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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
ballotpaper 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:
 Increased support, for a candidate who would otherwise have been
elected, should not prevent their election;

 a. Later preferences should not count against earlier preferences;
 b. Later preferences should not count towards earlier preferences;
 If no second preferences are expressed, and there is a candidate who
has more firstpreference votes than any other candidate, that candidate
should be elected;
 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
(d1) 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 nonelected candidates need to
sacrifice the smallest proportion of their votes.
Under STV(SS), each nonelected candidate in turn is tested to see what
proportion of the votes of the other nonelected 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
nonelected 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 nonelected candidates are ranked in 'order of
electability', which forms the basis on which candidates are elected or
suspended. All the nonelected candidates are subclassified at the start as
'contending'. There are two further subclassifications, 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 nonelected 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 nonelected 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 nonsuspended
candidate gets the seat instead. In this part, nonelected candidates are
subclassified 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 viceversa. 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.
Conclusion
As specified above, the system appears to be longwinded: there are possible
shortcuts, 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.
References
 D R Woodall, An Impossibility Theorem for Electoral Systems,
Discrete Mathematics 66 (1987) pp 209211
 Michael Dummett, Towards a More Representative Voting System: The
Plant Report, New Left Review (1992) pp 98113

 B L Meek, Une nouvelle approche du scrutin
transférable, Mathématiques et Sciences Humaines, No.
25, pp 1323 (1969)
 B L Meek, Une nouvelle approche
du scrutin transférable (fin), Mathématiques et
Sciences Humaines, No. 29, pp 3339 (1970).
 C H E Warren, Counting
in STV Elections, Voting matters, No. 1, pp 1213 (March 1994)
Annex: STV(SS)  Detailed Instructions
The first part
 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.
 Classify every nonelected 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 ballotpaper 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
nontransferable.
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 nontransferable.
 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.
 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.
 If there are N elected candidates, end the count. Otherwise, go
to Step 2.
The second part
 Classify every nonelected candidate as contending and set every
candidate's KV to 1.0. Repeat the following procedure until either the
highestranked contending or protected candidate and the elected candidates
have Q or more votes each, or there are only N nonsuspended
candidates.
 a. Examine each ballotpaper 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
nontransferable.
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 nontransferable.
 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.
 Classify as elected the highestranked contending or protected candidate.
 If N candidates are elected, end the count: otherwise, go to Step 2.
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