Game Theoretic Model and Its Macro Implications

In the previous sections, we found out that even after controlling the strength and experiences of the players, batters do have a significant advantage when facing opposite-handed pitchers. In this regard, game theory can be used to illustrate how the OH advantage influences the percentage of left-handed and right-handed players in MLB (Goldstien & Young, 1996; Flanagan, 1998), but the assumptions we made here

are different from the others. First, both studies assumed that the distribution of the players will make the expected payoff against opponent left-handed and right-handed players equal. However, such proposition only works when both the batters and pitchers are sent to the field on a random basis. As a matter of fact, they are not. Since the pitching rotation is highly predictable, the batting lineup is definitely not assigned at random. Moreover, both studies used the average performances of the lateral groups to estimate the expected payoffs. This would be a proper method if players are drawn from normal distributions which have mean values equal to the average performances. As a matter of fact, they are not, either. Only the best players can enter the league, so, in theory, all the best players should be already playing in MLB. If a team wants to increase its proportion of left-handed players, it must select from players inferior to the current league level. Similarly, if a team wants to cut its proportion of left-handed players, it must cut the least performed players. Using the average performance would make the estimation biased.

In our game model, the goal of the teams is to recruit players under a roster limit, and then to control the playing time of the left-handed and right-handed players to achieve the highest payoff, given how many left-handed players are there in their rival teams. Therefore, a sub-game perfect Nash equilibrium of the numbers of left-handed players can be derived.

First, we specify the payoff function between a batter and a pitcher. In team u, a left-handed batter i has an individual status level of (which is equal to

in equation (1)), and a right-handed batter i’ has an individual status level of , but the year dummy, the defensive position, and the random disturbance term are neglected for the time being. A batter may face pitcher j in team v, with an individual effect of (which is equal to

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in equation (1)) if left-handed, or if right-handed. Players of various skills can be recruited by the team. For convenience, let for all , and

for all . We also assume that:

 There are no switch hitters.

 Team u is able to observe the fixed effect of the potential players they can recruit, which is for left-handed pitcher for all j, for right-handed pitcher for all j', for left-handed batter for all i, and for right-handed batter for all i'..

 Team u can also observe the fixed effect of the potential players of opponent team v,

The teams undergo five stages in this game model. The five stages are:

Stage 1: Team manager decides pitching rotation. It is a practice in MLB that teams announce the starting pitchers for their next few games, so the pitching rotation is quite predictable. Furthermore, since starting pitchers need several days to rest and prepare for their next game, they cannot be easily substituted without giving the replacement starting pitcher some days to prepare and the team will have to announce the replacement in advance too.

Stage 2: The batting lineup adjustments will be made according to the pitching rotation at every match. Since the starting rotation for the opponent is known information, the manager will send his players with the highest expected payoff to play against the specific pitcher.

Stage 3: Starting batters confront starting pitchers in the match.

Stage 4: During the match, both teams substitute their players in order to gain the highest expected payoff for the subsequent plays. Players will be substituted because of injury, weariness, or simply because they are having a bad day.

Stage 5: The outcome of the match is finalized, and the winner is decided.

From equation (1), the individual payoff function for the batter-pitcher confrontation in stage 3 and 4 is as follows.

Batter vs. Pitcher by Handedness Expected Payoff bi,u (LH) vs. pj,v (LP) bi,u (LH) vs. p’j’,v (RP) b’i’,u (RH) vs. pj,v (LP) b’i’,u (RH) vs. p’j’,v (RP)

In the above expression, we allow OH advantages ( and ) to be different for left-handed and right-handed batters. In order to simplify the analysis, we also assume that OH advantage is constant among players with different fixed effects. Allowing OH advantage to vary across different players will not affect the following analysis. If a left-handed batter is batting against a left-handed pitcher, the batter’s expected payoff would be his fixed effect, , minus the pitcher’s fixed effect, , while the pitcher’s expected payoff would be minus . However, if the left-handed batter is facing a right-handed pitcher instead, the batter will gain an OH advantage, γ, in addition to the fixed effects, while the pitcher will take an OH disadvantage of γ.

Specifically, 4 kinds of situations may occur during a match:

1. Starting batter vs. starting pitcher, where the batter is arranged to extract the highest expected payoff against the scheduled starting pitcher.

2. Substitute batter vs. starting pitcher, where the batter comes up from the bench to replace the starting batter who didn't do well to gain the highest payoff against the pitcher.

3. Starting batter vs. substitute pitcher, where the pitcher comes up to gain the

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maximum payoff for the next few plays or even just one play.

4. Substitute batter vs. substitute pitcher, where both sides try to counteract their opponent’s strategy, or just to let the players gain some experience after the match is almost decided.

Situation 2 to 4 are more complicated because the affecting factors include not only the skill level of the player and the OH advantage, but also the unpredictable shocks that temporarily affect the players such as luck or tiredness, so our analysis focuses on situation 1.

In the first situation, since the batters are all facing the same pitcher, the batter’s expected payoff lies only in his own skill and the OH advantage, which is:

where = 1 if the batter's batting hand is different from the pitcher's pitching hand and = 0 if otherwise.

Let , where is the maximum of number of batters in a batting lineup. The strategy of the team is to maximize the expected payoff for the batting lineup. If the opponent pitcher is right-handed, the expected payoff of starting all the lineup with left-handed batters would be:

However, if your best right-handed batter is so good that he satisfies:

then it will be strictly better to send one more right-handed batter. As a result, there will be a batter who satisfies:

which means that it is still better to send the right-handed batter than a left-handed batter, but the batters weaker than will not have his chance. The

team's expected payoff will be: and the payoff for the team against a left-handed pitcher will be:

From equation (2), if there is an increase in , k' must decrease in order to satisfy the equilibrium. Similarly, from (3), an increase in will cause a decrease in k. The interpretation is that the optimal percentage of batters facing left-handed and right-handed pitchers only depends on the magnitude of the OH advantage and the distribution of the batters' fixed effect, and nothing else. A change in the percentage of left-handed pitchers will not affect this optimal strategy. If there is a change in the OH advantage which made the left-handed batter more favorable when playing against right-handed pitchers, then it will be better to send more left-handed batters when facing right-handed pitchers.

The strategy for the team's starting batting lineup for a whole season will be:

83 while the chance of a right-handed batter to play is:

How the percentage of the left-handed and right-handed batters is decided can be elaborated as follows.

1. If , and should be adjusted according to their skill level only. In this case, according to equation (2) and (3), and are adjusted to the point that the fixed effects of these two players are equal. Since there are 8 times more people using right hand than the people using the left hand in the real word, for every left-handed people with a given skill level, there should be 9 right-handed people as well. Therefore, will be equal to about 9 times of . Moreover, the starting lineup when facing a left-handed pitcher will be the same as facing a right-handed one, which means or . The optimal percentage of the left-handed batters will be the same as that in the general population, which is about 10%, and will not be influenced by the percentage of left-handed pitchers they are facing.

2. If , then and should depend on the magnitude of and .

Both and will be smaller than the situation under , because the weaker batters can get some benefit from the OH advantage. In this case, a change of the opponent left-handed pitcher percentage will affect the batting chances of both left-handed and right-handed players. An increase in will increase and decrease

The interpretation of the batters' equilibrium is this: teams have two different lineups against left-handed and right-handed pitchers. There are some regular players who have positions in both lineups, and some marginal players who can only play against left-handed or right-handed batters. Without the OH advantage, some of these batters will not even have a chance to play, but with the OH advantage, these batters are able to face pitchers with opposite handedness and gain a better payoff than a stronger batter who uses the same hand as the pitcher and has no OH advantage. If there are more left-handed pitchers, the lineup against left-handed pitchers will be arranged more often, thus right-handed marginal batters will have more chance to play, while left-handed marginal batters will have less playing time. However, such change in the proportion of left-handed pitchers will not change the numbers of regular and marginal players within the lineup. The numbers will stay the same, only that the lineup against the left-handed pitchers will be used more times in the season.

Now, in stage 1, the expected payoff for a left-handed pitcher c will

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The team wants to maximize the payoff of the starting rotation:

expected payoffs against the marginal batters are the same:

batters become more favorable against right-handed pitchers, the teams send more left-handed pitchers. On the contrary, when the right-handed batters are more advantageous against left-handed pitchers, the teams send less left-handed pitchers.

From (2), (3), and (4), the equilibrium number of both left-handed and right-handed batters and pitchers for every team can be derived, given the population distribution of the players' talent pool. First, the team chooses and according to the OH advantage and . Then, the opponent team chooses and according to and the OH advantage. The choice of and is independent of and , but there will be two different batting lineups against left-handed and right-handed batters.

If there are more left-handed pitchers pitching, then the lineup against left-handed pitchers will be used more often, and more right-handed batters will be batting.

A t-test shows that when a team faces left-handed pitchers, the batting chance for the left-handed batters is significantly less than when the team faces right-handed pitchers, which serves to reject that the left-handed and right-handed batters are randomly sent to play, because if they are, their batting chances should not be significantly different. However, another t-test also shows that the batting chance of left-handed batters is significantly larger than 0, which validates our basic assumption of the game model above that teams let some good left-handed batters face the left-handed pitchers even if they don't have the OH advantage.

Table 8 shows the relationship between a starting batter's percentage of facing a left-handed starting pitcher and percentage of the overall percentage of left-handed pitchers faced by his team. Batters are grouped into three levels according to their fixed

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effects from the lowest to the highest year by year. Based on our explanation above, the marginal batters are more inclined to face pitchers with opposite handedness to extract his OH advantage. The results in Table 8 tell us that this is indeed true. When the team faces more left-handed pitchers, the strongest left-handed batters will have more chance to play than the average or weakest players, while the average and weakest right-handed batters will have more playing time than the strongest right-handed batters.

Table 8: The percentage of left-handed pitchers faced by a batter and the percentage of the team's chance of facing left-handed pitchers.

Batter i P(LP, t) Batter i' P(LP, t)

P(Team vs. LP, t) 0.773*** 1.136***

(0.0492) (0.0662) P(Team vs. LP, t)* Average 1/3 -0.123*** 0.148***

(0.0199) (0.0263) P(Team vs. LP, t)* Weakest 1/3 -0.162*** 0.147***

(0.0230) (0.0286)

Constant -0.0473*** 0.0601***

(0.0134) (0.0182)

Observations 2,557 4,643

R-squared 0.089 0.078

Robust std. errors in (); *** p<0.01, ** p < 0.05, * p < 0.1.