The Astros cheated. That’s not in dispute. The search for just how much the banging scheme helped the team, however, is ongoing. Rob Arthur got the party started. Tony Adams chronicled the bangs. Here at FanGraphs, Jake Mailhot examined how much the Astros benefited, which players were helped most, and even how the banging scheme performed in clutch situations. In a recent press conference, owner Jim Crane downplayed the benefit, saying “It’s hard to determine how it impacted the game, if it impacted the game, and that’s where we’re going to leave it.” It’s a rich literature, and not just because it’s fun to write “banging scheme” — but I didn’t want to leave it there.

I thought I’d take a different tack. All of these studies are based on reality, and reality has one huge problem: it’s so maddeningly imprecise. You can’t know if we captured all the right bangs. You can’t know if the system changed, or if it had details or mechanisms we didn’t quite understand or know about. And even when everything is captured right, those sample sizes, those damn sample sizes, are never quite what you need to feel confident in their results.

If we simply ignore what actually happened and create our own world, we can skip all that grubby, confusing reality. Imagine, if you will, a player who makes perfectly average swing decisions and achieves perfectly average results on those decisions.

Let’s further stipulate, while we’re far off into imaginary land, that pitchers attack our perfectly average batter in a perfectly average way. For each count, they’ll throw a league average number of fastballs, and those fastballs will be in the strike zone at — you guessed it — a league average rate. The same is true for all other pitches — with cut fastballs included in “all other pitches” in this analysis.

For example, here’s what a 3-2 count looks like for our generic batter:

Average Behavior in 3-2 Counts Type Frequency Take% Swing% Foul/Swing Whiff/Swing In Play/Swing wOBAcon Fastball, In Zone 34.8% 9.0% 91.0% 42.3% 12.0% 45.7% 0.417 Fastball, OOZ 22.5% 55.4% 44.6% 44.3% 24.8% 30.9% 0.34 Other, In Zone 22.0% 9.3% 90.7% 39.1% 17.7% 43.2% 0.408 Other, OOZ 20.7% 48.6% 51.4% 32.3% 41.7% 26.0% 0.295

Since we know the rate at which each event happens as well as the wOBA value of each result, we can work out the expected wOBA for a 3-2 count: .366. You might notice, if you were inclined to dig, that actual wOBA in 3-2 counts in 2019 was .377. The reason for this discrepancy is another thing I’m assuming: I’m using an automatic strike zone. Every taken pitch in the zone is a called strike, and every taken pitch outside of the zone is a ball.

On 3-2 counts, this isn’t true in real life. Nearly 20% of pitches taken in the strike zone were called balls by the umpire, while only 5% of pitches outside the zone were called strikes. But to make the math easier, we’re using the exact rulebook definition, as defined by the Gameday strike zone.

Now that we know the 3-2 outcomes, we can work out the 3-1 outcomes, because any strike on 3-1 results in a 3-2 count, and we already know the wOBA value of a 3-2 count. Working backwards in this way, I found the value of every count. From 0-0, our hypothetical average batter produces a .322 wOBA, slightly higher than the actual average major league line but close.

Next, let’s start banging. Our imaginary cameraman in the stands sends his imaginary feed to the dugout hallway, where imaginary staffers bang on an imaginary trashcan. Our batter is now simply taking every pitch that isn’t a fastball. 3-2, for example, looks like this:

Fastballs Only, 3-2 Counts Type Frequency Take% Swing% Foul/Swing Whiff/Swing In Play/Swing wOBAcon Fastball, In Zone 34.8% 9.0% 91.0% 42.3% 12.0% 45.7% 0.417 Fastball, OOZ 22.5% 55.4% 44.6% 44.3% 24.8% 30.9% 0.34 Other, In Zone 22.0% 100.0% 0.0% n/a n/a n/a n/a Other, OOZ 20.7% 100.0% 0.0% n/a n/a n/a n/a

But something’s gone wrong. The wOBA of a batter who automatically takes everything but fastballs and behaves normally on fastballs is only .309. This batter knows what’s coming, and he’s doing worse!

Why is that? It comes down to behavior. Just because you know a pitch isn’t a fastball doesn’t mean that taking is optimal. In two strike counts, our hypothetical batter is far too passive. You don’t get extra credit for spitting on a curveball if it’s over the plate. So let’s revise our sketch. Now he’ll treat fastballs normally, and also treat non-fastballs in the strike zone normally. Is this a stretch? Maybe. But for the sake of argument, let’s say that after hearing a bang, our batter can work out that a given pitch is going to bend into the strike zone. Now 3-2 looks like this:

Idealized Sign-Stealing, 3-2 Counts Type Frequency Take% Swing% Foul/Swing Whiff/Swing In Play/Swing wOBAcon Fastball, In Zone 34.8% 9.0% 91.0% 42.3% 12.0% 45.7% 0.417 Fastball, OOZ 22.5% 55.4% 44.6% 44.3% 24.8% 30.9% 0.34 Other, In Zone 22.0% 9.3% 90.7% 39.1% 17.7% 43.2% 0.248 Other, OOZ 20.7% 100.0% 0.0% n/a n/a n/a n/a

This type of sign-stealing is phenomenally more valuable. Our hitter, who as you’ll remember has average results on contact and whiffs just like any mere mortal when he swings, now has a .372 wOBA. That’s like turning Adam Frazier into José Altuve.

Where does the added value come from? There’s a small boost to production on contact (because putting a pitch outside of the strike zone into play is generally a bad result), but it’s mostly strikeouts and walks. In a 3-2 count, an average batter strikes out roughly 27.5 % of the time. Our bang-enabled batter checks in at a delicious 20.2%. After 2-2 counts, an average batter strikes out 38% of the time. Golden boy checks in at 28.6%. He strikes out roughly as often, from a 2-2 count, as the average batter does on 3-2!

The walks are the same story — 44.6% walks out of a 3-2 count is better than Joey Votto has done for his career, and our guy doesn’t even have a good eye! In fact, taking breaking balls outside of the strike zone automatically produces a strikeout rate of 14.8% with an otherwise average player. That’s an extreme case, sure — but it shows how valuable knowing what’s coming can be.

My simulated sign-stealing diverges from reality in many ways. Most notably, the “never swings at breaking balls out of the zone but always swings at breaking balls in the zone” rule would never work in practice. Many of those pitches are only inches apart, and knowing a pitch isn’t a fastball isn’t the same as knowing exactly where it will cross the plate.

Even if the benefit is only half as pronounced as this model, though, it’s still a huge edge in a sport where everyone is looking for tiny edges. And though my swing decision rule gives the batter too much credit, my estimates for production on contact might be too conservative. After all, an average hitter might deliver better-than-average results when hitting a fastball that he knows is coming. Some percentage of the production on contact in every bucket is the result of hitters who were fooled but still put the ball in play, which wouldn’t happen if the batter knows what’s coming. That’s difficult to model, but it’s clearly worth something.

What was most instructive to me about this exercise is where the value comes from. The wOBA edge in 3-2 counts is real, but it’s less than 50 points. After 2-0, it’s even smaller than that. Lots of value is hidden in the first two pitches of a plate appearance. On the first pitch, the perfectly-informed batter goes down 0-1 45% of the time, as opposed to 48% for a regular batter. How about 0-1? A regular batter goes from 0-1 to 0-2 42% of the time. Sign-stealing lowers that to 34%. The average batter, after being ahead 1-0, gets to 2-0 37% of the time. In our hypothetical, that climbs to 42%. It’s not just that our bang beneficiary does better in every count; he also gets to the better counts more often.

The point is, the value of stealing signs doesn’t have to be obvious. It’s not about crushed home runs in surprising fastball counts or even really about loud contact. It’s merely an accumulation of small edges, the Votto-ization of average batters. Being a little ahead in every count stacks up to produce a huge effect.

Knowing this, it’s easier to understand why looking for the edge the Astros got has been so challenging. It’s mostly in not swinging and counts not achieved, and as we saw from the hypothetical player who only swung at fastballs, incorrectly laying off a pitch in the strike zone is pretty harmful. It doesn’t take many crossed signals, bangs that keep batters from swinging at fastballs, to do some damage to the bottom line.

I don’t mean to say that there was no benefit to the scheme, or that unless they were perfectly accurate it didn’t provide a competitive advantage. In fact, the potential gains are large. But without knowing the full extent of the system, or whether any alternative or additional methods were used, I’m not surprised that we have trouble isolating the benefit in the data.

Someday, we’ll know more about the exact particulars of the Dark Arts scheme. Until more evidence comes out, though, I hope it’s helpful to see the theoretical benefit of sign-stealing, to give an idea of the theoretical advantage to be gained, and to see why baseball has come down so strongly against it.

This article has been updated to correct a typo in the first table.