By now, we all know about Erick Aybar’s run-in with Justin Verlander last Sunday and the alleged social norm against bunting with a no-hitter in progress. At first, I adamantly believed that the norm was ridiculous, as it essentially demands a batter sacrifice his team’s probability of victory to appease the pitcher. Given my background in game theory, I set out to prove this formally.
Turns out, I was wrong. If the fielders and batters are all playing optimally, then there is a set of strategies such that the batter never bunts while simultaneously maximizing his team’s win percentage. However, this requires the third baseman to play where he would normally be, as if there wasn’t a no hitter in progress.
I thought this would be a simple model to solve. However, after a couple of hours staring at a computer screen, I realized I was wrong. Nevertheless, I want to keep this post as straightforward as possible and keep the math to a minimum. If anyone has concerns, I can respond to them in the comments section. Eventually, I will turn this into a full-scale manuscript to send to SABR, so I will appreciate any feedback.
If we look at bunting generally, there are three types of individuals to consider. First, there are the David Ortizs of the world. Against these guys, defenses completely ignore the bunt because the hitter is so immobile. I will not be considering this type of players in this post.
Second, there are hitters that are so awful (or premium) that they are going to bunt regardless of what the defense does. Jeff Mathis jokes aside, I again will not consider this type of player in this post.
Third, there are legitimate athletes who can both swing away and bunt, and the defense must be prepared for both. This is the type of player that I am interested in. Erick Aybar fits the bill, so I will discuss the fielding dilemma accordingly.
When Aybar is at the plate, the third baseman needs to select a fielding position. Let’s diagram it like this:
So the third baseman must select a position along a continuum between 0 and 1. The 0 position represents the ideal area to field a bunt. If he moves any closer, he will be less able to field a bunt or a swing. (Both will just go right by him.) The 1 position represents the ideal area to field a hit. Similarly, if he moves any further away from that position, he will be unable to throw out the runner on a hit or a bunt. As such, the continuum between 0 and 1 represents all of the plausible fielding positions for the third baseman.
As we can observe during any given at-bat, the fielder selects a position before the pitch is thrown. Consequently, the batter has an advantage here—he can condition his choice between swinging and bunting off of where the third baseman is. If the third baseman is too far back, he can bunt; if the third baseman is too close, he can swing away.
How does the fielder respond to such a disadvantage? Well, suppose he is really far away. Then the batter optimally bunts. So the fielder moves closer to the plate. As he approaches the 0 position, he is better able to field the bunt.
Likewise, suppose he is really close. Then the batter optimally swings away. So the fielder moves further away from the plate. As he approaches the 1 position, he is better able to field full-on hits.
These two pieces of information have an important implication. There is a unique position where the batter becomes indifferent between bunting and swinging. Let’s call that position X. My claim is that X is the optimal fielding position for the fielder.
To see why, we simply have to go through the previous logic one more time. If the fielder selects a position closer to 0, then the batter always swings, since swinging is better than bunting against that fielding position. Note that the batter does better swinging when the fielder moves away from X and closer to 0 than when the fielder is at X. In turn, that means the fielder is doing worse. As such, the fielder could improve by moving back to X.
The same is true the other way. If the fielder selects a position closer 1, then the batter always bunts, since bunting is better than swinging against that position. Note that the batter does better bunting when the fielder moves away from X and closer to 1 than when the fielder is at X. In turn, that means the fielder is doing worse. As such, the fielder could improve by moving back to X.
So we know the fielder optimally plays at position X. Because position X is the unique point that leaves the batter indifferent between bunting and swinging, it is optimal for the batter to do either. Put differently, the hitting team is equally likely to win if the batter swings away as if he bunts.
Interestingly, that means the norm against bunting with a no hitter in progress is reasonable—a batter can never bunt and it will not hurt his team’s chances of victory.
But note that this claim is predicated on the assumption that the fielder plays in position X. As soon as the fielder takes a single step backward, the batter’s best strategy is to bunt. So if the fielder assumes the batter will not bunt regardless of where he plays (because it would break the norm), then the batter has to bunt if he wants to maximize his team's win percentage.
I went back and watched replays of Sunday’s game, and it appears that Don Kelly was not playing in the X position—he appeared only a step in front of the third base bag. (Unfortunately, I could not find a screenshot of where he was playing earlier in the game, so I am speculating here.) It is as if Don Kelly was daring Erick Aybar to break the norm. As we know, Aybar obliged.
In any case, if Justin Verlander wants to get mad at somebody for Aybar bunting, it shouldn’t be Aybar—it should be Kelly. If he Kelly played another step in toward home, Aybar probably would not have bunted, and this ridiculous debate never would have started.