I’m writing this in October 2022. While I was writing my post, the Chancellor of the Exchequer resigned from his. I’m not going to redo the analysis if there’s another election, ok?

In several conversations with @NotAdric about elections and proportional representation, which are probably my favourite things in maths ((today, at least)), I’ve been pointed at a note from Littlewood’s miscellany that says:

  • Someone has claimed that if two parties receive votes in the proportion $p:q$ under first past the post, they can expect to win seats in roughly the proportion $p^3:q^3$;
  • Littlewood thinks this isn’t a great model and other suggestions work just as well.

It’s quite hard to come up with a decent model, mainly because general elections are fairly rate. However, we’ve had a few in the last decade or so ((just think what chaos under Ed Milliband would have been like)) and I figured that I could “improve” the data further still by working on regions rather than the UK as a whole.

So, let’s take the four general elections going back to 2010, split them up by region and see what we get.

I’ve made some decisions about data:

  • I’ll only consider Conservative and Labour vote tallies and seat counts because I’m lazy
  • I’ll exclude Scotland and Northern Ireland, to avoid small numbers like zero
  • Letting $r_v$ be the ratio of Conservative to Labour votes and $r_s$ be the ratio of seats, I’ll fit a curve of the form $\ln(r_s) = a + b \ln(r_v)$.

There’s a link to the data here.((I’ve collated it by hand, so am more than ready to believe there are errors; if you find any, let me know.))

The result

It turns out that the cubic model isn’t at all bad! It’s not perfect, but it’s not at all implausible.

The actual parameters are $a \approx -0.0913 \pm 0.0574$ and $k \approx 2.8694 \pm 0.1058$. The model $\ln(r_s) = 3\ln(r_v)$, the cubic law, lies – just – within the 95% confidence bound.

The upshot is that, as a regional heuristic – at least for the last few years, and at least for the two biggest parties in England and Wales – the cubic law holds up remarkably well.

Here, have a graph.

* Edited 2023-04-04 to put in the data and graphs that had been listed as “TODO” for over a week with nobody mentioning it. I also redid the analysis and found slightly different numbers pointing to the same conclusion.