Up: Issue 1 Next: Paper 5 Previous: Paper 3

Voting matters - Issue 1, March 1994

Counting in STV elections

C H E Warren

Hugh Warren is a retired aeronautical engineer, past member of the ERS Council and a present member of its Technical Committee.

Introduction

Whatever criticisms may be levelled against First-Past-The-Post as a system of voting, at least the system has the merit that, although the count may be conducted in many ways, all ways give the same result. The Single Transferable Vote is demonstrably a better system of voting, but the system has the disadvantage that the result depends upon how the counting is conducted.

Counts have been done in many ways, and in some peculiar ways by some well-meaning, but unversed, enthusiasts for STV. One of the commonest methods of conducting the count, and indeed the method that the Electoral Reform Society uses, is that given by Newland and Britton.[1] Their paper tells one how to conduct a count by their method, but not why they make many of the arbitrary decisions that they do. Woodall[2] has suggested that they are made for expediency - to simplify the count - and he goes on to propose another method, which he advocates whenever computer counting is used. As Woodall points out, his method would be prohibitively long with human counting. As Woodall also states, a differently worded but an exactly equivalent method to his had been proposed by Meek in 1969.[3],[4]

The object of this paper is, first, to consider some of the principles that are felt to be important in deciding upon a method for conducting the count, and then to go on and propose a method that meets these principles.

Principles

The first principle of the STV system is that election is by quota. A candidate is deemed elected when the vote assigned to him attains a given quota. The quota is chosen as the minimum vote which will not allow more than the required number of candidates to be elected. This is the Droop quota, and is the total valid vote divided by one more than the number of candidates to be elected.

The second principle concerns the transference of a voter's vote to the preferences later than his first preference. The voter needs to be assured that his later preferences will in no way upset the voter's earlier preferences. Equally a voter's later preferences should not be considered unless, in regard to each earlier preference candidate, either the voter has borne an equal share with other voters who have voted for that candidate in giving him the necessary quota, or that earlier preference candidate has been eliminated. The way in which Newland and Britton conduct a count does not meet this principle.

The third principle concerns the elimination of candidates. Unfortunately no-one appears to have proposed a principle in this regard. So what is usually done is that, when no candidate has a surplus above the quota, in order to allow the count to continue, the candidate whose vote is least is eliminated.

Method

If, after counting the first preference votes, the votes for one or more candidates exceed the quota, then the essential feature of the method proposed here is that these candidates are allowed to retain only part of the vote that had been expressed for them such as will give each candidate just the necessary quota. The part of the vote that the candidate retains is called the 'amount retained'. The voters who have voted for one of these candidates, for whom the amount retained is x1, say, then have an amount remaining of (1-x1), which is then transferred to the voters' expressed second preferences. If an expressed second preference has an amount retained of x2, say, and if x1+x2 is less than unity, then the voter still has an amount remaining of (1-x1-x2), which is then transferred to the expressed third preference, and so on. Proceeding in this way, the end of the first stage of the count is reached when some candidates have just the quota, whereas the remainder have varying amounts of vote less than the quota.

The candidate whose vote at the end of the first stage is least is eliminated. This means that, wherever his name appears on a ballot paper, it is 'passed over', and, in effect, all the later preferences are 'moved up one'. Elimination of a candidate will usually cause the votes for some other candidates to exceed the quota. The amount to be retained by each candidate is then reduced to such lower value as will give each candidate just the necessary quota. Voters who have voted for these candidates with reduced amount retained will then find that they have more vote remaining for transference to later preferences. Proceeding in this way, at the end of each stage of the count, some candidates will have just the quota, whereas the remainder will have varying amounts of vote less than the quota.

Eventually the number of non-eliminated candidates will be reduced to one more than the number to be elected. When the amounts to be retained are now recalculated so as to reduce each candidate's vote to the necessary quota, all candidates will have just the quota, so the one candidate who has an amount retained of just 1 is the one eliminated. The remaining candidates are deemed elected.

If at any stage a ballot paper does not contain sufficient preferences for transference to be made, then the balance of vote is ascribed 'non-transferable', and the quota is recalculated excluding the non-transferable vote.

The main question that the proposed method of conducting the count poses is: how does one decide upon the amount to be retained by each candidate at each stage? From what has been said, the amounts retained have to be such that, when the count is made, each candidate to whom an amount to be retained of less than 1 has been assigned achieves just a quota. The problem of finding the amounts retained, and the associated quota, is a mathematical one which is relatively straightforward, even if protracted, but which a computer can help to solve. Here we are concerned only with the principle, not with precisely how the task be done. However, it is not necessary for everyone to know how to assign the amounts retained. As Woodall[2] has exemplarily pointed out, it is only necessary for anyone to be able to check that the assigned amounts retained do in fact achieve the desired result.

References

  1. R A Newland and F S Britton, How to conduct an election by the Single Transferable Vote, second edition, Electoral Reform Society of Great Britain and Ireland (1976).
  2. D R Woodall, Computer counting in STV elections, Representation, Vol.23, No.90 (1982), 4-6.
  3. B L Meek, Une nouvelle approche du scrutin transférable, Mathématics et Sciences Humaines 25 (1969), 13-23.
  4. B L Meek, Une nouvelle approche du scrutin transférable (fin), Mathématics et Sciences Humaines 29 (1970), 33-39.

Up: Issue 1 Next: Paper 5 Previous: Paper 3