THE BOOK--Playing The Percentages In Baseball

Another assumption being made here is that all (or a great majority) of pitchers will use the same game theory to throw a curveball to FrenchyFrenchFrench with 2 strikes.  Some pitchers may be just as braindead and throw him strikes (5 hits in 44 ABs). 

In general, *some* pitchers must throw him strikes in an 0-2 count.  However, that doesn’t mean that *all* pitchers must sometimes throw him strikes.  Pitchers, especially ones not likely to face him many times in a given time period (interleague, relievers, old pitchers who won’t be around much longer, etc) shouldn’t worry about the batter getting a book on them because by the time they figure out what’s happening they won’t be facing them anymore.

What happens to the next pitcher or even the next game that the same pitcher sees him shouldn’t really be the focus right then.  The pitcher should throw the pitch most likely to get FrenchyFrenchFrench out *that AB*.  If he figures it out for next time, so be it, the pitcher regroups and finds another way to exploit the hitter.  In this specific case, even if Francoeur somehow lays off the 0-2 curve, his odds of getting a hit go up by less than 2% (11 for 83).

Game Theory is a fascinating topic (of which we have devoted a chapter of in The Book, written by MGL).

I believe (unsupported) that it is Game Theory that makes Maddux and Moyer look as brilliant as they do.

An interesting case is always Mike Piazza, who has never swung at a 3-0 strike pitch over a period of several years (150 or so pitches).  Never!  And yet, pitchers still get more called balls on him than with an average hitter (who swings at such a pitch almost 10% of the time).  Is it because Piazza has it in his head that he won’t do such a thing, or are pitchers so fearful of him that they never give him one anywhere close to the plate, that they’d literally rather walk him?

And of course, Piazza is a person, so that even if he has it in his head that he won’t swing, it may not stay like that, if pitchers start throwing pitches down the middle to him.  And if they do that, he might start swinging, and now pitchers don’t know what’s in his head.

The small sample sizes at certain counts leads to alot of uncertainty.  The aging, change in talent levels and wisdom of players, also adds more uncertainty.  The lack of control of pitchers adds more uncertainty (we don’t know where they intended the ball thrown, just where it actually ended up).

All in all, you can indeed always do something at 100%, as long as the other guy doesn’t figure out you are doing it at 100%.  In a case like Piazza, if he indeed wants to never swing at a 3-0 pitch, come hell or high water, while the rest of the league does it at say 9%, after 100 pitches, he’s at 3 SD.  We may be able to figure it out at that point.  So, the lesson is that it’s very hard to do something at 100% without the other guy being able to figure that out.

One complication in all this is that keeping batters “honest” is like a public good.  While it’s in the best interest of pitchers generally to keep Francouer swinging with 2 strikes, there isn’t as much incentive for any one pitcher to be the one who keeps Francouer honest by throwing him fastballs in the strike zone some percentage of the time.  Since batters are more likely to make adjustments generally ("don’t swing as much with two strikes") than against a particular pitcher ("don’t swing as much with two strikes against Maddux"), it would seem that pitchers wouldn’t necessarily find it in their interest to deviate much from the “perfect” pitch for any situation.  Perhaps I’m underestimating the amount of adjustments batters make against specific pitchers, though.

I believe this is a very difficult and tricky (and fascinating) concept to get one’s arms around.  I disagree with both Warren and Charles.  It seems as if any given pitcher can wait for the “other” guy to deviate from the “best” pitch, or that if you are only going to face a guy once or twice, why throw him that 10 or 20% pitch, or why worry about future PA’s, but IT DOESN’T WORK THAT WAY, and I can prove that!

In any given situation, no matter who you are, and no matter whom you face, you MUST throw a variety of pitches, even if the “bad” pitch is only thrown 10% of the time.  Now, pitchers don’t need to optimize (as if their opponents were game theory experts) their game theory strategies because batters do not generally optimize theirs, but still, no pitch can be thrown 100% of the time.

The variation a pitcher should stray from the “optimial” pitch is always going to be a function of the ability of the opposing hitter to “learn”.  A hitter with no memory should be thrown the pitcher’s best pitch every time, while facing extremely intelligent hitter requires quite a bit of randomness in pitch selection.

But even for a given batter, I would think there are different ways the batter can adapt to pitching strategies.  Francouer (for example), might continue swinging at two-strike pitches outside the zone as long as opposing pitchers given him a token few percent of hittable pitches.  But it’s not contradictory to say that pitchers can’t just throw Francouer pitchers out of the strike zone on *every* count.  Certainly, any major league hitter is going to stop swinging if they never see a pitch in the strike zone.  So even batters who appear to be poor optimizers are going to optimize to some degree.  Just because pitchers can’t throw their best pitch 100% of the time doesn’t mean that it’s in any individual pitcher’s interest to give Francouer a two-strike pitch in the strike zone.  While they get some of the gain from keeping Francouer honest, they don’t get all of the benefit, which should reduce their incentive to make those “bad” pitches.

There’s also the question of whether hitters are more likely to pick up on a pitcher’s patterns during an at-bat rather than over the course of multiple at-bats.  If a fair amount of a hitter’s learning occurs in the context of a single at-bat, then there’s less of a reason to keep a hitter honest on a pitch that is likely to end the at-bat, since the hitter is less likely to apply that knowledge to the next at-bat than to this one.  While this may be true to some degree, I’m not sure if that’s an accurate picture of how hitters learn, though.

MGL,

This is fascinating stuff, and I think much of what you’ve said is correct, but I think your conclusion isn’t quite as air-tight as your arguing.  Game theory assumes a rational agent who can reliably collect information and properly apply it in order to estimate an optimal behavior. The pitcher (one rational agent) and the batter (another rational agent) adapt to one another until they achieve an equilibrium. The problem with this type of model is that humans (most baseball players included) are not rational agents.

As tango pointed out, hitters arent ideal in gathering data: there is uncertainty stemming from small sample sizes, and unpredictable changes in hitter philosophy.  Additionally, there are psychological factors, such as imperfect memory, and biases, that add to the uncertainty. But more fundamentally, the brain simply may not perform the proper calculations to achieve the equilibrium game theory suggests is optimal.

For instance, consider a power hitter (Joe Slugger), who looks fastball in 0-2 counts occasionally (lets say 10%).  Joe knows the pitcher he’s up against (Sammy Splitter) has never thrown a fastballs in this count in his 10 year career.  Thus, because Sammy is perfectly predictable, a GT optimal behavior would likely involve never looking for a fastball.  The reality is, though, if Joe is being rewarded for hitting bombs off of other pitchers in 0-2 counts (or similar counts), the swinging behavior is reinforced.  Even though it isn’t rational, he will swing some percentage of the time. He will probably look disgusted right after the swing, as hitters often do, as the rational part of their brain catches up, and they realize they made a mistake.

The question is how good is the human brain is at learning the optimal behavioral response in these situations.  In many domains, theres a lot of evidence to suggest that we do a poor job of this.

Using game theory to explore the hitter-pitcher duel is fascinating, and I’m not arguing against it. I’d just argue that one should take any conclusions with a grain of salt, because formal GT models can’t explain all the variance in human behavior.

-cdm

I was all set to call bullsh-t on Barry Bonds.

``[Barry] Bonds said he had Randy Johnson cold,” reported Gaston. ``He told me, `I know what he’s throwing every time.’ “

Thanks to Retrosheet and Baseball-Reference, we know the answer:
http://www.baseball-reference.com/friv/bvp.cgi?n1=bondsba01&n2=johnsra05

On 58 PA, we get a BA/OBP/SLG of .333/.483/.600 (includes 2 IBB, 2 HBP).  That’s pretty much Barry’s late-season career line.  With the exclusions, it’s a wOBA of around .450.  In essence, he turned RANDY JOHNSON into Randy Johnson, league average pitcher whom Barry hits at a .450 clip.  (This performance is about 3 SD from the mean.)

``Bobby Abreu had my slider,” Timlin said. ``He told me I was tipping it after I got traded to the Phillies and became his teammate. But while I was with the Cardinals [2000-01], he had me. I had a pretty good slider in those years, but he always seemed to know to lay off that pitch.”

I guess we can call Timlin on it.  I’ll have to see if they faced each other in 1999 (interleague), but while with the Cards (which is what he claims), they faced each other TWO times, with a K and a fly to CF.

Retrosheet shows they didn’t face each other:
http://www.retrosheet.org/boxesetc/MU1_timlm001.htm

You’d think they may have faced each other in this game:
http://www.retrosheet.org/boxesetc/B06050BAL1999.htm

But, Rhodes faced three batters and walked them all, which could only be with Abreu as the third man.  Timlin likely faced 8 batters, meaning he didn’t face Abreu.

In the other two games in which they both appeared (which we can deduce that Timlin didn’t pitch to him), Abreu was hitless anyway.

So, Mike Timlin: “BullSh!t”.

Great post and comments!  Warren makes an interesting point in his post that it might not be in an individuals pitcher’s best interest to be the one to vary pitches, and just that some pitcher has to do it.  I think this makes sense if you are looking at the world of pitchers against a single batter.  However, when you think of a single pitcher against the world of batters, that single pitcher, at some point, has to vary his pitches.  Otherwise all batters will know what is coming from that pitcher.

Re:  Timlin, actually, all he’s claiming is that Abreu didn’t swing at the slider.  So I don’t think Retrosheet helps us with that.

GT says nothing about whether the players need to be or are rational or not.  GT is simply the study and analysis of strategies in certain types of contests.

The goal of a pitcher in the context of GT (or whatever you want to call it) is to optimize his strategy versus any given batter in any given situation, given that batter’s abilities and strategies.  Ditto for the batter.  That generally means that the pitcher should almost never throw any one pitch 100% of the time.  Likewise the batter should almost never look for any one pitch 100% of the time (otherwise the pitcher would never throw that pitch).

That’s really all there is to it.

A pitcher who waits for the “other guy” to vary his pitches, or who throws a certain pitch 100% of the time because he is “waiting” for the next PA to mix up his strategy, etc., is NOT, by definiton, optimizing his strategy, and that can be proved mathematically.

It is a little bit similar to not voting as a strategy (or non-strategy) because your vote has almost no chance of affecting the outcome of an election.

MGL:
“A pitcher who waits for the “other guy” to vary his pitches, or who throws a certain pitch 100% of the time because he is “waiting” for the next PA to mix up his strategy, etc., is NOT, by definiton, optimizing his strategy, and that can be proved mathematically.”

I don’t want to argue against this point strongly, because I think its generally correct.  But its important to emphasize that while your proof may be mathematical, unlike in physics, where a mathematical model can describe the real world incredibly accurately, mathematical models do not describe human behavior very well.

The GT “optimal” depends on the expected behavior of each of the participants.  The optimal will change drastically as a function of the dynamics of the players (consider how different the optimal strategy is in an AB against Mr. Swings-At_F*ing-Everything vs. Youk).

In order to model the players properly (i.e., predict the dynamics of his interaction; how the hitter will adapt to the pitcher and vise versa), you have to take into account a ton of psychological variables.  These variables are difficult to account for when applying a GT analysis. But if you don’t model them, then GT will not provide the “true” optimal behavior.

Essentially, regardless of the true, empirical value of different strategies (which can be mathematically determined), the “optimal” depends upon an understanding of why the players do what they do, and understanding the biases they have and the mistakes that is in their nature to make To the extent that you ignore these variables is the extent to which your estimate of the optimal strategy is flawed, regardless of the math.

I don’t want to argue either, and I think we are largely in agreement.  Just because you cannot predict with any particular degree of accuracy a person’s behavior, strategy, etc., does NOT mean that you cannot model it and come up with an optimal strategy.  “Optimal” simply means the one that produces the most desired result given the informattion you have or don’t have.

As well, I was not arguing for any particular result, only that it is rarely if ever correct for a pitcher to throw the same pitch 100% of the time regardless of the batter/situation.  It is a misconception that a pitcher can “wait” for someone else or for a future PA in order to mix up his pitches.  His optimal strategy, regardless of how well we can predict the strategy or behavior of the batter, is ALWAYS to mix up his pitches, and that applies to ONE PA or to the next 100.

By “mixup his pitches”, you likely mean to randomly select a pitch, based on the mean of each pitch. 

If Pedro say optimally throws, on a 0-0 count, a fastball 60% of the time, and 30% a breaking pitch, and 10% a changeup, his “randomizer” might say “breaking pitch” on that throw.

Say it’s now 0-1, and on those counts in which he already threw a breaking pitch, his mean might be: fastball 50%, breaking pitch 30%, changeup 20%.  His randomizer might say “breaking pitch” once more.

He’s now at 0-2, and on those counts in which he already threw two breaking pitches, his mean, for this kind of batter, might be fastball 20%, breaking pitch 60%, and changeup 20%.  His randomizer might say “breaking pitch” yet again.

He’s strikes him out.  3 pitches, 3 curveballs.  He in fact did “mix up” his pitches, but we can’t tell.

Yes, of course.  And even though you can “guess” a certain pitch (and location) 90% of the time occasionally, and it may seem like that is the ONLY pitch to throw in that situation, all good pitchers, and possibly even all pitchers, will in fact occasionally throw some other pitch, “just to keep the batter(s) honest.”

When you watch a pitcher pitch, it often seems like he throws the “right” pitch most of the time and the “wrong” pitch occasionally, when in fact he is doing the right thing overall and that “wrong” pitch is simply the 10% or 20% pitch.  This is the thing makes it hard to wrap your hands around GT and pitching.

Great Piece, MGL. As an econ grad, I can really appreciate the application that you have written here.

To some of the other posters here, how learning takes place, I would say something. Angels, aas a team realized that the Yankees are a very patient hitting team, and specifically reluctant to swing at the first pitch. This led to someone in the Angels pitching staff (Lackey?) once say that they use it as a strategy to throw a first pitch strike to the Yankees.

One thing that you did not mention, and what might have led to a confusion here, is that mistakes occur with respect to location, and not with pitch selection. So, if Frenchy is able to hit a fastball in the strike zone, it was probably not because of bad planning (in a sample size of 1, there is really no bad planning), but bad execution on the part of the pitcher. This does not invalidate the conclusions of game theory.

Of course, if you are perfect with execution, you can seemingly get away with violating this particular principle of game theory of randomizing your strategy (pitch type). The only outlier, I think, is Mariano Rivera, just as in many other cases. He has one pitch and impeccable control of that pitch. Doesn’t really matter if he tells people what is coming. But I would be surprised if, even he, doesn’t mix his pitches. He does throw a four seamer, I think.

"“Optimal” simply means the one that produces the most desired result given the information you have or don’t have.”

Sure. I agree with 99% of what you said. I was just emphasizing that even when you can prove mathematically that some bit of information should help the player, they aren’t necessarily capable of using it; they’re human. 

But, of course, quite frequently, they do use the information.  My personal favorite: May 10th, 2005. Kevin Millar vs. Octavio Dotel.  Millar said after the game that he watched Dotel throw 10 straight fastballs to Ramirez and Ortiz before his AB.  When he came to bat, he was looking fastball.  After 4 more fastballs from Dotel, he said he would have jumped out of his cleats if Dotel had thrown a breaking ball. But sure enough, Dotel threw a 15th fastball in a row… and fed the monster.

http://www.retrosheet.org/boxesetc/B05100BOS2005.htm

According to Inside Edge / STATS, of the 2719 Rivera pitches in their sample, 100% of them are fastballs.  50% of his pitches are thrown inside the strike zone.

Just taking a pitcher at random, Mussina also throws 50% of his pitches in the strike zone.  His distribution of pitches is 50% fastball, 25% curves, 15% sliders, 10% changeups.

Schilling is at 55% of his pitcher in the strike zone.  Distribution is 65% fastball, and an even spread otherwise.

While Rivera always throws a “fastball”, not all of his fastballs are the same.  His cut fastball (which has so much movement on it that it seems unfair to actually call it a fastball) is a very different pitch from his straight fastball (which, as a rough estimate, I’d say he throws around 20% of the time).  Even the guy with the best single pitch in baseball realizes he can’t throw it every single time.

GT is about the expectancy of each pitch as compared to what the batter is expecting.  Thus it is partly based on what a pitcher’s best (and worst) pitches are.  If a pitcher has a terrible curveball and a great fastball, he may never throw the curve even if that makes it easier for the batter to guess what pitch is coming.

In the Braves game last night, in the 9th inning Francouer was leading off with the Braves down by a run.  Borowski, the Marlins closer, started off by throwing a bunch a pitches out of the zone.  A couple were even close.  Francouer was not biting.  This is an example of how even the biggest free-swinger in baseball CAN lay off bad pitches, especially when he knows that everyone tries to throw him pitches out of the zone, especially with the game on the line.  Now when he got to 3-2, I would have thrown an off-speed pitch out of the zone to Francouer (not 100% of the time), but the pitcher threw like 3 or 4 pitches right over the plate and Francouer eventually hit a single (albeit a seeing-eye one).  I think that the risk of the walk (not good to walk the leadoff batter in a 1-run game) was better than the risk of the extra-base hit (really bad to give up a leadoff extra-base hit in a 1-run game), with Francouer at the plate, who obviously can crush pitches in the zone.

With more and more managers turning to statistics with their players, I think we will see an increase in the use of game theory in baseball soon!  Here is a cool article about how it could be used in the future:

http://www.mindreign.com/en/mindshare/Global-Economics/Game-Theory-e2-80-99s-Potential/sl35291137bp491cpp10pn1.html

I forgot how good a post this was!  And thanks for the link, Jimmy.  What I didn’t mention, and probably should have, was that the value of each pitch in each situation given the batter’s anticipation and response is part of the game theory equation.  And that the quality of each pitch is a factor in its value, which includes the location of course.  And I erred in my “treatise” by stating that the pitcher MUST never throw a certain pitch 100% of the time.  That is not correct.  It is entirely possible that the value of a certain pitcher to the pitcher might be higher even when the batter knows it is coming then any other pitch even if the batter is not expecting it or expects it only 1% or 10% of the time.  Examples should be obvious.  If I have a good fastball and no other pitch that I can throw in the strike zone, I will be forced to throw my fastball 100% of the time even though the batter knows it is coming.  Why do most pitchers throw a fastball 100% of the time with a 3-0 count to the opposing pitcher with 0 outs and no runners on base.  While the pitcher knows it is coming, the value of the fastball is greater (for the pitcher on the mound) than an off-speed pitch because of the chances that the off-speed pitch might be a ball (which is probably 50%).  The quality of the pitch in that situation does not matter at all, other than the location, because the pitcher at the plate is not swinging 99.9% of the time.

So, again, I made a mistake in the post. I should have said that game theory suggests a frequency for each pitch from 0-100%, inclusive, depending on the value of each pitch given the anticipation of the batter.  We can define exactly what those percentages are for each pitch in each game situation, given the quality of the pitch and the weaknesses and strengths of the batter, assuming that the batter responds with perfect game theory strategy (such that we are in a Nash equilibrium).  However, it is probable that the batter will NOT respond with perfect game theory strategy, in which case, we, as a pitcher, adjust our strategy.  However, if we don’t know anything about the batter, other than his physical tendencies, our frequencies are per-determined and fixed.  It is possible, maybe even probable again, that the average response by a batter is pretty far from optimal, so that if we don’t know anything about a batter, we DON’T assume that he will use perfect game theory strategy.  For example, I think that we can safely assume that most or all batters, and certainly the average batter, looks for a fastball too much in almost all counts, because it is the “macho” thing to do.  For example, I was watching the Cubs/SD game the otehr night with Dempster on the hill for the Cubs.  Dempster loves to pitch backwards.  He will throw an offspeed pitch (slider or changeup) more often than the typical pitcher in fastball counts and vice versa.  The Padre batters were constantly looking fastball in those counts (incorrectly so) and missing the off-speed pitches (and swinging at the ones out of the zone) by a mile.  Consequently, Dempster got like 10 or 11 K’s in 7 innings I think.  Apparently they are either slow learners (many of them are young or inexperienced) or they are afraid to look off-speed and then look really bad on a fastball.

Ah, give me an entire team for Spring Training and I’ll guarantee 5 extra wins between working with the managers, coaches, pitchers, and hitters. At least!  Hear that Tango?  Mariners?

Tango, you got it right the first time. The pitcher seeking an optimal strategy will never decide to throw one type of pitch 100% of the time if he has multiple pitches to choose from.

The reason why is not game theory, its learning theory.  The value of each type of pitch type is uncertain and nonstationary. Because of the uncertainty, you can never completely reject any pitch as sub-optimal.

If the value of a pitch is uncertain, then the only way that a pitcher can be considered an optimal controller is if (as time goes to infinity), all options have some probability of being selected.

The probability may quickly become 99.98% that he throws a fastball, but it can never hit 100% without sacrificing the claim to being an optimal controller under uncertainty.

cdm, that is me, not Tango, but I was wrong and you are wrong.  There are hundreds of pitches a pitcher can throw (OK maybe not 100’s), but he only throws 3 or so.  That invalidates your theory.  Lots of pitchers mess around with a knuckle ball.  Why do they not throw one more than 0% of the time?  Because the value of it is worse than any other pitch regardless of how often they are thrown.  Why do almost all pitchers never throw a screw ball?  They CAN throw one, but it would be a terrible pitch.  Once a certain pitch becomes so bad that even if the batter rarely if ever expects it and even if that makes all the other pitches less effective, you throw that bad pitch exactly 0% of the time.

The same is true for the sac bunt, which is completely analogous to the pitch selection situation.  Bunting or not involves game theory.  But with Prince Fielder at the plate, his hitting is so good, and his bunting is so bad that even with the third baseman playing back in a “sac bunt situation” it is still correct for him to hit away 100% of the time.

Same with pitching. 

cdm, let’s say that I had a great curve ball - a major league hook, but my fastball were straight and 70 mph.  How often would I throw that pitch, according to game theory tenets?  0% of the time.  The batter would completely ignore that as an option on the other pitches, so it would not affect the value of any of my other pitches.  And when I did throw it, it would get hammered even if the batter were looking for the curve ball 100% of the time.

Here’s another one.  I have a fastball and my curveball is decent but I never throw it for a strike. Never.  Can’t control it.  How often would I throw it?  0% of the time.  Even if I could only throw it for a strike 10% or 20% of the time. I would throw it 0% of the time, because whatever gain I got with the other pitches, if any, would be more than offset by the terrible results from the curveball that is almost always a ball.

Again, the fact that all pitchers do not throw every type of pitch known to man invalidates your contention.  If your contention were true, then every pitcher would throw every pitch that they have ever thrown.  The reason why pitchers only throw 2-4 pitches is that they throw their worst ones 0% of the time.  Why is that?  Is that incorrect? According to what you said, that IS incorrect.  You said that no pitch should ever have a 0% frequency of occurrence.

Whoops, sorry MGL.

OK.  Well, the theory is correct, but I agree that its application is complicated.  I think your right. Because much of the experimenting with pitches can be done outside the game, when there is less cost/gain, a control system could assign a probability of 0 to a given pitch in a limited context (i.e., in ML games) and remain optimal. Only because it assigns a non-zero probability to that pitch in other contexts.

The question becomes: is there anything that can be learned by throwing a pitch in a ML game that can’t be learned in a bullpen session?  If so, then your right: you can throw a pitch 0% of the time and be optimal. But if you do learn something about a pitch in a game that you can’t learn in a bullpen session, then optimality would demand you throw the pitch with some non-zero frequency. I can see arguments for either side, but its practically irrelevant because we’re talking about 100% vs. 99.9998%.

I was just pointing out that this problem is further complicated since it sits at the interaction between learning theory and game theory. You really do need both.

You are right about the learning aspect of throwing pitches in a game situation. I never thought about that. It might be correct to throw a pitch more often than game theory would dictate if a pitcher needs to learn how to throw that pitch better. Of course that only applies occasionally to a pitch or pitches that a pitcher is adding to his repertoire or learning to throw better.  It does not apply to a pitch that a pitcher is not planning to incorporate into his game plan.  Rivera is never going to throw a curveball in a game because he is not going to add that pitch to his repertoire (presumably, at this time). Similarly, most pitchers are not going to practice throwing a screw ball or knuckle ball in a game situation.  Thus they will never throw one in a game.