Most of this web-site is now archive, superseded by e-book versions, currently on Amazon. In old age, my last purpose is to alert the general public to my invention of Binomial STV. This is a modification and improvement of the single transferable vote, the worlds most effective voting system, and bane of political careerists, who cheat it relentlessly, because it takes away their safe seats and replaces incumbency with democracy, in voting systems.
Varieties of Binomial STV
How "Binomial STV" works: count automation
Another hand count election test for automation
Hand count of Binomial STV (instructions)
Certificate of works sought
Welsh Constitution and Democracy consultation 2022
Australian preference voting
Peace
STV legitimacy: zero order STV
First order STV and the Harmonic Mean quota
STV^1: one election count and a symmetrical exclusion count
Keep values for all candidates, both in election and exclusion
Conventional STV has many varieties, so has Binomial STV many versions,
ranging from the relatively simple first order STV, adequate for voting
purposes (STV^1).
In the first place, Binomial STV is a legitimation of traditional STV, which,
technically speaking, is STV to the power of zero (STV^0). In line with the
terminology of the binomial theorem, it just means, conventional STV deals with
one variable (whether an election or an exclusion/rejection) or is a uninomial
system, as distinct from the binomial case, with two variables, election and
exclusion.
It does not mean that traditional or conventional STV is wrong; it just means
that it is a more limited case than Binomial STV.
Binomial STV looks like any ranked choice vote or preference vote. The difference is that as the preferences exceed the number of seats, they start to count against the prefered candidates. So that a last preference may count as much against the candidate as a first preference counts for the candidate.
The simplest case of Binomial STV is first order STV (STV^1). It consists of one election count and one symmetrical exclusion count. That is, it is a consistent or non-contradictory one-truth system. It is especially reserved for voting, because the numbers are usually not too great, on this little planet.
The mass data involved, with the representation of the exponential growth of human knowledge, might call for more sophisticated versions of Binomial STV, involving higher orders of representation. In accord with the bnomial theorem, the term for higher orders of STV, nth order STV, might be: STV^n
Starting from the simplest case of Binomial STV, that is, first-order STV, there are various options. It is easiest to follow conventional practise for the quota, which is the Droop quota.
Contrary to accepted belief, this is no better than the original Hare quota.
The Droop quota is just more convenient for small constituencies. The Droop
quota is just the minimum proportional representation. While the Hare quota is
the maximum proportional representation.
The Droop quota became more fashionable, not because it is better than the Hare
quota, but because politicians dislike the greater competition of large
constituencies.
In fact, both Droop and Hare have democratic deficiencies: Droop, because it can elect candidates with a statistical insignificance of more votes than rival candidates; and Hare, because it requires too much deference, from the voters, to the elected majority candidate, to work.
Hence, the harmonic mean of these minimum and maximum proportional representations is the most democratic quota: V/(S + ½). (The harmonic mean is the average of two harmonic series.) Since I discovered this average, I called it the harmonic mean quota. To call it after myself, as I did not, in the case of Binomial STV, would have been less meaningful, and less useful or memorable.
Later, I found that the harmonic mean quota corresponds to the St Lague divisor system, the most equitable of divisors. It is less well known as the Webster apportionment of seats in Congress, for which it was adopted on two occasions, but this comparison is more suitable because divisor systems, like St Lague, generally apply only to parties, not affording personal representation.
The harmonic mean quota only works for constituencies of at least four or five seats. But this is required for the proportional representation to be minimally democratic, or represent something approaching all the people. Therefore the HM quota is advised, even tho the Droop quota is a more convenient default for small constituencies. Single member districts, however, are only half a democracy, no matter what the count rules for arriving at that result.
The quota sets the number of votes that each candidate has to win to be elected. In Binomial STV, the same quota also sets the number of votes, that determine whether a candidate will be excluded.
STV transfers surplus votes to next prefered candidates. Binomial STV uses
Meek method for this purpose. More-over, it employs Meek method to exclude, as
well as elect, candidates (which not even Meek method does).
If a candidate has both an election quota and an exclusion quota, the balance
of a candidates votes decides election or exclusion. (It pays to be nice.)
With Binomial STV, the election quota count is followed by an exclusion quota count, in which the candidates are excluded, if achieving an exclusion quota. Because it is a symmetrical count to the election count, any exclusion votes received after a candidates exclusion, by quota, are also counted (increasing the decisiveness of their exclusion and exclusion keep value).
In both counts, all the abstentions are counted. If a quota of abstentions is reached, a seat remains unfilled, acting as an incentive for better candidates. A blank paper or carte blanche is equivalent to NOTA or none of the above. Any preference from first to last may be an abstention. Where the abstentions fall, early or late in the preferences, determines how much the voters want to elect or exclude candidates, so that the relative importance, of election or exclusion of candidates to the voters, is known.
Meek has a useful way of recording the election of candidates, which Binomial STV extends or makes extra use of.
With Meek, candidates are elected on a "keep value" of a ratio of one or less. The keep value is the quota divided by votes for a candidate, including those they may obtain after election to a quota.
Binomial STV also extends Meek use of the keep value, in two ways: a candidate has an exclusion keep value, as well as an election keep value; and keep values are calculated for all candidates, not just those who reach an election quota.
So, first order STV conducts an election count and then an exclusion count. To arrive at a final result for the candidates, these counts are averaged. In this case, invert the exclusion count values for candidates, to serve as a second-opinion election count. The two counts average is their geometric mean, presumably because taking the average or representative value of two non-linear distributions, Droop and Hare harmonic series. The concept of the average is required, not only for statistics in general, but for purposes of democratic representation.
Note that dividing the election count by the exclusion count, considered as keep values, cancels out the quota from both the ratios, which is convenient for calculation.
All candidates with (geometric) average keep values of one or below, are elected, the candidate having the smallest average keep value being the leader. Candidates with average keep values, of more than one, are excluded.
So far, two averages have been considered for a more representative result: the harmonic mean quota and geometric mean keep values. The latter performs arithmetic mean calculations in its indices, for a multiplication by the geometric mean.
There is yet to consider the case, where more than one candidate has a surplus of votes over the quota. In this case, the average size of their surpluses is used, rather than, say, the candidate with the biggest (quota) surplus, who may not be representative. This calculation requires a third average, namely the arithmetic mean.
This average comprises the third of the four basic representatives or averages of distributions. They are: the arithmetic mean, the harmonic mean, and the power arithmetic mean or geometric mean.
This leaves a fourth average which did not even have a known name but may be
called the power harmonic mean. This may be used to average higher orders of
Binomial STV. Hence FAB STV: Four Averages Binomial STV. (As described in my
book of that title.)
(Binomial STV of any higher order, or the nth order, is symbolised STV^n.)
It is possible to do a hand count of Binomial STV. It is essentially an adaptation of the best part of a computer count, namely Meek method, which is more accurate than a hand count. But it means giving up the best part of Meek method, whose virtue is to go on counting surplus votes for a candidate, even when they have already surpassed the quota. Nevertheless, previous hand-count versions of STV did without this advantage, without obviously deficient results. The same virtue holds: a large constituency with many seats is more proportional and gives highly accurate representation of first or high preferences, for individual candidates.
Thus, the hand-count first order STV procedure stops counting a candidates preferences, when the quota has been reached, just as traditional STV hand-counts do, to simplify calculation. But it is still a Binomial STV with one election count followed by one exactly similar exclusion count. Unlike traditional STV, including Meek method, the "last past the post" exclusion of candidates, when the surplus votes run out, is not needed.
All abstentions are counted (proportionally) so the Meek method of reducing the quota with exhausted preferences is also not needed.
or preference vote chooses candidates in order of choice, 1, 2, 3,…, up to the number of candidates.
A Binomial STV ballot looks like another ranked choice vote, singling out prefered candidates, in an order of choice. Unlike traditional STV, any preference, that exceeds the number of seats or vacancies, does not help to elect, but to exclude, that candidate.
For example, if there are 12 candidates, competing for 6 seats, preference 7 will count slightly against that candidate. Preference 12 may count as much to exclude a candidate, as preference 1 counts to elect a candidate.
Any preferences, from first to last, may be left unfilled. A completely unfilled ballot or carte blanche (blank paper) is equivalent to None Of The Above (NOTA). Unlike traditional STV, abstentions are counted, towards an unelective quota of votes.
A database, of all the (logged in) voters ranked choices, including abstentions, is compiled into a list, whose number of rows comprises the total number of votes.
A proportional count is used for this basic (first order) Binomial STV.
The elective proportion of votes, or quota, is: total votes divided by one more than the number of seats. Or: votes/(seats plus one), which abbreviates to: V/(S+1).
Abstentions are proportionally counted, like the votes for any other candidate. Abstentions may have the alias "Nemo" or No-one. But abstentions may fill from zero cells to every cell, on a row of preferences, which equals the number of candidates. Whereas, actual candidates may be prefered only in zero cells or just one cell in the row of preferences. A quota of abstentions leaves a seat unoccupied. A quota of votes for a single candidate, leaves a seat occupied. Voters may prefer none, or one candidate, up to all the candidates. In so far as number order is not maintained, as by a missing numeral, the further preferences of the vote should count as abstentions.
All the votes are counted to establish the total vote (given by the length of the list of preference votes). All the first preferences are counted for their respective candidates, including any first preference abstentions (for “Nemo”) if applicable.
The first column, of an election count table, is named, at the top, candidates, followed by a vertical list of the candidates names. The second column, named, at the top, first preferences, lists the number of first preferences, which each candidate gains, trailed by a cell for (abstentions) candidate “Nemo.” Below is a cell for any invalid votes. And the foot of the column should sum to the total vote.
If there are no candidates with first preferences, in surplus of a quota, the election count halts. And if no candidates vote is equal to the quota, there is no provisional election of any candidates.
[Unlike traditional STV counts, there is no “last past the post” elimination of candidates, after election surpluses run out.]
A (provisionally) elected candidates surplus vote is transferable to next prefered candidates, by (what statisticians call) “weighting in arithmetic proportion” (a.k.a. Gregory method): all an elected candidates votes, greater in number than (or surplus to) the quota, are transferable, in proportion to the (voting pattern of) next preferences of all that elected candidates vote, called the “total transferable vote.”
Any surplus to the total transferable vote minus the quota is the “surplus value” of an elected candidates vote.
The ratio of the surplus value to the total transferable vote, equals the “transfer value” of each next preference. (In the first instance, a surplus transfer is of second preferences, to an elected candidates first preferences).
(Surplus value)/(total transferable vote) = 1 –[(quota)/(total transferable vote)] = transfer value.
In a third, or further, column, the transfer value is multiplied by the
number of next preferences for each candidate. These surplus votes, for next
preferences, should sum, in this case, to the surplus value, at the foot of the
column.
Each candidates surplus transfer of votes are added, in a fourth, or further,
column, to their existing votes. In the surplus transfer, from an elected
candidates vote, the elected candidate cell is tabulated, at just the quota of
votes. This serves as an arithmetic check that the column of votes, after
surplus transfers, still sums to the same total vote, at the foot of the
column.
The surplus transfer procedure should be iterated with respect to every surplus voted candidate. And more than one surplus transfer for a given candidate, may be necessary, if the surplus vote is greater than the quota.
The candidate with the bigger surplus has prior transfer. A further surplus transfer is counted if the surplus is more than a quota in size. Surplus transfers may bring another candidates votes, into a surplus to the quota, resulting in that candidates own surplus transfer of votes.
The exclusion count is (completely symmetric) iteration of the election count, with the preferences reversed. (The database, of all the voters ranked choices, is reversed from first to last preferences, into last to first preferences, counting the preference order of cells, not from left to right, but from right to left.)
The election count elects candidates. The exclusion count excludes candidates.
A candidate, who is both elected and excluded, may be called “Schrödinger’s candidate” (a term from Forest Simmons, after the unfortunate quadruped, Schrödinger’s cat, deemed both dead and alive, in quantum theory). If the ratio of exclusion votes divided by election votes (“the quotient”) of a candidate, is unity or less than unity, that candidate is elected. Should there be two or more Schrödinger candidates, vying for a lesser number of vacancies, the candidate(s) with the smaller quotient below unity is deemed elected.
Generally, the lowness of candidates quotients establishes their order of precedence.
(The quotient is the square of the geometric mean, but taking the square root of that average (or statistical representation of voters by candidates) does not alter that order of precedence.)
with 12 voters, 5 candidates and 3 seats.
Using minimal proportional representation of the Droop quota: 12/(3+1) = 3 votes, for an election or exclusion of a candidate.
Table 0:
12 voters preferences, 1 2 3 4 5, for 5 candidates a b c d e:
| 1 | a | b | c | d | e | a | a | a | b | b | b | c |
| 2 | b | c | d | e | a | c | b | b | a | c | c | b |
| 3 | c | d | e | a | b | d | d | c | c | a | d | d |
| 4 | d | e | a | b | c | e | e | e | d | d | a | e |
| 5 | e | a | b | c | d | b | c | d | e | e | e | a |
Table 0, of 5 candidates 60 voter permutations. No abstentions are shown, marked by dashes ( - ) which may occur randomly, instead of any given preference. (Abstentions are less likely in small-scale votes.) Every voter may have from zero abstentions to all abstentions, 12 here, equivalent to None Of The Above (NOTA). If a quota of abstentions is reached, one of the seats remains unfilled, encouraging better candidates.
Table 1: The distribution of preference orders:
| Candid -ates |
Prefs 1 |
2 | 3 | 4 | 5 |
| A | 4 | 2 | 2 | 2 | 2 |
| B | 4 | 4 | 1 | 1 | 2 |
| C | 2 | 4 | 3 | 1 | 2 |
| D | 1 | 1 | 5 | 3 | 2 |
| E | 1 | 1 | 1 | 5 | 4 |
Candidates A and B are elected each with 1 vote above the quota of 3 votes.
Stage 1: Election count A and B, 2 votes surplus transfer. We know from
table zero, that 3/4 of a vote go from A to C and that 1/4 of a vote goes from
B to C, equaling 1 vote, in all. Therefore C takes the third seat, with 2 + 1 =
3 votes.
Binomial STV takes after Meek method in transfering surplus votes to already
elected candidates. The Keep value of A was 3/4. But 1/4 of a vote transfers
from B to A. So the new (election) keep value for A is 3/(4 + 1/4) = 12/17.
This gives A a slightly lower keep value than B, and therefore a slight
elective lead over B, which has a lead over C.
Stage 2: Exclusion count:
The exclusion count, which is the exact same as the election count but with the preferences reversed (in table zero) is always carried out with Binomial STV, because it confirms election or exclusion of candidates. If a candidate elected on a quota has few exclusion votes; less than an exclusion quota, that reinforces their election. Not otherwise. It pays to be nice.
It is not necessary to note, for this election, that candidate E is excluded with a surplus above the quota of 1 vote. Looking at the four 5th preferences for E, on table zero, 3/4 of them go to candidate D, and 1/4 to A. This gives A a small exclusion vote, slightly more than B but nowhere near an exclusion quota of 3 votes. The next exclusion preference goes from A to D again. D already has 2 exclusion preferences (as do all the other candidates) which means D is (almost on principle) excluded with (the best part of an extra exclusion vote) and an exclusion quota of 2 + 1 (approx) = c3 votes.
If a candidate has both an election quota and an exclusion quota, the balance of their vote, the Quotient of exclusion votes to election votes decides whether the candidate is finally elected or excluded.
32 voters for 8 candidates in 8 orders of choice.
| voters | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 8 | C | B | D | A | - | - | - | - |
| 6 | H | F | G | E | - | - | - | |
| 4 | - | - | - | - | G | H | F | E |
| 3 | D | B | A | C | F | H | - | - |
| 2 | C | E | G | A | H | B | D | F |
| 4 | F | G | H | E | A | B | C | D |
| 1 | D | E | B | A | H | F | G | E |
| 3 | A | D | E | F | H | G | C | B |
| 1 | G | A | F | B | C | H | D | E |
| total: 32 |
The database is a table of the voters permutations of preference for the candidates. Since there are 8 candidates, the possible number of preference permutations is factorial eight, a huge number. The dashes signify abstentions, which, unlike a candidate, are allowed in more than one cell per row. Abstentions would most probably curb the multiplicity of permutations. Even so, this simplistic example contains a mere nine permutations, which is drastically unrealistic, but leaves the arithmetic work, at managable levels, for only one person to make-up an example and calculate the resulting count.
This basic (first order) Binomial STV is itself a simplified procedure. Tho
still not simple, it should be practical as a hand count, at least by a trained
team of returning officers, when conducted on a large scale.
Whether one untutored old man could graphically program this binomial count, in
nocode, is another matter.
Several versions of traditional STV have been coded by professionals into
electronic counts. The republic of Ireland still uses the original random
sample surplus transfer STV hand count. Cambridge Massachusetts has followed
this essential model into an electronic count. Except for general elections,
Northern Ireland uses surplus transfer (Gregory method) weighting in arithmetic
proportion, by professional hand count. But Scotland uses an electronic count
version, for local elections.
Meek method STV is a specialised computer count, which New Zealand uses for its
health boards and a few local elections. But no conventional STV method uses a
rational exclusion count, with their rational election count. In fact, no
official election methods, in the world, use a rational exclusion count.
In Binomial STV, the quota is the same for both the election count and the exclusion count. This remains true for more than a first order Binomial STV, described here. But the more advanced version innovates a modified quota.
Quota, Q, of candidate election, per seat, is the total vote, V, divided by
one more than the number of seats, S. Or, Q = V/(S+1). Therefore, for 32 voters
electing candidates to four seats, Q = 32/(4+1) = 6.4
A candidate is elected on reaching a quota of 6.4 votes.
| Candidates | First preferences | 3.6 surplus transfer from C to B & E (see database) |
Votes after surplus transfers |
| A | 3 | 3 | |
| B | 0 | (8/10)*3.6 = 2.88 | 2.88 |
| C | 10 | 6.4 | 6.4 |
| D | 4 | 4 | |
| E | 0 | (2/10)*3.6 = 0.72 | 0.72 |
| F | 4 | 4 | |
| G | 1 | 1 | |
| H | 6 | 6 | |
| Abstentions: "Nemo" |
4 | 4 | |
| Invalid votes | - | - | |
| Total votes | 32 | 10 | 32 |
Table one shows only one candidate elected to 4 seats. A standard deviation
test shows whether any other candidate is not significantly below the election
threshold. A standard deviation, SD, is given by the square root of [the total
votes multiplied by an arithmetic mean (set at the quota ratio), and multiplied
by (unity minus that ratio). In other words, in this case, SD = square root
[32*(1/5)*(4/5)] = 2.26. Candidate H, with 6 votes, only 0.4 short of the
quota, is well within one standard deviation of the quota. However, no other
candidate is within one standard deviation of the quota.
If it was previously agreed beforehand to take this statistical significance in
consideration, candidate H, as well as C, could be considered elected. This
still leaves only two out of four seats filled.
| Candidates | Last preferences |
Prior abstention surplus transfer to H (see database) |
Further abstention surplus transfers to A & E |
Consecutive surplus transfers to D & G |
E surplus transfer 2.343 to F; to G & D |
D surplus transfer 0.3165 |
Maximum exclusion votes |
| A | 0 | (8/14)*6.4=3.657 | 3.657 | ||||
| B | 3 | 1 | |||||
| C | 0 | 0.3165 | 0.3165 | ||||
| D | 4 | (8/14)*4.071 =2.326 |
(1/6)*2.343 =0.3905 |
6.4 | 8.743 | ||
| E | 6 | (6/14)*6.4=2.743 | 6.4 | 3.5616 | |||
| F | 2 | (4/6)*2.343 =1.562 |
1.562 | ||||
| G | 0 | (6/14)*4.071 =1.745 |
(1/6)*2.342 =0.3905 |
2.1343 | |||
| H | 0 | (3/17)*6.4=1.129 | 1.129 | ||||
| Abstentions: "Nemo" |
17 | 17-6.4=11.6 + (14/17)*6.4=5.271 |
16.871-6.4 =10.471 |
6.4 |
17 | ||
| Invalid votes | - | - | - | - | - | - | |
| Total votes | 32-17=15 | 17 | 16.871 | 10.471 | 8.743 | 6.7165 |
Binomial STV introduces the novelty of counting abstentions. (Abstentons counting is necessary to determine the balance of voters desire to elect or exclude candidates.) This example is marked by a much greater desire of voters (17 of their last preferences) not to exclude candidates, than (4 of their first preferences) not to elect candidates.
The exclusion count has nearly three quotas of abstentions. In other words,
nearly three "Nemos" or No-ones are excluded. Correspondingly, only one actual
candidate, E, is definitely excluded. None of the other candidates come close
to being excluded, by a test of statistical significance. It must be
remembered, however, that this simple example of voting is conducted at the
extreme lower limit of numbers for any statistical test to be at all
significant.
Even so, it was still necessary to conduct the count to four decimal places
("argument four") for consistency in the count.
The main reason why this example was not fully elective of all four seats is that the voting pattern did not follow a (random or) normal distribution. The typical random voting pattern for 32 voters would assign to candidates approximately the distribution: 0, 1, 5, 10, 10, 5, 1, 0. In the above election example, there was only one 10 (C) and no fives. Had there been this sort of distribution, two candidates would have been elected with surpuses, which would probably have helped the other two candidates with 5 first preferences, to fill all four vacancies.
Unlike traditional STV, Binomial STV does not assume that any line-up of candidates will fill all the vacancies, if they lack substantial support.
(This letter was a well enough written response but it was before I learned to slant my explanations less steeply. I appreciated later that FAB STV: Four Averages Binomial STV was not necessary for elections. Basic (first order) Binomial STV, one election count with one exclusion count, would do it. All Four Averages came in with higher orders of STV for the representation of a growing mass of data, not just voters. I've written e-books on this and other Binomial STV research.)
In terms of current electoral knowledge, the best place to start with FAB STV is where it departs from Meek method.
Experts recognise Meek to be the best STV count, because it does not stop counting any further preferences for already elected candidates. It does this by computer count flow chart, to update an elected candidates keep value. This is the ratio of the quota to a candidates total transferable vote. If a candidate reaches no more than the quota of votes, their keep value is unity. Surplus preferences diminish the candidates keep value to below unity.
(Every voter has one vote. Transferable voting allows this vote to be split up into fractions between several most prefered candidates, in order of choice. The keep value plus the transfer value always sums to one vote.)
FAB STV departs from Meek in extending the keep value for elected candidates, with surplus votes, also to unelected candidates, in deficit of the quota. This allows for a universal keep value comparison of popularity between the candidates.
The aim is to do away with the ad hoc or arbitrary (or “non-monotonic”) elimination of candidates, when the surplus votes happen to run out.
The elimination of candidates is given its own rational count of exclusion, similar to a rational count of election.
One can justify this by the truth doctrine, that there is only one truth, not two truths. Thus, one must count exclusions the same way one counts elections. It can be called the symmetrical count requirement. (By the way, this requirement refutes all hybrid electoral systems, with contrary counts.)
In FAB STV, the exclusion count is counted exactly the same way as the election count, but with the preferences reversed. Consequently, a voters least preferences will count as much to excluding candidates, as their greatest preferences count towards electing candidates.
It is not necessary for voters to prefer, positively or negatively, all the candidates. Blank preferences count towards a NOTA quota, which would leave a seat unfilled.
Once the computer has calculated all the candidates election keep values and calculated all the candidates exclusion keep values, the two sets of keep values can be averaged for the final overall keep values of each candidate.
This is done by converting the exclusion keep values into a second set of election keep values, simply by inverting them. The geometric mean is taken of the election keep values and the inverted exclusion keep values.
This completes a basic Binomial STV. It is actually first order binomial STV. Higher-order counts and three more averages, besides the geometric mean, are possible, tho not needed for the general run of elections.
Please let me familiarise you with the (simplified) binomial STV hand count.
It starts just like the traditional STV hand count. Count all the first preferences. If there are any surpluses, transfer them, until there are no more surpluses.
Calculate the keep values of the elected candidates, Meek method fashion. There is no elimination of candidates, when the surpluses run out. And there is no continued counting, of an already elected candidates preferences, to update their keep values. That of course is the distinctive contribution of Meek method, requiring a computer count. Elected candidates will all have keep values of unity or less.
Unlike Meek, also calculate the keep values of the remaining candidates in deficit of a quota. These will have keep values of greater than unity. You have now completed the binomial STV election count. Still to come is the binomial STV exclusion count. This is conducted exactly like the election count. It is just that the exclusion count is conducted with the preferences in reverse order, starting with the last preferences.
The exclusion count will also give you keep values, for candidates in surplus and deficit of a quota. However, the meaning of the exclusion count is the exact opposite to the election count. Candidates with exclusion keep values of one or less are deemed excluded. Candidates with exclusion keep values of more than one are deemed not excluded.
Neither the election count nor the exclusion count have the final say in who is elected or excluded. It is the average keep value of the two counts that determines the final decision as to who does or does not gain a seat.
Averaging the election and exclusion keep values is made possible by inverting the exclusion keep values, to make them a makeshift alternative set of election keep values. The quickest way to do this is simply to divide each candidates election keep value, by that candidates exclusion keep value. And then to take the square root, of this ratio, for the final keep value. This result completes the operation of averaging by the geometric mean.
The lowest keep values, which should be fractions of less than unity, determine who fills the seats.
Leaving the ballot paper completely blank is equivalent to nota, none of the above. If enough voters have left enough blank preferences, then a seat may remain unfilled. That explains why there is no Meek method reduction of the quota with exhausted preferences. Because, exhausted preferences are still counted, under binomial STV.
That completes the (first-order) binomial STV hand count.
I seek a certificate of works, from some organisation, not necessarily a normative endorsement, to show that my invention of Binomial STV works or is a practical proposition. First order STV (STV^1) whose coding I hired, should be sufficient for voting: a logical one-truth system of one election (by the relevant part of Meek method) and one exclusion (counted the same way or symmetrically, without the criticised "last past the post" exclusion count).
And count all preference abstentions to know, by whether early or late in preference, the relative importance for election and exclusion of candidates to the voters. The mathematical significance of positive, negative and neutral choice counting is one complete dimension of choice that can be part of a scientific model of understanding.
Higher orders of STV might systematically represent the exponential growth of human knowledge.
It looks as tho I have kept my discovery of ”Binomial STV” to myself but it is the wilfully ignorant nature of politics that has produced that impression. A truthful voting system should have only one set of rules, not contradictory sets. Which is true? Election and exclusion of candidates should be counted the same way, because positive and negative choice are the same in principle, not justified by different sets of rules; a mistake which every voting system in the world makes.
But politics knows nothing of the progress of knowledge. Any number of institutions, official and reputably independent, have any number of superficial objectives. Sometimes they make bogus claims they are “revolutionary.” The civil servant, Sir Patrick Nairne described the 1997 New Labour antics at gaining public attention for constitutional reform: "a colossal display of catastrophic cosmetics." Without going into details, the so-called independent Jenkins voting commission somewhat justified that description. Sir Kier appears to be another New Labour “pragmatist.” For well over a century, Parliament has been putting incumbency before democracy. First Past the Post was about a 70% safe seat system. (Of MPs it was remarked: You have a job for life, don't you?)
Meanwhile, Labour pushed regionally a doubly safe seat system (AMS/MMP), fundamentally undemocratic, denying voters the right to reject candidates, according to the Richard report (recommending replacement by STV).
This is an opportune moment to point out that a decision to use a closed party list for the election of the Welsh Parliament is not going to politically engage the Welsh people. This system is an abandonment of representative democracy, for manifesto referendums, run by party bureaucracies, dependent on their leaders, and not the Welsh people.
Representative elections utilise our common knowledge of character, gained from our social lives, and guaranteed to engage the public. – Provided that representation is effective. Pioneer Australian activist, Catherine Helen Spence, who met John Stuart Mill, called the Hare system “effective voting.”
It is true, as Enid Lakeman repeatedly said, that the personal element in the Additional Members System is “illusory.” Single members are a system of monopolies on personal representation, the minimum possible.
The Richard commission counted less than a handful of Members not returned by the Additional Member System, concluding AMS denies voters the fundamental democratic right to reject candidates.
The McAllister report refers to academic evidence, that diversity of representation requires a minimum of four, and up to seven, member constituencies for proportional representation. Cited by Joe Rogaly, in Parliament for the People, Wnston Churchill said: I would rather be one fifth of the Members for the whole of Leeds, than one Member for a fifth of Leeds.
Churchill urged proportional representation, in his reply to the 1950 Kings Speech.
The Irish constitutional convention recommended a minimum of five member constituencies, with the single transferable vote.
Official Welsh reports, Sunderland, Richard and McAllister, as well as expert committee testimony, have repeatedly endorsed this STV system.
The Sunderland commission, on Welsh local elections, said it was “an insult” to claim that the Welsh people would not be able to vote by number order of choice. The Welsh government recently allowed Welsh councils, but not voters, to adopt STV, as in Northern Ireland and Scotland.
Wales has maximally empowered the state with an increase from 60 to 90 Members (in the gift of the parties) but minimally empowered the voters, with no more than an option for transferable voting, “the super-vote” (in the gift of local councils).
On British Euro elections, Richard Wood MP (later Lord Holderness) got up, in Parliament, on the tin legs of a crashed Spitfire pilot, and said it was “an insult to the intelligence of the British people” to refuse them a vote: 1, 2, 3,….
A government proposed Regional List was an x-vote for individual candidates on an open party list. When challenged, in Parliament, the Home Secretary had to admit that the open list could elect a party candidate with no personal votes.
Such is the degradation of regard for Members of Parliament, and, by the same token, representative democracy.
A closed party list only makes matters worse.
The Regional (open) List would have split the nationalist vote, in Ulster Euro elections. Because, the peace party (SDLP) and the war party (Sinn Fein) would not have shared a list. But the single transferable vote allowed nationalists both party candidates, in their orders of preference, proportionally representing, with a seat, the nationalist third of Ulster.
May 2025
An outstanding abuse remaining in Australian elections.
It may seem unfair to pick on Australia, one of the few countries to recognise that voting is necessarily preference voting. Other countries admitted this, with the concept of wasted votes and strategic or tactical votes but don't take the (relatively) honest Australian course. However the situation is complicated by Australian use of compulsory voting. I don't know whether it can ever be justified but I do know that a vote for none of the above or not some of the above is compatible with compulsory voting.
This would be the simplest remedy for the situation in the Australian Lower House, where compulsory voting still obliges the compulsory use of all preference votes, which means that ultimately voters are compelled to vote for a party candidate to which they are opposed. This breaches the fundamental democratic right to reject candidates.
My invention of "Binomial STV," its basic voting version, with one election count and one exclusion count, exactly like it, would be another way to remedy the same abuse. Some candidates may get both an election quota and an exclusion quota. In this case, the balance of the two decides whether the candidate be elected or excluded. The candidates, in the best position, are those who have an election quota but fall short of an exclusion quota. In other words, they are popular with one group but not unpopular with any other group. It pays to be nice.
This reform has other consequences, some of which are profound.
Peace is not achieved because of false ideas of democracy, which are not peaceful but provoke conflict. Anglo-American "democracy" is, as JS Mill and Lani Guinier said, "the tyranny of the majority". An unrepresented minority seeks to break away from a majority to form its own smaller majority tyranny (often with its own breakaway unrepresented minority of a minority).
So, why don't we have the temerity to improve democracy to something more like Mill said was the rule of all the people, approaching two centuries ago, and Guinier advocated something like two decades ago? Because it is institutionalised. Keir Starmer had the intelligence to realise we must address safe seats, which leave career politicians unaccountable. But when it came to taking power and a monopoly of power, at that, First Past The Post elections were "right" (for him). The Houseof Commons is actually a house of monopolies.
Many electoral reformers so-called want to jump out of the frying pan into the fire, with party lists. These are so-called proportional reprsentation, where party proportion is the obsession, at the expense of individual representation and responsibility, which you have to have for peace with the voters.
In two respects, the obsolete state of unreformed or corruptly reformed
voting shows. The universal contemptuous admission there are "wasted votes" and
"tactical" votes is an admission of the prejudice and intolerance against
preference voting in 98% of the worlds voting systems.
A 100% of them fail to realise that elections and exclusions are but positive
and negative versions of the same thing, and require the same procedure, making
them all contradictory two-truth systems. I invented a one-truth voting system,
as my no doubt small but I hope significant and worthy to be considered
contribution.
The under-representation of women has been solved by the best-kept secret in
electorl reform. Before 1979, the British NationL Health Service returned all
white male General Practitioners First Past the Post, to the General Medical
Council. From 1979, the single transferable vote (the Hare system, essentially
Cambridge Mass.) proportionally represented women, immigrants and specialists.
The reason we don't have this system in our general politics is because career
politicians want incumbency not democracy. Politics is so stupid, not because
people are stupid but because it is dedicated to selfish ignorance rather than
knowledge of how to do things.
This is a disease that is spread thru-out society. Never have we had so many
institutions pretending to do so much, doing so little, apart from each other,
presumably. Or how would they otherwise justify their existence? The
universities are a closed shop without any apparent sense of responsibility to
the general public. Admittedly, the hierarchical organisation of society turns
people into a mass of over-looked subordinates.
Liberating democracy from Impossibility Theorem.
Binomial STV and the Harmonic Mean quota.
The Four Loves: romance, friendship, affection and
charity.
(From C S Lewis)
Sigmund Freud and CG Jung:
seeking the whole man thru a democracy of ones selfs.
The moral sciences as the ethics of scientific method.
Science is ethics or "electics."
A synthesis of deterministic & statistical world-views.
Relativity theory and election method.
The determined dishonesty of atomic energy.
Nuclear deterence threatens its own world war crime.
Index to some early (unrevised) pages in simpler English spelling.
En français: (In French)
Modèle Scientifique du Procès Electoral. (Scrutin Transferable: Single Transferable Vote.)
Getting Ideas: How I dreamt my life away
and survey of this site.
Simpler English spelings Home page of some early versions to these pages. (More consistent than conventional English, tho superseded by my later spelling reforms.)
Relegated pages (or junk-yard).
Three e-books on electoral reform & research may be down-loaded free, in epub format. Down-loading the Calibre program enables reading e-books in many formats.
The Commentaries series. (Three more e-books available.)
Help notes on self-publishing electronic books.
The works of HG Wells came out of copyright, 70 years after his death. Wells might have been dismayed to find that he is still ahead of his time, not least on electoral reform:
World peace thru democracy: HG Wells neglected third phase.
My e-books include works on my friend, the poet and novelist Dorothy Cowlin
May 2025.
e-mail: voting[at]ukscientists[dot]com
Copyright © Richard Lung.
This Democracy Science site began in Spring 1999 at a former venue.