Boyd's World-> Breadcrumbs Back to Omaha-> Theoretical Winning Percentage, Part I About the author, Boyd Nation

Theoretical Winning Percentage, Part I

Publication Date: August 22, 2000

What the world needs now is another folk singer, like I need a hole in the head. -- Cracker, Teen Angst

Zero, One, or Infinity

Opinions vary as to how many ratings systems for college baseball the world needs. Those opinions are almost completely split, however, between zero and one, which means that the ISR's either completely fill the world demand or completely overrun it. Nonetheless, there's another idea I've been bouncing around my head for a while, and I've decided to go ahead and code it up and see what happens.

Part of the reason why doing this is a risky endeavor is that there's nothing particularly wrong with the ISR's, as far as anyone can tell. Because there's no way to know exactly what the "correct" ordering of teams is (in actuality, there probably is no "correct" ordering, in part because of the reasons we discussed last week), all we can do is look at the results, see if they look reasonable, and try to minimize the angels-on-pinheads-type arguments.

The one nit that I have with the ISR's is that, as a part of the process, they basically use an average measure for considering strength of schedule, and that can be a less than accurate measure for some fringe cases like Miami. That's a minor thing, and I'm still not sure if it needs adjusting for. A bigger philosophical question that I have about them involves the fact that there's no particular mathematical reason that they work; they just do as a byproduct of the process. That bugs the purist in me.

The Underpinnings of Theoretical Winning Percentage

This section's a bit math-heavy, but, come on, what are you using those brain cells for right now, anyway?

If you took a team and magically let them play an infinite number of games against every other team, or at least a few hundred so the random bounces all evened out, they would win a certain percentage of these games. I call that percentage their Theoretical Winning Percentage, or TWP. If we did this with everyone, we would have the truest possible measure of team quality. There would still be cases, of course, where teams that weren't as good would be unusually suited to take advantage of a better team's weaknesses, but we'd understand that when it happened.

Now, for any given game, there's a formula, given an accurate winning percentage for each team, for how likely one team is to win. If a team's winning percentage is P and their opponent's winning percentage is Q, then the likelihood that they win is

              P x (1 - Q)
        P x (1 - Q) + Q x (1 - P)

(Save that formula somewhere; it tends to come in really handy at times.)

Now, we can't actually go out and play an infinite number of games, but we can pretend that we did and try out different values for the results. What the TWP algorithm does is to pick starting values for each team, see how likely the actual on-the-field results are for those values, and then adjust the values until we get the set of TWP values that best explain what actually happened on the field (in mathematical terms, we're trying to maximize the product of the probabilities of all results). What comes out is a number between 0 and 1 which represents what portion of their games against all teams each team would have won.

Results for the 2000 Season

This is running a bit long, so I'm going to wait until next week to fully take a look at the results that I get out of this algorithm. However, just to give you a taste for what the results look like, here's the top 20 from the 2000 season, with their ISR rank thrown in for reference:

Rank ISR  TWP   W   L  Team

  1   1  0.962  56  10 South Carolina
  2   3  0.945  52  17 Louisiana State
  3   2  0.937  50  16 Stanford
  4   5  0.934  53  19 Florida State
  5   8  0.924  50  16 Georgia Tech
  6   7  0.922  51  18 Clemson
  7   6  0.918  44  15 Arizona State
  8   4  0.915  44  20 Southern California
  9  12  0.901  46  17 North Carolina
 10  14  0.899  44  23 Florida
 11   9  0.896  42  16 Baylor
 12  10  0.895  48  18 Houston
 13  13  0.892  48  17 Nebraska
 14  18  0.891  41  20 Mississippi State
 15  17  0.890  41  20 Auburn
 16  11  0.886  45  21 Texas
 17  16  0.885  40  24 Alabama
 18  15  0.882  47  20 Louisiana-Lafayette
 19  26  0.877  39  19 Miami, Florida
 20  23  0.875  41  20 Wake Forest

In other words, South Carolina would have won 96.2% of their games if they had played everyone in the nation, for example. I'll go more in depth on this kind of thing next week, but almost everyone moves a spot or two, and most folks don't move much more than that. That provides a nice confirmation of the ISR's, while the few that do differ a bit may provide some insight into the behavior of the two algorithms.

Boyd's World-> Breadcrumbs Back to Omaha-> Theoretical Winning Percentage, Part I About the author, Boyd Nation