Quantifying “Good” and “Bad” Pitches

I found Jeff’s recent post on Jake Arrieta fascinating, because he goes into a game and pulls out Arrieta’s eight worst pitches from that game. This is something I’d never really thought deeply about before. We all know what bad pitches look like, right? An 0-2 fastball down the heart of the plate, a hanging slider, a pitch in the dirt on a full count, sure. But can we quantify this? Is there a way to say mathematically (in a way that makes some sort of sense) whether one pitch was better than another? Follow me beyond the jump and I’ll share some thoughts about how we might do this.

Consider a batter who has just stepped to the plate. Can we talk about his “expected OPS” in this situation? Sure we can — we can say, on average, what a major league baseball player’s OPS will be, so we can assign that to the player. The league average OPS this year is somewhere around .700, so we can say that we “expect” a hitter to have a .700 OPS in that plate appearance (which is of course impossible — but this is “on average”, remember). Now suppose we have that same hitter in a 1-0 count. This tips the scales a bit toward the hitter, and we can find out what the expected OPS is in this situation; let’s say it’s something like .720. On the other hand, the hitter might get into a 0-1 hole, which would lower the expected OPS to something like .650.

The point is, at each stage, we can assign what we might like to call a prior probability on what the batter is going to do in a given at-bat, and then when new information comes in, we update that probability. More to the point of this article, the pitcher is making good pitches if he is driving this expected OPS down.

Let’s now fix the parameters of a pitch. We have the handedness of the batter and the pitcher, the type of pitch thrown, and where the pitch ended up. The first two are known to the pitcher and he can control (to some extent) the last two. How do we determine if it was a good pitch? Here are the basic parameters of what I’m proposing:

1. Take the expected OPS of the situation before the pitch (prior expected OPS)
2. Determine the probability p_i of the possible outcomes of the pitch
3. Determine the expected OPS o_i of the batter given each possible outcome
4. Take the sum over p_i * o_i for all i, and call it the updated (posterior expected) OPS.
5. Assign to the pitch a value equal to the prior expected OPS minus the posterior expected OPS.

This might sound complicated, so let’s take an example. Suppose we have a righty pitcher and a righty hitter, and a 0-0 count. Let’s assign the hitter the .700 prior OPS from above (again, the actual number will be different once we actually run the numbers). Now let’s say the pitch was a fastball more or less right down the middle. For this pitch and any other, there are basically five possible outcomes — assume for this pitch we have calculated their probabilities as follows:

• Called or swinging strike: 40%
• Ball in play: 30%
• Foul ball: 20%
• Ball: 10%
• HBP: 0%

If the pitch is a strike (or fouled off) the batter will be in a 0-1 count, meaning the expected OPS will be .650 by the approximation above. If it’s a ball the count will be 0-1, raising the expected OPS to .720. If the ball is in play — I haven’t gone over this case yet, but we should be able to calculate the expected OPS of this particular ball in play from historical data — let’s say the expected OPS is .900. This would result in a posterior expected OPS of

0.4 * .650 + 0.3 * .900 + 0.2 * .650 + 0.1 * .720 = 0.732

Since the posterior expected OPS is greater than the prior OPS, this was not a particularly good pitch, and it gets a value of 0.700 – 0.732 = -0.032.

All right, this is the framework of what I am proposing.  As written above, I don’t even think it would take too long to run these numbers and to start quantifying good and bad pitches.  But before I do, I need to ask some questions to the sages here at FanGraphs, such as:

1. Has this already been done?
2. Is this even worth doing?
3. OPS is not the best stat for doing this with.  Something along the lines of linear weights would be better.  Suggestions?
4. What should I consider when making these models?
5. Is anyone interested in helping with R coding or Pitch/FX data for this?

Thanks for reading.