#### Introduction

In last week’s post, we explored **pitch decisions** and the **influence of the count** on those decisions. For example, pitchers are most likely to throw off-speed pitches when they are ahead in the count. Also, the location of the pitch depends on the count. For example, on a 0-2 count, it is common for a pitcher to throw a low off-speed pitch or a high fastball. In contrast, when the pitcher is behind in the count, he is likely to throw a fastball in the middle of the zone. By use of density estimate graphs, we saw some interesting pitch location patterns for individual pitchers.

Here we focus on the **outcome of the pitch **and how that outcome varies as a **function of the pitch location.** From the pitcher’s perspective, there are different ways to think about a desirable pitch outcome. During the plate appearance, the pitcher gains with a pitch resulting in an additional strike or loses with a pitch resulting in an additional ball. A pitch that ends the plate appearance with a strikeout or other out is a gain and a pitch that is put in-play for a hit is a loss for the pitcher. One way of measuring a desirable pitch outcome is by use of expected runs, specifically the runs gained (on average) from that pitch. We’ll review some of the material from the “Balls and Strikes Effects” chapter of Analyzing Baseball with R. Then we’ll use that material together with information about the runs value of different end-of-PA events to define pitch values. Once we have assigned a value to each pitch during the 2019 season, we can use graphs of a smoothed fit to understand the locations of the regions about the zone where a particular type of pitch is effective.

#### Review of Balls and Strikes Effects

In Chapter 6 of ABWR we explain how to measure balls and strikes effects. Using Retrosheet play-by-play data, we first compute (from Chapter 5 work) the runs value for each plate appearance outcome that changes the state (runners on base, number of outs and runs scored). Next, we define variables c01, c10, c11, etc. that indicate if the count for a plate appearance passes through the respective counts 0-1, 1-0, 1-1, etc. By averaging the run values over each of these indicator variables, we obtain the mean run value passing through each possible count. I’ve displayed these mean run values in the following figure using 2019 season data — these are similar to the values displayed in ABWR, Figure 6.2 from data from an earlier season.

We also can compute the runs value of each possible end-of-PA event (out, single, walk, HBP, strikeout, etc) by averaging the run values. We learn for example, that an out loses on average 0.28 runs and a single and home run, gain on average 0.46 and 1.38 runs, respectively. Note that these run values are averages and don’t depend on the current situation of runners and outs.

#### Pitch Value

Once we’ve computed the runs value for each possible count and each end-of-PA event, then the value of a pitch is simply the change in runs value

Pitch Value = Runs Value (new count or end of PA event) – Runs Value (old count).

For example, suppose the count is 1-1 and the pitcher throws a ball, changing the count to 2-1. The value of that pitch (using numbers from the above figure) is

Value = Runs Value (2-1) – Runs Value (1-1) = 0.035 – (-0.015) = 0.050

Here a pitch adding a ball has a runs value of 0.05 favorable to the hitter. Suppose a batter hits a home run on an 0-2 pitch. The value of this pitch is

Value = Runs Value (HR) – Runs Value (0-2) = 1.38 – (-0.103) = 1.48.

Note that the pitch value of this home run exceeds the average HR runs value since the home run was hit on a 0-2 pitch.

#### Graph of Pitch Values

Now that we have assigned values to all pitches, we are interested in seeing how the pitch value varies across the zone. I have already illustrated the use of the CalledStrike package in an earlier post to display patterns of smoothed fits of different batting measures, say launch speed or swinging rates, and so it is straightforward to use similar functions to display patterns of smoothed pitch values over the zone. Basically, one fits a generalized additive model where the pitch value is represented as a smooth function of the plate_x and plate_z variables, one uses the fitted model to predict the pitch value over a grid, and then graphs the fitted values by a filled contour graph.

Usually we think of desirable pitch locations from the pitcher’s perspective. So we will consider the negative of the runs values, so a **positive pitch value is advantageous to the pitcher**. The values of the two pitches above (a ball on a 1-1 count) and a home run on a 0-2 pitch will be recoded as -0.050 and -1.48 respectively. The color scheme in the plots below will be orange for locations desirable for the pitcher and yellow for locations desirable to the hitter.

#### Values of Four-Seamers

To begin, here is a contour graph display of the smoothed pitch values of all four-seam fastballs thrown in the 2019 season. Since the pattern of the values depends on the pitcher and batter sides, there are four displays, one for each combination of sides. **Orange indicates a region advantageous to the pitcher** and** yellow indicates an advantage to the hitter**. For a left-handed hitter (left side of the graph), for pitchers of both sides we see that a fastball is beneficial outside and high. We see a similar pattern of orange (high and outside) for right-handed hitters. It is interesting that southpaws also seem to have positive value for fastballs thrown inside to right-handed hitters. Clearly it is not desirable to throw a fastball in the middle of the zone.

#### Values of Sliders

Here is a similar display for all sliders thrown by both sides of pitcher to both sides of batter. A right-handed pitcher (bottom row) wants to throw the slider to the outside, either low outside or high outside. For southpaw pitchers (top row), they want to throw their sliders outside to left-handed hitters and high outside to right-handed hitters. It is interesting that there appears to be some advantage for a leftie to throw low and inside to a right-handed hitter.

#### Values of Changeups

Here is a visual display of the pitch values of changeups. Generally effective changeups are ones that are low and outside. Looking at the right-right confrontation (bottom-right), note the large white circle in the middle indicating that poorly located sliders tend to have negative consequences.

#### R Work?

I have outlined all of my work creating the pitch values and graphing the smoothed values over the zone on my Github Gist site. There are several new functions in the CalledStrike package — the function `compute_pitch_values()`

will compute the pitch values for a Statcast dataset and the function `pitch_value_contour()`

will produce the type of filled contour plots shown here.

#### Further Reading on Pitch Value

The notion of pitch value has been around awhile in sabermetrics so there is a good literature on the subject. Here are some older references that I found including the first one by my ABWR coauthor Max (by the way, there is a graph in Max’s post of pitch values of sliders that resembles the one presented here).

- Pitch Run Value and Count by Max Marchi, The Hardball Times
- Searching for the Game’s Best Pitch by John Walsh, The Hardball Times
- Pitch Type Linear Weights by Steve Slowinski, FanGraphs
- Run Value by Pitch Location by Dave Allen, The Baseball Analysts