I have probably had more arguments with people about bunting than any other aspect of baseball, which is a bit silly since bunting isn't really something I care too much about. I used to have this argument mostly with people who loved the bunt, but now I find myself having bunt arguments the other way -- even though I generally hate the bunt. For some reason, some people think it's never OK to bunt for a non-pitcher. So after having the same argument AGAIN I decided to post my thoughts online so that I could say, "I don't have time to argue with you, just google 'a robot manager's defense of bunting.'"
Most of the anti-bunt sentiment derives from a great statistic called Win Expectancy (WE) . WE is an empirical statistic, derived from inning, base situation and score -- an old idea which goes back to the 60's. Tangotiger's chart, derived from years 1970-1999 (I believe) shows (based on Retrosheet data) the winning percentage of the home team in any of the 27 possible base/out inning situations over those years. In every case, even tied in the bottom of the 9th,the winning percentage decreases going from "man on 1st, nobody out" to "man on 2nd, one out".
Yet, I stand here saying that even a robot manager would bunt occasionally for a non-pitcher in certain situations.
How can that be?
My reasoning is based on decision theory. People who base their argument on Tango's chart ignore the fact that baseball events are not independent. Most of the reasoning for "never bunt," is flawed because the comparison is flawed. You should not be comparing the WE of "man on first, nobody out" to "man on second, 1 out," but the WE of all possible outcome states, such as "man on 2nd, 1 out" and "man on first, 1 out," etc. After all, the win expectancy of "man on 1st, nobody out" includes wins in which that man was bunted over to 2nd; it's kind of strange to compare just these two numbers.
Consider the situation in which a pitcher is up. Why is it a forgone conclusion to bunt even if the base-out situation says that the win probability is lowered by a successful result? It is because something has to happen, and that something, i.e. bunting, results in an outcome that is preferable to the WE of other situations, given their likelihood.
If a player has a 0% chance of causing any situation better than "man on 2nd, 1 out," it would be entirely ridiculous to choose to do anything other than bunt.
Let's get robotic.
Suppose the Mets are at Citifield, bottom of the 9th game tied, with a man on first, nobody out. The WE of this according to the chart above is .715. The question now is whether to bunt him over. Robot manager looks at the WE chart and sees that if the batter does not make an out, the resulting WE`s are 1, .816 and .928. If he makes an out, the resulting probabilities are .637, .703, .830, and .532 for a double play. (I`ll ignore getting thrown out at the plate, since there`s no one out.)
If we assign probabilities to each of these events, and add them together we`ll have the expected WE. If that expectation is lower than .703, then robot manager will bunt.
Here`s a simple (and probably poor) model: basically, anything good happening is dependent on not making an out without advancing the runner. The average WE of the good events, (essentially saying they're all equally likely), is .8935 -- lets call this our reward (R). The average of the bad events is .6240, lets call this punishment (P). Let the simple probability of not making an out (O) be our only transition probability. The outcome is then just the model O*R + (1 - O)*P. How high does O have to be in this model to make bunting a wash? A little algebra says that O = .293. If the probability of making an out without advancing the runner is roughly greater than 70%, then you should bunt -- which is the reason why nobody complains about pitchers bunting, and which you know intuitively already.
It's important to note that O is not the same as OBP, although here it's probably a fine proxy. There are situations in which batter vs pitcher may effect the O estimate -- maybe the closer is lights out strikeout artist, and the probability of striking out is extremely high -- who knows what went into robot manager's decision to bunt algorithm?
What is clear is that it is almost definitely not OK to bunt in anything but late innings within one run, and it usually not OK to bunt -- an average hitter will be able to advance the runner in most cases. A dumb manager might be told, hey, never ever ever bunt, and that would probably work just fine.
It is, however, not true that it is never OK to bunt. It is fine to bunt in certain situations, but it is dependent on both the win expectancy and the ability of the batter to avoid outs without advancing the runner. It is certainly defensible to order a sacrifice when you can show evidence that this probability is sufficiently low. E.g, Lucas Duda, at this point, might bunt based on his OBP. I'm not going to actually recommend that he do it, but if robot manager asks him to bunt, then clearly he saw something he didn't like about Duda's swings that night against the pitcher.
I just don't want to hear other people saying things like it's never OK to bunt based strictly off WE.