The 3-0 Count Dilemma

While it might not appear so, baseball games constantly portray economic thought, such as in the mathematical model of game theory. There are many ways game theory takes place, but a classic example is the prisoner’s dilemma. Imagine a police officer is interrogating two suspects of robbing a bank together. The police officer has some evidence to put them in jail, but a confession would go a long way. Each suspect is contemplating confessing to the crime. If both suspects keep quiet, they will each receive five years in jail. If one suspect confesses and the other keeps quiet, the one who kept quiet will receive 20 years in jail while the suspect who confessed will receive just one year. If both confess, they each receive 10 years in jail. The logical choice for each suspect is called the dominant strategy. The end result, or the combination of each suspects decision, is called the Nash Equilibrium. By using game theory, we come to the conclusion that each suspect should confess to the crime, meaning they will each get 10 years in prison. I won’t go much into why this is the case, but feel free to research more about game theory and the Nash Equilibrium on your own.

What does this have to with baseball? We can think of each pitch as game theory, with each suspect as the pitcher and batter. Instead of confessing to a crime, the pitcher is contemplating throwing a ball in the strike zone while the batter is contemplating swinging. While the prisoner’s dilemma has a Nash Equilibrium, not only does a pitch to a batter not have a Nash Equilibrium, but the combination of decisions is constantly changing. If the batter’s dominant strategy is to swing, then pitchers will throw more balls outside the batter’s reach. If the pitcher’s dominant strategy is to throw a ball, then the batter will take more pitches.

We could observe this thought process for every pitch thrown. However, let’s look at one type of pitch: 3-0 counts. If you are the batter, it might seem obvious to take the pitch. The worst-case scenario is you end up with a 3-1 count. If you are the pitcher, it might seem obvious to throw an easy strike. You do not want to walk the batter, and you know the batter doesn’t want to swing and risk giving you an easy popup to get out of good count. So I guess the batter should take every pitch and the pitcher should throw the ball right down the middle every time.

However, it does not always work like that. The most popular example of this thought process at work was when Fernando Tatis Jr. took Juan Nicasio deep to right center for a grand slam on August 17, 2020.

I’m not going to get into Chris Woodward’s opinion on this play because I feel his logic is wrong. However, the Padres said that Tatis missed the take sign. Should he have taken the pitch? He hit a grand slam, so the answer is likely no, but you should not judge your decision based on the result. You should judge it based off the information beforehand. Tatis is one of the best hitters in baseball and Nicasio is not a great reliever. His four-seamer gets up to only 93 mph and all the pitches in his arsenal have little movement. The pitch was not in a perfect spot for Tatis, but he is good enough to crush the ball over the right-center wall.

However, every batter is not as good as Tatis, and every pitcher is not as average as Nicasio. In order to measure how batters and pitchers fare during these occasions, let’s look at the weighted On-Base Average (wOBA) for 3-0 counts.

Baseball is a game of adjustments. Let’s say pitchers tend throw more pitches in the strike zone on 3-0 counts while the batter tends to take the pitch. Then the batter will adjust and swing at the pitches in the strike zone. If they start to swing more, then the pitcher will adjust and throw the ball out of the strike zone, because the batter will start swinging at bad pitches. However, the batter will adjust and take more pitches because they are difficult to hit. Then the pitcher will adjust and throw pitches right down the middle for an easy strike, and we are right back to where we started. This is obviously a simplified version of how the batter and pitcher will think, but it helps us visualize the thought process.

We can divide the possible outcomes on a 3-0 pitch into 4 distinct categories:

1. Pitch is in the zone, batter swings
2. Pitch is out of the zone, batter swings
3. Pitch is in the zone, batter takes
4. Pitch is out of the zone, batter takes

The two wOBA values that are important for analyzing the data are the value of a walk and the value of a 3-1 count. During the 2020 season, a walk had a wOBA value of 0.70. The average wOBA of a pitch that a batter takes that is in the strike zone is 0.416. We get this value from the wOBA of batters on a 3-1 count or 3-2 count. We do not only use the value from a 3-1 count because a 3-2 count is still a possible count that can arise from taking a 3-0 pitch in the strike zone.

For the 2020 season, here is how the wOBAs looked for each scenario. The average wOBA on a 3-0 count regardless of where the pitch is or whether or not the batter swung was 0.514.

Before we go any further, there are a few things that need to be cleared up. Foul balls are considered contact. The wOBA would be higher if we didn’t consider foul balls, but I felt they should be considered contact. Also, the last two wOBA values look a little strange. Theoretically, if the ball is in the zone and the batter takes, the wOBA should be 0.416. However, some umpires called these pitches balls, so the wOBA is a tad higher. Also, the wOBA for a pitch that the batter takes out of the zone is 0.655. It should be 0.70, but the umpires call some of these pitches as strikes.

If we want to evaluate what the batter and pitcher should do, we also need to analyze how their patterns change throughout the season. How does the wOBA of a pitch in the zone versus a pitch out of the zone change throughout the season? How does the wOBA of a pitch taken versus a pitch swung at change?

First, let’s take the viewpoint of the batter. The batter only has control over whether he swings or takes. Here is how wOBA changes throughout the season for a pitch taken versus a pitch swung at. The average wOBA on a 3-0 is also shown for relativity.

Near the beginning of the season, it seemed like the batter should swing at the pitch. They increased their wOBA by around 0.2 points if they swung at the pitch versus if they took. However, the wOBA drastically decreased by the first week of August and stayed at around 0.47 for the rest of the season while the wOBA for a pitch taken hovered around 0.52. One possible explanation could be a rise in swing rates. If the batter is swinging a lot, then the pitchers might tend to throw more pitches out of the zone. But that’s not the case the here. At first, the swing rate is volatile, but then it flatlines around the time the wOBAs flatline. This is most likely because of the law of large numbers. From the numbers at the end of the season, it sure looks like the batter should take the pitch. If they do, they can expect their wOBA to be around 0.05 points higher.

Now let’s look at the viewpoint of the pitcher. They only have control on where the pitch goes. Obviously he can’t be certain whether or not a pitch will be thrown in the strike zone, but we will use the wOBAs for pitches thrown in the zone versus out of the zone. The average wOBA on a 3-0 offering is also shown for relativity.

There is a little bit of volatility in wOBA near the beginning of the season, but this could be explained by a small sample size. There is a great difference between a pitch in the zone versus a pitch out of the zone, and each line doesn’t change much throughout the season. A pitch in the zone has a wOBA of 0.435 while a pitch out of the zone has a wOBA of 0.644. The pitcher has a much riskier decision to make than the batter. The difference in their respective decisions is a wOBA value of 0.21, while the batter’s decision has a difference of 0.05. This makes sense given the excellent position the batter is in. A 3-0 count is terrible for the pitcher, so the outcome of their decision puts them in much greater risk. Also, they don’t have as much say in their decision as the batter. A batter can choose if he swings or not, while a pitcher can try to put the ball in the strike zone, but he can not be certain. A pitcher in a 3-0 count likely doesn’t have their best stuff, so the outcome of their pitch will likely have a greater variance. In short, someone with less control over the outcome will face a greater risk.

Let’s look at a combination of pitches swung at in the zone, swung at out of the zone, taken in the zone, and taken out of the zone.

It makes sense that the outcome with the highest value is when the batter takes a pitch out of the zone. This is a good decision by the batter, even if the ump decides the give lenience to the pitcher. Taking the pitch is definitely the safe bet, because at worst, you will face a 3-1 count on your next pitch. An expected wOBA of 0.416 is a good position to be in. While it is probably a safer decision to take the pitch, it is a good idea to mix in a few swings to keep the pitcher guessing. If the batter is a pretty good player and the pitcher has been struggling, it is a good idea to give the batter the green light. The batter should look for one pitch in one spot and if he gets it, swing away.

The pitcher has a much riskier decision to make. If he throws an easy pitch down the middle, he is at risk of letting up a big hit. The answer is not black and white. You should not throw an easy pitch down the middle and you shouldn’t throw a waste pitch out of the zone. The pitcher should approach the upcoming pitch the same way he approaches any other pitch. Fortunately for him, there is a chance the umpire will be on his side if the pitch is borderline. Out of the 2,563 pitches that were taken on a 3-0 count, 223, or 8.7% of those, were incorrectly called. Out of those 223, 163 of them, or 73%, were strikes. If the ump is going to make a mistake, it will likely be in favor of the pitcher. Out of the 1,053 pitches that were taken out of the zone, 15.7% were called strikes. Even if a pitcher narrowly misses his spot out of the zone, there is a slight chance the umpire will give him the benefit of the doubt.

This situation is always going to benefit the batter. An expected wOBA of 0.514 on a 3-0 count is a pretty good spot to be in. As for the pitcher, you cannot give the batter an easy pitch because it might end up in the bleachers, but even if the pitcher misses his spot, the umpire might feel generous.

Did Tatis make the right decision? It was low and away and not down the middle, but Tatis managed to clobber it out of the park for a grand slam. After the game, Padres manager Jayce Tingler said that Tatis was given the take sign. This is the safe decision, but Tatis proved that swinging was the right decision in the moment. Given that he is one of the best hitters in baseball and Nicasio does not generate a lot of velocity, you would probably give Tatis the green light there in the future. However, as more players swing at 3-0 counts, more pitchers will throw balls out of the strike zone, and now we are right back to where we started.

Statistics courtesy of Baseball Savant. Graphs made using RStudio.