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Voting matters - Issue 14, December 2001

STV with Symmetric Completion

Simon Gazeley

Meek's[1] formula for STV differs from manual systems in significant ways which have been explained by Hill[2]. These differences make Meek more acceptable to many than manual STV, but it means that a computer is necessary for any but the very simplest Meek counts. I believe it is possible to improve manual STV without either losing the ability to do it manually, or introducing some unintended unacceptable effect. The current ERS rules[3] are taken as a starting point in formulating the changes proposed, and will be referred to as N-B.

When a candidate has a surplus, N-B transfers the `parcel' of votes which gave rise to that surplus - ie, the votes which that candidate received most recently. Note that the ballot-papers will all be of the same value, which can be 1.0 or less. The papers in the parcel are sub-divided into transferable votes (those on which a subsequent preference has been expressed for a candidate who is not yet elected or eliminated), and non-transferable (those on which all the candidates for whom a preference has been expressed are either elected or eliminated). If the total of transferable votes at their present value is less than or equal to the surplus, they are all transferred at that value to the voters' next preferences, and sufficient of the non-transferable votes are left with the elected candidate to preserve that candidate's quota with no surplus; any non-transferable votes over and above the quota are put to the non-transferable pile. If the total of transferable votes is greater than the surplus, a new value is calculated for each transferable vote such that when all of them are transferred at that value, their total value is equal to the surplus, and the elected candidate is left with the quota.

This procedure in effect shares out the non-transferable votes among the continuing candidates in the proportions of the transferable votes, and can give a result which I consider perverse. Consider the following count for two seats, adapted from one devised by David Hill:

     Case 1
   A       60
   AB      60
   CD      51
   DC       9
The quota is 60, so A gets the first seat. N-B ignores the 60 voters who expressed no preference after A. It transfers the 60 AB votes at full value to B, who now gets the other seat. On the other hand, Meek transfers all the votes credited to A, in this case at a value of 0.5. Thus B gets 30 of the AB votes, while 30 of the A votes go to non-transferable. The new total of effective votes is now 150, making the new quota 50. C, with 51 votes, has attained this new quota and gets the second seat.

Now suppose that the 60 A voters had in fact expressed second preferences, three for C, the rest for B. Votes would be:

     Case 2
   AB      117
   AC        3
   CD       51
   DC        9
In Case 2, the N-B count is identical to the Meek count. A gets the first seat, but this time all the votes credited to A are transferred at a value of 0.5, leaving A with 60. B gets 58.5 of the transferred votes and C gets 1.5, increasing C's total to 52.5. Now, nobody other than A has the quota, so we eliminate D. C's total of votes now goes up to 61.5, more than the quota, so C gets the second seat. Comparing Cases 1 and 2, we see that the additional 57 votes on which the second preference is for B are counteracted under N-B by just three voters whose second preference is for C.

Owing to the habit of many voters of not casting preferences for all candidates, the total number of votes credited to candidates tends to decline as the count proceeds. This is countered in some rules by requiring the voters to cast preferences for all candidates, forcing them to register preferences they do not feel and perhaps cannot justify. This means that in N-B counts, the final candidates to be elected often have less than a quota. As the quota is higher in these cases than it needs to be, the opportunity is lost to transfer as many surplus votes as could have been transferred if the quota had been lower from the beginning but still attainable by only as many candidates as there are seats. In a Meek count, the quota is recalculated at every stage to take account of the votes which become non-transferable and all surpluses over each successive value of the quota are transferred. Thus, the only criterion for election in a Meek count is attainment of the quota.

It is reasonable to presume that a voter who does not rank all the candidates is indifferent to the fates of the candidates left unranked, and therefore does not wish the vote to favour any of the unranked candidates over the others. As the example above clearly shows, N-B can give second and subsequent preferences more votes than the voters are presumed to have intended them to receive. Note that the A voters have no right to feel aggrieved; if they had wanted to cast further preferences, they were perfectly entitled to do so. However, the CD voters are certainly entitled to protest that the 60 A votes were treated by N-B in effect as AB votes, thus denying the second seat to C.

In a manual count, the option of reducing the quota as in Meek is not available, as the count would have to be restarted at every change of the quota. The other option is to share among the continuing candidates the votes which would otherwise have been non-transferable, treating them as if they had in fact been cast as equal lowest preferences for the candidates concerned. Following Woodall[4], I shall call this `symmetric completion'. To those who are against symmetric completion on the grounds that it is never justified to award any part of a vote to a candidate for whom no preference has been expressed, my response is that symmetric completion treats all short votes alike and does not give too much weight to surplus votes on transfer. In both these respects, it is superior in my view to N-B.

With symmetric completion, the numbers of votes credited to the continuing candidates will usually be greater than they would have been under N-B, especially at the later stages. This means that there will be a tendency for more surpluses to be available for transfer, and therefore for more voters' preferences to be taken into account. Applying symmetric completion to Case 1 above, we get at the first stage

   A       120
   C        51
   D        9
The quota is 60, and A is elected. A's votes are all transferred at a value of 0.5 to next preferences: the 60 AB votes go to B, who now has (60 × 0.5) = 30 votes, and the 60 A votes go equally to B, C, and D, who each get (20 × 0.5) = 10 votes. Votes are now:
   A       60
   B       40
   C       61
   D       19
and C gets the second seat.

Implementing STV with symmetric completion (STV-SC) would entail some changes to the N-B procedure. This is best illustrated by an example. Six candidates are contesting three seats, with votes:

   A        59
   AEFB     66
   B       172
   BCAE     12
   C       112
   CABD     86
   D        11
   DFEA    195
   E        33
   EDCF    148
   F        21
   FBDC     85
The quota is 250. As no candidate has the quota, F, with fewest votes, is eliminated. As in N-B, the 85 FBDC votes are transferred to B. Although STV-SC puts the 21 F ballot-papers to the non-transferable pile, it does not put the 21 F votes to non-transferable, as all votes in STV-SC are transferred. Instead, we call these 21 votes on which no further preferences are expressed `dividend votes', because they are divided equally among the continuing candidates, in this case 21/5=4.2 to each. The number of dividend votes is calculated as the difference between the total of votes currently credited to candidates and the original total of valid ballot-papers; a running total is kept against each candidate's name of the number of dividend votes (s)he has received, and the stage at which they were gained. Effective votes at stage 2 are:
   A       129.20
   B       273.20
   C       202.20
   D       210.20
   E       185.20
Now, the sum of A's votes and B's surplus is less than the votes credited to E, the candidate in last-but-one place. Under N-B rules, and therefore under STV-SC rules, the transfer of B's surplus is deferred, and we eliminate A at once. The 66 AEFB votes go to E, the 59 A papers to non-transferable. The total of votes credited to the candidates is now 936.80; the 63.2 dividend votes are awarded equally to C, D, and E, 21.06 to each. Votes are now:
   B       273.20
   C       223.26
   D       231.26
   E       272.26
We now transfer B's surplus, as that is the larger. The most recent parcel received by B contains the 85 transferred FBDC votes, plus A's share of the 21 dividend F votes, making 89.2 in all. We now transfer the 85 FBDC votes to D and the 4.2 F votes to C and D @ 23.2/89.2=0.26. As this boosts D's total above the quota, we end the count.

The only criterion for election in STV-SC, as in Meek, is attainment of the quota. To cater for rounding errors in transferred votes, the number of dividend votes is recalculated at each stage as the difference between the original total (in this case, 1000) and the total of the votes credited to candidates after all transferable votes have been transferred; the number of dividend votes awarded to each continuing candidate is truncated if necessary to two decimal places. As the total of the votes credited to the candidates is the same after each stage as it was after the previous one (except perhaps for rounding error), surpluses can arise at any point, giving the voters concerned a greater opportunity than under conventional N-B to influence the subsequent course of the election.

Should symmetric completion be imported into Meek? The answer is emphatically no. Woodall[4], using an example provided by David Hill, has shown that quota reduction in Meek is preferable to symmetric completion, even though Meek himself was equivocal on the point. The purpose of this paper has been to show that, given the practical constraints of a manual count, symmetric completion can deal with a problem that may arise in N-B without in general substituting one that is as bad or worse.


  1. B. L. Meek, A New Approach to the Single Transferable Vote, Voting matters. Issue 1 (1994), 1-6 (Paper 1), 6-11 (Paper 2)
  2. I D Hill. Meek style STV - a simple introduction, Voting matters. Issue 7 (1996), 5-6.
  3. Robert A. Newland & Frank S. Britton, How to Conduct an Election by the Single Transferable Vote, Electoral Reform Society, 3rd Edition, 1997.
  4. Douglas R. Woodall, Properties of Preferential Electoral Systems, Voting matters. Issue 3 (1994), 8-15.

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