Brian Meek is now at the Computing Centre, King's College London, Strand,
London WC2R 2LS. This article was originally published in
*Mathématiques et Sciences Humaines*, 13, No 50 1975 pp23-29.

After I wrote the two papers describing what has since become known as
'Meek's method' - published (in French) in *Mathématiques et
Sciences Humaines* in 1969 and 1970, and republished in English in *Voting matters* No.1 - I went on
to write a third paper, which the same journal published (in English!) in
1975. Some people have been aware of the existence of this third paper, and
this led to a request that it too be republished in *Voting matters*. I
have no objection to this being done, but it is important to stress that its
status is quite different from the other two.

The first two papers present my analysis of STV counting, and how it can be made as accurate as possible. The method totally accepts the basis of STV as it is, and does not alter or challenge its fundamental assumptions at all. (It does seem to challenge some people's own assumptions about STV, but that's not the same thing at all!) As such, 'Meek's method' was always intended as a practical method for conducting an STV count, albeit an expensive one at that time - far less so now, of course. Years later, David Hill, Brian Wichmann and Douglas Woodall demonstrated beyond question that it is a practical method, and earned my eternal gratitude for so doing.

This third paper does not have that status at all. It is in fact no more than an academic exercise, exploring an issue which arises from time to time in the literature on aggregation of individual preferences. It demonstrates that a method of taking account of intensity of preference is possible. This is very far from advancing it as a practical method for implementing an election.

I have never regarded it as a practical method. I do not advocate its
adoption, and I shall be very annoyed if anyone attempts to present it as
(say) 'Meek's proposal' or otherwise imply that I advocate its use. It
should not even be linked to 'Meek's method' (e.g. by alleging it is an *
extension* to my method), at least without very careful qualification.
The reason is that 'Meek's method' is STV, whereas the process described in
this paper is *not* STV. (It is certainly not a 'single' transferable
vote, for a start.) The way that votes are cast and interpreted is quite
different from STV.

To be sure, the vote *counting* shares some similarities, but that is
only because the same logic that led to the invention of STV and to the
Meek method has been applied to the aggregation process. The individual
votes being aggregated are, however, not STV votes. The consequence is
that the result can end up very far from STV, as the paper itself clearly
shows.

So the paper should be read for what it is, a mathematical demonstration that individual preferences can be fairly aggregated while still taking intensity of preference into account, and not as a suggested practical method for conducting elections. If that is done, there should be no misunderstandings. A voting system, derived from the STV (Single Transferable Vote), is described which includes intensity of preference while avoiding difficulties due to inter-personal comparison of utilities. It is shown that this system allows the voters some control over the method used to aggregate their preferences.

The vote counting procedure begins by normalising all the weights
*w _{}*ij which the

*c* being the number of candidates. This is the key,
as we shall see later, to the avoidance of troubles due to inter-personal
comparison of utilities, since it ensures that as far as possible each voter
has an equal say in the voting procedure.

The count proceeds by summing all the weights for all the candidates, i.e. calculating

*v* being the number of voters. Thereafter the count
proceeds much in the same way as in the single transferable vote, as
modified by the proposals in two earlier papers[1],[2]. An STV-type quota is calculated according to the formula

where *s* is the number of seats to be filled and

is the total vote, and the brackets indicate that the integral part is to be
taken. *q* is the minimum number such that, if *s* candidates have
that number, any other candidate must have less than that number.

(In practice it is likely that working will be to fractions of votes -
say three decimal places, in which case the "+1" in the formula for *q*
is replaced by "+0.001", or equivalently the weights *w*_{ij}
are normalised to sum to 1000 for each voter and the formula for* q* is
unaltered.)

The count may proceed by one of two steps. If no *W _{}*j
exceeds

If, however, a candidate, say *y*, has *W _{}*y greater
than

Counting continues by the application of one or other of these rules until
the requisite number *s* of candidates are elected. Once elected and
allocated the quota *q* the weights for that candidate are of course
not included in the recalculation. This makes the procedure somewhat simpler
than in the modified form of STV described in [1].
However, if all of a voter's choices - i.e. those candidates he has
allotted a positive weight - are eliminated, the quota *q* has to
be recalculated as in [2] so that this undistributable
vote is not included; similarly, when all a voter's choices have been
elected and allotted recalculated weights, the residue is non- distributable
and also must be subtracted from *W*. Recalculation of the quota does
of course imply recalculation of the weights of elected candidates, and an
iterative procedure as described in [2] can be used to
obtain the new *q* and *w _{}*iy to any desired accuracy.

Lest this be regarded as too trivial an example, it is often the case in committee that the collective choice for chairman is X, even though a majority prefer Y, simply because a substantial minority strongly object to Y. Any theory of voting which does not allow for intensity of preference is certainly incomplete, and any voting system which does not permit its expression cannot be wholly satisfactory.

The present system is based on two principles: that the only person who
can gauge the intensity of his preferences is the voter himself; and that
as far as possible each voter should contribute equally in the choice of
those elected. In a multi-vacancy election (*s* > 1) there is more
than just a single choice involved, and so it makes sense to allow a voter
to express his preference intensities by contributing all his voting power
to the choice of one candidate, or to share this power between the choices
of different candidates. Of course, it is possible to regard an
*s*-vacancy election as a single choice from the
* ^{n}*C

- STV
Let 1 > e > 0. Let the voters order their choices 1-e, e-e

^{2}, e^{2}-e^{3},..... e^{c-2}-e^{c-1}, e^{c-1}. Then the closer e is to 0 the closer the actual voting process becomes equivalent to STV. For example, suppose there are 5 candidates and e = 0.01. A voter's choice will be in the proportions 0.99, 0.0099, 0.000099, 0.00000099, 0.00000001, counting 99% for his first choice. If his first choice is eliminated, the four lower votes remain, and total 0.01. These have to be renormalised to add up to 1, and so are multiplied by 100 to give 0.99, 0.0099, 0.000099, 0.000001. A similar argument applies to votes transferred from elected candidates. - Single non-transferable vote
This, trivially, is when the voter gives 1 to his first choice and 0 to all the others.

- Simple majority with multiple vote
Here the voter gives 1/

*s*to each of*s*candidates, or perhaps 1/*k*to each of*k*candidates,*k*<*s*. These are special cases of giving a to*k*candidates and b to*c*-*k*candidates, where a*k*+b(*c*-*k*) = 1 giving a weighting between a more preferred and a less preferred group. - Cumulative vote
In this case the voter gives a

_{1}, a_{2}, .... a_{k}, to*k*candidates respectively, such thatFor an exact analogy to the cumulative vote each a

_{i}must be a multiple of 1/*s*.

Such a voting system would require a more than usual sophistication on the part of the voter. This being so, one can consider a further sophistication. The choice of voting procedure is one of immense importance in the democratic process, and no system is wholly stable wherein a substantial minority is dissatisfied with the voting procedure in current use. The required consensus may either be achieved through ignorance or habit, or by general agreement that a system is fair even though another may be advantageous to many, perhaps even a majority. In situations where there is awareness of and controversy about the different properties of voting systems, the present system offers a possible way out of deadlock. For, if most voters use the system in one of the ways described in the last section, then the election will be largely determined according to that voting procedure. Looking at it from the point of view of parties, each party can urge the voters to use the method they favour of filling in the ballot forms. However, it is a weakness in this area that voting systems are so often argued about in terms of fairness to parties or candidates, seldom in terms of fairness to voters. The present system, whose main fault is its complexity, has the virtue of that fault in that each voter can specify as precisely as he wishes the way his vote is to be counted, without this being imposed by others on him or on others by him. Most voting systems allow some such flexibility; the virtue of this system is the much greater precision with which the sophisticated voter can specify his wishes, without his being able by strategic voting to exercise more influence on the final result than is implied by his actual possession of a vote.

- B L Meek, Une nouvelle approche du scrutin
transférable I: égalite de traitement des électeurs et
technique à rétroaction utilisée pour le depouillement
des votes,
*Mathématiques et Sciences Humaines*, 25, 1969, pp 13-23. Reproduced in*Voting matters*, Issue 1, pp1-6. - B L Meek, Une nouvelle approche du scrutin
transférable II: le problème des votes
non-transférables
*Mathématiques et Sciences Humaines*, 29, 1970, pp 33-39. Reproduced in*Voting matters*, Issue 1, pp7-11.