How many % of hit points is one point of AC worth?

[Mikko Kallio]

Hey all,

I'm planning to write a number of articles about monster design on my blog later this summer. As is stated in the Bestiary, "creating a monster is part science and part art", and I'd like to crowd-source your opinions on the science part of monster creation.

Let's start with something (fairly) simple: hit points and how they interact with other things that keep a monster alive (such as AC, DR, and saves). Everyone who's designed monsters for Paizo or 3PPs (or home games) probably knows the Monster Statistics by CR table and that you should stay reasonably close to the target values if you want the CR to be accurate. However, it's ok compensate for low hp with a high AC, for example. But how much higher or lower should the AC be, mathematically speaking, if the hp deviates from the baseline by a particular percentage (if all other things are equal)?

* How much higher/lower should a monster's AC be if it's hp are 20% higher/lower than the target value? Does the amount change with higher CRs? If it does, is the change linear?
* What about DR? How many hp or points of AC is DR 5/10/15/20 worth and how does it change with the CR? How much are the different types (DR —/cold iron/silver/adamantine/magic/etc.) worth?
* What about saves? Or a miss chance of 20% or 50%?

Thoughts, opinions? If you can show some math (and graphs!) to back it up, that's even better!

Thanks!

-Mikko / A Sword for Hire

[Nawtyit]

Let's just look at two fairly common CR 6 monsters, the will-o'-wisp and a very young red dragon, compared to the 'average' from the table.

Will-o'-Wisp: AC 26/26/16, HP 40, Saves +3/+12/+9, natural invisibility, immune to magic
Red dragon: AC 18/11/17, HP 59, Saves +7/+6/+8, immune to acid, paralysis, sleep
Average: AC 19, HP 70, Saves +9 good/+5 bad

The WoW is on average 40% harder to hit, but has 68% the HP over the dragon. The dragon has lower AC, HP, and saves, but is immune to three things. The WoW has higher AC and saves, but much lower HP and is invisible and immune to magic.

Monster design is a delicate dance between offence and defense that also should take into account what the PCs should have as far as gear and magic at that level.

[kestral287]

To keep things equivalent it should depend on hit rates. Which is a pain because it shifts over time.

The ACs can be plotted to track, say, a Fighter's hit rate with each attack. If a Fighter is hitting the creature 50% of the time, then a two point loss in AC (equivalent to increasing hit rate by 10%) should increase HP by 20% to stay equivalent:

.5X=.6Y
5/6X=1Y

Where X and Y are the percentage of damage that the Fighter deals with each swing. Since the Fighter's damage should remain constant (he's the same guy), HP has to rise by 20% (one fifth of 5/6 is 1/6, giving you the 1Y).

However, if the monster has a very low AC, such that the Fighter hits 80% of the time...

.8X=.9Y
8/9X=1Y

Then you only increase HP by 12.5%

You can run the same thing in reverse of course.

Miss Chance should be handled similarly.

Assuming a 20% miss chance and the base hit rate is 50%, you're effectively decreasing accuracy by 10%. (1-[.2+.5-[.2*.5]] is how I express it; there's probably a cleaner mathematical solution). So, the Fighter now hits 40% of the time:

.5X=.4Y
1.25X=1Y

Reduce HP by one-fifth.

Again that'll vary between different ACs and different degrees of miss chance but we see the theory.

For DR, the degree that a DR type matters should depend on CR. DR 5/Adamantine is a lot at CR2 and nothing at CR20, as by CR20 we expect PCs to be able to handle that. As with hit rate, ideally we'd establish the baseline of when we expect the Fighter to get each +X weapon (or the actual material) and deal with it then. A pure DR/- (or a DR to something odd like Glass, which should be handled the same way) is actually easiest.

If we assume that in the average four-man party, each monster will be attacked thrice per round for two rounds apiece, then we can figure out the average hit rate of those attacks and adjust accordingly. For simplicity, even though it's dead-wrong we'll give all of those attacks the same 50% hit rate.

So, we expect the monster to take six attacks, three of which will hit. A DR 10 should thus be equal to 30 HP.

Saves and HP don't typically overlap. You could maybe do this for Ref saves, against a Caster level*D6 spell, and scale the DCs. But it doesn't work well for debuffs, so that's... a nightmare of an exercise.

[Mikko Kallio]

Nawtyit wrote:

Monster design is a delicate dance between offence and defense that also should take into account what the PCs should have as far as gear and magic at that level.

I agree, calculating a monster's CR is a sum of great many things. However, I don't think these two monsters are particularly good examples of CR balance, the w.o.w. in particular is notorious. Just because they're in the Bestiary doesn't mean they're balanced for their CR or that the designers even consulted that table back in the day. A lot of 3.5 baggage was inherited with monsters that appear in the first Bestiary even though a lot of things were changed.

What I'm trying to say is that you won't find the answer to my OP question just by looking at existing bestiary entries.

[Cheapy]

I'm very interested in seeing where this goes, having done a numerical comparison of DPR for an optimized and non-optimized CRB only fighter, as well as many other number crunches.

That said, I think this will in many ways depend on the optimization level of the party. AC is meant to fall behind attack bonuses at a certain point, which means HP is the main deterrent against death that monsters will have. Things like mischances help a bit, of course.

[Mikko Kallio]

Cheapy wrote:

That said, I think this will in many ways depend on the optimization level of the party.

I guess I should have given a bit more background info -- on my blog I mostly write about freelancing, the RPG Superstar contest, and RPG design in other similar contexts where you cannot know who'll be using the monster. So, for the purpose of this discussion, assume that the monster's hp, AC, DR, and miss chance ratio should be balanced for a Paizo product.

[kestral287]

Serpent wrote:

So, for the purpose of this discussion, assume that the monster's hp, AC, DR, and miss chance ratio should be balanced for a Paizo product.

Use Valeros as the hypothetical Fighter for the plotting of hit rates.

[Charon's Little Helper]

There are really too many variables to make a definite % statement.

For example: if the wizard's Sleep Spell works on them - neither one really matters much.

In the above Will o' Wisp example (it is known for very high AC) - it's one of the only creatures I'd feint against without Improved Feint. A single character with a good bluff score will make them FAR easier to deal with - dropping their AC down to 16, and their CMD to 14 every other turn. Once their AC & CMD are dropped that much, a well-timed Dirty-Trick (worth eating an AOO) - can blind them pretty easily and make them even easier to hit for EVERYONE in the party - likely killing them in a turn.

How do you balance that combo against their lowish HP? *shrug*

[Berti Blackfoot]

I use anydice.com to calculate attack bonus vs AC and average damage.

I found this one someone made, which lets you plug in AC and hit bonuses and crit etc. and the resulting number is the average damage per round.
Then your comparison would be for each AC and hp level, how long will the monster last.

You can create several lines with different values for AC and compare them, I find the Summary screen is the easiest way to do comparisons like this.

http://anydice.com/program/5e0b

[Mikko Kallio]

kestral287 wrote:

(math)

Thanks for the calculations! The bit about monsters with very low ACs is particularly interesting--the amount of extra hp you get for each point of AC should decrease the farther below the target value you go.

You're probably right about saves vs hp; saves and SR would be a more meaningful comparison because spells are one of the most common reasons a monster has to attempt a saving throw.

[Mikko Kallio]

Charon's Little Helper wrote:

There are really too many variables to make a definite % statement.

For example: if the wizard's Sleep Spell works on them - neither one really matters much.

In the above Will o' Wisp example (it is known for very high AC) - it's one of the only creatures I'd feint against without Improved Feint. A single character with a good bluff score will make them FAR easier to deal with - dropping their AC down to 16, and their CMD to 14 every other turn. Once their AC & CMD are dropped that much, a well-timed Dirty-Trick (worth eating an AOO) - can blind them pretty easily and make them even easier to hit for EVERYONE in the party - likely killing them in a turn.

How do you balance that combo against their lowish HP? *shrug*

I mostly agree and I'm aware that the usefulness of both AC and hp are situational: some abilities target AC but don't cause hp damage while some abilities cause hp damage but bypass AC or allow a save. However, effects that target AC and cause hp damage are very common, so I do think it's useful for a designer to know how much AC corresponds to how much hp on average. "On average" is the key phrase here, your example of feinting vs a w.o.w. is a statistical outlier.

Also, a mathematical formula doesn't have to apply to 100% of situations for it to be useful.

[kestral287]

Serpent wrote:

kestral287 wrote:

(math)

Thanks for the calculations! The bit about monsters with very low ACs is particularly interesting--the amount of extra hp you get for each point of AC should decrease the farther below the target value you go.

You're probably right about saves vs hp; saves and SR would be a more meaningful comparison because spells are one of the most common reasons a monster has to attempt a saving throw.

Yeah, Saves and SR are a much better overlap. CMD's really in the same boat as saves, in that it's a key defense but it lies more-or-less on its own.

In a world where I actually had sufficient free time (ha!) I'd love to take the theories behind the numbers and try to figure out if Pazio does have some consistent assumptions in things like this.

[PIXIE DUST]

There is also the thing that it is not really linear... Its more like a bell curve.

At extremely low AC's, a single point change is pretty much negligible (a the most extreme the chances of being hit are the same). When you start getting to the mid range AC levels, a single point can very much change chances of being hit. Then you get to the other extreme and suddenly a 60 AC is no different than a 100 AC. Even beyond that, at certain levels the difference between needing a 19 to be rolled to be hit is not much different than only getting hit on 20s.