After showing that the at-bat Markov model can be used to predict and explain pitcher performance across a season of data, a logical next question is whether or not this model can be useful in the decision-making process revolving around whether or not to pull a starting pitcher from a game. In a baseball game, a team has one pitcher pitch until he grows tired or until the coach decides that the team would be better off
moving forward with a pitcher from the bullpen.
There are many factors and strategies that influence whether or not a coach should, and will, pull a starting pitcher from the game. One of these factors is certainly expected performance for the remainder of the game. To determine if the at-bat Markov model can be useful in predicting a pitcher’s future performance within a game, I ran a step-wise regression using data for every game, for all 90 pitchers, in 2015. The dependent variable is actual on-base percentage for all innings after the fourth inning for each game, OBP>4. The explanatory variables are the 112 lettered entries in M≤4,M calculated only using data from the first four innings, plus actual on-base percentage for the first four innings, OBP≤4. I include OBP≤4 in the regression because I assume that the best predictor of a pitcher’s opponent OBP in innings 5–9 of a given game will be opponent OBP in innings 1–4 of that game, and I wish to test if any of the Markov transition probabilities are useful if OBP≤4 is already used.
I ran a stepwise OLS regression in Stata using a hurdlep-value of 0.05. The results of this regression are displayed in Table 14. The adjusted R2 for this regression was quite low at 4.1%. Fourteen entries of M are statistically significant in the regression output.
Many of these variables have coefficients that make sense intuitively. For example, the coefficients on F and H are negative. These coefficients imply that all else equal, converting more (1,0) and (1,1) counts into (1,1) and (1,2) counts early on in the game corresponds to a lower OBP in the second half of the game. However, some of the coefficients do not make as much intuitive sense. For example, the coefficient on AO is negative, but AO is the probability of hitting the batter with the (1,0) pitch. All else equal, this implies that hitting more batters — generally a sign of a lack of location control — early in the game corresponds to a lower OBP later in the game. This effect is likely not reliable due to the rarity of this event. In conclusion, while
Variable Coefficient (Std. Err.) BP -1.184∗∗ (0.232) AK -0.122∗∗ (0.044) BG 0.633∗ (0.312) AB 0.108∗ (0.047) AF 0.068∗ (0.033) F -0.887∗∗ (0.178) Y -0.877∗∗ (0.192) H -0.119∗ (0.053) AP 2.016∗ (0.849) J -0.805∗∗ (0.177) BY 1.586∗ (0.649) AO -3.064∗∗ (1.165) BO 0.332∗∗ (0.124) OBP≤4 0.113∗∗ (0.036) Q 0.146∗∗ (0.040) Intercept 1.121∗∗ (0.176)
Table 14: Output of the stepwise regression of OBP>4 on the lettered entries of M and
OBP≤4. Note: ∗ denotes significance at the 5% level, ∗∗ denotes significance at the 1% level.
it is probably helpful to monitor some of these values, especially F and H, the low
R2 and the slightly counterintuitive directionality of some of the coefficients likely indicate that the at-bat Markov model does not have much promise for predicting late-inning OBP or determining whether or not to pull a starting pitcher.
9
Conclusion
My goal of this thesis was to explore and test rigorously the at-bat Markov model initially proposed by Katz thirty years ago. I have shown that a visual comparison of the matrices for two players reflects known differences in each pitcher’s skill level and pitching style. In addition, a visual analysis of changes over time in a player’s Markov chain helps diagnose the causes of changes in performance. Furthermore, I have shown that over the timeline of one season, the at-bat Markov model predicts
strikeout percentage, pitches per inning, and walk plus hit percentages that are very close to actual statistics, and this precision indicates that outputs of the model given varying inputs will be accurate.
My model suggests that the effectiveness of taking a first pitch or throwing the ball outside the strike zone on an (0,2) count depends on the pitcher. In general, taking the first pitch is more successful against weaker pitchers than stronger pitchers, while throwing an (0,2) waste pitch benefits strikeout pitchers slightly more than pitchers who pitch to contact. However, these general trends cannot be applied as blanket rules; for example, there are many strikeout pitchers who would benefit from throwing fewer waste pitches. Neither of these results take into account the potential changes in batter temporal and spatial calibration that could result from a player employing one of these strategies.
More generally, I find that, on average, pitchers should be increasing their usage of waste pitches on (0,2) and (1,2) counts in order to reduce expected opponent on-base percentage. Furthermore, these waste pitches should often consist of off-speed pitches. I find that expected opponent OBP is less sensitive to pitch selection decisions compared to pitch location decisions, and the general trends in decision sensitivity are more pronounced for pitch location compared to pitch selection.
I believe that if MLB teams examine the outputs displayed in Tables 9 through 12, and if these pitchers perform in 2016 like they did in 2015, pitchers could actually reduce their opponents’ on-base percentages, albeit by small absolute amounts. For some pitchers, namely those who currently make worse decisions, the potential reduction in OBP is larger. Appendix A displays a relative ranking of decision error for all 90 pitchers included in this study. While it would be unrealistic to expect that pitcher decision errors could be reduced to 0%, my research shows that some players have fundamentally flawed strategies on certain counts. For instance, based on 2015 data,
Masahiro Tanaka’s expected opponent OBP starting from (0,2) is more than three times higher if he throws a waste pitch on (0,2) compared to if he throws a strike on (0,2), yet he decided to throw a waste pitch 77% of the time! Although I find that the majority of players should be increasing their usage of waste pitches on (0,2), thus agreeing with traditional strategy, there is an obvious overuse of the waste pitch among pitchers who it does not benefit.
The latter portion of this thesis moves on from pitching strategy and determines how a pitcher’s transition matrix is related to his actual performance. I find that both the full transition matrix, as well as the matrix only representing transitions from one count to another, are highly related to performance as measured by WHIP and xFIP. Although this model may not have much power to predict in-game performance, it could serve as a useful tool in categorizing pitchers and determining the validity of performance across several games or a season.
All of these results are subject to limitations mentioned throughout this thesis. The most important limitation is the lack of incorporated game theory feedback. Future research using this model could incorporate a game theory perspective by constructing a payoff matrix for player decision probabilities and modeling how batters learn from pitchers and adapt to changes in pitcher strategy. Furthermore, this model interprets baseball only as a series of one-on-one events; it does not really model a baseball game in the sense that it does not account for numbers of outs and runners on base.
Pitching in baseball is a mix of science and art, and there are too many unique factors that come into play to draw conclusions about exactly what a given pitcher should be doing on each pitch. However, I believe that if MLB teams incorporated this model in their analysis, they could improve the scouting and evaluation processes used to build a team and then help these players improve, on average, by making better decisions.