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):


-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:


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:


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.


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.

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Jeff 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 three FSWA Awards including on for his MASH series. In his first season in Tout Wars, he won the H2H league. Follow him on Twitter @jeffwzimmerman.

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