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Voting matters - Issue 8, May 1997

Measuring proportionality

I D Hill

When you can measure what you are speaking about and express it in numbers, you know something about it, but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind. Lord Kelvin.

It is important to consider what the problem actually is, and solve it as well as you can, even if only approximately, rather than invent a substitute problem that can be solved exactly but is irrelevant. Anon.

I agree with the first of those quotations but I agree much more strongly with the second one. As Philip Kestelman points out in a recent article, if we are to talk of proportional representation, and to claim that one aim of STV is to achieve it, it is desirable that we should have some idea of how to measure it and thus be able to detect the extent to which one system or another is able to achieve it.

Many indices have been proposed for the purpose, of which Kestelman prefers the Rose index, or Party Total Representativity (PTR) as he renames it. While differently formulated, the various indices all seem to have similar effects, usually placing different elections in the same order of merit even if the numbers that they assign are very different. They mostly depend, in one way or another, on the differences between percentages of votes by party and percentages of seats by party. It seems a little odd when considering a multiplicative type of thing, like proportionality, to use an additive type of measure, but this does overcome some difficulties that might otherwise arise when parties get zero seats.

A correlation measure

There is an additional measure that is rather different from all these, mentioned by Douglas Woodall as having been proposed by Dr J E G Farina and depending on the cosine of an angle in multi-dimensional space. This is not a concept with which the general public would feel easily at home, but the measure does turn out to be closely associated with the statistical measure known as the correlation coefficient, and many people seem to feel happy that they know what correlation means (even if, in fact, they do not). However the ordinary correlation will not do, because it measures whether points tend to be grouped around a straight line, but not all straight lines give proportionality.

For example with votes of 200, 400 and 600 and the proportional 2, 4 and 6 seats we get a correlation of 1.0, but the non-proportional 3, 4 and 5 seats equally get 1.0 as those points also fall on a straight line. To get a suitable measure we also need to include the same numbers over again, but negated. Thus 200, 400, 600, -200, -400, -600 with 2, 4, 6, -2, -4, -6 gives a correlation of 1.0 as before, but 200, 400, 600, -200, -400, -600 with 3, 4, 5, -3, -4, -5 gives only 0.983 demonstrating a less good fit.

The fatal flaw

If going for any of these measures, I like the last one best, but they all have one fatal flaw - they depend only upon party representation and only upon first preference votes. It is possible to use them upon features other than formal political parties if there is enough information available on those other features, which usually there is not. Kestelman does so, but this is rarely done, while how to extend them to deal with anything other than first preferences does not even seem to be discussed. They therefore, to my mind, fall within the terms of the second quotation in my heading, as the substitute problem that is irrelevant.

It is true that, in many elections, voting is mainly in terms of party, and that most people's party allegiances will be detectable in terms of their first-preference votes, but I object to those who say that all we need to know about an electorate is to be found in those things. I much more strongly object to any suggestion that voters ought not to vote cross-party if they wish, or even should not be allowed to do so.

It often helps discussion to look at an exaggerated case, even though it is far removed from what normally happens in practice. An example that I have used before concerns 9 candidates: A1, A2 and A3 from party A; B1, B2 and B3 from party B; C1, C2 and C3 from party C. The election is for 3 seats and the votes are, say,

50% A1 B1
50% A1 C1
If a system elects A1, A2 and A3 the above measures will all say that it has done well - with 100% of the votes for party A and 100% of the seats for party A. Yet nobody actually voted for A2 or A3 at any level of preference. From that election STV would elect A1, B1 and C1, the candidates whom the voters mentioned, yet such measures will all say that it has done badly. While I believe that a measure of proportionality, if we can find a suitable one, would be a good thing I am not prepared to accept as useful any measure that cannot deal sensibly with that case.

Minor parties and independents

A further difficulty with all these measures occurs if there are a number of minor parties (and/or independent candidates), none of which get enough votes to be entitled to a seat. If each of them is put into the formula as a separate entity, you get one answer, but if you put them together as "others" you may get a very different answer because that number of votes for a single party would have been worth a seat (or more). Such minor parties are likely to be so divergent that to elect any one of their candidates to represent all their voters would be quite unsatisfactory.

STV's proportionality

STV's proportionality comes from what Woodall calls DPC for "Droop proportionality criterion". This says that if, for some whole numbers k and m (where k is greater than 0 and m is greater than or equal to k), more than k Droop quotas of voters put the same m candidates (not necessarily in the same order) as their top m preferences, then at least k of those m candidates will be elected. In particular this must lead to proportionality by party (except for one Droop quota necessarily unrepresented) if voters decide to vote solely by party. Anti-STVites may argue that this is not altogether relevant because people may not vote like that, but they cannot have it both ways - if voters are not concerned solely with party, and do not vote solely by party, then measures that assume that only party matters must be wrong.

The STV argument is that it will behave proportionately, as defined above, so long as voters do vote solely by one thing, whether that is party or not, but if (as is usual) voters have a mixture of aims and motives it will adjust itself to match what they do want to a reasonable degree. Looking at how it works suggests that it must do so, but I know of no way of proving it. What I find obnoxious is to find those who oppose it saying that it cannot be guaranteed to do so, and therefore wanting instead some system that does not even attempt it.

Furthermore STV gives the voters freedom to show their true wishes, major party, minor party, independents, sole party or cross-party, by sex or race or religion or colour of socks, or whatever they wish, whether others think that a sensible way of choosing or not. Even if it did not give a reasonable degree of proportionality as well, it would be worth it for that freedom and choice. Party proportionality is a bonus, not the be-all-and-end-all. It may be that "when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind" but can we measure love, or aesthetic pleasure, or scientific curiosity? Perhaps there would be some advantages if we could measure them, but our inability to do so does not in the least affect our conviction that they are things worth having. Let us continue to seek a useful measure, but not be bound by imperfect ones.

First-preference measures unsatisfactory

Even within strictly party voting, the first-preference measures are unsatisfactory. Consider a 5-seater constituency and several candidates from each of Right, Left and Far-left parties. Suppose that all voters vote first for all the candidates of their favoured parties, but Left and Far-left then put the other of those on the ends of their lists. If the first preferences are 48% Right, 43% Left, 9% Far- left, all the measures will say that 3, 2, 0 is a more proportional result than 2, 3, 0. Yet STV will elect 2, 3, 0 and that is the genuinely best result, because there were more left-wing than right-wing voters. There is no escape by comparing with final preferences, after redistribution, instead of first preferences. That is merely to claim that STV has done well by comparing it with itself. Our opponents may sometimes be dim, but I doubt whether they are dim enough to fall for that one.


I remain of the opinion that a measure of proportionality is very much desired if we can find a suitable one, but we know of none, and an unsuitable one may be worse than useless. What do others think?


  1. P Kestelman. Is STV a form of PR? Voting matters, 1996, Issue 6, 5-9.
  2. D R Woodall. How proportional is proportional representation? Mathematical intelligencer, 1986, 8, 36-46.
  3. D R Woodall. Properties of preferential election rules. Voting matters, 1994, Issue 3, 8-15.

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