I just wanted to post a short follow-up to a post from a few days back, Controlling the Zone in order to make what might appear to be a completely unreasonable assertion. Those are, after all, the best kind.
Umpires absolutely should be biased to give pitchers with good control a wider strike-zone. If an umpire does not give a pitcher with good control a wider strike zone, then he is being unfair.
The basic principle is this: if a pitcher has better control, then before you even see the pitch you should guess it will be a strike. If you see a borderline pitch that could go either way, you will be correct more often if you err on the side of calling strikes. That may not be too convincing, so let's do better.
Let's simplify this and look only at the lateral location of the pitch. Figure 1 shows a hypothetical distribution of pitches for a given pitcher. We'll assume that the distribution is normal along this dimension (this assumption is false for real pitchers, but that doesn't matter for our purposes here). In blue, we see the majority of pitches (60%) that fall between -10 and 10 inches from the center of the plate. These pitches are "true" strikes; they actually crossed the plate. The area in red represents the 40% of the pitches that fall outside the zone. These are "true" balls.
Of these true balls, many appear to the umpire to cross the plate in the strike zone. That is, just the fact that the umpire is not perfect leads to some misclassifications. The green area in figure 3 reflects the true balls that are called strikes. In this figure, 16% of the actual balls are called strikes because of this error. But this isn't a bias; this error term will apply to all pitchers, regardless of their skill level.
The bias in the umpires perception comes in if he is trying to maximize his own performance, that is, make the fewest mistakes. The perceived distribution of pitches in figure 2 and 3 show how they would be classified if they were each considered in isolation. But we have a lot more information: we know the overall distribution of pitches. We know that a pitch closer to the center of the plate is more likely than a pitch outside. Therefore, our optimal guess, given the information and uncertainty that we have, is shown in figure 4. The green distribution in figure 4 shows the perceived location of the actual balls after the umpire takes his prior knowledge into account.
Nearly 40% of the "true" pitches are now being classified as strikes (that's OK, some of the strikes are going to be misclassified as balls). Figure 5 shows the source of the misclassification. The area in red is the error caused by measurement error, the noise in the umpires perceptual system that causes him to be inherently uncertain. The area in green is caused by his priors, which will change depending on the context. If he faces a good pitcher with great control, the umpire's prior distribution should be very tight, with many strikes. If he faces Joel Zumaya, the umpire's prior distribution should be much more even (or even inverted, so that he is biased to call a ball).
Failing to take the context into account will result in impaired performance. The umpire would get more pitches wrong. If an umpire takes this "bias" into account, he is actually being as fair as he can be. If he did not use this bias, he would actually be unfairly biased against the pitchers with better control. What is fairness? Here, we would want the umpire to mistakenly call a true ball a strike as often as he calls a true strike a ball. If the umpire does not update and apply a prior based on the context, he is being unfair by this definition: when judging a good control pitcher, he will misclassify more true strikes as balls than vise versa.
Hence my initial claim: Umpires absolutely should be biased to give control pitchers a larger strike-zone.
This might be able to explain why the strikezone is a
All of that said, it's entirely unclear how an umpire should construct his prior, or what experiences should be used as a basis. Should it be based on a pitchers history? History with that umpire? The performance of that pitcher that day? The performance of all pitchers that day (not too unreasonable if the process is automatic)? The hypothesis becomes hard to test because the prior could be constructed in a number of very different ways.