If you’re not British, or you live under a stone somewhere, then you may not have heard about one of the most extraordinary sporting stories ever. Leicester City, a football (in the British sense) team that last year only just escaped relegation from the top division, has just won the league. At the start of the season you could have bet on this happening at odds of 5000-1. Just 12 people availed themselves of this opportunity.
Ten pounds bet then would have net me 50000 pounds now, so a natural question arises: should I be kicking myself (the appropriate reaction given the sport) for not placing such a bet? In one sense the answer is obviously yes, as I’d have made a lot of money if I had. But I’m not in the habit of placing bets, and had no idea that these odds were being offered anyway, so I’m not too cut up about it.
Nevertheless, it’s still interesting to think about the question hypothetically: if I had been the betting type and had known about these odds, should I have gone for them? Or would regretting not doing so be as silly as regretting not choosing and betting on the particular set of numbers that just happened to win the national lottery last week?
Here’s a possible argument that the 5000-1 odds at the beginning of the season were about right, or at least not too low, and an attempted explanation of why hardly anybody bet on Leicester. If you’ve watched football for any length of time, you know that the league is dominated by the big clubs, with their vast resources to spend on top players and managers. Just occasionally a middle-ranking club has a surprisingly good season and ends up somewhere near the top. But a bottom-ranking club that hasn’t just been lavished with money doesn’t become a top club overnight, and since to win the league you have to do consistently well over an entire season, it just ain’t gonna happen that a club like Leicester will win.
And here are a few criticisms of the above argument.
1. The argument that we know how things work from following the game for years or even decades is convincing if all you want to prove is that it is very unlikely that a team like Leicester will win. But here we want to prove that the odds are not just low, but one-in-five-thousand low. What if the probability of it happening in any given season were 100 to 1? We haven’t had many more than 100 seasons ever, so we might well never have observed what we observed this season.
2. The argument that consistency is required over a whole season is a very strong one if the conclusion to be established is that a mediocre team will almost never win. Indeed, for a mediocre team to beat a very good team some significantly good luck is required. And the chances of that kind of luck happening enough times during a season for the team to win the league are given by the tail of a binomial distribution, so they are tiny.
However, in practice it is not at all true that results of different matches are independent. Once Leicester had won a few matches against far bigger and richer clubs, a simple Bayesian calculation would have shown that it was far more likely that Leicester had somehow become a much better team since last season than that it had won those matches by a series of independent flukes. I think the bookmakers probably made a big mistake by offering odds of 1000-1 three months into the season, at which point Leicester were top. Of course we all expected them to fall off, but were we 99.9% sure of that? Surely not. (I think if I’d known about those odds, I probably would have bet £20 or so. Oh well.)
3. Although it was very unlikely that Leicester would suddenly become far better, there were changes, such as a new manager and some unheralded new players who turned out to be incredibly good. How unlikely is it that a player who has caught someone’s eye will be much better than anybody expected? Pretty unlikely but not impossible, I’d have thought: it’s quite common for players to blossom when they move to a new club.
4. The fact that Leicester had a remarkable escape from relegation at the end of last season, winning seven of their last nine matches, was already fairly strong evidence that something had changed (see point 2 above). Had they accumulated their meagre points total in a more uniform manner, it would have reduced the odds of their winning this season.
The first criticism above is not itself beyond criticism, since we have more data to go on than just the English league. If nothing like the Leicester story had happened in any league anywhere in the world since the beginning of the game, then the evidence would be more convincing. But from what I’ve read in the papers the story isn’t completely unprecedented: that is, pretty big surprises do just occasionally happen. Though against that, the way that money has come into the game has made the big clubs more dominant in recent years, which would seem to reduce Leicester’s chances.
I’m not going to come to any firm conclusion here, but my instinct is that 5,000-1 was a very good bet to take at the beginning of the season, even without hindsight, and that 1000-1 three months later was an amazing chance. I’m ignoring here the well-known question of whether it is sensible to take unlikely bets just because your expected gain is positive. I’m just wondering whether the expected gain was positive. Your back-of-envelope calculations on the subject are welcome …