One of the sports most amenable to game theory analysis is baseball. This is due to a lot of factors, including the vast number of isolated, 1v1 encounters (on every pitch thrown) and the relative ease with which results can be quantified for each player (as opposed to say, football, where it’s difficult to measure the impact a defensive lineman has). The batter has two decisions on a given pitch, take it or let it rip, and the pitcher might have 2, 3, or 4 decisions depending on their pitch repertoire. This means, for a given at-bat, there should be a Nash Equilibrium for the pitcher when choosing what pitch to throw. The batter has a chance to react to a pitch, so there’s not necessarily an equilibrium for them.
The question is, do we expect most pitchers to naturally gravitate towards their own Nash Equilibrium as they realize which pitches work at which frequencies? We’ve seen that football coaches, when deciding to call run vs. pass plays in short yardage situations, have roughly the same success rate with both types of plays, a product of finding the Nash Equilibrium of that situation. Have most pitchers found their own balances naturally, or do a lot of them have room for optimization?
This article from fivethirtyeight suggests that no, many pitchers have not fully optimized their pitching decisions, and would benefit from throwing certain pitches more. Take R.A. Dickey, for example, who throws a knuckleball, one of the weirdest pitches in the game. Despite throwing the pitch a staggering 87% of the time, analytics actually suggests he should be throwing it even more than that. This was calculated using FanGraphs data of pitch type, potency, and the amount of success opposing batters have against it.
Is this analysis actually valuable? There are strong points for and against the value of analysis like this. On one hand, one standard deviation of optimality of pitch choice over the entire population of pitchers results in one-quarter of a standard deviation less ERA, which is an expected, although small correlation. However, small sample size often plagues analysis of pitcher data, as there are so many possible situations they can face. A pitcher’s optimality has very little indication of what their optimality will be next season. All said and done, though, there is definitely something to be gained by applying game theory to pitch selection.