October 6, 2005

Crooked Numbers

Crappy Odds

by James Click

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It's a crapshoot.

That's the easy way to explain the playoffs, but more so since the advent of divisional play and Billy Beane's now famous line, "My [means of building a baseball team by continually taking advantage of tiny inefficiencies in the market] doesn't work in the playoffs." Saying that the playoffs are a crapshoot is the easy way of saying that small sample sizes render the minute differences between playoffs teams largely moot. But largely does not equal entirely.

Take the Cardinals and Padres series. Assuming that each team's true probability of winning a baseball game is their winning percentage for the season, that gives the Cardinals a .617 percentage and the Padres a .506 chance. Matching those two teams, we can estimate the Cardinals' chances of winning the game by a shorthand version of the log5 method as .500 + .617 - .506 = .611. Conversely, the Padres' odds of winning a single game between the two teams are .389. This doesn't mean that the Cardinals have a 61% chance of winning the series for several reasons.

The primary reason is that it's not a one-game playoff, but rather a five game series. To estimate the chance of the Cardinals winning the series, we can use a method similar to Pascal's Triangle or Khayyam-Pascal's Triangle or n choose k. However, because the series ends as soon as one team wins three games, we'll use a slightly different arrangement. Assuming that the Cardinals have the same odds of winning each game, the odds of them sweeping the Padres before the series begins is (.611)^3 = 22.8%. Because there are three ways for them to win three of four (win-win-loss-win, win-loss-win-win, loss-win-win-win), the odds that they will win three of four is 3 * (.611)^3 * (1-.611) = 26.6%. The odds of winning in five games is similar: 6 * (.611)^3 * (1-.611)^2 = 20.7%. Adding those three numbers up gives the Cardinals a 70.1% chance of winning the series. While their chances of winning a five-game series are higher than their chances of winning a single game over the Padres, the Redbirds will still lose nearly three of every ten times they play a five games series.

The second reason the Cardinals likely don't have a 70% chance of winning each game is because of home field advantage. Home teams win about 54% of their games over the course of the season, so that gives the Cardinals a .660 winning percentage at home and .562 on the road. The calculation to determine the Cardinals' odds of winning the series gets rather complicated now because there are both home winning percentages and road winning percentages, but in the end, St. Louis comes out with a 71.8% chance to take the series, compared to their 70.1% chance when ignoring home field advantage. Intuitively, this seems about right because the Cards only get a single extra game at home. Assuming that additional 1.7% chance to win the series is similar in other series--we'll get to those in a minute--all the hand-wringing going on in the AL over Buck Showalter pulling his starters against the Angels in the last game of the season seems a little silly. That's not to say that a 1.7% chance is worthless, but only once in every 59 series will that make the difference.

Finally, there is one other factor that can alter a team's winning percentage: the starting pitcher. In particular, teams typically only start their top three or four starters in the post-season, so instead of using a team's actual winning percentage as their odds of winning each game, we can use their winning percentage when their top three or four starters take the mound. Here's how teams have done behind their top three starting pitchers as measured by SNLVAR:

TEAM    W    L    W%   Full W%   Diff
SLN    66   31   .680   .617     .063
ATL    54   28   .659   .556     .103
NYA    48   28   .632   .586     .046
ANA    59   36   .621   .586     .035
BOS    59   36   .621   .586     .035
CHA    60   38   .612   .611     .001
HOU    57   43   .570   .549     .021
SDN    47   38   .553   .506     .047

Amazingly, the Astros top three starters, who are the third best Big Three since 1972 by SNLVAR, only increase their team's winning percentage by 21 points. Given the rather poor W-L record of Roger Clemens and, to a smaller extent, Andy Pettitte, that low percentage isn't terribly surprising, but the Astros' inability to turn over 675 innings of 2.43 ERA into more than a .570 winning percentage is astounding. Certainly there are some problems with assuming that the winning percentages above are the true winning percentages of the playoff teams--most notably, some teams are using a four man rotation--but they give us a decent idea of which teams benefit from dropping the back of their rotation.

Plugging those winning percentages into the equations (and of course keeping the home field advantage adjustments), the Cardinals now come out as a 74.6% favorite to win the series. Obviously, one Jake Peavy start and two Peavy ribs later this isn't still the case, but going into the series, even considering their regular season record, home field advantage, and an increased advantage based on a shorter rotation, the Cardinals still only come out as a 74.6% favorite to win a five game series.

As for the other matchups, the Braves come out as a startlingly high 68.1% favorite over the Astros, due mostly to Houston's low winning percentage when the Big Three pitches. It's hard to see that kind of trend keeping up, but the Astros have shown an impressive ability to squander those games all season long. In the AL, the Yankees come out as a 50.6% favorite and the Red Sox enjoy a healthy 50.2% chance to win their series.

If those numbers are a little too close for comfort, consider just how little things would change in a seven game series. Here are the four playoff matchups and the odds of the "favorite" winning a five game series versus a seven game series:

Matchup  5-game Series  7-game Series
SLN-SDN      74.6%          77.5%
ATL-HOU      68.1%          70.3%
NYA-ANA      50.6%          51.2%
BOS-CHA      50.2%          50.7%

Expanding the division series to seven games does little to increase the odds of the true favorites, clearly not enough to justify beginning to expand the playoffs towards an NHL-style season-long second season.

When the games are actually played, there are many other factors in play--the strength of lineups, injuries, availability of bullpen pitchers--that affect the odds of a team winning a game. Perceived upsets in short series are explained as a favorite running into a "hot team" or a few "bad bounces." But even without considering if teams are truly under- or over-playing their seasonal winning percentages, the favorite is any postseason series isn't likely to exceed the 74.6% chance the Cardinals have to win just their first series, not to mention two more after that against stiffer competition. And when it comes to teams as evenly matched as the two AL series, the postseason is, to a large extent, a crapshoot. Which is, of course, why it's so much fun to watch.