Author Archive

Juiced Ball Economics

Look, another article about the juiced ball! I know, I know. At this point, the narrative of the juiced ball has been written from many angles. (For anyone reading this who is somehow not familiar with that narrative, Jay Jaffe’s piece from June is replete with graphs, tables, and external links.)

As fans, this narrative is interesting, as it contextualizes the product we see on the field. It feels like there are more home runs — hey, there are more home runs! It’s human nature to wonder why. Baseball writers have tough jobs, and they need something to write about. Many baseball writers have tried to tackle the “why” to satisfy both reader curiosity and their editor’s article quota.

While this narrative is interesting for fans, for teams, it’s of massive importance financially. Teams have payroll budgets, and presumably, some of those teams will have room in those budgets for free agents this offseason. Teams also have players under contract who are eligible for arbitration, on whom the team must place valuations in advance of submitting a figure to arbitrators or deciding to non-tender the player. In short, teams have financial decisions to make, and they rely on all the information at their disposal to make them. Wasted dollars represent opportunity cost more than anything, in a league where advantages are razor thin and random variance plays such a key role in success or failure from year to year. Read the rest of this entry »

Let’s Talk About That Michael Lorenzen Appearance

You may have heard that the Reds are approaching their bullpen a bit differently than other teams this season. The Reds aren’t expected to be particularly good this season, and as such, they are a bit more free to experiment.

One recent game highlighted the new-school approach of manager Bryan Price. In the third inning of Monday’s game, Brandon Finnegan started the 3rd inning with a 5-run lead and proceeded to implode, loading the bases before walking the first run of the game in. With no outs recorded yet in the inning, Finnegan was primed to give up several more. Price pulled the trigger on a highly unusual move: He went to the bullpen in favor of Michael Lorenzen, one of his better bullpen arms.

This decision was lauded by quite a few writers and pundits, including those here at FanGraphs. Craig Edwards used it as the impetus for examining the overall usage in the Reds bullpen so far this year, and Ben Lindbergh and Jeff Sullivan called it out in their latest “Effectively Wild” episode. The emphasis, in both cases, was on the decision to bring Lorenzen into the game. Which was a great decision! It was weird! It was wonderful! Most importantly, it worked!

There’s another aspect of this Lorenzen appearance that shouldn’t go overlooked, though. After Lorenzen worked the 3rd inning with great success, he stayed on for the 4th inning, in which he maintained their 5-1 lead. He retired the side in order with 10 pitches in the 4th, having thrown 14 in the 3rd. The Reds tacked on another run in the top of the 5th, and Price stuck with Lorenzen again for the bottom of the inning, now with a 5-run cushion. Lorenzen, once again, set down the side in order, this time on just 8 pitches. With 32 total pitches on the day, Price elected to turn to a lesser pair of arms in Cody Reed and Wandy Peralta to finish out the game (although not before allowing Lorenzen to lead off the top of the 6th at the plate).

While the 3rd inning represented a quintessential high-leverage situation, the 4th contained much less leverage, and the 5th, still less. The numbers bear this out: In the third inning, Lorenzen faced three batters in situations commanding a Leverage Index of 2.68, 2.66, and 2.53. The total Leverage Index of these three batters was a whopping 7.87. By contrast, the total LI associated with Lorenzen’s work in the 4th and 5th innings was 2.40. The six outs Lorenzen got in those innings weren’t as important, cumulatively, as the least important hitter in the 3rd inning!

(Click the graph for an interactive version)

Price was rightfully lauded for bringing one of his best pitchers into one of the most critical moments of the game. That’s only half of the equation, though. Knowing when to take a key reliever out of the game, in the context of the season as a whole, is just as important as knowing when to put him in.

As Edwards rightly notes, Andrew Miller is only on pace for about 88 innings this season. Andrew Miller threw 74.1 IP last season, and it was the most he had ever thrown in his tenure as a full-time reliever. It’s not as though the Yankees or Indians were trying to limit his usage — it’s that a reliever, any reliever, has a limit to the number of innings (and more appropriately, the number of pitches) they can throw in a season without breaking down or losing their effectiveness.

The question, then, is how to maximize the value of these innings. Lorenzen threw 2 innings and 18 pitches that he, quite possibly, didn’t need to throw. He consumed 2.40 “units” of leverage in the process. The next day, he was (quite predictably) unavailable. Price, faced with a close/late game situation, had to throw Peralta in the 7th inning of a one-run game, where he retired the top of the Pirates’ lineup in order, but consumed 4.22 “units” of leverage — 75% more than Lorenzen did in those two innings the day before.

This isn’t to say that “perfect” bullpen usage is achievable. The nature of the game is to guess when the situation you’re faced with will be the most important in the remainder of the game, or the remainder of the series, or the remainder of the homestand. In some cases, a more important, later, closer, more tense situation will arise in the same game, and you’ll have used your most effective bullets. In other cases, you’ll have used a pitcher in a big spot one day, and he’ll be unavailable in an even bigger spot the next day. In still others, the team will go on a run of 4-5 close games in a row, and lesser parts are needed to fill the surplus of close/late innings.

CIN 040317 - 041217
(Click the graph for an interactive version)

But the concept of “perfect” bullpen usage must start with the recognition of constraints, and an approach that optimizes the total leverage that a pitcher can consume within those constraints. It’s not enough to pick the right person for the job when the job is hard; it’s also necessary to pick the right person for the job when the job is somewhat easier, so that the right person for the next hard job is available. Michael Lorenzen did the hard job, but he also did an easier one, and as a result, wasn’t available for the next hard job.

When Do Managers Use the Hook?

For the uninitiated, this piece heavily relies on my previous work around refining the inning/score matrix to quantify bullpen usage, and more recently, using RE24 to adjust the score differential for the base/out state in cases where the pitcher is not entering into a “clean” inning.

In that most recent piece, I concluded by alluding to a sort of “leaderboard” for base/out state adjustments. One hypothesis that you might have – certainly, one that this author had – was that we might see elite non-closers at the top of the list, implying that those pitchers are being brought in with runners on base more often than usual. Although closers are generally among the most highly-regarded relief pitchers in the game, the managerial status-quo has been to use closers almost exclusively in the “clean inning” state entering the 9th. Thus, while closers might not lead in terms of score adjustments due to inherited runners, an elite setup man certainly might.

Without further ado, here’s what that leaderboard looked like in 2016.

Largest Average Negative Score Adjustments
Player Team # Apps Mean Adj. Score Mean Adj. Inn Score Diff Inn Diff
Colton Murray PHI 24 -2.30 6.90 -0.22 0.15
Chaz Roe ATL 21 -0.73 7.57 -0.21 0.11
Gavin Floyd TOR 28 0.54 8.04 -0.21 0.11
Dean Kiekhefer STL 26 -1.78 7.59 -0.21 0.13
Alex Wilson DET 62 0.18 6.97 -0.19 0.13
Carl Edwards CHC 36 1.31 7.84 -0.19 0.15
James Hoyt HOU 22 -1.77 7.26 -0.18 0.26
Jordan Lyles COL 35 0.68 7.34 -0.18 0.09
Tommy Layne NYY 29 0.83 7.49 -0.17 0.25
Matt Bowman STL 59 1.08 7.28 -0.17 0.06

So… this isn’t exactly what I thought I’d find. There aren’t any closers in this group, but there really aren’t many top-flight middle relievers, either. If anything, this group came in when the team was tied or trailing more often than not. What’s going on here?

What we can’t discern is whether mid-inning appearances tend to be high-leverage affairs. There are most certainly cases where long men are used in the middle of the 4th inning to relieve an ineffective starter. That situation isn’t interesting in a vacuum; but it may be interesting to know what portion of those mid-inning appearances are of this low-leverage variety, and which are of the high-leverage variety.

One way that we can answer this question is to stratify qualifying relief pitchers by their average inning when entering the game. To accomplish this, let’s define a “closer” as a pitcher with an average inning of 8.5 or higher, and a “middle reliever” as a pitcher with an average inning between 7 and 8.5. Then we can look at the percentage of appearances for each group which were not “clean” innings.

(Click the graph for an interactive version)

As you might expect – even if you vehemently disagree with the practice – closers very rarely enter the game mid-inning. 85-90% of their appearances come in clean innings. Middle relievers, on the other hand, come into the game at the start of an inning closer to 60-65% of the time. That number has been on the rise recently, which seems a bit odd, or at least, at odds with what we’ve seen in the postseason recently (more on that in a bit).

Some small percentage of the time – the area between the lines of the same color – pitching changes are made with 1 or 2 outs in the inning but with no one on base. This is probably not optimal: The pitcher coming into that situation has an easier-than-average job, as they’re essentially getting a shortened inning to work through. If a guy like Dellin Betances can face 300 batters in a season, why waste 20 of them on situations that are easier than average?

The orange lines represent a subset of the overall middle relief group where the team in question is either tied or has no greater than a 3-run lead, in either the 7th or 8th inning. These are situations of high importance and leverage. An effective manager might be employing mid-inning pitching changes more often in these situations in order to limit damage and preserve leads.

Yet, this subset isn’t very different than the overall middle relief group. Whatever difference exited in 2012 and 2013 has been eroded in the last few years, as part of a general trend: Mid-inning appearances in the regular season are becoming less common.

As a final step, let’s contrast this picture of usage with an analogous graph on postseason appearances. We’ll maintain the same definitions of “closer” and “middle reliever” for consistency.

(Click the graph for an interactive version)

Chaos! This graph looks more disorganized than the regular-season version, but then again, the postseason is more chaotic in general. We’re dealing with smaller samples and we can’t put too much faith into these trends. That said, two things stand out when comparing postseason usage to regular-season usage:

  • Closers are no longer treated as a special species. Even through 2014, closers were entering postseason games in clean innings about 80% of the time. In the postseason! When the managers are paying attention! When there are high-leverage situations at every turn! But in the past two seasons, closers have been used increasingly with runners on base – in fact, even more so than middle relievers have in close/lead situations during that time. Again, small samples, but this screams efficiency. If your closer is your most effective weapon, you should be using him with runners on base and a late lead, instead of using your second-most effective weapon instead.
  • Middle relievers have been used more often in “matchup” situations. 2014 and 2016 stand out in this regard, and it probably has something to do with guys named Bochy and Maddon representing large shares of the sample in those years. Recall that the gap between the dotted and solid lines of the same color represents the frequency of “1+ out, 0 on” appearances. Those gaps are huge in 2014 and 2016! While mid-inning appearances among all classes of pitchers were highest in 2016, that’s not the case at all for “men on base” appearances, which were more or less in line with historical norms. This represents an increase in match-up-based thinking, not leverage-based thinking.

These graphs look different, and they probably always will. Teams have relatively fewer resource constraints in the bullpen come October. They have more days off between games, and fewer games to budget resources for in the future.

That said, there’s been no carryover at all from the wild, and relatively new, bullpen management seen in the postseasons of 2015 and 2016. Constraints will limit the extent to which managers can call upon their best arms with runners on base late in games, but it would be hard to imagine that a status quo which holds the closer for the 9th inning almost 90% of the time can’t be improved upon in some way. Teams have spent more on bullpens, but they haven’t figured out how to use them any more efficiently in the regular season, and the differences we’ve witnessed in the postseason show that they’re only getting it about half right, even when it matters most.

Adjusting Appearance Data for Base-Out State

So far, we’ve developed some mathematical principles for visualizing appearance data for relief pitchers, and for measuring how apart they are. The goal has been to say something about how pitchers are being used, not only in a vacuum, but in the context of the way in which the team has chosen to divide up its relief innings for the season. We’ve only partially gotten there so far, but today let’s take a slight detour to ask: Is the underlying data conveying the most useful information?

Inning and score differential at the time of entering the game are the critical data elements in answering questions related to usage. The numbers and tables in my previous articles all focused on using these two elements. Here’s an example of the underlying data being used, in the form of three Daniel Hudson appearances which appear identical.

Three (Similar?) Daniel Hudson Appearances
Date Player Season Inning Score
6/28/2016 Daniel Hudson 2016 8 1
8/20/2016 Daniel Hudson 2016 8 1
9/21/2016 Daniel Hudson 2016 8 1

Inning and score differential are critical; however, as data elements are concerned, they are somewhat raw. Fortunately, those aren’t the only data elements we can look at. The next-most impactful data, I would argue, is the base-out state at the time that the pitcher enters the game.

Let’s establish a baseline: It’s the norm for relief pitchers to enter the game in a clean inning (no outs, no runners on base). Among pitchers with 20+ relief appearances in 2016, this was the situation in 68.1% of appearances. That’s a very high percentage, considering that there are 24 base-out states. It’s also very intuitive when we think about the game. Among other reasons, pitchers need time to warm up, and mostly, they do so while their own team is batting. It’s also the only base-out state which is guaranteed to happen every inning.

It would be atypical – and therefore, interesting – for a pitcher to be used frequently in other base-out states. Moreover, we should be giving credit to pitchers who are being used in that way. An appearance where a pitcher enters with a four-run lead but the bases loaded should not be viewed in the same way as an appearance where a pitcher enters with a four-run lead in a clean inning. More than likely, the manager has two different pitchers in mind for each of these scenarios.

Adjusting the inning is easy: Credit partial innings in the event that the pitcher enters with more than zero outs in the inning. This will bump the inning component of every pitcher’s “center of gravity” up a bit, giving credit to players for working slightly later in the game when called upon mid-inning. (Note: we could also define terms in a different way, and say that a pitcher who enters in a “clean” 9th inning is actually entering at inning 8.0, as 8 innings have been recorded prior to his entrance; however, this makes the resulting metric less intuitive.)

Adjusting the score differential doesn’t seem as straightforward at first, but fortunately, we can use the concept of RE24 to accomplish this. Given that entering in a clean inning is the default status, we will make no adjustment to the score differential for a given appearance if the pitcher entered in a clean inning. For any other base-out state, we will add or subtract the difference between expected runs in that base-out state and expected runs in a clean inning state (0 on, 0 out).

Let’s return to the three appearances shown above. As you might have guessed by now, they are not identical. Rather, they illustrate the importance of adjusting for base-out state.

Three Daniel Hudson Appearances (in greater detail)
Date Player Inning Score Outs Bases Adj. Inn. Adj. Score
6/28/2016 Daniel Hudson 8 1 0 ___ 8.00 1.00
8/20/2016 Daniel Hudson 8 1 0 123 8.00 -0.82
9/21/2016 Daniel Hudson 8 1 2 _2_ 8.67 1.16

If you were to ask Daniel Hudson to recall what he could about these three appearances, he’d probably feel very differently about each of them (if he remembers, anyway). In the first case, he’s coming into a clean 8th inning, protecting a one-run lead. It was a situation he found himself in with some regularity in 2016, prior to assuming the closer’s role.

The second situation is an absolute bear. Jake Barrett has allowed a leadoff single to lead off the inning, and poor Steve Hathaway, who shouldn’t be touching this game situation with a 10-foot pole at this point in his career, has subsequently allowed a double and a walk to load the bases. Hudson has been brought in to protect a one-run lead with the bases loaded and nobody out. The opposing team has an expected run value of 2.282. While technically Hudson has been given a lead, it’s one that he would be hard-pressed to keep, even if he does everything right. The reality is that this appearance is associated with an expectation that Arizona will trail by the end of it – as you can see on the play-by-play log, the Padres have a 70.6% win probability at this point. It would be silly to give this appearance the same treatment as the first two. (Hudson, by the way, does a masterful job of escaping this situation without surrendering the lead!)

The third case is the one I want to focus on. Rather than a clean inning, Hudson was asked to get the third out of the 8th inning, with the tying run standing on second base. While the Leverage Index at the time of entry for this appearance is higher (3.50) than in the first instance (2.17), Hudson actually has an easier job: He needs just one out instead of three, and the opposing team is expected to score fewer runs in this situation, all else being equal. In the “clean” 8th inning, he can be expected to give up 0.481 runs, while in the two-out, runner-on-second situation, he can be expected to give up just 0.319 runs. Moreover, the chance of scoring at least one run – presumably the more important question where one-run leads are concerned – is also lower in the “higher leverage” situation. (This doesn’t even account for the batter, Hector Sanchez, who is hardly Wil Myers at the plate, and is probably inferior to the 4-5-6 hitters in the Phillies lineup, as well.)

This brings up an important distinction between leverage and run prevention. Leverage Index, certainly, is an important tool. What it measures, however, is variance in win probability for a single at-bat. Managers rarely have the luxury of giving their pitchers one-batter appearances in the regular season. Even the notoriously fleeting Javier Lopez averaged nearly three batters per appearance in 2016. Managers must therefore determine how to maximize the value of relief appearances as a whole, not just at the time when the reliever is entering the game. Leverage Index shows how much variance can arise from the current plate appearance, but a manager may very well be better served having their best pitcher throw the entirety of the 8th inning, rather than having him get the third out in a situation that commands high leverage but still has relatively low run expectation.

Next time, we’ll look at how base-out state adjustments impacted the raw inning-score matrix data in 2016, to draw conclusions about which relievers were used most often in high-pressure, mid-inning situations, and whether that sort of usage aligns with what we’d expect from an optimal manager.

Quantifying Bullpen Roles: The 2016 Season

Author’s Note: This is the second of a two-part article, both of which are intended to stand on their own. The first introduces terminology and a mathematical framework used to derive statistics; the second uses these new ideas to draw conclusions which are hopefully intriguing to the reader. If you need it as a reference, you can refer back to the first article (here).

Below, I’ll use some metrics – average and weighted-average Euclidian distance between relievers – to look at the 2016 season. Ideally, we’d like to be able to associate a covariate with these metrics. That is, we’d like to be able to say “bullpens with lower weighted-average distances are (blank),” where we fill in the blank with some common-sense concept or truism about the way we know the game to work. Short of that though, maybe we can just get an understanding of why the bullpens at either extreme have found themselves there.

So, without further ado, here are the bullpens of all 30 teams as sorted by weighted average Euclidian distance in 2016.

2016 WAED Leaders

How can we interpret this? There’s no real obvious trend here: there are “good” and “bad” bullpens on both ends of the table, along with “good” and “bad” teams. At the extremes are good case studies, though: A subpar Phillies bullpen on a subpar Phillies team, a solid Orioles bullpen on a solid Orioles team, and of course, the Cubs. What can we learn from looking at them in more detail?

The 2016 Phillies Bullpen: An Ode to Brett Oberholtzer

Most people reading this know how the Phillies season went last year. They were supposed to be bad. Then, briefly, they appeared to be good. People did what they could to explain why the Phillies appeared to be good, including looking at their overachieving bullpen. As it turns out, the Phillies were bad after all. Baseball is fun.


The Phillies being bad explains part of what you see above. They tended to employ a lot of guys in the middle innings when they were already behind in the game. That’s a product of circumstance, and not an indictment of those guys. Elvis Araujo, Severino Gonzalez and Colton Murray weren’t great pitchers, and it’s sort of odd to have three of those guys rotating into your bullpen at various points in the season. Then again, the Phillies were bad, and those three guys were young, and they could afford to give young guys longer runs than a competing team could have.

There are those three guys, and then there’s Brett Oberholtzer, a slightly older, more experienced pitcher, whose MLB time before 2016 was mostly as a starter. He can be considered the quintessential mop-up guy in 2016. He’s way over there to the left – in fact, he had the lowest average score differential when entering the game out of any relief pitcher in 2016. Here’s what his inning-score matrix looked like:


This doesn’t even do Brett Oberholtzer justice, though. Here’s a histogram of score differential by appearance that puts it into context.


Oberholtzer made 26 appearances for the Phillies in 2016, and most of them were in garbage time. Then, there was the one appearance where the Phillies actually led when he came into the game. It was the 10th inning, and most of the Phillies bullpen had already been spent. Pete Mackanin had little choice but to bring Oberholtzer in to protect a one-run lead in the 10th. Which he did, earning a save. Brett Oberholtzer has no “regular” mode, no “normal” days. Baseball is wonderful. Baseball is weird.

Getting back to the Phillies bullpen as a whole: It’s not so atypical outside of Oberholtzer and an abundance of negative-score pitchers. Jeanmar Gomez was used in a fairly typical “closer” role, with Hector Neris and Edubray Ramos in higher-leverage setup roles. This all seems to comport with how we think of modern bullpens.

The 2016 Orioles: A Well-Oiled Machine

The Orioles had a very effective bullpen by most measures in 2016. Certainly, it helps to have Zach Britton churning out ground ball after ground ball, but overall the group was very effective, registering a league-leading 10.22 WPA for the season (with second place not being particularly close). Their 53 “meltdowns” were also fewest in the league. This was a playoff team, largely because of their bullpen. That is to say, this is a very different team than the 2016 Phillies.

That said, there are some similarities here.


The general shape is the same, although the Orioles were giving their bullpen a lead more often than the Phillies. One striking similarity is the presence of a “mop-up” guy, in this case, Vance Worley. Worley logged an impressive 64.2 innings in just 31 relief appearances. He was also never given the ball with a lead of less than six (!).


Worley soaked up a lot of innings for the O’s, and he did so in a rather effective way, ending with an ERA of 3.53 – a number which, while partially luck-driven, probably doesn’t suffer from quite as much inherited-runner variance as the average reliever. He created his own messes, and was allowed to clean them up, because Buck Showalter mostly thought the game was over anyway. The overall structure of a bullpen may be related, by necessity, to the depth that the starting rotation can get on a regular basis.

One item of interest here: The unweighted average distance is actually higher in the O’s bullpen than in the Phillies bullpen. When weighting by inverse variance, the Phillies show an even larger average distance, while the average distance narrows for the Orioles. This speaks to more rigid roles, particularly for the setup guys. Darren O’Day was very seldom called upon when the team was behind (four out of 34 appearances, none when trailing by more than three runs), whereas Hector Neris was used a bit more fluidly (18 out of 79 appearances, five appearances when trailing by five or more runs). There may again be a team effect at work here: Maybe the Phillies found themselves needing to get Neris work more often during long losing streaks, and were set on throwing him on a certain day regardless of score.

The 2016 Cubs: An Embarrassment of Riches

If you’ve been under a rock or are currently time traveling, this may shock you: The Cubs were really good last year. They even won the World Series! The Cubs!

OK, with that out of the way, this graph is going to look quite different than the previous two.


Did the Cubs ever not have a lead going into the seventh inning? Well, yes, I assure you that they did. Multiple times, in fact! However, they didn’t do it often enough to give anyone in their bullpen a “mop-up” role, or anything that resembles one. Look at that graph! The Cubs had Aroldis Chapman and Hector Rondon, and then they had seven other guys hanging out in the O’Day / Neris / Brad Brach neighborhood of the graph. What’s going on here?

There’s another thing that’s different about the Cubs which can help explain this. A lot of members of their bullpen have very high variances by score. Whereas O’Day, Neris and Brach have score variances in the single digits, many of the Cubs relievers have score variances north of 10. Take another look at the score variances in the Phillies and Orioles bullpen. Double-digit numbers are typically reserved for long men, mop-up guys, and lower-leverage relievers. Here’s Justin Grimm, who represents this pretty well:


Maybe this was a conscious decision by Joe Maddon, matching up in high-leverage situations with different arms. Maybe this was simply a necessary decision to keep everyone fresh in the face of repeated high-leverage situations: If you have late-game leads for five or six consecutive games, the same three arms can’t be used in all of them. It’s not as if Justin Grimm was used a lot in these situations, and no one would refer to him as a “high-leverage reliever.” He did have a dozen or so appearances in the high-leverage areas of the graph, though, and that’s not nothing.

You can chalk this up to the Cubs being really, really good in 2016, and likely, there’s some merit to that. But it also probably doesn’t tell the whole story. Out of 279 relievers with 20 or more appearances in 2016, only 18 of them had an average inning of 7 or later, an average score differential of 1 or more, and a score variance of 10 or more. Five of those 18 were on the Cubs. The Nationals, Rangers, Red Sox and Dodgers – all good teams in their own right, if not quite as dominant as the Cubs – had one such player each. The Indians had none.

It’s safe to say that Joe Maddon managed his bullpen differently than any of these teams in 2016. It’s also hard to argue with the results.

Quantifying Bullpen Roles: The Math

Author’s Note: This is the first of a two-part article, both parts of which are intended to stand on their own. The first introduces terminology and a mathematical framework used to derive statistics; the second uses these new ideas to draw conclusions which are hopefully intriguing to the reader. If you’re not into math, you can skip to the second article (here) and refer back to this one as needed.

Recently, I wrote about the inning-score matrix, and how we could refine the concept to put a finer point on when and how certain relief pitchers are used. Statistical oddities and outliers are always fun topics of conversation, and certainly, appearance data can give us that.

But can it give us more than that? I don’t care so much that Will Smith was used differently after he was traded or that Brett Oberholtzer was the closest thing to a true mop-up man in the game last year – OK, actually, those things are really interesting too – so much as I care to define how managers are employing bullpens. This may not even give rise to why managers are doing what they’re doing; it’s difficult to attribute intent when looking at numbers abstracted away from the human elements of the game. However, the decision to bring a specific relief pitcher into the game is a conscious one by the manager, largely influenced by game situation. To that end, appearance data can also be aggregated by team — and, if what we care about is the managerial decisions that give rise to bullpen roles, we should really be focused at the team level.

To gain insight into, and ultimately quantify, how bullpens are constructed, we need to define a few concepts. As we go through, I’ll do my best to explain the concept that we’re trying to quantify in baseball terms, before diving into the nuts and bolts of how I’m quantifying them.

Concept 1: Center of gravity

Your personal center of gravity is probably around your belly button – it’s the point at which half of your mass is above, half is below, half is left, half is right.

In addition to their physical centers of gravity (which they work so hard on, Bartolo Colon notwithstanding), relief pitchers have another “center of gravity”: the one at the center of their inning-score matrix. The inning-score matrix has two dimensions (score differential on the X-axis, inning on the Y-axis), and each appearance can be plotted in these two dimensions.

If we treat all appearances equally, a reliever’s center of gravity can be defined as the average inning and score when entering the game. This tells us a great deal about how the pitcher is being used on its own. For example, without looking at the names, you can probably guess which of these guys was a high-leverage reliever in 2016 and which was a mop-up guy.

Player A: Vance Worley; Player B: Zach Britton

The center of gravity is a snapshot of a player’s role. It doesn’t tell you everything – you can’t pick out a lefty specialist, for example, or a guy whose game situations changed drastically over the course of a season. In fact, in the latter case, a player’s center of gravity for an entire season may actually be misleading. Still, it’s the most information you can get about the player’s usage in a couple numbers. We’ll think of it as where the player “lives” in the inning-score matrix.

Concept 2: Euclidian distance

If you’re not a math person, ignore the word “Euclidian.” This is just “distance” in the way you think about it in everyday life. If I have two points in space, a straight line between them has a distance, and in layman’s terms, we’d say that the size of that distance constitutes “how close” or “how far apart” the two points are. Mathematically, for two points with coordinates (xi, yi) and (xj, yj), the Euclidian distance between them can be calculated as:

ED formula

A bullpen lives in the two-dimensional space that we used to define center of gravity: For every appearance a member of the bullpen makes, there is an inning (y), and there is a score (x). In this space, each member of the bullpen has a center of gravity. As such, we can say the two pitchers in our earlier example were far apart, but that these two are close together:

Player A: Shane Greene; Player B: Justin Wilson

In fact, you can start to look at entire bullpens graphically, in order to form an image of how the bullpen is constructed. Our “twins” from above are easy to pick out when we do this:


Nice to look at, and the trend makes intuitive sense: guys who pitch later in games are generally also trusted with leads. But how can we use it to compare bullpens? We need metrics to quantify what we’re seeing above, to describe how similar or dissimilar the roles are in a bullpen. Then we can compare that to other bullpens and give context to how a team is managing their pen relative to the rest of the league.

Concept 3: Average Euclidian distance

The simplest thing one could do would be to sum the distances of the lines connecting each player’s center of gravity. This has the disadvantage of being biased: Bullpens which have more qualifying players will have more dots to connect and, therefore, more total distance.


Naturally, we can calculate an average of these distances instead. This requires us to know how many unique distances there are between distinct pairs of relievers. We can deduce this logically: From the first of n relievers, there are (n – 1) lines, connecting that reliever to all the others. From the second reliever, we’ve already drawn the line to the first reliever, so we can draw (n – 2) more lines, connecting him to the remaining relievers … and so forth. Thus, for n relievers in a bullpen, there are (n – 1) + (n – 2) + … + 2 + 1 distances between them, and we can calculate the average Euclidian distance as:

AED Formula

This looks intimidating, but the numerator is really just the sum of all the distances of all the lines that we drew. The denominator is the number of lines that we drew. Voila: an average!

Concept 4: Weighted-average Euclidian distance

You may be tiring of all this talk about Euclidian distance. It’s important, though, to take this one step further. To use the average distance between all members of the bullpen as a basis of comparison is to make the assumption that all relievers are created equal – that, if you’re a fan of the Indians, you care about the distance between Kyle Crockett and Dan Otero as much as you do about the distance between Bryan Shaw and Cody Allen. You probably don’t, and that makes sense – the former duo isn’t nearly as important to the makeup of the Indians’ bullpen as the latter. We should, therefore, be emphasizing certain relievers and the distances associated with them.

How do we characterize certain members of a bullpen as important, numerically? We could weight them by, say, the average Leverage Index at the time they entered the game; players who are trusted in critical situations are surely more important, right? The issue with this idea is that leverage is highly correlated with the inning and score – in fact, it’s derived from them. Weighting by Leverage Index would tell us that players in a certain area of the graph are more important to team success. This is intuitive and not very interesting.

What do we want to measure? It might be interesting to know how rigid or fluid a team’s bullpen is; that is, do they have a “seventh-inning guy” or a “mop-up guy” who is consistently called on in certain situations? In this case, we want to give more weight to relievers who have lower variance by game situation when entering the game. If the manager gives someone a highly-specific role by inning and score, that reliever is important insofar as the structure of the bullpen is concerned. That may not translate to how important they are with respect to the outcome of games, but presumably, that reliever has a fixed role because they have a skillset that in some way lends itself to his residence in a certain part of the graph.

Fortunately, the concept of inverse-variance weighting is an established mathematical concept. The idea is that players with lower variance by inning and score should be weighted more heavily. In short, this works in three steps:

  1. For each pair of players, divide the Euclidian distance between them by the sum of score and inning variances associated with their centers of gravity;
  2. For each pair of players, divide 1 by that very same sum of score and inning variances;
  3. Divide the sum of results of (1) by the sum of results of (2).

Mathematically, this looks like this:

WAED Formula

Portrait of a Modern Bullpen

If you’re still with me, you may be wondering what the use of all this is. Let’s summarize what we’ve done so far:

  • The average Euclidian distance between members of the bullpen tells us how clustered or spread out that bullpen is as a whole.
  • Using a weighted average refines that metric in order to emphasize members of the bullpen that have well-defined, rigid roles – usually a closer and a setup man or two, but sometimes a surprise as well.

We can summarize a bullpen with these metrics and a plot of all members of a bullpen (as represented by their centers of gravity). Here’s how the 2016 Marlins bullpen looks in a snapshot. The 2016 Marlins have been chosen because they were a very average bullpen in terms of performance as well as structure, on a very average team overall. I couldn’t find anything at all that stood out about them.


We can use this framework to compare bullpens going forward: Which teams have very large distances between relievers? Which are more clustered? Which are oriented differently? We can not only compare bullpens within a single season, but also how bullpen structures have changed over time across the league. We can explore whether the structure of a bullpen is consistent from year to year on a single team, or if certain managers have ways of managing their bullpens which consistently show up in the data associated with their teams. There are a lot of exciting possible applications.

And of course, we can point out statistical oddities along the way. Why wouldn’t we?

Exploring Relief Pitcher Usage Via the Inning-Score Matrix

Relief pitching has gotten a lot of attention across baseball in the past few seasons, both in traditional and analytical circles. This has come into particular focus in the past two World Series, which saw the Royals’ three-headed monster effectively reducing games to six innings in 2015, and a near over-reliance on relief aces by each manager this past October. It came to a head this offseason, when Aroldis Chapman signed the largest contract in history for a relief pitcher. Teams are more willing than ever to invest in their bullpens.

At the same time, analytical fans have long argued for a change in the way top-tier relievers are used – why not use your best pitcher in the most critical moments of the game, regardless of inning? For the most part, however, managers have appeared largely reluctant to stray from traditional bullpen roles: The closer gets the 9th inning with the lead, the setup man gets the 8th, and so forth. This might be in part due to managerial philosophy, or in part due to the fact that relievers are, in fact, human beings who value continuity and routine in their roles.

That’s the general narrative, but we can also quantify relief-pitching roles by looking at the circumstances when a pitcher comes into the game. One basic tool for this is the inning/score matrix found at the bottom of a player’s “Game Log” page at Baseball-Reference. The vertical axis denotes the inning in which the pitcher entered the game, while the horizontal axis measures the score differential (+1 indicating a 1-run lead, -1 indicating a 1-run deficit).


From this, we can tell that Andrew Miller was largely used in the 7th through 9th innings to protect a lead. This leaves a lot to be desired, however, both visually and in terms of the data itself. Namely:

  • Starts are included in this data. This doesn’t matter for Miller, but skews things quite a bit if we only care about bullpen usage for a player who switched from bullpen to rotation, such as Dylan Bundy.
  • Data is aggregated for innings 1-4 and 10+, and for score differentials of 4+. In Miller’s case, those two games in the far left column of the above chart actually represent games where his team was down seven runs. This is important if we want to calculate summary statistics (more on this in a bit).
  • Appearances are aggregated for an entire year, regardless of team. This is a big issue for Miller, who split his time between the Yankees and Indians last year, as there is no easy way to discern how his usage changed upon being traded from one to the other.

To address these issues, I’ve collected appearance data for all pitchers making at least 20 relief appearances for a single team in 2016. We can then construct an inning/score matrix which is specific by team and includes only relief appearances. Additionally, we can calculate summary statistics (mean and variance) for the statistics associated with their relief appearances, including: score and inning when they entered the game, days rest prior to the appearance, batters faced, and average Leverage Index during the appearance. This gives insight into the way the manager decided to use that pitcher: Was there a typical inning or score situation where he was called upon? Was he usually asked to face one batter, or go multiple innings? Was his role highly specific or more fluid?

So let’s start there – and in particular, let’s see if we can identify some relievers who had very rigid roles, or roles that simply stood out from the crowd. To start, here are the relievers who had the lowest variance by inning in 2016.


No surprise here: Most teams reserve their closers for the 9th inning, and rarely deviate from that formula. What you have is a list of guys who were closers for the vast majority of their time with the listed team in 2016, with one very notable exception. Prior to being traded over to Toronto, Joaquin Benoit made 26 appearances for Seattle – 25 of which were in the 8th inning! The next-most rigid role by inning, excluding the 9th inning “closer” role, was Addison Reed, who racked up 63 appearances in the 8th inning for the Mets, but was also given 17 appearances in either the 7th or 9th. In short, Benoit’s role with the Mariners was shockingly inning-specific. I’ve also included the variance of the score differential, which shows that score seemed to have no bearing on whether Benoit was coming into the game. The 8th inning was his, whether the team really needed him there or not.


Speaking of variance in score differential, there’s a name at the top of that list which is quite interesting, too.


Here we mostly see a collection of accomplished setup men and closers who are coming in to protect 1-2 run leads in highly-defined roles (low variance by inning). We also see Matt Strahm, a young lefty who quietly made a fantastic two-month debut for a Royals team that was mostly out of the playoff picture, and a guy who Paul Sporer mentioned as someone who might be in line for a closer’s role soon. Strahm’s great numbers – 13 hits and 0 home runs surrendered in 22.0 innings, to go with 30 strikeouts – went under the radar, but Ned Yost certainly trusted Strahm with a fairly high-leverage role in the 6th and 7th innings rather quickly. With Wade Davis and Greg Holland both out of the picture, it’s not unreasonable to think Strahm will move into a later-game role, if the Royals opt not to try him in the rotation instead.


This next leaderboard, sorted by average batters faced per appearance, either exemplifies Bruce Bochy’s quick hook, or the fact that the Giants bullpen was a dumpster fire, or perhaps both.


This is a list mostly reserved for lefty specialists: The top 13 names on the list are left-handed. Occupying the 14th spot is Sergio Romo, which is notable because he’s right-handed, and also because he’s the fourth Giants pitcher on the list. The Giants take up four of the top 14 spots!

While they never did quite figure out the right configuration (or simply never had enough high-quality arms at their disposal), certainly one could question why Will Smith appears here; the Giants traded for Smith who was, by all accounts, an effective and important part of the Brewers’ pen. The Giants not only used him (on average) in lower-leverage situations, but they also used him in shorter outings, and with less regard for the score of the game.


Dave Cameron used different data to come to the same conclusion several months ago. Very strange, considering that they had not just one, but two guys who already fit the lefty-specialist role in Javier Lopez and Josh Osich. Smith is back in San Francisco for the 2017 season, and it will be interesting to track whether his usage returns to the high-leverage setup role that he occupied in Milwaukee.

This is a taste of how this data can be used to pick out unique bullpens and bullpen roles. My hope is that a deeper, more mathematical review of the data can produce insights on how bullpens are structured: Perhaps certain teams are ahead of the curve (or just different) in this regard, or perhaps the data will show that there is a trend toward greater flexibility over the past few seasons. Certainly, if teams are spending more than ever on their bullpens, it stands to reason that they should be thinking more than ever about how to manage them, too.