# The Hardball Times

## The limits of baseball

by Josh Weinstock
December 30, 2011

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.

You can read more of Josh's work at FanGraphs, Beyond the Box Score, and itsaboutthemoney.net. Josh welcomes discussion through email and twitter. You can reach him at josh82093 at gmail dot com and on Twitter @J__Stock (two underscores).