# Should Lidge Have Thrown More Sliders to Tex and A-Rod?

Actually, he didn’t throw any (to them), but the question remains the same. Like the sac bunting question that I addressed a few days ago, the question is not easy to answer because it involves game theory.

Let me start by addressing the issue of a “pitcher’s best pitch.” There is no such thing as a “pitcher’s best pitch!” I know that sounds like blasphemy and it sort of depends on what we mean by “best,” but it’s true. How can I say that? Because a pitcher’s goal against any given batter in any given situation is to throw all of his pitches in a certain proportion (for example, in situation A, I, the pitcher, might throw my fastball 60% of the time, my curveball 25% of the time, and my change up 15% of the time) such that the average result of each pitch (measured any way you want, but ideally in win expectancy) is the same, and such that it doesn’t matter what pitch or pitches the batter is “looking” for. Again, you’ll notice a lot of similarities between this and my sac bunt discussion from a few days ago. And, since a pitcher should in the long run (he doesn’t have to of course) face an unbiased sample of batters and game situations, the average value of all his pitches should be roughly equal. If the value of all a pitcher’s pitches are equal, how can he have a “best pitch?” He can’t – sort of.

How can that be when clearly one pitcher’s fastball is better than another’s, one pitcher’s slider is better than another’s, etc? Because the quality, as measured in the results, of each of a pitcher’s pitches depends on two things: One, the quality of that pitch in a vacuum, as in if we were scouting a pitcher and we said, “Let’s see your fastball,” and two, the percentage of time he throws that pitch in any given situation. The “better” the pitch in a vacuum, the more he throws that pitch, such that all of his pitches eventually have the same value in any given situation. Obviously, the “better” a certain pitch is (in a vacuum), the more he will throw it (depending also on how many other pitches he has).

So, if you define “best” or “good” as the quality of a pitch in a vacuum or if the batter knows that it’s coming (or he thinks that all pitches have an equal likelihood of coming), then yes, pitchers do have a “best” pitch. But, if we define the quality of a pitch as the value of its average result in a game situation, then all pitchers’ pitches are of the same “quality.”

Let me give you an example: Let’s say that Lidge has an average fastball but a very good slider, and that’s all he throws. And let’s say that we have another pitcher with a great fastball and only an average slider. You would be tempted to say that Lidge’s “best” pitch is his slider and that the other pitcher’s “best” pitch is his fastball. And if we put a batter at the plate and told him what was coming, Lidge’s slider would outperform the other guy’s slider and the other guy’s fastball would outperform Lidge’s fastball. Even if we didn’t tell the batter what was coming but he assumed an equal likelihood of receiving each pitch, the results would be the same.

But, in an actual game situation, Lidge is going to throw more sliders than the other guy and the other pitcher is going to throw more fastballs than Lidge (and batters will know that), such that the value of Lidge’s fastball and slider will be exactly the same – ditto for the other pitcher. Overall whose average pitch value will be the same is not self-evident. That depends on who is the better pitcher overall.

You may still be tempted to think that what I’m saying is impossible. Surely the value of Lidge’s slider is going to be better than the value of his fastball in any given situation or in some situations at least. That is actually slightly true (if something can be slightly true – like being partially pregnant). There may be situations where it is correct for Lidge to throw fastballs and nothing but fastballs (such as a 3-0 count to the opposing team’s pitcher). In that case, the value of the fastball would be greater than the value of the slider and thus it is correct to throw the fastball 100% of the time even though the batter knows it is coming 100% of the time. Less likely, it may be correct to throw a slider 100% of the time, in which case the value of the slider has to be greater than the value of the fastball even if the batter knows that the slider is coming. But, most of the time it is not correct to throw one pitch or another 100% of the time, in which case, by definition, the value of all the pitches you throw must be equal (again, given the batter and the game situation, including the count of course). If for example, you threw a fastball and slider each 50% of the time, but the value of the slider were greater than the value of the fastball, then you should be throwing the slider more than 50% of the time, right? Once you do that, the batter can anticipate the slider more often such that the value of the slider will go down and the value of the fastball will go up. You and the batter will keep doing this kind of “jockeying” until you reach an equilibrium such that the value of both pitches are exactly equal. In reality, this equilibrium (presumably) exists at all times without any jockeying, since these confrontations have been going on for over 100 years. This is called the “minimax theory” in statistical decision (game) theory and in fact there is an interesting academic paper on exactly what I am discussing by Kovash and Levitt (http://www.nber.org/papers/w15347.pdf).

So while the notion of whether a pitcher actually has a “best pitch” is one of semantics, the important thing to remember is that if a pitcher throws each pitch the optimal percentage of time, the value of each of those pitches, in any given situation, must be exactly the same (the authors of the above study found that that wasn’t the case and concluded that pitchers threw too many fastballs in general, which may or may not be true as there were numerous methodological problems with their study). The other important thing to remember is that in any given situation a pitcher must throw each pitch a certain percentage of time, from 0 to 100%, and that it is rare for that percentage to be 0 or 100% (like it might be on that 3-0 pitch to a pitcher). The reason of course is that if a batter knows that a certain pitch is coming 100% of the time, that pitch is not likely to be that effective and other pitches are going to be more effective. There obviously are exceptions to this rule.

Some pitchers, like Mariano Rivera, throw the same pitch almost all the time. But even he throws the occasional slider and he actually throws two different fastballs. As well, he throws his cutter in different locations, which is the equivalent of throwing different pitches. But, as I said, if a pitch is more effective than any other pitch even when the batter knows it is coming, you are forced to throw that pitch all the time. If Mariano were to throw a curveball (I assume that he can), its value would be less than that of his fastball/cutter even if the curveball were a complete surprise. That is why he doesn’t throw a curveball. If I were able to throw a 105 mph fastball it is likely that its value would be greater than any other pitch even you were looking for that fastball 100% of the time; therefore I would have to throw the fastball all the time.

Anyway, getting back to the title question, “Should Lidge have thrown more (some actually, since he threw none) sliders to Tex or A-Rod?” There is no way of knowing the answer to that. When a pitcher is taking his signs from the catcher there are generally several pitches that he can throw, depending on his repertoire, the batter, the count, and the game situation. And he must decide the optimal percentages, again, such that one, it doesn’t matter what the batter is looking for, and two, the value of all of those pitches is the same. Of course with Damon on third base, the value of the slider includes the chance of the wild pitch, so presumably he must throw fewer sliders than if Damon were not on third base. Also, it is possible that even if the batter knows (or thinks) that a fastball is coming 100% of the time that the value of the surprise slider is less than that of the value of the predictable fastball because of the threat of the wild pitch. That is unlikely, I think, but it is possible. If that were true than he would have to throw all fastballs.

But what if that were not true, which is probably correct (especially when the pitcher, like Lidge, has an excellent slider and not a great fastball)? Then he simply throws fewer sliders than he normally would (with no runner on third). Let’s say that he is supposed to throw 2/3 fastballs and 1/3 sliders to those batters in that situation. IOW, that those are the optimal percentages such that the value of each pitch is going to be exactly equal for each count (obviously those percentages will change with the count, but for now, we’ll assume that they don’t). Well, on the first pitch to A-Rod, Ruiz and Lidge flip a mental coin such that “heads” or “fastball” comes up 2/3 of the time, and “tails” or “slider” comes up 1/3 of the time. Say heads comes up. O.K. he throws a fastball. Remember that Lidge is operating perfectly optimally as long as he keeps flipping that mental coin. There is nothing he can or wants to do differently to improve his team’s chances of winning the game (other than executing those pitches of course). Keep in mind that location is part of the pitch repertoire, but we’ll ignore that as well. Now he gets ready to throw the next pitch so he flips another mental coin. It comes up heads again, so he throws another fastball. Again, he is doing exactly what he is supposed to be doing. Now he is about to throw pitch #3. Tim McCarver would probably say something like, “He surely has to throw a slider now after 2 straight fastballs.” No! If that were his thinking then the batter would know that a slider is likely coming and all of a sudden the slider would have less value than the fastball. Remember that we said that the slider and fastball have the same value when they are thrown in that 2-1 ration. He must continue to use that 2-1 ratio when making his pitch selection (again, in reality that ratio might change, but not so much because of the prior sequence but because of the count). In fact, if Lidge thought that A-Rod was thinking a little like McCarver, which is possible, he might even be more likely to throw another fastball – he might change that ratio from 2-1 to 70% fastball or something like that. Anyway, let’s say he flips another mental coin on that 3rd pitch and it comes up head again. Another fastball. The 4th pitch? Heads, another fastball.

So he has now thrown 4 fastballs in a row, but he is acting perfectly optimally. On each pitch he has a 2/3 chance of throwing a fastball (or whatever the actual ratio would be in a real situation). But sometimes heads can keep coming up. That is the way it is supposed to be. All possible permutations of coin flips must be possible otherwise the batter can gain an edge because the pitcher is being too predictable. If the pitcher is unwilling to throw a 4th fastball after 3 fastballs in a row, such as if McCarver were calling the game, the batter would know that a slider was likely on the next pitch, which would be bad news for Lidge. In fact, the chances of 4 fastballs in a row in that situation where fastballs should be thrown 2/3 of the time, is almost 20%. So 1/5 of the time, A-Rod is going to see 4 fastballs in a row even if Lidge is throwing optimally and plans on throwing a slider 1/3 of the time on every pitch. Yet it looks to the naked eye that Lidge is just throwing all fastballs and that he is NOT pitching optimally when in fact he is.

So how can we evaluate pitch selection from watching a small sample of a pitcher’s repertoire, say against one or two batters? I am afraid we can’t. We would have to do one of two things: One, find out from the pitcher and catcher what those percentages were on each and every pitch, or two, observe those percentages over a large sample of batters and situations and try and compare them to what we think is optimal.

So, did Lidge throw Tex and A-Rod too many fastballs? Unless you have not been paying attention, I have no idea and neither should you.

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