Baseball is a game of match-ups. In any given at-bat, a wide range of factors come into play. Handedness, pitch types, and the umpire are just a few of the more prevalent variables. However, especially important is the count. Pitchers and batters employ different strategies based on small changes in the count. Consider the graphs below, which I created in an effort to get a glimpse into the pitcher-batter match-up. These data are only from right-handed pitchers to right-handed batters.

This graph shows the locations where batters swing the most and where pitchers throw to the most. The blue contours (circles) represent where batters swing the most, with locations inside the smaller blue circle indicating a swing rate of at least 40 percent and locations within the larger blue contour indicating where batters swung at least 20 percent of the time. The red contour indicates locations where pitchers threw to at least 12 percent of the time. That may seem arbitrary, but if I set the threshold for pitchers to be any larger, no contour appears in the 0-2 graph. The dotted box indicates the strike zone, and the graph is from the catcher’s perspective.

You should also see a small green line. I found the two sets of coordinates where pitchers threw to the most and where batters swung the most, and the green line segment connects to each of these points. The length of the line, or the distance between the two points measured in feet, corresponds to the area in the top left of the graph with text that says “distance.” I should note that this line is really more of an approximation given the method of calculating pitch density. The text that says “IWZ” is an abbreviation for “in wide zone percentage.” If the graph reads “IWZ: 0.589”, then a pitch was thrown within a wide zone 58.9 percent of the time in that count. By “wide zone” I am referring not to the rulebook strike zone, but to an approximation of the actual called strike zone (the dotted box).

I don’t know a whole lot about game theory, but it seems to me that batters don’t swing enough in 0-0 counts. This is because pitchers throw a decent amount of strikes and a lot of fastballs to open at-bats, but batters really are pretty passive compared to other counts. As you can see, the pitcher and batter contours overlap a decent amount, meaning that pitchers are throwing the ball where batters like to swing.

This time area within the blue contour represents locations where batters swing at least 50 percent of the time. As you can see, the 50 percent contour in this graph is very similar to the 20 percent contour in the 0-0 graph.

Area within the blue contour represents locations where batters swing at least 50 percent of the time. All graphs use a 50 percent swing contour except for the 0-0 graph (20 percent and 40 percent) and the 3-0 graph (10 percent). As will all other graphs, area within the red contour (pitch location) represents locations where pitchers threw to at least 12 percent of the time.

The zone has the largest distance (longest green line), which is expected. Pitchers really like to locate down and away, but they also vary location a lot, making the red contour very small. Batters really expand their strike zones to protect.

Area within the blue contour represents locations where batters swung at least 50 percent of the time. Compare this contour to the contour of the previous two graphs. They all are 50 percent swing contours, but the contour in the 1-0 count has a much smaller area, meaning that the swings are much more concentrated. Of course this is to be expected, because batters aren’t worried about protecting the zone in this count.

This count is pretty neutral.

This count is pretty similar to 0-2, except less extreme. Pitchers throw more strikes than in 0-2, leading to a higher swing rate as well. Note that the distance figure is again large, though still smaller than the distance in 0-2. The 50 percent swing contour once again completely envelopes the strikezone.

As expected, this is a pretty extreme hitters’ count. The 50 percent swing contour is once again very compact, and the red pitcher contour is pretty much within the strike zone. The distance figure of .308 feet is very small; remember, the value that we saw in the neutral counts (0-0 and 1-1) was just a little less than .7 feet, or a little over eight inches. This means that in this count the pitchers often threw the ball to where the batters swing the most.

What’s interesting about this plot is that pitchers seem to be throwing the ball to the same locations as in the 2-0, as evidenced by similar looking contours and in wide zone rates. The big change here is the batters. Presumably cognizant of a looming 2-2 count, batters expand the zone a lot compared to 2-0 and swing rate increases from 42 percent to 60 percent.

Notice that the 50 percent blue swing contour balloons once again.

Here the blue swing contour represents only locations where batters swing at least 10 percent of the time. As you can see in the top left of the graph, batters swing just 7.5 percent of the time, by far the lowest amount in any count. The batters can smell the walk at this point, and they really tighten up their approach.

Interestingly, the red pitcher contour almost completely envelops the swing contour. Note that the distance value here of .459 is larger than in pitcher-favorable counts like 2-2, suggesting that distance is not such a great measure of pitcher advantage (although still less than the distance found within neutral counts). A contributing factor is that this count had only 1,564 total pitches in my data set, which is a much smaller sample size than the other counts.

Also of importance here is that while the 60.5 percent in wide zone rate is very high, it’s actually lower than the in wide zone rates of the following counts: 1-0 (60.6 percent), 2-0 (62 percent ), 2-1 (61.4 percent), in addition to 3-1 and 3-2 counts which we will see later. This does not make sense. Given how rarely batters swing on 3-0 counts, this should be a free strike for pitchers. There is no reason why any other count should have a higher in wide zone rate.

It is possible that pitchers are either afraid of giving up a 3-0 home run on a meatball or have a lapse in concentration in this count. Of course that assertion relies on the assumption that batters wouldn’t wise up and start swinging more at all of these grooved pitches. Alternatively, I could suggest that batters should be more aggressive in this count. Again, the sample was small here, so that may play a role. I also have not removed intentional walks from the data, so that may have a small influence.

Here the blue swing contour returns to 50 percent. The red pitcher contour is 12 percent, as always.

Ah, the 3-2 count, subject of much discussion. A few years ago Dave Allen posted this thought provoking analysis of the 3-2 count. Dave said this about 3-2 counts:

Consider the two counts 2-2 and 3-2. In both counts the penalty for taking a strike is the same—a strikeout—but the benefit from taking a ball is greater at 3-2. Taking a ball at 3-2 results in a walk, while taking a ball at 2-2 just brings the count full. If a pitch is right on the border of a strike/ball a batter has more incentive to take that pitch at 3-2 than 2-2.

Everything here is correct and makes sense. But when you think about the pitcher’s mindset, the difference between the counts becomes more apparent. There is much more of a consequence to throwing a ball in a 3-2 count than in a 2-2 count, so pitchers adjust and throw more strikes. Specifically, in 3-2 counts pitchers have an in wide zone rate of 62.5 percent compared to a rate of 51.2 percent in 2-2 counts. The batters are adjusting their strategy because they are also aware of the pitcher’s mindset in this situation. On average, a pitch taken in a 3-2 count results in a strikeout more often than a pitch taken in 2-2 count, so batters swing more (a similar explanation is provided in the comments).

However, it’s still likely that batters swing more than they should in 3-2 counts, which is shown in the post. In this count they swing 75 percent of the time, which is exactly 10 times as much as they swing in 3-0 counts! Also of note here is that the distance value is the smallest of all counts, suggesting that pitchers are throwing the ball pretty close to where batters swing the most.

**Finishing Thoughts**

This post was originally meant to visually juxtapose the pitcher’s and batter’s strategies in each count, but it got a little out of control. The distance measure turned out to not be very helpful. A more interesting relationship is that between swing rate and in wide zone rate. In the most extreme pitcher-favorable counts, swing rate is much higher than in wide zone rate, and in the most favorable batter counts, swing rate is much lower than in wide zone rate. This relationship seems to do a pretty decent job of describing the polarity of the batter pitcher match-up except for 0-0 and 3-2 counts, where the relationship breaks.

**References & Resources**

*PITCHf/x data via MLBAM through Darrel Zimmerman’s pbp2 database and scripts by Joseph Adler/Mike Fast/Darrel Zimmerman

*Dave Allen’s post

Dave Studeman said...

Hey Josh, I just want to say that I really enjoyed this. Nice overview. You probably didn’t get questions or comments because it was so self-sufficient.

Josh Weinstock said...

Andy,

The umpire’s interpretation of the strike zone does change with the count, in addition to other situations. Checkout J-Doug’s work at Beyond the Box Score: http://www.beyondtheboxscore.com/2011/1/5/1915896/benefit-of-the-doubt-the-cruel-umpire

Mike Fast said...

Yeah, I think this is one of those articles which I will want to go back to as a reference six months or a year from now when I am writing about something else. Or that I will point other people to when they have questions.

Josh Weinstock said...

Thanks, Mike.

Gary Yeh said...

Hey Josh, Can I translate this great article into traditional chinese, and post it in my blog??

Andy Everett said...

Great article Josh! One boundary I’d like to see added is what the umpires call as a strike. That is does an umpire’s interpretation of the strike zone alter with the count.

Josh Weinstock said...

Thanks, Dave. Much appreciated.