The first time I studied defensive alignments, I was so proud of my analysis that I started a blog just to show that work.

Nevertheless there was something in that study that left me unsatisfied; I even expected critics for that, and they duly arrived.

What was wrong (or needed at least a big improvement)? My analysis showed an efficient alignment to prevent balls in play from becoming base hits.

This can be accepted in a world where batting average is the queen stat, thus neither here at THT nor by a team trying to gain an edge to win one more ballgame.

The critics I was expecting suggested that, for example, ground balls going through the infield down the line have a different impact than rollers up the middle; thus improving your out/hit ratio doesn’t necessarily imply you are improving your run prevention. In this article I’ll show a way to take this issue into account.

Let’s take Derek Jeter‘s 2008 ground balls.

In the following chart the x axis represents the batted ball trajectory angle (from -45° down the left field line, to 0° up-the-middle, to 45° down the right field line); the y axis shows in a continuum the density of ground balls hit by Jeter.

I gave a brief explanation of the density plot a couple of weeks ago (*References and Resources* section); anyway, as you may have guessed, the higher the line, the more ground balls hit at the corresponding angle.

You have surely noticed the four dotted lines on the graph. I will refer to them as the *infielders’ average positioning*, and I obtained them as follows:

I selected all the ground balls hit by right-handed batters (should I consider a lefty instead of Jeter, I’ll take the ground balls hit by left-handed batters) with the bases empty and resulting in an out by one of the four infielders. Then I averaged the trajectory angle for outs made by the third baseman and considered the resulting value as the average position of third basemen. I repeated the same process to get the average positioning for shortstops, second basemen and first basemen.

I chose the empty bases situation because infielders shouldn’t be shifted for other reasons than the batter’s spray pattern. I can’t wait for the day when we’ll have the actual positioning of the fielders on every batted ball, but for the time being I consider the values I got as reasonable estimates.

Have another look at the density plot. Where do you think the fielders should play Derek Jeter? If you believe the dotted lines should cross the (blue) density line in places where it peaks, you are quite right. Thus the middle infielders should play The Captain close to their average positions (maybe a little farther from the bag), the third baseman should move way toward the hole leaving the line unguarded and the first baseman, having no peak to cover, should cheat toward his right as far as he can and still get to the base in time to receive the assists from his peers.

Right? Well, no. Again, we are minimizing the proportion of ground balls by Jeter that go through the infield for a base hit. How can we minimize his run production?

The following chart will give us some help.

This one represents (on the vertical axis) the run value of every ground ball hit by a right-handed batter and fielded by an outfielder. On the x axis we have again the trajectory angle of the batted ball. In other words, we are looking at the run values of ground balls gone through the infield. As we would expect, we have the highest values at the corners, while there’s not much variation over the rest of the field, except for a little hump in the left/center gap.

Now we just have to multiply one chart by the other to get what we are after. To be honest, density is not probability, thus a transformation is needed, but I believe nobody wants to hear about integrals.

The resulting chart has a shape very similar to the density plot. Wait! All the smoothing, charting, integrating, multiplying to discover we could just have used the first chart? If that’s the case, it hasn’t been a waste of time anyway. If preventing base hits on grounders is equivalent to preventing runs on grounders, that was something that needed to be proved.

Anyway, we have looked at just one player, one known to spray the ball to all directions. I ran the same kind of analysis for a bunch of hitters and I’m going to report an interesting case in the paragraphs that follow.

Ryan Howard is a power lefty who hits home runs to the opposite field as well as to the pull side. His grounders, on the contrary, tend to cluster on the right side of the diamond, and so do infielders of teams playing against him. Here is Howard’s density plot for grounders.

Where would you put your four infielders? (The dotted lines show the average positioning against lefty hitters). Don’t base your answer on this chart.

Let’s look at the run value of grounders hit by left-handed batters gone through the infield.

It’s a mirror image of the one for righties. Again, highest values on the corners and a hump, this time in right center. Howard hits a significant portion of grounders close to the right field line, thus the chart that results after smoothing/integrating/multiplying shows something new.

Now we have a couple of peaks on the right instead of just one. Okay, so the first baseman should place himself toward the line, the second baseman in the hole (closer to a first baseman’s usual position), the shortstop over the keystone, and the third baseman in the left side hole. (Basically this is how opposing ballclubs play Howard.)

*Note: while placing fielders exactly on the peaks may be a good rule of thumb, we should note that there isn’t always symmetry along the peaks’ lines. Thus, the optimal positioning would be slightly off center.*

Now. we know that for some hitters it’s worth to do some extra work to devise an optimal defensive alignment.

I’m not done on fielders’ positioning yet. Finding the optimal alignment would require taking into account the range of the glove men and their different abilities going to either side. It’s not an easy task, but it can be done. Moreover, I would like to quantify (in runs prevented) the impact of moving a fielder to a better position.

…and then there is the issue of the batter’s willingness and ability to hit against the shift and all the cat-and-mouse games that would follow…

**References & Resources**

Data from 2008 MLBAM Gameday.

I owe you some explanations about the weighted charts. Basically I’ve cut the field into 0.2 degree slices from foul line to foul line. For each slice I calculated the probability of a ground ball going through the slice, multiplied by the expected run value of a ground ball that reaches the outfield through that slice, multiplied by the total number of batted balls for the batter . If you sum these products across all the slices you get the run value of the batter’s ground balls if none of them are converted into an out by an infielder.

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