Combining Baseball and Game Theory

Baseball, at its heart, is a game between pitcher and hitter, and this game is particularly susceptible to game theory analysis. Over the next few weeks, I want to try to start figuring out ways to break down this game theory, and perhaps understand the game better. I'm not entirely sure how this would work, but I have some ideas.

To get the basic idea of how a game theory interpretation of an at-bat would go, let's look at an extremely simplified version of the pitcher-hitter confrontation. Suppose that on any given pitch, the pitcher has two choices: throw a strike, or throw a ball; and the hitter also has two choices: swing, or don't swing. Suppose that if a hitter swings at a strike, they make solid contact and get a hit, and if he swings at a ball, he whiffs. This would create a payoff table looking something like this:



The hitter, unsurprisingly, benefits from swinging at strikes and not swinging at balls, whereas the pitcher benefits by getting the hitter to swing at balls and not swing at strikes.

This is all unsurprising, but when used in a much more detailed model, this could be a tremendous tool for analysis. What I hope to do is, by analyzing pitch-f/x data, come up with accurate utility numbers to plug into payoff matrices for pitchers, and then with these utility numbers, find Nash Equilibria for the pitchers against league-average hitters (or against specific hitters). These Nash Equilibria would give us the frequency at which the pitchers theoretically should throw each of their pitches, which could then be compared to the frequency they actually do throw these pitches.

To give an example of how this could be used, consider this article from Hardball Times, where Josh Kalk looks at pitch sequencing and finds, among other things, that throwing the same pitch twice in a row is often effective. One possible explanation for this is that hitters just don't expect to see the same pitch twice in a row; indeed, under the section on curveballs Kalk says, "The main exception appears to be curveball/curveball, which appears surprisingly good. Hitters must not be expecting a second curveball. Maybe they got in a hole early and then when they laid off the first curveball they were expecting the pitcher to come back with something hard."

However, if pitchers always followed their curveball with another curveball, hitters would never be fooled, because they'd expect the second curveball. Clearly, there is some frequency at which it is best to follow a curveball with another curveball, and a game theory model has the potential to find this frequency.

Hopefully I can wring enough good data out of pitch-f/x to get some real conclusions out of a game theory model.