Given the nine starting players, in what order should they bat? Traditional guidelines such as ìthe leadoff man should be a good base stealerî, ìnumber two should be a contact hitter who can hit behind the runnerî, ìbat your best hitter thirdî abound. Due to computational complexities, there have been few studies that analyze the batting order question from a quantitative viewpoint. This article discusses what I believe is the most comprehensive mathematical and statistical approach to lineup determination. The models and the methods used to develop them are described, and some resulting principles of batting order construction are presented. Finally, the models are applied to the 1991 AL division winners and compared to the batting orders employed by the teamsí managers.

The material presented here is an expanded version of the talk I gave at SABR XXI in New York during July, 1991. I have written several pieces on using Markov models applied to baseball; readers wanting more information may write to me [1018 N. Cleveland St., Arlington, VA 22201].

The study utilizes two mathematical/statistical models: 1) a Markov process model that calculates the long-term average (often called expected) runs per game that a given lineup will score, and 2) a statistically derived model that quantitatively evaluates the suitability of each of the nine players in each of the nine batting order positions. Data for the second model were generated by numerous runs of the Markov model. Hence, we see that the Markov model underlies the entire analysis.

THE MARKOV PROCESS MODEL

The Markov process model is based on the probabilities of moving from one runners and outs situation to another, possibly the same, situation. These probabilities, which depend on who is batting, are called transition probabilities. For example, one such transition is from no one on and no outs to a runner on first and no outs; and the transition probability is that of a single, walk, hit batsman, safe at first on an error, catcher interference, or striking out and reaching first on a wild pitch or passed ball. The Markov model employs matrix algebra to perform the complex calculations. However, once all the requisite probabilities have been determined, the matrix formulation enables the remaining calculations to be carried out without much difficulty.

It is important to note that assumptions made in determining the transition probabilities have an enormous influence on the the batting order results presented later. The goal is to choose a realistic set of assumptions, but, as always, some simplifying assumptions are quite helpful. Moreover, some of the assumptions are open to alternatives, the particular ones employed being a matter of judgment or study objectives. The key assumptions for the current analysis are:

1) Players bat the same in all situations. For this study, each playerís 1990 full season data was used to determine how he would bat.

2) All base advancement, outs on the bases (including double plays), wild pitches, passed balls, balks, etc. occur according to major league average probabilities.

3) Stolen base attempts are permitted with a runner on first only.

4) Only pitchers attempt sacrifice bunts.

5) Overall 1990 pitcher batting is used for all pitchers.

6) Small adjustments to hit and walk frequencies are made in certain situations. In particular, there are more walks and fewer hits when there are runners on base and first base is not occupied.

Data for 2) and 6) are derived from combined AL and NL data for the 1986 season. I used this season because I had extracted the needed data from the Project Scoresheet database for a prior study. Since this is a time consuming operation, I decided not to repeat it using 1990 data. Comparable data for several seasons would be better, and I may do the computer work on the entire Project Scoresheet database covering 1984-91. However, I doubt that the essential results and lineup optimization models derived would be affected very much.

The first assumption is the most critical and most controversial. One of its consequences is that the differences in expected runs between batting orders tend to be relatively small. A previous, less extensive, study that incorporated situational performance assumptions (e.g. certain players hit better with runners on) showed much larger differences in expected scoring. I plan to explore various alternative assumptions about performance levels in future batting order studies.

Base advancement on hits certainly is not uniform since it depends on runner speed and where the particular batter tends to get his hits (e.g. the percentage of singles to left, center, or right). However, I did not have the data needed to incorporate such effects. Data availability also prevented batter specific double play modeling.

The stolen base try restriction does not have a large effect because over 80% of steal attempts occur with a runner on first only. The restriction to this case greatly simplifies the computations and is not likely to affect comparisons between batting orders. Sacrifice bunt tries are not included for non-pitchers because they are game situation specific and reduce overall scoring, contrary to the study objective of finding the highest scoring lineups.

DATA FOR THE STATISTICAL MODELS

The Markov model was used for two primary purposes. One purpose is to evaluate a specific batting order by calculating its expected runs per game. In this way, alternative lineups can be compared. The second purpose is the generation of data for use in the statistical models. For each of the 26 major league teams in 1990, 200 ìbatting rotationsî were chosen at random. A batting rotation consists in specifying the order in which the players will bat by establishing who follows whom, but a rotation does not become a lineup or batting order until the leadoff hitter in the first inning is specified. Each batting rotation corresponds to nine lineups, one for each possible leadoff batter. The Markov calculations have the property that the computations needed for one lineup are also sufficient for the other eight lineups corresponding to the same batting rotation. There is nothing special about the choice of 200; it was a function of the computing power available to me and the amount of time I could spend on this phase of the study. More, as usually is the case for statistical analyses, would have been better.

Thus, the Markov model computed the expected runs per game for 1800 ìsemi-randomlyî (a made up concept since only the batting rotations are chosen at random) generated batting orders incorporating the nine most frequent players, one for each position. One property of the 1800 lineups is that each of the nine players hits in each batting position exactly 200 times.

The next step was to select the best lineups for each team from the 1800 tested. I used two definitions of best. The first is obvious: select the ones with the highest expected runs per game. The second definition is more subtle. Each batting rotation will have one lineup that scores the best, and this lineup may or may not be one of the highest scoring lineups out of the 1800. Call the highest scoring lineup for each rotation, a maximal lineup. The reason a maximal lineup, which may not be a particularly high scoring lineup overall, is of interest is that it can reveal advantages to batting certain players in certain positions although the overall scoring is held down by the batting positions of other players. Since there were 200 maximal lineups, one for each rotation, I decided to use them and the 200 highest scoring lineups as the basis for the statistical analysis. I did not determine how many of the maximal lineups were also in the 200 highest scoring.

Within each set of 200 best lineups, I computed how often each player hit in each batting position. For example, Wade Boggs leads off in 21% of Bostonís highest scoring lineups. (This value, the highest on the team, means that Boggs is a good first hitter since the average is 100%/9 = 11.1%) In this way, each player has a rating for his suitability for each batting order position.

For each player, I computed scores in 21 offensive measures relative to the group of nine starting players on his team. The offensive measures are batting average; on base average; slugging average; slugging average modified by counting walks as singles and SF as AB (which is the relationship of on base percentage to batting average); extra base average [=SA-BA, also called isolated power]; runs created per game; frequency per plate appearance of each type of hit, walks (including hit by pitch), and strikeouts; relative frequency of each type of hit (i.e. the percentage of players hits that are singles, doubles, …); percentage of plate appearances that are not walks or strikeouts (which measures putting the ball in play); secondary average [ = (TB-H+BB+SB-CS)/AB, a Bill James idea]; run element ratio [ = (BB+SB)/(TB-H), another Bill James idea]; steal attempt frequency [ = (SB+CS)/(1B+BB)]; and stolen base success percentage [ = SB/(SB+CS)]. No claim is made that the set of measures chosen is complete or perfect, just that it covers all the significant aspects of offensive performance.

I used two measures of player performance relative to the team: 1) percentage above or below the team mean in the category, and 2) the z-score, which is the number of standard deviations above or below the mean. By using z-scores, I am not claiming any of the these distributions is normal (given that there are only nine values for a team in each offensive category, the distributions are almost certainly not even approximately normal); I am just using z-scores as a measure of relative performance.

REGRESSION ANALYSIS

In the next phase, I applied regression analysis using the playersí batting position ratings (e.g. Wade Boggs 21% batting first) as the dependent variable and their relative scores for the various offensive measures as the candidate independent variables. For each batting position there are 236 data points,óone for each of the nine players on the 26 teamsóused in the regression estimates. Because there were two measures for batting position ratingsóone based on the highest scoring lineups and one based on the maximal lineupsóand two measures of relative offensive performanceópercentages above or below the team mean and z-scores, there are four possible categories of models that can be derived. I tested all four, as described below, decided on the one that seemed to yield the models with the best statistical properties, and focused on that one. The best combination from the first round of testing was highest scoring rather than maximal lineups as the basis of the dependent variable and z-scores for the independent variables.

To do the regressions, I used the stepwise regression procedure in the SHAZAM statistical package with a 10% significance level required for variables to enter or leave the equations. One equation is estimated for each batting order position, and the estimates are done independently. Since the nine batting position values for a given player must add to 100%, I experimented with some joint estimation techniques. However, they did not yield significantly different models from the independent estimates, so I used the independent estimates throughout this study. After performing stepwise regressions for each of the four categories of models described in the previous paragraph, I restricted further investigation to the highest scoring/z-scores category.

For this first set of regressions for highest scoring/z-scores models, the r2 values ranges from a high of 0.914 (#9 position) to a low of 0.580 (#6). It is no surprise that the best fit is obtained for the #9 position because of the inclusion of NL teams with pitchers that bat. The number of independent variables in these equations range from a low of 4 (#2,#4) to 12 (#9). Overall, I judged this to be good and workable set of models. Three candidate variablesóhome runs per plate appearance, run element ratio, and stolen base success percentage (which is highly correlated with steal attempt frequency)ódid not enter any of the nine model equations. The variables most frequently in the equations were runs created per game (in 7 equations, all but #4 and #5) and modified slugging average including walks (in 6, all but #2, #5, #7).

The offensive performance measures that are the basis of the independent variables are not truly independent, and several measure similar player performance characteristics. Since the models usually included several such variables, often with opposite signs, I decided to see if a smaller set of independent variables could yield models with r2 values almost as high, but which lend themselves to more sensible interpretations. After examining the equations and the correlation matrix of the candidate independent variables, I restricted the candidates to the following nine: on base average (OBA), slugging average (SA), extra base average (EBA), BB/PA, K/PA, 1B/H, HR/H, ball in play percentage (INPLAY), steal attempt frequency (SBTRY).

The resulting set of models had r2 values from 0.885 (#9) down to 0.607 (#5) and 0.434 (#6). With the exception of #6, the decline in r2 is not a major concern. In order to improve the model for the sixth position, I added RC/G to set of candidate independent variables for that equation only, which improved its r2 to 0.557. The number of independent variables ranges from 3 (#3,#4,#7) to 7 (#9). Each candidate variable appeared in at least one of the model equations. The table that follows summarizes the models; a plus sign before the variable means high scores are best for the particular batting order position, and a minus sign indicates the opposite. There are numerical values, the model equation parameters, which are not shown, associated with each variable in the table. These values determine the relative importance of the variables.

I also did some regression analyses using each of the leagues separately because I wanted to see if the DH rule affected the models. In general, the statistical propertiesógoodness of fit and significance levels of the parametersówere poorer for the models based on the separate leagues. Also, I was not able to interpret the models in a way that could answer the DH question. I suspect that I need more and better data to do this analysis. More in that teams from seasons other than 1990 should be included, and better in that more than 200 batting rotations should be calculated to determine the player/batting position scores. Additional candidate independent variables should also be considered. Due to time constraints, I did not pursue these models further, but this is a topic worth further investigation if for no other reason than the feeling of some AL managers that the number nine hitter should considered as a second leadoff hitter.

GENERATING LINEUPS BASED ON THE BATTING POSITION MODELS

Once the batting position model equations are in hand, for a given team, we can compute a value in each of the nine batting order positions for each player. These values can be positive, meaning the player is better than average for the particular lineup position, or negative, which has the opposite meaning. These scores serve to rank the nine players for each lineup position and also to identify the best position for each player. The next step is using those values to find one or more high scoring lineups. Things would be easy if the best position for each player was the highest rating for that position on the entire team. This occurs, for example if Wade Boggs best spot is leadoff and the highest scoring leadoff man on the Red Sox is Boggs; Jody Reedís best spot is #2 and the Soxí best #2 is Reed; etc. However, such is rarely the case. Due to the nature of the models, it is common for the player with the best leadoff score to also have the best #2 score and a high #3 score. Also, the scores on the ends of the lineup (#1, #2, #8, #9) tend to be more extreme, both on the high and low sides, than the scores in the middle. This reflects the modelsí emphasis on the importance of having high on base average hitters at the top of the order, which is discussed later.

What we need is a method of assigning players to lineup positions so that total model scores from the assignments is high. This is a well known Operations Research topic known as an assignment problem. Fortunately, this type of problem can be solved used several methods, some of which are easy to implement on computers and run quickly. I chose an algorithm that not only finds the best possible assignment, but also finds the top n assignments, where n can be specified. For the purposes of this study, I set n equal to five. For each set of batting positions modelsóone based on the full set of independent variables and one based on the reduced setóI found the five highest assignments for a team, which were always quite close in total batting position values. These lineups were fed into the Markov model to find the expected runs per game. The lineup with the highest expected scoring was usually one of the top three solutions to the assignment problem, but the best solution did not seem to have an advantage over the next two. In some cases, a comparison of the expected scoring and the batting order differences among lineups led me to formulate a lineup with even better expected runs per game that was not in the five solutions to the assignment problem.

For each of the 1990 major league teams, I compared the expected runs of the best lineups found using the models described in the table with the best found using the models based on the full set of candidate independent variables. For 3 AL and 6 NL teams, the full variable models had a slight advantage (about 1-2 runs a season), and for 4 AL and 2 NL teams, the reduced variable set models had a similar advantage. For the rest of the teams, the two sets of models were virtually the same. Because the smaller variable set models are easier to comprehend, the discussion in the next section is based on those models.

INTERPRETING THE MODELS

Due to the nature of the regression process, it can be misleading to draw conclusions about individual variables without considering the context provided by the entire set of variables. One example is the -SBTRY for the leadoff position. This is the fifth most important variable (its weight is about 10% that of OBA, which is by far and away the most important characteristic of a good leadoff hitter). Even so, does it mean that other things being equal, which they never are, it is better to have a leadoff hitter who doesnít try to steal? It might, but it also may just be the regression distinguishing certain slow effective leadoff hitters based on the Markov model, Wade Boggs for example. Additional statistical analysis, which I have not yet gotten to, could determine if one or two specific players are the cause of the -SBTRY.

The less important explanatory variables often play a role of emphasizing or modifying the more important ones. For example, the -INPLAY in the #1 and #2 positions serves to emphasize BB/PA. If I wanted to try to find the best set of variables for each position, I would try to build these two models without INPLAY. To illustrate the idea of modification, the -EBA in #2 balances the +SLUG and +OBA. Often players with high OBA have high BA and above average SLUG since slugging average incorporates batting average. The negative EBA in effect puts more weight on the OBA and less weight on power. A more interesting instance is the -HR/H in the model for #4. Does this mean that the clean up hitter shouldnít hit homers? No, what it means is that among players with high slugging averages, it is better to have one who does not get his slugging average mainly from home runsóa Dave Kingmanóbut instead has a good batting average and hits a fair number of doublesóan Eddie Murray.

The model equations, which are not shown, can be interpreted to characterize the desirable abilities for each batting order position:

1) Getting on base is everything. To much lesser extent, home run hitters should not lead off. Stolen base ability is irrelevant.

2) Similar to the leadoff hitter, but not quite as crucial to get on base; some power is also desirable.

3) Should have fair power, be able to draw walks, and not strike out much.

4) Highest slugging average; also has a good on base percentage and is not necessarily the best home run hitter.

5) Good power; secondarily puts ball in play (i.e. does not walk or strike out a lot).

6) Hardest spot to characterize and probably least critical. Probably want to use player who doesnít fit well in other positions. Base stealing ability is a small plus.

7-9) Decreasing overall abilities as hitters as characterized by on base percentage and measures of power hitting.

One clear result from this and prior studies is the importance of having the right batters at the top of the order. This follows from the finding that most of the difference in expected runs between high and low scoring lineups using the same players occurs in the first inning. In particular, the leadoff batter must have a high on base percentage. Also, the second hitter must be good. The practice of leading off a fast runner who can steal bases, but doesnít get on base much, and putting a weak hitter ìwith good bat control who can bunt or hit behind the runnerî second is a perfect prescription for a lower scoring batting order.

APPLYING THE MODELS

To see them in action, consider what these models say about the 1991 ALCS teams, Toronto and Minnesota. Batter performance is based on full season 1991 data, and no righty-lefty splits are used. The lineups used by the teams were against right handed starting pitchers. Before Joe Carter was hurt in game three, Cito Gaston used the batting order:

1) D. White, 2) R. Alomar, 3) J. Carter, 4) J. Olerud, 5) K. Gruber, 6) C. Maldonado, 7) L. Mulliniks, 8) P. Borders, 9) M. Lee.

The Markov model expected runs per game for this lineup is 4.739. This value is about 0.5 higher than Torontoís 1991 actual of 4.222 runs per game. That the Markov values are higher than the actuals is to be expected for several reasons. The most important are: 1) the players listed are generally better than the substitutes who play for various reasons; 2) sacrifice bunt attempts, which decrease overall scoring, are not included in the Markov model; 3) relief pitchers brought in with men on base or to face particular hitters can reduce late inning scoring; and 4) a good team usually loses more innings in games won at home than it gains in extra inning games, but the Markov value is based on nine complete innings per game.

The highest scoring lineup found by the models is:

1) Mulliniks, 2) Olerud, 3) Maldonado, 4) White, 5) Alomar, 6) Carter, 7) Gruber, 8) Borders, 9) Lee

The Markov value for the above lineup is 4.795 runs per game, which is about 9 runs per 162 game season more than Gastonís, a difference that should be worth one extra win. (Keep in mind that differences in expected runs between lineups are small due to the assumption that each playerís batting is the same in all situations.)

Mulliniks should lead off because he has an on-base average (OBA) of .364, the highest in this group, and little power. White, in contrast, has an OBA of .342 and the second best slugging average (.455, Carterís is .503), so he should not lead off despite his stolen base ability. The major surprise is that Carter bats sixth. The batting position equations score him as best on the team in the third, fourth, and fifth spots, but Maldonado, White, and Alomar rate so low as sixth, that Carter is put there instead. Tests using the Markov model showed its makes virtually no difference if Carter bats fourth and White and Alomar fill the five and six slots in either order.

Minnesotaís Tom Kelly employed the following order in the four games against right handed starters:

1) D. Gladden, 2) C. Knoblauch, 3) K. Puckett, 4) K. Hrbek, 5) C. Davis, 6) B. Harper, 7) S. Mack, 8) M. Pagliarulo, 9) G. Gagne.

The Markov process expected runs per game is 5.383 for this lineup, which is higher than the Twins 1991 average of 4.790 for the reasons given previously.

The best model generated lineup is:

1) Hrbek, 2) Davis, 3) Mack, 4) Puckett, 5) Harper, 6) Gagne, 7) Gladden, 8) Pagliarulo, 9) Knoblauch.

The Markov value of the model lineup is 5.431, about 8 runs higher than Kellyís, which might yield one more victory. Clearly, the model result flies in the face of ìconventional wisdomî, but one reason for building models is to gain new knowledge. Perhaps the best thing is getting Gladden out of the lead off spot because his 1991 OBA of .306 is by far the worst among the nine players. I am never ceased to be amazed by managers who are so fascinated by speed that they forget players canít steal first base! Davis and Hrbek have the two highest OBAs, and the model takes advantage of this by loading the top part of the order. One reason Davis with a slugging average of .507 can bat second is that Mackís at .529 is even better. Knoblauch is an interesting case because the model values him highest at either the top or bottom of the order. However, on this team, he is best suited to the bottom because his OBA is far from the best.

One important factor not considered is what assumptions, if any, the managers make about batting performance by their players. If I knew such, those levels could be put into the models, and then we could judge better how well the managers constructed their batting orders.

Those with computer baseball games that will automatically play hundreds or thousands of games may find it interesting to enter the 1991 data for these two teams and then compare the scoring of the lineups shown above for a large number of games. I would be interested in seeing how the results of the simulations compare with the Markov calculations.

As a test of how well the models work, I compared lineups found by the models with lineups used by the teams in 1990. For each team, I tabulated the number of times each player started a game in each batting order position. From this information, I constructed one or more typical lineups for each team. Some teams did not really have anything close to a set lineup, and others platooned certain fielding and batting order positions. In all cases, I developed batting orders that were typical of those used by the managers and that reflect their thinking. Using the Markov process expected runs calculations, I compared the best team lineup with the best lineup found by either of the two sets of models óone using all the candidate independent variables and one using the reduced variable set described above. The table shows the extent to which the model did better than the major league managers

ADVANTAGE IN EXPECTED RUNS OF MODEL OVER MANAGERS

               Approx.
Runs/Game      Runs/162G       AL              NL         
.095 - .105       16           1 (White Sox)
.085 - .095
.075 - .085       13                           1 (Phillies)
.065 - .075       11           1               2
.055 - .065       9.5          2               1
.045 - .055       8            4               2
.035 - .045       6.5          1               2
.025 - .035       5            1               3
.015 - .025       3.25         2
.005 - .015
-005 - .005       0            2 (Bos, Milw)   1 (SF)