Introduction.
Binomial STV example.
First order Binomial STV.
Second order Binomial STV.
Politicians and the public ignore the fact that election method is a science. Representative elections and referendums are an undoubted democratic right. But unininformed opinion is not helpful. That applies to the contemptuous prejudice and wilful ignorance of the Establishment duncers, who desperately fought and frauded for their lives against the dud system of the Alternative Vote, in the 2011 UK referendum, which was the only system alternative they themselves pretended to offer.
We all know what the verdict on Ptolemaic versus Copernican astronomy would have been, if the Holy Inquisition urged the congregational flock, who is right, the wisdom of the Church Fathers or that upstart Galileo.
That comparison with electoral reform is perhaps unfair on the medieval church. Opponents of electoral reform are so reactionary, they have not discovered the division of labor. Their mentality is of a prehistoric gatherer society, in which everyone can gather a basket of berries. And they think the voting method that is simple enough for everyone to understand, should remain primeval, like the simple plurality count.
Antiprogressives confound the practise of voting (which is easy to understand) with the science of the count (which is hard). Everyone can and does make lists of ranked choices. But some mean spirits want to exclude everyone but the politicians from making those lists. Hence, the party list systems that infest the worlds socalled democracies. Many socalled electoral reformers, by advocating party list systems, for the simple plurality voting countries, merely want to hand over the right to rank choices to the parties and not the people.
All elections, all over the world, are carried out under a secret embarrassment, that specialists call premature exclusion. Elections exclude candidates, without really knowing whether they are the least popular candidates.
This was first pointed out, by Condorcet, two centuries ago, one of the founders of election science, during the French Enlightenment. His remedy, which I’ve discussed before, is still being actively followed. But it is not the approach used here, which is to combine the traditional proportional count of a preference vote with another proportional count of the preference vote reversed. My Binomial STV matches election counts with exclusion counts.
Binomial STV is a complicated version of an electoral system, that is already too often disparaged as complicated. Then again, so was the Alternative Vote (AV) so reviled, in the 2011 UK referendum, because it offered a ranked choice, instead of the illiterate Xvote for the serfs.
The big difference of Binomial STV, from previous versions is that until now, STV has only been a count of the most prefered candidates. That is to say an election count but not an exclusion count. All the most prefered candidates have the values of their surplus votes, over the quota, transfered to their next preferences. When there are no more surpluses to transfer, the unlucky candidate, who happens to have the least votes at this stage, is excluded.
Binomial STV is more sophisticated than that, because it has an exclusion count, based on the voters preferences, in reverse order. That means that the voters have a real incentive to put candidates they really dislike, right at the end of their ranked choices, instead of simply abstaining on the final available choices of candidates.
So it would be fair to say, that, if STV is "love voting," then Binomial STV is "lovehate voting." A “lovehate relationship” is a colloquial expression in English, so I have used this familiar phrase to familiarise the public with the basic idea to an extra complicated and sophisticated election system, Binomial STV or, to give it a fuller technical description: Abstentionsinclusive keep valueaveraged Binomial STV.
(Most of the rest of this essay will be for specialists in election counting procedures.)
Shakespear gives a lovehate relationship, in Much Ado About Nothing. The title is in itself not a bad assessment of the vanity of human turmoil. Hector Berlioz adapted the play to his last opera, Beatrice and Benedict. Its classical style is like a last journey back to a pure and clear highland stream. Scarcely a chord hints at Berlioz the grandiose, in stately procession thru the chromatic canyons of his scores.
I must admit that I had misgivings about the catchphrase, "lovehate voting" because there is obviously too much hate in the world. There is too much fearmongering negative campaigning in elections and referendums. Nevertheless, it can be positive to vote negatively against negative candidates.
Binomial STV allows political anger to be powerfully, but peacefully, articulated. This might give popular grievances a more constructive expression.
If the positive vote of a preference count cannot decisively elect candidates to the required number of seats, then the extra information, provided by an exclusion count, may help arrive at a more fully rational conclusion, as to what is the balance of public opinion.
Previous versions of STV degenerate to the socalled Alternative Vote, when there is not more than one seat in the constituency, and so it is not possible to transfer a surplus of votes from one elected candidate, to help a next prefered candidate win a second seat. Proportional representation is not possible in single member constituencies.
(Those who want to combine a single member system with a party proportional system, called the Additional Member System or Mixed Member Proportional system (AMS/MMP), are trying to have their cake and eat it. That is to the voters loss of genuine personal and proportional representation.)
Binomial STV will even work in singlemember contests. (And this is one of the reasons why I would like to see the system operationalised and tested, to see if it works more assuredly well than existing methods.)
Binomial STV is able to work in single member constituencies, as well as making STV work better in multimember constituencies, by extending the use of the readjustable keep value, first introduced, by Brian Meek, to Meek STV. This records the keep values of candidates in surplus of a quota. Whereas Binomial STV also records keep values of candidates in deficit of a quota.
The following example shows how Binomial STV works on an election, in which the numbers have been deliberately simplified. Thanks to Kristofer Munsterhjelm for the link to the article,
http://www.votingmatters.org.uk/ISSUE20/I20P1.PDF
in Voting Matters, called Meek versus Warren, by I. D. Hill and C. H. E. Warren.
Even this skeleton of an election was quite a labor of love, with Binomial STV. Like Meek method, which it greatly elaborates, it is strictly for a computer program. As someone, educated long before the first personal computer, who never acquired coding skills, I hope some programmer, ideally people conversant in the Meek STV computerised election count, will adapt it to Binomial STV.
Binomial STV extends the use of the readjustable keep value to quotadeficit candidates, as well as quotasurplus candidates, and combines an exclusion count with the normal Meek election count. Indeed, Binomial STV has higher orders of systematicly qualified election and exclusion recounts.
In one respect Binomial STV is simpler than Meek STV, in that it abandons progressively lowering the quota, as voters stop ranking the candidates. For, Binomial STV counts all the abstentions, as well. A wholly blank ballot paper returned under Binomial STV, counts as None Of The Above.
Abstentions, coming after voters have no more preferences for candidates, have to be counted, because otherwise the reverse preference count gives primary importance to merely the voters last preference, which it does not have, unless it is the last possible preference.
Table 1, below, gives all the ranked choices of 39996 voters, electing three out of five candidates. There are factorial five, which equals 120, permutations of preference for five candidates. And among so many voters, most, if not all, of those permutations, probably would be on the ballot papers. The authors, of the article, from which this election example is taken, confined themselves to just five permutations, for ease of calculation, over a fine point of difference about election method, that is not pertinent to this essay.
Voters per ranking.  Rankings of candidates. 
10000  ABCXX 
100  AEXXX 
10000  BDXXX 
9998  CXXXX 
9898  DXXXX 
Total votes: 39996 
Suppose the election is for three of the five candidates, A, B, C, D, E. The "candidate" X stands for the abstentions from stating a further preference for one of the five candidates. If the abstentions (for "candidate X") reach an elective quota, then one of the three seats must go unfilled.
The proportion of the total votes, that each candidate needs to be elected, the (Droop) quota is: total votes, divided by one more than the number of seats:
(39996)/(3+1) = 9999.
First order Binomial STV conducts a count of most prefered candidates, like
Meek STV, using its key concept of the readjustable keep value. But it extends
the use of the keep value from candidates reaching a quota, with a surplus of
votes, to the rest of the candidates still in deficit of a quota.
The keep value is the ratio of the quota over a candidates total vote. With
Meek method, the keep value is only counted when a candidate has a surplus over
the quota. With Binomial STV, all candidates have a keep value, even if their
total vote remains in deficit of the quota.
Also, unlike Meek method, Binomial STV has no readjustment of the Droop quota downwards, as voters cease to express further preferences. On the contrary, Binomial STV counts all abstentions, as an integral part of the voters total preferential information.
Unlike Meek and other previous methods of STV, Binomial STV does not resort
to excluding candidates with least votes, when there are no more surplus votes
to transfer. That is exclusion by happenstance.
Instead, Binomial STV counts, of candidates keep values, still short of a
quota, let them live to fight another count, when they might exceed the quota.
What finally counts, with Binomial STV is the overall average keep values of
the candidates.
First order Binomial STV goes on to conduct, in exactly the same way as the preference count, a reverse preference count. The resulting unpreference keep values of the candidates are inverted, to provide another source of preferential information, from the voters, about the candidates.
The keep values, from the counts of the preference vote and of the inverted unpreference vote keep values, are then averaged with the geometric mean, to arrive at an overall result.
Quota = 39996/(3+1) = 9999.
Cand idates 
1st prefs. 
1st pref. keep values 
1st to 2nd prefs. transfer A to B 10000 votes @ tranfer value 1/100; B to D 10000 votes @ transfer value 1/10000. 
2nd to 3rd pref. tranfer 10099 votes B to C @ transfer value 100/10099 
3rd to 4th pref. transfer 10098 votes C to X @ transfer value 99/10098 
Final keep values: 9999 quota divided by candidates maximum vote. 
A  10100  .99  9999  9999  9999  9999/10100 = .99 
B  10000  .9999  (A>B: +100; B>D: 1; so:) 10099. 
9999  9999  9999/10099 = ,990098 
C  9998  1.0001  9998  (B>C +100) 10098 
9999  9999/10098 = .990196 
D  9898  1.0102  (B>D: +1.) 9899 
9899  9899  9999/9899 = 1.010102 
E  0  (A>E: +1.) 1 
1  1  9999/1 = 9999 

X  0  0  (C>X: +99) 99 
9999/99 = 101 

All votes  39996  39996  39996  39996 
A candidate achieves election with a keep value of one or less, because that
is the quota divided by an equal number of votes, or more, to the candidate.
Ihe preference count alone is essentially conventional (Meek) STV. Here, it has
decisively elected candidates A, B and C, all with keep values of less than
one, to the three seats. There is no need to go on from this traditional
Uninomial (preference count only) STV to (Preference and Unpreference counts)
Binomial STV.
If the former is "love voting" and the latter is "lovehate voting," one might
say that love has primacy over hate.
The only reason I go on to complete the first order Binomial STV count is to
show how it works.
Voters per ranking.  Rankings of candidates. 
10000  XXCBA 
100  XXXEA 
10000  XXXDB 
9998  XXXXC 
9898  XXXXD 
Total votes: 39996 
Cand idates 
1st unprefs. 
2nd to 3rd unpref. transfer 10000 votes from X to C @ transfer value .75 
3rd to 4th 
4th to 5th unprefs. transfer X to C 9998 votes & X to D 9898 votes @ transfer value .60794016 
Final keep values: 9999 quota divided by candidates maximum vote. 
A  0  0  0  0  9999/1* 
B  0  0  0  0  9999/1* 
C  0  7500  7500  +6078.1857 sums to: 13578.1857 
.7364018 
D  0  0  6923.0059  +6017.3917 sums to: 12940.3976 
.7726965 
E  0  0  69.2306  69.2306  144.4316 
X  39996  32496  25503.76  13408.1861  .25 
All votes  39996  39996  39996  39996 
* Binomial STV always requires at least one vote for a candidate, considered as the candidate voting for onesself. Otherwise, the keep value for a candidate with no votes would be infinite, which would spoil the calculations.
Real elections follow a more or less normal distribution of votes for candidates, so that you can always expect a few votes finding all the candidates, whether with respect to popularity, or unpopularity, as table 4 measures. The 9999 keep values showing total lack of unpopularity, for A and B, are vastly overblown, in this artificial example. Normally, we could expect to find at least a handful of voters to single them out for especial disfavor, and this would soon bring down an inflated unpreference keep value.
Candidates  Preference keep values 
Inverted Unpreference keep values. 
Geometric averaged keep values 
A  .99  .0001  .00994987 
B  .990098  .0001  .00995037 
C  .990196  1.35795437  1.159587 
D  1.010102  1.294169177  1.14335 
E  9999  .006923692  8.32 
X  101  4  20.1 
A keep value is the ratio of the elective proportion of votes, or quota, to the maximum number of votes for the candidate, achieved during the course of the count. A keep value of one means that the candidate has reached the elective quota.
The completion of the first order Binomial STV count has deprived candidate C of election, with a keep value of over unity, meaning the ratio of the quota over C's votes is greater than one, because C's votes over the preference and unpreference counts average less votes than the quota. What is more, candidate D now has a slight lead over C, tho also not quite at the quota.
This is not surprising because the Unpreference count brought a lot more
information into play by way of abstentions, which were transferable first to
C, the only third prefered candidate, and by the same token, the only third
unprefered candidate.
This position of candidate C was a case of what you gain on the roundabouts you
lose on the swings.
So, it is of interest to find out how second order Binomial STV qualifies the first order procedure.
You will find, in this first order, as well as the second order, Binomial STV, that this particular example greatly skews some keep values, as for candidates A and B.
This simplistic example is practically a statistical impossibility. It is extremely likely that there would be at least a handful of votes, for the most unpopular candidates, which would be enough to dramaticly moderate their keep values.
Then again, Binomial STV is a like and dislike system (lovehate voting) in which last preferences may more significantly affect the outcome, and so are more likely to be made.
In terms of the binomial theorem, first order Binomial STV can be expressed algebraicly as:
(p + u).
This factor is actually to the power of one, but the unity power is usually omitted. However, the next order of binomial expansion is the factor to the power of two, or the square of the factor:
(p+u)².
The expansion of the factor is:
(p+u)² = pp + up + pu + uu.
The expansion is a noncommutative algebra because "up" and "pu" stand for different operations. That is to say different counts.
There are two preference counts, based on the first order preference count. One of these qualified preference counts redistributes the votes of the most prefered candidate. The other qualified preference count redistributes the votes of the most unprefered candidate. Then the two so qualified preference counts proceeed. A geometric mean (GM) is taken of the candidates respective preference keep values.
Exactly the same procedure is followed for the two qualified unpreference counts, to arrive at geometric average unpreference keep values for each candidate.
These unpreference keep values are then inverted to provide another statistical estimate of preference. The averaged inverted unpreference keep values are then averaged with the average preference keep values, for an overall average of keep values that is the second order Binomial STV result.
We start with the two preference counts, refering to Table 1. First
by redistribution of most prefered candidate (A) votes.
Candid ates 
1st prefs.  Redistrib. A's votes 
1st to 2nd pref. 
2nd to 3rd pref. surplus transfer to X of 4999.5 votes from C & 4899.5 votes from D 
Keep values as quota divided by candidates maximum votes 
A  10100  9999/10100 = .99 

B  10000  20000  9999  9999  9999/20000 = .49995 
C  9998  9998  9998+ 5000.5= 14998.5 
9999  9999/14998.5 = .6666666 
D  9898  9898  9898+ 5000.5= 14898.5 
9999  9999/14898.5 = .67114139 
E  0  100  100  100  9999/100 = 99.99 
X  0  0  0  9899  9999/9899 =1.010102 
All votes  39996  39996  39996  39996 
Next, the second qualified preference count, the
redistributing the votes of the least prefered candidate, E. Due to the extreme simplicity of this example, candidate E has no surplus votes to redistribute. (Binomial STV differs from previous transferable voting systems that take a trailing candidates votes to give to other candidates, and dismiss him in midcount.)
I also have to remind myself, let alone everyone else, that the most
unprefered candidate is likely to have a significant number of votes, to be
redistributed , with Binomial STV, where unpopularity counts, albeit
adversely.
Here, the (second order) unpreferencequalified preference (UP) count does not
differ from the (first order) preference count (P). So we simply refer to Table
2 keep values, when it comes to averaging the keep values for the final result
of the second order count.
Now for the two qualified unpreference counts.
redistributes the votes of the most unprefered candidate, C, in an unpreference count.
Candid
ates 
1st & 2nd least prefs. 
Transfer 10000 votes from X to C @ transfer value 29997/ 39996 = .75. Redistrib. to B. 
3rd to 4th pref. transfer 10000 X votes to D; & 100 to E @ transfer value 22497/ 32496 = .6923006 
4th to 5th pref transfer 9898 votes from X to D @ transfer value 15504.764/ 25503.764 = .60794022 
(UU) keep values (quota divided by candidates maximum votes) 
A  0  0  0  0  9999* 
B  0  7500  7500  7500  1.332 
C  0  0  0  0  9999/7500 = 1.332 
D  0  0  6923.0059  6923.0059 + 6017.3923 = 12940.398 
9999/ 12040.3923 = .7726965 
E  0  0  69.2301  69.2301  144.4316 
X  39996  32496  25503.764  19486.3718  .25 
All votes  39996  39996  39996  39996 
* Binomial STV always assumes at least the candidate gives one vote for himself.
The final 2nd order count,
redistributes the votes of most prefered candidate, A, in the unpreference count. All A's votes are first preferences, so votes for A cannot go anywhere, in a reverse preference count, and all the candidates keep values are the same as in the original first order unqualified unpreference count (U): Table 4.
Having now obtained all four second order STV counts, it only remains to take their average, for the final overall Binomial STV count results.
PP: pref. qualif'd pref. keep values (Table 6) 
UP: unpref. qualif'd pref. keep values (Table 2) 
GM pref. keep values 
UU: unpref. qualif'd unpref. keep values (Table 7) 
PU: pref. qualif'd unpref. keep values (Table 4) 
GM unpref. keep values 
Inverse GM unpref. keep values 
Overall GM** pref. keep values 

A  .99  .99  .99  9999  9999  9999  .0001  .00995 
B  .4995  .99009  .49499  1.332  9999  115.407  .00010  .0084 
C  .66666  .99019  .66013  1.332  .73640  .990397  1.0097  .8164 
D  .67114  1.0101  .82356  .77269  .77269  .772697  1.2942  1.0323 
E  99.99  9999  99.9  144.43  144.43  144.432  .00692  2.6312 
X  1,01,01  101  10.100  .30769  .25  .277353  3.6055  6.0347 
**Overall GM: (Square root of Geometric Mean preference keep values multiplied by inverse Geometric Mean unpreference keep values)
The last column of table 8 gives the final result of the second order Binomial STV count. The decision is the same as the traditional STV count, which was given in table 2, without any further qualification. Candidates A, B and C are elected. All candidates with keep values of one have reached the elective quota. Keep values of less than one mean that the candidate was elected with votes to spare.
This concludes the second order Binomial STV count.
Richard Lung.
16 january 2016.