for measuring permissible deviation of unelected candidates' keep values, if they are to benefit from good "not unpopular" keep values.


Sidney Siegal. Non-parametric statistics for the behavioral sciences.Chapter 3. ( An example of many texts refering to measurement in research methods. It cites S S Stevens, 1946. On the theory of scales of measurement. Science, 103, 677-680. )

J F S Ross. Elections And Electors. Ch. 7. Eyre & Spottiswoode. 1955.

I D Hill. Sequential STV. Voting Matters- Issue 2, Sept. 1994.

Richard Lung.

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