Game Theory: End of Game Pitch Location
Over at The Book Blog, Mitchel Lichtman commented that Astros pitcher Mark Melancon, should not have thrown an inside fastball on a 2-2 count with two outs in the bottom of the ninth to the Diamondbacks’ Paul Goldschmidt. Instead the pitcher should be looking to throw a ball on the outside part of the plate. I decided to take a look at the location and results of similar pitches and the effect on the Win Probability Added (WPA) of the game.
Lichtman reasoned that the Astros were ahead by one run and the last thing a pitcher should do is pitch the ball inside. He stated that a batter is more likely to hit a home run on an inside pitch, which Goldschidmt did (+0.489 WPA). The hit led to the Diamondbacks tying the game with two outs in the bottom of the ninth. They went on to win in extra innings.
Instead, the pitcher should have been throwing the ball to the outer part of the plate where a hit is more likely, but not a devastating home run.
To find which location is best to pitch towards, I divided the the strike zone into 16 areas and looked at the results of every pitch on 2-2 count and a right handed pitcher throwing to a right handed batter. Then, I took the percentage chance of an event happening and here are the chances of a home run being hit:
Note: The box is the strike zone from the catcher’s/umpire’s perspective and is composed of 1ft by 1ft squares from -2 ft to +2 ft across the plate and from 0ft to 4 ft high.
Home Run%
| 0.1% | 1.1% | 1.1% | 0.1% |
| 0.4% | 2.1% | 1.1% | 0.1% |
| 0.4% | 0.7% | 0.3% | 0.0% |
| 0.0% | 0.0% | 0.0% | 0.0% |
As it can be seen, a right-handed hitter does have a better chance of hitting a home run on the inner part of the plate. The other key for not allowing a home run is to keep the ball down.
The problem with just looking at home runs is that other events could happen, such as getting the final out of the game. That could sway the decision on where to pitch. I took all the events in each zone and then weighted them to the WPA gained or lost for the event*. Using WPA values from the “The Book”, here is the WPA matrix from the same situation (negative value is good for the pitching team, positive is good for the hitting team):
WPA
| -0.004 | -0.010 | -0.010 | -0.007 |
| -0.009 | -0.004 | -0.011 | -0.011 |
| -0.008 | -0.011 | -0.016 | -0.012 |
| -0.003 | -0.012 | -0.013 | -0.008 |
The best position for the pitcher to put the ball is low and away in this situation. The one position the pitcher wants to stay away from is higher and inside.
Now, it is important to remember, that there was no one on base, so the home run is the event the pitcher wants to stay away from. Now, if the bases are loaded with 2 outs, the key for the pitcher is to get the last out. A single, hit by pitch or home run all cause a loss. Here is the percentage chance that an out will occur as a result of the pitch:
Out%
| 17.0% | 39.9% | 38.6% | 18.1% |
| 26.2% | 45.2% | 48.5% | 26.9% |
| 21.0% | 40.1% | 44.0% | 24.6% |
| 4.8% | 25.2% | 25.8% | 15.1% |
With the bases loaded, the pitcher will want to keep the ball high in the zone. The hitter will likely swing at the pitch having a higher chance of striking out or hitting into an out.
Another possible scenario is that there is a runner at third with the same 2-2 count and two outs. In this case, the pitcher wants to stay away from a hit. A walk would be fine since there are two empty bases for the batter to occupy. Here is the matrix for the chance of a batter getting a hit:
Hit%
| 2.3% | 8.8% | 8.5% | 2.7% |
| 4.6% | 16.0% | 14.0% | 5.0% |
| 1.8% | 9.6% | 8.4% | 2.2% |
| 0.0% | 0.9% | 0.8% | 0.1% |
Now, the key is to keep the ball low in the zone. It doesn’t really matter if it inside or outside.
Conclusion:
At the end of the game, it is really important for a pitcher to try for certain results (an out, a hit, etc). In this case Mark Melancon should have kept the ball low on the outer half of the plate in order to limit the chances of a home run.
* I gave a called ball an WPA value of 0. WPA is only based on outs and the base runners. The change from a 2-2 count to a 3-2 would probably be fairly small.
Jeff, one of the authors of the fantasy baseball guide,The Process, writes for RotoGraphs, The Hardball Times, Rotowire, Baseball America, and BaseballHQ. He has been nominated for two SABR Analytics Research Award for Contemporary Analysis and won it in 2013 in tandem with Bill Petti. He has won four FSWA Awards including on for his Mining the News series. He's won Tout Wars three times, LABR twice, and got his first NFBC Main Event win in 2021. Follow him on Twitter @jeffwzimmerman.
I don’t know if aggregate data is really appropriate here. It just feels like a case were there are too many variables. How does the hitter hit inside pitches? What’s the pitcher’s best pitch? How’s the pitcher’s location today? If I have a guy who’s capable of spotting a good fastball under a guys hands I don’t think I worry so much about him pitching inside. Likewise if a guys up there who’s never shown the ability to turn on a hard inside pitch.
I was just looking at the inside outside data. The problem with getting too many variables is that there is no data eventually.
I may look at more situations like this in the off season.
This may be a case were chipping away is best. Maybe limit your data to guys with good fastballs. See how they perform by location. Then do the same for hitters who hit fastballs well. Seems like eventually you’d get a very good picture.
Either way the data is interesting and very in-depth for an in-season article.
It seems to me that the sequence of pitches is important. One of the reasons a pitcher would go inside is because the previous pitches were outside and the hitter is adjusting to the outside pitches.
Was the pitch intended to be an inside pitch? Or was it meant to be much more inside and it wandered further out over the plate than intended?
These are just two extra variables (sequence and intended location) that would complicate studying this.
Exactly. A post entitled “game theory” should consider that: if there’s a simple optimal strategy, then a competent batter would use backward induction to guess the pitch and plaster it. It’s all about the sequence: sending the pitch where and how it’s not expected. That’s where Melancon (and his catcher) failed, and where everyone will fail from time to time.