deGrom-Theoretical Optimality in Two-Strike Counts

Today, I’m looking into something that doesn’t require much explaining. Well, that’s not quite accurate. I’m looking into a situation that’s so good for the pitching team that in our minds, we go ahead and write it off. That doesn’t mean it’s not interesting, though; it can just be hard to see why it’s interesting, which is why I’m writing about it. That’s right: let’s talk about when Jacob deGrom gets ahead in the count.

When the best pitcher in baseball has the advantage on a hitter, that hitter doesn’t do well – a real shocker, that one. With two strikes, deGrom turned batters into sub-pitcher-hitting-level zeroes in 2021:

Those aren’t typos. When deGrom hit two strikes before three balls, he struck out roughly three-quarters of the batters he faced and walked almost none. Survive until 3-2, and you stood a chance – he had a 12% walk rate and a mere 52% strikeout rate after 3-2 counts – but for the most part, facing deGrom with two strikes is a one-way ticket back to the bench.

You might think that it all comes down to his nigh-unhittable slider, but that isn’t quite the case. I looked at all the pitches deGrom threw to opposing righties in 0-2, 1-2, and 2-2 counts – righties so that we’re comparing apples to apples and all three counts to bulk up the sample size. He threw a nearly even split of fastballs and sliders, with the occasional changeup thrown in for good measure:

We’ll come back to the changeups at the end, but for now, let’s toss them out. Imagine that deGrom has only a two-pitch mix in these counts: fastball and slider. When batters took a pitch, they did well against the slider (naturally enough) and poorly against the fastball. I’m using Statcast’s run values for the remainder of this article, because reducing success and failure to a single number works really well for what I’m trying to do here. The results look like so:

When they swung, the opposite happened:

It’s too simplistic to say that batters should swing when a fastball is coming and take when a slider is coming, but at least for today’s article, I’m mostly going to live in that simplistic world. Let’s say, for the sake of argument, that before each pitch, deGrom decides to throw a fastball or a slider, and the hitter independently decides to swing or take. If that were the case, the matrix of outcomes would look like this:

The bind that deGrom puts hitters in is the same as the one all pitchers put all hitters in, only magnified. Taking is good against breaking pitches, which frequently start out in the strike zone before breaking out of it. Swinging is good against fastballs, which are easier to make contact with. But make the wrong choice, and you’re asking for trouble.

For now, we’ll leave deGrom’s pitch mix static and throw out the changeups. He threw 89 fastballs and 86 sliders in these counts last year, which we’re going to approximate as 50/50. The math gets pretty easy in that case – if you always swing, you’ll put up -3.4 runs of value (per 100 pitches) half the time and -10.4 runs half the time, an average of -6.9 runs per 100 pitches. If you always take, you’ll either accrue -4.5 runs (fastball) or positive 1.8 runs (slider), an average of -1.35 runs per 100 pitches. That’s way better. Maybe batters should just always take against deGrom – an idea Devan Fink wrote about last year, believe it or not.

Let’s say that you choose a strategy of always taking. The optimal counter, from deGrom’s perspective, would be to always throw fastballs. But if deGrom chooses to throw all fastballs, then you should always swing. But if you’re always swinging, he should throw all sliders. But if he’s throwing only sliders, you should only take… it’s recursive, though some might argue that deGrom has solved the puzzle by poisoning both cups and not caring what his opponents do.

To get a game-theoretical solution to this extremely contrived puzzle, you need to use something called a mixed strategy. That means randomizing between your two options and playing each some percentage of the time that isn’t exploitable by the other person playing the game.

The classic example is rock-paper-scissors. If you pick a strategy that’s anything other than playing each option a third of the time, your opponent can pick a counter-strategy that beats you. Not everything is so easy to calculate, but the same principle — leave your opponent with no good options — drives the math behind mixed-strategy optimality. You can work out the math from that Wikipedia link if you’d like, but in the interest of not turning this into a game theory paper, I’ll just tell you that the solution is as follows: deGrom should throw fastballs 91% of the time and sliders 9% of the time. The batter (you, in my above hypothetical) should swing 47% of the time and take 53% of the time.

That’s an interesting outcome, because the best square in the grid for deGrom comes from throwing a slider, and the solution calls for very few sliders. Why? Taking all the way makes a slider more or less useless, because deGrom mostly doesn’t throw it in the zone – that’s why it’s the only quadrant with a positive run value for the hitter. Against his 50/50 blend of fastballs and sliders, the optimal strategy is easy: take 100% of the time. Batters don’t do much against fastballs even when they swing, so all swinging does is turn his slider from a pitch that frequently misses the zone into a guaranteed out. In fact, until deGrom throws 91% fastballs, the upside of correctly taking a slider instead of swinging at it is so high that batters should theoretically take all the time.

Of course, baseball doesn’t work that way in real life. For one thing, hitters don’t decide whether or not they’ll swing before the pitch is released. A more realistic strategy might be something like “defend the plate” or “sit fastball.” Likewise, deGrom has more options than just fastball or slider. Without even getting into his changeup, he can throw either pitch for a strike or a ball – and the ball doesn’t always go where he wants it to go, of course.

Let’s consider a meaningfully more complicated grid, with some values that I made up for the purposes of this article:

I’m not saying that all of these values are right, but I think they’re reasonable approximations. If you’re protecting the zone and deGrom throws a fastball for a ball, that’s a relatively easy take. If you’re sitting fastball and he throws a slider that breaks out of the zone, same deal. Meanwhile, if you’re trying to protect the plate and deGrom throws you a slider that starts over it and breaks away, you’re toast. Likewise, if you’re sitting on a fastball and he throws a slider for a strike, better luck next time, buddy. I’ve cleverly set the numbers up so that if he uses each option a quarter of the time, the expected value of each strategy is the same.

With only two batter options and four pitcher options, this isn’t a solvable grid; there’s no strategy the batter can pick that doesn’t give deGrom a best option. But let’s diverge slightly from game theory and approach it from deGrom’s perspective. He has several options, but let’s pick out some pitch mixes that will make batters indifferent between protecting the plate and sitting on a fastball:

These are basically nonsense numbers, but there’s some pleasing logic to them. If deGrom throws a lot of fastballs in the zone, he should pair them with sliders in the zone to catch batters who are trying to only swing at fastballs off guard. Likewise, if he’s throwing a lot of fastballs for balls, he should pair them with sliders that miss the zone, so that batters who are looking for pitches that start over the zone swing through them. Those pitches pair off of each other well – each punishes the batter for picking a strategy that is effective against its opposite.

None of this is actually how baseball works, but for me, it’s a useful abstraction of how pitchers and batters should be thinking. It’s easy to wonder why deGrom doesn’t throw his slider all the time in two-strike counts. It gets better results, after all. But if you think of the world in a game theoretical frame, he should actually be throwing fewer sliders, and batters should be swinging far less often than they do. In reality, he’s splitting the difference between “throw more sliders because they’re unhittable” and the theoretical equilibrium, which seems like a solid choice. As for that sprinkling of changeups? You can throw them in the grid, too, and figure out how many he should use, but the fact that he throws them infrequently but gets great results suggests that they’re doing a good job of punishing hitters for sitting on his other pitches.

Likewise, “pair pitches that work well against each other” is a nice theory, but how much should they be paired, and why? If you build these behavioral grids and spend plenty of time making sure the numbers all check out, you can work out what the right relationship between fastballs for strikes and sliders off the plate should be – or between selling out for a fastball and taking all the way, or whatever options you’re trying to decide between.

Should batters take every pitch against Jacob deGrom? No, they should not. But it’s fun to think about what they should do, and what he should do knowing what they should do, and so on and so on. That’s why I wrote this article – because it’s a timeless encapsulation of the mind game between pitcher and hitter. If you’re a pitcher, consider what your pitch mix means for the hitters facing you. And if you’re a hitter, never fall for one of the two classic blunders: starting a land war in Asia, or going against Jacob deGrom with a strikeout on the line.