"The most important pitch in baseball is strike one." – Attributed to many.
"Just be careful who hears you say that…" – RAC
There's no question that, for a pitcher, getting a first-pitch strike and a 0-1 count is much better than a first-pitch ball and a 1-0 count. It's not complicated to figure out why, and so as a result, coaches from Little League to the big leagues tell pitchers that they should get ahead of the hitter. Great. How?
Well, that's a little more complicated, primarily because the confrontation between batter and pitcher is a game within a game. When I say it's a game, I mean that in the proper sense of the word. The batter/pitcher matchup is a game of strategy, bluffing, and sometimes, outright guessing. It's like poker, with projectiles. Like in poker, there's an element of the unknown. The batter doesn't know what the pitcher is going to throw. The pitcher doesn't know what the batter is looking for. We have entered the wonderful world of game theory. Game theory is an area of study that looks at these types of situations which involve strategic thinking. One person's actions affect the other person's. Game theory is the sort of thing to which one could devote , but here we'll stick to some basics.
It's not entirely true that the pitcher and batter have no idea what the other will do. Both have doubtless studied the scouting reports and are perhaps drawing on their previous interactions with each other (and in other situations) to figure it out. Anything to gain an advantage. Before we continue, let me vastly oversimplify the possible things that might happen as the pitcher goes into the windup. It'll make things a little easier to digest. We'll assume that the pitcher has one pitch, a fastball, and that he can throw it either down the middle or out of the strike zone. The batter, for his part, can either swing or not (at least that's real). If it's one of those right-down-Broadway pitches, he will hit it. If he swings at the ball out of the zone, he'll swing and miss.
Now, since we're playing with the boundaries of reality, let's make the pitcher psychic (or Greg Maddux; same thing.) He knows, even before he winds up, that this batter will take the first pitch, no matter what. Logically, he will throw a fastball down the middle. Even if we gave him back his full arsenal of pitches, he'd probably do the same thing. (What usually happens on a 3-0 pitch?) Why mess around with breaking stuff which might go flying all over the place when you have a gimme strike? If you know that poor predictable Bart will go with rock…
Both batter and pitcher are trying to gain what's called a second-move advantage. Imagine how easy rock-paper-scissors would be if the other player had to go before you. This is the reason that pitchers bother with breaking balls, off-speed pitches, and deception. A good curve breaks close to the plate, so that up to the last moment, when the batter is trying to decide whether or not to swing, he's not sure if it's a good idea or not. If he can pick up the spin on the ball out of the pitcher's hand (as Ted Williams was alleged to be able to do) and project its course from there, he has a second-move advantage in that the pitcher has already made his move and the batter knows what it is.
Back to our overly simplified universe where the pitcher has two pitches (down-the-middle fastball or fastball outside). Let's assume that the pitcher is not Maddux and that the batter is not Williams. Both batter and pitcher are essentially guessing. We're in a simultaneous move game. What should the pitcher do, since he’s the one who has to throw the ball? (Note: the question is not "What will he do?" We'll take that up in a moment.) Game theory says that he should take a look at the rewards that go with each choice from the perspective of what the other person in the game might do, in this case, the batter. We can, with some ease, come up good estimates of what it's worth to a batter to hit the ball, or to have strike one or ball one. Those particular calculations are for another day. It's enough here to say that we could figure it out with a little elbow grease.
Warning! Mathematical Equations Ahead!
Instead, let's make up some numbers. For the pitcher, a batted ball is worth -12 (it doesn't matter 12 of what, I'm picking numbers for ease of calculation.) A strike is worth +4. A ball is worth -8. For the batter, you change the signs.
So, if the pitcher picks "down the middle," then the pitcher's rewards are:
Batter's swing % * -12 + (1 – Batter's swing %) * 4
If the pitcher picks "outside," then the pitcher's rewards are:
Batter's swing % * 4 + (1 – Batter's swing %) * -8
If our pitcher is playing the game correctly, he'll be looking to see which side gives him a better outcome (assuming that he has a decent idea of what the batter's tendency to swing is). So, if "down the middle:" will give him a better outcome, he'll go with that 100 percent of the time. The problem is that since the batter will see multiple pitches, it won't take him long to figure out that he should change his swing percentage to compensate. As you might imagine, a cat-and-mouse game would ensue, and the batter would want to find a good balance where neither "down the middle" or "outside" gave better results, so that the pitcher was constantly guessing. So, we can set both of those sides equal to each other (using "p" to stand in for "batter’s swing %"):
-12 * p + 4 * (1 – p) = 4 * p + -8 * (1 – p)
p = 12/28 = 42.9%
If our batter swings 42.9 percent, he'll have the pitcher guessing. We could go through a similar set of calculations for the pitcher. Because the numbers are all the same—just in the opposite direction—for the pitcher, it works out to the same number. At that point, the two sides have played each other to what's known as a Nash equilibrium, where neither one has an advantage. Again, the number is the product of made-up inputs, but the important thing to know is that there is an equilibrium, and that baseball is the sort of game where there is going to be strong natural pressure to reach that equilibrium.
That's how it plays out in theory. What about reality?
First off, pitchers have more than two selections for a pitch; this isn’t an Atari 2600, and the outcomes aren't so cut and dried (batters do hit home runs on balls out of the strike zone and swing and miss at balls down the middle). But we're looking at first-pitch strikes. A pitcher does have a bit of a tradeoff in locating his pitches. The closer to the heart of the plate he locates the pitch, the more likely it is to be called a strike if the batter takes it, but the more likely it is to be hit if the batter swings. If he tries to paint the corners, he runs a higher chance of the pitch being called a ball, but the batter isn't as likely to swing and probably won’t make as good of contact as he otherwise would.
And Then the Pitching Coach Messes Everything Up…
The calculus that goes into determining the equilibrium of how often and when a pitcher should throw a certain type of pitch is complex (not impossible), but still there is still an equilibrium.
Now, what happens when the pitching coach approaches this pitcher who is at equilibrium and reminds him that, "you need to be getting ahead of the hitters, because the most important pitch in baseball is strike one."
The pitching coach means well. He thinks that he's just offering some words of encouragement and a truism about baseball. He may have completely messed everything up. Consider for a moment that the pitching coach is in a position of authority, and he is making a semi-direct request of the pitcher. The pitcher, honestly wanting to make his pitching coach happy (or perhaps not wanting to make him angry), considers how he will go about this.
In one little sentence, the pitching coach has introduced a bunch of different factors into the pitcher's mind. One is the fact that humans do not weight all information equally. The fact that the pitching coach said this very recently means that it's more likely that this will be more available in the pitcher's mind. The request also came from an authority figure, and there is always a good deal of pressure to obey authority, especially because he will be watching and he holds a certain amount of power over the pitcher's future.
The pitcher, if he is wise and he has reached an equilibrium in this delicate dance with the batters whom he faces, will ignore the pitching coach. But there's another force acting in his mind that might throw things off. It's natural that if I know that I'm going to be evaluated, I’m going to try to take as much control over the process as I can. Let's go back to our universe in which there were two types of pitches. There are two ways that I can get a strike: throw an outside pitch and hope that the batter swings or throw one down the middle and hope that the batter doesn’t swing. In throwing a ball down the middle, if the pitcher gets the strike, he can proudly claim that "I did that." If he throws a pitch outside, he is passively waiting for the batter to make a mistake. While this type of strike counts just as much, American (and especially male American) culture values being active rather than passive. All of these forces will likely lead our pitcher to begin to come into the zone more.
This may be exactly what he needs. If he's too far away from the equilibrium point in the direction of being too timid, this may fix him up. But if he's at the equilibrium point, it's going to knock him off that balance. He may indeed get more first-pitch strikes, but it may come at the cost of a bunch more balls in play, perhaps hard-hit balls. The tradeoff may not end up being all it was cracked up to be.
A Warning for Sabermetricians
There are always unintended consequences. Murphy's Law clearly states this. What may strike some people as counter-intuitive is that at no point did the pitching coach say anything that was incorrect. It really is a good idea to get a first-pitch strike and there are studies which prove it. However, putting this information into the wrong hands can have disastrous results.
There's a certain unspoken assumption in sabermetrics that if we simply spread the results of our studies far enough, someone with decision-making authority will read them and fix whatever problem we’ve identified. The problem is that information is not always perceived by everyone in the same manner, and even if it is, attempting to change human behavior is a task that must take into account a huge number of variables. It isn't that easy.
Special thanks to BP’s in-house professor of game theory (literally), Matt Swartz, for helping me out with calculating that Nash equilibrium.