# One Pitch, Optimally Speaking

As I’ve chosen topics to research and write about over the past few months, I’ve let an obsession creep into my writing catalog — I’m fascinated by pitcher and batter behavior in 3-0 counts. Whether it’s three-pitch strikeouts after falling behind 3-0, Ronald Acuña aggressively hacking on 3-0, or even just Brandon Belt sneaking in a bunt, I can’t get enough of the goofy ways baseball gets distorted in that most extreme of counts.

What’s so fun about 3-0 is that context matters. For a lot of baseball, looking at things in a context-neutral fashion is the best way to analyze it. A double is a double is a double, and it doesn’t make sense to treat one with the bases juiced in a tie game differently than you would a leadoff double in the first inning when you’re assessing a player. Hitters have little control over balls in play, and absolutely none over who’s on base when they come to the plate. Pitchers, likewise, can’t control sequencing — that’s why concepts like wOBA and FIP do a better job predicting future results than RBI and win/loss record (or, fine, ERA).

But one place context does matter is the count. The world of 3-0 counts is only barely related to 0-2 counts. A pitcher’s arsenal is limitless at 0-2, constrained mostly to fastballs on 3-0. Conversely, a batter has no choice other than to defend the strike zone on 0-2, whereas 3-0 opens up myriad possibilities. That context is what makes the realm of 3-0 counts so fascinating to me. Today, I thought I’d take a theoretical approach to the subject.

To do this concept justice, I’m going to make a quick detour into game theory. Game theory is a branch of economics that deals with optimal decisions in the context of multiple-participant games. The most famous example is the Prisoner’s Dilemma, where making optimal decisions produces suboptimal results, but let’s look at a quick example in baseball.

Imagine a simplified game. It’s a 3-2 count. The batter has some unique traits — if he swings at a ball in the strike zone, he hits a home run 100% of the time, and if he swings at a pitch outside the strike zone, he always misses. That leaves us an outcome matrix like this:

Three True Outcome Strategy
Strategy Swing Take
Strike Home Run Strikeout
Ball Strikeout Walk

In this situation, neither pitcher nor batter has a strictly dominant strategy. Without some idea what your opponent is going to do, you can’t know what you should do. Another example of this is rock-paper-scissors, where you can’t describe an optimal strategy without considering your opponent’s behavior.

Let’s apply this thinking to a specific baseball situation. On June 19, Lucas Giolito’s day got off to a rough start. Kyle Schwarber greeted him with a double on the first pitch of the game. After a strikeout of Kris Bryant, things got dicey again — Giolito fell behind Anthony Rizzo 3-0. Just like that, we have our situation: 3-0, man on second, one out, no score.

For our purposes, we’re going to replace Rizzo with a perfectly league average batter. Before a single pitch was thrown in the at-bat, the run expectancy in this situation was 0.7136 runs. In other words, an average team would score 0.7136 runs on average after bringing a batter to the plate with a man on second and one out. By 3-0, though, the situation got even worse for Giolito. Plugging in the results of major league batters after 3-0 counts and converting those situations to run expectancy, on average, 0.8545 runs would score.

Now that we have our baseline, it’s time to figure out what the pitcher and batter can do to affect it. The pitcher’s options are fairly limited. Pitchers throw more than 95% fastballs in this situation, which is close enough to 100% that we can just ignore the chance of other pitches. All the pitcher can decide is his zone rate. There’s a decent range here. 61.5% of pitches on 3-0 are in the strike zone league-wide, which is the pitcher’s baseline option. There’s also the option of being incredibly aggressive — on 3-0 counts to opposing pitchers, that rate balloons to 68%. That’s the lay-it-in-there level. On the other extreme, there’s treating 3-0 like any other pitch — fastballs have a 52.1% zone rate overall per Baseball Savant.

The batter’s options exist on a similar spectrum. He could swing literally never, like Matt Carpenter. He could swing an average amount — batters swing at 15% of strikes and 4.2% of balls on 3-0. He could go all out like Acuña and swing at, say, 60% of strikes and 16.8% of balls (four times as likely to swing at either). The point is, there are options.

The reason to swing 3-0 is that you get to hit a fastball that’s likely to be a strike. Put a ball in the strike zone in play on 3-0, and major league batters have a cool .506 wOBA over the past two years, better than Mike Trout. That’s the brass ring — connect on a 3-0 strike, and you’re going to do damage. It’s not all sunshine and lollipops, of course — you might foul the ball off or whiff. Those aren’t even that bad, though, because there’s no difference between a swinging strike and a called strike.

The real downside to swinging 3-0 comes on out-of-zone swings. Pitchers just miss sometimes, and passing on a walk to swing at a ball outside the strike zone is devastating. If you miss it, you’ve turned a walk into a 3-1 count, but if you hit it, things are even worse. There’s just not much hay to be made on putting an out-of-zone pitch into play — even on 3-0, batters have produced only a .337 wOBA when they make fair contact since 2015 (I expanded the years I was looking at to bulk up the sample size for out-of-zone swings).

Now that we have our strategies, I worked out how they interact. I applied the in-zone and out-of-zone swing rates from the batter’s strategies to the zone rate the pitcher chooses, worked out the run value of each event (walk, ball in play, 3-1 count) using a run expectancy table, and averaged across the likelihood of each event to work out the run expectancy from each interaction.

One quick note: I made some adjustments to the value of a ball in play based on how much the pitcher was filling the zone. For the highest strike rate, I used outcomes from pitches in the very heart of the zone, because if you’re selling out to throw a strike, the batter will get a lot more of those to hit. For the lowest zone rate, I used the value of a 2-0 ball in play to simulate that the pitcher is working more towards the periphery than they do on 3-0:

3-0 Strategy, Man on Second, 1 Out
Strategy No Swings Average Aggressive
52.1% Strikes 0.8691 0.8651 0.8532
61.5% Strikes 0.8553 0.8532 0.8381
68% Strikes 0.8457 0.8472 0.8519

What does this mean? For the batter, there’s no dominant strategy. If the pitcher avoids the zone or just throws an average number of strikes, the best decision is to not swing at all, though there’s basically no difference between not swinging at all and swinging an average amount if the pitcher throws 61.5% strikes. If the pitcher is flooding the zone, however, the batter’s best decision is to swing aggressively — contact is more valuable and walks are less likely, which tips the equation. Neither strategy is always correct in a vacuum.

For the pitcher, we can say a little more. If the batter is swinging aggressively, the pitcher should throw an average number of strikes, and otherwise he should flood the zone. There is one interesting point, though — there’s no situation where avoiding the zone makes sense. No matter what the batter’s tendencies are, you can always do better by throwing an average number of strikes than a below-average number. We call this a strictly dominated strategy — there’s no reason for the pitcher to ever select it, because there’s no option the batter can choose that makes avoiding the zone better than throwing an average number of strikes.

If neither player has an option that is best regardless of what their opponent chooses, we’re left with what’s called a mixed strategy. Think of rock-paper-scissors again — the optimal strategy is to play each choice a third of the time at random. Anything else will be exploited. The pitcher and batter both need to select some mix of behavior such that the other side’s choices are equivalent.

I’ll spare you the mathematical details, but this equation can be solved. The pitcher should sell out for strikes around 59% of the time and pitch like a regular pitcher would on 3-0 the other 41% of the time. The batter has multiple optimal sets, but they all involve being all-out aggressive between 50-65% of the time, splitting the remainder of the time roughly evenly between only taking and swinging normally. The more the batter is all-out aggressive, the more he needs to offset it by taking all the way — at almost exactly 60% aggression, taking and behaving normally are perfectly balanced:

Optimal Pitching Mix
Strategy 52.1% Strikes 61.5% Strikes 68% Strikes
Percent 0.00% 59.20% 40.80%

One Optimal Batting Mix
Strategy No Swings Average Aggressive
Percent 20.15% 20.15% 59.70%

What this means, in practice, is that pitchers are probably not being aggressive enough on 3-0 counts. To avoid being exploitable, they need to be filling the zone at all costs more often than they do now. Batters, by the same token, are probably behaving optimally now given what pitchers are doing — as you can see from the table above, there’s almost no difference between swinging normally and not swinging if the pitcher is throwing a league-average amount of strikes. If pitchers start getting more aggressive, though, batters need to go up with a swing-heavy mindset reasonably often to keep pitchers from getting the upper hand.

I happen to think this conclusion is pretty neat. Pitchers aren’t being aggressive enough avoiding walks. If they start to behave that way, though, batters will have to counter by swinging more. The fact that doing either all the time is exploitable also makes complete sense — that cat-and-mouse dynamic is obvious, and it’s a good sign that the theoretically correct answer takes that into account.

The situation, by the way, changes a lot with the context of the game. If there are two outs instead of one, walks hurt the pitcher much less, which makes the grid look like this:

3-0 Strategy, Man On Second, 2 Outs
Strategy No Swings Average Aggressive
52.1% Strikes 0.4321 0.4362 0.4485
61.5% Strikes 0.4268 0.4335 0.4534
68% Strikes 0.4231 0.4301 0.4648

Just like that, there’s a dominant strategy, a natural equilibrium. The batter should swing aggressively, and the pitcher should throw a normal fastball without regard to count. The reasoning behind this is pretty clear — the baserunner you allow via a walk is a lot less dangerous with two outs than with one, while contact is at a huge premium for hitters, as it will often drive the run in from second base.

The point of this analysis isn’t to say that these answers, these strategy blends, are the immutably correct decisions. There’s some estimation going into my run values, and the margins are small on each individual pitch. The three options I gave each player aren’t exhaustive — they’re just an approximation. Baseball is far more complex than this. There’s a tendency, though, to want one side to make a huge behavioral shift without thinking about the other side. Ronald Acuña swings like a madman on 3-0, and it’s fun to wonder whether that’s good or bad.

But it’s a false choice. Swinging more or less means nothing without considering what pitchers should do in response to that behavior. Hitting isn’t a static game played by batters, and pitching isn’t a static game played by pitchers. The two interact, and in situations as extreme as 3-0 counts, the way they interact can be hugely meaningful.

Ben is a writer at FanGraphs. He can be found on Twitter @_Ben_Clemens.