If there's been a trend over the last few years that has caught on like wildfire, it's been the rise of the short starter. Not the 5-foot-9 starter who wishes he was a little bit taller or wishes he was a baller. The Tampa starter. The guy who goes 4 innings and 18 batters and then leaves. In 2015, the Rays began experimenting with the concept, although the Rockies were even trying a four-man rotation with pitchers limited to 75 pitches in 2012. But it was in the hands of the Rays that the idea found its heart. The team that always seemed to be a little bit further ahead of the curve than anyone else had it all figured out two years ago.

It's been impressive that the Rays have managed to keep coming up with these ideas, with the exodus of talent from the front office (not to mention now-Cubs manager Joe Maddon). With former execs running the Dodgers, Mariners, and Expos and a few other saber-savvy teams starting to copy the model, it's worth asking: Where did the Rays find these guys? Matt Andriese was no one's idea of a major-league starter, and maybe he isn't. But he got down-ballot support for the Cy Young Award for perfecting the role he was given.

Did the team have some sort of magic formula? Or a super-computer? I reached out to new Rays GM Peter Bendix to ask him. He had a very enlightening response: " ."

The problem, as the Rays seemed to have perceived it, was that in the dark ages, there were only two roles a pitcher could fill. He could either be good at throwing 100 pitches or good at throwing 15. If a pitcher didn't fit into one of those two boxes, he was a failure. If he didn't have "let it all hang out" stuff to throw in the late innings, he was considered, at best, a cannon-fodder arm. If he couldn't last past 60 pitches and hold it together on a regular basis as a starter, he was considered fringy and flawed. But what if a guy could give you 60 good pitches? Not overpowering closer stuff for 60 pitches, but 60 pitches that are as good or better than what the average starter gives you. Shouldn't that guy have a job? Why don't we write a job description for him? Especially if he's on the metaphorical junk heap somewhere?

Heading into the All-Star break next week in Miami, the Rays are in the same place they always seem to be: contending for a playoff spot on a shoestring budget. Can we crack their secret recipe for success?

Warning! Gory Mathematical Details Ahead!

I used Retrosheet data from 2012–16 and looked for a very specific type of pitcher. I looked at all plate appearances pitched by starters in those years and looked to see whether they ended in an on-base event (walk, hit, HBP). I determined the hitter's overall OBP for that year, the pitcher's overall OBP given up, and the league OBP against starters. Using these, I created a control variable for how likely the plate appearance was to end in an on-base event using the log-odds ratio method.

I then calculated the pitch count for each plate appearance before the plate appearance began (i.e., Smith has thrown 54 pitches so far today). I also calculated the square of this number (i.e., 10^2 = 100).

For all pitchers who faced at least 200 hitters as starters in a given year, I constructed an individual logistic regression for that particular pitcher-year. This would give us the "shape" of the pitcher's tendencies in giving up on-base events. Is he a guy who slopes upward as the game goes on? Is he a guy who is vulnerable early, but eventually settles down (but then if you get him back up above 90, the line turns again?) Is he pretty stable throughout the game? If we make the assumption that he's always facing a league-average hitter (of course that isn't actually true in reality, but it gives us a good place to start), we can figure out what we would expect his OBP to be, based on his overall numbers, and then adjust it for the pitch count.

For each pitcher (in each year), that gives us a regression that provides a projected OBP at each pitch count (again, assuming that he was facing a league-average hitter).

The regression has the form:

( A * pitch count^2 ) + ( B * pitch count ) + regression constant + projected OBP in natural log of odds ratio form = projected OBP for this plate appearance in natural log of odds ratio form.

Since we're assuming that the pitcher is facing a league-average hitter (average against a starter, because reliever cumulative stats are usually better), let's look for the pitch count where the pitcher is exactly at a league-average expectation.

Now, the equation reads

( A * pitch count^2 ) + ( B * pitch count ) + regression constant + projected OBP in natural log of odds ratio form – league average OBP in natural log of odds ratio form = 0

The last three terms can simply be added/subtracted to produce one number, and now the equation has the form of

Ax^2 + Bx + C = 0.

All we need now is the quadratic formula for solving for X.

Some pitchers were always, no matter their pitch count, above the league average (we call these guys "uh ohs") and some are always below the league average (we call them… well, since we're talking about the Rays, Chris Archer). Or at least, some guys had solutions for X that said that they were below league average from pitch -137 to pitch 265. I know medical science has advanced a bit in the past few years, but 265 pitches is a little much. I cut the bounds at 0 (for obvious reasons) and 100 pitches.

It's not that the Rays would (or should) turn away a pitcher who can go seven innings and pitch better than average. Even a couple of years ago, I found 23 guys who had an "effective" range between the first 35 pitches of the game and the first 75 pitches of the game. Several of them had overall OBPs allowed that were over the league average, but this didn't tell the whole story. If the league believes that a pitcher is below average and a team can create a role for him where he is above average, then the team has discovered a point of arbitrage.

I looked to see how well this "effective range" remained consistent over time. (Players who were always above 100 were given a score of 100, those who never made it below league average were given a score of zero.) Over the five years in sample, the intraclass correlation (it's like a year-to-year correlation, but with more than two data points) for the sample was .31. That's moderately high. It's not great, but we call home runs a true outcome for pitchers for a correlation around this magnitude. It means that this method isn't a sure-fire way to identify guys who might fit the role, but it's a good place to start a list.

On top of that, it means that there are guys in MLB who fit the bill, mostly because it's really hard to find guys in general who are great—or even average—for 100 pitches. What the Rays did was to take a look at the assets they had on hand. If there are 20-something of these guys already in the majors, how many more are lurking in the minors? They had Chris Archer, Matt Moore, and Drew Smyly, but needed to fill spots four and five in their rotation. They had pitchers in the system who fit the bill of being able to throw 50 good pitches. That made them a good candidate to go three to four innings or, as the Rays seem to prefer, 18 batters.

On some nights, that's not going to be enough to get it done. The team might need more length. Because these guys are starters by training, they have the built-up endurance to sustain six innings, even if they won't be six glorious innings. But on some nights, when the bullpen is rested or there's a forthcoming day off, you can get away with a four-inning start, so why ask a guy to go beyond what he's actually good at doing when you don't have to? And, as the Rays discovered, since there aren't rules saying that these guys have to actually start in the first inning, you can find guys who embrace (or quietly tolerate) being used both as a "starter" or "reliever" and you can create a pitching staff with three "real" starters and play mix-and-match with three or four swingmen and some regular relievers. When there's always a guy around with a starter's arsenal, a demonstrated ability to get lefties and righties out at a pace that is better than league average, and who can throw three good innings (if not six), it makes it easier to extend the "one-inning guys" a little past their bedtime.

Why not make the hole square?

All good market inefficiencies come in the form of people clinging to an idea despite the fact that there's nothing really keeping that idea in place other than the fact that they are clinging to it. Baseball had been under the spell that there are two kinds of pitchers and if a pitcher didn't fit into this rigid binary system, he was junk. Instead of trying to fit the square peg into the round hole, why not make a square hole? What the Rays discovered was that by cleverly manipulating the 25-man roster, it was possible to have fresh relievers available to cover the required workload, and it was possible to take a spot (the fourth and fifth starter spots) that teams normally wrote off as a home for below-average starters and squeeze above-average performance out of it, at least some of the time.

If we assume that the average fourth and fifth starter are worth two wins combined over the course of a season, and that your average second and third starters, who we assume are a bit above average, are worth four to five wins combined, and we can get guys slotted for the fourth and fifth starter roles (even if that means more than two pitchers in terms of who staffs those slots/innings) and get them to perform like second and third starters even half the time, then those spots could be worth a win to a win and a half more in value and can be staffed by guys whom everyone else was throwing on the garbage pile.

The Rays cleared (a lot of) extra value simply by thinking differently. Now that the rest of the league is starting to catch up, it's going to be interesting to see what Bendix and the rest of the Rays come up with to stay ahead.