Gordon Brown is in trouble over an open mic gaffe, but the general picture in the British election campaign hasn’t changed much: polls still show the three parties evenly matched, with the Conservatives in the lead but well short of what they need for an absolute majority.

The Liberal Democrat vote seems to be holding somewhere in the neighborhood of 30%, but even optimistic reports of their chances suggest they will get only about half that share of the seats. Single-member districts disadvantage a party with widely spread support, and first-past-the-post voting means they get no countervailing advantage from preferences.

As Peter Kellner pointed out last week, there is an upside for the Lib Dems, albeit a distant one. If they can get well ahead of the other parties, their evenly spread vote means that they will pick up seats very quickly. A vote of say 42% would give them a larger majority than it would either of the other parties. That won’t happen this time, but the threat of it happening in the future might set people thinking in the Labour and Conservative parties.

Again, I recommend playing with the BBC’s seat calculator to see how striking these effects can be.

Calculators such as this are based on the assumption of uniform swing: that is, that if the Labour vote nationally falls, for example, from 36% to 30%, then a seat that gave them 48% last time will come down to 42% — and conversely for the other parties. No one imagines that this will happen exactly, but the idea is that deviations from uniformity will roughly cancel out.

Nate Silver at fivethirtyeight.com has challenged that assumption, suggesting instead a calculation based on proportional swing: in other words, that a national movement against Labour from 36% to 30% should be thought of as losing one-sixth of their vote, so that a 48% seat would be expected to come down to 40%, not 42%.

He shows that the consequences of this would be noticeably better for the Lib Dems and worse for Labor.

Silver points out that the assumption of uniform swing leads to absurd results in extreme cases, including votes greater than 100% or less than 0%. That’s true. But the assumption of proportional swing has its own logical problem, since it implies that over time, all seats would drift towards the average, becoming more marginal with each swing back and forth — which doesn’t in fact happen.

Research in Britain and Australia, going back to David Butler’s work in the 1940s, has shown that uniform swing has a better predictive record. But it’s admittedly far from perfect, and a genuine three-party contest — with its expanded scope for tactical voting under first-past-the-post — will certainly test its limits.

The best approximation this time will probably be somewhere between the uniform swing and the proportional approach. Alternatively, you could just pick numbers out of a hat, since either way the relationship between votes cast and seats won will be tenuous at best.

Peter Fray

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