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Monday, August 31, 2009
Bunting and game theory (this is really interesting stuff)
In another thread, I talked a little about bunting (for a hit) and game theory. I want to talk a little more about that. What I am going to say applies to bunting for a hit and bunting for a sacrifice.
Because the defense has to set itself before the offense has to decide what to do, essentially “showing his hand,” we really don’t have true “game theory” involved in the decision making process, at least for the offensive team. Technically, game theory involves all kinds of strategies in a contest, including when one side or the other has to “show their hand” first, but traditionally, when we speak of game theory in a contest, we usually refer to strategies that must be decided before each contestant gets to see the other contestants’ strategies. To wit:
Let’s say that the offense and defense had to set their strategies in baseball, with regard to the bunt, before they saw each other’s “hand.” The defense would position itself in such a way that it didn’t matter whether the offense bunted or not (for a hit or for a sacrifice) and the offense would bunt a certain percentage of the time (from 0 to 100) such that the defense could not take advantage of them either. If they both did that, the game would be in a state of equilibrium called the “Nash equilibrium.”
That is not what happens in baseball though, of course. First the defense sets up and then the offense gets to decide what to do. So what SHOULD happen? Simple.
The defense should set up in a way that it doesn’t matter what the offense does. That is the optimal position for the defense. That should change of course, depending on all kinds of circumstance - score, inning, pitcher, base runners. Now, the only time that might not change is when the defense is playing as far back as it can (for a player who is such a bad bunter or such a good hitter, or some combination thereof) or it is playing up as far as it can, for the opposite reason.
Now, once that defense is set optimally, which it should be, the offense can do anything it wants or doesn’t want, and it should not change their win expectancy. That bears repeating and bolding:
If the defense is playing optimally, and there is no reason that it shouldn’t (even if it does not know EXACTLY what optimal positioning is, neither should the other team, unless I guess, there is a batter or two that the defense is not familiar with, which would be poor advanced scouting), then it does not matter what the offense does. They can bunt 0% of the time or 100% of the time and it won’t change their WE!
That SHOULD be the case in baseball. But it is not of course.
Keep in mind, that in order for the defense to know where to position itself optimally, it doesn’t even have to know how often each batter bunts in various situations. All it needs to know if how good a hitter they are, how good of a bunter they are, and how fast they are when they bunt. Of course knowing how often they bunt in various situations can help them with that information, but it is by no means necessary.
Now, since teams and managers by no means understand this (Tango, do the Mariners? If not, they should!), there is a good chance that at least sometimes the defense does not position itself optimally. If that is the case, what should the offense do? The answer is interesting! Take a second to see if you can figure it out for yourself…
OK, time is up!
If the defense is not playing optimally in any game situation (for the bunt hit or bunt sacrifice), then the batter is supposed to bunt or not bunt 100% of the time!
There is no game theory involved for the batter (and his manager) in terms of “mixing it up.” That would only be true if the batters had to make his decision before the defense set up. That is never the case.
Of course the defense can try and decoy the batter and then move up or back before the batter makes his decision. Whether the batter can change that decision in mid-stream, I don’t know. Let’s assume that the defense always plays their hand and then the batter can respond.
So, to summarize all that, the defense should be playing in such a way that whether the batter bunts or not does not change the WE of the game in all situations. Obviously they should be constantly moving around unless they are playing all the way back or all the way in on certain batters. If the defense does this, then it makes no difference to the batter what he does. Even the best bunter in the world, if the defense is playing him properly, can choose to NEVER bunt! He would not be making a mistake! If the defense is not playing optimally, which is often the case, then the batter is supposed to either bunt or not bunt 100% of the time.
Now, here is one of the kickers. Let’s say that the defense is playing too far back in any given situation. In other words, the batter is supposed to bunt 100% of the time. What will happen if he does? The defense will move up, right? Into a position which is presumably more optimal. So the batter will be teaching the defense to play more optimally. So what do they do? Well, they don’t bunt all the time. They bunt just enough so that the defense moves up a little but not too much. Do they revert to pure game theory and bunt at a percentage such that it does not matter what the defense does? No! If they do that, then they no longer have an optimum advantage - either that makes the defense play optimally and the game is in equilibrium or if they do that and the defense does not respond optimally, then they have to modify those percentages again. So if the defense is not going to play optimally, the offense should mix it up, but not at the “correct” percentages (the percentage that would be correct of the defense played optimally AND they didn’t know where the defense was playing before they made their decision). They know that the defense is going to adjust and they will live with that as long as they don’t adjust so far as to make the defense play optimally. And if they adjust all the way to the “correct” percentage and the defense is still out of position, then they keep going. For example, if it is correct for me to bunt 25% of the time and the defense is playing too close, I might bunt 15% of the time. If they move back a little but not to an optimal position, I might bunt 17% of the time (the 15% was too little). If they start playing optimally at any point, I started bunting too infrequently. If I get to 25%, which is optimal for me if they are experts, and they still are not playing optimally, then I have to move off that number, at least a little. So, I’ll never be on that number (25%) unless they are experts, either intentionally or accidentally. It is possible to be a perfect game strategist accidentally.
In poker, it is a well-known strategy among good players that if your opponent calls bluffs too often that while you should never bluff against an opponent like that you have to still bluff some of the time otherwise he will start to call your bluffs less frequently and eventually he will accidentally settle on an optimal strategy. Same thing with bunting and baseball. If the defense is out of position, you make your percentages such that while you will automatically move the defense towards a more optimal position, you don’t move him that much.
How much do you move him? Trial and error and a knowledge of your opponent. For trial and error, you change your strategy (from the optimal one) and you see much he moves. Let’s say again that the defense is playing too far back given your batter and the game situation. It is correct of course to bunt 100% of the time, and in fact, you would do that if that were the only time you were playing this opponent and no other future opponents were watching or had access to any data. But, once you bunted, say, 2 or 3 times in a row (or however many), your opponent will likely wise up and play more optimally. If he happens to move to an exactly optimal position, you have screwed yourself in the future of course. (Then again, if he is a dumb opponent, you might be able to move him back by never bunting again until he does.) So what do you do? Say that game theory says that it is correct to bunt 50% of the time, if you had to make your decision before you saw his and he was a prefect strategist. Well, you might bunt 25% of the time and see what he does. If that moves him to an optimal position, you have bunted too often. If he does not move at all, then you bunt a little more. If he moves a little but still short of the optimal position, you have to decide if it is better to leave him where he is and continue to bunt 25% of the time, or it is better to bunt a little more knowing that he will move in a little more, but hopefully not to the optimal position.
Now, what about the batter? What if you are the defense and you know all this stuff, so that you know exactly where to play optimally? You might position yourself optimally and then you don’t care what the defense does. They can bunt 100% of the time or 0% of the time and you don’t give a hoot. You might be playing halfway in with Willy Taveras at the plate (a notorious bunter) and if he never, ever bunts again, you don’t give a hoot, you will never change your position. But, that is ONLY if you think or know that the offense is a perfect or near-perfect strategist. What if they aren’t?
Well, now you can do what the offense should do and through trial and error and knowledge of your opponent, you can figure out where to play that would not have been optimal if the batters were perfect strategists. You now have to figure out where to play given that the offense is making mistakes. Which they will of course.
So let’s say that you start out in an “optimal” (either with no knowledge of the batter’s tendencies or you think or know he is an expert strategist) position so that it does not matter what the batter does. Now, let’s say that the batter starts to bunt or not bunt at a percentage that does not seem to be what he would do if he were playing optimally and had to make his decision without seeing your decision first? Is he really an expert and is trying to trick you? If he is, then you simply stay where you are. If he is just playing incorrectly, then you have to move. You’ll have to experiment and find out. Say, he starts bunting too much. You move in a little. Remember that you are now out of an optimal position if he were an expert strategist. You now open yourself up to be taken advantage of. He can now hit away all the time and gain an edge on you that he didn’t have when your were playing back!
But, what if he isn’t an expert? You move in a little thinking that perhaps he is bunting too much and will continue to do that. Now, just like when the batter started bunting more or less than when he started out, of course once you move in, he is going to bunt less automatically. If he now bunts a lot less, then you just screwed yourself by moving in and you just got suckered by an expert strategist or he accidentally got an edge on you simply because you moved in and he decided to bunt a lot less often. So just like the offense, if you think that the batter is not an expert strategist, you should change your position from the “optimal” one and see what happens. The batter will adjust of course and that adjustment will always be in the optimal direction. If he settles right on or near that optimal position even though he started out NOT optimally, then you have moved too much and accidentally corrected him. You have to move just enough so that even when he corrects himself, he is still not playing optimally. Now you have gained something. If you don’t move at all and he is not playing optimally, you gain nothing by staying in that optimal position. If you change your position and you force him to play optimally or even worse, to take advantage of you being out of position, then you made a bigger mistake than staying where you were in the face of him playing sub-optimally. So again, you move a little - just enough so that while he automatically plays more optimally, he does not go all the way towards that optimal position.
So, if you play in a certain optimal position against Taveras such that his bunt and his hit away yield the same WE, and his optimal bunt percentage is 25%, if he bunts more than 25% of the time, you move in. You don’t know whether he is decoying you to try and get you away from the optimal fielding position or he is not an expert strategist and is just bunting too often. If, when you move in, he starts to bunt 25% of less, you just screwed yourself. If he still bunts more than 25%, then you have gained over not moving at all.
But what if he started out at 40%, you moved in and he is now bunting at 30% (when 25% is optimal for him)? Did you move too much? Maybe and maybe not. Maybe moving in less and having him bunt 35% is better. You have to figure that out. It depends in how incorrectly he is “willing” to play. The more incorrectly a player or his manager is willing to play, the more you can moved off that optimal position and gain an edge. For example, if a manager is absolutely intent on sacrificing late in a tie game with one of his non-power hitters at bat, then even if the optimal position is for you to play 60 feet from home plate, you might as well play 45 feet (or whatever the max is) if you think that he is still going to bunt.