How much of Leicester City's championship was luck?
How much of Leicester City's run to the Premier League championship was just luck? I was curious to get a better gut feel for how random it might have been, so I wrote a simulation.
Specifics of the simulation are in small font below. The most important shortcoming, I think, was that I kept teams symmetrical, instead of creating a few high-spending "superteams" like actually exist in the Premier League (Chelsea, Manchester United, Arsenal, etc.). Maybe I'll revisit that in a future post, but I'll just go with it as is for now.
Details: For each simulated season, I created 20 random teams. I started each of them with a goal-scoring and goal-allowing talent of 1.35 goals per game (about what the observed figure was for 2015-16). Then, I gave each a random offensive and defensive talent adjustment, each with mean zero and SD of about 0.42 goals per game. For each season, I corrected the adjustments to sum to zero overall. I played each game of the season assuming the two teams' adjustments were additive, and used Poisson random variables for goals for and against. I didn't adjust for home field advantage.
At the beginning of the season, Leicester City was a 5000:1 longshot. What kind of team, in my simulation, actually showed true odds of 5000:1? We can narrow it down to teams with a goal differential (GD) talent of -4 to -9 for the season. In 500,000 random seasons, here's how many times those teams won:
tal #tms ch odds
-9 166135 20 8307
-8 168954 25 6758
-7 171272 26 6587
-6 173327 22 7879
-5 175017 53 3302
-4 177305 61 2907
1032010 207 4986
In 500,000 seasons of the simulation, 1,032,010 teams had a GD talent between -3 and -9. Only 207 of them won a championship, for odds of 4,985:1 against, which is close to the 5000:1 we're looking for.
Even half a million simulated seasons isn't enough for randomness to even out, which is why the odds don't decrease smoothly as the teams get better. I'll maybe just go with a point estimate of -8. In other words, for Leicester City to be a 5000:1 shot to win the league, their talent would have to be such that you'd expect them to be outscored by 8 goals over the course of the 38-game season. Maybe it might be 7 goals instead of 8, but probably not 6 and probably not 9. (I guess I could run the simulation again to be more sure.)
Leicester City actually wound up outscoring their opponents by 32 goals last year. Could that be luck? What's the chance that a team that should be -8 would actually wind up at +32? That's a 40 goal difference -- Leicester City would have had to be lucky by more than a goal a game.
The SD of goal differential is pretty easy to figure out, if you assume goals are Poisson. Last season, the average game had 1.35 goals for each team. In a Poisson distribution, the variance equals the mean, so, for a single game, the variance of goal differential is 2.70. For the season, multiply that by 38 to get 102.6. For the SD, take the square root of that, which is about 10.1. Let's just call it 10.
So, a 40-goal surprise is about four SDs from zero. Roughly speaking, that's about a 1 in 30,000 shot.
If we were surprised that Leicester City won the championship, we should be even *more* surprised that they went +32. In fact, we should be around six times more surprised!
Why are the "+32" odds so much worse than the "championship" odds? Because, on those rare occasions when a simulated -8 team wins the championship, it usually does it with much less than a +32 performance. Maybe it goes +20 but gets "Pythagorean luck" and winds lots of close games. Maybe it goes +17 but the other teams have bad luck and it squeaks in.
If you assume that a team that actually scores +32 in a season has, say, a 3-in-10 chance of winning the championship, then the odds of both things happening -- a -8 talent team going +32 observed, and that being enough to win -- is 1 in 100,000. Well, maybe a bit less, because the two events aren't completely independent.
The oddsmakers have priced Leicester City at around 25:1 for next season. That's a decent first guess for what they should have been this year.
Except ... in retrospect, Leicester should probably have been even better than 25:1 this season (you'd expect them to decline in talent next year -- they have an older-than-average team, they may lose players in the off-season, and other teams should catch on to their strategy). On the other hand, MGL says oddsmakers overcompensate for unexpected random events that don't look random.
Those two things kind of cancel out. But, commenter Eduardo Sauceda points out that bookmakers build a substantial profit margin into a 20-way bet, so let's lower last season's "true" odds to 35:1, as an estimate.
According to the simulation, for a team to legitimately be a 35:1 shot, its expected goal differential for the season would have to be around +16.
Taking all this at face value, we'd have to conclude:
1. The bookies and public thought Leicester City was a -8 talent, when, in reality, it was a +16 talent. So, they underestimated the club by 24 goals.
2. Leicester City outperformed their +16 talent by 16 goals.
3. And, while I'm here ... the simulation says a team with a +32 GD averages 74 points in the standings. Leicester wound up at 81 points. So, maybe they were +7 points in Pythagorean luck.
One thing you notice, from all this, is how difficult it is to set good odds on longshots, when you can't estimate true talent well enough.
Suppose you analyze a team as best you can, and you conclude that they should be a league average team, based on everything you know about their players and manager. (I'm going to call them a ".500 team," which ignores, for now, the Premier League scoring asymmetry of three points for a win and one point for a draw.)
You run a simulation, and you find that a .500 team wins the championship once every 770 simulated seasons. If the simulation is perfect, can you just go and set odds of 769:1, plus vigorish?
Not really. Because you haven't accounted for the fact that you might be wrong that Everton is a .500 team. Maybe they're a .450 team, or a .600 team, and you just didn't see it.
But, isn't there a symmetry there? You may be wrong that they're exactly average in talent, but if your analysts' estimate is unbiased, aren't they just as likely to be -8 as they are to be +8? So, doesn't it all even out?
No, it doesn't. Because even if the error in estimating talent is symmetrical, the resulting error in odds is not.
By the simulation, a team with .500 talent is about 1 in 940 to win the championship. But, what if half the time you incorrectly estimate them at -8, and half the time you incorrectly estimate them at +8?
By my simulation, a team with -8 GD talent is 1 in 6,758 to win. A team with +8 talent is 1 in 157. The average of those two is not 1 in 940, but, rather, 1 in 307.
If you're that wildly inaccurate in your talent evaluations, you're going to be offering 939:1 odds on what is really only a 307:1 longshot. Even if you're conservative, going, say, 600:1 instead of 939:1, you're still going to get burned.
This doesn't happen as much with favorites. In my simulation, a +30 team was 1 in 5.4. The average of a +22 team and a +38 team is 1 in 4.7. Not as big a difference. Sure, it's probably still enough difference to cost the bookmakers money, but I bet the market in favorites is competitive enough that they've probably figured out other methods to correct for this and get the odds right.
Anyway, the example I used had the bookies being off by exactly 8 goals. Is that reasonable? I have no idea what the SD of "talent error" is for bookmakers (or bettors' consensus). Could it be as high as 8 goals?
For the record, the calculation of SD(talent) for 2014-15 (the season before Leicester's win), using the "Tango method," goes like this:
SD(observed) = 22.3 goals
SD(luck) = 10 goals
SD(talent) = 19.9 goals
For a few other seasons I checked:
2015-16 SD(talent) = 19.9
2013-14 SD(talent) = 27.8
1998-99 SD(talent) = 18.3
In MLB, the SD of talent is about 9 wins. How well, on average, could you evaluate a baseball team's talent for the coming season? Maybe, within 3 wins, on average? That's a third of an SD.
In the Premier League, a third of an SD is about 6 goals. But evaluation is harder in soccer than in baseball, because there are strategic considerations, and team interactions make individual talent harder to separate out. So, let's up it to 9 goals. Offsetting that, the public consensus for talent -- as judged by market prices of players -- reduces uncertainty a bit. So, let's arbitrarily bring it back down to 8 goals.
That means ... well, two SDs is 16 goals. That means that in an average year, the public overrates or underrates one team's talent by 16 goals. That seems high -- 16 goals is about 10 points in the standings. But, remember -- that's just talent! If luck (with an SD of 10 goals) goes the opposite direction from the bad talent estimate, you could occasionally see teams vary from their preseason forecast by as many as 36 or 46 goals.
Does that happen? What's the right number? Anyone have an idea? At the very least, we now know it's possible once in a while, to be off by a lot. In this case, it looked like everyone underestimated the Foxes by (maybe) 24 goals in talent.
In light of all this, bookmaker William Hill announced that, next year, they will not be offering any odds longer than 1,000:1. When I first read that, I thought, what's the point? If they had offered a thousand to one on Leicester City, they still would have lost a lot of money, if the true odds were 35:1.
But ... now I get it. Maybe they're saying something like this: "A Premier League team with middle-of-the-road talent -- one that you'd expect to score about as many goals as it allows -- has about a 1 in 1,000 chance of winning the championship. We're not confident enough that we can say, of any bad team, that they can't change their style of play to become average, or that they haven't improved to average over the off-season, or that they've been a .500 team all along but we've just been fooled by randomness. So, we're never again going to set odds based on an evaluation that a team's talent is significantly worse than average, because the cost of a mistake is just too high."
That makes a certain kind of sense. And the logic makes me wonder: were the odds on extreme longshots always strongly biased in bettors' favor, but nobody realized it until now?
(My previous post on the Leicester City is here.)