Computing Dragon CRs from first principles

[MaxWilson]

I'm saying the wound deals more damage every time it's inflicted, and CR is calculated assuming all hits hit and all saves fail (then you modify for accuracy later).

If every round you are stabbed with the glaive and every round you fail the save, and we ignore the beard, then on every round you will take 8 slashing damage, but the wound damage accumulates. So you take 5 DPR round 1 from your wound, then 10 on round 2 - at the end of round 2, you have taken 15 wound damage, not 10. At the end of N rounds, you have taken wound damage quadratic in N, not linear: 2.5N^2 + 2.5N (the 2.5 is due to the base damage being calculated as 5).

But the damage per stab is linear: (2.5N^2 + 2.5N) / N = 2.5N + 2.5.

And in actual practice it's

(2.5MT + 2.5M) / N

where M is your chance of being hit and then failing a DC 12 Con save each round, and T for Tolerance is the lesser of N and the number of rounds you're going to let yourself bleed before seeking healing or number of rounds to kill the Bearded Devil. T is typically approximately 1, so the damage per stab is approximately 5M, call it 2ish points of damage for an average PC.

In actual play, Bearded Devils do less damage than similar CR 3 monsters like a Red Fang of Shargaas or a Githyanki Warrior. 14 damage with a DC 12 Con save to get another d10 or so is less damage than 32 damage up front. (14+2ish < 32.)

TL;DR Bearded Devil damage per hit is linear in T rounds and T tends to be small, IME.

[Yakk]

They have stated that the DMG CR table is a simplification of a more complex internal spreadsheet they use to calculate CR.

It being off by a bit is not surprising, especially by 1 or less. Being off by quantization and rounding means it was as correct as the simplification permits.

Converting the DMG into an unquantized system isn't all that tricky. Like, the dOCR/dDPR is a constant number.

Subtracting proficiency from attack/save bonuses gives you a target stat modifier (while monsters do vary from Prof+stat based math, that is rare enough presuming that is a typo is reasonable).

All else being equal, 4 points of AC generates 1 higher CR. Same for 4 points of attack attribute bonus.

So take (ATK attribute/4) + (AC/4) and prof from attack mod away from the DMG table, and you'll find the d3roundDPR/dCR and dHP/dCR remain pretty constant in the 1 to 19 CR range.

If you take them at their word on how saves modifies effective AC, we already have 4 AC/CR, so divide that by 4 for the CR bonus.

Even this probably does not match their internal spreadsheet, but it removes most of the quantization and rounding artifacts you guys are running into.

[quindraco]

But the damage per stab is linear: (2.5N^2 + 2.5N) / N = 2.5N + 2.5.

And in actual practice it's

(2.5MT + 2.5M) / N

where M is your chance of being hit and then failing a DC 12 Con save each round, and T for Tolerance is the lesser of N and the number of rounds you're going to let yourself bleed before seeking healing or number of rounds to kill the Bearded Devil. T is typically approximately 1, so the damage per stab is approximately 5M, call it 2ish points of damage for an average PC.

In actual play, Bearded Devils do less damage than similar CR 3 monsters like a Red Fang of Shargaas or a Githyanki Warrior. 14 damage with a DC 12 Con save to get another d10 or so is less damage than 32 damage up front. (14+2ish < 32.)

TL;DR Bearded Devil damage per hit is linear in T rounds and T tends to be small, IME.

Yes, that's right. That's what makes it geometric: it's an infinite number of terms with a constant ratio for increasing the terms. Because it starts small and exhibits infinite growth, it needs many rounds to build up. The larger the ratio, the more rapidly it grows.

On the nth round of combat, if the glaive hits half the time and the save is failed half the time, the target has been hit about (n-1)/2 times and therefore has stacked the wound about (n-1)/4 times. That means on this round alone, they'll take 5.5(n-1)/4 + (8.5+5.5/2)/2 damage. See that n term in there? That's crazy dangerous and makes the total damage superlinear, since the damage itself is linear in the number of rounds.

For example, on the 9th round of being stabbed, the target suffers, in expectation, 16.625 damage. On the 17th round, 27.625. And so on. And the wound damage, RAW, ignores all resistances and invulnerabilities. That's for 50% failure; if the save DC and the melee accuracy are raised to more respectable levels by a DM just trying to follow the recommended rules for scaling a foe's CR up, the increase in the ratios from 1/2 will have an explosive effect. If you change both ratios from landing 1/2 the time to 2/3, the damage on round 9 alone is 27.67. Now may be a good time to point out that Bearded Devils reliably swing with advantage under magical darkness conditions.

Did a quick search of the MM. Here's who can do this infernal wound trick in it:

Bearded Devil: +5 to Hit, DC 12 Con save, Damage 5.5. This is on a carried weapon which may fall into PC hands. CR 3. Side note: Medicine check is DC 12, and the devil's beard can stop magical healing from stanching the wound if it sinks both hit and save.
Horned Devil: +10 to hit, DC 17 Con save, Damage 10.5. This is on its tail. CR 11. Side note: Medicine check DC 12.
Nycaloth: +9 to hit, DC 16 save, Damage 5. This is on its claw. CR 9. Side notes: Medicine check DC 13. The Nycaloth can teleport and turn invisible at will, and can also cast mirror image at will. It can cast Darkness at will and has 60 foot blindsight.

The Bearded Devil numbers might be too small, but I would 100% expect in practice for the Nycaloth to be far more dangerous than the Horned Devil, as it's likely to be far more competent at dragging a fight out, and the wound damage needs to time to become explosively large.

[MaxWilson]

Yes, that's right. That's what makes it geometric: it's an infinite number of terms with a constant ratio for increasing the terms. Because it starts small and exhibits infinite growth, it needs many rounds to build up. The larger the ratio, the more rapidly it grows.

On the nth round of combat, if the glaive hits half the time and the save is failed half the time, the target has been hit about (n-1)/2 times and therefore has stacked the wound about (n-1)/4 times. That means on this round alone, they'll take 5.5(n-1)/4 + (8.5+5.5/2)/2 damage. See that n term in there? That's crazy dangerous and makes the total damage superlinear, since the damage itself is linear in the number of rounds.

But... it's not crazy dangerous in normal cases because n is small. Fights don't last an infinite number of rounds. If n goes from 1 to 3ish, the glaive wound will inflict only about 4 damage total. That isn't super dangerous, in fact it's anemic compared to the actual up front damage that e.g. an Orc Fang of Shargaas does, up front, with advantage on the attack roll.

I feel like you're doing the equivalent of applying Quicksort to a list of length 3, instead of Selection Sort, just because Quicksort is faster for lists as size approaches infinity. Telling you what you probably already know, just in case: selection sort is far better at exploiting local caches, and does less total copying, making it faster on small lists--how big the list has to get before Quicksort is better depends on hardware details.

For a party with no magical healing, would a hundred bearded devils with infinite reach on their weapons be scarier than a hundred Githyanki Warriors or Orc Fangs of Shargaas? Sure, probably. But two or three Bearded Devils who have to melee you? Nope, two or three Githyanki or Shargaas Fangs are worse.

Most dangerous weapon in the game? Not by a long shot.