Love-hate (and indifference) voting: Abstentions-inclusive keep-value averaged first and second order Binomial STV example.

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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.

Anti-progressives 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 so-called democracies. Many so-called 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 X-vote 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 "love-hate voting." A “love-hate 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: Abstentions-inclusive keep value-averaged Binomial STV.

(Most of the rest of this essay will be for specialists in election counting procedures.)

Shakespear gives a love-hate 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 catch-phrase, "love-hate voting" because there is obviously too much hate in the world. There is too much fear-mongering 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 so-called 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 single-member 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 multi-member constituencies, by extending the use of the re-adjustable 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.

Binomial STV example.

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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,

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 re-adjustable keep value to quota-deficit candidates, as well as quota-surplus candidates, and combines an exclusion count with the normal Meek election count. Indeed, Binomial STV has higher orders of systematicly qualified election and exclusion re-counts.

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.

Table 1: Artificially simplified election with only five distinct rankings by 39996 voters.
Voters per ranking. Rankings of candidates.
10000 ABCXX
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.

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First order Binomial STV conducts a count of most prefered candidates, like Meek STV, using its key concept of the re-adjustable 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 re-adjustment 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 over-all 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 over-all result.

First order Binomial STV count one: the preference count (P).

Quota = 39996/(3+1) = 9999.

Table 2: preference count (P)
1st pref.
keep values
1st to 2nd prefs.
A to B
10000 votes
@ tranfer
value 1/100;
B to D
10000 votes @
2nd to 3rd
pref. tranfer
10099 votes
B to C @
3rd to 4th
pref. transfer
10098 votes
C to X @
Final keep
9999 quota
divided by candidates maximum
A 10100 .99 9999 9999 9999 9999/10100
= .99
B 10000 .9999 (A->B: +100;
B->D: -1; so:)
9999 9999 9999/10099
= ,990098
C 9998 1.0001 9998 (B->C +100)
9999 9999/10098
= .990196
D 9898 1.0102 (B->D: +1.)
9899 9899 9999/9899
= 1.010102
E 0 (A->E: +1.)
1 1 9999/1
= 9999
X 0 0 (C->X: +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 "love-hate 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.

First order Binomial STV count two: reverse preference or unpreference count (U).

Table 3: Reversed table 1, for Unpreference count in table 4.
Voters per ranking. Rankings of candidates.
10000 XXCBA
10000 XXXDB
9998 XXXXC
9898 XXXXD
Total votes: 39996

Table 4: Unpreference count (U).
2nd to 3rd
10000 votes
from X to C
@ transfer
value .75

3rd to 4th
10000 votes
from X to D & 100 votes X to E @
transfer value

4th to 5th
X to C
9998 votes
& X to D
9898 votes
@ transfer
Final keep
9999 quota
divided by
A 0 0 0 0 9999/1*
B 0 0 0 0 9999/1*
C 0 7500 7500 +6078.1857
sums to:
D 0 0 6923.0059 +6017.3917
sums to:
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 ones-self. 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 over-blown, 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.

Table 5: Binomial STV result of averaged first order counts of Preference and Unpreference.
Candidates Preference
keep values
keep values.
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 (love-hate voting) in which last preferences may more significantly affect the outcome, and so are more likely to be made.

Second order Binomial STV

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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:


The expansion of the factor is:

(p+u) = pp + up + pu + uu.

The expansion is a non-commutative 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 over-all average of keep values that is the second order Binomial STV result.

Second order Binomial STV

We start with the two preference counts, refering to Table 1. First

the preference-qualified preference count (PP)

by re-distribution of most prefered candidate (A) votes.

Table 6. (PP) Preference-qualified preference count.
1st prefs. Redistrib.
A's votes

1st to 2nd pref.
transfer of B's
surplus votes,
10000 each to C & D @
transfer value

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
A 10100 9999/10100
= .99
B 10000 20000 9999 9999 9999/20000
= .49995
C 9998 9998 9998+
9999 9999/14998.5
= .6666666
D 9898 9898 9898+
9999 9999/14898.5
= .67114139
E 0 100 100 100 9999/100
= 99.99
X 0 0 0 9899 9999/9899
All votes 39996 39996 39996 39996

Next, the second qualified preference count, the

unpreference-qualified preference (UP) count

re-distributing the votes of the least prefered candidate, E. Due to the extreme simplicity of this example, candidate E has no surplus votes to re-distribute. (Binomial STV differs from previous transferable voting systems that take a trailing candidates votes to give to other candidates, and dismiss him in mid-count.)

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 re-distributed , with Binomial STV, where unpopularity counts, albeit adversely.
Here, the (second order) unpreference-qualified 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.

The unpreference-qualified unpreference (UU) count

redistributes the votes of the most unprefered candidate, C, in an unpreference count.

Table 7. (UU) Unpreference-qualified unpreference count.


1st & 2nd least
Transfer 10000
votes from
X to C @ transfer value
39996 =
Redistrib. to B.
3rd to 4th
X votes to D; & 100
to E @ transfer value
32496 = .6923006
4th to 5th pref
transfer 9898
votes from
X to D @ transfer
= .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 =
= .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,

the preference-qualified unpreference count (PU)

re-distributes 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 over-all Binomial STV count results.

Table 8. 2nd order Binomial STV averaged keep values.
PP: pref.
pref. keep
(Table 6)
UP: unpref.
pref. keep
(Table 2)
GM pref.
UU: unpref.
keep values
(Table 7)
PU: pref.
keep values
(Table 4)
GM unpref.
keep values
Inverse GM unpref.
keep values
GM** pref.
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

**Over-all 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.

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