The accuracy of a game theory analysis of baseball
Baseball has long been considered America’s national pastime. On the surface it is a team sport: nine players playing for each team at any given point of the game, with even more players waiting to be substituted in to catalyze a swing in momentum or retain a late game lead. But beneath the surface the main aspect of the game of baseball, the interaction between the pitcher and the batter, boils down to a two player game. The pitcher has several different pitches in his arsenal, no doubt some of them more effective than others. The batter needs to react to whatever pitch is thrown at him and decide whether or not to swing. There are various outcomes, each favoring either the pitcher’s team of the batter’s team. In this sense of the game, the at bat between a pitcher and a batter can be analyzed using game theory. While many assumptions (which I will discuss) need to be made, the analysis can lead to interesting strategies from each player. Even more interesting is the comparison between these idealized strategies and what actually takes place during real baseball games. This comparison leads us to see how truly complex the game of baseball has become, and how truly complex our analysis would have to become to accurately model the game America loves.
The first assumption that we have to make to analyze baseball using game theory is that there are only two players present during the at bat. This however, is not the case in reality. The batting team has a batter at the plate and a coach in the dugout who is more often than not feeding the batter signs as to how he should proceed. For the sake of our analysis, we will assume the batter knows everything his coach is going to tell him, and consider the batter to represent his entire team. The pitching team has much more going on. We will assume the pitcher most likely has around 4 different pitches that he could deliver to the batter (there are some pitchers who throw almost exclusively one pitch). he could throw a fastball, a changeup, and two different types of breaking balls, probably either a slider or a curveball. For the sake of our analysis, we will only consider a pitcher with only two pitches to choose from. Our pitcher can throw a fastball, or he can throw an offspeed pitch (this will encapsulate changeups, sliders, and curveballs). This assumption is well justified, as professional batters typically guess ahead of time whether they are going to see a fastball or any of the three offspeed pitches. While their preparation for the fastball is generally unique (they need to be prepared to swing faster), their preparation for the other three is often similar (they need to wait and read the break of the pitch). Now that we have defined both sides’ possible responses, it is possible to start to assemble our payoff matrix.
Player 1 is the pitcher, Player 2 is the batter. Player 1 can throw a fastball (F) or an offspeed pitch (O), and Player 2 can guess F or O as well. Using John’s Walsh’s data from an article he wrote in 2007, we can assemble the payoff matrix in the following manner. We will assume the payoffs for each player are the probabilities that their choice will end in a favorable outcome. Likewise, we will average the probabilities for all the offspeed pitches and consider them as one choice, O.
We will consider the pitcher’s outcome to be the sum of the probabilities: p(called strike), ⅔*p(foul ball), p(swinging), and ½*p(in play). We are assuming that ⅔ of the time, a foul ball results in a strike against the batter (if the batter has 0 or 1 strikes currently), and that ½ of the balls put in play result in outs. We will consider the batter’s outcome to be the sum of the probabilities: p(ball), ½*p(in play). We use one of the same assumptions we used above, assuming ½ of the balls put in play result in a hit for the batter. The last large assumption we will make is that if the batter guesses wrong, his payoff is half of what it would be if he guessed right.
The payoff matrix looks as follows:
Now if we analyze this matrix, it becomes clear that there is no pure Nash Equilibrium. Therefore, we must look for a mixed strategy Nash Equilibrium. If we run the procedure to find mixed strategy Nash Equilibrium, we come up with the following values for p (the probability the pitcher will throw a fastball) and q (the probability the batter will guess fastball):
(p , q) = (.5428 , .5143)
Our results tell us that the pitcher will pitch a fastball about 54% of the time, and the batter will guess fastball about 51% of the time. Comparing this to the values that John Walsh collected over about 310,000 pitches we can see the accuracy of our idealized situation.
According to the article, a fastball was thrown about 59% of the time, and an offspeed pitch was thrown about 40% of the time (the remaining 1% most likely is due to the omission of rare pitches not often thrown). Our estimation yielded fastball being thrown 54% of the time and offspeed being thrown 46% as the optimal way for the pitcher to act. Our estimation that the batter should guess fastball 51% makes sense when we look at Walsh’s other data. A fastball being thrown results in a home run about 3.7% of the time, more than any other pitch. Likewise, the batter’s average when a fastball is thrown is .330 (meaning they got a hit 33% of the time), which is also higher than any other pitch.
There are a couple of factors that could have caused this discrepancy. One factor is the consideration that a pitcher often throws fastballs when he needs to throw a strike, regardless of what the batter may guess. For example, on a 3-0 count (3 balls and 0 strikes thrown thus far in the at bat), a fastball was thrown 84% of the time according to Walsh’s data. The fact that pitchers can “count” on their fastball to be more accurate is a factor we did not include in our analysis. Likewise, our analysis did not take into account the fact that different pitches are more skilled at throwing certain pitches. If a pitcher has an amazing curveball but a subpar fastball, he is more likely to throw the curveball than another pitcher with different talents.
Another set of factors that our analysis did not take into account is how our payoff matrix would change over the course of the game. As a pitcher progresses through the game, his stamina drops and consequently the effectiveness of his pitches drops. Therefore, if we ran the payoff matrix again at a later point in the game, we could shade down his payoffs by some multiplying factor delta 1. Likewise, the more times a batter faces a pitcher, the more accustomed to his pitching the batter becomes. Therefore, we could increase the batter’s payoffs later in the game by some multiplying factor delta 2. This would change the payoffs and thus change the outcome of our analysis.
An idealization like this one, while it does shed some light on baseball player behavior, underlines the fact the baseball is a complex game, full of human error, emotion and talent. No matter how accurate your assumptions are, there is no way to precisely model the outcome of an at bat, simply because the outcome always depends (at least a little) on the talent of the players involved. When you consider the fact that there are nine players playing for each team at a time, the idea that we could capture all of those players’ tendencies, talents, emotions, etc. is a ludicrous one. The uncertainty that will always remain in baseball is what makes the game great and will keep fans watching for a long time.
http://www.hardballtimes.com/main/article/fastball-slider-changeup-curveball-an-analysis/