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Voting matters - Issue 1, March 1994

Computer counting in STV elections

D R Woodall, Department of Mathematics, University of Nottingham

Douglas Woodall is Reader in Pure Mathematics at Nottingham University.

Reprinted [with minor corrections] from Representation 23 (1983), 4-6.

The Single Transferable Vote is by far and away the fairest form of electoral system. Nevertheless, when the counting in STV elections is carried out by hand, rather arbitrary decisions have to be made in order to simplify the count, and these introduce anomalies. Although small in comparison with anomalies present in other electoral systems, these anomalies may affect the result, and are certainly annoying to the purist.

The biggest anomaly is caused by the decision, always made, not to transfer votes to candidates who have already reached the quota of votes necessary for election. This means that the way in which a given voter's vote will be assigned may depend on the order in which candidates are declared elected or eliminated during the counting, and it can lead to the following form of tactical voting by those who understand the system. If it is possible to identify a candidate W who is sure to be eliminated early (say, the Cambridge University Raving Loony Party candidate), then a voter can increase the effect of his genuine second choice by putting W first. For example, if two voters both want A as first choice and B as second, and A happens to be declared elected on the first count, then the voter who lists his choices as 'A B ...' will have (say) one third of his vote transferred to B, whereas the one who lists his choices as 'W A B ...' will have all of his vote transferred to B, since A will already have been declared elected by the time W is eliminated. Since one aim of an electoral system should be to discourage tactical voting, this seems to me to be a serious drawback.

If, on the other hand, one agrees that surpluses will be transferred to candidates who have already reached the quota, then one has to do something to avoid the never-ending transfer of progressively smaller surpluses between two candidates. Whatever strategy one adopts, it is bound to introduce other anomalies, albeit smaller than the one already described.

If the counting is carried out by computer, however, no such arbitrary decisions are necessary, as the never-ending transfer can be carried out to completion, or at least until the surpluses remaining to be transferred are less than (say) a millionth of a vote. The resulting procedure is described in the next paragraph in a different way. It is comparatively simple in concept, and the undoubtedly long calculations are all safely hidden inside the computer.

The counting is divided into rounds, in each of which one candidate is eliminated. In each round of the elimination, a scaling factor is assigned to each candidate, representing the proportion that will actually be credited to him out of the votes potentially available to him, in such a way that:

  1. a candidate who has already been eliminated in a previous round is assigned scaling factor 0, so that no votes will be credited to him in the current round;
  2. a candidate whose fate is undecided at the end of the current round is assigned scaling factor 1, so that all the votes potentially available to him are credited to him; and
  3. a candidate who by the end of the current round has at least the quota of votes necessary for election (and so is certain to be elected) is assigned a scaling factor less than or equal to 1 so that the number of votes credited to him is brought down exactly to the quota.
The candidate with the smallest number of votes is then eliminated, and the process is repeated until the number of candidates remaining is equal to the number of places to be filled.

For example, suppose that, in a given round of the counting, candidates A and B are certain of election and have scaling factors of two thirds and three quarters respectively, and candidates C, D and E have already been eliminated in previous rounds, whereas the fates of the remaining candidates remain undecided. Then a voter who lists the candidates in the order C, A, D, B, E, F will, in the current round, have none of his vote assigned to C. The whole of his vote will be passed down to A, who will retain two thirds of it. The remaining third of his vote will be passed over D and down to B, who will retain three quarters of it (that is, one quarter of a vote). The twelfth of a vote that is still unassigned will be passed over E and down to F, who will retain all of it.

The calculation of the scaling factors, which would be prohibitively long to do by hand, could be carried out quite easily by computer. However, once the computer had done the work, it would be possible to check by hand that the computer was correct; certainly this would take no longer than carrying out the whole count by hand as at present.

(This situation is not unusual in mathematics. Suppose, for example, that you were asked to find a number x between 1 and 2, accurate to seven places of decimals, such that (say) x(x(x(x(x+1)-4)-3)+3)+1 = 0.

You would find it very tedious to do so by hand, even with the aid of a pocket calculator. Suppose, however, that a computer were to do the work and tell you that the answer is 1.6825071; then it would take you only a few minutes to check that the computer was correct.)

The size of computer required would depend on the size of the electorate, on the number of places to be filled and, to a lesser extent, on the number of candidates. In the case of an election with both a very large electorate and a large number of places, it might even be impossible to carry out the calculations in a reasonable time with the present generation of computers.

However, for parliamentary elections, there would be no problem: the calculations could be done quite easily even on a mini-computer.

Since proposing the above method, I have learnt that it is not new; a differently worded but exactly equivalent method was proposed by Brian Meek in 1969.[1],[2] I hope it will be possible to agree that, whenever computer counting is used in STV elections, this method should be used.


  1. B L Meek, Une nouvelle approche du scrutin transférable, Mathématics et Sciences Humaines 25 (1969), 13-23.
  2. B L Meek, Une nouvelle approche du scrutin transférable (fin), Mathématics et Sciences Humaines 29 (1970), 33-39.

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