Editor’s note: This post contains considerable math. If you’re the sort interested merely in conclusions, scroll to the bottom.

Introduction
In my previous article, I developed a model for calculating the WAR threshold for qualifying offers, the threshold above which it makes sense for teams to hand out a qualifying offer to a player. In order to keep the math to a reasonable level, I limited myself to linear and exponential functions to model the probabilities that a player accepts the offer, declines the offer, and signs with another team. In this article, I extend the model to use a more rigorous class of functions called sigmoids. This should yield a better model, while still keeping the math to a reasonable level.

Analysis
Sigmoids are a class of functions that have “S” shapes, and which asymptotically approach two y-values as x approaches negative infinity and positive infinity. The most well known sigmoid is S(X) = 1 / (1 + exp(-X)) :

This function satisfies the fundamental requirements of the P(rejects) function, that P(rejects) approaches 0 for low values of X, and approaches 1 for high values of X. We can generalize this function to S(X) = 1 / (1 + exp(-(a+bX))). Note that the previously graphed function is the special case of this generalization where a=0 and b=1.  In an intuitive sense, “b” controls the sharpness of transition between 0 and 1, and the value of X where S(X) = 0.5 is X = -a/b.

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In absolute terms, we know that strikeouts are at an all-time high. We see it in box scores, talking heads consistently discuss and lament the phenomenon on broadcasts, and in truth, it’s been going on for years. We’re left to wonder and analyze where the ceiling is for this trend, and exactly where the line between passable and unacceptable strikeout totals for batters begins and ends. For pitchers — whose velocity is a main factor in the increased strikeout numbers — going to work must be that much more enjoyable. And, in 2015, it was most enjoyable in terms of strikeouts for the rotations of the Chicago Cubs and Cleveland Indians.

If we look at strikeout rates for individual team seasons over the course of baseball history, no one struck out batters at a higher clip than the rotations of the 2015 Indians (24.2%) and 2015 Cubs (23.9%). That isn’t really surprising given the strikeout trend of recent years, but in mid-June of last season, the Indians’ rotation was actually on pace for the third-highest league-adjusted strikeout rate since 1950. At that point in time, they were striking out a historic rate of batters in a historic strikeout period, which is the sort of thing that tends to lend itself to positive team results. It didn’t, but most of that wasn’t the rotation’s fault (hello, team defense!), and the ridiculous strikeout pace didn’t quite continue into the second half of the season.

In the end, they finished as the 41st-best league-adjusted strikeout rotation, which really isn’t too bad: they ended up striking out almost 27% more batters than the league average starting rotation in 2015. Here’s the 2015 Indians compared to the best 15 league-adjusted strikeout rotations since 1950 by K%+ (percentage points above average compared to a given year’s league average strikeout rate):

In case you’re wondering, the 2015 Cubs finished 105th-best, with a K%+ rate of 120. Also not bad, but it’s illustrative of just how many strikeouts a team has to amass to make a run at breaking the record. And so I wondered: what strikeout rate would it take in 2016 to break the league-adjusted rate? And do the Cubs or Indians (or another rotation) have any realistic shot at breaking it?

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Previous editions: Baltimore / Boston / Chicago AL / Chicago NL / Cincinnati  / Cleveland / Colorado / Detroit / Houston / Kansas City / Los Angeles (AL) / Miami / Minnesota / Milwaukee.

Yeaterday, lead prospect analyst Dan Farnsworth published his excellently in-depth prospect list for the New York Mets. In this companion piece, I look at that same New York farm system through the lens of my recently refined KATOH projection system. The Mets have the 23rd-best farm system in baseball according to KATOH.

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I’ll begin by saying I’m not sure what value all of what’s to follow actually has. I know that’s about the least-compelling way to begin a blog post, but I just want that to be very clear. TangoTiger’s recent Building a Better WAR Metric series on the site jumpstarted an idea I’d been kicking around in my head for a while. It’s an idea that mostly exists because I’ve seen people on the internet say they’d want to see something like it, and, at the very least, it could serve as a talking point for another constructive discussion about WAR, and any constructive discussion about WAR is a good thing, because we all admit it’s far from perfect and constructive discussions usher progress.

People don’t really have beef with the offensive side of WAR, I don’t think. As far as sabermetric stats go, wRC+, and therefore wRAA, are about as infallible as they come. Tough to argue with the outcomes of history. I don’t see too many quibbles with the base-running numbers, partially because I think most people think they do a good job, but also because they don’t move the needle much either way, and there’s bigger fish to fry. Some people aren’t fans of the positional adjustments — both the assigned weights, and the entire concept of including them. I’m in the camp that firmly believes in the idea of the positional adjustment, but, like anything, the formula for the weights could always be looked at to see if it could be improved in any way, and Jeff Zimmerman’s work on this topic last offseason was a great place to start.

But the bigger beef, beyond the positional adjustments, is of course defense. Anyone will admit this is the weak link of WAR. It’s probably the weakest link of sabermetrics, as a whole, in 2016. And mostly, what it boils down to is, we know that defensive metrics don’t stabilize until the sample spans roughly three years, or 3,000-ish innings. Meaning, the single-season data is subject to noise, and if we wanted to draw any conclusions from it, we’d have to regress it. Despite that, single-season WAR is powered by noisy, unregressed, single-season defensive metrics. That’s the crux of the beef with WAR.

So, some folks have suggested that the defensive component of WAR ought to be regressed in some way, in an effort to strip out some of the noise that comes with single-year defensive data, or to better capture a defender’s true performance. I think there are a number of flaws in this general line of thinking, but there are a number of flaws with the way it’s being done now, too, so let’s humor one another.

Both ZiPS and Steamer use multiple years of data, giving more weight to the most recent seasons. Multiple years of data to weed out noise? Check. Both also incorporate some form of “scouting” information: Steamer regresses toward the results of the Fans Scouting Report, ZiPS searches for keywords in actual, physical scouting reports and uses those as a means for regression. Eye test? Check. Blend all that together and factor in some aging curves, and you’ve got yourself as good an idea of any player’s true-talent defensive ability as you’re going to find. Sort the Fld column here and I think you’ll agree that these numbers pass the eye test with flying colors.

So let’s imagine a world where, last year, every player performed exactly to their true-talent defensive ability. Everyone hit the same, everyone ran the same, everyone had the same amount of playing time, but defensively, we knew exactly what everyone’s true-talent ability was worth, and no one varied from it.

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Ruben Amaro had a reputation in Philadelphia. To many, the only evaluation tools he trusted were his scouts’ eyes. Basically, he was an old-fashioned — if not backwards-thinking — general manager.

The extent to which that’s accurate is debatable. Amaro wasn’t necessarily cutting edge — Matt Klentak, who replaced him as Phillies general manager, is clearly more analytical — but the perception was skewed. Amaro attended Stanford and learned from Pat Gillick, so his intelligence and knowledge base are anything but slight.

That’s not to say he didn’t make errors in judgement over his tenure. He made several, which is part of the reason he was relieved of his duties last September. Amaro is now with the Red Sox, having made an atypical move from high-ranking front-office executive to first-base coach.

On Sunday, Amaro took a few minutes to shed some light on his days as a decision-maker. The role of analytics in the evaluation process formed the crux of our conversation.

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Amaro on analytics: “You can’t ever deny the numbers. That’s true for every GM and every baseball person, regardless of whether you’re ‘old school’ or ‘new school.’ When a scout walks in, the first thing he does is pick up a stat sheet and look at what the player does and what he’s been doing. The numbers don’t lie.

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I write some version of this post every spring, and each time, I’m more excited at first than I am as I get more involved. Whenever it comes back to me, I always like the idea, but then eventually I remember it just isn’t that important. It certainly isn’t something people are keeping in mind all season long — no one really worries about the standings until, I don’t know, July, and the league-wide landscape in baseball is pretty even, relative to other professional sports. Most fans operate under the assumption the schedules are more or less even, and they nearly are. Differences are subtle.

But, you know, differences are there, and remember that this is an MLB environment that considers a win on the free-agent market to be worth something like $8 million. Every single win is important, in some sense, and because the schedules aren’t truly identical, there’s no harm in examining the projected advantages and disadvantages. Acknowledging from the outset that this is all based on projections, and that you don’t agree with all the projections, let’s quickly go over the various schedule strengths.

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Ben and Sam banter about a T-Pain tweet, two famous phrases, Ray Searage, and the final results of their contracts draft, then answer listener emails about the Cubs, a baseball broadcast from Better Call Saul, redistributing salaries, and more.

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9:00	Paul Swydan:

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Every year, we go through spring training, and every year, we mostly ignore it, so, every year, we get asked what, if anything, really matters out of these preseason contests. For me, the answer has remained the same. As hitters go, it’s difficult to find substance, although you might be able to read into any newfound power to the opposite field. That’s what tipped me off a few years ago to the coming emergence by Michael Saunders. It’s a little easier to get into pitchers, and while it can be fun to track any progress by newly-adopted pitches, it mainly comes down to velocity. There’s not really any “faking” velocity. Any velocity spike warrants attention. Any velocity drop warrants different attention.

It’s simplistic, sure, and it can be a little annoying, because some pitchers are still building up their arm strength, and spring-training velocities aren’t widely available. If you focus on velocity, though, you have the best chance of keeping signal separated from noise. You have a decent chance of not being deceived, and with all this in mind, I’ve already seen one particularly encouraging note. When the Astros signed Doug Fister, he was something of a reclamation project. He might already be most of the way fixed.

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