THE BOOK--Playing The Percentages In Baseball

Wednesday, May 05, 2010

Changing WPA to credit reaching base and eventually scoring

It’s the bottom of the 9th, 2 outs, bases empty, down by 1 run.  What are the chances the home team will win the game?

Well, we can look at the historical record and see that they win .035 times per game.  We can look at a Markov model and see a .042 win%. You can even get your hands dirty and try to figure it out yourself.  The chance of scoring 2 runs or more in that situation has historically been 2.2%.  And the chance of scoring exactly 1 run (and sending the game into extra innings, with 50/50 shot of winning) was 4.5%.  So, you get 2.2% plus half of 4.5% for a total of 4.45%, or .045 win%.

For purposes of this illustration, I’m going to use .044 chance of winning with bases empty, two outs, down by 1, bottom of the 9th.

Here are the chances of winning after each event, along with the frequency of each event:
freq win% event
0.665 0.000 out
0.100 0.090 bb
0.160 0.090 1b
0.045 0.140 2b
0.005 0.170 3b
0.025 0.530 hr

So, 66.5% of the time, you make an out, and the game is over.  You hit a HR 2.5% of the time, and your chances of winning is 53%.  And so on.  The overall average, in this illustration, is .044 wins.  So, if you hit a double, your chance of winning goes from .044 to .140, or a gain of +.096 wins.  So, if Mike Sweeney hits a double, we assign +.096 wins for that particular double.  In these cases, we are uninterested what happens after Sweeney gets the double.  The next batter can make an out or a HR, and Sweeney is still assigned +.096 wins.

But, what if we DO care about the next batter?  So, now we have a new chart:
freq win% event
0.665 0.000 out
0.260 0.044 Reach base, doesn’t score
0.050 0.383 Reach base, ends up scoring
0.025 0.530 hr

The weighted sum is still .044 wins.  So, if Sweeney reaches base, but doesn’t score, he basically had no value.  The chances of winning before he came to bat was .044, and so, his WPA (after the fact) was 0, which is why the win% assigned (after the fact) remained at .044.  If he reaches base, and scores, we give him .383 - .044 = +.339 wins.  What we did here was account for the fact that him scoring is more valuable than not, and it depends on the batter after him.

So, for those who think that we need to change the credit for getting on base, based on whether the runner eventally scores or not, this is how you do it.

But, do you really want to do this?  Do you really want to decide whether to give Sweeney +.000 or +.339 based on whether he is eventually driven in or not by a future batter?  Or do we simply want to give him credit for reaching on base, regardless of what the next batters do?

This is what you have to decide as a reader and analyst.  You have to answer that question first.  And once you answer that question, then I have given you the path that you need to follow to get the answer.