Let’s say an NFL team starts the season 3-1. You have to bet on their overall record this season (16 games) without knowing any more information than this. Where do you put your money?
It’s easy to say 12-4. Extrapolating three wins and a loss over a 16-game season gives us 12 wins, four losses.
But if you said 12 wins, you’re making a big mistake that will cost you a lot of money in the long run. While plenty of teams start out 3-1, only a handful each year win 12 games or more.
If the team had been 4-0 to start the season, you probably wouldn’t have predicted them to go 16-0. But with anything less extreme than that, the temptation is to make the big mistake of extrapolating their record exactly into the future.
Now try this one
Every year, some baseball player starts out on a ridiculous tear for the first two weeks of the season, and we joke about how he’s on pace to hit 120 homeruns.
But if you had to make a futures bet on how many homeruns he would hit that year, what would be your best guess at the actual number?
Nobody in their right mind would say 120. Anything over 60 or 70 would be crazy. You’d probably expect him to have a very good season, but not a historical one. And you’d probably be right.
Reversion to the mean
The force at work here is known as reversion to the mean. It’s a property of independent random events which dictates that after the observation of an “extreme event,” the next event is more likely to be closer to the mean than farther away from it.
We want to make a guess at a football team’s true ability to win games, in terms of winning percentage. We’ll never know exactly what that number is; randomness in the outcomes of their games limits us to making only an educated guess at it.
To better understand reversion to the mean, let’s first look at a case where randomness is the only factor.
Coin flippers—an extreme example
Suppose we have 10,000 people flipping coins together. All flip at the same time. If someone flips heads, he advances to the next round. If someone flips tails, he’s eliminated from the game. What happens?
After the first round, about 5,000 people have flipped heads, so they’re still in the game. After the next round, only about 2,500 are left.
Fast forward ten rounds or so, and we’re left with just a couple lucky flippers who have gotten heads every single time.
If you want, you can call them expert coin flippers. But if we were to play again, would you expect those same guys to make it to end again?
Of course not. Their apparent ability to flip coins was an illusion, entirely due to luck. We’d give them the same chances as anyone else in the next game.
In other words, it’s far more likely that they’ll perform around the average next time than it is that they’ll do even better than their exceptional performance the first time. Because their performance was based on luck alone, they will revert to the mean.
[As a commenter pointed out, this coin flipping example is often used in Fooled by Randomness to demonstrate survivorship bias.]
Back to football
When a team starts out 3-1, we’re tempted to estimate their ability to win at 75%. But is it really that high?
We don’t know for sure. Part of what we’ve observed in their 3-1 record is due to skill, part is due to randomness. It’s impossible for us to say how big a role each factor played.
It’s possible that they’re only a 60%-winning team in the long run, but that luck has been on their side so far. By the same token, it’s possible they’re a great team, and that their only loss was due to a few bad bounces.
However (and this is a big however), we must suspect this team of being lucky. If you’ll admit that luck plays at least a small part in the outcome of a football game—and how can you not—then we have to view the teams that start off hot with some of the suspicion with which we viewed the lucky coin flippers.
In short: The fact that they’ve been successful makes it more likely that luck was on their side than that it was against them. Once you accept that, then the only logical conclusion about their future performance is that they’re more likely to perform worse than they are to perform better.
In a given NFL season, you might see six, seven, or eight teams start out 3-1. A few will finish the season at 12-4. A few (maybe) will finish even better than that, but most will finish worse. And that’s reversion to the mean at work.
(For the more mathematically inclined, here’s a more precise explanation of reversion to the mean. Warning: integrals!)
How you can use reversion to the mean
There was nothing specific to winning teams about the above discussion. Reversion to the mean also happens with bad teams. A team that starts off 1-3 or 0-4 has likely been on the receiving end of more bad luck than good, and we can expect them to improve in the future.
So if bad teams aren’t as bad as they look, and good teams aren’t as good as they look, the message is simple: take the underdog early in the season. Good handicappers won’t be fooled by the early appearance of good and bad teams, but most of the public might. And large numbers of people certainly have the ability to influence the line and give too many points to the underdog.
Similarly, when making futures bets on end-0f-season wins, homeruns, touchdowns, or whatever, remember that most things will end up less extreme than they start. The lines will already account for this to some extent, so they’re very temping bets if you don’t remember that teams and players will, in general, revert to the mean over time.
As shown by the example of the homerun hitter in baseball or the team that starts off undefeated, we tend to intuitively understand reversion to the mean in the most extreme cases. We don’t predict that anyone will hit 120 homeruns or that many teams will go undefeated, for example. For this reason, I suspect that there’s less value in betting on mean reversion in extreme cases, since everyone already has a good idea that it’s hard to continue such extreme performance.
Want more?
I’ll leave you with a question.
Last post I wrote about the Gambler’s Fallacy, where we established why a roulette wheel that has landed on black several times is in no way “due” to hit red because of the streak of blacks. How does this reversion to the mean not contradict this?
Be the first to answer correctly and you win…the envy of the dozens (yes, DOZENS) of Thinking Bettor readers everywhere. Come on, what more could you ask for?
The reversion to mean doesn’t contradict the Gambler’s fallacy because in both cases, the past results do not premeditate the future results.
In the same way that each and every future spin on a roulette wheel has an equal chance of landing black or red – even if the past 5 spins have been black – the reversion to mean also somewhat discounts the past results and their impact on the future. If a team wins 5 games in a row, it is possible that it will win 16 in a row in the same way that it is possible for a roulette wheel to spin black 15 times in a row. But it is also very likely that it will not.
Sure, in the sports example, it is not a clear cut 50/50 chance like flipping a coin or spinning a roulette wheel. There are other factors such as talent, health, and strategy. But overall, no team in a competitive league with any major sense of parity has been able to win 100 games in a row.
So, in a sense, the Gambler’s fallacy not only fails to contradict the reversion to mean, but it actually supports it. You cannot say with certainty that the team which just won 5 games in a row will lose it’s next game based on its unlikely past 5 games, nor can you say with certainty that it will win it’s next game based on it’s hot streak.
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For the record, if you were going to bet on the Houston Texans record after a 3-1 start, you should look at the records of every team it has played and those it will be playing in the future to determine where it will finish.
Assuming the Texans lost to a 3-1 team, beat two 2-2 teams, and beat a 1-3 team with their future games lining up against one 4-0 team, three 3-1 teams, four 2-2 teams, three 1-3 teams, and one 0-4 team… I would say:
-They will go 5-3 or 4-4 against the 4-0 (L), 3-1 (W,L,?) & 2-2 (W,W,W,L) teams.
-They will go 3-1 against the 1-3 and 0-4 teams.
Without doing the exact calculations, I would put my money at 11-5 or 10-6 based on this type of schedule. If you held me to one number I’d have to analyze the scores and situations more closely.
Chad Kettner recently posted..Blogosphere Roundup
Chad, thanks for commenting. And your answer is good. especially the part that says “the Gambler’s Fallacy [really, the fact that the events are independent]…actually supports it.” That’s true, because the reason for mean reversion is simply that events near the mean are more common than events far from the mean. So reversion to the mean actually follows from the statement “events are independent and extreme events are rare.”
As you point out, it’s much more clear cut when we’re talking about a random variable with known distribution like the spin of a roulette wheel, rather than trying to estimate unknown parameters like team abilities. And a lot of the added complexity, is left out of my post, even from a strictly theoretical standpoint where we aren’t worried about strength of schedule, injuries, and all that extra real-world stuff.
Your example about the Houston Texans is actually the very problem that motivated me to create a program for betting on NFL games. When you look at the schedule they’ve played and their upcoming games, you run into a problem: Maybe they lost to a 3-1 team, but how do we know how good THAT team is? We need to evaluate who they played, just like we’re doing for the Texans. But what if they’ve played someone who the Texans played? How do we then evaluate that team without knowing about the Texans, the original team we’re trying to evaluate? Anyway the program I wrote solves this problem in an interesting way. One day I’ll dust it off and post about it or post its predictions here.
Haha – awesome. Have you ever checked out ESPNs AccuScore Simulator? It’s interesting – because they have all the statistics in the world available, yet it still falls short much of the time.
Which is why I love sports… it is so unpredictable.
Chad Kettner recently posted..Blogosphere Roundup
Oh… and I’d be really interested to hear about your program.
Chad Kettner recently posted..Blogosphere Roundup
Here’s what I did when I used to gamble on football regularly…
I spent 5 months working for Electronic Arts to make the Madden NFL 2003 football game. In the process, I took an existing body of knowledge on football and increased it with more stats than anybody ever needs to know about the game.
Then, I crushed both my fantasy leagues and 3 pick the winner leagues (1 with a spread, 2 straight up wins).
A few years later, I got married, my wife went to law school, and I stopped getting cable and couldn’t watch the games unless I traveled.
(Now that over the air has switched to digital and I got a new TV, I can at least watch the CBS games and the odd NBC or ABC game. Used to only be able to watch that odd NBC or ABC game.)
My football knowledge has declined to the point that I no longer play in the fantasy leagues since I can’t watch enough games in a season to really enjoy it. And if you aren’t having fun, why gamble?
Blaine, what did you do for Electronic Arts? That seems like it’d be fun. Did you get the job just so you’d get all that info for gambling?
I always wondered how EA would do with betting football if they just simulated games between two CPU players. And then I think they actually started doing that for ESPN. Not sure if anyone kept long-term stats on how they did, but that would be interesting.
I was the database manager for the quality assurance team.
Worst job ever…I’d go back to working at a gas station before I went back there.
If you remember the letter from a developer’s wife, which was an open to letter to EA on the working conditions of the company that appeared while I was working there, it was pretty accurate.
A few months of full time work…and then a few months of 16-20 hour days 6-7 days per week. I’m not a big fan of leaving work at 2 or 3 in the morning and coming back at 8 or 9.
At least traffic wasn’t too bad on my way home.
Your coin flipping example reminded me of something called ‘Survivorship Bias’ that Nassim Taleb explains in his book, “Fooled by Randomness”, where he used a similar example to explain how financial stock-pickers with extraordinarily great records of success may be nothing more than lucky that can be explained as much by randomness as skill. Interesting to see the principal applied elsewhere! Great job with your blog!
Tim/Dad, survivorship bias (or at least, the idea that companies and fund managers who appear skilled might be just lucky) and mean reversion are actually very closely tied in financial markets. There are lots of mean-reversion trading strategies, where you sell stocks or funds that have performed well, and buy those that have performed poorly, on the hunch that they are over/under valued because some degree of their extreme results was luck.
But I think the term survivorship bias really applies when the losers leave the market, and for that reason we neglect to examine their results. So it looks like everyone who plays wins. That I don’t think has anything to do with mean reversion.
No argument there. But to continue the analogy… if someone flips tails, he’s eliminated from the game just like an ordinary stock-picker that doesnt get it right often enough to appear credible. Only the “expert” coin flippers (the survivors) remain in the game. The losers are not likely to be evaluated with all those surviving expert coin flippers around “selling their expertise”.