Monday, September 07, 2009

Magic Number in a Slightly Different Form

As you know, the magic number is the combined total of wins and opponent losses necessary for a team to clinch a playoff spot. If we let g = number of games in season, then the magic number (M#) can be calculated as:

M# = g + 1 - W - oL

where "o" denotes opponent.

Of course, there's a big difference between having a M# of 5 with 10 games to play and having a M# of 5 with 20 games to play. We could look at what I will call Magic Percentage (M%)--the percentage of game outcomes that must go a team's way in order for them to clinch. In this case, game outcomes include both the team in question's games and the games of their opponents.

Suppose that the race between the Alphas and the Bravos is shaping up like this, with ten games left for each team:


The Alphas' M# is 9. Again, that means that a combination of nine Alpha wins and Bravo losses will clinch the division. Since each team has ten games left, there are twenty total game outcomes outstanding, and 45% of them (9/20) must go the Alphas' way. Therefore, their M% is 45%. Since I always feel compelled to write out a formula, here it is:

M% = (M#)/(2*g - W - L - oW - oL)

Suppose that the race tightens up a bit over the next five games, and with five to play the standings look like this:


The Alphas have trimmed their M# to five, but their M% has increased to 50, as they now need half of the game outcomes to go their way.

One might be tempted to say that the M% is better than the M#, because it puts the magic number in context...a M# of one with 20 outstanding game outcomes is a sure thing (M% = 5%); with one game left it's a nailbiter (100%). But the M% can be just as misleading--a M# of one with two games left is 50%, but a one game lead after the first day of the season will produce a similar percentage.

So the M% is not a magic bullet by any means. You still need to consider it in conjunction with the number of games remaining, just like you do with the plain old M#. All that I've done is express the number of favorable outcomes required as a rate rather than a total.

The M% only works in relation to one set of leader and pursuer. As I pointed out before when talking about variations on games behind, when multiple teams are in the race, the results of your team are more important than the results of either of the other teams considered independently, because your victories help your position against both of your opponents. So if the Alphas have a two game lead over both the Bravos and the Charlies, one can't look at their M% and state that if X% of game outcomes go their way, they will win. You would need two M%s, one with respect to each opponent.

Through games of Sunday, here are the standings for the top two teams in each division, along with the leader's magic number and magic percentage (the magic percentage for the pursuer is simply the complement of the magic percentage for the leader, and is thus omitted.  This is an embarrassing oversight on my part; they are not complements.  The magic percentage only gives you the percentage of favorable outcomes to win the division outright.  A tie scenario is not taken into account, and thus the M%s for two adversaries do not sum to one):

If you assume (falsely, of course, but it's usually not too large of a distortion at this stage of the season) that both teams have an equal number of games remaining, then one way to look at the M% is that it is half of the minimum W% needed to clinch the playoff spot even if the opponent goes undefeated--or half of the maximum W% your opponent can play, even if your team goes winless.

All of the above is just spitballing. If you are serious about estimating a team's playoffs chances, then playoff odds approach as implemented by Baseball Prospectus and others is the way to go. However, that involves estimating team quality and running thousands upon thousands of simulations. The freak show stat here is intended to be one you can figure out with nothing more than a calculator and your morning paper, err, the standings on Baseball-Reference, and a way to put a little twist on the familiar magic number.

1 comment:

  1. "Toffer Peak", commenting on Beyond the Box Score, says he'd prefer to see (M#)/(team games remaining), which would be the W% the team in question must achieve in order to assure themselves the title. He feels that M% is unintuitive.

    I certainly agree that M% is a little clumsy--just look at my phrasing as evidenced by "game outcomes outstanding". I'm certainly not tied to any particular formulation or insistent on the need to contextualize M# in any way other than considering it along with games remaining. I prefer M% because I understand what it means and it takes into account all outstanding games, leader and pursuer, but there's no accounting for taste.


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