Up: Issue 1 Next: Paper 3 Previous: Paper 1

Voting matters - Issue 1, March 1994

A New Approach to the Single Transferable Vote

Paper II: The problem of non-transferable votes

B L Meek, Computer Unit, Queen Elizabeth College, London

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.


The feedback counting method used for Single Transferable Vote elections, developed in an earlier paper, is extended to cover situations in which there are non-transferable votes. It is shown that present counting methods, on the other hand, may not satisfy the condition that the number of wasted votes be kept to a minimum in such situations. The extension of the method to permit voters to give equal preferences to candidates is also described.

1. Introduction

In an earlier paper[1] (hereafter referred to as Paper I) the Single Transferable Vote (STV) system of voting was considered from the point of view of certain conditions, the main one being that as far as possible the opinions of all voters are taken equally into account; it was shown that present STV counting methods do not satisfy this condition. A 'feedback' counting mechanism was suggested which would overcome this problem. In Paper I, however, we confined ourselves only to the cases where, whenever a vote is rescrutinised for transfer, a next preference is always given. In this paper we shall show how the feedback method can be extended to cope with situations where no such preference is available. We shall here adopt the reverse procedure to Paper I; we shall consider the application of the feedback mechanism to these cases first, and only then discuss present counting methods in the light of the conditions.

2. Rules for vote-casting

Even within the same voting system major differences can be made simply by changing the rules governing what constitutes a valid ballot. For example, in a multiple-vacancy election by simple majority where each voter has one independent vote for each vacancy, the result can be totally different if the voter is forced to use all of his votes (in effect to vote against his favourite candidates) instead of using only some.[2] In STV the equivalent requirement would be that all candidates should be listed in preference order. However, in the simple-majority case distortions can arise in that some votes may not be genuine, having only been added in order to make up the correct number; in STV a voter may only wish to express his preferences for a few candidates, being indifferent to the remainder. Normal STV practice is in fact to accept as valid any ballot showing a unique first preference; thereafter the voter may, optionally, give further preferences for as many or as few of the remaining candidates as he wishes. In STV the feedback mechanism could be applied as it stands simply by declaring as invalid any votes which do not give preferences for all candidates or (relaxing this somewhat) declaring invalid during the progress of the count any vote encountered for which a next preference is required but not available, and then restarting the count. However, it would clearly be more satisfactory not to impose additional restrictions on the voter if this can be avoided.

3. Extension of the feedback method

We recall here the two principles of the feedback mechanism stated in Paper I: Since transfers are only made from eliminated or elected candidates, non-transferability only arises when all the marked candidates are eliminated or elected. The simplest case to consider is that when all the marked candidates are eliminated. By Principle 1, such a ballot has to be treated as if those candidates had never stood; and hence as if the ballot is invalid. This implies that the total T of valid ballots is reduced; this in turn implies that, on the elimination of any candidate, if non-transferable ballots occur the feedback should include the recalculation of the quota, using the reduced value of T.

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

Method (a) is based on the view that the voter regards the unmarked candidates as of equal merit, which is why he has not given preferences. The second method is based on the view that the voter's action is a partial abstention; he has not sufficient knowledge of these candidates to judge between them, and prefers to leave the choice to the other voters. It should be noted that the two methods are not equivalent; in the first the totals of the unmarked candidates, in particular the non-eliminated ones, are raised equally, whereas in the second the quota increases the proportions transferred from the elected candidates, and the increase in the votes of non-elected candidates will vary according to these values. For the moment we shall resolve the (apparent) dilemma by making the (apparently) arbitrary decision to adopt the second method. The prima facie case for this is that in general some unmarked candidates will be elected candidates, and hence the adoption of the first method will in any case involve the recalculation of the quota. However, the real justification will appear in section 6, when it will be shown that the dilemma need not, in fact, exist at all.

4. Current STV practice

Current STV procedure in dealing with non-transferable votes involves different rules in different circumstances. The main rules are
  1. If a vote is not transferable from an eliminated candidate, it is set aside; such votes play no further part in the count.
  2. 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.
In Paper I we considered STV from the point of view of three conditions. Condition (C) we shall discuss later; the others were It is clear at once that, when there are non-transferable votes, condition (B) cannot be satisfied even by the feedback counting method unless recalculation of the quota is included, for otherwise candidates at a later stage of the count, when a number of non-transferable votes have accumulated, need less that the original quota to be elected. Indeed, if as many as q votes become non-transferable, it is impossible for the last elected candidate to achieve a full quota. We saw in Paper I that condition (A) is satisfied when there are no non-transferable votes. When votes do become non-transferable these have to be added to the 'wasted' total W, and the formula in Paper I becomes

W <= T/(S + 1) + T0

where T0 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 - T0. Thus with recalculation of the quota we have

W ' < T '/(S + 1) + T0 = W - T0 /(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 T0 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.

5. Recalculating the quota

It can be seen that in recalculating the quota and having to apply it in retrospect to candidates already elected, the same difficulties occur as in the simple feedback situation, without non-transferable votes, described in Paper I. We consider first the case of an elected candidate. If some of his votes are non-transferable, the appropriate proportion is subtracted from the total vote, and the quota recalculated. the reduction in the quota makes more of the elected candidate's votes surplus, which increases the proportion for transfer; this increases the non-transferable proportion to be subtracted from the total, which further reduces the quota, and so on. The equations to be solved are

q = [(T - p1×t10)/(S + 1) + 1] .....(Equation:1)

t1(1 - p1) = 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), t1 the total for the elected candidate, p1 the proportion he transfers, t10 the total vote for the candidate not transferable to others, and q is the quota.

These two equations can be solved easily for p1 and q by equating the expressions for q; however, if there is more than one elected candidate the iterative method of finding the pi, 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 p1 = 0 gives the original value of q. Equation (2) then gives a first value of p1 > 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 p1, and so on. Thus we have a monotone increasing sequence of values for p1, 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 = [Tn/(S + 1) + 1]

which must be evaluated for q first in each iterative cycle. Tn = Tn(p1,p2,....,pn) is the total available for transfer in each case; for n = 1, 2, 3 it is given by

T1 = T - p1×t10

T2 = T - {p1×t10 + p2×t20 + p1×p2(t120 + t210)}

T3 = T - {S1 pi×ti0 + S2 pi×pj×tij0 + S3 pp2×pt(123)0}

In these formulae tij...k0 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 Tn gives the expression for Tn-1.

6. Equal preferences

In section 2 we discussed briefly the effect of different validity rules on otherwise identical voting systems. The usual STV counting procedures depend on the existence at each stage of a unique next preference, the only deviation allowed being, as we have seen, that the absence of further preferences does not make the vote as a whole invalid. It is standard practice to accept as valid a vote with a unique first preference, and to accept further preferences provided one and only one is marked at each stage; if no, or more than one, next preference is given at any point, all markings at and past this point are ignored.

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:

Here we are interpreting this condition in a particular way not discussed in Paper I: the STV voting rules not merely encourage but force a voter to vote other than according to his preference in the restricted sense that, e.g. if he rates two candidates as equal first he is not allowed to vote accordingly, but must assign a preference order between them which may well be arbitrary. In view of the importance of first preferences in STV, this is undesirable. A voter is similarly forced to make an unreal ordering of candidates to which he is indifferent if, for example, he has listed his real preferences but wishes to give the lowest ranking to a candidate he particularly dislikes. This kind of voting is very common.

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]

7. Concluding remarks

Two distinct problems arise in the development of a voting system; the information with regard to the choices which is required from each voter, and the way in which this information is to be processed to arrive at "the social choice".

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.

References and notes

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. And, indeed, necessary information!
  6. K Arrow, Social choice and individual values, 2nd edn, Wiley 1962. For what is meant by "in a certain sense" see Paper I.
  7. See B L Meek, Electronic voting by 1975?, Data Systems July 1967, p 12.
  8. 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.

Up: Issue 1 Next: Paper 3 Previous: Paper 1