Me, last night on Bluesky:
All the local polling that’s been published, and the messaging from both Labour and the Greens, has suggested an incredibly tight result in Gorton and Denton, along the lines of 29%/28%/28% (with the Greens probably, but not at all definitely, in the lead). The betting market on G+D, on the other hand, still had the Greens at 2/1 on when it closed.
If you think of (say) 2/1 odds as an estimated 1/3 probability of winning, and then treat that as a probability of getting any given vote (which it isn’t), then the Greens should get the lion’s share of the Lab/RefUK/Green votes, and end up closer to 50% than 28%.It should, in other words, be Not Even Close. The result will either be an awful warning against taking political betting seriously or an awful warning against ignoring it.
Unless RefUK win, of course, in which case it’ll just be an awful warning.
Gorton and Denton, this morning:
Green Hannah Spencer 14,980 40.6% +27.4% Reform Matt Goodwin 10,578 28.7% +14.6% Labour Angeliki Stogia 9,364 25.4% −25.4% 5.3% the field (-16.6%)
Does this vindicate everyone who was citing the betting odds during the campaign, as against (e.g.) this guy?
Well, it does a bit – but with a couple of fairly major qualifications, and some actual maths. To which I now turn.
The mathematics of odds (easy part)
We think of probabilities in terms of fractions – a ‘one in three’ chance, etc. Percentages, although they sound more scientific, are fractions too – we’ve just all agreed to say ‘70%’ and ‘29%’ instead of ‘seven in ten’ and ’29 in a hundred’.
Odds are different. It’s not just that odds are quoted against something happening – ’10 to 1′ is a very bad bet, not a near-certainty. More importantly, when you give the odds against something you’re always quoting two probabilities: the likelihood of it happening and the likelihood of it not happening. “1 in 3” (probability) looks like a numerator and a denominator, because it is: it’s a one-third chance. 3/1 (odds) looks like a numerator and a denominator, but it isn’t: it’s one chance of the event happening and three chances of it not happening. Essentially what you see are two numerators with an implicit denominator, which is the sum of the two: 3/1 means that there’s one chance out of (3+1=4) that the event will happen and 3 chances out of (3+1=4) that it won’t.
The odds on the Greens winning Gorton and Denton, at close of play last night, were 2/1 on: 2/3 probability that they would win vs 1/3 probability that they wouldn’t; call it a 66% chance of winning. Reform were quoted at 5/2 (=2/7=28% chance) and Labour at 4/1(=1/5=20%).
The second and third of these, in particular, don’t look too far adrift from the actual voting percentages of 40%, 28% and 25%. If those percentages were also probabilities, they would equate to odds of 3/2, 5/2 and 3/1.
But there are qualifications.
The qualifications
The most obvious qualification is that the percentages of actual votes cast aren’t probabilities. This isn’t because of some ontological distinction between a competitive estimation of probabilities and the proportions in a set of outcomes, though. If the bookies had been taking bets on how a random individual punter in Gorton and Denton would vote, the odds would represent the bookies’ estimation of how the votes would fall (to see this, imagine scaling up from an x% probability of an individual voter voting for a given party to the probabilities for 100 of those voters, or a thousand, or the whole constituency).
But the bookies weren’t taking bets on a random individual vote, what with it being hard to decide whether to pay out on a bet like that; the bet was on who would win. To make the betting markets track actual preferences, we have to assume that it’s mostly people in the constituency who are betting, and that they’re mostly betting on the party they want to win. We have to assume that the market’s not being distorted deliberately, and that ‘smart’ betting by ordinary punters – people betting on the winner as a sure thing; people hedging their preference by betting on the runner-up – will largely get washed out in the noise. Generally these aren’t crazy assumptions, to be fair, but they do call for a bit of scepticism. (It could be argued that a high-profile election such as this is more likely to have people gaming the betting market, but this is speculative; apart from anything else, a high-profile election is also more likely to have ordinary punters putting a bet on.)
The other factor for scepticism is the obvious fact – frequently cited as if it completely demolished the case for paying any attention to betting odds – that the bookies want to make money. This is less of a knock-down argument than it at first appears. Suppose that the bookies’ finger-in-air estimate is that Labour are probably going to lose. Being motivated by the desire to make money (who knew), they will give Labour good odds, making a promise of a big payout that they think they won’t have to honour. Fiendish! This is all right as far as it goes, but that isn’t very far. Put it this way: how good will they make the odds on Labour? What if there’s another party standing which the bookies also think is going to lose – how good an offer will they make on that party? If it were just a matter of “A is going to win, let’s encourage lots of people to vote for B and C”, the odds on B and C would always be 100 to 1 against. But bookies can’t see the future any more than anyone else, so they set the odds by balancing the lure of a phantom payout on loser party B against the probability that B might actually win. Which is more or less where we came in. Galaxy-brain take: the bookies set the odds in order to make money, and that’s what makes the odds informative.
Where’s my ‘factor for scepticism’ gone? Here it is: the thing to keep in mind isn’t that bookies want to make money, it’s more specifically that they don’t want to lose money. What’s going to happen to the odds on a favourite? They certainly aren’t going to get any longer; the last thing the bookies will want to do is encourage more punters to bet on it. The lifespan of a betting market in a lot of sporting events is measured in minutes, but over the course of a long-running betting market like this one a favourite can get really solidly established; the bookies can collectively reach a point where there really isn’t any doubt which way the market’s going, suspending all judgment on whether the said market tracks the outcome. And, once you’re past that point, there’s no point offering to throw good money after bad. If a given outcome looks better than evens – more likely to happen than not – the odds cited can get arbitrarily short. The Greens finished the campaign trading at 1/2 on – 66% chance of winning – and those were long odds by the standards of the campaign; at one stage their odds were down at 10/3 on, 77% chance. But the shortness of those odds weren’t (just) an estimate of the likelihood of a Green win, let alone the likelihood of an individual punter voting Green; beyond a certain point, that really was the bookies wanting to make money, or rather wanting to avoid paying out more than they had to.
The mathematics of odds (slightly harder part)
Odds make an unexpected appearance when you’re studying statistics, specifically in the form of odds ratios. The idea of an odds ratio is that you sometimes want to compare the likelihood of two pairs of outcomes; in clinical trials, for example, one way to compare a treatment group and a control group is to compare the odds of an infection (say) occurring, as against it not occurring, in the treatment group with the same pair of odds in the control group. The comparison is straightforward, and involves treating the sets of odds as if they were fractions; you divide one by the other, which you can do by multiplying the first number in the first pair by the second in the second pair and vice versa. For example, the ratio of odds of 3/1 against to odds of 4/1 against is (3×1/4×1) = 0.75.
As I mentioned earlier, the actual vote distribution can be translated into odds, if you squint and make a mental apology to your Stats lecturer. And, as I also mentioned earlier, the vote was expected to be a lot closer – and with more scattering to minor parties such as Rejoin EU and the Conservative Party – with both fag-packet estimates and one actual poll converging on the region of 30%/28%/28% as between the Greens, Labour and RefUK.
The analysis I’m about to embark on hits a snag straight away in the shape of Reform UK. Not only were the betting odds on RefUK – 5/2 – almost identical to the 28% in the polling and the ‘too close to call’ estimates throughout the campaign; they were both almost identical to the vote share their unlovely candidate actually got. The bookies were dead on, but so – in the solitary case of Reform UK – were the parties. So the odds ratios comparing RefUK’s betting form with the result, and comparing the polling with the same result, are both 1: an exact match.
But let’s press on anyway.
| Odds | Est. | = | Actual | = | OR (odds) | OR (est) | |
| Greens | 1/2 | 30% | 7/3 | 40% | 3/2 | 3 | 0.64 |
| RefUK | 5/2 | 28% | 5/2 | 28% | 5/2 | 1 | 1 |
| Labour | 4/1 | 28% | 5/2 | 25% | 3/1 | 0.75 | 1.2 |
Left to right, you’ve got: the final odds against that party winning; the polling estimate, followed by the odds to which this figure is equivalent; the actual vote share, followed by the odds to which that figure is equivalent; the ratio of odds to outcome; and the ratio of the polling estimate to the outcome.
We can see that the polling underestimated the Greens substantially. The betting odds wildly overestimated them, for reasons touched on earlier, but the bookies did at least put them substantially out in front. Since Reform UK rather unsportingly got almost exactly the same vote share everyone expected, we’ll pass over them (although not without pointing out that the bookies didn’t get them wrong). Lastly, the polling slightly overestimated Labour, while the bookies slightly overestimated them.
With the caveat about what happens to the odds when a favourite is locked in, I’d say that the betting odds ended up being a reasonably good predictor of the result in Gorton and Denton. The bookies’ implicit forecast wasn’t dramatically worse than the result predicted by polling, and it did have the additional merit of suggesting that it wasn’t, in fact, going to be a three-way scrap for first place. I can’t conclude this post better than by quoting myself again:
All the local polling that’s been published, and the messaging from both Labour and the Greens, has suggested an incredibly tight result in Gorton and Denton, along the lines of 29%/28%/28% (with the Greens probably, but not at all definitely, in the lead).
If the bookies were right, on the other hand,
the Greens should … end up closer to 50% than 28%.It should, in other words, be Not Even Close. The result will either be an awful warning against taking political betting seriously or an awful warning against ignoring it.
(Although I will admit that 40% isn’t much closer to 50% than 28%!)
Workers’ Playtime 