We were tasked with fine-tuning the strike zone of Robot Baseball to maximize the probability of an at bat reaching a full count, and finding this maximal probability. The most straightforward way to do this was to work backwards from full count down to the start of an at bat, for a given strike zone probability p, determine the Nash equilibrium of pitcher’s probability to throw a strike and the batter’s probability to swing.

An interesting symmetry to simplify the work: the solver might notice that the outcome of a pitch is symmetric with respect to the pitcher’s choice and the batter’s choice. This means the probability of the pitcher throwing a strike and the probability of the batter swinging will be the same in any scenario (both players want to choose a probability that makes the opponent indifferent between their two options, otherwise their move is exploitable and will reduce their expected points (or for the pitcher increases the opponents expected points)).

Once we work backwards to the starting count of zero balls and zero strikes, we can compute the probability of reaching a full count and then adjust p in order to maximize it. Happily we are asked for the maximal height of the function, which is easier to estimate than the location of that maximum (the p value that achieves it). Careful calibration finds the maximum probability of reaching full count to be 0.2959679934… (achieved at approximately p=0.2269732…).

Other stats of interest: optimal baseball pitchers have an impressive 68% strike out rate and admit 21% walks. Batters hit home runs at 11% of plate appearances, and have a slash line of .139/.318/.556.

Congrats to this month’s solvers!