US Electoral Maths – Seeing the funny side

[ This piece was written before the 2016 election…
Suffice to say, I’m not seeing the funny side any more. ]

Back in November 2012, I attempted some mathematical stand-up comedy for Bright Club, the public engagement scheme where academics talk about their research. Although I wasn’t actually working on elections or voting, the US election was topical at the time (and I had followed it obsessively), so I was asked to do a piece on it:

[If you’re wondering about the ending, the comedian Fin Taylor was hosting the evening.]

The maths is essentially a crude application of the idea behind the Banzhaf power index: considering the probability that an individual vote will change the outcome of an election and comparing this probability between different voters. All the numbers I give are the result of genuine calculations, though I would have to admit that my methodology is pretty dubious.

Firstly, the simplifying assumption that voters in each state chose their candidate according to probabilities proportional to the final vote totals (essentially assuming that the outcome in each state was drawn from a binomial distribution with an enormous number of trials) is pretty ridiculous and is guaranteed to lead an astronomically small measure of voting power, such as that given for Utah.*

Secondly, citing a figure for the voting power of an individual voter is a bit of a red herring anyway. For any electorate numbered in the hundreds of thousands or more, the chance of any single vote changing the outcome of an election is obviously going to be tiny. You’d be a bit worried about the fairness of the system if that weren’t the case. The important thing is the comparative power of different voters, so my later comparison of voting power between Utah and Florida is more sensible than talking about the raw numbers (though it is still significantly exaggerated by the binomial assumption).

There were two main reasons that I wasn’t too worried about using a rough-and-ready approach to the maths in this case. The first is that, hey, it was meant to be comedy. I wasn’t being peer reviewed, I was trying to appeal to the crowd that had turned up on the night, and they weren’t there to check my working.

The second reason is the more critical one, though. Even if I had done the maths more carefully, the overall thrust of the results would have been the same. The electoral college system for US elections leads to huge disparities in voting power between voters in different states.

As it happens, the superb data journalism site actually have done the maths more carefully (here) and they reckon that a voter in Florida has over twenty-five times the power of a voter in Utah in 2016.** OK, so not the enormous differential that I quoted,*** but still a sizeable difference, which could be eliminated if the president was chosen by simply comparing the vote totals across the country.

While it may be the case that a popular vote majority is almost always reflected by a majority in the electoral college, I don’t think this fact constitutes a defence of the system, since on those rare occasions when a presidential election is very close, it is the high power voters in the tipping point states that make the difference – and they are certainly not guaranteed to tip the balance in favour of the candidate with the most votes overall.

Of course, the situation in the UK is not much better. The first-past-the-post system**** used in British general elections also leads to huge disparities in voting power. For example, a voter in the Liverpool Walton constituency could increase their voting power by a factor of 170, just by crossing the Mersey to Wirral South. You can hear more about that in my interview with Carl Cullinane of Democratic Dashboard, in Episode 38 of The Global Lab (from June 2015).

* A slightly better approach might be to divide the electorate into partisans (whose choice is pre-determined) and swing voters, who could go for either candidate with a certain probability; or to have a continuum of voter types with different probabilities of supporting each candidate; or (probably most sensibly) to consider one voter whose power you wish to quantify and to treat all others in a state as an amorphous blob, modelling their vote distribution using a model based on historical trends.

** At time of writing (the numbers change regularly as new polls come in), the fivethirtyeight polls-only model gives the relative voting power of a voter in Florida as 2.5, while the figure for Utah is “<0.1”, so the factor could actually be somewhat greater. I’m not sure exactly what their methodology is, but I would guess that it is based on a) the relative probability that a state’s vote is split precisely fifty-fifty (based on their models) and b) the proportion of their 20,000 simulations in which the winner of that state would determine the winner of the election.

*** In my defence, the electoral landscape has shifted slightly since the 2012 election, making Utah less rock solid than it was for the Republicans. From the fivethirtyeight data, it looks like the most impressive comparison in 2016 would be D.C. against New Hampshire.

**** Is it just me, or is “first-past-the-post” a painfully abstruse way to describe an extremely simple concept. Why not the “most-votes-wins-the-seat” system?

Thomas Oléron Evans, 2016

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