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It's March, and I've gone mad

I don't care much for March Madness.  But since Carl Bialik, the Wall Street Journal's "Numbers Guy," chose to write about it today, I thought I'd forget about baseball for a second and check it out.  Bialik reports that all sorts of companies--from sportsbooks to pizzerias--are offering enormous prizes if you correctly predict the outcome of the NCAA men's college basketball tournament bracket.  No, not pick the winner; to win, you've got to nail all 63 games.

Bialik is quick to point out that nobody's about to win a $10 million prize:

A look at the odds of winning shows why companies are willing to risk such valuable loot. Filling out a perfect bracket means predicting the outcome of 63 games. If each game were a true toss-up, that would mean your chance of perfection is a mere one in two to the 63rd power, or one in nine million trillion (yes, million trillion -- there are no tidy terms for numbers this large). Put another way, you are about 60 billion times more likely to win the multistate Powerball lottery.

For reference (this will come in handy later on), one in two to the 63rd power (2^63) is equal to 1.08 * 10^-19, or .000000000000000000108.  Approximately.  Some of the article is great reportage on those unreachable prizes.  The rest is analysis of just how likely success really is.  Of course, every game isn't a toss-up--not in baseball, and not in NCAA hoops.  

In fact, many NCAA basketball games are much closer to "sure things" than any MLB game will ever be.  Bialik points out that "A No. 1 seed has never lost a first-round game to a No. 16 seed, though several came close this year; and No. 2 seeds rarely lose to No. 15 seeds -- just four times in 88 games."  In other words, just about everybody will get those games right, and if you do, you substantially increase your odds of reaching perfection.

But how close can you get?  A series of math professors chime in with variations of the above theme, using seedings to predict victories, or theorizing that a really skillful picker could be right 70% or 75% of the time.  Even taking the most optimistic of those projections, your odds of filling out a perfect bracket are 1 in 150 million: about 7 * 10^-9, or .000000007.  Not very good.

Still interested?  Good.  I'm no math professor, but we're going to try applying this to baseball.  

I'm going to give away the ending and say this: if you're in a baseball-related industry want to make big headlines, feel free to offer a $1 billion cash prize for anyone who can correctly predict the outcome of their favorite team's games for the season.  No one's going to come very close.

However, let's say you wanted to try.  If, like in the NCAA hoops contests, you had to fill out your MLB season "bracket" before Opening Day, you'd be in quite a bind.  You wouldn't know who's pitching for each team each day, you wouldn't know who'll be injured and when, you couldn't predict mid-season acquisitions, and depending on how early the bracket had to be submitted, you might not even know the exact construction of your team's roster.  

You might take a different approach, but my instinct would be to start with each team's pythagorean records from the previous year.  Then, using those records, come up with the likelihood of a team's victory by using Bill James's log5 method.  For example, let's say my Milwaukee Brewers had a pythagorean record of .650 last year, and they're going to play the lowly Chicago Cubs, who had a pythag of .380.  The log5 method suggests that the Brewers would win about 3/4 of the time.  Sounds about right, I think.

Now, you wouldn't want to just pick the better team every time.  When predicting a college hoops tournament, that may be the best approach, but if you were predicting the outcome of a season's worth of games for the best team in the league, that method would lead you to predict a record of 162-0.  That may give you the best chance of being right for any single game, but there's no chance you'd win.  So, how do you predict losses?

Again, you could do this in a number of ways, but here's mine: for each game, figure out the probability your team will win, as described above.  Then, use a random number generator (RNG) to come up with a number (or a sequence of numbers) between 0 and 1.  In the game above, the probability of the Brewers winning was 0.75.  If the RNG is providing truly random numbers, you'll get a number under .75 about 3/4 of time, and above .75 the rest.  So, if the RNG spits out 0.67 for a Cubs-Brewers matchup, go with the Brewers.  If it returns 0.87, defy common sense and pencil in the Cubs.

No sane person would do this for very long by hand.  I didn't even start by hand, but that's only because I don't think anymore--I let the Python programming language think for me.  Using the 2004 game log provided by Retrosheet.org, I could run some tests.  I used 2003 Pythagorean winning percentages and the method I've laid out so far, and determined just how likely it is to perfectly predict the outcome of a season's worth of games.

For the 2004 Milwaukee Brewers (only 161 games, so we might have a chance), the probability of prediction perfection is 2.78 * 10^-48.  Since I like zeros, I might as well spell it out: .000000000000000000000000000000000000000000000000000000000278.  Yeah, baby.  If you want a reference point, earth consists of 10^50 atoms.  So you'd have to fill out a lot of brackets to get it right.  The Brewers are no anomaly: I thought that much better teams might be substantially easier to predict, but the Cardinals come in at 1.0 * 10^-47, and the Yankees at 6.25 * 10^-48.  

Surely we can do better than that, no?  Well, um, maybe.  Let's suspend reality a bit and say that we had the magical power to see the season before it happens.   Well--not every game, but let's say, rather than basing our predictions off 2003 pythags, we go with 2004 pythags.  Thus, we take into account the changes that teams make from year to year.  In truth, a savvy fan probably could make some of those predictions, so this example is plausible.  Maybe it'll simulate what a savvy fan might do to have a shot at predicting a season's worth of games.

In a word, no.  It won't.  It improves your chances at nailing the Brewers 2004 season by about 25%.  Now--congratulations!--your odds are 3.97 * 10^-48.  The Yankees season prediction improves by a factor of 10, to 5.66 * 10^-47, and the Cardinals projection improves by nearly a factor of 100, to 9.28 * 10^-46.

I've got one more method up my sleeve.  Let's say you got to make your picks like the touts in Vegas do, announcing them the day of, or day before games throughout the season.  This way, you'd know who's starting for each team, who's hurting, who's hot, and who, tragically, has been exiled to Kansas City.  Armed with all this knowledge, you'll be able to make much smarter picks.  Right?

The best (somewhat simple) way to simulate a method like that is to use the official Vegas moneyline.  Moneylines are basically representations of the expected probability of a team winning.  If a team is favored at -200, that means you have to bet $200 to win $100--a smart bet if you think that team's chance of winning is greater than 0.67.  Any moneyline can be converted into a winning percentage, so we have a record of what pretty smart people thought would happen in several seasons' worth of past games.  So, instead of using pythagorean records and the log5 method, we'll use the winning percentages suggested by Vegas.

With giant computers and great minds working on it, and with millions of dollars at stake, do our odds of predicting perfectly increase?  No.  (You saw that one coming, didn't you?)  In our saga to predict the vicissitudes of the 2004 Brewers season, we can now increase our odds by a little more than double, to about 1.0 * 10^-47.  Our success rate with the Yankees approximately doubles as well, but--hmm, did someone say giant computers and great minds?--the odds for the Cardinals only increased about 50% from the first method, when we used 2003 pythags.  That might be an anomaly, but it also reflects the randomness inherent in baseball.

I wish I could wrap this up with an important conclusion that logically follows from all this, like that American League teams should draft more college first basemen, or that Billy Beane has a track record of overvaluing Mexican pitching.  But it's really an elaborate way of looking at just how much chance plays a role over the course of a long season.  Remember that the math professors improved their chances at filling out a perfect NCAA bracket by finding "locks?"  In a MLB season, there are no locks.  

Using the log5 method, it's very rare to find a probable winner above 70% or below 30%.  Vegas moneylines almost never cross a magic line in the vicinity of +/- 300 (~75% implied probability), and they only get that far when the Devil Rays visit Yankee Stadium and think Seth McClung is their answer to Randy Johnson.

Since I could make my computer do all the work, I wanted to cap off the project with a giant experiment: what are the odds of correctly predicting the results of an entire season's worth of games, for all of Major League Baseball?  I ran the numbers for all three of the methods discussed above.  '03 pythags were relatively useless, with odds over a million times steeper than either of the other two.  Surprisingly, '04 pythags were a tiny bit better than the betting-line method.

To send you off, I thought you'd enjoy seeing the exact probability of successfully predicting an entire MLB season's worth of games with the best method I've yet devised.  Here you go: .000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
0000001076.  

Or, as my simulator responded the first time I asked it to do this: 0.

Update [2006-3-23 16:27:3 by Jeff]: One thing about Vegas lines that I want to clarify. If you've ever bet a moneyline, you know that the favorite and underdog lines aren't the same -- the Brewers wouldn't be -200 and the Cubs +200. Because bookies plan on making some money off you, there's the "vig": in this example, it'd be something like -200 and +180. IOW, you put down $200 to make $100 on the Crew (predicting a 66.7% chance of winning); you put down $100 to make $180 on the Cubs. (predicting 35.7% chance).

Thus, the actual prediction Vegas is making is the mean of those two numbers: in this case -190/+190: 65.5% / 34.5%. That's the number I used for these simulations.

One more caveat about Vegas moneylines: it's only a convenient fiction to say that they are purely a prediction of the likelihood of each team winning. They are not. That prediction is a very large component, but they also take into account the perceived conventional wisdom (what will the betting public think about the game?) and what the betting patterns will be on the game (how are people likely to put money on it?).

Despite all that, Vegas lines are probably the best fast-and-dirty way to figure out the likelihood of a team winning a specific game. To do much better, you'd have to take into account the starting pitchers, the starting lineups, the home park, maybe the home plate umpire, and regress some unknown amount. It'd still be hard to consistently beat the Vegas line by much.