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Voting matters - Issue 16, February 2003

What sort of proportionality?

I D Hill

In pure mathematics proportionality is a well-defined concept, but that is because we can always go into fractions whenever necessary. For proportionality within voting systems we are restricted to whole numbers in those elected for each party (using "party" in the general sense of any relevant grouping of the candidates, not only in the sense of a formal political party). Under such circumstances it is in many cases not at all easy to say whether one result is more nearly proportional than another. This is particularly so where some parties, quite correctly, get zero seats, while none get zero votes.

I agree with Philip Kestelman[1] that none of the measures that he discusses is perfect. I agree also that the comparative answers that they produce are so similar that, if using any, we might as well settle on one of them. But as I have said before[2] they are all fundamentally flawed in basing their calculations on first-preference votes only, and this can be very misleading, particularly where there is a substantial amount of cross-party voting for successive preferences.

However there is an additional point to be considered, even where first preferences do give full information on party popularity, there being no cross-party voting at all. Under such circumstances it could be the rule that if n is the minimum value, across parties, of votes per seat, then any party with at least n votes must get at least 1 seat, any party with at least 2n votes must get at least 2 seats, any party with at least 3n votes must get at least 3 seats, and so on. Given the restriction to whole numbers, and that some parties may get zero seats, what could be more proportional than that? Yet none of the measures that Kestelman considers meets that rule.

For simplicity, consider the case of only 2 parties and only 2 seats to be filled. Suppose the votes are 70 for party A and 30 for party B. We can at once rule out the option of giving both seats to party B, but is it better to give both to A or one to each?

Suppose we allot them as 1 to each. Then n = 30 / 1 so party A with more than 2n votes must get at least 2 seats and the rule is violated. Suppose we allot them as both to party A. Then n = 70 / 2 and the rule is satisfied for party B does not reach 35 to be worth a seat. Yet every one of the measures that Kestelman considers says that 1 to each is a better answer than both to party A. To my mind that shows all those measures to be unsatisfactory. I regret that I do not know of a better alternative, but to do without a measure is preferable to using a defective one.

If anyone doubts that both to party A is the better answer, let them assume that there had been only 3 candidates and votes 36 A1 A2, 34 A2 A1, 30 B. The measures all say that to elect A1 and B, or even A2 and B, is preferable to A1 and A2, which is surely nonsense.

However, I am grateful to Philip Kestelman for the suggestion that we might, perhaps, say that to elect A1 and B is more party-representative, while to elect A1 and A2 is more candidate-representative. There might be something in that.


  1. P Kestelman, Quantifying representativity. Voting matters, Issue 10, 7-10. 1999.
  2. I D Hill, Measuring proportionality. Voting matters, Issue 8, 7-8. 1997.

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