Percentage of Left-handed Players Explained by OH Advantage 87

We have shown in the previous sections the existence of the OH advantage, and also verified the manager’s strategic adjustment on hand-specific proportion of the players according to the OH advantage. The next logical question to ask is: how many percentage points of the left-handed players come from the OH advantage?

Again, we adopted the same model specification in column (5) of Table 5, but left

out the batter’s PA effects. We obtain the fixed effects of the left-handed and right-handed batters who started at least one game during 2000 and 2012 from our estimation. Then the fixed effects of the players are adjusted by the player’s experience year by year. We also ruled out switch hitters, because they enjoy the OH advantage from both sides of the plate.

Recall from equations (2) and (3) that

From Table 5, we know that . Since by the assumptions in section 5, the teams are able to observe all the fixed effects of the batters and then choose the best ones, it will not serve our purpose to make a simulation and draw samples from the distributions of the players. Instead, we set up an estimation that would allow the teams to observe the full distribution of the players and recruit the best players first. An estimation of the fixed effects of the batters with respect to the numbers of the batters can be derived by regressing each batter's fixed effect on his rank in the year. The estimations would be:

where Batter Rank is the yearly fixed effect ranking of left-handed batter i or right-handed batter i' in the league. These two estimations imply the good players are signed first, and the numbers of the batters in the league reflect the fixed effects of the average and marginal batters. By substituting the fixed effect estimations into the equilibrium, the optimal percentages of the batters can be obtained:

89

Simplifying the above equations leads to:

There are currently 30 teams in MLB. Each team has an active roster limit of 25 players¹³. Suppose 13 of the 25 players are batters, which means there should be 390 batters in the whole league. Substituting = 390 into the above equations, and results would be k = 0.399, k'=0.536. Thus, the percentage of left-handed batters who starts against pitchers with both handedness should 39.9%, while the percentage of right-handed batters who start against pitchers with both handedness should be 53.6%.

The percentage of the players whose chance of playing depends on the opponent pitcher's handedness will be 1 - 0.399 - 0.536 = 0.065.

With the same method, the optimal percentage of the left-handed pitchers can also be obtained. From equation (4),

Same as the above, an estimation of the relationship between the fixed effects of the pitchers and their ranking gives us an estimation of the optimal percentage of left-handed pitchers. However, as mentioned above, the fixed effects of the pitchers cannot be estimated in the same model with the batters' fixed effects due to multi-collinearity. Therefore, the pitchers' OPS against the same-handed batters in the previous season multiplying the coefficient in column 5 in Table 5 (0.0802) is used as an approximation of the pitchers' fixed effects. The estimations of the relationship between pitchers' fixed effects and rankings are:

13 The active roster limit will increase to 40 after September 1st each season, but here we only discuss the regular 25-man roster limit..

Suppose all the 30 teams in MLB uses a 5-pitcher starting rotation, that is,

. Substituting the numbers of , and the pitchers' fixed effect estimations into equation (4) gives us , and . From this ratio, the proportion of left-handed batters is 0.399 + 0.73*0.065 = 0.446.

In order to estimate the excess left-handed players in MLB, we need to find out three different ratios of left-handed players under different assumptions. 1) If there is no OH advantage, then . In this case, the proportion of left-handed batters and pitchers should be the same as the general population, which is 10%. 2) If the OH advantage exists and is the only consideration for the managers in adjusting the plate appearances, the above estimation shows that the optimal equilibrium should be 44.6%

for the left-handed batters and 27% for the left-handed pitchers. 3) The final scenario concerns the numbers from the MLB data. After ruling out the switch hitters and focusing on starting players only, the left-handed batters take up 37.1% of the plate appearances while the left-handed pitchers take up 27%.

The above analysis implies that while the percentage of left-handed pitchers is at the optimal, there should be more left-handed batters in MLB. There are some reasons for the lower percentage of the actual left-handed batters. The first reason is the defensive side. Catchers, second-basemen, third-basemen, and shortstops are usually right-handed players because they can pass the ball more easily to the first base than the left-handed players. Thus, an average right-handed shortstop may have a position in the regular starting lineup, while a good left-handed outfielder has to compete with other left- and right-handed players.

Also, a left-handed pitcher is more likely to succeed than a left-handed batter. In

91

addressing the absence of left-handed catchers, Bill James (2001) emphasized the biggest reason for having no left-handed catchers is “natural selection.” He remarked that catchers “need good throwing arms,” but asked, “If you have a kid on your baseball team who is left-handed and has a strong arm, what are you going to do with him?” (p.

41) It is most probable that this kid will be trained as a left-handed pitcher. By the same token, this reasoning may be applied to other position players. Left-handed players may either choose to face competition from both left-handed and right-handed batters in the first base, outfield, and the DH, or they can give up batting and become a pitcher instead.

7. Conclusion

In this paper, we tried to explore the commonly accepted OH advantage in the baseball field by using statistical method, and determine how this OH advantage created an imbalance on the employment rate of left-and right-handed players. Our regression results show that in MLB, the OH advantage for left-handed batters explains about 15%

of their OPS, while for right-handed batters, it only explains about 7% of their OPS.

Such difference may lend support to the fighting hypothesis, in view of the advantage of the minority of left-handers in the baseball games. There seems to be no such advantage for switch hitters. On the other hand, pitchers throw better to batters of the same handedness, as compared to opposite-handed batters. We then used OH advantage and game theory to estimate the optimal percentage of left-handed players in MLB.

However, our approach is different from Goldstein and Young (1996) in that we assume the teams can observe the talent of every possible player and sign the best ones. The estimation of our model shows that while our estimation of the percentage of the left-handed pitchers is close to the number in MLB, there should be more left-handed

batters playing in MLB.

Although our model partly explained the in-field management decision of sending which kind of batters to the plate, it is far from perfect. For example, we can't explain why the left-handed batters have a larger OH advantage. We also completely omitted the effect of defensive characteristics. In the baseball field, catchers, second-basemen, third-basemen, and shortstops are usually right-handed players. Plus, we didn't put too much emphasis on switch hitters. Neither did we distinguish the right-pitching left-handed batters with the natural left-handed batters, simply because the results are mostly the same when we split them apart. We also ruled out the substitute batters and the relief pitchers in the theoretical model, because they would make the model way too complicated to analyze. Moreover, although OPS is the most accepted baseball stat when evaluating player performance, using OPS as the dependent variable may give too much weight to base hits and too little to walks. There might be ways to make improvements in these aspects. However, this model not only explains how the batters and pitchers are arranged in accordance with the OH advantage to play against the opponent’s deployment, but also serves as a fair estimation to the practical operations of the teams.

93