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’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.

* 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.