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Voting matters - Issue 2, September 1994

STV with successive selection - An alternative to excluding the lowest

S Gazeley

Simon Gazeley is a civil servant working in Bath. He is a member of Council and of the Technical Committee of the ERS.

The problem with current STV systems

A feature of STV which is not shared by other preferential voting systems is election on attaining a certain number of votes (the 'quota'). If the number of candidates who have a quota of first preference votes is insufficient to fill all the seats being contested, those which are left are filled by candidates whose quotas contain votes which have been transferred from other candidates. These transfers take two forms: of surpluses above the quota for election from candidates who are already elected, and of all the votes previously standing to the credit of candidates who have been excluded in accordance with the rules. When it is necessary to withdraw a candidate from contention, all versions of STV currently in use exclude the one who has fewest votes at that time. It is contended that the consequences of this rule in conventional STV formulations can be haphazard and therefore unjust in their effect. Consider the following count:
          AD  35
          BD  33
          CD  32
There are here 3 separate and substantial majorities: against A, against B and against C. The only thing that all the voters agree on is that D is preferable to two out of the other three candidates; yet STV excludes D first, however many seats are being contested. Unfairness and anomalies such as this arise because candidates are excluded before the full extent of the support available to them has been investigated. Even though every ballot-paper may have the same candidate marked as the next available preference, that candidate will not survive if they do not have enough votes now.

An even more serious consequence of the 'exclude the lowest' rule is that it is possible for voters to assist their favoured candidates by withholding support rather than giving it. Consider the following election for one seat:

          AC  13
          BC   8
          CA   9
Having been excluded, B's votes go to C, who now has an absolute majority and gets the seat. But suppose that two of A's supporters had voted BC instead:
          AC 11
          BC 10
          CA  9
Now C is excluded first and A gets the seat.

Is it possible, then, to remove this anomaly without introducing another? The answer, unfortunately, is 'no'. Woodall[1] proposed that every count under any reasonable electoral system should have the following four properties:

  1. Increased support, for a candidate who would otherwise have been elected, should not prevent their election;
  2. If no second preferences are expressed, and there is a candidate who has more first-preference votes than any other candidate, that candidate should be elected;
  3. If the number of ballots marked X first, Y second plus the number marked Y first, X second is more than half the total number of ballots, then at least one of X and Y should be elected.
He then proved that no such system can be devised.

We have already noted that current STV systems can (but usually do not) fail on Woodall's first property; this is the failure that in Dummett's[2] eyes precludes consideration of STV as a possible option for public elections in the UK. As no system can have all four properties, a price for having one has always to be paid in terms of lacking at least one other. Under the system proposed below, some counts (but by no means all) may fail to have Woodall's first or second property, but all will have the other two. Whether the price is worth paying is a question to which no definitive answer can be given: it is ultimately a matter of personal preference.

STV by successive selection (SS)

The object of exclusion in current STV formulas is to release votes from one candidate to be transferred to others so that one or more of them will get a quota. STV(SS) retains the transfer of votes from candidates who are not yet elected, but differs from present STV systems in that no candidate is permanently withdrawn from contention. When it becomes necessary to release a candidate's votes, that candidate is 'suspended' (withdrawn temporarily) after being identified as the one whose election to the next vacant seat would be least appropriate.

Manual STV systems need to keep within reasonable bounds both the time taken to count an election and the scope for human error and this need can give rise to anomalies. Meek[3] and Warren[4] have devised schemes without these anomalies for distributing votes which would be impracticable using manual methods. STV(SS) is designed (but not yet programmed) to be run on a computer using either of these schemes, but only one should be used in any one election.

In addition to Woodall's four properties, every count under a reasonable system would have the property that of a set of d or more candidates to which d Droop quotas of voters are solidly committed, more than (d-1) should be elected; if the set contains fewer than d candidates, all of them should be elected. According to Dummett, a group of voters are 'solidly committed' to a set of candidates if every voter in the group prefers all candidates within the set to any candidate outside it. STV(SS) and other STV formulas achieve proper representation of sets of candidates by withdrawing from contention candidates who have less than a quota of votes and by transferring surplus votes from those candidates who have more than a quota.

The principle underlying STV(SS)

STV(SS) is predicated on the proposition that when no surpluses remain to be transferred, there is only one candidate (barring ties) who is the most appropriate occupant of the next seat. Appropriateness depends among other things on who has been elected already: if Candidate X is the 'most appropriate' and Candidate Y is the 'next most appropriate' at any given point, it does not follow when X is elected that Y is now the 'most appropriate'. The next candidate to be elected is the one who can command a quota and for whose election the other non-elected candidates need to sacrifice the smallest proportion of their votes.

Under STV(SS), each non-elected candidate in turn is tested to see what proportion of the votes of the other non-elected candidates have to be passed on in addition to the surpluses of the elected candidates to give them the quota. Of those who can command a quota, the candidate who requires the smallest proportion of the others' votes is the 'most appropriate' to be elected next. The process is best illustrated by an example. Consider the following votes for one seat:

            A  49
            BC 26
            CB 25
No candidate has a quota, but instead of excluding the lowest we test each candidate in turn to see which is the 'best buy'. Let us test A first. The quota is 50 and B and C have 51 votes between them; we therefore change their Keep Values (KVs: see the Annex for further details) from 1.0 to 50/51 (0.9804). At the second distribution the votes look like this:
            A   49.0000
            B   25.9708
            C   25.0096
The new total of votes is 99.9804, making the quota 49.9902. A still has not got the quota, so the count proceeds. The final distribution looks like this:
            A   49.0000
            B   24.8216
            C   24.1784
At this point, we record the fact that the common KV of B and C is 0.8020. If we now test B, we find that the final common KV of A and C is 0.5152; when we test C the common KV of A and B is 0.5050.

At first sight, A seems the obvious choice to get the seat: however, if A were to be successful, Woodall's fourth property would be lacking. No candidate should be elected who cannot command a Droop quota of the votes which are active at the time of their election. If we remove C from contention (C is 'least appropriate' as the other candidates had to give up the greatest proportion of their own votes to secure C's quota) and redistribute C's votes, B now secures a Droop quota and is elected.

But why make the selection on the basis of the other candidates' final KVs? The reason is that these represent the degree of support that exists for the proposition that a given candidate should be added to the set of elected candidates. Suppose that some of the votes in an election were cast as follows:

            AC   54
            BC   45
(there may be other candidates and other votes, but these need not concern us) and that it is necessary for 33 of these votes to be passed from A and B to C. This is achieved by setting the common KV of A and B at 0.6667 {\153} A and B have to pass on 0.3333 of the current value of each incoming vote to secure C's quota. But suppose the votes had been
            ABC 54
            BAC 45
the other votes and candidates being the same. This time, to give 33 votes to C, the common KV of A and B has to be 0.4226 i.e. 0.5774 of the current value of each incoming vote has to be passed on, over 1.7 times as much. The lower a candidate is in the order of preference of the average vote being considered at any point, the lower the common KV of the other non-elected candidates has to be in order to give that candidate a quota.

How STV(SS) works

STV(SS) has two parts: detailed instructions to the computer are given in the Annex. What follows is a general description and explanation of their functions.

The first part

In the first part, the non-elected candidates are ranked in 'order of electability', which forms the basis on which candidates are elected or suspended. All the non-elected candidates are sub-classified at the start as 'contending'. There are two further sub-classifications, namely 'under test' and 'tested'; only one candidate at a time is under test. The object is to ascertain for the candidate under test what proportion of the votes of the contending and tested candidates it is necessary to pass on to give them the current quota. Each non-elected candidate in turn is classified as under test. If a candidate under test is classified as elected, the first part is repeated.

When the candidate under test and the elected candidate have Q or more votes each, the candidate under test has recorded against their name the common KV of the contending and tested candidates: this is that candidate's 'electability score'. When all the non-elected candidates have been tested, they are ranked in descending order of electability score: this ranking is for use in the second part. An electability score of 1.0 indicates that the candidate needs to take no votes from other unelected candidates to get the quota, so there is no reason not to classify that candidate as elected at once.

The second part

In the second part, the next candidate to be elected is identified on the basis of their ranking from the first part and their ability to command a Droop quota of votes. The highest candidate in the ranking is elected as soon as it is shown that they can command a Droop quota of currently active votes. If the highest candidate cannot, the second highest non-suspended candidate gets the seat instead. In this part, non-elected candidates are sub-classified as 'contending', 'protected' (contending candidates become protected when they get a quota) and 'suspended'; they are all classified as contending at the start. Suspended candidates have a KV of 0.0. At the end of the procedure, all the candidates' KVs are reset at 1.0.

Contending candidates are suspended in reverse order of ranking: protected candidates cannot be suspended before the next candidate is classified elected. The fact that a candidate has a Droop quota of currently active votes now does not necessarily indicate that they will achieve one at a subsequent stage and vice-versa. The rankings obtained in each pass through the first part are crucially dependent on which of the previously contending candidates was elected in the preceding second part.

An example

Let us see how STV(SS) works on the examples on page 1:
     Count 1    Count 2 
     AC  13      AC  11
     BC   8      BC  10
     CA   9      CA   9
In Count 1, the ranking is A (the common KV of the other two candidates would be 0.7962), C (0.7143) and B (0.2023), so B is suspended first and C gets the seat. The Count 2 ranking is C (0.7143), A (0.6311) and B (0.2929); B is once more the first to be suspended so C again gets the seat.


As specified above, the system appears to be long-winded: there are possible short-cuts, but these would obscure essentials and have been excluded.

STV(SS) is a logical system which is submitted as a contribution to the continuing debate on what the characteristics of the best possible system might be. Refinements are necessary (for instance, a way of breaking ties has to be devised), but there is here the basis for a debate.


  1. D R Woodall, An Impossibility Theorem for Electoral Systems, Discrete Mathematics 66 (1987) pp 209-211
  2. Michael Dummett, Towards a More Representative Voting System: The Plant Report, New Left Review (1992) pp 98-113
    1. B L Meek, Une nouvelle approche du scrutin transférable, Mathématiques et Sciences Humaines, No. 25, pp 13-23 (1969)
    2. B L Meek, Une nouvelle approche du scrutin transférable (fin), Mathématiques et Sciences Humaines, No. 29, pp 33-39 (1970).
  3. C H E Warren, Counting in STV Elections, Voting matters, No. 1, pp 12-13 (March 1994)

Annex: STV(SS) - Detailed Instructions

The first part
  1. If there is any candidate for whom no voter has expressed any preference at all, treat every such candidate as having withdrawn. If fewer than (N+1) candidates remain, end the count; otherwise, set the ranking of every remaining candidate to equal first.
  2. Classify every non-elected candidate as contending and repeat the following procedure until there are no contending candidates left:
  3. If no tested candidate has an electability score of 1.0, rank the tested candidates in their existing order within descending order of electability score and go to Step 5. Otherwise, classify as elected every tested candidate whose electability score is 1.0.
  4. If there are N elected candidates, end the count. Otherwise, go to Step 2.
    The second part
  5. Classify every non-elected candidate as contending and set every candidate's KV to 1.0. Repeat the following procedure until either the highest-ranked contending or protected candidate and the elected candidates have Q or more votes each, or there are only N non-suspended candidates.
  6. Classify as elected the highest-ranked contending or protected candidate.
  7. If N candidates are elected, end the count: otherwise, go to Step 2.

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