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

          AD  35
          BD  33
          CD  32

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

          AC 11
          BC 10
          CA  9

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. • a. Later preferences should not count against earlier preferences;
   • b. Later preferences should not count towards earlier preferences;
3. 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;
4. 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.

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)

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)

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

            A   49.0000
            B   25.9708
            C   25.0096

            A   49.0000
            B   24.8216
            C   24.1784

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

            ABC 54
            BAC 45

How STV(SS) works

The first part

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

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

     Count 1    Count 2 
     AC  13      AC  11
     BC   8      BC  10
     CA   9      CA   9

Conclusion

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.

References

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
3. 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).
4. C H E Warren, Counting in STV Elections, Voting matters, No. 1, pp 12-13 (March 1994)

Annex: STV(SS) - Detailed Instructions

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:
   • a. Set every candidate's KV at 1.0 and select a contending candidate to be the candidate under test.
   • b. Examine each ballot-paper in turn and distribute the value of the vote in accordance with the voter's preferences and the KVs of the candidates as follows:

     Either

     • i. The Meek Formulation. Offer the vote to each candidate for whom the voter has voted in order of preference expressed. Multiply the fraction of the vote which has not yet been allocated by the KV of the candidate to whom it is being offered, and allocate that proportion of the vote to that candidate. Any part of the vote left over after all the candidates for whom the voter has voted have received their share is non-transferable.

       or

     • ii. The Warren Formulation. Offer the vote to each candidate for whom the voter has voted in order of preference expressed. Award to each candidate in turn a fraction of the vote equal to that candidate's KV; if the fraction of the vote remaining is less than the KV of the current candidate, award all that is left to that candidate. Any part of the vote left over after all the candidates for whom the voter has voted have received their share is non-transferable.
   • c. Calculate the quota according to the formula Q=V/(N+1), where V is the total number of votes credited to all the candidates and N is the number of seats being contested.
   • d. If the elected candidates and the candidate under test have at least Q votes each, go to Step e. Otherwise, calculate new KVs for all the candidates as follows:
     • i. For all the elected candidates and the candidate under test, multiply the current KV by Q and divide the result by that candidate's current total of votes.
     • ii. Multiply the common KV of the contending candidates and the tested candidates by (V-(E+1)Q)/T, where E is the number of candidates elected so far and T is the total of the votes credited to the contending and tested candidates.
     If any new KV exceeds 1.0, reset it at 1.0. Go to Step b.
   • e. Record the common KV of the contending and tested candidates against the name of the current candidate under test; let this be that candidate's 'electability score'. Classify that candidate as tested.
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.

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   The second part

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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.
   • a. Examine each ballot-paper in turn and distribute the value of the vote in accordance with the voter's preferences and the KVs of the candidates as follows:

     Either

     • i. The Meek Formulation. Offer the vote to each candidate for whom the voter has voted in order of preference expressed. Multiply the fraction of the vote which has not yet been allocated by the KV of the candidate to whom it is being offered, and allocate that proportion of the vote to that candidate. Any part of the vote left over after all the candidates for whom the voter has voted have received their share is non-transferable.

       or

     • ii. The Warren Formulation. Offer the vote to each candidate for whom the voter has voted in order of preference expressed. Award to each candidate in turn a fraction of the vote equal to that candidate's KV; if the fraction of the vote remaining is less than the KV of the current candidate, award all that is left to that candidate. Any part of the vote left over after all the candidates for whom the voter has voted have received their share is non-transferable.
   • b. Calculate the quota according to the formula Q=V/(N+1), where V is the total number of votes credited to all the candidates and N is the number of seats being contested. Classify any contending candidate with Q or more votes as 'protected'.
   • c. If any candidate has more than Q votes, calculate a new KV for each such candidate by multiplying their present KV by Q and dividing the result by their present total of votes. Otherwise, suspend the contending candidate who is ranked lowest.
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.