This March, the MLB regular season was right around the corner, and in preparation for the SABR Analytics Conference the Cardinals analytics department gave us the opportunity to investigate and present on MLB's new three-batter-rule, which stipulates that any reliever entering the game must pitch until having faced three batters, or until the end of the half-inning. Four long months later, we're still yet to see the this new rule in action - though we should soon. This article breaks down the analysis performed by myself, along with would-be SABR teammates Owen Ricketts, Tory Farmer, and Adam Akbani.
To approach this project we built a Markov chain Monte Carlo simulation, playing out the results of relief pitching appearances in different late game scenarios - 784 of them to be exact. This accounts for every permutation of pitcher handedness/the handedness the first three batters, as well as every base/run/out state in the 8th or 9th inning with the home team pitching and leading by one run or tied. Using Baseball Prospectus' run expectancy matrix, and Tom Tango's win expectancy matrix, we assigned expected run and win outcomes to each of these scenarios to provide a baseline for our simulation. We also compiled the average outcomes from the most used left-on-left and right-on-right relief pitchers from 2017-2019, and used that as the baseline for our simulation. More detail and the code itself can be found on the GitHub here.
For each of these 784 scenarios, we simulated the outcome 2000 times, either until the end of the inning was reached, or until the pitcher had faced three batters. At the end of each individual simulation, we assigned a run value, and in the most revised version, a win value based on the outcome. The win value was the win expectancy for the home team wherever the simulation 'left off', and the run value was however many runs were scored (if any), plus the run value of the remaining runners on base/due up if there were fewer than three outs. Here's a concrete example:
Runner on 2nd, 1 Out
Top of the 8th inning, up by 1
RHP facing three LHB
In this trial, let's say the pitcher records a strikeout, then gives up two singles, leaving off a tie game with two outs and runners on the corners. The win value added is ending win expectancy of the situation (0.493) minus the initial win expectancy of the scenario (0.702), resulting in -0.209 wins added. Because one run scored and the remaining situation has an expected run value of 0.5182, this results in 1.5182 runs added. Complete this same scenario 2000 times for each row and average these calculations, then do the same for each of the 784 rows, and you have the results.
After compiling the averages for all of the scenarios, the results indicate fairly minimal gains in both wins and runs, even accounting for these high leverage situations. This matches up with some previous exploration on the subject, and is visualized below.
As you can see, this simulation skews slightly towards pitching, meaning that the average scenario is worth about 0.015 more wins for the pitching team in our simulation than in real life. First - this is pretty small, and would amount to under than a full win over the course of the new 60- game season. Second - it's almost definitely explained by our selection of pitchers, those who survived to pitch the most in 2017-2019 are better than the average that these baseline expectations are taken from.
Sidebar: One avenue for future exploration with this project would be implementing different stat lines instead of the generalized LHP and RHP we created. As expected, the RHP was less subject to platoon splits compared to the LHP, which informs the results later on.
As a result of this, the better way to find the meaning in the data is to compare simulated results to those of the opposite handed pitcher, which is plotted below.
Here the margins are even finer, with a few outliers in the range of 1/20th of a win. These scenarios are where, in a pinch-hitter free, one sided game, bringing in a relief pitcher brings the most benefit to the pitching team.
These are all incredibly extreme examples, where using a RHP is obviously quite favorable, but there are also some instances where bringing in a pitcher to face 2/3 batters of the opposite handedness is slightly advantageous, as also covered by Fangraphs' Craig Edwards. The margins for each of these decisions are in the hundredths of wins, however.
Ultimately, even when accounting for the possibility of pinch hitting, the margins become even slimmer, as diagrammed below. In conclusion, there are distinct benefits to bringing in relief pitchers in crucial situations to get the first batter out, but overall the effects of the three batter rule are small, even at a time when baseball's insignificant minutiae doesn't feel extra insignificant.
Special thanks to Isaiah Berg, Kevin Seats, and Matt Bayer of the St. Louis Cardinals for giving us feedback and for providing the premise for this project.
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