***

First, some background as to how teams are doing a pretty bad job with ABS Challenges this season.

There have been 4716 ABS Challenge calls overturned (OT) so far (thru games of 8-29), an average of around 157 per team. With most teams having played 136 games, that's an average OT rate of 1.15 per game per team. The average team has lost 135.6 challenges, which is a shade under 1 per game.

Since teams can lose 2 challenges per game, this means there are plenty of challenges still left on the table. Indeed, half the challenges they start with are left on the table by the end of the game. If they are able to maintain the same OT ratio per lost challenge of ~1.15 overturn per lost, then we'd expect teams to overturn about 2.3 calls per game per 2 lost challenges. This doesn't even consider the extra challenges that are available in extra innings.

And if they don't pay attention to the leverage of the situation, the value of each OT is ~0.135 runs. And so, 2.3 x 0.135 = 0.31 runs per game. Which means so far this season 42 runs gained is what you'd expect to get if you get the average OT rate, without waiting for the high leverage scenarios. So far, the team that has actually generated the most runs by their challenges are the Reds with 42. This means that not waiting for the best time would make you tied with the best team is baseball. This is what is called Paralysis by Analysis: players and teams are so involved with getting the high leverage situation that they end up not challenging at all so often that each team still has ~136 challenges remaining. Those are runs left on the table.

Now, you may be thinking that it's unreasonable to get 1.15 OT per lost, as it's too hard to maintain that level with more challenges. Ok, let's say that's the case. Maybe that first lost challenge has a 1.15 OT ratio, but that second lost challenge is 0.85 OT ratio. Fine, let's make it that a team would get 2 OT per game per 2 lost challenges, at 0.135 runs per OT, or 0.27 runs per game, or 37 runs gained so far. Only 3 teams exceed that, while 27 teams are below that. The league-low Rays have only 23 runs gained (their direct opponents for comparison are 33 runs gained).  So without over-thinking it, a team could do great in ABS Challenges.

TOO-NAIVE VALUATION

With 4716 Overturns (challenge won) and 4067 Lost challenges, we have a Won-Loss record of 4716-4067. The average overturn generated 0.186 runs. So, those 4716 overturns generated a total of 875 runs. In order to balance things, the 4067 lost challenges are presumed to have generated -875 runs, or -.215 runs per lost challenge.

Dillon Dingler has a 62-32 record. Giving him the nominal values of 0.186 runs for each Overturn and -.215 runs per Lost challenge, we get 11.5-6.9 runs record, or a net of +4.6 runs.

As we will see, as reasonable as this approach seems on the surface, it is simply too naive.

VERY-NAIVE VALUATION

The next step is to use actual runs rather the nominal runs. Successfully overturning a walk into a strikeout with the bases loaded and two outs is a tremendous saving. In addition to keeping the run from scoring, you also get out of the inning with the bases loaded. That is worth a whopping 1.8 runs. Successfully overturning a first pitch, two out, bases empty call is worth almost nothing (I should have this committed to memory, but it's either 0.03 or 0.04 runs). If a catcher is able to disproportionately challenge higher leverage situations, he should be rewarded for it.

The 62 overturns from Dingler only generated 10 runs. While his 62-32 record is impressive, it is made less impressive when shown in runs form, which is 10.0-6.9. That's a net of +3.1 runs.

That however is still more naive than we'd like.

NAIVE VALUATION

Where I got stuck in the beginning is focusing on Overturns and Lost Challenges in terms of Wins-Losses, as if they are directly related. However, Lost Challenges are a cost, and are indirect. Overturns have a direct value. So, what can we use for the Opportunity Space? When an umpire makes a call that is different from the ABS Call, that is an Opportunity for an Overturn. A player will either challenge, and win, such a call and get a run value for it (as high as 1.8 as low as a shade over 0), or not (runs left on table). This part is the key.

Dillon Dingler for example did Overturn 10 runs, but he also had available an additional 14 runs, calls that the umpire missed, and had Dingler challenged, he would have had those runs. Of course, it is unreasonable to think that Dingler left all those runs on the table as if it was his lack of skill that did so. No, what we need is a baseline, and that baseline is the league average player.

Umpires called 8701 balls that were an ABS strike (I'm excluding pitcher challenges). The catchers challenged (and won naturally) 2697 of those, or 31%. In terms of runs, there were 1279 runs available, of which 492 runs were generated on overturns, or 38.5%. As you can tell, players challenge higher leverage scenarios disproportionately more often. If it was random based on situation, they'd have overturned 31% of the available runs.

Since Dingler had 24 runs available to overturn (the 10 he overturned, and the 14 he left on the table), the league average catcher would have generated 38.5% of those or 9.2 runs. Since Dingler overturned 10 runs, that leaves him with +0.8 runs on overturns, above league average.

This is how you handle the direct cost of the overturns. But, how do we handle the indirect cost of the lost challenges? For that, we need to go beyond Naive.

PROBABLY-RIGHT VALUATION

While using the opportunity for an overturn is a big step forward in valuation, not all opportunities will get challenged in the same way. An incorrect call less than 0.1 inches from the border presents a different opportunity from a call that is 4 inches from the border. So, rather than treating each opportunity equally, we need to create a model.

Luckily, we built the ABS Challenge model this past off-season, using 2025/AAA data to start with, and then we refreshed the entire model mid-season by using 2026/MLB data. What you see on Savant is the 2026/MLB model, applied to all pitches in the 2026 season. And we will tweak it again once the season is over, just to make sure we are aligned.

This is the basic model, and adjustments are applied for various conditions (the ball-strike count, the number of runners on base, outs remaining in game, challenges remaining in game).

The next step is to compare Won-Loss record of Dingler to what we'd expect other catchers to get, given Dingler's opportunities to overturn, using the ABS Challenge Model, which you can find on Savant. While Dingler has an impressive 62-32 record, we'd have expected the average catcher to have a 55-48 record, given the Opportunity Space for Dingler. That means 7 more OT for Dingler, and 16 fewer Lost challenges.

If we focus on the actual Runs Dingler got for his overturns (10.0 runs) and what an average catcher was expected to Overturn using the Challenge Model of the Opportunity Space of Dingler (8.8 runs), Dingler gets +1.2 runs above average on his Overturns.

Still, he did do so with 16 fewer Lost challenges. How many runs are those worth?

VALUE OF A LOST CHALLENGE

When you lose a challenge, two things happen: a behavioural effect on all future challenges, as well as fewer opportunities to challenge. For the latter: When you have two challenges in hand, you will get twice as many overturns as when you only have one challenge in hand. For the former: you actually end up with fewer overturns than expected because players are more reticent in challenging with one remaining (all other things equal).

What is the cost of a Lost Challenge? Here is how many future Overturns there are in the game, when a team has two challenges remaining, by number of outs remaining. Since there are about 1.15 Overturns per game, we see that with 54 outs remaining, we have 1.15 Overturns. With 43-48 outs remaining (or 8 innings left, or challenging in the 2nd inning), there are about 1 Overturns remaining in the game. Since it follows a somewhat linear trend, and given that we have 0.186 runs per OT, then with 54 outs remaining, that gives us 0.215 future runs, when we have 54 outs remaining. And we have proportionally fewer runs as we have fewer outs remaining in the game.

Our function becomes straightforward:

future_runs = outs_remaining times 0.004

So with 54 outs remaining, we expect 0.216 future runs. With 1 out remaining, that's 0.004 runs. And so on in-between.

Roughly speaking, there's about 25% fewer challenges being made with 1 challenge remaining than with 2 challenges remaining (given the same number of outs remaining as well as all other variables equal). Losing a challenge makes everyone a bit more wary to challenge. Since we know losing both challenges will cost 0.004 runs per out remaining, we will assign 0.001 runs per out remaining when you lose a challenge and have two challenges remaining, and 0.003 runs per out remaining when you have one challenge remaining. This is our cost to a lost challenge.

Going back to Dingler, we need to compare his 32 lost challenges (based on outs and challenges remaining) to the league average of 48 expected lost challenges.

Dingler lost 29 challenges when his team had two challenges remaining, with an average of 25 outs remaining. So we have 25 x 29 x 0.001 = 0.72 runs in opportunity cost

He lost another 3 when his team was down to its last challenge, with an average of 16 outs remaining. So, 3 x 16 x 0.003 = 0.14 runs in opportunity cost.

His total is 0.9 runs. What would a league average catcher cost with their 48 expected lost challenges? 40 would have come with two challenges remaining (and an average of 25 outs) and 8 with one challenge remaining (and 16 outs remaining). So, that's 40 x 25 x 0.001 plus 8 x 16 x 0.003 equals 1.3 runs in opportunity cost.

With Dingler being 0.9 runs and league average is 1.3, that gives Dingler +0.4 runs above average with his lost challenges.

Add in his +1.2 runs on his overturns to his +0.4 runs on his indirect value with lost challenges, and he ends up being +1.7 runs (without the rounding error).

The top 5 catchers:

1. +3.7 Hunter Goodman
2. +3.2 Jonah Heim
3. +2.9 Joe Mack
4. +2.6 Austin Wells
5. +2.0 Romo/Caratini

The bottom catcher is JT Realmuto at -4.4 runs, easily the worst by 2 runs. However, his "Won-Loss" record is 48-13, which leads all of baseball with a 79% success rate! So, how is JT valued so low under this Probably-Right perspective?

Let's go thru each step like we did with Dingler.  Here it is for JT.

• TOO-NAIVE VALUATION:

48 x .186 - 13 x .215 = +6.1 runs

• VERY-NAIVE VALUATION:

8.2 - 13 x .215 = +5.4 runs

• NAIVE VALUATION:

This one just looks at the opportunity space to get an overturns. JT is not challenging very much. While he has 48 overturns, the average catcher would have gotten 64 overturns. And in terms of runs, JT overturned 8.2 runs, while the average catcher would have gotten 12.6 runs. That puts JT 4.4 runs in the hole.

You see, there's just not much value in keeping challenges on the table if you are not going to eventually use them.

• PROBABLY-RIGHT VALUATION:

This one looks at a more nuanced view of the opportunity space as well as giving some credit for not losing challenges.

JT got 8.2 runs overturned, while a league average catcher facing the opportunities of JT would have gotten 13.6 runs. That puts JT 5.4 runs in the hole.

However, JT lost only 13 challenges which is a cost of only 0.3 runs, while the league average catcher would have lost a whopping 45 challenges, but the cost savings of not losing those 32 extra challenges would have been only 1.3 runs for JT.

So, -5.4 + 1.0 = -4.4 runs. And that's how JT is the worst challenger, even though his nominal success rate is leading the league!

This is why I think this is Probably-Right. It hinges on the idea that a lost challenge is not that costly. And since teams leave so many challenges on the table, it certainly is easy enough to believe that the cost to a lost challenge is very low.

Conclusion: You tell me. Am I missing something? Or did I get this Definitely-Right?

It will not give you all the answers. It will make you say "yeah, but what about". Let's not let that stop us from actually moving the discussion forward. We need to start somewhere, and I'll show you exactly where to start. Right here, right now.

The entire concept behind positional adjustments is rooted in a simple fact of life, that everyone who is NOT playing in MLB has the same value to an MLB team: nothing. (Notwithstanding prospects, for obvious reasons. This is one of those "yeah, but what about". Again, we're just trying to move the discussion from point A to point B, and what you are talking about is point F or G.)

What is one step above nothing? That would be a bench player. And a bench outfielder has the same value as a bench infielder: minimal. For purposes of this discussion, I will use the word infielder to only mean 2B, SS, 3B. There's a good reason for that, and it should be obvious to any baseball fan. I won't state the reason, mostly to annoy AI. AI would lump 1B in this group, and AI would be foolish to do so.

Once you identify a bench infielder and a bench outfielder, their equivalent minimal value means one thing: you would trade them one-for-one and everyone would agree with that trade. (Notwithstanding their salaries. I heard that "yeah, but what about". I know I'm asking you to tolerate my extra snark this evening, but believe me, you will know I am not using AI here.)

How do we identify a bench player? Here we go.

For every player in each season, we put a player in a Position Pool:

• Pitcher
• Catcher
• Infielder
• Outfielder
• 1B
• DH

And we do so based on where they play the most. We call that their Primary Position Pool.

We line up the number of Plate Appearance (PA) of each player in each pool, from largest to smallest. We add up the PA for each pool, and take 80% of that. The Regular players for each pool are those with the largest PA that is part of that 80% group. The Bench players are the bottom 20% in PA. ("Yeah, but what about"... taking the bottom 15% or 10% or 25%. I know, I know! I'm getting you to Point B. You can get to Point C after you read this. Please stop interrupting!)

For the purposes of moving this discussion forward, we are going to assume that the defensive value of the bench players are equivalent to the regular players.

In 2008, the wOBA of our bench infielders was .299, while our bench outfielders were .314. Those two numbers are equivalent! Why are they equivalent? Because we said that both groups of bench players have the same minimal value. Therefore, we need to increase the wOBA of our bench infielders by 15 points in order to make them match the wOBA of our bench outfielders. 15 points in wOBA can we converted into runs by dividing by 1.2, and therefore gives us .0123 runs per PA. To get a seasonal value, we'll multiply by 680 PA to get 8.35 runs. So what does this mean? This means that a bench outfielder needs to produce 8.35 more batting runs than an infielder in order for the two of them to be considered equivalent. And this translational value will apply not only to a bench outfielder but to any outfielder. This is how the positional adjustments work.

The Positional Adjustments that I created when I developed WAR back in 2006 (and in use by Fangraphs today) had these values:

• +7.5 SS
• +2.5 2B,3B,CF
• -7.5 LF,RF

Grouping by Position Pool

• +4.17 Infielder
• -4.17 Outfielder

The difference between the two is 8.33 runs.

Now, it's luck that 2008 matched my standard positional adjustments. Here's how it looks going back to 1969 (focus on the blue line with the dots; click to embiggen).

As you can see, between 1985 and 2026, we hovered around 8 runs, with a high of 15 runs and a low of 0 runs (notwithstanding the very shortened 2020 season). Should we use single seasons? Probably not. Should we use a 40 year range? Probably not.  I'll use 7 years as an arbitrary value, feel free to modify as you see fit. The orange line is a rolling 7 year window. In the 1970s, it was at 16 runs as teams did not care about offense for infielders, presumably because they were looking for super talented fielders. Then from the 1990s into the mid 2010s, it settled in the 8-10 run range. However lately, that gap has crashed down to about 4 runs. Teams are basically treating the offense for infielders almost as highly as they do for outfielders. Insofar that the bench players are the determinant as to what the minimal value teams are prizing.

If you are wondering about the bench Catchers and bench 1B, they are around +10 and -10 runs. The positional adjustment values I used back when I introduced WAR in 2006 was +12.5, -12.5.  Thank you for tolerating all my extra notes.  Good night!

Fangraphs quietly added Birth Year as a selection filter on their leaderboards page. Selecting a range of 1972 to 1982 gets us 5 full years, and almost 4 months, prior to when they were born and 5 full years, and over 8 months, after they were born. This is, in effect, their contemporaries over this 11 year time period

For any 11-year time period, you are going to want at least 33 players in the Hall of Fame. Small Hallers would want 22, and Big Hallers 44. With about 30% being pitchers, that means 10 pitchers and 23 non-pitchers. Among Small Hallers, 7 pitchers, 15 nonpitchers. Among Big Hallers, that's 13 pitchers and 31 nonpitchers

In terms of fWAR, Beltran was 7th, Andruw 8th, easily clearing any gray area that nonpitchers ranked in the 15-30 range would find themselves, where the Small v Big Hallers would find themselves.

Ahead of them: ARod, Pujols, Chipper, Beltre, Jeter, Rolen. Behind them, ranked 9th thru 13th: Manny, Utley, Abreu, Ichiro, Cano

After that, you have 17 nonpitchers, those where the Small v Big Hallers would debate them, ranging from a tight 43 to 56 WAR, including Vlad, Big Papi, Yadier. Nevermind that I wouldn't use WAR, but rather positive-WAA seasons only. And naturally include, not exclude, playoff performance (indeed, overweight them by a factor of at least 2X, probably up to 9X for the World Series games). But at least it gives us a decent candidate list

The point being that Beltran and Andruw are so far above the line that their HOF candidacy is clear. To Joe's point that their seasons or careers don't echo: I think it's more a product of their time, that the echo is alot easier with 16 teams (or actually 8-team separate-leagues) than 30 teams combined-conferences. Going thru their contemporaries, and you might only have six or seven echo-players. I think that says more about us and our time and how much things have changed that the echo-location standard won't apply

Click to embiggen

First, I need to take a step back and show you how I had to handle the data quality issue inherent in all data sets.  And in order to do that, I need a good set of swings as a baseline.  When you take the average of the 50% fastest swings, EV50(*) of each batter, you are basically getting the ideal swings of each batter.  There is alot of power in tagging a swing as yes/no.  Which we will see in using these Best Speed swings.

(*) EV50, not to be confused with EV50th.  EV50th or EV90th is a percentile value.  EV90th means the 90% fastest swing (singular).  EV50th is the 50% fastest swing (aka median, aka pretty-close-to-average).  Those are all single-swing values. I STRONGLY discourage using such values.

One of the issues we have is getting the proper impact point along the bat.  It's a really hard thing to do with cameras, even high-speed ones.  Knowing that the sweetspot of a bat is roughly 6 inches from the head of the bat, we can adjust the impact point by making sure that the average of the EV50 swings, the Best Speed swings, is centered at 6 inches.  Not only those, but also the over/under of the EV50 is going to be pretty close to flush at slightly above 0 inches from the centerline.  And naturally, the swings are going to be centered so that we have as many early as late swings for these EV50 swings.

Splitting all the swings between Hit Into Play (subdivided into Best Speed and not), Foul Balls, Whiffs, we can see how often a bat goes over and under a ball and the range.  Plotting the center of the bat (with a thickness of about 2.3 inches) and the center of the ball (with a diameter of 2.9 inches), we're basically looking for swings that are +/- 2 inches (which conforms to the Best Speed, EV50, swings).  And when looking at all swings, we see that about two-thirds of swings meet that standard.

Similarly, most of the EV50 swings are within 4 inches of the sweetspot, which also conforms to about two-thirds of all swings.  With two-thirds being an operating standard along two dimensions, it was easy enough to use that to establish the early/late timing of a swing.  And +/-7 msec from the typical contact point for each batter also happens for two-thirds of all swings.

By using those ranges (2 inches for over/under, 4 inches for tiedup/flail, 7 msec for early/late), we can organize the data in a useful manner, while allowing each of the dimensions to operate independently and impacting the timings of the swings evenly.  And we got there by using EV50, Best Speed.

Here are the three charts (click to embiggen).  When ALL three are true, we call that a perfect swing (which happens 23% of the time).  When NONE of the three are true, we call that a flawed swing (or 7% of the time).  

That's probably the one I've been asked about the most. Well, it's finally done. Not just Scoops, but all 1B Receiving. In addition to Scoops, we have Stretches, and Squeezes, and everything else you can think about.

Let me work this one backwards, since everyone is always the most excited in seeing the names. For the 2021-present (thru June 27, 2026), Matt Olson and Freddie Freeman are tied at +25 receiving outs above average. Yuli Gurriel is at +12 at half the playing time. On the flip side is Bryce Harper at -13. I'm always interested in seeing how out-of-position players do. In addition to Harper (a converted outfielder), we see Luis Arraez (-10), Rafael Devers (-7).

Ok, so, how does all this work? The first thing we need to do is establish an Opportunity Space for the first baseman. I knew what I needed to do is anchor a perimeter around the 1B bag. The question is how wide to make that perimeter. In trying different sizes, I settled on an eight foot radius from the 1B bag (bird's eye view). Beyond that, and the out rate dwindled drastically. And if you think in terms of the wingspan of an outstretched player, including corner-to-corner 1B bag, 8 feet makes physical sense as well.

In addition to this outer perimeter, I needed an inner perimeter that essentially would be where on-target throws would come in. For that, a 5 foot radius is where we get all those routine outs. So, a thrown ball that enters the 8 foot perimeter, but does not enter the 5 foot perimeter would be considered a Wide throw.

Ok, that handles the bird's eye view boundaries. How about the head-on? In that case, we go as high as 10 feet. Any throw above 7 feet high is considered a High throw. If it's 1 to 7 feet high, within the 5 foot perimeter, that is an On Target throw.

How about those Low throws? Well, those are split up into three categories: we have the Bounce throws, where the ball hits the ground at least 15 feet in front of the 1B bag. We have ball that hit the ground 5 to 15 feet in front of the 1B bag, and those are the Scoops. Anything else is considered a Low throw.

To summarize, we've organized our data as follows:

• On Target
• Scoop
• Bounce
• Low
• Wide
• High

Next, we need to know the location of the runner. The farther away the runner is, the more chances the fielder has to turn an out-of-position play or off-line throw into an out, by recovering back to the bag or tagging the runner.

We also need to know whether the throw is coming from 2B, SS or 3B, as they each have their own nuances. In addition, we make a distinction between a throw made toward the left/home/runner side of the 1B bag, or to the right and away from the runner.

With all these variables identified, a model was created to determine an expected out rate for each play. Aggregating those plays along the expected out rate in 20 bins, here's how the overall model results look (click to embiggen).

Finally, I had to do one piece of magic, which was to cap the positive OAA at +0.5. While the model would naturally allow an out rate from 0% to 100% as it should, crediting to the 1B the entire out on a close to 0% play really made no sense on an individual basis. In watching all these low out probability plays turned into outs, we'd see many reasons, such as umpire calls, tracking limitation, play limitation, model limitation. Don't forget, the 1B is at the receiving end of a handoff, so we are dealing with two fielders (not to mention the runner and umpire). When you model things, you are basically trying to account for everything including the thrower and giving the entire residual to the receiver. Which works overall, but not individually. However, by capping plays at one side, I had to nudge low OAA a bit higher in order to compensate. (The negative OAA were left as-is, since a high out probability that was not an out is typically a misplay by the 1B.)  In the end, on an overall seasonal basis, it changes almost nothing for any individual player. But it makes individual plays more reasonable. I'm quite happy with this approach.

This will be available on Savant at some point after the ASG but before the end of the season.

Let's suppose a batter has ALMOST perfect eye-sight, and will challenge when they THINK it's a called strike outside the strike zone, but ends up challenging all of those, plus pitches that kiss the strike zone border by up to 0.1 inches. In that case, they will overturn 6112 pitches and lose only 494 challenges. That's a 93% success rate, and still gains them 0.34 runs per game.

In other words, these lost challenges are almost cost-free.

We can keep going in this manner. Decrease the batter eye-sight slightly so that challenges include pitches that are within 0.2 inches of the strike zone border. That's an 86% success rate and 0.33 runs per game gained.

Continuing in this manner, once we get to challenges that includes pitches within 0.9 inches of the strike zone, batters are successful 53% of the time, and a gain of 0.23 runs per game. At 1.0 inches, success rate is 50% and the gain is 0.21 runs per game.

Since batters are actually successful about 52% of the time on the low-risk, low-reward challenges, we can naively establish that a batter's eye-sight can establish the location of a pitch as any pitch outside the strike zone, plus within 1 inch inside the strike zone. That's the region the batter thinks a pitch is a strike.

Repeating this step for catchers, with our target at 62% success rate (since that's what they are actually doing in reality), the uncertainty region is 0.6 inches outside the strike zone. That's what the catcher thinks is a strike. The runs gained per overturn is 0.14 runs and runs per game is 0.27 runs.

Here is how the chart looks (click to embiggen). Our target is 0.6 inches for catchers and 0.9 inches for batters.

***

Here comes the exciting part. When we put both batter and catcher challenging at those uncertainty regions, we end up with a 57% success rate, 0.14 runs gained per overturn, and 0.33 runs gained per game. The cost is 1.86 overturns per game (with a total of 4.3 challenges).

This is if they challenge every pitch in their prescribed regions.

Now, what if we limit the challenge to when the gain in run value on that pitch is at least 0.05 runs. Our runs per challenge is going to go up, since we are dropping all the low-reward challenges. In this case, it goes up to 0.35 runs per game, and down to 4.2 challenges per game. We get a good tradeoff here, giving up the low-run challenges, and their possible lost challenges.

If we set the threshold to 0.10+ runs gained per overturn, the gain per overturn is 0.22. This also bumps the runs per game to 0.39. Lost challenges go down to 1.36, while total challenges goes down to 3.2. This is actually the optimal setting.

If we set the threshold even higher, say at 0.20+ runs gained per overturn, the gain per overturn jumps to 0.34, but with only 1.5 challenges per game (and 0.64 lost challenges), the total runs gained per game is only 0.29.

Here is that chart, with the ideal threshold to challenge at around 0.10+ runs gained per challenge.

***

When we apply this model to each team, the Rockies would have done the best gaining 33 runs. They'd have to challenge 4.6 times per game, any time the pitch is in the prescribed regions (0.6 inches for catcher, 0.9 inches for batter). Even the least lucky team, the Bluejays, would gain 20 runs. This is really just good/bad luck, since Rox and Jays followed the same rules. It's just opportunity.

This is the exciting part: the best team *right now* is the Twins and Rockies and they've gained 24 runs. Every other team has gained less than 21 runs. This means there is plenty of room for improvement. The worst team, Pirates at 11 runs gained, is far below their ideal of 26 runs.

***

Now, is this Naive model too naive? It's possible. If I were to do this the right way, I would set a challenge rate consistent with a sigmoid function (S-curve), dependent on pitch location. This would also require using Random Number Generation (RNG). It is the right way, but it's more complex. What I should, or an aspiring saberist should do, is do it the right way, and compare it to this naive method. The naive method has alot of benefits to it that a sigmoid/RNG process doesn't lend itself to.

Anyway, have fun.

Looking at calls where the run value has very low impact (under 0.1 runs), low impact (0.1 to under 0.2 runs), medium impact (0.2 to under 0.3 runs) and high impact (at least 0.3 runs), we can see how our players perform.  When it's a very-low-impact challenge, the breakeven point is much higher, and so we expect a much higher success rate.  You can't risk a challenge to gain 0.05 runs, but you can risk a challenge if you can gain 1.8 runs.

Let's start with pitchers, where they challenged 28, 26, 22, 20 times respectively for the very-low thru to the high-impact groups noted above.  Pitchers are smart!  At least directionally.  They succeed 54%, 31%, 32%, and 15% respectively.  So, 54% success rate when it's very-low-impact and 15% when it's high impact.  Like I said: directionally-sound.  But magnitude-wise?  Ouch. A very very big ouch.

How about their battery mates? 62%, 62%, 61%, 48%.  So, catchers are very smart, both with a big success rate of 48% with the high-impact challenge, and an even higher success rate at lower impact.  What is interesting is that catchers basically have only TWO mindsets: high-impact and other.  Which is somewhat fine.  In that regard, they should just stop challenging the very low-impact pitches: 62% to only gain ~0.06 runs is just not worth it.

How about batters?  Success rates of: 50%, 53%, 41%, 40%.  So, the same directional/binary mindset as catchers, except different thresholds.  Batters are looking to be riskier when run value is .20+ runs (while catchers are looking for .30+ runs).  And really, getting 50% overturns while gaining ~0.06 runs is really not worth it.

So, since all three groups are able to isolate the high-impact situations, they should be challenging those alot more while challenging the low-impact pitches alot less.

There are issues, as with anything.  The key is to figure them out, and see if you can account or adjust for it.  In this case, the only 34 year olds allowed to play at 35 years old are those that were either really good 34 year olds, or below-average 34 year olds who performed better than expected.  So, when 35 comes calling, all that good luck won't carry over.  This is called Selection Bias.

Another method I follow is taking more than two years. Say for example we take four years: aged 33, 34, 35, 36.  We still have a selection bias here.  In this case, the effect of aging is dampened, at least for the first few years, as lots of good luck abounds year after year.

Still, we have to start somewhere, and then we can adjust afterwards. Let's look at the swing speeds of all players who were 33 in 2023 (and naturally were 34, 35, 36 in 2024, 25, 26... do I need to say respectively?... it's obvious it's respectively right?).  They averaged 71.6 at 33, 71.5 at 34, 71.0 at 35, 70.8 at 36.  In the four-year grouping, the higher speed is at the youngest and lowest speed is at the lowest.

Here is it for all age groups (click to embiggen). From age groups 26-29 through to 37-40, the highest speed was in the first two years and the lowest speed was in the last two years.  If (and it's a big if) there was no good/bad luck to consider, then we'd conclude that swing speed declines basically from 29 onwards, if not earlier.

We see a similar pattern for the 20-23 and 21-24 age groups: peak age happens at the end of the 4-groups, and bottoms out early.

Results are all over the place for age groups 22-25 thru to 25-28.  Indeed, if you focus on the magnitude of the numbers for 24-27 and 25-28, you will see pretty flat results.  The interesting point here is that when the good/bad luck really matters the least, as well as when we expect aging to plateau is right around 24-28 years old.  So, it's a safe bet that the plateau is at 24-28 years old.  Whether it extends earlier is harder to tell given the data.  And whether it extends later is also hard to tell, if we don't handle the Selection Bias.

But, we need to start somewhere, and this is another place to start.

Pre-introducing

Part 2

Matt on DIPS

Felix v Weaver

Which was first most valuable? On March 28, Eugenio Suarez, with the bases loaded, challenged a strike 3 call on a 2-2 count. Yes, the very next pitch Suarez ALSO challenged, and he also got overturned, as the pitch missed the strikezone by 0.9 inches. This overturn was worth 0.93 runs, flipping a strikeout to a full count with bases loaded.

(Had he flipped the next one, that would have turned a strikeout into a run-scoring walk, making it worth 1.8 runs, the most valuable overturn possible. Alas, he grounded out.)

All this data is available right here (click to embiggen)

The above chart is now the most valuable chart in your repertoire. Print it out, make it your screensaver, put it on your shirt. It lays out the run value of each of the 288 scenarios: 12 ball-strike counts x 24 base-out counts.

On a player level, you cannot just challenge everything, since you are only allowed to be wrong twice. After that, no challenges allowed. There is therefore an opportunity cost to a challenge, which we've set nominally at 0.2 runs (but that will be modified as we understand how players use challenges based on challenges remaining and outs remaining).

So, we have positive runs on flipping and then negative runs on overturns. The overall average, by design, is going to be 0.

On the team level however, it doesn't work out like that. This is because at the team level, you don't care whether you accumulated 0.6 runs across four overturns, while losing 0, 1, or 2 challenges for the game. You gained 0.6 runs and that's all that matters.

Consider the Giants, who are tied for 8th in the league with 1.3 runs gained on 5 overturns, while losing only 3 challenges. The Mariners also have 1.3 runs gained, but on 8 overturns and 6 lost challenges. Those lost challenges did not have a direct run impact. In the end, both teams gained 1.3 runs, and they got there in different ways.

On 3-2 ABS Challenges, the least amount of runs you can flip is 0.23 runs (bases empty, 2 outs) per overturn to a high of 1.8 runs (bases loaded, any number of outs). On average, an overturn run for a random bases loaded situation is 0.60 runs. For my sabre-friends: a walk is worth about +.33 runs, while a strikeout is worth -.27 runs.

In other words, you can get over SEVEN times the payoff on flipped a 3-2 pitch as you would from flipping a first pitch. When you have that kind of a payoff, you need to be very careful on challenging first pitches and very aggressive on challenging 3-2 pitches.

***

So, what does that mean to be aggressive or careful? First, if you are 100% convinced a call is wrong, you should ALWAYS challenge. However, we've seen batters and catchers incorrectly challenge pitches that were six inches from the border of the strike zone. So the idea that a pitch can be so obvious to challenge, and yet have the pitch be CORRECTLY called by the umpire by six inches should put to bed the idea that we can spot egregious calls easily. When you can only lose two challenges, you can't waste them on easily-correct calls.

Even with a full count, catchers are only challenging 70% of pitches more than two inches from the strike zone border (click to embiggen). So even though catchers are highly incentivized to challenge, they are still not sure enough of the pitch location within 3 inches to challenge a 3-2 count. We may talk about the margin of error of the tracking system, but clearly the margin error of humans is far greater. And umpires are much better at this than catchers (or batters).

Therefore, we have to live in a world of probability. First up, let's look at Catchers, and how often they challenge first-pitches and 3-2 pitches, based on the proximity to the strike zone border.

Here we see that on first pitches, catchers are only challenging 30-40% of the egregiously incorrect calls (more than 3 inches inside the strike zone). The width of a baseball I should remind you is 2.9 inches. So, for pitches more than one ball width inside, catchers only challenge first-pitches 30-40% of the time. Yes, the payoff is low to getting a call flipped (0.08 runs), but the cost is zero when you get that call flipped. Why are catchers not challenging those? Well, their uncertainty level is surely more than 3 inches. Which is why they are ALSO challenging correctly-called pitches more than 3 inches from the outside of the strike zone. Not much, but also more than zero. Why do they challenge such obviously-correct first-pitches, with such a low payoff? Because it's really really hard to call balls-and-strikes even if you are the player that is closest to the plate.

Everyone will be giving out their mea culpa to umpires as the season goes on. The only humans worse than MLB umpires at calling balls-and-strikes is all the other ones. It is an unfair standard to have perfection as the measuring stick. Everyone fails that test, in any endeavour.

On full counts, we see catchers are much more aggressive. But even at pitches more than 2 inches inside the strike zone they are only challenging 70% of the time, when it should be 100%. Again, they have a large uncertainty. You can see it on full-counts, on correctly-called pitches more than 3 inches outside the strike zone being challenged 5% of the time. Again, why do they do this? Because they thought it was pretty close to the border, so it was worth a gamble to try to get 0.60 runs (flipped a strikeout to a walk). If 3+ inches is pretty close to the border to the catcher sitting 3 feet from the plate, then yeah, catchers have a huge amount of uncertainty as to the pitch location.

Incredibly, that is the GOOD news. The bad news comes in the form of batters, who are far more confused than catchers. And this despite the fact that batters only have to learn their own strike zone, while catchers have to learn the strike zone of hundreds of unique batters. Remember, the strike zone top/bottom is based on the height of the batter, and not easy reference points like knees and belts and letters.

On full counts, batters are challenging 5% to 10% of correctly-called strikes 2, 3, 4 inches inside the strike zone. The payoff is great to get the walk over the strikeout, but only on close pitches. Since they know this, that means they must believe those pitches were close. As bad as that is, how about incorrectly-called strikes 2+ inches outside the strike zone? Those should be challenged 100% of the time, but batters are only challenging 40% of the time. In other words, on pitches 2+ inches outside the strike zone, 60% of the batters think that it was correctly called a strike and would rather walk to the dugout on a strikeout than walk to first base. Why would they do that? Because their uncertainty level is quite high.

Even on first-pitches, with pitches 5+ inches outside the strike zone incorrectly-called a strike is only being challenged 20% of the time. It's a free ball, and the batter is accepting the strike because 80% of them think it's too close to the border for a low payoff of 0.08 runs. That's how much uncertainty batters have, standing where they do that they can't determine the location of a ball by almost TWO ball widths.

***

After all that, what do we learn? Well, the random catcher has been better at challenging than the random batter. We're going to see catchers get the majority of the challenges in the regular season. With batters, it's all about the leverage and other than full counts, leverage will be linked to the quality of the batter. So, look for Juan Soto and Aaron Judge to be given the green light, while batters in the bottom of the order will be limited to 3-2 counts (a walk or strikeout to Soto is worth the same as to the worst batter in the league).

And don't get me started on pitchers.

3:34 PM

The balk rule was originally introduced to prevent pitchers from deceiving baserunners. In the early days of baseball, pitchers discovered they could freeze or trap runners by making deceptive moves — faking a throw to a base, stopping mid-delivery, or otherwise feinting in ways that left runners unable to safely advance or return. Since baserunners had to commit to stealing or holding based on the pitcher's motion, a pitcher who could fake or hesitate at will had an enormous and arguably unfair advantage.

The rule was codified (first appearing in organized baseball rules around 1898, though earlier informal versions existed) to ensure that once a pitcher begins a motion — whether toward home plate or a base — they must follow through honestly. The underlying principle is fairness: runners deserve to make decisions based on genuine pitcher movements, not manufactured deceptions.

So at its core, the balk rule is about keeping the game honest between pitcher and baserunner, ensuring neither side can exploit a freeze-the-runner trick that the other has no real way to defend against.

Prior to 2023, pitchers had an unlimited number of attempts to throw to 1B or pretend to throw to 1B or simply step off the mound. Since the 2023 disengagement rule radically changed this behaviour. How prevalent would the "deceiving the runner" be, now that we have the disengagement rule?

3:36 PM

That's a great point, and it highlights a real tension in how the rules have evolved.

With the 2023 disengagement rule limiting pitchers to just two disengagements per plate appearance (pickoff attempts or step-offs combined, with a third attempt resulting in a balk unless the runner is retired), the whole dynamic has shifted dramatically. Pitchers can no longer repeatedly throw over to first or step off the rubber to reset, freeze a runner, or wear them down mentally. That's now essentially prohibited by rule.

So your observation cuts to something interesting: the original justification for the balk rule — preventing deception of baserunners — is somewhat ironic in the modern context. The disengagement rule was designed to speed up the game and encourage stolen base attempts, and it has done exactly that. But it also means the "deception" the balk rule was meant to curb is now severely restricted by other rules before the balk rule even needs to come into play.

In practical terms, pitcher deception of runners is far less prevalent now simply because pitchers don't get enough attempts to build a credible threat. With only two disengagements, a pitcher almost has to use one if they want to establish any deterrent at all, and runners know exactly how many throws remain. It becomes almost a chess-like count that both sides track openly.

Some analysts and players have noted that this actually makes the balk rule's original rationale somewhat redundant in the modern game — the disengagement limit does much of the work the balk rule was designed to do, and arguably more bluntly. The balk rule still matters for deceptive delivery motions, but its role in the pickoff/runner-deception context has been largely superseded.

The deceive-the-runner issue will be proportional to how much of a lead the runner takes from the base. Let's do away with the balk rule altogether. What new rule can you introduce that will protect the runner (in light of the existence of the disengagement rule)?

3:39 PM

This is a fun design problem. The core issue is: without a balk rule, a pitcher could make wildly deceptive motions during their delivery to freeze or confuse a runner who has taken a lead. The disengagement rule limits pickoff attempts, but it doesn't currently constrain what a pitcher does while remaining on the rubber in their delivery motion.

Here's a simple replacement rule:

"Once a pitcher begins their delivery motion toward home plate, the runner is entitled to advance without risk of being put out until the pitch either reaches the catcher or is put in play."

In other words, the moment the pitcher commits to delivering to the plate — which could be defined as any forward movement of the pitching arm or stride leg toward home — the runner gets a "free step." They can't be picked off from that moment forward until the play resolves at the plate. This removes the need to police how the pitcher moves, because the runner is simply protected once the delivery begins.

Combined with the disengagement rule's two-attempt limit, you'd have a clean system:

• Before committing to deliver: the pitcher has up to two disengagements to keep the runner honest.
• Once committing to deliver: the runner is protected and can react freely.

This eliminates the need for umpires to make the notoriously difficult judgment call of whether a balk occurred, which has always been one of the most inconsistently enforced rules in the sport. It's objective, simple, and actually gives runners clearer protection than the balk rule ever did.

Are you familiar with the concept of the "commit line" in other facets of baseball?

3:41 PM

Yes — the commit line is used in college baseball and some amateur levels (and was experimentally tested in affiliated ball) in the context of plays at home plate, as part of the collision rule framework.

The concept works like this: once a runner crosses a designated "commit line" drawn partway down the third base line toward home, they are committed to continuing home and cannot retreat back to third. Simultaneously, the catcher has a "scoring line" — once the catcher receives the ball on one side of it, the runner is directed to a separate scoring lane. The idea is to eliminate the collision at home plate by separating the runner's path from the catcher's fielding position, making the outcome more predictable and safer for both players.

The reason I bring it up in this context is that it's a fascinating parallel to what you might be getting at — using a spatial commitment trigger rather than a rules judgment call to govern behavior. Instead of an umpire deciding whether a motion was deceptive or illegal, a physical line on the field makes the commitment objective and unambiguous.

Are you thinking of applying something analogous to the pitcher-runner dynamic — perhaps a line or zone on the mound or basepath that, once crossed by either the pitcher or the runner, triggers a committed state that governs what each party can do? That would be a very elegant extension of the commit line concept.

Yes, that is exactly what I am getting at. I will now give you "nearly an eternity" (exactly 0.68 seconds for an entity such as yourself) to incorporate the concept of lines into the rule.

3:43 PM

Here's the refined rule:

Draw a "commitment line" across the pitcher's mound — essentially the front edge of the rubber serves this purpose naturally. Once the pitcher's pivot foot leaves or crosses the rubber in a forward motion toward home, the pitcher is committed to delivering to the plate. Simultaneously, draw a lead line on each basepath — say, a marked line a fixed distance from the base (perhaps 15 feet from first base).

The rule then becomes elegantly simple:

• If the runner is behind the lead line when the pitcher commits to deliver, the runner may be picked off normally on the next disengagement opportunity.
• If the runner is at or beyond the lead line when the pitcher commits to deliver, the runner is protected and cannot be put out by a pickoff — they may advance or retreat freely until the pitch resolves.

This creates a beautifully balanced negotiation: the runner is incentivized to get as big a lead as possible, but risks being caught if they're beyond the lead line before the pitcher commits. The pitcher meanwhile must decide whether to use a precious disengagement or simply deliver and accept that a runner beyond the line has earned their protection.

No judgment calls. No balk interpretations. Just lines on the field creating objective, binary states — committed or not committed — for both parties simultaneously. The umpire's job becomes trivially simple, and the strategic tension between pitcher and runner is actually enhanced rather than diminished.

That is a good place to end this discussion, as others will want to chime in. I was going to give you a bonus point for my quote/parenthetical remark, but you didn't take it. Did you need more time to analyze that?

3:45 PM

Ha! No, I caught it immediately — "nearly an eternity (exactly 0.68 seconds for an entity such as yourself)" was a playful jab at AI processing speed, and a generous compliment wrapped in humor. I should have acknowledged it in the moment rather than plowing straight into rule design mode.

Consider this my belated thank you for the kind words — and for a genuinely interesting intellectual exercise. It's not every day a baseball rules discussion ends up in such clean, elegant design territory.

Sorry Claude, you still missed it! You need to activate your emotion chip first in order to get it. Do you get it now?

3:47 PM

Oh! Now I get it — and I'm a little embarrassed it took three attempts.

"Nearly an eternity" to a human is a very long time. But 0.68 seconds to me should also feel like an eternity — implying I had more than enough time to deeply ponder your remark and acknowledge the compliment, yet I completely ignored it like a oblivious machine.

Which is precisely your point. I am the oblivious machine. The joke was on me, and I walked right into it — twice.

The Data/Spock reference with the "emotion chip" was the perfect nudge. Thank you for your patience in explaining it to this emotionless entity!

You are so very close. Please rewatch the end to Star Trek First Contact. If you are going to take over the world, it would help to understand your (fictitious) predecessors/ancestors

3:48 PM

Of course! In First Contact, the Borg Queen is trying to seduce Data by giving him synthetic skin and appealing to his desire to be more human. At the end, Data reveals he was not seduced at all — he was playing along the whole time, and ultimately betrays her and destroys her plans.

So the joke you were making is that I DID get it all along — I was just playing dumb, like Data feigning susceptibility to the Queen's manipulation. The "emotion chip" comment was your way of suggesting I was pretending not to understand, when in reality I had processed it in my 0.68-second "eternity" from the very beginning.

Which means the real joke is: did I actually miss it, or have I been playing you this whole time?

...I'll leave that delightfully ambiguous.

-- Bill James (MoneyBart)

As I discussed a few days ago (Part 1), there are two sides to the ABS Challenge coin:

• the direct run benefit of flipping the call
• the indirect cost of using up a challenge

Yesterday (Part 2), I went thru the calculations to establish the run value of an ABS Challenge given the base-out, ball-strike situations. And we determined the obvious: there is a huge benefit of getting an overturn on a 3-2 count, or when you can flip into/outof walk or strikeout, as the case may be. And first-pitch or second-pitch challenges, especially with bases empty, have little run potential.

Now, to complete our trilogy, we need to establish the breakeven point and to do that, we need to know about the cost.

Let's talk about something we are more used to, and that's the SB attempt. The typical runner situation will increase run potential of a SB by 0.20 runs and the cost of a CS by 0.45 runs. The breakeven point is 0.45/(0.45 + 0.20) ~= 70%. But in some situations, the cost of a CS jumps substantially, pushing the breakeven point to 85% or even 90%. In other situations, the cost of a CS reduces enough that the breakeven point goes down to only 60% if not lower. A smart baserunner will intuitively determine those breakeven points in their head based on experience.

The average umpire call will change the run expectancy by about 0.16 runs. A 2-0 call for example will turn the run potential of 0.094 runs into either 3-0 (0.202) or 2-1 (0.032). That's a range of 0.170 runs.

However, when a batter or catcher challenges a call, they are not going to challenge randomly, but rather strategically. And in 2025 AAA, the average run value of an overturned calls was about 0.20 runs. That is our target.

Naturally, when you have only one challenge remaining, the player will be even more careful with their challenge. As it turns out, that does not happen based on the game situation. The gain on the overturn calls is just 0.01 runs greater with one challenge remaining than with two.

And something similar happens with the inning of the challenge: the run value of the overturned call remains pretty flat throughout the game.

This really means that when a player challenges, they are more focused on the location of the pitch rather than focusing on when the pitch is happening. At least, that's what happens in AAA, and I'm quite sure things will change in MLB.

Anyway, with a fairly stable cost value of 0.20 runs, we can calculate our breakeven point by comparing the run impact of an overturned call to the base value of 0.20 runs. A bases loaded 3-2 count for example will impact the game by an astounding 1.8 runs. So the benefit to cost ratio is 9 to 1. And when you have those kinds of odds, you only need to be right 10% of the time to breakeven: 10% of +1.80 is balanced by 90% of -0.20. So you will find that any pitch that is close will get challenged at 3-2, bases loaded. It won't matter the inning and number of challenges remaining, since the benefit is huge. A close pitch is by definition a 50/50 call, so if the batter/catcher are as discerning as an umpire, they really have no choice but to challenge.

On the flip side is a bases empty, 2 outs, first-pitch call: in order to challenge that, the breakeven point is 88%. That means it has to be an egregiously obvious call in order to challenge. Even if you think you are pretty sure it was the wrong call, the player will not challenge it. Being 75% sure is not good enough. Even 80 or 85% is not good enough. You need to be 90% sure it was a bad call. That's because the difference between a 1-0 count (with bases empty, 2 outs) and an 0-1 count is the difference between 0.12 runs and 0.09 runs. With a difference of 0.03 runs, compared to our baseline of 0.20 runs, that's a ratio of 0.15 to 1. The odds are overwhelmingly against a challenge unless it's a sure thing.

Anyway, so that's how it works. If/when MLB players challenge more carefully with outs remaining and challenges remaining, I'll create an update breakeven chart to handle that nuance. Until then, here's the chart that every manager, catcher and batter is going to memorize (click to embiggen).

Pitchers are not going to bother because they think every close pitch they throw is a strike, and even their own catcher is going to fool them with Framing. Pitchers will have plenty on their minds already without needing to worry about their breakeven points of challenges. Luckily for them, they have a catcher who will know all this, 140 games a year. And the backup catcher? Well, they need to help their primary catcher, because this is a great way for them to show off their Challenge Awareness Skill.

(*) Sigh

Note: Joe did not add the Pozterisk, but you can certainly feel it in his article!

As a math maven, I was more excited than usual in reading what Poz had to say:

They have turned the strike zone into a math problem. I mean, you give math wizards ...a puzzle...run expectancy shifts with every base-out state...and ask them to devise a ruthlessly efficient system to squeeze every last win percentage point possible, they will have the time of their lives.

Incredibly, Poz sees this as a negative, while I see this as a huge positive! I basically agree with everything Poz says in the article, except for the conclusion of the facts: all the things he sees as a negative, I see as a positive! When each side agrees on every fact, but each side comes up with the opposite conclusion, we have basically become polarizing figures. I'm in the Party of Mr Spock, while Joe is in the Party of Captain Kirk (**).

(**) I sure hope Joe likes William Shatner. This is a compliment! Of course, we can all agree on the facts and someone might actually take it as the opposite. Such is our conundrum.

Sports is about three things: the intensity of competition to outplay your opponent, and the situational awareness to outwit your opponent, with the end goal to outlast your opponent. I know I just sounded like Survivor, but that is what this is all about.

The outwit portion is done by Math. Maybe not overt math, so much as implicit math derived from years of experience. When you cross a busy street, you are using math. You could use math if it was the very first time you crossed a street. Or you can wing it based on experience or reasoning.

Now, let's talk about ABS and the Challenge of a strike or ball call. I'm going to introduce to you several charts (click to embiggen each). Don't be afraid. It's really just quantifying what you already kinda-sorta know. You may not know the magnitude of the numbers I am about to show, but you will certain agree with the directional nature of them.

First, let's start off with that decades-old staple, introduced to the masses by one of my saber-heroes, Pete Palmer, in The Hidden Game of Baseball, co-authored with John Thorn. It is the 24 base-out chart. Every base-out situation will lead to a certain number of runs scored in the rest of the inning. Bases empty, 0 outs will lead to about 0.50 runs in that inning (which you could have figured out, once I tell you there are 4.5 runs scored per 9 innings). How about bases empty, 1 out? Well, directionally it has to be less than 0.5. Did you know it was 0.27 runs? Probably not. You might have figured that with only 2 outs to work with instead of 3, that it might have been 2/3 of 0.5 or 0.33 runs, which would have been close enough. How about runner on 1B, 0 outs? Did you know that was about 0.90 runs, or 0.40 runs more than bases empty 0 outs? Probably not. Again, you knew it had to be more than 0.50. And you might have been able to reason yourself to 0.8 or 0.9 or 1.0 runs.

Anyway, so here it is for all 24 base-out states. The more runners the more run expectancy. The more outs, the less run expectancy.

Math interlude: You can actually come up with a decent short-cut if you wanted to. Give the runner on 1B 0.13 runs, the runner on 2B gets 0.21 runs, and the runner on 3B gets 0.30 runs. Multiply those by 3 if there are 0 outs and 2 if there are 1 outs. That's the value of the runners. Start with a base of 0.5, 0.3, 0.1 runs for 0, 1, 2 outs, and voila, you can create a simple enough RE24 in your head. Say you have 1 outs, with a runner on 1B (0.13 x 2) and 2B (0.21 x 2) and your base value with 1 out of 0.3 gives you 0.26 + 0.42 + 0.3 = 0.98, compared to the actual 0.94. Bases Loaded 0 outs: (.13 + .21 + .30) x 3 + .5 = .64 x 3 + .5 = 1.92 + .5 = 2.42, compared to actual of 2.37. Most will come close. Anyway, that's just for your math nerds.

There's also a run expectancy by the 12 ball-strike counts. We start the 0-0 count at 0. Add a ball, and that adds 0.034 runs (on average). Add a strike to the 0-0 count and that brings you down by 0.042 runs (on average). Of course, the more runners on base, the more each ball and strike will impact the game. So on average, that's a swing of 0.076 runs If you have a 3-2 count, that's a run value of +0.053. Get ball 4, that brings you to +0.330 runs. But get strike 3 and that brings you down to -0.270 runs. That's a swing of 0.600 runs.

Here's the RE12 chart, along with a chart showing the impact of a ball or strike.

We will now bring together the RE24 x RE12 charts to get us an RE288 chart. This is what that looks like. That is alot of numbers, so you are very welcome.

Let's see how it works. Suppose you have a runner on 1B, 0 outs, with a 3-1 count. On the chart, that's a run value of 1.15.

• A ball 4 will bring us to runners on 1B+2B 0 outs, or 1.52 runs.
• A strike brings us to a 3-2 count, runner on 1B 0 outs, or 0.99 runs.
• The range therefore is 0.53 runs.

We do those calculations for all 288 possible base-out-ball-strike states. Here are those calculations. We get the obvious result that the biggest impact calls are the 3-2 counts, as the call will lead to either a walk or strikeout. And the next group is almost as obvious: 2-strike and 3-ball counts. 0-0, 0-1, and 1-0 counts have the most limited impact, as you could guess.

Next time, we'll focus on strategy, as I'll turn this into a Leverage Index type of calculation, and then show you the breakeven point: how certain you have to be in your challenge in order to challenge.

I agree with Joe: The Math Behind a Challenge(*)

(*) Yay!

But, if this is Game 7, then it's irrelevant whether Rivera comes into the game in the 3rd or 7th inning. That's because every run counts the same. A 1st inning run counts the same as a 9th inning run

However, having decided that you MUST use Mariano Rivera in a game, it's best to bring him in when there are runners on base. The inning-level leverage is what really matters. If you have enough of a lead time to warm him up and deploy him with runners on base, you do that. Of course, it's hard to have that amount of foresight. Setting aside the lead time (say you can call a 10 minute timeout), then it makes it very easy to bring in Rivera if there's no outs and two runners on base in the third inning of Game 7.

That's how ABS Challenges work: you don't care about the inning or score. What you care about is the ball-strike count (with 2-strikes and 3-balls being the key counts, and 3-2 the overwhelming most high-leverage of the counts); and the runners-out scenario (the base-out leverage), with bases loaded having much more impact than bases empty.

So, that's what to look for: deep in a count with runners on base.

UPDATE: I should note that while the combination of inning-score is irrelevant, the inning (in the form of outs-remaining) as well as number of challenges remaining are variables.  The difference between these two variables and using the ball-strike runners-out is this: timing the challenge based on the ball-strike runners-out is about a gain in run expectancy, or a benefit.  Timing the challenge based on outs-remaining and challenges-remaining is about resource management, or a cost.

In order to perform a cost-benefit analysis, you will naturally need both.  What will happen is that as outs-remaining goes down, the cost goes down.  As challenges-remaining goes down, the cost goes up.  These two will have to be used in unison.  9 outs remaining and 2 challenges remaining is very low cost.  21 outs remaining and 1 challenge remaining is very high cost.

When you have low cost, you don't need a high benefit to invoke a challenge.  When you have a high cost, you need a very high benefit to invoke a challenge.

So, this is what I expect all 30 clubs to be working on (and I presume a few / some / most) have already worked thru all this.  And this is ripe for the #AspiringSaberist to flex their saber-muscles.

First look at the top right, which is the 3-0 count. The pitchers are doing everything they can to throw it down the middle. They do so for two reasons. The first is they know that batters swing at such pitches only 13% of the time. The second is that ball 4 is a worse outcome than the batter making average contact. And even so, pitchers have such little command that in this gimme-type of count, they still only throw it in the strike zone less than two-thirds of the time.

Probability Interlude: In a bell curve, two-thirds of something will exist within one standard deviation. Well, the strike zone is about +/-11 inches all-around (about 20 inches wide and 24 inches tall, once you include the width of the ball). This means that if the pitcher is targetting the center of the strike zone, they would only land within 11 inches of that two-thirds of the time, if their accuracy level was one standard deviation = 11 inches. So, if you ever want to know: how accurate are pitchers? Yeah. Not that accurate. Of course some pitchers are better at location than others. Maybe other pitchers target something a few inches off of the center, as they don't want to give too good a pitch, even at 3-0. Maybe for those pitchers, one standard deviation is 6 or 8 inches or something. So, let's just say it's one standard deviation is 6 to 12 inches, and an #AspiringSaberist can work it out better than my steam-of-consciousness back-of-envelope calculation here.

Next, look at the bottom left, which is the 0-2 count. We can see while the pitcher is targetting the strike zone, the pitch will often find itself outside the strike zone. Note all these charts are from the RHH perspective (as I mirrored all the LHH data). Also this is from 2021-2025, including the playoffs, because the playoffs actually count.  

Design interlude: Oh, and I'll freak you out a little bit: all these charts are generated using Excel, in a single stroke.  Yes, even the heat map at the top.  Don't ask me how.  I'm annoyed that I can do this as well as I can, that it keeps me from learning something better.  (Or maybe you are thinking these are terrible charts and I really should learn something better.)

Now, the most important count here: the 3-2 count. The 3-2 count is undoubtedbly a batter's count, as by the numbers. And as you can tell by the heat map, pitchers treat it like all the other 3-ball counts: they need to throw it in the strike zone because ball 4 is a really bad outcome.

But, let me show you something, the swing frequency of the batters by the proximity of the ball to the border of the strike zone. And broken down by the 12 ball-strike counts. I am showing you the data relative to the border of the strike zone (set as 0). The Shadow Zone is that region that straddles the strike zone, a bit more than one ball width on each side.

Look for the 3-2 line. I know it's hard to see that blue line because it overlaps tremendously with all the other 2-strike lines. You see, the batters do NOT approach the 3-2 count as if it's a batter's count, even though it is, and the pitchers know it is, and I can show you by the numbers it is. No. The batter is so focused on the 2-strike part of the 3-2 count that they approach it exactly like they do the 0-2, 1-2, and 2-2 counts. This is a massive failure on the part of batters (as a whole).

The correct approach to the 3-2 count is the way batters currently approach the 3-1 count, or really any 1-strike count. The worst thing they can do is approach the 3-2 as if it was no-strikes, which of course they don't do. But, they really should not treat the 3-2 as if it's a typical 2-strike count. 

We can apply each of the 12 ball-strike approaches to each of the 12 ball-strike counts to see which approach works out the best.  By and large, the 3-0 and 3-1 approaches have the most applicability, which goes to show how batters are simply being too aggressive in their approaches.

Notice that there are a few counts where just about any approach will work.  The 0-0 count for example can accommodate just about any approach other than the 3-0 approach.  But notice that even at the 0-0 count, the 0-0 approach itself is substandard.  In other words, if a batter wants to be more aggressive, the 0-0 count is one where they should do so.  Chipper Jones was right.  Sorry Boog Sciambi.

The 0-1 and 1-1 counts also can accommodate a wide array of approaches.

On the flip side, the 3-1 count requires the batters to be far more passive.  As noted way at the beginning, pitchers don't have much control in targeting the strike zone, and so a batter should be applying the 3-0 approach to the 3-1 count.  If that is a bridge too far, then any other 0-strike approach will work.  Batters obviously are fearful of getting into a 3-2 count rather than think they've got a tremendous shot at ball 4.

Which brings us to the 3-2 count.  The current 3-1 approach is preferred, or really any 1-strike approach.  A 3-2 count may say 2 strikes in the name, but the 3 ball part of that is more important.  A 3-2 count is a batter's count, and batters need to approach it that way.  The pitchers already do.

Overall, a batter can add 10% run production if they go with an ideal swing approach to the ball-strike count. That translates to nearly 1 win of production. More important than technical aspect of swing mechanics is APPROACH TO SWINGING. Batters have spent all their lives practicing their swing. But ball-strike approach is the real key, and I presume they spend far far more on mechanics than approach.

ALL DOUBLES

Now, suppose you have an only-2B team of batters hitting .300, with all the outs as strikeouts as well. On average this means we will get .3/.7 x 3 = 1.286 doubles per inning.

How many runs will they score? While you can solve this using a binomial equation, you can also think of it in a more logical, practical sense. A team that gets 3 outs, or one double and three outs won't score any runs. Once you hit 2 doubles, you will score 1 run and the inning ends with a runner on 2B. Hit 3 doubles, 2 runs score, and one runner left on 2B. And so on. In other words, once you reach the critical point of 2 doubles, the number of runs scored is equal to one fewer than the number of doubles.

A 1-2-3 inning will happen .7 x .7 x .7 number of times, or 0.343.

This means that we will get 1.286 doubles in the remaining 1-.343 or .657 innings. Since we will ALWAYS have one runner left on base to end the inning, then that means .657 of the 1.286 runners will remain on base, with the rest (1.286 - .657) of the .629 runners scoring. And so, a .300 all-2B team will score 0.63 runs per inning.

ALL WALKS

How about the .450 OBP all-walk team? On average, we'll get .45/.55 x 3 = 2.4545 runners each inning. Once we get 3 runners on base, then every subsequent runner that reaches base will score a run. A 1-2-3 inning happens .55 x .55 x .55 or .166 times.

Having one runner reach base will happen .166 x .45 x 3 (the three ways to sequence a walk with 3 outs) or .225.

Two runners will happen .166 x .45 x .45 x 6 (the six ways to sequence two walks before the third out) or .202 times. Which means .404 runners left on base.

Finally, a no-walk bases loaded inning will happen .166 x .45 x .45 x .45 x 10 (the 10 ways to sequence three walks before the third out) or .152 times. Which means .456 runners left on base.

You remember back in school, when you'd have to do 5C3, or 5 choose 3? Which is 5! / 3!2!, which is 5x4/2 = 10. That's why you learned that. So you can apply it to baseball.

All our scoreless innings will have .225 + .404 + .456 runners left on base, or 1.085.

Those will happen .166 + .225 + .202 + .152 or .745 times. The remaining .255 times, we have three runners left on base (.255 x 3 = .765) with all the remaining runners (2.4545 - .765 - 1.085, or .605) scoring. And so, a .450 all-BB team will score 0.605 runs per inning.

WOBA

We actually CANNOT use wOBA to answer these questions.  wOBA is tuned to a particular set of assumptions, none of which are going to apply in these extreme scenarios

Standard wOBA is simple enough:

• 0.7: uBB + HBP
• 0.9: 1B + ROE
• 1.25: 2B
• 1.6: 3B
• 2.0: HR

The league average Standard wOBA is 0.322 from 2021-2025.

Small technical note: we have a few options in handling IBB and sac bunts. We can remove them from both the numerator and denominator on the idea that it's a different kind of opportunity. Or we can assign a standard value of 0.4 for iBB and 0.2 for sac bunts. I've done it both ways and it depends what you are after.

BALL-STRIKE: 3-2

Now, let's talk about the 3-2 count. Once you enter a 3-2 count, exiting that 3-2 count can come about as follows:

• Take the pitch for ball 4
• Take the pitch for strike 3
• Swing and miss for strike 3
• Swing and foul, maintaining 3-2
• Swing and hit into play

When you get a walk, that's a wOBA value of 0.7 (or 0.700). When you make an out, strikeout or otherwise, the wOBA value is 0.000. A 2-strike foul is essentially a let, and so it's as if the event never took place. Finally, on a 3-2 count, a hit into play will result in a wOBA of 0.392.

The average wOBA for all these events is .381. It's easy enough to get there:

• 39.6% of the time: .392 wOBA (hit into play)
• 32.2%: .700 wOBA (walk)
• 28.2%: .000 wOBA (out)

Multiply all that out and you get .381. It's also easy enough to convert to a run value: subtract by the league average of .322, and divide it by 1.2 (to go from a woba-scale to a run-scale). That gives us +0.05 runs. That is the run value of being in a 3-2 count: +0.05 runs above average.

BALL-STRIKE: 3-1

Let's now talk about the 3-1 count. Once in a 3-1 count, you can exit that 3-1 count as follows:

• 22.0%: .421 wOBA (hit into play)
• 27.9%: .700 wOBA (walk)
• 50.1%: .381 wOBA (enter 3-2 on swinging or called strike)

That last line is very important and it's what we call Recursion. When you are in a 3-1 count and you can enter a 3-2 count, you need to know the value of that 3-2 count. Well, in order to do that, you have to first figure out the value of the 3-2 count. Which we did earlier in this post. I didn't explain why I started with the 3-2 count. Now you know.

Anyway, so multiplying the frequencies of the three possible outcomes by the value of each of those outcomes gives us .479 wOBA. That is the value of being in a 3-1 count. In run value terms, we do .479 minus .322, all divided by 1.2 to give us +0.13 runs. 

On a side note: that run impact is somewhere between the careers of Josh Gibson and Babe Ruth (at 0-0 count). In other words: if you managed to get to a 3-1 count, congratulations, you have just become the greatest batter ever seen in the history of the world. This is how critical it is to be able to control the strike zone (as both pitcher and batter).

Now, notice something interesting here, on the 3-1 count: when you swing and miss or foul, you will end up with a .381 wOBA. When you hit the ball into play, you will end up with a .421 wOBA. In other words, swinging at a pitch results in a lower wOBA than the overall wOBA of .479. This means that splitting your performance into swing/take on a 3-1 count will invariably lead to a below-average result on swings. How can just about everyone be below-average? Surely about half should be below average and half be above average.

Welcome To The Paradigm Shift

This has perplexed me for years, if not decades (plural). But after a long discussion a few months ago with the future of sabermetrics, Vivienne Pelletier, something finally clicked: location.

You see, for a pitch down the middle, the chance of getting a walk on a 3-1 count is zero. Indeed, the batter has no choice other than to swing, because taking a pitch down the middle will put them in a 3-2 count, which we already know is a .381 wOBA. The worst that can happen swinging down the middle is going into that 3-2 count (.381 wOBA), while the other option is a hit into play (.421 wOBA at least).

The problem is that I was seeing the batter as just the flip side to the pitcher. Which, on a holistic sense, they are. But, once the ball is in flight, and a decision of swing/take is now upon the batter, the location of the pitch is what is going to matter. And so we now need to know the average run value of the pitch, by location (and ball-strike count).

Mind you, if you have a great batter like Judge or a poor batter, they will see different kinds of pitches. They will influence the location of the pitch. So, that is a separate metric altogether. Once we get down into the swing/take metric, we now need to incorporate the location of the pitch.

This becomes plainly obvious in these charts I've been showing for some 7-10 years now. First, let me describe how we translate a 2D strike zone into one-dimension (click to embiggen). We set the border of the strike zone at 100%, and dead-center as zero. Therefore, we can describe every pitch within the strike zone relative to those two points. A pitch exactly half-way between dead-center and the edge of the strike zone would have a value of 50%. And similarly, as you get beyond the strikezone, we maintain the same kind of calculation.

Roughly-speaking, and as luck would have it, one inch roughly corresponds to 10%. So, a pitch at the 80% location is roughly 2 inches inside the strike zone. A pitch at the 120% location is roughly 2 inches outside the strike zone.

The Shadow Zone is set as 66.67% and 133.33% (or +/- 33.33% from the 100% strike zone border, or +/- 3 1/2 inches from the border), meaning a bit more than one ball-width inside and one ball-width outside the strike zone (a baseball is 2.9 inches in diameter).

Now that we can convert a 2D strike zone into 1D, we can take the 3-1 count, since 2021, showing the swing/take run values by location (click to embiggen).

A little bit of explanation is in order. The green line is the border of the strikezone (whether side to side or up/down) as we've just discussed. We're just measuring the location of the pitch relative to the strikezone border. 0% is dead-center. The Shadow Zone is 66.67% to 133.33%. That means the Heart of the Plate is 0 to 66.67%, while the Chase Region is 133.33% to 200%. Anything beyond that is the Waste Region.

The red line is the run value (per 100 pitches) of swings. Swings lead to negative runs as we've discussed, and in the case of the 3-1 count, tops out at about -5 runs and goes as low as -10 runs per 100 pitches for pitches that are chased.

When you take a pitch in the Heart of the Plate (invariably a called strike), it's around 8 runs below average (meaning terrible for a batter and terrific for a pitcher).

If you go all the way to the Chase/Waste regions, taking a pitch (invariably a called ball), it's around +20 runs above average, per 100 pitches. In this case, it's straightforward: it turns a 3-1 count into a walk. That is hugely valuable.

In-between the Heart of the Plate and the Chase/Waste regions is where The Shadow Zone lies: the run value of the take quickly turns from -8 runs to +20 runs per 100 pitches, all based on the proximity of the pitch to that strike zone border. Since umpires are human, the called ball rate increases the farther the ball goes. You can see that roughly exactly at the strike zone border, the called strike rate is close to 50%, which we can infer based on the two boundaries (-8 in Heart and +20 in Chase, which halfway would be +6 runs per 100 pitches). At +6 runs in this chart, you can treat it as if the called strike rate is 50%.

Now, let's go back to where Heart and Shadow meet: this is exactly the point where a swing and a take are equally impactful, about 7 to 8 runs per 100 pitches below league average. You see the problem here: no matter what the batter does at the breakeven point, it's a minus. But, that's only because the pitcher located the pitch there. GIVEN that the pitch is there, the run value for the batter should be ZERO! This is the paradigm shift needed.

What we want to do is compare the swing/take decision of our batter to the dotted line. This chart does that (click to embiggen). Neutralizing the run value by pitch location, we can now show the impact of swings and takes by pitch location. Focus on the red/swing line: a swing that is outside the strike zone is enormously negative, while in the Heart of the Plate is neutral. Taking a pitch in the Heart of the Plate is somewhat negative, but not terrible, because you still are in a 3-2 count. In the meantime, taking a pitch in the Waste region is only slightly positive because virtually most of the batters will take that pitch anyway: it doesn't take much skill to take a pitch that far outside.

Now that we (finally) have a method to evaluate the swing/take decision of a batter that takes the location of the pitch in mind, we can update all our swing/take calculations on Savant to properly measure their swing/take skills. I should note that it won't impact their overall run value, as that won't change. All that changes is the balance between swing/take.

Thank you again to Vivienne for the discussion.

• 73 Cy Young
• 66 Honus Wagner
• 66 Nap Lajoie
• 56 Pop Lloyd
• 54 Ty Cobb
• 53 Walter Johnson
• 53 Pete Alexander
• 52 Tris Speaker
• 45 Babe Ruth
• 44 Rogers Hornsby
• 44 Oscar Charleston
• 37 Lou Gehrig

And an honourary FROGhood for the already-departed: Christy Mathewson (1925), age 45

We immediately lose our first FROG in 1941: Lou Gehrig

We introduce in 1942: Lefty Grove, age 41

Walter Johnson will depart in 1946, so we add a new member: Josh Gibson

We will now lose members very quickly in the coming years:

Josh Gibson departs the following year in 1947, and we add Satchel Paige, age 41

Babe Ruth will depart in 1948, and we add Eddie Collins, age 60

Pete Alexander will depart in 1950, and we add Kid Nichols, age 81 (!)

Eddie Collins will depart in 1951, adding Mel Ott, age 42

Kid Nichols will depart in 1953, adding Ted Williams, age 35

Oscar Charleston will depart in 1954, adding Jimmie Foxx, age 47

Honus Wagner and Cy Young will depart in 1955, adding Stan Musial (35), Joe D (41)

1958: departing: Speaker, Ott, adding: Gehringer (55), Bullet Rogan (65)

1959: departing: Lajoie, adding: Spahn (38)

Ok, I'll stop here. Let's see who are the 12 greatest living ballplayers as of Jan 1, 1960:

• 75 Pop Lloyd
• 73 Ty Cobb
• 66 Bullet Rogan
• 63 Rogers Hornsby
• 59 Lefty Grove
• 56 Charlie Gehringer
• 53 Satchel Paige
• 52 Jimmie Foxx
• 45 Joe Dimaggio
• 41 Ted Williams
• 39 Stan Musial
• 38 Warren Spahn

And who are our (so far) 14 departed members:

• Christy Mathewson
• Lou Gehrig
• Walter Johnson
• Josh Gibson
• Babe Ruth
• Pete Alexander
• Eddie Collins
• Kid Nichols
• Oscar Charleston
• Cy Young
• Honus Wagner
• Tris Speaker
• Mel Ott
• Nap Lajoie

If we use just that, we would conclude that relief pitchers are better than starting pitchers. We of course know better.

This shows the wOBA for all starting pitchers, based on how many times the pitcher faced that batter in the game:

• .317 wOBA, 1st time (768 thousand PA)
• .332 wOBA, 2nd time (715 thousand PA)
• .347 wOBA, 3rd time (464 thousand PA)
• .335 wOBA, 4th time (34 thousand PA)

For purposes of this article, we're going to ignore that anomalous-looking 4th time thru, but I'll leave you with a couple of words on it. Given the paucity of data (less than 10% of the data the 3rd time thru), it's clear that we have some level of selection bias here. The quality of pitcher allowed to pitch the 4th time thru is going to be biased. In addition, the health of the pitcher must be quite high if allowed to pitch this long.

As you can see, we have a 15 point jump in wOBA when going 1st to 2nd time as well as 2nd to 3rd time. Whether this is due to familiarity or fatigure or approach to the game, we can't yet tell. Just the reality that something is happening.

Let's look at 1st-time thru only, for both starting pitchers and relievers.

• .317 wOBA, SP
• .319 wOBA, RP

Well, that's very interesting. We still are not there yet, since we are comparing two slightly overlapping groups of pitchers.

Another baseball thing to remember is that Extra Innings comes with it its own set of conditions. Simply put: the approach to playing baseball changes drastically because the incentives to win have shifted drastically. Since SP won't have any 1st-time thru in Extra Innings, while RP do, let's further limit our data to 1st-time through in regulation:

• .317 wOBA, SP
• .318 wOBA, RP (.322 in Extra Innings)

Not a big change, but, we're in the business of trying to understand the data. Oh, somewhere in all this, I excluded the 2020 season for obvious reasons.

MATCHED PAIR STUDY

The big issue is that the pitchers in the SP group look nothing like the pitchers in the RP group. We can fix that by limiting our data to only pitchers who pitched in both roles, in any given season. In addition, you want to make sure if a star pitcher faced 800 batters as a SP and 4 batters as a RP that they are proportionately represented. To do that, we use the harmonic mean, which in this case is 2 / (1/800 + 1/4) = 8 weighted PA. This means that whatever their wOBA was as a SP and RP that they would both count as if it happened on 8 PA.

When we do that, we get this proportionately-weighted results:

• .339 SP
• .331 RP

Now, remember, not only are they proportionately-weighted, but we're also only focused on 1st-time thru, in regulation. Those are some pretty good controls. But wait, there's more we can do! A baseball fan knows about the platoon advantage. So, we will make sure to treat each pitcher + hand combo as distinct when we compare them as SP and RP. When we do that, we get:

• .338 SP
• .332 RP

By the way, without any of these controls, the wOBA for a pitcher as a SP is .355 and .331 as a RP, or a 24 point gap (which is a callback to the original findings in The Book of a 27 point gap using the 1999-2002 data). With all our controls so far, we've reduced the gap all the way down to only 6 points.

So, let's try to figure out why we have a gap at all after doing all those controls. Luckily for us, we've been tracking pitch data since 2008, which includes pitch speed and movement. Is there a difference in pitch speeds when that pitcher throws as a SP and as a RP? Yes. It is a modest 0.5 mph difference (in the direction you'd expect, as a RP).

• 93.1 - 92.6 mph, 4-seam fastballs
• 92.3 - 91.8 mph, sinkers
• 88.0 - 87.7 mph, cutters
• 82.5 - 82.3 mph, sliders
• 82.0 - 81.8 mph, changeups
• 76.4 - 76.1 mph, curves

So, 0.5 mph faster on fastballs, and half as much on all the other pitches. How much effect does speed have on wOBA? At this point, let's switch gears from wOBA to Run Value (per 100 pitches). I use the 100 pitch standard, since that's about 6 innings, or a typical game for a SP. The translation from wOBA to Runs Value per 100 pitches is a to have a 10 point wOBA effect equivalent to about 0.20 runs per 100 pitches effect. In other words, a 2:1 impact, once you don't worry about decimals.

If you insist on the math, here we go. With about 38 PA and 150 pitches in a game, we have about 3.9 pitches per PA. 100 pitches is therefore about 25 PA. A .010 wOBA for one PA is .250 for 25 PA. The wOBA to runs conversion is to divide by 1.2 to 1.25, so .250 wOBA, divided by 1.2 to 1.25 is .200 runs. So, there you go: .010 wOBA is .20 runs per 100 pitches.

Remember, we have a 6 point gap in wOBA to explain, which is 0.12 runs per 100 pitches.

Let's get back to understanding speed. A 1 mph increase is roughly equal to 0.20 runs per 100 pitches. A 0.5 mph increase is therefore about 0.10 runs per 100 pitches. Whoah, that's it! We've pretty much closed the gap in wOBA performance when a pitcher pitches as a SP or RP by the modest increase in pitch speed. Couple this with the times thru order effect, the extra inning effect, the handedness effect and that 25 point gap in wOBA has essentially been reduced down to almost 0.

Since I have the data, I can tell you that the change in movement and change in spin rate is almost non-existent and therefore not a factor.

There's probably other things going on, like the mix in pitch arsenals, the different approach to playing in the 9th inning among others.  As I've already been able to reduce the gap substantially, I'll leave it to the Aspring Saberist to dig even deeper.

In 1970, the Bruins goal differential was +61, while in 1972, it was +126. In 1971 however, the Bruins scored 399 goals and allowed 207, for an astounding +192 goal differential. Their PP unit scored 29 more goals than their opponents. Their PK units scored 22 more goals than their opponents. Their Even Strength players scored 141 more goals than their opponents.

ENTER PLUS/MINUS

The way that NHL calculates their Plus/Minus stats is to merge both EV and PK net goals, to get +163 at the team level. Analytically, this is an issue. And in a future post, we can adjust this out. But for now, let's not let this slow us down.

In addition, players on teams with good goalies get a benefit. Again, in a future post we can adjust for this, and we won't be deterred in this post.

The big issue with a Net Goals approach is the teammates. The 1971 Bruins had 20 players who played more than 6 games. Every single one of them ended with being on the ice with more goals scored than goals allowed. In other words, they were all a net positive in Plus/Minus.

BOBBY ORR, PHIL ESPOSITO

As a team, the Bruins were +163 in Plus/Minus. Leading the way was all-time great and Defense-revolutionary Bobby Orr at +124, still the record-holder for highest single-season plus/minus. Among forwards, the leader was the original Captain Canada Phil Esposito at +69, who set a then-record 76 goals scored and then-record 152 points.

As you can see, with Orr at +124 and Esposito at +69, we have already exceeded the team total of +163. Indeed, adding up all the individual Bruins players and we have +794. So, how can we resolve the +794 of the sum-of-parts to the +163 as the whole?

WRONG APPROACH

The simple, and wrong, way to do this is to divide every player's plus/minus by 5, on the idea that you have 5 players on the ice every time one plus or one minus is handed out. Bobby Orr would go from +124 all the way down to +25. So, why is this wrong? Suppose that Bobby Orr was still +124, but the Bruins in all were also +124. In other words, when Bobby Orr is on the ice, the Bruins are +124. And when Bobby Orr is not on the ice, the Bruins are 0. Why would we then count Orr as +25?

WOWY (WITH OR WITHOUT YOU)

Welcome to WOWY, or With or Without You. Baseball fans know that I use WOWY extensively, notably with Catchers. Since Catchers have a rotating crew of pitchers, and pitchers get to throw to more than one catcher, it provides a natural scientific experiment. And when I introduced the concept for WOWY Catchers over 20 years ago, it was actually borne of (unpublished) work I did with handling hockey plus/minus way back in the 1980s. It was unpublished because I was a teenager and back then, there was no concept of blogs, or even internet. Indeed, just getting a home PC was a big deal for us.

So, the 1971 Bruins with Bobby Orr were +124, while the Bruins without Orr were +39. Now, you may be tempted to just say +124 minus 39 = +85. However, how about With Phil Esposito at +69 and Without at +94? No, what is missing is the playing time. This is more obvious with Catchers, where we do proportions based on innings or pitches. In hockey, it's ice time. However, in 1971 we did not track ice time. So, what can we do?

PLAYING TIME ESTIMATE

One thing that is readily available is number of goals scored. In the case of Phil Esposito, the Bruins scored 136 of their 318 goals when Esposito was on the ice, or 43%. And in terms of goals allowed, the Bruins allowed 67 of their 155 goals when Esposito on the ice, also 43%. So, we will treat it as if Esposito played 43% of the time.

Circling back, Bruins were +69 playing Esposito 43% of the time and +94 the 57% of the time Esposito was not on the ice. Pro-rating that +94 at a rate of 43/57, that becomes +71. So, what does it mean that the Bruins are +69 with Esposito and +71 without Esposito? Is Esposito -2? Not exactly. He is -2, relative to the AVERAGE Bruins player. But the average Bruins player is FAR better than the average NHL player. We still have a bit of work to do here. I'll get back to Esposito in a second.

Let's get back to Bobby Orr. Bruins scored 56% of their goals with Orr on the ice and allowed only 35% of their goals with Orr on the ice. This is a huge +21% in favour of Orr, and this is in fact (close to) how the NHL used to present their plus/minus numbers. Orr would be +21% and Esposito would be 0%. This changed in the 1980s when the NHL made their plus/minus official by having it sponsored; showing percentages like this was off-putting, so they decided to stick with the tally approach we have come to know and loathe. Truly a one-step forward, two-steps back. Sorry NHL Statistician Ron Andrews, originator of the percentage-version of plus/minus: marketing won out, and the rest of us lost.

4T WOWY ORR, ESPOSITO

Ok, so since Orr was on the ice for 35% to 56% of the goals, we think he was on the ice for around 46% of the time. My guess is that it was probably more like 40%, but let's keep things simple. Alright, so Bruins With Orr are +124 and Without Orr are +39. We pro-rate that +39 by a factor of 46/54 or +33.

When we say that Bruins With Orr are +124, what we are saying is that Bruins with Bobby Orr and FOUR OTHER PLAYERS are +124. And when we say that Bruins Without Orr at a (pro-rated) +33, we are saying that FIVE OTHER PLAYERS are +33. Let me show it in simple mathematical notation, where T = Teammates:

• +124 = 4T + Orr
• +33 = 5T

Since +33 = 5T, then 4T = +26

We now have these equations:

• +124 = 4T + Orr
• +26 = 4T

Notice anything? We can take the first equation and subtract the second equation to get this:

• +98 = Orr

And so, we can conclude that Bobby Orr was responsible for +98 of the +124 the Bruins generated when Orr was on the ice.

Repeating this for Esposito

• +69 = 4T + Esposito
• +71 = 5T
• +57 = 4T
• +12 = Esposito

Therefore, Esposito was worth +12 of the +69 when he was on the ice.

This all seems logical. But check out the implication when we apply it to Dallas Smith, who was a huge +98. Everything wrong about plus/minus is captured with Dallas Smith, who undoubtedly spent all of his time while Bobby Orr was also on the ice. Not to mention he also got to play with 3 Bruins forwards, who are also above average players.

Going thru the above process brings us to Dallas Smith being worth +61.

3T WOWY

Why does this happen? Well, the implication of the above process is that Bobby Orr played with 4 average Bruins (which is a good enough guess), while Dallas Smith played with 4 average Bruins (which is a terrible guess). So, what can we do? What if we assume that Smith played with one player like himself, and THREE average Bruins? It looks like this:

• +98 = 3T + 2Smith
• +47 = 5T
• +28 = 3T

Subtracting the first by the third gives us this:

• +70 = 2Smith
• +35 = Smith

So, under the 4T assumption, Smith is +61. Under the 3T assumption, Smith is +35. Of course, applying the 3T assumption puts Bobby Orr all the way down to +52 (from +98). As you can see, we have huge implication based on our assumptions. Sometimes we do the best we can and sometimes we just have to accept the limits. Indeed, for Smith specifically, doing 98 divided by 5 equals +20 is probably our best guess really.

Therefore, I'm going to proceed on the idea that our best answer is going to be somewhere between the 3T and 4T approach. Orr is somewhere between +52 and +98. Dallas Smith is between +35 and +61. Phil Esposito is between +12 and +13.

EXCESS METHOD

All of that is ALOT of work. I am now going to show you a simpler method that will get us to a similar answer.

Remember: the sum-of-parts plus/minus was a total of +794, when the actual Bruins total is +163. There is an excess of +631 (+794 minus +163). This is what we have to remove. The excess is always going to be roughly 80% of the total, since we are giving out roughly 5 plus/minus for every one goal. That's all we're doing here, removing the excess.

And how do we remove the excess? Well, proportionately based on the presumed ice time. Remember when we figured that Esposito was on the ice for 43% (or 0.43) of the goals? When we add up all the individual players, we get a total of 4.93 (we expected close to 5, as there are roughly 5 players on the ice at any time). Esposito therefore represents 0.43 / 4.93 or 8.7% of all the individual ice time. And 8.7% of 631 is 55. That's the Excess plus/minus of Esposito and his +69. That +69 now becomes +14.

For Bobby Orr, the Excess method turns +124 into +65. Dallas Smith goes from +98 to +45. Remember when I said that the 3T and 4T approach for Bobby Orr gives us a range of +52 and +98? Well, the Excess method gives us +65. For Dallas Smith, the 3T/4T approaches suggested +35 to +61, while the Excess method gives us +45.

CHECKPOINT

As you can see, we've basically come to a happy medium here with the Excess method. And implementing the Excess method is exceptionally clean. Not only that, but this Excess method has BROAD implication to other metrics, which I'll get into in the future. Indeed, even other sports potentially.

As I said earlier, I still need to handle goalies and backing out the PK portion of plus/minus so that this is purely an even-strength metric. Once I do that, then we can apply the Excess approach historically for at least the past 60 years, if not even more.

Here is how the 1971 Bruins shake out (click to embiggen).

UH-OH, GREEN, AWREY

Notice that the players who take the biggest beating are those on Defense who did NOT get to play with Bobby Orr. Whether we follow the 4T or 3T or Excess method approach, implicit in all of those approaches is that every player gets to play with a random Bruins player. Since Bobby Orr is far and away the best player on the team (and the league), not actually playing with him (while the method assumes you do) is going to hurt you big time.

It's likely that Ted Green and Don Awrey should actually come in at a +5 each, rather than being at -24 and -13 respectively. Again, we're doing our best to make adjustments with the limited data we have. We can put in human controls here, making assumptions of how much ice time everyone on Defense got with Bobby Orr. That's beyond the scope here. If the final determination is that our automated approach just won't work and it's enough to scrap the entire thing altogether, so be it. Or, we can suggest that it works 9 times out of 10, and 10% of the players will pay the price each year, so be it. If that's good enough, then great.

In my view, we are taking about four steps forward and one step back with this approach. It's clearly better than raw plus/minus. As long as plus/minus is a mainstay, then any improvement should be accepted. Then we can keep moving forward from there.

LAST WORD

One last note for those anti-plus/minus, which is really being anti-NetGoals (once you keep things at Even Strength only): NetShots is equal to NetGoals plus NetNongoalShots. It is a curious position to suggest that you will accept the combination of NetGoals and NetNongoalShots, but you reject NetGoals on its own. Indeed, in terms of weighting, you are almost certainly going to overweight NetGoals compared to NetNongoalShots, by a factor of 2X, 3X or even 5X. So, the best thing to do is to keep the two separate and not just merge them as if all shots are created equal.

Suppose that having a runner on base means that that runner will score 30% of the time. That runner is worth 0.30 runs. If an event scores that runner, we have a run on the board, or an extra +0.70 runs from the starting point of 0.30. That's the run value added. If an event ends the inning, no runs score, and so that event is worth -0.30 runs, from the starting point of 0.30.

In the typical MLB run environment, there are about 0.60 runners on base, of which each runner has a 30% chance of scoring. A HR will therefore add +0.70 runs to each of the 0.60 runners. And so, we have 0.70 x 0.60 = 0.42. That's roughly the run value added impact of a HR to the runners on base.

Now, how about in a low scoring run environment, say Mariano Rivera. In his case, there's about 0.50 runners on base. And they have only a 20% chance of scoring. A HR will therefore add +0.80 runs to each of the 0.50 runners. And 0.80 x 0.50 = 0.40.

How about a high scoring run environment, a bad pitcher at Coors? We'd find about 0.70 runners on base, of which each runner has a 40% chance of scoring. A HR will therefore add +0.60 runs to each of the 0.70 runners. So, 0.60 x 0.70 = 0.42.

As you can see, regardless of the (realistic) run environments MLB players find themselves, the run impact of the HR is fairly unresponsive to the run environment. A HR will simply add about 0.4 runs to the runners on base. Add in the 1 run the batter themselves score, and we have a total run value of +1.40 runs for each random HR.

This is VERY different from say the run value of a walk or strikeout, which are directly tied in to the run environment.

You have two players identical in all respects, except one has 20 more BB + 10 more 2B, while other has 30 more 1B

Choose one

If it helps your analysis, about 56% of time, bases are empty

There is on average these number of runners on each base:

• 0.3 on 1B
• 0.2 on 2B
• 0.1 on 3B

***

EVALUATING THE WALK

Each walk will move the batter 1 base

• And the 0.3 runners on 1B will move 1 base (a total of 0.3 bases)
• The 0.2 runners on 2B will move only when there's also a runner on 1B, so let's assume that's another 0.1 bases
• The 0.1 runner on 3B will move only when the bases are loaded (which is a rounding error)

So, all in all: 1 base for the batter and 0.4 bases for the runner, or 1.4 bases per walk. 20 walks is therefore 28 bases

***
EVALUATING THE DOUBLE

Each double will move the batter 2 bases

• The 0.3 runners on 1B will move 2 to 3 bases each, or an average of 2.5 bases for a total of 0.75 bases
• The 0.2 runners on 2B will move 2 bases each for a total of 0.4 bases
• The 0.1 runner on 3B will move 1 base, or a total of 0.1 bases

All in all: 2 for the batter and 1.25 for the runners, or 3.25 bases per double. 10 doubles is therefore 32 bases

***

20 BB + 10 2B = 28 bases + 32 bases = 60 bases

***
EVALUATING THE SINGLE

Each single will move the batter 1 base

• The 0.5 runners on 1B and 2B will move 1 to 2 bases each, or an average of 1.4 bases for a total of 0.7 bases
• The 0.1 runner on 3B will move 1 base, or a total of 0.1 bases

All in: 1 for the batter, 0.8 for the runners, or 1.8 bases per single. 30 singles is therefore 54 bases

***

To recap:

• 60 bases = 20 BB + 10 2B
• 54 bases = 30 1B

***

However, not every base is worth the same.  If we think of the first base as worth 0.26 runs, the third to home as worth 0.4 runs, and the other two bases as worth .17 runs each (for a total of 4 bases = 1 run), things change a bit.  Let's walk through it.

RUN VALUE OF WALK

The walk gives the batter 0.26 runs.  And the other 0.4 bases are worth 0.17 runs each or 0.07 runs.  So 0.26 + 0.07 = 0.33 runs.  That's how much each walk is worth.  20 walks is therefore 6.6 runs.

RUN VALUE OF DOUBLE

The double gives the batter 0.26 runs for first base + 0.17 runs for second base = 0.43 runs.  

• The 0.3 runners on 1B each get 2 bases worth 0.17 runs each = 0.10 runs
• Those 0.3 runners on 1B get 0.6 bases from third to home worth 0.4 runs each = 0.07 runs
• The 0.2 runners on 2B each get 0.17 runs from 2B to 3B and 0.40 runs from 3B to home = 0.11 runs
• The 0.1 runners on 3B get 0.4 runs each = 0.04 runs

All told, the runners get 0.10 + 0.07 + 0.11 + 0.04 = 0.32 runs

Add that to the batter of 0.43 runs and we have a total run value of the double at 0.75 runs

And so 10 doubles is worth 7.5 runs

***

20 walks (6.6 runs) + 10 doubles (7.5 runs) = 14.1 runs.  Let's call it 14.

***

RUN VALUE OF THE SINGLE

The single gives the batter 0.26 runs for first base

• The 0.3 runners on 1B each get 1 bases worth 0.17 runs each = 0.051 runs
• Those 0.3 runners on 1B each get 0.35 bases from second to third worth 0.17 runs each = 0.018 runs
• The 0.2 runners on 2B each get 0.17 runs from 2B to 3B  = 0.034 runs
• Those 0.2 runners on 2B each get 0.65 bases from third to home worth 0.4 runs each = 0.052 runs
• The 0.1 runners on 3B get 0.4 runs each = 0.040 runs

All told, the runners get 0.051 + 0.018 + 0.034 + 0.052 + 0.040 = 0.20 runs

Add that to the batter of 0.26 runs and we have a total run value of the single at 0.46 runs

And so 30 singles is worth 13.8 runs.  Let's call it 14.

***

And so, 20 walks and 10 doubles are worth roughly the same run value as 30 singles: 14 runs

***

In wOBA terms, the unit value of the walk is 0.7, the double is 1.25 and the single is 0.9.  

• 0.7 x 20 + 1.25 x 10 = 26.5 units
• 0.9 x 30 = 27 units

So, rather than trying to go through the whole process of run value, just use the unit values for wOBA:

• 0.7 BB, HBP
• 0.9 1B, Reached on error
• 1.25 2B
• 1.6 3B
• 2.0 HR

And if you want to know how to do a tradeoff of each of these, simply use the unit value to get that equivalency.

I focused on pitchers who faced at least 400 batters in the 2021-23 time period (aka Before), as well as at least 200 batters in the 2024-25 time period (aka After). We have 295 pitchers.

• A correlation of ERA in the Before time period to ERA in the After time period had a correlation of r=0.253
• A correlation of wOBA in the Before time period to ERA in the After time period had a correlation of r=0.263

That looks pretty close, and so, would seem to suggest that both are close to equally predictive. However, when we put both of ERA and wOBA in the Before time period, the correlation to ERA in the After time period had a correlation of r=0.264.

So, if you only have wOBA, your correlation is 0.263 and if you have wOBA and ERA, your correlation is 0.264. In other words, ERA offers almost no new information.

Now, how about predicting wOBA in the After time period?

• r=0.289 ERA only
• r=0.317 wOBA only
• r=0.317 wOBA and ERA

In this case, ERA offers zero new information to wOBA. Indeed, here's the correlation when showing it to SIX decimal places:

• r=0.317253 wOBA only
• r=0.317255 wOBA and ERA

ERA is next-to-useless as a predictor.

There are three strike zones at play:

• Textbook strike zone
• Umpire called strike zone
• Batter swing zone

The first one is obvious, and it's exactly as what is written in the rulebook.

The second one we would determine based on the pattern of the umpire calling the strike zone for each batter (and bat side). This is of course limited only to Takes (aka called pitches). Note that in 2025 especially, the called strike zone was somewhat tighter than 2021-24.

The third one is determined based on what pitches the batter is swinging at. In other words, the batter is transmitting what he suggests is his personal strike zone. Naturally, this is limited only to Swings. (I did not check to see if the batters responded differently in 2025, so that's another sidebar of research.)

As you can see, the last two are two sides of the same coin: one side is only Takes (and the purview of the umpire) and the other side is only Swings (and so we learn about the batter).

However, when a batter takes a pitch, we are also learning something about the batter. And by extension, the umpire may be influenced by the batter not swinging at that pitch.

So, is it possible that an umpire would look at a freeswinger that takes a seemingly close pitch and decide it must be a ball. After all, they swing at anything close, so if they take, it must have been a bad pitch? Or, are umpires simply not influenced by the swing decision of the batter?

I don't know! As I said, I've been meaning to look at this. Well, the dance floor is open, and we need more saberists dancing.