## 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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### 35 Responses to “Should Lidge Have Thrown More Sliders to Tex and A-Rod?”

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1. TheUnrepentantGunner says:

Out of curiosity, Mike Marshall when he was with the Dodgers (as referenced in the immortal Ball Four) believed in basically random pitch distribution as well.

The question then becomes how do you randomize? It’s extremely hard for even someone very bright to randomize data in their head (too much of a tendency to avoid long runs of one pitch vs another), and there are few cues readily available. Maybe something alphanumeric with the players name, so if you want to throw your slider 2/3 times, fastball 30% of the time and some small percentage fo the time your changeup, maybe letters A-I = fastball (with c as a changeup) and letters J-Z= slider..

So then pitching to alex rodriguez first time through = fastball, slider, fastball, slider, slider, slider.

but that is really poor, because again in different counts the odds change, and the idea of using alphanumeric seeding is absurd. maybe use of a stopwatch? i dont know how you actually implement this, but the theory is absolutely beautiful.

• Bryz says:

Don’t forget that you’d have to mix in pitch location as well, unless you want that to be random as well.

It’s probably because I’m in a statistics class right now in college, but when I heard “random selection” I instantly thought of a random number table like in my textbook. Using this http://www.gifted.uconn.edu/siegle/research/Samples/RANTBLE.JPG if you want 50% fastballs, 30% sliders, 20% change-ups (or whatever) then assign numbers 0-4 for fastball, 5-7 for slider, and 8 and 9 for change-up. Then just go down every row and that’s how you form your pitch selection. So if you went down the first row in the table, you’d get a pitch sequence of:

SL, FB, FB, FB, FB, FB, FB, FB, FB, CH, CH, SL, SL, FB, SL, FB, FB, SL, FB, CH

12 (60%) FB, 5 (25%) SL, 3 (15%) CH

You unfortunately get a run of 8 FB in a row, but just like flipping a coin, you’re going to get a significant run of heads in a row every once in a while as well. Just increase the sample size, and eventually you’ll get your preference of a 50-30-20 distribution.

• Max says:

Also, Lefties would have different optimal pitch percentages than Righties. You would have to keep track of your optimal tendencies for both lefties and righties over an extremely large sample.

Pitches also may be more effective on a visual level when used in a particular sequence. An A.J. Burnett curveball after a fastball may be more effective because the curve looks like a fastball as it starts toward the plate, and then darts downward so late that the batter can’t hold up his swing. If the batter hadn’t seen the fastball before the curveball they might be geared for a slower pitch which would give the batter more time to see the curveball is in fact a curveball and not a fastball and will dart downward and out of the strike zone for a ball which should not be swung at.

I think this analysis is great and game theory in baseball is extremely interesting for discussion. Please let me know if you think anything I stated was confusing and/or erroneous.

• TheUnrepentantGunner says:

Max: i think you are on to something. Most of what Michael Lichtman is writing supposes that you are assuming the batter is savvy, or at the very least is giving nothing away with his stance.

There are some cues though that maybe make the entire discussion completely irrelevant, such as the batter maybe moving up 18 inches in the box. Now if the batter is only mildly sophisticated, this will tell you something. Namely, that he is struggling to hit your splitter/sinker etc, and may well be expecting another one. This of course is an excellent time to throw a fastball.

Now if the hitter is truly sophisticated, they may step up, knowing that the pitcher might know what that means, and then sit fastball. Alternately through film study you could see other decisive tendencies in batters, and at least take a chance that the batter won’t change their approach.

It all reminds me of the Simpsons, when Bart plays Rock Paper Scissors, confidently thinking that “nothing beats rock”. Lisa of course is thinking “bart always throws rock”

In situations like that, you can throw your randomized probabilities, if you had them, out the window.

2. Patrick says:

It’s almost frustrating to think that something that turned out so poorly was very possibly approached in just the right way.

Which isn’t the same thing as saying Lidge wouldn’t have had better results had he mixed in some sliders (it’s very possible he would’ve), but… As MGL deconstructed above, his approach should be such that 4 fastballs is just something that happens.. And isn’t even a bad thing.

Very important to keep them from knowing what’s coming next.

Though this leaves out things like the batter getting used to a pitch, and wanting to change up the look the batter is getting. I’m sure MGL is aware of that and either knows it doesn’t really matter or just decided to leave it out.

But the idea is that the value of a fastball decreases with each fastball thrown in sequence, IE, the third fastball in a row is better than the fourth, etc, because the batter gets “used” to it, zeroes in on it, whatever.

I’m not sure it’s right – Though this thought should be testable by using Pitch(fx) data. It certainly seems at first blush like it might be an important component in deciding pitch value, important enough to modify the simple 30/70 (or whatever) split described here on a “how many of pitch X have been thrown in sequence” basis.

FX, say your fastball/slider mix should be 70/30 normally, but each fastball in a row is less useful than the one before.

If pitch 1 is a fastball, then the mix for pitch 2 might be 60/40, if pitch 2 is a fastball as well, then pitch 3′s mix might be 50/50… And so on.

Just food for thought!

• Andy says:

Well, he did mention the hypothetical 2-1 ratio wouldn’t really be static. It would change with the count. But that’s an interesting thought, I guess.

And while this might be an approach that pitchers ought to use, is this really what they’re told? “In an X-X count with this many outs after this many innings, throw this pitch Y% of the time?” I was never aware of this being how pitchers/pitching coaches operated. Really fascinating article.

3. longgandhi says:

Nice idea, but it doesn’t really work. Lidge’s slider is his best pitch. Your theory assumes that if he keeps throwing it that the Yankee batters will eventually figure out how to hit it and render it no more effective than his fastball. Unfortunately, they do not have an endless number of chances to figure it out. In fact, unless they are Joe Sewell and can foul off an endless number of pitches, their opportunities are quite limited given that Lidge would likely pitch only one inning and not nearly enough for them to familiarize themselves with it to equalize it with his fastball. It was his best chance to get hitters out.

• JDSussman says:

What MGL is saying is that every pitch’s value isn’t independant of the pitcher’s other pitchers. A large part of a pitch’s value is the proprotion in which it is offered compared those other pitches.

Because those pitches are equally as valuable given a proper proportion they all need to be thrown. However, what MGL is ending with is there is a random variation of the selection of pitches thrown within the proportion. And the 4 FB thrown doesn’t mean he should have been throwing more sliders.

• JDSussman says:

MGL, if that is completely wrong feel free to tear me apart lol

4. taylor says:

I think you are ignoring the fact that A-Rod and Texiera destroy fastballs. Lidge’s fastball is not his best pitch AND those two hitters are very good at hitting the fastball. The saying is something like, you don’t want to get beat on a pitch that isn’t your best, Lidge was playing with fire throwing those pitches, and I think its telling that each at bat only lasted 4 fastballs.

Also I think you’re forgetting the human element, this wasn’t a vacuum, this was game 4 of the World Series. Not to mention what Damon had done to his psyche. I don’t know game theory can properly describe what was happening there, I think it was just baseball.

• PhD Brian says:

No he is not. The ratio of sliders to fastballs adjusts with the batters abilities. If Tex kills fastballs then you throw him fewer fastballs, but you do not stop throwing him fastballs. Almost every player in baseball kills a pitch if they know it is coming. If Tex knows Lidge is always going to throw a slider (100%), then it probably does not matter how good a slider lidge can throw because Tex will hit it nearly every time. Accordingly, if Tex is 100% certain Lidge will never throw him a fastball, then even the worst thrown fastball becomes a decent pitch.

Pascuel Perez was not the best pitcher ever, but he used to throw this weird lob of a pitch like once a game that really embarrassed hitters. I forget what the pitch was called. But it looked like a pitch any 3rd grader could hit out of the park, but since it usually came at a totally unexpected time the batter would just take it almost every time. Perez could throw a low 90s fastball for a strike, and would over and over. then he would lob that weird third grade batting practice pitch and the batter would just stand there. It always made me laugh. Problem with perez is sometimes he fell in love with making batters look stupid and he would sometimes throw that pitch a second or third time in the game. If he did then it was treated the way it deserved and boom! The pitch only worked because it was a complete surprise. As soon as one guy was embarrassed all the other players would keep an eye open for it and kill it. Moral: a third grade batting practice pitch can be very effective if thrown about 1% of the time. On the flipside, almost no pitch is effective if major league players know you will throw it every single time.

5. MGL says:

Great comments so far, at least the first 3.

“Though this leaves out things like the batter getting used to a pitch, and wanting to change up the look the batter is getting. I’m sure MGL is aware of that and either knows it doesn’t really matter or just decided to leave it out.

But the idea is that the value of a fastball decreases with each fastball thrown in sequence, IE, the third fastball in a row is better than the fourth, etc, because the batter gets “used” to it, zeroes in on it, whatever.

I’m not sure it’s right – Though this thought should be testable by using Pitch(fx) data.”

That is 100% correct. It MAY be correct that after 1 fastball or 3 fastballs that a batter is “used” to seeing the fastball and therefore another pitch like the slider goes up in value (of course once you throw it more it has the same value as the fastball). As you correctly say, we don’t know that to be true but we can find out (presumably) with the pitch f/x data.

But, even if that is true, all that means is that we have to adjust the percentages. For example, let’s say that in a certain situation the 2-1 ratio of fastball to slider is correct. It might be, if there is anything to the “used to” or “change the batter’s line of sight” theory, or whatever you want to call it, that after one fastball, the optimal percentages are now 64/36 rather than 67/33 and that after 3 fastballs it is 60/40. I highly doubt that the percentages will change all that much. (BTW, the authors found that after throwing pitch A, pitchers are more likely to throw pitch B than pitch A even after adjusting for the count. “This suggests that either there IS something to the “used to” theory or that pitchers are making a mistake. I think it may be both. Because, as Gunner says above, it is very difficult for a human being to randomize their decisions, so they tend to “change gears” more often than is random.)

Even if that is the case (that pitchers should “switch gears” to some extent), we still should see runs of 3,4,5 or more fastballs (or any pitch) in a row. So again, we can never know if the pitch sequence to a batter is correct or not unless the pitchers throws one pitch that we think should never be thrown, which is rarely the case (such as if he throws a 3-0 curve ball to a pitcher leading off an inning when we know that he cannot control the curve ball as much as he can control the fastball – there is obviously a near 100% chance that the pitcher is not swinging).

Again, if it is indeed correct to throw more off-speed pitches after a run of fastballs because the hitter is “used to” seeing fastballs, the hitter will know that, so it has to be correct given the fact that the hitter now will look for a different pitch more often than if the previous pitcher were a random pitch. Again, that might be the case. We don’t know. It sounds plausible, but we don’t know for sure. And as I said, if it is true, it is not going to change the percentages all the much. Imagine that you just threw 3 fastballs in a row and the count is 2-2. And imagine that at a 2-2 count, your usually (it is correct to) throw your fastball 50% of the time and your curve ball 50% of the time. If there is something to the “used to” theory, surely you can’t now throw a curve ball 70% of the time instead of the usual 50% of the time at a 2-2 count. If the batter knows a curve ball is coming 70% of the time, even though he was just used to seeing fastballs 3 times in a row, the value of that curve ball is surely going to be less than the value of the fastball. Again, I would say that the change in percentages should not be more than a couple of percent or so. And I’m not really sure it is anything (or if it is it may be de minimus).

6. MGL says:

Sorry, I meant the first 4 (comments are good). #5 and #6 have no idea what I am taking about I am afraid.

7. Todd says:

I think that you are making the assumption that there is a strong, if not perfect, correlation between the effectiveness of a pitch and the batters knowledge that it is coming. I am not sure that this is the case.

Some batters may sit on one pitch and protect the others, but I think when A-Rod is locked in, he is not playing games, he just stands at the plate and is ready to react to whatever is presented to him.

Some pitches are simply harder to hit than others. He could tell A-Rod that a slider is coming, and A-Rod may still have a harder time centering the ball and getting a hit than if he threw a fastball when A-Rod did not know what was coming.

That being said, I often think that it would be an effective strategy for a catcher to have a die with him and make it appear to the batter that the pitch selection is random.

I guess what you may be saying is that you could create a grid, with all the pitches a batter could throw on both the x and y axis, and in the boxes put the effectiveness given what the batter is expecting. Then the game becomes balancing throwing a pitch sequence that controls what the batter is expecting and still providing the most effective pitch for that situation – that is the game theory.

8. Neil says:

What about Johnny Damon’s theory? He suggested that Lidge laid off the slider because, once Damon had reached third, even an effective pitch (in terms of the batter missing it) would have a much higher chance of skipping away from the catcher than would a fastball, allowing Damon to score. At the very least, that possibility had to be in Lidge’s (and Ruiz’s) mind.

• Wally says:

And it was probably in the batter’s mind. So the batter was probably looking for more fastballs, making the slider a more valuable pitch. Which means you have two competing factors, the slider loses value because of the WP possibility, but gains value because the batter isn’t likely to look for it. Its pretty hard to just guess what that means for the usage percentage or which factor wins.

9. Wally says:

So how about the ol’ reverse phsycology trick where the catcher tells the batter what pitch is coming (wasn’t this in Major League?). Now the hitter can’t possibly think the catch is actually telling him the real pitch right, but what if that’s what the catcher wants him to think. I love shit like this.

But I do believe Todd brings up a good point that may be worth an extra parameter in the model, but it doesn’t change the basic principle. Say A-rod genuinely does just recognize and react to pitches and does not guess at a specific pitch very often. Then you might be temped to just through your best pitch all the time because he isn’t looking for anything in particular. But if you started treating him like that, he’ll surely adjust and assume you’re just going to through your best pitch. So we’re just back to where MGL put us. Then if, say Cano, guesses on fastballs a lot and just tries to foul off breaking balls, then you might be tempted to through more breaking balls. But, if Cano is smart he’ll adjust and stop thinking fastball so much. So again we’re back where MGL put us. And it just turns into a batter specific usage percentage. So if a batter tends to look for certain pitches, through him more pitches of a different kind and only change that once he changes.

Oh and about how well humans can randomize, how often do you see people go back to back with the same choice in “rock-paper-scissors,” much less 3 times?

10. Toffer Peak says:

The theory is solid enough but in practice it has not been true. Since 2002 Lidge’s wSL/C has been 2.03 and his wFB/C has been below average at -0.62. Sure Pitch Type Values are context and fielding dependent but with that large a sample size they should be reasonably stable. Wouldn’t this suggest that either hitters are still not recognizing Lidge’s Sliders or he is pitching inefficiently? Or am I misinterpreting the data or the argument?

11. MGL says:

Toffer, as I said, a pitcher does not necessarily have to have the same value for all his pitchers overall, only in any given situation. For example, let’s say that a pitcher only faces two batters, A-Rod and Jose Molina. And let’s say that to A-Rod, you throw 40% fastballs and 60% off-speed. The value of those pitches has to be the same to A-Rod. Say it is +2.00 (runs per 1000 or whatever they use at Fangraphs). And say to Molina, you throw 60% fastballs and 40% off-speed, and the value of both those pitches is -2.00. While those are optimal percentages to each batter, the overall is 1.6 for the fastball and -.4 for the off-speed. Then there are count issues as well.

So while the value of a pitcher’s different pitches overall do not have to be the same, they should probably be close. If they aren’t, then yes, it is likely that the pitcher or the batters are not doing the proper thing.

I don’t now off the top of my head what the SD of those pitch values are, but I would guess that even in several seasons for a reliever like Lidge, there is a decent amount of noise in those values.

But given such a large disparity, someone IS probably doing something wrong.

12. Mike Green says:

The key points are that:

1. the optimal distribution of pitches from Lidge’s perspective varies with the hitter (especially handedness), game situation and count, and
2. you cannot really talk about pitch type from pitch location independently.

Rodriguez kills fastballs, and particularly poor ones, and doesn’t kill sliders. It may be that the optimal rate in the situation overall would be 50-50. You still can get four fastballs or four sliders in a row as a reasonably possible distribution.

13. Hardy says:

Good article. But there is one mistake that was also (in a slightly different fashion) in the sac bunt article:

> 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.

If A-Rod has an out of equilibrium belief, pitch values will be different. In that case, Lidge should use the better pitch (the fastball) 100%.

14. MGL says:

“If A-Rod has an out of equilibrium belief, pitch values will be different. In that case, Lidge should use the better pitch (the fastball) 100%.”

Yes, for that one instant. But if he wants to preserve that mistake against A-Rod and everyone else he faces in the future, and surely he does, he cannot throw the fastball 100% of the time!

Mike, I don’t know that your #2 is a “key point” (it is true of course – a “pitch” includes the type AND the intended location), but yes, you are 100% correct in what you wrote.

• Hardy says:

> But if he wants to preserve that mistake against A-Rod and everyone else he faces in the future, and surely he does, he cannot throw the fastball 100% of the time!

Again, like in the hit-or-bunt example, this is certainly possible. But optimal behavior against out-of-equilibrium is not easyli solved without further information / assumptions. It now becomes a matter of the difference in WE between the optimal and the current behavior of the batter, how the learning happens etc.

And regarding the situation at hand: Do your really think that you will gain enough value in the future to throw the inferior pitch in Game 4 of the WS with a LI of 5?

15. kds says:

Great article, lots to think about.
If you are implying that in practice pitchers are close to equalibrium in deciding how to mix up their pitches, I think that it is unlikely. Partly this is based on how poorly people do in economic markets and other games. Also, I have seen very little evidence that teams are trying to include game theory in their decision making. ( The analysis of bunting in the Book and elsewhere does not suggest to me much team sophistication in these matters.) How far beyond, “this is a fastball count,” are many teams?
As others have said, I think it is hard to come up with true randomness in picking which pitches to throw in a situation. In practice you might have to have all pitches called from the dugout, because I think you might only be able to do this with computer assistance. You would want to know, “what to throw A-Rod on a 1-1 pitch having thrown him 2 fastballs, with runners at the corners and 2 out” It wouldn’t be too hard to program such a device, add a random number generator and you’re ready to play.

16. Max says:

Pitches also may be more effective on a visual level when used in a particular sequence. An A.J. Burnett curveball after a fastball may be more effective because the curve looks like a fastball as it starts toward the plate, and then darts downward so late that the batter can’t hold up his swing. If the batter hadn’t seen the fastball before the curveball they might be geared for a slower pitch which would give the batter more time to see the curveball is in fact a curveball and not a fastball and will dart downward and out of the strike zone for a ball which should not be swung at.

I think this analysis is great and game theory in baseball is extremely interesting for discussion. Please let me know if you think anything I stated was confusing and/or erroneous.

17. MGL says:

max, yes you may be right about the “sequencing of pitches.” We really don’t know but it sounds plausible. Another reason why the pitch f/x data is the “holy grail.”

kds, I am more confident that pitchers throw more optimally than batters think optimally. There are many reasons why batters may anticipate the fastball more than they should. One of the primary reason is that batters are “taught” to look fastball and adjust to the off-speed. While there are sound reasons for that, I don’t think that is a good enough way to teach batters the proper method of anticipating pitches. Another primary reason is that batters look foolish when they look off speed and get a fastball. Batters do not like to swing really late at a fastball. It is not a “macho” thing to do. Looking foolish on an off-speed pitch is not such a bad thing to a batter’s ego. One more reason why I think that batters look for fastballs too often or too much is that many batters tend to gear up for one pitch or another rather than gearing up for percentages. Like McCarver, they somehow think there is only one correct pitch they are going to get. If you are only going to gear up for one pitch, which pitch are you going to gear up for in most counts? The one that pitchers throw 70% of the time – the fastball. Consequently, pitchers can throw 65% fastballs, or 60% fastballs or even 50% fastballs, and batters may still be completely gearing up for the fastball, incorrectly so.

As kds says, most baseball people do not understand the basic concepts of game theory (again, think McCarver, who played for 20 years) and they are not taught it. Therefore, it is unlikely that they play optimally, although as I said, I do think that pitchers throw more optimally than batters anticipate the pitches. If for no other reason, pitchers are selected at least partially on their pitch selection ability. Batters are not (although it obviously helps).

• Jesse says:

Brad Lidge threw his slider 49% of the time this season and 56% of the time last season, so your 2-1 ratio is off right from the start (though I suppose you could be arguing that he throws it too much in general). Given that he threw the fastball around 50% of the time this year, the probability of throwing eight in a row is .0039 (or roughly 1/3 of 1%).

Also, Josh Kalk has studied the effect of throwing the same pitch in the same location twice in a row and and second fastballs (both 4-seamers and 2-seamers) grade out terribly. See the discussion at http://www.hardballtimes.com/main/article/doubling-up/

Finally, as others have pointed out, Alex Rodriguez absolutely murders fastballs and this is known throughout the league to the extent that he saw them just 59% of the time this season. Mark Teixeira does the same and saw just 55% fastballs. Compare that with Derek Jeter (64%) and Johnny Damon (65%), two good hitters (respective wOBAs of .390 and .376) who have faced many of the same pitchers.

Even outside of doubling up pitches and the fastball-hitting tendencies of the hitters, it is highly improbable that Lidge would have thrown eight straight fastballs (again, a probability of 0.0039).

So, did Lidge throw Tex and A-Rod too many fastballs? I have been paying attention and I paid attention to the game and the answer is yes. Tim McCarver is a stooge, but even he wouldn’t ask a pitcher with a below average fastball to throw eight straight of them to hitters with wFB/C of 2.19 and 1.86, respectively.

18. neuter_your_dogma says:

One thing we can agree on – Lidge threw Tex and A-Rod too many hittable fastballs.

19. Oscar says:

I would LOVE to see a pitcher try to truly randomize their pitch selection. Find some linguistic example: for example, make a rule that before every pitch, you choose a word “at random”. If the third letter is a vowel or y, you throw a slider, otherwise fastball. That’s close to 30%, and it’s hard to subconsciously choose a word based on its third letter.

• neuter_your_dogma says:

Are any of us truely able to think a random word? I tried, and the only word I keep seeing is “slider.”

20. MGL says:

The best way to think of a “random” word is to look around you and choose an object. The problem of course is that you have to randomize each pitch around a certain mean which changes from pitch to pitch. That is not an easy task to say the least. You are almost better off not randomizing and just throwing the 70% (or whatever the number is) a lot, and then just mixing in some other pitches to the best of your ability.

Jesse, my 2-1 ratio was just for illustration purposes and had nothing to do with Lidge or A-Rod. The correct ratio versus A-Rod might be 3-2 slider with no one on third base and 1-1 with a runner on third. Whatever it is, one sequence that is POSSIBLE if he is throwing optimally and adhering to the correct numbers whether they are 1-1 or 3-1 (in favor of a slider) is 4 fastballs in a row or 8 fastballs in a row. Eventually in similar situations (given enough of those situations), he will throw 8 fastballs in a row (even though that is exceedingly unlikely in any one situation) without it being a mistake. That was my point.

However, I should have said, given your point, that if we see a pitcher throw a sequence that is unlikely, given the fact that we know that pitchers DO make mistakes and do not always use optimal strategies, it is much more likely that he was not using the proper percentages and did in fact make a “mistake.”

• Jesse says:

mgl — I agree with you that it’s theoretically possible that Lidge wasn’t avoiding the slider at all costs, but I would say the evidence overwhelmingly supports the conclusion that he was. I guess what I took issue with from the original article (and thanks for writing it by the way) was that you closed it with “I have no idea and neither should you.” I think we can probably agree that it’s incredibly unlikely that the probability of Lidge throwing a slider on any given pitch was anywhere near as high as it should have been.

If he was planning to throw the slider once every four pitches, there would still only be a 10% chance of him throwing eight straight fastballs, and I think he probably should have thrown it more frequently than that, anyway.

21. intricatenick says:

Like in all minimax debates – this valuation assumes the situation is at equilibrium. A scientific analogy is dissolved CO2 concentration in soda versus its pH. The only way to solve the simultaneuous equations is to assume a state of equilibrium has been reached. There is no way to prove the equilibrium exists – it must be assumed. The word “optimal” is something of a useful fiction like “true talent level”.

It’s a fiction because you can’t measure it – your measurements imply it. In physics, this equates to things like “centrifugal forces”, which only arise due to the setting of reference frames.

22. MGL says:

“guess what I took issue with from the original article (and thanks for writing it by the way) was that you closed it with “I have no idea and neither should you.”

Right, I should have said that we can never be 100% certain whether he was making a mistake or not, but given the Bayesian problem I described in my last post, it was probably much more likely that he made a mistake than that he just happened to throw 4 fastballs in a row of whatever it was. I definitely stand corrected.

• Jesse says:

no worries, glad to see we agree on it!