Do Errors Count as Baserunners for Fatigue?

Rich Capellini recently posted the following question to the Season Ticket Baseball Facebook Group:

With respect to pitcher fatigue, the rules define “baserunner” as a hit, walk or hit batter. I’m curious as to why an error doesn’t contribute to fatigue. Clay…?

My answer:

Big Picture: It doesn't matter. The effect on the sim of counting errors as baserunners would be neglible. If it makes more sense to you to count an error as a baserunner, go for it!

Smaller Picture (i.e. getting into the weeds): "Fatigue" isn't just about how tired the pitcher's arm is. It's about creating game mechanics that make the engine behave the way I want it to. And that makes managers (human or "auto") behave the way they realistically would in that situation.

It usually takes 5+ pitches to strike someone out. And most home runs happen in the first 3 pitches of an at bat. So why does giving up a home run cause more "fatigue" than striking someone out? It's because I'm not just measuring muscle fatigue. I'm trying to reflect whether the pitcher is "losing it". If they are, then the game should apply a penalty that reflects this. And that penalty should also encourage the manager to bring in a relief pitcher — because that's what usually happens in real life.

Imagine the Red Sox are leading 5-2 going into the bottom of the 8th with Pedro Martinez on the mound.

Scenario #1: Pedro strikes out the first two batters. Grady Little leaves him in to try to get the third out.

Scenario #2: Pedro walks the first batter and hits the second batter. Grady Little goes to the bullpen.

Scenario #3: Pedro gives up a double to the first batter and a home run to the 2nd batter. Grady Little goes to the bullpen.

If Fatigue was just a measure of batters faced, there would be no difference between these three scenarios. But in real life, there is a difference. Season Ticket reflects this difference by making his Fatigue higher in Scenarios #2 and #3 than in Scenario #1.

Now, imagine Scenario #4:

Pedro starts the 8th inning facing Derek Jeter. His first pitch is a fastball that paints the outside corner. His next is a knee-buckling curve ball that falls in for strike two. His 0-2 pitch is a perfect changeup that drops out of the zone but Jeter gets his bat on it and grounds it short. Nomar Garciaparra fields it and throws it over Kevin Millar's head for an E6.

Bernie Williams comes to the plate with Jeter on first. He works the count to 2-2 then Pedro spots an inside fastball just above the strike zone (what Eckersley would call "high cheese"). It jams Bernie for a soft fly ball to short left field, but Manny Ramirez drops it! Jeter advances to 3rd and Williams ends up at safe at 2nd on a 2-base error by Manny.

Does the manager pull Pedro?

I don't actually know the answer to this question. I would like to collect data from the last 100+ years about when managers pull their starters in the middle of the inning. Specifically, what are the stats for the last batter before the manager goes to the pen. I know from watching a lot of baseball that a manager is far more likely to pull a pitcher after they allow a home run than after they strike someone out. But I don't have good data on whether they are likely to pull the pitcher after a batter reaches on error. I also don't know if it varies by manager, or by era.

That, to me, is the main factor here. If you think that a manager is likely to pull their starter after a batter reaches on an error (ROE), then it probably makes sense to count the ROE as a "baserunner".

This data exists (in Retrosheet). But I haven't taken the time to collect it and analyze it, and I'm not aware of anyone who has. In the meantime, I'm happy to admit three things:

1) I don't know whether an ROE should count as a baserunner.
2) The "real" answer is probably somewhere in the middle (i.e. an ROE should cause less "fatigue" than allowing a walk or hit, but more fatigue than getting the batter out).
3) The effect is tiny, so either answer is fine.