Moves and countermoves: The game theory of 3-0 pitches

If you're a baseball fan, you're at least somewhat familiar with game theory, whether you know it or not. Wikipedia defines it as "the study of strategic decision making," which is accurate, if a bit obtuse. In baseball, game theory most often takes the form of trying to anticipate what the opposing player or manager is likely to do, and taking actions to counter those tendencies. Stacking a lineup with lefties against a righthanded starter is one example, and emphasizing outfield defense over offense in a game against a team prone to fly balls is another.

Probably the place game theory is applied most is during an at-bat. When pitchers decide where and what pitch to throw, they can't just throw the pitch they think is their best. They also have to consider how likely the batter is to swing, his ability to make contact, the damage he can do upon contact, and more, for each type of pitch and each potential location. The batter also has to consider the pitcher's skillset and how it interacts with his own, and both have to consider that their opponent is taking their tendencies into account! This leads to constant adjustments on the part of both batter and pitcher.

Game theory explains why, for example, not every first pitch of an at-bat is a fastball. If the batter doesn't know what's coming, throwing a hard pitch with a high probability of being called a strike is probably the best choice, but if a pitcher starts every at-bat that way, batters will start sitting fastball and making lots of loud contact. A similar thought process can be applied to 3-0 pitches. On the one hand, a walk is a very bad outcome for the pitcher, so he really wants to throw a strike. But a walk is a great outcome for the hitter, and swinging runs the risk of making an out, or turning the fourth ball into the first strike, so he is very likely to take a pitch. But if the batter takes every pitch, he'll watch some hittable pitches go by, and 3-1 is a much worse count to be in than 3-0. On the other hand, if the pitcher throws too many meatballs, though, hitters will start swinging, because of the increased chance of an outcome like this:

So you have strong, opposing incentives in 3-0 counts, pushing pitchers and hitters back and forth on how many strikes to throw and how often to swing, with each tendency informing the other. Does the data show any clear tendencies on the part of either pitcher or hitter?

Using the PITCHf/x leaderboard tool available at BaseballSavant.com, I found the number of 3-0 counts for each qualified hitter of 2014, their swing rate in those counts, and the rate of called strikes when they didn't swing. None of these include pitchouts, automatic balls or strikes, or intentional balls. 46 different players didn't swing at a single 3-0 pitch in 2014, in a total of 1,012 opportunities. The highest swing rate, by a large margin, was Carlos Gomez's 39%, or 7 of 18 opportunities. He was followed by Aramis Ramirez (31%, 5 of 16), Hanley Ramirez (30%, 7 or 23), and Albert Pujols (30%, 8 of 27). The weighted average of the entire group was 8.99%, and the median was 5.76%.

The rates of called strikes on 3-0 counts also varied widely, though this is a little different than swing rates, because even if pitchers mean to throw 100% strikes, they will miss some amount of the time. (As proof, consider that Jon Lester, current owner of a 53.7% strikeout rate and a wRC+ of negative-92, has one walk in his career.) The highest rate was 90% for Marcell Ozuna, or 19 of 21, and 90% for Elvis Andrus, or 27 of 30. The lowest were 38% for Jay Bruce, 6 of 16, and 39% for Howie Kendrick, or 11 of 28.

The expectation is for pitchers to consider the hitter's tendency to swing on 3-0 and adjust their strike rate accordingly. Is that actually the case?

Not really. There are lots of players with zone numbers seemingly totally uninformed by their swing rate. For example, Howard Kendrick, he of the second-lowest 3-0 strike rate, also has a 0% swing rate on 3-0 counts. There is a slight downward slope to the data, but as shown by the correlation coefficient in the chart above, 3-0 swing rate alone does a very poor job explaining the variation in 3-0 strike rate.

Obviously not all hitters are equal, though, and pitchers must consider the basic characteristics of the hitter as well. Pitchers are not that worried about what will happen if, say, Ben Revere makes contact, regardless of if it's on a grooved fastball, so they might pound the zone to him on 3-0 even if he swings at an above-average rate. Similarly, Giancarlo Stanton is a good candidate to do a lot of damage on a grooved pitch, so pitchers will probably not throw him a meatball, even if the scouting report says he basically never swings on 3-0.

What happens if we consider ISO, a decent proxy of "damage on contact", as well as 3-0 swing rates? When including that in a linear regression, the R-squared rockets up from .03 to .04, which is basically totally unpredictive, but both 3-0 swing rate and ISO have p-values around .1, which is a mild suggestion that there might be a real relationship. That said, there are clearly many, many other factors impacting pitcher behavior in 3-0 counts.

This really shouldn't be a surprise; Chris Teeter found that a number of situational factors influence how often a batter swings on 3-0, including the inning, the score, and the base-out state. It stands to reason that pitchers are considering those same factors when throwing a 3-0 pitch, in addition to the power of the opposing hitter and their swing tendencies. This regression only considers how pitchers react to batters, but the batters are reacting to pitchers as well, and their overall swing rate might not be predictive of their actual swing probability in a given at-bat. Finally, it's important to remember that these swing and zone rates are being calculated from a fairly low number of "trials", as it were, so there is a lot of noise in the data as well.

The regression could be run again, with variables to account for all those factors, but that's moving further from game theory and toward detailed projection. In a lot of ways, this analysis demonstrates the weakness of game theoretic models in predicting specific rates and behaviors. They tend to look like a simplified version of real life, and don't capture much of the nuance that determines the outcome of a given situation.

Where they are helpful, however, is understanding why players change their strategies, what opponent moves they react to, and what the next move for the opponent might be. For example, in 2014, Mike Trout struggled mightily on fastballs high and inside, and pitchers began to throw there at a higher rate as the season went on. He appears to have taken advantage of that predictability so far in 2015, and has almost literally stopped whiffing on those pitches and is doing a lot of damage in that part of the zone. Game theory won't be able to predict the exact distribution of pitches to Trout for the rest of 2015, but it does let you anticipate what the next adjustment will be. We probably won't see pitchers throw high and inside so frequently for much longer, and given that Trout probably had to give up some coverage of another part of the plate to increase his contact in this area so much, pitchers will probably settle on a different part of the zone.

At its core, game theory is about adjusting, and anticipating your opponent's adjustments, and anticipating that they're anticipating your adjustments, and so on. Understanding that process might not be very helpful at the micro level of predicting swing or zone rates, but it is helpful for understanding why the rates are where they are.

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All statistics courtesy of Baseball Savant and FanGraphs.

Henry Druschel is a Contributor at Beyond the Box Score. You can follow him on Twitter at @henrydruschel.