A recent discussion on another forum spurred me to make this argument, and I thought it was interesting enough that I'd post it here for reaction. The discussion revolves around whether a baseball player can decide to 'let a ball go' that they could have fielded because they believe another defender can get it *and* have an easier throw to first base. The argument revolved around whether defensive metrics which give Pujols credit for his range are flawed, because his range is actually a detriment in that he fields balls that the 2B could get, but makes for a much harder throw to the pitcher covering.

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And, of course, 1B are taught to get everything they can, and when they can't to retreat to 1B for a throw to 1B on a grounder fielded by another player. Not that you really need to be 'taught' this. Even if we tried to 'teach' player to differentiate between the batted balls they can get versus those they can get but should let someone else get (the slow dribbler excepted) it wouldn't matter...because they simply aren't going to be able to do it most of the time. There simply isn't enough time. If you think you can get to it you try for it. That's undoubtedly the decision rule 99% of players use on 99% of plays.


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Yes, all players have time on a slow grounder to consider whether another player can get to it. We don't need to keep talking about slow grounders since players will almost always let the player whose zone that ball was hit to field it. AP included.

As far as making the determination as to whether or not to let the ball go, it is easier for the middle infielder because in most cases the other middle infielder is in their perephrial vision. For 1st baggers and 3rd baggers, this is only sometimes the case. Alot of the time they have no vision of the nearest fielder because that fielder is *behind* them. So they have to guesstimate based on pre-hit positioning.

From simple game theoretic terms, I don't think it is ever rational to assume the other fielder will get the ball when there is a sufficiently non-zero probability that the other fielder will not, in fact, field the ball. On a slow dribbler in the other player's zone, let's say there's a .01% chance of the fielder not being in position to make the play (he falls down, etc.). And let's say there is a .05% chance of making an error on the play if I (the 1B) make the play. In this instance, it's rational, from a utility-maximizing stand point, to go ahead and retreat to 1B and presume the 2nd baseman will field the ball without even checking to see if he can.

Now let's consider the question of whether or not to try for a ball at the outside of a 1B range (which overlaps with the outside of the 2B range). So we assume from the outset that there is no P = 1 situation--where we *know* that another fielder could make the play. Our chance of making an error in fielding the ball is still .05%. And the chance of the 2B making an error *if he gets to it* is .01%. So, if we knew before hand that both of us would make the play, clearly we could let the 2B make the play and hence reduce the chance of making the error. But we *don't* know that. Let's add an additional assumption: Let's assume that the probability of my fielding a ball is 99.9% when it is hit directly at me, and that probability monotonically decreases to zero as the trajectory of the ball moves off my centerpoint in either direction (as a 1B, of course, my range is censored by the foul line to my left). That point where my probability of fielding the ball goes to zero is the limit of my range. Given this, the decision on whether or not to try for a ball is a simple calculation:

If P(FB1B) >= [P(E2B) - P(E1B)], attempt to field the ball.

If P(FB) < [P(E2B) - P(E1B)], retreat to 1B.

In words, if the probability of fielding the ball is greater than or equal to the difference between the probability of the 1B making an error on the play and the probability of the 2B making an error, then the first baseman should field the ball. If the probability of fielding the ball is less than the probability of the 1B making an error on the play (minus the probability of the 2B making that error), then the first baseman should retreat to first. As should be apparent, the decision rule here is almost *always* going to be to try and field the ball. And if we assume bounded rationality (IOW if we assume that players are not probability calculators but rather their probability decisions are 'fuzzy')...then it may *never* be the case that it is rational to retreat to 1B on a ball hit within his range where he cannot be sure another fielder will get it.

Let's look at it from the manager's standpoint. What does he want his defenders to do when a ball is put in play? Naturally, he wants them to increase the chance that an out is recorded. So what should he teach his fielders to do? Naturally, they should do anything that increases the chance that an out is recorded. So let's consider the following case: A ball is hit sharply on the ground between the 2B and 1B. The following probabilites hold:

If neither player attempts to field the ball, the probability of a hit is 1.

If the 2B tries for the ball, he has a 20% chance of fielding the ball.

If the 1B tries for the ball, he has a 20% chance of fielding the ball.

If both players try for the ball, there is a joint probability of 30% that the ball will be fielded. IOW, 10% of the time when the both try for it, the 1B gets to it and the 2nd baseman wouldn't have. 10% of the time the 2B gets to it and the 1B wouldn't have. 10% of the time they both would have gotten to it (and thus the closest fielder makes the play but either would have recorded the out). 70% of the time, neither will get to the ball and it will be a hit.

What should a manager teach? Well, if he teaches the 1B to retreat to 1B when there is a non-zero chance the 2B will get it, then the probability of a batted ball recorded as an out in our above scenario declines by 10%. He loses out on the times the 1B would have made the play and the 2B wouldn't have. Now let's consider an error. Let's say that when the 2B fields this ball there's a 1% chance of a throwing error and when the 1B fields the ball there's a 5% chance of a throwing error (note, I think that greatly exceeds the actual probability of making that error. If a 1B makes 4 throws a game to the pitcher covering first, that would mean that he makes 648 throws in a season. If he makes an error 5% of the time on those throws, that would mean 33 throwing errors in a season on just throws to the pitcher covering first). Even then, it is rational to teach your first baseman to try to get everything you can get to. Because the probability of an out being recorded teaching that is 26%.

P(out) = U [P(1Bout), P(2Bout)] - U [P(1Ber), P(2Ber)

P(out) = 30% - [5% - 1%] = 26%

The chance of an out being recorded otherwise is 19%

P(out) = P(2Bout) - P(2Ber)

P(out) = 20% - 1% = 19%


D.GOOCH