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# The limits of baseball

In calculus there is a concept called a limit. As described by Wikipedia:

the concept of a “limit” is used to describe the value that a function or sequence “approaches” as the input or index approaches some value.

Many measures of athletic performance have a limit. Every few years someone runs the mile more quickly than anyone had before. But while we are continually running the mile faster, it would be unthinkable to run the mile in negative time, or even no time. This means that there is a lower bound which our mile times are approaching, but won’t quite reach. It’s hard to say exactly what that limit is, or when we will reach it, but we know it exists.

We can also observe this phenomenon in baseball; pitchers throw much harder than they used to just a few years ago. Most estimates have the average fastball velocity as increasing more than 1 mph in the past few years. While it’s possible that teams are simply emphasizing velocity more than they used to, it also seems likely at least some of this velocity increase is real. But at some point, pitchers will stop throwing harder; without this upper bound, baseball players would eventually average infinite miles per hour.

I firmly believe that there is a higher level of competition in the major leagues now than ever before. Players are bigger and stronger, pitchers throw harder, and teams are smarter. As Ben Lindbergh wrote, many teams now employ intelligent, “sabr-savvy” GMs. This makes any discussion of “would a star in the past be a star now?” problematic. Thanks to ESPN broadcasts, I used to listen Joe Morgan occasionally engage in this discussion. He used to say that stars in the past could be stars now, but current stars may not have been as great in the past. Essentially, Morgan implicitly argued that players used to be better. His reasons, if I remember correctly, used to focus on how ballparks are smaller now, and that players are protected more.

I ardently disagree. Say we estimate that in the past 10 years, pitchers are on average capable of throwing 1 mph harder than they were before. If we extrapolate this rate, then 50 years ago pitchers threw on average around 86 mph, which seems reasonable, although I wasn’t around then. Baseball is a game in which every small change can have a large impact. Just hit the ball a fraction of an inch off the sweet spot, and you may weakly ground out or pop out. It stands to reason, then, that an increase from 86 to 91 mph is of an extreme magnitude.

We have seen evidence that the level of competition in the majors has improved a lot, both in terms of players and in terms of the evaluation methods used. And despite our tendencies to cast star athletes as immortal, we know that they are human, bounded by flesh and bones. If baseball continues to improve, then at some point, all players will be nearly equal in talent and all teams will employ optimal strategies—the limit.

What would such a world look like? And what does baseball today tell us about what it would be like?

Pitchers would certainly be different. In recent times we have seen pitchers model themselves after Roy Halladay; both Charlie Morton and Brandon McCarthy significantly revamped their mechanics and repertoire to imitate the Phillies right hander. And part of the changes they implemented have become more widespread in baseball. Both pitchers focused on developing a cutter and two-seam combination, and with good results. The cutter especially is a pitch that has increased in usage in recent years, and it’s not hard to see why. The cutter is most often used in two ways; like a fastball, or a slider. When used like a slider, its benefits are especially clear. Sliders are notorious for causing pitching injuries, while cutters are generally considered to be less strenuous. And when used as a fastball, the cutter helps to keep batters guessing with a similar velocity pitch.

The Roy Halladay model may grow in popularity. It doesn’t require a pitcher to have very good stuff, but to be able to generate ground balls by throwing a cutter and two-seamer to each side of the plate. His mechanics are also considered very clean, and should be able to be imitated by many pitchers. But of course not every pitcher would be the same. That doesn’t make sense for two reasons. Firstly, unless all pitchers are identical anatomically in the future, different pitchers have different optimal sets of mechanics and repertoires. Secondly, it would be a terrible decision from a game theory perspective, offering no variety in looks to the batter.

Change-ups may also grow in popularity. Teams may decide that they could save a tremendous amount of resources by eliminating injury-inducing sliders, which would help them get more innings out of their starters. And while much less strenuous, change-ups are not necessarily any less effective; last year two of the pitches with the highest whiff rates (swing-and-misses pitches) were Ryan Madson‘s and Cole Hamels‘ change-ups. Change-ups have another big advantage as well; most have a very limited platoon split, while most sliders have significant platoon splits. Of course change-ups are not perfect. They largely rely on deception be effective; if the batter recognizes the pitch, it’s basically just a slow fastball. Tt is a “feel” pitch, and given the obvious advantages, would probably be much more popular today than it actually is if it were easier to throw effectively.

But to be honest, I have no idea what the future will bring. Maybe pitchers will throw a two-seam, cutter and change-up with greater frequency than they do now, or maybe everyone will just throw knuckeballs. But the recent proliferation of sabermetrics in the management styles of major league teams serves as a bit of reminder of this ominous limit in the future. We can see a limit where, because all teams are so similar, major league baseball is not much more than a complex coin flip; a true game of inches, if it isn’t already now. How far away is this limit and when we will get there? It’s hard, if not impossible, to know. But we are certainly approaching it.

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tc
Well, where to begin… I guess with the general approach which is to emphasize the physical aspects of the game,which are of course the part that sabre metrics can most easily measure. I suspect that what Joe Morgan was contending was that modern players weren’t mentally tough enough to be as successful in his era. One frequently hears from older players how the next generation is not as “tough”. Morgan probably would have mentioned pitchers like Bob Gibson and Don Drysdale, guys who snarled and threw inside. Another part of the same equation, something you frequently hear from older players… Read more »
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Todd Boss
Completely agree.  Morgan is a fool (no surprise there; we’ve been listening to him call games with fellow fool Tim McCarver for years). Babe Ruth, were he to get into the batter’s box today, probably looks at a modern day pitcher with a low 90s slider and mutters to himself, “I can’t hit what I can’t see.”  Which is why we always need to measure players in the context of their era, using such adjusted stats.  But we already knew that right?  Why havn’t we had a .400 hitter since the 40s?  Because the game has changed radically, hitters face… Read more »
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tc
Todd, I don’t know if Morgan is a fool, I never listen to him, but most “old” players tend to idealize the era they played in, especially against the next generation. I’ve been listening to that kind of thing for 60 years now, it’s always the same. As to Feller, during his peak he was throwing 300 innings per season. He was above or just slightly below that for 5 seasons. Nolan Ryan pitched 300 innings once and was at 299 one other year. And he didn’t do it consecutively. Reason tells us that if Feller were trying to throw… Read more »
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tc

One other thought in this notion of trying to build in expectations based on past performance. Shifting to another sport, when Michael Jordan began his NBA career many veteran players remarked at his high energy performances that if he kept that up his career would burn out very quickly. That was based on the way they had to play an 80 game regular season. So much for predictions.

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ChuckO
I don’t know how you can conclude that pitchers threw on average at 86 mph fifty years ago. Even if one accepts your claim that they throw one mph faster than they did ten years ago, that does not necessarily mean that they threw five mph slower fifty years ago. In fact, what evidence we have would tend to suggest that the hardest throwing pitchers threw at pretty much the same velocities as modern-day hard throwers. Check this out for examples, http://www.baseball-almanac.com/articles/fastest-pitcher-in-baseball.shtml Note that the so-called speeding motorcycle test had clocked both Walter Johnson and Bob Feller (obviously not in… Read more »
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tc
I’m not a mathematician but I’m fairly certain that in any particular set of stats the standard deviation fluctuates from one period of time to the next. Since a standard deviation in any case needs interpretation and additional facts, even in one category, fielding avg. for example, what the standard deviation would tell you would not be the current talent level of fielders. Even an accumulation of stats would only tell you what the researcher “thought” were key samples. End result would be a very subjective standard deviation. Since it seems that the commentary on this theme appears to have… Read more »
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bob b.
One thing I’ve been wondering about is that for many years the general trend in baseball stats (well, the distribution of said stats I suppose) has been for a DECREASE in the standard deviation (well, if i’m remembering correctly) but recently I believe I’ve heard that it is INCREASING. And back before it was increasing I remember reading that it was decreasing because of the INCREASE in skill (overall). So… I was wondering if the increase in standard deviation (assuming this is true) means that the overall talent level is decreasing? So I guess officially I’m asking two main questions:… Read more »
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David P Stokes

While I don’t doubt that the overall level of play has generally increased over time, I’m don’t necessarily buy that pitchers on average are throwing harder than 50 years ago.  Pitchers nowdays don’t use full windups and really high leg kicks as much as pitchers in the past, and while I’ll admit that I’m not an expert on the matter, it would seem that those things increase velocity somewhat.  Also, throwing harder is better, all else being equal, but everything eles isn’t necessarily equal, which is why Steve Dalkowski never even made the majors, much less the Hall of Fame.

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Paul E
Men of merit exist in every generation, but mankind in general prefer the meritorious of their own generation….. 1)  The DH sucks…let’s get the 38 -43 year old hangers-on and relics off the field 2) The Selig era – expansion, expanded playoffs, and steroids – sucks. I witnessed in back-to-back visits to Camden Yards in the mid-to-late 1990’s the two longest 9 – inning contests in history thanks to intentional walks andching around 30 HR middle infielders on roids 3) It all went downhill when Reggie Jack-off-son stood in the batter’s box and watched his Detroit All Star game HR… Read more »
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Marvin Leibowitz
A very interesting article by Josh Weinstock .  Congratulations on very interesting topic.  The limits of baseball.  The idea of a limit, or wall has been around for quite a while.  There is no doubt a wall, but Josh reaches, I think, a very wrong conclusion.  For Josh, and for all interested in this concept I refer you to an article firstbpublished in 1986, written by the great Stephen Jay Gould, paleontologist, evolutionary biologist, and historian of science.  His great passion was baseball.  His only genetic disorder was a that he was 2nd generation Yankee fan.  He wrote a great… Read more »
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Marvin Leibowitz
Congratulations to Josh Weinstock on a very interesting topic.  The limits of baseball.  The idea of a limit, or wall has been around for quite a while.  There is no doubt a wall, but Josh reaches, I think, a very wrong conclusion.  For Josh, and for all interested in this concept I refer you to an article published in 1986, written by the great Stephen Jay Gould, paleontologist, evolutionary biologist, and historian of science.  His great passion was baseball.  His only genetic disorder was that he was 2nd generation Yankee fan.  He wrote a great number of baseball articles one… Read more »
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Norman
“I have always maintained that the juiced ball had more to do with McGwire, Sosa and Bonds (and all of those lesser hitters who suddenly became sluggers) than the “juice” that they were injecting. Just like after the Black Sox scandal, MLB looked to inject interest in the game after the disastrous strike. Bye-bye baseball, it’s out of here.” I think there is a lot to this. In cricket (a sport I only casually follow) the governing bodies have been much more conscientious about not monkeying the ball and more resistant to rule changes. What you see there is that… Read more »