I've Seen That Before
While a pitcher's stuff diminishes over the course of game, the effects I found were relatively small. So why do batters gain an edge over pitchers as the game goes on? Well, baseball is a game of adjustments. Batters get their timing down and start picking up the ball out of the pitcher's hand. All that good stuff.
The first time a batter faces a curveball, he might be caught off-guard. That’s why pitchers throw predominantly fastballs the first time through the order. And that’s why batters do so well the third time they face a pitcher. They’ve seen most of his repertoire, and are able to recognize the curve. As the saying goes, “Fool me once, shame on you. Fool me…you can’t get fooled again.”
First, here is the average run value per 100 pitches based on the number of times a batter has seen a given type of pitch. I include all data points for which I have approximately 1,000 pitches.
This chart indicates that a batter facing a fastball from the same pitcher for the 12th time will perform better than a batter facing a pitcher's first fastball. Chances are, however, that batters who face 12 fastballs are better from those who only face a few. One way to get around this bias might be to take the difference in run value between the 11th fastball and 12th fastball. This method, called the delta method, allows you to compare apples to apples as each change in measurement is at least composed of players from the same sample. This produced the following chart:
The magnitude of the results is enormous, if the results are to be believed. A batter facing a changeup for a fifth time is expected to perform over five runs per 100 pitches better than he performs the first time he saw the changeup. That's pretty much the difference between the best and worst hitter in the league. Unfortunately, I have to say that I don't think the delta method is the way to go here, and I'm not sure how to fix my sampling problems. Batters who face at least three changeups have a rv100 of 0.2 on the third changeup, but they only have an rv100 of -1.1 on the second change. This is a delta of 1.3 runs. Meanwhile, batters who face at least four changeups have an rv100 of -1.3 runs on the third change and 0.3 on the fourth, another huge delta of 1.6 runs. This would mean that batters perform three runs per 100 pitches better on the fourth changeup they see than on the second. The oddity here is that batters who face at least three changeups are above average on the third changeup, but batters who face at least four changeups are well below average on the third changeup. I think what this means is that once pitchers get burned on a given pitch, they quit throwing it to that batter the rest of the game. I don't know how to solve for these biases.
I went on and produced the same two charts, except this time at the at-bat level instead of the game level.
Batters who face seven fastballs in an at-bat are good, in that they are able to work the count. Meanwhile, pitchers who throw five sliders in an at-bat are good, in that they are either ahead in the count or can locate their breaking balls.
Using the delta method:
No pitch gains in effectiveness after its been thrown once already in an at-bat. This finding was applicable at the game level as well. However, there are differences between the at-bat and game level. Off-speed pitches such as the changeup and curveball lose more value than fastballs during the game, given an even distribution of pitches. But in an at-bat, off-speed pitches do not lose as much effectiveness as fastballs when they're repeatedly thrown. It makes sense to me that changeups are the worst pitch to show multiple times to the same batter throughout the game, since the success of changeups is built on deception. Yet I'm not sure why changeups don't lose as much effectiveness in an at-bat once thrown multiple times as fastballs do. I think it has something to do with the count in which they're thrown and the theory of the out pitch.