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Voting matters - Issue 1, March 1994

A New Approach to the Single Transferable Vote

Paper I: Equality of Treatment of voters and a feedback mechanism for vote counting

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

Brian Meek is now at King's College London.

With some differences in presentation, the paper was originally published in French in Mathématiques et Sciences Humaines, No 25, pp13-23, 1969.


It is shown that none of the counting methods so far used in single transferable vote elections satisfies the criterion that all votes should, as far as possible, be taken equally into account. A feedback method of counting is described which does satisfy this criterion within the general limitations imposed by the STV system. This counting method, though very laborious for manual counting, would be feasible in automated elections.

1. Introduction

While the preferential voting system known as the Single Transferable Vote (STV)[1] has been criticised on various grounds, the following advantages claimed for it do not seem to have been seriously challenged: It is the purpose of this and a subsequent paper to consider (A), (B) and (C) from a decision-theoretic viewpoint, within a single constituency; it will be shown that (A), (B) and (C) in fact do not hold in present STV procedures, but may be made to hold, within certain overall limitations, by appropriate modification of the counting method.

2. The wasted vote

An essential feature of an STV election is the 'quota'. If there are s vacancies to be filled, the quota q is the smallest number such that, if s candidates have q votes each, it is not possible for an (s+1)th candidate to have as many as q votes. Thus if the total votes are T, then T-sq < q, but T-s(q-1) >= q-1, whence q = [1+T/(s+1)], where the square brackets denote 'integer part of'.

Candidates with more than q votes are elected, and have their surplus votes transferred according to the next preferences marked; if there are no such candidates, the bottom candidate is eliminated and all his votes so transferred. Repeated application of these rules ensures that at the end of the count s candidates have at least q votes each and so the total wasted vote w satisfies w < T/(s+1).

Given s and T, it is clear from the definition of q that condition (A) is satisfied provided the next preference at each transfer is always given. It is possible for the above inequality, and hence condition (A), to be violated, if w is increased by the addition of votes which are non-transferable because no next preference has been indicated. In this paper we shall assume that this does not occur; it will be shown in a second paper that it is possible still to satisfy (A) in such cases by modifying the definition of q.

3. Equality of treatment

The discussion of condition (A) shows that, in general, there will be some wasted votes, except in the trivial cases when s >= T. It is therefore not possible under STV to guarantee that all votes will be taken equally into account (e.g. votes with first preferences for runner-up candidates), although all are taken indirectly into account when calculating the quota.[2]

Within this obvious limitation, attempts have been made to eliminate possible sources of inequity of treatment by various modifications of the counting rules. Such sources include:

  1. the choice of which votes to transfer from the total for a candidate who has exceeded the quota
  2. errors introduced by taking whole-number approximations to fractions of totals for transfer - particularly in elections with small total vote
  3. calculation of the proportion for transfer from an elected candidate on the basis of the last batch of votes transferred to him, and not on his total vote.
The common way of overcoming difficulties (1) and (2) is to use the variant of STV known as the Senate Rules. Each vote is divided into K parts (usually K = 100 or 1000) and each part treated as a separate vote (of value 1/K) with identical preference listings.

Difficulty (1) is overcome by transferring the appropriate proportion of each divided vote, while the method clearly reduces the errors involved in (2) by the factor 1/K. If K=10**n this is simply working to n decimal places. The value of K has only to be increased until the errors are too small to affect the result of the election.[3] The method is equivalent to transferring the whole vote at an appropriately reduced value, and it is this interpretation we shall use from now on.

Difficulty (3) is slightly more technical, and warrants further explanation. Suppose at some stage a candidate has obtained x (<q) votes. By transfer from another (elected or eliminated) candidate he now acquires a further y votes, where x+y >= q. His surplus is now z=x+y-q. It would appear that his x+y votes should now be transferred, with value reduced by the factor z/(x+y).

It is, however, common practice for only the y votes to be transferred, with value reduced by the factor z/y. The reason for adopting this procedure is simply the practical one, in a manual count, of reducing as much as possible the rescrutiny of ballots for later preferences. However, neither this nor the argument that 'the difference is unlikely to affect the result' are particularly relevant to a decision-theoretic discussion, though we shall return to the practicability problem later.

Of more importance here is the argument 'in STV a vote only counts for one candidate at a time, and should count for the first preference where possible'. If accepted, this would of course also render difficulties (i) and (ii) irrelevant, and the Senate Rules unnecessary; the first part of it is in fact sometimes used as a 'proof' that STV satisfies condition (B). But even without the Senate Rules the statement is false; however the surplus votes are chosen for transfer, it is the existence of the untransferred votes which makes the transferred votes surplus. A vote not only counts directly for one candidate; it can indirectly affect the progress of the count, the pattern of transfers, and ultimately the election or non-election of other candidates.[4]

It is this fact which is at the root of the failure of STV to satisfy condition (B).

In the specific situation described above, the candidate achieves election not only because of the accession of the y new votes, but because of the existence of the x previous votes; hence for condition (B) to be satisfied, all x+y votes should be transferred at the appropriate reduced value.

However, there is yet a fourth difficulty, one which does not seem to have been recognised hitherto.

Let us suppose that of y votes to be transferred, y/2 are marked next to go to candidate A, and y/2 to candidate B. Let us further suppose that A has already been elected; under STV the y/2 votes which would otherwise go to him are transferred to the next candidate marked (assumed C in every case) provided that that candidate is not also already elected. Thus y/2 go to B, and y/2 to C. The inequities are plain; the votes for A which enabled the y/2 to go to C rather than A had no say in their destination, while C obtains these votes at the same value as B receives his. Suppose these y votes were originally first-preference votes for a candidate D, now eliminated; those who voted for A next and then C at least have had their second choice elected, while those who voted next for B have not - yet these votes go, under STV, to both B and C at full value.

In section 6 we shall describe a counting mechanism which overcomes all these difficulties.

4. Making the most of one's vote

Any system which contains wasted votes contains at least some element of incentive to vote in other than his preferred way; the case for (C) in STV is that it is difficult for a voter to be sure (rightly or wrongly) that his vote will be wasted, both because the number of wasted votes is relatively small, and because the wasted votes are those for the non-elected but non-eliminated candidates - i.e. of the stronger, not the weaker, runners-up. However, it is also possible for voters to take advantage of the features of STV described in section 3, provided they are sufficiently well informed, by voting in a sophisticated manner. This is most easily shown by an example:

Let T=3599, s=3, q=900, and the unsophisticated first-preference votes for the six candidates A, B, ... F be as follows:

     A      B      C       D       E     F
  1020    890    880     589     200    20
In this case the 120 surplus votes of A divide 60 to B, 20 to C, 40 to D and the elected candidates are A, B and C.

Suppose there are 170 voters who above voted A, D, C ... It is known that the second-preference votes of F will go to C, and of E to D. Then the sophisticated way for these 170 to vote is F, A, D, C,... in order to prevent A from being elected on the first count.

     A      B      C       D       E      F
   850    890    880     589     200    190
On the elimination of F, his original 20 votes go to C, and the 170 sophisticated votes return to A. However, the 120 surplus is now taken entirely from this batch (see (3) in section 3) and goes to D. C having no surplus, E must be eliminated and D is elected.

A different type of sophisticated voting is given below: T=239, s=2, q=80.

Unsophisticated case: C and A elected:

   C,A,B...      C,B,A...     B,A....     A,B.....
    120            80           31          8
Sophisticated case: C and B elected:
   C,A,B...      C,B,A...    E,B,A...     B,A....     A,B.....
    120            50          30          31           8
It seems to be a new result that sophisticated voting is possible in STV, though it is well-known that it can occur in other voting systems and considerable work has been done on decision processes using a games-theoretic approach. Black [5] in his discussion of STV does mention the possibility of 'an organised minority (perverting) the use of the system' but only in connection with a candidate with just the quota on first preferences who is rated last by the rest of the electorate. STV supporters would claim that if a candidate can obtain a quota this ipso facto entitles him to be elected, particularly if he gets the quota on first preferences, and it is certainly difficult to understand what Black means by 'pervert' in this context.

5. Other considerations

At this point we shall mention some other aspects of STV, mainly in order to define the limitations of the present discussion. Proper treatment of the points raised in this section are well outside the scope of the present work, and is the subject of a projected further, more general paper.

The conditions (A), (B), (C) discussed so far were chosen simply because they seem to be specific to STV among constituency-type systems in parliamentary elections. However, other conditions could be applied, notably those specified by Arrow in his General Possibility Theorem.[6]

As STV elections are multi-vacancy, the preferences between candidates listed by the voters do not as they stand represent an ordering of independent alternatives, and so Arrow's analysis is not directly applicable. The deduction from the voter's ordering of candidates of his ordering of the actual independent alternatives (the possible subsets of the set of all candidates who might actually be elected) is by no means straightforward. Nevertheless, at some stage of the count the process reduces to electing one candidate to one remaining vacancy, and so the consequences of the theorem, and the Condorcet paradox, cannot be escaped. Using the alternatives as they stand, even though they are not independent, STV clearly satisfies Arrow's conditions 1, 4, and 5. The condition 3 of independence of irrelevant alternatives is not satisfied, nor is condition 2 (the positive association of social and individual values). This can be seen from the above analysis.

A related point, and probably the strongest decision-theoretic argument against STV, is the fact that a candidate may be everyone's second choice but not be elected. This difficulty is not overcome by the feedback method, and it does not seem to the author to be possible to do so while retaining a system which would be recognisably a 'single' transferable vote.

Virtually all other discussion of STV, both for and against, seem to have been about political and not decision-theoretic considerations.

For example, Black[5] does discuss STV from what he terms the 'statical' point of view, but although he does express some disquiet about the 'heterogeneity' involved in STV (basically, that some votes count for first preferences, others for second or later preferences), he does not go into the problem in detail and concludes 'in spite of those drawbacks (STV) has merits ... it is not difficult to see why many people, regarding it purely as a statical system, (Black's italics) should hold (it) in esteem'. The italicised phrase is to introduce other, 'dynamical' arguments against STV.[7] Black does not discuss the conditions mentioned here; though the germ of the idea of inequity is contained in the word 'heterogeneity'; in fact as section 3 shows, the heterogeneity which worries him is more apparent than real, and the feedback method described in section 6 eliminates what there is. Nor - oddly - does the 'everyone's second choice' problem, even though this is closely connected with the doubts mentioned at the end of the last section.

6. The feedback process

One of the criticisms of STV which is often made is that its rules are too complicated, and are not derived from principles which can be simply stated. The above discussion shows that this is not surprising; the rules are in many cases little more than rules of thumb, designed for practical convenience rather than theoretic merit. The feedback process, however, is derived from simply-stated principles: Principle 1 is the one which leads to the feedback mechanism. For, suppose a voter marks his ballot A, B, C,.. and A is eliminated, the ballot, by Principle 1, is henceforward treated as if it read B, C,.. on the assumption that if A had not stood at all, the voter would have ordered the other candidates as before and B would have been first preference [9]. But suppose that B has at an earlier count reached the quota. Then this ballot must now be treated as an original first preference for B; that is, according to Principle 2, the same proportion of this vote must be retained by B as for the others, passing the rest to C (instead of the whole vote going to C as in previous methods). However, this will mean that the total retained by B is now greater than the quota. Thus the proportion of B's votes to be retained must be recalculated, and will in fact drop - in other words we must go back to the beginning, with A now eliminated. This is the feedback process.

Note that the proportion of each of B's votes to be transferred is increased by this accession of support; B's supporters have a say in the transfer of the extra surplus, since it is their existence which has made it surplus. All support for B is now treated equally, being divided proportionately to leave him with exactly the quota.

Consider now the effect of Principle 2. The transfer of B's vote may lead to another candidate, D, being elected. All votes, new and old, for D, have now to be divided, leaving D with the quota and distributing the rest to the next non-eliminated candidate. Some ballots may have B, another elected candidate, as next candidate. Under previous rules, only continuing (i.e. non-eliminated and non-elected) candidates can receive transfers. Now these votes are regarded as extra support for B: he takes the proportion allotted him by D, retains the proportion that he keeps of all he receives, and transfers the rest - now the third marked candidate. Formerly the third candidate would get all of the proportion transferred by D (see (iv) section 3).

It can be seen that B will once more have more than the quota if he does not again reduce the proportion which he retains. However, the increased proportion transferred may in part go to D who will therefore have to reduce the proportion he retains. This will react back on B, and it is clear that we have an infinite regression. However, it is also clear that the proportions for transfer do not increase without limit, there being only a finite total surplus available from B and D, who must each retain a quota. The problem is in fact a mathematical one of determining the proportions to be retained by each which will leave them both with a quota, taking into account the extent of mutual support. If pB is the proportion B transfers, and pD that which D transfers, supporters of both B and D have their votes transferred to third preferences at value pB×pD. Those putting B first have 1-pB retained by him and pB×(1-pD) retained by D; those putting D first have 1-pD retained by him and pD×(1-pB) retained by B.

We now, as examples, give the formulae for the proportions for transfer in the cases of 1, 2, 3 and 4 elected candidates:

One candidate


This is the same formula as before, except that t1 now contains all effective first-preference votes for the candidate, including those obtained from eliminated candidates, who by Principle 1 are now ignored. The proportion p1 is recalculated every time t1 is increased by the elimination of a candidate.

Two candidates

The first elected candidate has t1 first preference votes, of which t12 have the second elected candidate as second preference. Hence p1×t12 are passed on to that candidate. Similarly p2×t21 are received from the second candidate. Thus



Three candidates

The votes received by candidate 1 are now his first-preference t1, second-preference p2×t21 from candidate 2 and p3×t31 from candidate 3, and third-preference p2(p3×t321) from candidate 3 (1st), 2 (2nd) and p3(p2×t231) from candidate 2 (1st), 3(2nd).



Two similar formulae hold, obtained by cyclic permutation of the suffices.

Four candidates

The formula now is:

where dashed summation indicates summation over all permutations of (234); there are three similar formulae.

The extension to any number of candidates is straightforward. It should be noted:

  1. The formulae for n candidates may be reduced to those for n-1 candidates by eliminating the nth equation and putting pn=0 in the others;
  2. Full recursion is not necessary on the elimination of a candidate if none of the totals or subtotals in the formulae in use at that stage are changed as a result.

7. Calculating the proportions

It can be seen that one of the difficulties involved in the feedback process arises from the need to calculate the proportions for transfer. However, a simple iterative procedure enables this to be done to any required accuracy. We shall take as the simplest example the position with two elected candidates, where the equations to be solved are, as above:
  1. (t1+p2×t21)(1-p1)=q
  2. (t2+p1×t12)(1-p2)=q
In these equations only the pi are unknown. Suppose we guess a value of p2 which is too low; then (1-p1) will be too large in equation (1), that is p1 will also be too small. If we substitute this in equation (2) it will similarly give a value of p2 which is too low.

The total vote for the two candidates is t1+t2; for them both to be elected t1+t2 >= 2q. Suppose the strict inequality holds; in a non-trivial case t12, t21 are both non-zero. Further, at least one of t1, t2 is greater than q; assume it is t1. If we put p2=0 in (1) we can solve for p1, giving a value p1 > 0. This p1 is the proportion to be transferred if candidate 1 were the only elected candidate; thus t2+p1×t12 > q or candidate 2 would not be elected. If the equality holds, candidate 2 only just gets the quota and so p2=0 from equation (2); thus the equations are solved.

If the strict inequality holds, we get a value of p2 > 0 which is too small. Substituting in (1) increases the coefficient of (1-p1) and hence increases p1; the new value of p1 is increased (but is still too low). Substitution in (2) gives similarly an increased, but too low, value of p2. Thus the iterative process gives monotonically increasing sequences of values p1, p2 bounded above, which hence tend to limits which are the solutions of the equations. A cycle of iterations which leads to two successive sets of values the same to the given accuracy is taken as the approximate solution required. Note that the approximate values may be slightly smaller than the exact ones, but this is exactly what we want; otherwise too much of the support for the candidate concerned would be transferred and he would be left with less than the quota. The process can also be easily shown to work in the limiting case, t1+t2=2q.

It is clear that the success of this iterative procedure depends on the fact that all the quantities in the totals (the coefficients of (1-pi) in each equation) are non-negative, and that therefore it will work for any number of equations provided they are solved cyclically in order of election - this condition being necessary to avoid getting negative values of pi. Since the counting process can only increase the totals of support for elected candidates, it is also clear that the pi for those candidates can only increase as the count progresses;[10] thus it is safe to take as starting values of the pi the ones obtained at a previous stage, putting pi=0 initially for newly-elected candidates only (in which case, as mentioned above, the equations reduce to the ones at the previous stage and hence will yield, at the beginning of the iteration, the same answers).

It can be shown fairly simply that the convergence rate of the iterative process is likely to be unsatisfactory only when both of the following conditions hold; that all the pi are small, and the cross-totals tij etc, are as large as possible. This would not cause difficulty even on the rare occasions on which all these conditions were satisfied, since the occurrence of slow convergence can be detected in advance and allowed for, while at a later stage in the count some at least of the pi are likely to rise sufficiently to accelerate to the true convergence satisfactorily.

8. Conclusions

It is obvious even from the above example that the feedback process is a much more laborious method of arriving at a result than any at present in use; in a full-scale election with thousands of ballots to scrutinise, it would be very lengthy indeed. However, even the present methods are sufficiently lengthy to make it worthwhile using computers to help in the counting,[11] and if this is done, then complex counting methods are no problem.

It may be argued that the actual results of any election would be different so infrequently that the additional complication is unnecessary. This is a matter for conjecture, or preferably, for further investigation. However, the method has been tried out in two cases, once using figures obtained by a quasi-random process, and once in an actual STV election. In both, there were differences in the candidates elected.[12] Particularly since STV supporters lay such emphasis on the criterion of equality of treatment (condition (B)), it would seem worthwhile in automated counting to adopt the feedback method.

To sum up, the feedback method does satisfy the criterion, subject to the limitations imposed by the basic STV system - i.e. the theoretical minimum of wasted votes, and the elimination of candidates. There is one further limitation not so far discussed, imposed by the voters themselves if they take advantage of the possibility allowed by STV of listing only some of the candidates in preference order. The extension of the feedback method to cover this is dealt with in Paper II; it turns out that the extension also, as a bonus, allows voters to express their views much more accurately than under previous STV methods.[13]

References and Notes

  1. For a complete description of STV see E Lakeman and J Lambert: Voting in Democracies (Faber and Faber 1955). (The current edition in 1994 is E Lakeman: How Democracies Vote (4th edition, Faber and Faber 1974).)
  2. This is nevertheless more than can be said for some common voting systems, such as the simple majority system.
  3. This cannot, of course, cope with the case of exact equality, where some other method has to be used, if only drawing of lots.
  4. To argue, in connection with a transferable system, that a vote should where possible not be transferable, seems inconsistent, particularly in view of the strong arguments put forward by STV supporters against the single non-transferable vote system, where an elector may choose only one from a list of candidates even though more than one are to be elected. See Lakeman and Lambert, op, cit. [1]
  5. Duncan Black: Theory of Committees and Elections (2nd edition, Cambridge, 1963, pp 80-83).
  6. K Arrow: Social Choice and Individual Values (2nd edition, Wiley 1962).
  7. The case for the other side may be found in Lakeman and Lambert, op cit.[1]
  8. The similarity of this principle to Arrow's condition of independence of irrelevant alternatives is obvious. However, the interdependence of the alternatives here means that the condition is not in fact satisfied.
  9. This innocent-looking assumption is open to major criticism. Full discussion is outside the scope of this paper; it is hoped to include this in the projected more general paper mentioned in section 5.
  10. Clearly Arrow's condition 2, the positive association of individual and social values, is now satisfied by the non-independent alternatives.
  11. For a feasibility study in general terms, see P Dean and B L Meek: the Automation of Voting Systems; Paper I; Analysis (Data and Control Systems, January 1967, p16); Paper II; Implementation (Data and Control Systems, February 1967, p22), and B L Meek: Electronic Voting by 1975? (Data Systems, July 1967, p12) - the date in the last source referring to the UK. For a description of the actual use of computers in STV elections in the United States, see Walter L Pragnell: Computers and Conventions (The Living Church, 20th August 1967, p12).
  12. For obvious reasons the work on the actual election cannot be made public!
  13. These papers are the result of a problem posed by Miss Enid Lakeman, Director of the Electoral Reform Society, London; the author wishes to thank her for her encouragement in the progress of the work. Thanks are due also to Professor W B Bonnor, Mr Robert Cassen, Mr Peter Dean, Mr Michael Steed and Professor Gordon Tullock for valuable discussions, correspondence and advice.

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