Former reform examples of Reversible STV, and the Re-transferable Vote.

• Hand count example one of Reversible STV.
• Reversible STV manual count: Example two
• Example three.
• Conclusion.
• The Re-transferable Vote as a proportional progression from STV and reversible STV.
• From STV to rSTV to RV.
• Keep values of deficit candidates, as well as surplus candidates.
• Hand count example of retransferable voting.

Archive note:

This page contains examples of a first reform proposal, Reversible STV, and a prototype, called the Re-transferable Vote, of Keep-valued transferable voting as with later Binomial and Condorcet counts.

Reversible STV.

Hand count example of Reversible STV.

Suppose a club of 32 members elect a committee of three by Reversible STV. And there are five candidates. That means there are factorial five or 120 possible permutations of preference. No more than 32 permutations are possible, in this case, because there are only 32 voters. The returning officer makes a table of how many voters there are for each permutation. To make the count easy, I have supposed there are only eight permutations with the following votes each. See table 1.

The permutation row numbers, perm 1, perm 2 etc, are for the returning officer to track any given transfer of votes according to voters' next preferences.

Reversible STV is set out into separate count tables for election and exclusion. The returning officer starts table 2, the election table, by adding up how many first preferences go to each candidate. Candidate C has more votes than the quota and is declared elected in stage 1. The value of C's surplus is shared in proportion to C's voters' next preferences, which each have a transfer value, calculated as (12 - 8)/12 = 4/12 = 1/3 of a vote.

Table 1 shows 12 first preference voters for C fall into four different permutations. 1 voter, for perm 4, and three voters, for perm 5, both make D their second choice. So D gets 4 transferable votes at 1/3 transfer value each, or 4/3 votes.
Perm 6 transfers 5 x 1/3 = 5/3 votes from C to A. Perm 7 transfers 3 x 1/3 = 1 vote from C to B. This transfer of C's surplus is shown in table 2, stage 2. Adding the surplus transfer, to the remaining candidates' previous tally, is enough for B to reach a quota and be declared elected ( in stage 2 continued ).

Tables of the Reversible Single Transferable Vote by Election and Exclusion Counts.

There is one more candidate to elect to the third available seat but no more surplus votes to transfer. At this point, conventional STV would exclude the candidate who happens to be last past the post. That candidate may be relatively more prefered than his present position shows. Reversible STV gives such candidates a reprieve.

Table 1 of permutations is considered in reverse to see which candidate first achieves an exclusion quota of least preferences. Table 3, the exclusion table starts with a count of the last preferences per candidate. For instance, reading from the right of table 1, perms 1, 2, and 7 show that B is the last preference respectively of 2, 5, and 5 voters, totalling 12 voters.

B is already elected. But B's surplus vote can be transfered to the next least prefered candidates. B's unpopularity happens to be the same size ( 12 votes ) as C's popularity, so B's surplus transfer value also happens to be 1/3.

Therefore, perm 2 and perm 6 each see 5 next preferences, at 5 x 1/3 = 5/3 votes value, go, respectively to D and E.
Perm 1 sees 2 x 1/3 = 2/3 votes go from B to C. Note that C is already elected but cannot be passed over in the exclusion count. This would be anomalous because B, also already elected, had to achieve an exclusion quota, before exclusion votes, in the form of surpluses, could be passed on to next least preferences.

The transfer of B's surplus least preferences is not enough to bring another candidate to the exclusion quota. So Reversible STV has to resort to a less rational method called "contingent exclusion". ( My later reforms, Binomial STV and so forth, avoid this contingency. ) Tho this is only on the scale of an ordering of candidates with more or less votes, it is more broadly based than the usual last past the post in hand-counted STV or Meek's method.

Simple STV only takes account of the fewest elective votes, currently held ( as in table 2 stage 2 continued ), to exclude a candidate. Reversible STV also takes into account, negatively, the current least preferences ( as shown in table 3 stage 2 continued ). The positive elective votes and the negative excluding votes are summed. The candidate with the least sum is excluded.

Table four contingently excludes D with an overall sum of positive and negative votes coming to -1 1/3. D's vote can now be re-distributed, in table 2, stage 3. Table 1 perm 1 transfers 2 votes from D to E. The other 4/3 votes, D picked-up from C's surplus vote, are tracked to perms 4 and 5. Of that 4/3 votes, 1/3 belongs to E, as next preference in line on perm 4, and 3/3 belongs to D, as next in line on perm 5.

The 2 1/3 votes from D to E are enough to take E over the quota. E is the third candidate to be elected to the three vacancies. See table 2 stage 3 continued.

The Reversible STV count tables show that candidates C, B and E are elected.

Note that in the above example, it so happens that candidate D would have been excluded by the traditional STV count, because D happened to be last past the post, when one of the candidates had to be excluded. The traditional hand count would have been like the first table, except that D was not excluded by virtue of the second table telling that D was the first candidate to reach an exclusion quota or be contingently excluded.

This simple example shows that Reversible STV can involve considerably more calculation than traditional STV.

Examples two and three, of the same voters' preferences, for two and three seats.

Table 5, above, gives the voting information, which enables a count to be conducted on the rules of the single transferable vote. My web page on a simple example of an STV count used the traditional means of excluding candidates. When no more surplus votes could be transfered, the candidate with least votes, at that stage, was excluded.

The returning officer writes out a table with new columns for each stage of the count. With traditional STV, when a candidate is excluded and his votes re-distributed to other candidates, the officer simply writes up another column on the table of the count.

A reversible STV recommends a different procedure, with two tables. Table 6 of the STV election count and, for a reversible STV, table 7 of an STV exclusion count.
Note how the two tables are cross-referenced, showing where the count may have to resort to exclusion, if no further election is possible at a given stage, and then back to the election count.

Example two of a Reversible STV two-seat count of the single transferable vote for election and exclusion.

For 110 voters electing 2 representatives. ( See table 6. ) The election quota is 110/(2+1) = 37 ( rounded up to the nearest whole number ). This is also the exclusion quota for table 7, if required.

The above two tables of the reversible STV count show that candidates D and E are elected. ( It may be of interest that these were the respective results of Borda's and Condorcet's methods. )
A few comments might be in order. One might think it perverse that candidate A with 36 votes, only one vote short of a quota, is the first to be excluded. But it must be remembered that this is not an actual election. Rather it is a pattern of preferences devised by critics of voting methods to show their fallibility.

One has to bear in mind that the way voters distribute their preferences are subject to probability. It is not too probable that a candidate, like A, who has much more first preferences than any other, is least prefered by all the other voters. In other words, actual elections are not too likely to turn up many of these odd results because voters are not too likely to vote in the odd ways dreamed up by critics of voting methods.
To say probable preference patterns dont matter, in assessing voting method, might be a bit like saying statistics doesnt matter in physics.

Such an example was specially dreamt up by critics so that it could go any way by changes of the rules. So, naturally, this version of STV conveys this contrived election as a closely fought contest. The point is that Reversible STV gives greater leeway against critics' objections, such as to non-monotonicity.

Example three of Reversible STV for a manual count for three seats.

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Reversible STV applied to the manual count, the same ballot is considered as electing three candidates. The manual count of Reversible STV is shown below.

The Reversible STV count result differs from traditional STV. When the latter runs out of surplus votes to elect further candidates, the candidate who happens to have the least votes, at that stage of the count, is eliminated. By that procedure, E would have been eliminated in this count. But Reversible STV requires a candidate to have reached a quota of least preference votes, before exclusion is allowed. E was not that quota-eliminated candidate and, in fact, E went on to take one of the three seats to be won.

Conclusion.

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Actually, it makes no difference to this simplistic example's result whether STV was conducted by the manual count or the systematic computer count called Meek's method, as normally it would. Likewise, it makes no difference to this result whether Reversible STV was applied to the manual count or Meek's method. But it did make a difference between Reversible STV and STV: it happened that the candidate, normally excluded, was not the first to reach an exclusion quota and went on to reach an election quota.

Reversible STV prevents a candidate being excluded, to re-distribute his votes to next preferences, just because he happens to have the least votes, when the surplus votes run-out for transfering to next prefered candidates. Instead, the candidate, who first achieves an exclusion quota, as least prefered candidate, is excluded, if not already elected.

The Re-transferable Vote as a proportional progression from STV and reversible STV.

After-note:

The so-called "Re-transferable Vote" has most of the essential features of my subsequent reformed transferable voting without being definitive. So, this section also is archive, rather than an introduction to method improvements.

From STV to Reversible STV to Retransferable Vote.

Reversible STV claimed to be an improved version of the single transferable vote. So, why a "retransferable vote"? By the way, RV is, like STV, still a single vote but Ive dropped the term "single" as being slightly pedantic. I suspect it originates from the days when the cumulative vote was also in experimental use in politics. This gives a voter several votes which can be distributed like so many points to the candidates. Giving all the points to one candidate was known as "plumping". Despite being the single transferable vote, its re-introduction to Northern Ireland in the 1970s apparently confused one voter, who told a politician he had given all his votes to him!

What is Retransferable Voting? Reversible STV was designed to reduce a problem with STV, the traditional transferable vote. But it created a new problem of its own, which RV turns from a problem into a further improvement in method.

STV was itself a reform of voting method that dates back to the middle of the nineteenth century. Several significant improvements have been made since then. In the later part of the twentieth century, criticisms of STV often had more to do with the way candidates are excluded, than STV's way of electing them. STV elects candidates when they have received a sufficient proportion of the votes in a multi-member constituency. Some candidates get more votes than this required proportion, or quota. The votes for a candidate, who has a surplus over the quota, are all transferable in accord with those voters' next preferences, at a ( fractional ) value proportionate to the size of the elected candidate's surplus.

Often, some seats remain to be taken, when there are no more surplus votes to transfer. Then the traditional method used with STV is simply to exclude the candidate with least votes, and transfer or re-distribute those votes to their next preferences. But this is an arbitrary thing to do. A candidate is excluded just because that person happens to have the least votes when the surplus votes run out.

To ensure a higher standard of excluding a candidate, early in 2004, I suggested that candidates not be excluded without reaching an exclusion quota. This was calculated in the same way as the usual election quota. The only difference is that all the candidates' preferences are considered in reverse order. STV starts by counting each candidate's first preferences, with any surplus votes transferable to next most prefered candidates. When the surpluses run out, Reversible STV differs from STV. The candidate Last Past the Post is not necessarily excluded. Instead, the exclusion depends on whether a candidate, not already elected, has enough last preferences to reach the quota ( the so-called Droop quota ) now used also for exclusion purposes.
That is to say, reversible transferable voting applies the same rational standard to excluding candidates as traditional STV only does to electing candidates.

But it turns out that there are two ways to carry out a proportional count for exclusion. At first, I assumed that one was correct and the other not. This caused me some confusion, as both alternatives had something logical to be said for them. I now believe my assumption was wrong. ( It's unwarranted assumptions that often get in the way of finding things out. ) My example of Reversible STV used an exclusion count based on the principle that one transfered surpluses of least prefered candidates towards the next least prefered, just as one transfered greatest preferences to next greatest preferences for election.

But there is another possible mode of exclusion. Suppose a candidate has already been elected. That candidate can no longer be excluded. Therefore, that candidate has zero votes for exclusion, tho probably the last preference of a number of voters. Those voters cannot exclude their least prefered candidate but their preferences, negatively considered, could count towards excluding their next least prefered candidate. Since there is no question of an already elected candidate having to achieve an exclusion quota, there is no question of transfering a surplus of least preferences, at a fractional value ( the transfer value ) in proportion to the size of some surplus. In other words, the already elected candidate's least preferences are transfered or re-distributed at full unit value to the next least prefered candidates. Of course, the normal ( "Senatorial" ) rules of transferable voting still apply in the case of surplus least preferences from a candidate who achieves more than a quota of unpreference and has not already been elected.

These ambiguities facing a Reversible STV revealed not one or two or three but four logicly possible modes of voting transfer. Hence, a "Retransferable Vote". The Re- in Retransferable means "back" or "again". In this reformed system, the transferable vote is transfered forward and back according to greatest and least preferences, then it is transfered forward and back, again, according to least and greatest preferences, respectively. Combining these four different ways, of processing the preferential information, should make for less erratic, more representative counting results.

( This logic, here explained, was not the one I was to adopt as Binomial STV's systems of re-counts but it did make me aware of other possibilities. )

The two sets of count are an election of preference qualified by an exclusion of unpreference and an election of unpreference qualified by an exclusion of preference. The first set of counts stands for the election of "electively prefered" candidates, qualified, if necessary, by the exclusion of the unprefered or the "exclusively unprefered". Then, the second reversible count stands for the "electively unprefered" candidates, qualified, if necessary, by the "exclusively prefered".

Use of the terms, liked and disliked, for prefered and unprefered, is avoided. Obviously, many voters do dislike some candidates and might like to have a power of excluding as well as electing, that reversible STV or retransferable voting would give. However, elections are really about suitability of candidates for office, not about whether they are likable or not.

Electively prefering candidates merely refers to STV's standard proportional election of candidates in accord with voters' preferences. The reverse of this is exclusive unpreference, which starts from the votes in reverse order of least preference or unpreference, for purposes of the candidates' negative election or exclusion.

The term "electively unprefered" may convey a kind of perverse admiration for a work that is outstandingly bad, like an unconscious comedy or a caterwauling tune. "Golden Turkeys" were awarded once for worst SF movies ever made. There are fan clubs for them, such is the enjoyment they give. An older term for golden turkey is Booby prize. Both phrases, by ironic use of the word, golden, and the word, prize, intimate a mock election of the least suitable.
The bird called the booby is an amazingly good flier and an all the more amazingly bad lander. One feels that it deserves a prize as much for its inept foot-work as its apt wing-work. A booby prize may amount to a sort of positive exclusion or election in its own right, in the sense that the purpose of the count was not so much to lose the chaff to find the wheat as to cherish a seemingly worthless by-product. That is an "elective unpreference."

Lastly, an "exclusive preference" is the reverse of an election of unpreference, described in the previous paragraph. When there are no more surplus votes to choose-out who are the most unprefered, then the voters' preferences are reversed from order of least preference to order of greatest preference, for the purpose of excluding the most prefered. For instance, some SF films were judged "much too good" to be considered as golden turkeys and were therefore excluded.

Thus, a count by the retransferable vote goes something like this. To begin, the votes are added according to how many voters share a given order of choice for the candidates. This is set out in a table of the permutations of preference, that the voters happen to have chosen. I call this table 0 ( that is table zero ) of voters' permutations. Table 1 is the preference election. If and when table 1 cannot fill all the vacant seats, table 2, for unpreference exclusion, takes over. Table 3 is the unpreference election, which may need the back-up of table 4 for preference exclusion.

Keep values of deficit candidates, as well as surplus candidates.

The four tables represent logicly possible variations on how to conduct the count. Each count assigns a relative measure of the candidates' impact on the voters as a whole. This measure is called the "keep value". It complements the transfer value, already met above. Every voter has one vote. This unit value usually gets more or less fractioned out among the candidates that the voter has ranked on the ballot paper, with a 1, 2, 3, etc order of choice. This fractioning merely means that once the voter's most prefered candidate has achieved a quota, part of the vote may be transfered to help elect the next prefered candidate. Thus the full power of that vote is successively approached, instead of being partly wasted if it went to only one prefered candidate.
The transfer value is the fraction of one vote that goes to help a next prefered candidate. The keep value is the remaining fraction of one vote that stays with the already prefered candidate, who did not need all of it, to achieve a quota.

Retransferable voting requires some modifications to the book-keeping of the STV count, whether by a traditional hand count or by a more thoro computer count. It is not that existing means of recording the keep value in an STV count are wrong. It is just that they do not make full enough use of the preferential information supplied by the voters, for the purposes of counting a "retransferable vote".
When a candidate achieves more votes than needed for a quota, that candidate's keep value becomes smaller than one. The bigger the surplus, the bigger the transfer value, and, correspondingly, the smaller the keep value. A candidate, who had votes exactly equal to the quota, would have a keep value of one. The transfer value would be zero. During the course of a count, a candidate who has not yet achieved a quota is still given a keep value of one. But that only means that the candidate gets all of any transferable votes coming.

Current STV counts do not estimate keep values for candidates who end up with less than a quota. Yet to leave them all, with keep values of one, does not do justice to the fact that these candidates, in deficit of a quota, will be in deficit by different amounts. How then to calculate the keep values of deficit candidates, as well as surplus candidates?
The keep value plus the transfer value equals one. Say, in a multi-member constituency, a candidate needs a quota of 60 votes to win a seat. Say one candidate wins 80 votes. All those 80 voters get a proportionate say in the transfer of that 20 votes surplus to their next prefered candidates. That proportionate say is estimated as ( 80 - 60 )/80 = 20/80 = 1/4 of a vote. This is the transfer value of all those 80 voters' next preferences. So, the keep value, assigned the surplus candidate, is 1 - 1/4 = 3/4.

That, so far, is standard practise in the STV count. But retransferable voting also estimates the keep values of deficit candidates. Suppose, in the above example, that a candidate ends up with 20 votes, which is 40 votes in deficit of the quota. Therefore, we may say this candidate has a certain size deficit or negative surplus of votes to transfer. The calculation of the transfer value is ( 20 - 60 )/20 = -40/20 = -2. Since the keep value plus the transfer value equals one, then this deficit candidate's keep value is one minus minus two, which is one plus two equals three. Or, 1 - -2 = 1 + 2 = 3.

The point about retransferable voting is that there are four proportional counts, rather than just the one of STV, or just the two of my previous innovation of Reversible STV, which retransferable voting elaborates. That means that, with RV, any candidate has a chance to make up for a poor keep value on one count ( say the preference election, which is essentially the STV count ) with better keep values on the other three counts. The merit of this is a consistently proportional count, in deciding the collective order of choice of the voters.

Retransferable voting is exemplified in the next section.

Hand count example of Retransferable Voting.

Suppose a club of 32 members elect a committee of three. And there are five candidates. That means there are factorial five or 120 possible permutations of preference. No more than 32 permutations are possible, in this case, because there are only 32 voters. The returning officer makes a table of how many voters there are for each permutation. To make the count easy, I have supposed there are only eight permutations with the following votes each. See table 0.

The permutation row numbers, perm 1, perm 2 etc, are for the returning officer to track any given transfer of votes according to voters' next preferences.

The returning officer starts table 1 for preference election, by adding up how many first preferences go to each candidate. Candidate C has more votes than the quota and is declared elected in stage 1. The value of C's surplus is proportionly shared between all C's voters' next preferences, which each have a transfer value, calculated as (12 - 8)/12 = 4/12 = 1/3 of a vote.

Table 0 shows 12 first preference voters for C fall into four different permutations. 1 voter, for perm 4, and three voters, for perm 5, both make D their second choice. So D gets 4 transferable votes at 1/3 transfer value each, or 4/3 votes.
Perm 6 transfers 5 x 1/3 = 5/3 votes from C to A. Perm 7 transfers 3 x 1/3 = 1 vote from C to B. This transfer of C's surplus is shown in table 1, stage 2. Adding the surplus transfer, to the remaining candidates' previous tally, is enough for B to reach a quota and be declared elected ( in stage 2 continued ).

Counting tables of the Retransferable Vote.

There is one more candidate to elect to the third available seat but no more surplus votes to transfer. At this point, conventional STV would exclude the candidate who happens to be last past the post. ( That would be candidate D with only 3.33 votes. ) That candidate may be relatively more prefered than his present position shows. Retransferable voting gives such candidates a reprieve.

Table 0 of permutations is considered in reverse to see which candidate first achieves an exclusion quota of least preferences. Table 2, the unpreference exclusion table starts with a count of the last preferences per candidate. For instance, reading from the right of table 0, perms 1, 2, and 7 show that B is the last preference respectively of 2, 5, and 5 voters, totalling 12 voters.

B is already elected in table 1. Table 2 is with reference to table 1, so B cannot be excluded. Instead, B's voters have to be content with making their next least preferences count towards exclusions.

Therefore, perm 2 and perm 6 each see 5 next preferences, go, respectively to D and E.
Perm 1 sees 2 votes go from B to C. C now has 9 votes and would have been excluded but for being already elected. But D and E have now reached the exclusion quota without being elected already. This only leaves A as not yet excluded. By this process of exclusion, A is the only candidate remaining eligible for the third and last available vacancy.

Reversible STV used essentially the two above counting tables as an example, which would elect candidates C and B, with A being elected by default, as the only other candidate still not excluded. I still believe that would be an improvement on STV. Later, in my confusion over the options, there actually are, to retransfer the vote, I adopted what is essentially table 3 below, as the reversed count in rSTV. Adopting the latter course was not so conclusive of a result. So, in that case, I had to introduce, above, a "contingent exclusion" procedure, which is merely an improved version of Last Past the Post -- since abandoned for Binomial STV counts.

As with table 2, table 3 starts with the last preferences. But now the object is simply to elect or determine the most unprefered candidates, not to use them ( as in table 2 ) to ease the way for all the most prefered candidates to be elected ( in table 1 ). Like table 1, table 3 is an election in its own right, albeit of unpreferences. Hence, the first stage is to transfer any surpluses of last preferences, as is done for candidate B with 4 votes above the quota of unpreference or unpopularity.

Table 3 refered to table 0 of the voters' preference permutations in their reverse order ( from right to left ) as did table 2. With Table 4 of preference exclusion, table 0 of preference permutations are considered by greatness of preference ( from left to right) as was the case with table 1. As table 2 of unpreference exclusion was counted with reference to table 1 of preference election, so table 4 of preference exclusion is counted with reference to table 3 of unpreference election.

At the top of the last column of table 4, it is merely noted that all C's surplus passes to D, thus eventually getting 8 votes ( and a keep value of 8/8 = 1 ). I know this without following the trail of preferences, because all the voters express, in table 0, a full set of preferences and there are no other candidates left for C's surplus to go to. ( I took a similar short-cut in the last column of table 2. ) One wouldnt be able to get away with this in most real elections, where not all people give a preference for every candidate. But this example is just a demonstration.

Table 5 gathers the keep values from the four count runs, shown in tables 1 to 4, and combines them for a ratio of preference to unpreference keep values, as explained in the key following table 5.

Key to table 5: Comments on combining the keep values of the retransferable vote:

1. Preference election keep values ( from table 1 ).
2. Unpreference exclusion keep values ( from table 2 ).
3. Unpreference election keep values ( from table 3 ).
4. Preference exclusion keep values ( from table 4 ).
5. Preference election times preference exclusion keep values: col. 1 x col. 4.
   Tables 1 and 4 were two ways to find the most prefered candidates. Table 1 does this by transferably electing from most prefered to next prefered candidates. Table 4 does this, by excluding from the least unprefered to the next least unprefered, which is not quite the same thing. Multiplying their respective keep values is more likely to give a more representative combined preferences' keep value.
6. Unpreference election times unpreference exclusion: col. 2 x col. 3.
   The same reasoning applies to combined unpreferences' keep values.
7. Dividing the preferences' keep values by the unpreferences' keep values means that the latter is an inverted multiple, in effect, turning the unpreferences' keep values into another index of preference. In this way, two pairs of keep values ( from the four tables of election and exclusion ) combine in a final index of preference for the candidates.

( After-note: In the interest of simplicity, I under-rated the importance of taking the square roots of combined and re-combined values, as I should have done. )

Suppose a candidate has a preferences' combined keep value of 1/2 and an unpreferences' combined keep value of 3/4. Then, the ratio, of preference to unpreference, equals 1/2 x 4/3 = 4/6. The relative lowness, of the preferences' to the unpreferences' keep value, is an index that the candidate in question is relatively more prefered than unprefered. And conversely, if the unpreferences' keep value was lower than the preferences' keep value.

Hence, the lowest keep value indices, in table 7, show the most prefered candidates. The three seats in the contest go to the three candidates E, C and A. Only those three are more prefered than unprefered. An obvious rule suggests itself that a candidate, more unprefered than prefered, should not be elected, even if that necessitates another election.

If the keep values, in the ratio, were both one, then the candidate would be as much unprefered as prefered. That is a neutrality of the voters towards the candidate. ( I over-looked, til later studies, the problem of voters being neutral to a candidate without wishing to so much as elect him. ) If there are a lot of candidates, voters may only wish to register preferences for the candidates they most like and unpreferences for the candidates they definitely do not wish to see elected.
With a Resumed Preference Vote, you could mark 1, 2, 3, etc in order of greatest preference. The ballot paper or computer screen tells you there are 58 candidates, mark 58, 57, 56, etc for any candidates you hope to exclude.