Brian Meek is now at King's College London.

*The original version of this paper was dated 21 March 1968 and was
published in French in Mathématiques et Sciences Humaines No 29, pp
33-39, 1970. This note, and note 8 in its present form, have been added in
this reprint.*

**Principle 1.**If a candidate is eliminated, all ballots are treated*as if that candidate had never stood*.**Principle 2.**If a candidate has achieved the quota, he retains a fixed proportion of every vote received, and transfers the remainder to the next*non-eliminated*candidate, the retained total equalling the quota.

The case of a ballot with marked candidates who are elected is less
straightforward. Suppose an elected candidate C receives a total *x* of
votes with no further preferences marked on them (any marked eliminated
candidates can, by Principle 1, be ignored). By Principle 2, C must pass on
a fixed proportion *p* of these, as all other, votes and retain the
rest as part of his quota. The difficulty arises because it is not clear to
whom these votes should be transferred.

If the difficulty were to be avoided by increasing the proportion
transferred of votes for which a next preference is marked, to enable all
*x* votes to be retained by C, this would clearly reintroduce
inequities of the kind Principle 2 was designed to eliminate. Not to
transfer the proportion at all would mean leaving C with more than the quota
(see also section 4). The two possible ways of strictly obeying Principle 2
are

- (a) to divide the otherwise non-transferable proportion equally between the remaining (i.e. unmarked and uneliminated) candidates; or
- (b) to subtract this quantity from the total
*T*of votes cast, and recalculate the quota with the new value.

- If a vote is not transferable from an eliminated candidate, it is set aside; such votes play no further part in the count.
- If the number of votes non-transferable from an elected candidate is not greater than the quota, those votes are included in the quota and only the transferable votes determine the distribution of the surplus. If the number is greater than the quota, then the transferable votes are transferred (at unreduced value), the difference between the non-transferable votes and the quota increasing the non-transferable total.

- (A) The number of wasted votes in an election (i.e. which do not contribute to the election of any candidate) is kept to a minimum.
- (B) As far as possible the opinions of each voter are taken equally into account.

*W* <= *T*/(*S* + 1) + *T*0

where *T*0 is the non-transferable total. However, this is derived from
a quota calculated on the total *T* and not on the total available vote
*T* ' = *T* - *T*0. Thus with recalculation of the quota we
have

*W* ' < *T* '/(*S* + 1) + *T*0 = *W* - *T*0
/(*S* + 1) < *W*

i.e. condition (A) is violated unless the quota is recalculated[3].

It is clear that rule (ii) above is an attempt to satisfy condition (A), but
it only does so at the cost of violating condition (B); for example, if a
candidate E is elected with *q* + *x* votes, *q* of which are
non-transferable, the *x* remaining votes will be transferred at
unreduced value to the next preference even though their earlier preference
for E has been satisfied. Further, the present rule that votes cannot be
transferred to an elected candidate (see Paper I) means that both by rule
(i) and by rule (ii) many whole votes may be declared completely
non-transferable, thus swelling *T*0 and *W* above, whereas the
feedback method allows each vote to count partly for the elected candidates
marked and only a fraction becomes non-transferable.

Thus, on two grounds, current STV counting methods violate condition (A). It could perhaps be argued that the feedback method cannot satisfy condition (A) unless method (a) rather than method (b) of section 3 is used when dealing with unmarked candidates. We shall discuss this point in section 6.

*q* = [(*T* - *p*1*×t*10)/(*S* + 1) + 1] .....(Equation:1)

*t*1(1 - *p*1) = *q* ............(Equation:2)

where, as in Paper I, *S* is the number of vacancies, *T* is the
total votes (now ignoring any which mark only eliminated candidates),
*t*1 the total for the elected candidate, *p*1 the proportion he
transfers, *t*10 the total vote for the candidate not transferable to
others, and *q* is the quota.

These two equations can be solved easily for *p*1 and *q* by
equating the expressions for *q*; however, if there is more than one
elected candidate the iterative method of finding the *p*i, described
in Paper I, will be needed, and it is convenient to discuss the extension of
the iterative process to include the recalculation of the quota in terms of
the simplest case, above. Equation (1) with *p*1 = 0 gives the original
value of *q*. Equation (2) then gives a first value of *p*1 >
0. Substitution of this value in (1) gives a new value of *q* smaller
than before; use of the new *q* in (2) gives a larger *p*1, and so
on. Thus we have a monotone increasing sequence of values for *p*1,
bounded above by 1, and a monotone decreasing sequence of values of *q*
bounded below by 0; these sequences must therefore tend to limits which are
the solutions to the equations. The convergence rate is satisfactory; simple
analysis shows that the errors are multiplied in each cycle by a factor
which is at most 1/(*S* + 1).

The process is extended to the case of *n* elected candidates by adding to the equations in Paper I the equation

*q* = [*T*n/(*S* + 1) + 1]

which must be evaluated for *q* first in each iterative cycle.
*T*n = *T*n(*p*1,*p*2,....,*p*n) is the total
available for transfer in each case; for *n* = 1, 2, 3 it is given by

*T*1 = *T* - *p*1*×t*10

*T*2 = *T* - {*p*1*×t*10 + *p*2*×t*20 + *p*1*×p*2(*t*120 + *t*210)}

*T*3 = *T* - {S1 *pi×ti*0 + S2 *pi×pj×tij*0 +
S3 *p*1×*p*2*×p*3×*t*(123)0}

In these formulae *tij*...*k*0 is the total transferable from
candidate *i* to candidate *j,* to ..., to candidate *k* but
not further; S1 denotes summing over *i*; S2 denotes summing over all
*i*, *j*, *i* /= *j*; S3 denotes summing over all
permutations of (123).

The reader will easily derive equivalent formulae for higher values of
*n*; putting *pn* = 0 in the expression for *T**n* gives
the expression for *T**n*-1.

For the simplest form of STV counting, involving the physical transfer of
ballot papers from pile to pile, the need for a unique next preference is
obvious. However, with the feedback method such a restriction is no longer
necessary, and indeed it is not necessary even with Senate Rules counting. A
vote can be marked A1, B1, C2, ... with A and B as equal first preferences
and credited at 0.5 each to A and B. If A is elected or eliminated the 0.5
is transferred at reduced or full value to the next preference — which
of course is B and not C. In effect, such a vote is equivalent to two normal
STV votes, of value 0.5 each, marked A,B,C... and B,A,C... respectively.
Similarly, if A, B, C are all marked equal first, this is equivalent to 6 (=
3!) votes of value 1/6 each, marked A,B,C...; A,C,B...; B,A,C...; B,C,A...;
C,A,B...; and C,B,A... . It is easy to see that this can be extended to
equal preferences at any stage, and that *K* equal preferences
correspond to *K*! possible orderings of the candidates concerned, each
sharing 1/*K*! of the value at that stage.

Such an extension of the validity rules enables us to resolve the dilemma between the methods (a) and (b) in section 3 of dealing with non-transferable votes. A voter who, at a certain stage, wishes his vote, if transferred, to be shared equally between the remaining candidates, can simply mark those candidates as equal (i.e. last) preferences. Thus the dilemma does not after all exist; both of the methods can be used, and the voter himself can determine which is to be used for his own ballot by the way that he marks it; failure to rank a candidate indicates a genuine (partial) abstention.

This extension of the validity rules also enables condition (C) of Paper I to be satisfied more closely. The condition was:

- (C) There is no incentive for a voter to vote in any way other than according to his actual preference.

Permitting equal preferences thus gives much greater flexibility to the voter to express his ordering of the candidates, and is thus a desirable reform whether the feedback method is used for counting or the Senate Rules retained.[4]

The first problem is mainly outside the scope of these papers, but has been touched on in the last section. It is a basic assumption of STV that the individual preference orderings of each voter is sufficient information[5] to obtain the social ordering, and the voting rule extensions described above follow naturally from this principle, and indeed bring STV more closely into line, in a certain sense, with the work of Arrow.[6]

The possible development of (preferential, transferable) voting systems which use further relevant information is the subject of continuing work.[8]

The second problem is the classical problem of decision theory. Assuming the basic STV structure, these papers have shown that the feedback method of counting is needed to satisfy the declared aims of STV as a decision-making procedure more consistently.

This improvement can be made without causing any more difficulty to the voter, and allows the counting procedure to be described by two simple principles instead of by a collection of rules, some of which are rules of thumb.

The disadvantage of the method is the need for many repetitive calculations, which for reasons of sheer practicality rules it out for manual counting except when the numbers of vacancies, candidates and votes are small. However, as pointed out in Paper I, an STV count is already a sufficiently tedious process for it to be worthwhile to use a computer, and the additions to the feedback method described in this paper would be simple to add to the computer program.

As E G Cluff has pointed out,[7] one advantage of election automation is that one is not restricted in the choice of voting system to what is practically feasible in a manual count. The feedback method can lead to different results from the Senate Rules in non-trivial cases, and is therefore a choice to be considered when the automation of STV elections is being implemented.

- B L Meek, A new approach to the Single Transferable Vote I: Equality of treatment of voters and a feedback mechanism for vote counting, Mathématiques et Sciences Humaines No 25, pp 13-23, 1969.
- It can in fact lead to the defeat of a candidate who is first choice of a majority of the electorate. It is depressing to note that a public election in England was held under precisely these rules as recently as 1964.
- This kind of inequity can be found most often in elections with large numbers of candidates and vacancies - e.g. for society committees - and can lead to disillusion with STV as a voting system which has little relation to its merits or demerits.
- The possibility of a voter sharing his first preference other than equally between a number of candidates would take us too far afield, into the realm of multiple transferable voting systems - the subject of continuing work on more general preferential voting systems. In STV the task of the voter is in comparison a straightforward one, in some ways made easier by allowing equal preferences.
- And, indeed, necessary information!
- K Arrow, Social choice and individual values, 2nd edn, Wiley 1962. For what is meant by "in a certain sense" see Paper I.
- See B L Meek, Electronic voting by 1975?, Data Systems July 1967, p 12.
- See also note 4. This further work was later published (in English) as: B L Meek, A transferable voting system including intensity of preference, Mathématiques et Sciences Humaines No 50, pp23-29, 1975.