D&D 5E RTU: a defensive counterpart to DPR

[Esker]

In DPR analyses it's common to combine to-hit and damage into a single number representing the expected damage done if we assume a certain level-appropriate monster AC; that is, we convert to-hit bonuses into damage bonuses. But I haven't seen many quantitative analyses on the defense side that consolidate the defensive counterparts of to-hit and damage, namely AC and HP, into a single defensive number. I realize that it's more complicated than this: some things (concentration saves, for instance) work very differently if you take lots of small hits vs one big one, etc., but then, most DPR analyses also ignore whether you're doing your damage in one big attack vs spreading it out over multiple attacks to the same target, even though this can matter, so on a first pass it seems reasonable to set those wrinkles aside.

Non-combat damage and things like concentration checks aside, the main reason HP and AC matter is that they govern how many combat rounds you last before going down. So the operative quantity seems to be your expected "RTU" (rounds-til-unconsciousness). A nice thing about this measure is that you can add together the RTUs of a party to figure out how many actual combat rounds it would take, on average (if they share the incoming damage equally), for them all to go down. You can also factor in sources of healing as additions to the HP pool when calculating a party's RTU.

For fixed AC, RTU will scale proportionally with HP, with the proportionality constant being a function of your enemies' DPR vs your AC. So we have RTU = HP / enemyDPR. But how does AC affect your RTU?

Well, enemyDPR is equal to baseDPH * toHit, where toHit is (21 - AC + atkBonus) / 20. So all together, assuming neither advantage or disadvantage, we have

RTU = HP / [baseDPH * (21 - AC + atkBonus) / 20]

For example, suppose you have 40 HP and the enemy's attack damage on hits is 20 per round, with a 60% chance to hit you vs your current armor class (let's say it's 15 and their attack bonus is +6). Their DPR against you will be 12, giving you, on average 40/12 = 3.33 rounds of getting attacked before you go down. So your RTU is 3.33.

If we set aside advantage and disadvantage, and assume a fixed enemy attack bonus, then each additional point of AC reduces enemyDPR by 5 percent of their base DPH; that is, it's linear. However, this doesn't translate to a linear effect on your RTU, since RTU is inversely related to enemy DPR.

The table below shows, for each additional point of AC, what the DPR of the enemy in the example above becomes (eDPR), what your RTU becomes vs this enemy, and, using AC 15 as a baseline, how many HP you would need to have to achieve that same RTU (eqHP). The columns labeled dRTU and dHP are the marginal increases in RTU and HP associated with the last point of AC.

AC  hit% eDPR   RTU  dRTU  eqHP   dHP
15   60    12  3.33  --    --      --
16   55    11  3.64  0.31  43.7   3.7
17   50    10  4.00  0.36  48.0   4.3
18   45     9  4.44  0.44  53.3   5.3
19   40     8  5.00  0.56  60.0   6.7
20   35     7  5.71  0.71  68.5   8.5
21   30     6  6.67  0.95  80.0  11.5
22   25     5  8.00  1.33  96.0  16.0
23   20     4 10.00  2.00 120.0  24.0

So we have a crude way of quantifying how much each additional point of AC is worth, in terms of RTU, and in terms of HP.

What we can see is that each additional point of AC is worth more than the one before, and moreover that the rate of change increases as well, with the second point of AC being worth 0.6 more than the first, and the third being worth 1.0 more than the second, etc. Going from AC 22 to AC 23 is worth over 6x as many HP as going from 15 to 16!

Now, a caveat, this doesn't factor in crits, which is important since as you go up in AC a higher percentage of the hits against you are crits; but it's easy to include them by just treating crit damage as a bonus to the to-hit roll: if the enemy has a 60% chance to hit and a 5% chance to crit, and half their damage is from dice, then their average damage is the same as if they had a 62.5% chance to hit. 55% to hit becomes 57.5%, etc. So crits don't change the shape of the curve; they just offset it a bit. Altering the enemy's base damage will just rescale these values; whereas changing their attack bonus will of course shift them (increase their to-hit by 1 and an AC of 16 becomes like an AC of 15 before, etc.)

I don't have any grand point to make here; just had been thinking about this and wanted to share it, since I don't know that I've seen a metric like this come up here.

 

[Tony Vargas]

Also missing saves vs damaging effects.
A lot of those are DEX save for 1/2.

 

[Esker]

Also missing saves vs damaging effects.
A lot of those are DEX save for 1/2.

Yup. My main goal here was to look at the value of AC in HP terms, but it'd be nice to expand it. DEX save bonus affects the number of times you can get hit by a fireball or the like, but it's hard to integrate into an average survival duration since these effects tend to be more damage than an attack, so you'd have to set a second damage parameter for those and make some additional assumptions about what proportion of the time you're making DEX saves vs getting attacked vs. AC. Certainly interested if you have ideas about how to do it.

 

[NotAYakk]

Dividing DPR by enemy HP is rarely done in 5e analysis. Here you divide by enemy damage per hit.

That isn't needed; you can scale HP by enemy accuracy without aslo scaling by enemy damage, and the value of AC wrt HP remains unchanged.

There is some utility in enemy-HP normalized DPR, in that is permits cross-level comparisons. Same here.

 

[Blue]

This works well for a front liner designed to stay and pound. I'm sure it could be expanded to deal with resistance and self-healing.

Attacks vs. saves would need to be worked in. Maybe a % per tier

But characters designed to attack from range, to disengage and get to hard-to-attack locations, and all sorts of play designed around tactics that aren't "let them attack me" are poorly modeled by this. It feels like the equivilent of BMI for talking about health - in specific cases it's on the mark, and in others it's misleading.

Still, it doesn't hurt to have a start of the conversation.

I have seen a better metric that DPR before, which basically assumed X foes of Y defenses and Z offense. So that your offense would help reduce attacks against you by dropping foes. Good when all you have is a white room and no team synergy to work with.

 

[Esker]

Dividing DPR by enemy HP is rarely done in 5e analysis. Here you divide by enemy damage per hit.

It was intended to be enemy damage per round if they hit on every attack; since my measure of durability is in rounds, not attacks. I guess I used the abbreviation DPH in one spot, but it really represented... what's the standard abbreviation? DPRh?

That isn't needed; you can scale HP by enemy accuracy without aslo scaling by enemy damage, and the value of AC wrt HP remains unchanged.

Yeah, enemy damage is just a constant of proportionality. But it felt like putting it in terms of rounds that you can last, rather than in HP, was more immediately interpretable.

There is some utility in enemy-HP normalized DPR, in that is permits cross-level comparisons. Same here.

I think that's what this is like, isn't it? Maybe that's what you're saying.

 

[Esker]

This works well for a front liner designed to stay and pound. I'm sure it could be expanded to deal with resistance and self-healing.

Yeah, if you are willing to make some assumptions about how often you encounter particular damage types, then incorporating resistance is easy. Self-healing is a bit trickier, seems to me, since you'd want to somehow factor in the opportunity cost of the action economy involved. But as a first approximation, taking a feature like second wind and just adding it to daily HP based on a standard 2 SR/day seems reasonable.

Attacks vs. saves would need to be worked in. Maybe a % per tier

Yeah, I'd be interested in references if there are commonly accepted numbers there. But you also need to know how much damage the save effects are doing vs the attack effects.

But characters designed to attack from range, to disengage and get to hard-to-attack locations, and all sorts of play designed around tactics that aren't "let them attack me" are poorly modeled by this. It feels like the equivilent of BMI for talking about health - in specific cases it's on the mark, and in others it's misleading.

In a single-character analysis, yes; cunning action disengage, for example is a survivability feature that isn't about AC or HP. But on a party level, usually someone is taking those attacks instead, and so adding up a party's RTU should still mostly work, I think? Of course, some kinds of tactical evasiveness, not to mention debuff effects that mostly serve to reduce the effectiveness of incoming attacks genuinely enhance the party's survival in a way that is difficult for this measure to account for. So maybe those things are the counterpart to control effects offensive side; i.e., things that don't directly do damage but improve the party's action efficiency.

Still, it doesn't hurt to have a start of the conversation.

Yeah, this is a very rough first approximation.

I have seen a better metric that DPR before, which basically assumed X foes of Y defenses and Z offense. So that your offense would help reduce attacks against you by dropping foes. Good when all you have is a white room and no team synergy to work with.

Team synergy is the huge elephant in the room that introduces a huge gulf between optimization analyses made in a vacuum and the actual optimization that you might do in a party inclined toward that sort of thing. When you have an actual party, your spell picks, feat picks, etc. are often not the ones that you'd think of as the optimal choices in a white room. I realize I'm not saying anything here that everyone doesn't know, but it's still worth saying out loud.

 

[NotAYakk]

KPR and FPR; "kills per round" and "fatalities per round"?

KPR/FPR is a power ratio (foes defeated per time defeated).

In 4e nDPR was DPR divided by (L+3) against an (14+L) AC or (12+L) NAD. Even level monsters had about 8 nHP. (N stood for normalized).

In 5e, similar math based off DMG CR guidelines could be used, or blog of holding interpolations.