D&D 5E Quantifying AOE impact

[Esker]

@Esker

Something is off though because the case of M=1 for the AOE should reduce to the above formula. However, it doesn't.

Is it possible that the /2 in the P(M^2)/2 formula should actually be K?

Ah, so in the first derivation, K had represented the ratio between a monster's hitpoints and the AoE damage per target, whereas in the second it became the ratio between a monster's hitpoints and the party's damage.

Let's use the second definition, and redo the first formula, also changing M to be the actual number of monsters, and ignoring the floor and ceiling bits in order to be more general (if a bit more idealized).

So now it takes K turns to kill each monster with no contribution from the caster, and so the first monster will get K turns, the second 2K, etc., for a total of K(1+2+...+M) monster turns in total, which is KM(M+1)/2.

The AoE doing PD damage to each target reduces each monster's HP from KD to KD-PD = (K-P)D, in other words, changing K to K'=K-P. Therefore, the monsters will now get K'M(M+1)/2 = KM(M+1)/2 - PM(M+1)2 turns, and the AoE has saved PM(M+1)/2 turns in total.

A single target burst doing PD damage to one target is shaving off P/K monsters worth of hitpoints, which is P rounds worth. This accelerates every monster's demise therefore by P rounds (I had this wrong before in my late-night delirium; assuming efficient spillover, the chance of pushing a monster's demise back a round doesn't depend on the monster's HP; it depends on the number of additional rounds worth of damage you're supplying), and so we've prevented a total of MP monster-turns. Now, reassuringly, this is a special case of AoE damage if there's only one monster.

The ratio in impact, therefore, between doing PD AoE damage to each of M monsters and doing PD damage to a single target is (M+1)/2; whereas scaling up single target by a factor of M would increase impact proportionally, so AoE damage is (M+1)/2M as efficient as an equivalent amount of single target damage.

This has @NotAYakk's derivation as a (sort of) special case: instead of fixing the rest of the party's baseline damage to one monster's worth of HP, we've allowed monster HP and background party damage to vary separately, with their ratio controlled by the constant K, and expressed the AoE damage in terms of the party's damage per round, rather than in terms of a monster's HP.

 

[NotAYakk]

@Esker

One interesting thing with the equation:

D/H * (M^2)

where
D = AOE Damage
H = Enemy HP
M = Number of enemies


In the game there is a loose relationship between M and H. In general (for a given party level) as H increases M will decrease and vice versa. Inverse Proportionality.

Let K be a constant, then

H = K/M (assuming for now that the proportionality is inversely linear)

That means D * M/K * (M^2) = D(M^3)/K

So I guess the question is, do HP trends tend to go H = X for a solo. H = X/2 for 2 enemies. H = X/3 for 3 enemies. Etc. Or does the game reveal some other trend?

CR~ (HP*DPR)^(2/3)
Sum(CR) is roughly constant.

That isn't quite true, as higher HP/DPR brings higher AC/Accuracy, which also boosts CR. (this is a modest effect however)

But ignoring that...

Monster with D DPR and H HP. Another set of monsters with 1/10 that value.

(D*H)^(2/3) = k (0.1 H*0.1 D)^(2/3)
1=k*0.01^(2/3)
k is 20ish

So 20 monsters with 1/10 the DPR/HP can roughly replace one big monster.

 

[FrogReaver]

Ah, so in the first derivation, K had represented the ratio between a monster's hitpoints and the AoE damage per target, whereas in the second it became the ratio between a monster's hitpoints and the party's damage.

Let's use the second definition, and redo the first formula, also changing M to be the actual number of monsters, and ignoring the floor and ceiling bits in order to be more general (if a bit more idealized).

So now it takes K turns to kill each monster with no contribution from the caster, and so the first monster will get K turns, the second 2K, etc., for a total of K(1+2+...+M) monster turns in total, which is KM(M+1)/2.

The AoE doing PD damage to each target reduces each monster's HP from KD to KD-PD = (K-P)D, in other words, changing K to K'=K-P. Therefore, the monsters will now get K'M(M+1)/2 = KM(M+1)/2 - PM(M+1)2 turns, and the AoE has saved PM(M+1)/2 turns in total.

A single target burst doing PD damage to one target is shaving off P/K monsters worth of hitpoints, which is P rounds worth. This accelerates every monster's demise therefore by P rounds (I had this wrong before in my late-night delirium; assuming efficient spillover, the chance of pushing a monster's demise back a round doesn't depend on the monster's HP; it depends on the number of additional rounds worth of damage you're supplying), and so we've prevented a total of MP monster-turns. Now, reassuringly, this is a special case of AoE damage if there's only one monster.

The ratio in impact, therefore, between doing PD AoE damage to each of M monsters and doing PD damage to a single target is (M+1)/2; whereas scaling up single target by a factor of M would increase impact proportionally, so AoE damage is (M+1)/2M as efficient as an equivalent amount of single target damage.

This has @NotAYakk's derivation as a (sort of) special case: instead of fixing the rest of the party's baseline damage to one monster's worth of HP, we've allowed monster HP and background party damage to vary separately, with their ratio controlled by the constant K, and expressed the AoE damage in terms of the party's damage per round, rather than in terms of a monster's HP.

So monster hp isn't actually a factor? That's unintuitive to me. Or is it that the monster hp factor has just been lumped into the K above?

 

[FrogReaver]

The ratio in impact, therefore, between doing PD AoE damage to each of M monsters and doing PD damage to a single target is (M+1)/2; whereas scaling up single target by a factor of M would increase impact proportionally, so AoE damage is (M+1)/2M as efficient as an equivalent amount of single target damage.

I love this bit. It makes perfect sense to me. A great comparative tool for single target vs multi target abilities!!

Also allows comparisons for divine smites and AOE abilities. And it's simple and elegant to use. While not perfect it will be a decent guide!

If you actually look at what is occuring this means

1 enemy = 1*1*(AoE Dmg) = single target damage for same impact
2 enemies = 2*(3/4)(AoE Dmg) = 1.5(AoE Dmg) = single damage for same impact
3 enemies = 3*(2/3)(AoE Dmg) = 2(AoE Dmg)
4 enemies = 4*(5/8)(AoE Dmg) = 2.5(AoE Dmg)
5 enemies = 5*(6/10)(AoE Dmg) = 3(AoE Dmg)

So essentially AoE Dmg (D) is equivalent to [1+0.5(M-1) ]*D

(***Excluding the case where the AoE kills the enemies outright)

Which means for most situations where you would use an AoE it would require 2 to 3 times the single target damage to have the same impact.

 

[Esker]

So monster hp isn't actually a factor? That's unintuitive to me. Or is it that the monster hp factor has just been lumped into the K above?

Nope; at least not if you measure impact in terms of the absolute number of monster turns saved. You could measure it as a proportion of the baseline number of monster turns, in which case the AoE that hits all of the monsters scales the number of monster turns by a factor of P/K, whereas single target damage scales the number of monster turns by a factor of 2P/(K(M+1)). But the relative efficiency measured this way still doesn't depend on monster HP, since K still cancels in the ratio.

I think the main reason AoEs are better against more smaller monsters is simply that you can usually hit more targets without increasing your resource cost, thereby increasing M. So your DPR goes up. The same can't be said for single target effects. So AoEs are more resource-efficient against smaller monsters because there tend to be more of them, and therefore you get more bang for your spell slot. Which is completely intuitive. But no, it's not the case that the impact per DPR is different.

 

[Esker]

I love this bit. It makes perfect sense to me. A great comparative tool for single target vs multi target abilities!!

Also allows comparisons for divine smites and AOE abilities. And it's simple and elegant to use. While not perfect it will be a decent guide!

If you actually look at what is occuring this means

1 enemy = 1*1*(AoE Dmg) = single target damage for same impact
2 enemies = 2*(3/4)(AoE Dmg) = 1.5(AoE Dmg) = single damage for same impact
3 enemies = 3*(2/3)(AoE Dmg) = 2(AoE Dmg)
4 enemies = 4*(5/8)(AoE Dmg) = 2.5(AoE Dmg)
5 enemies = 5*(6/10)(AoE Dmg) = 3(AoE Dmg)

So essentially AoE Dmg (D) is equivalent to [1+0.5(M-1) ]*D

(***Excluding the case where the AoE kills the enemies outright)

Which means for most situations where you would use an AoE it would require 2 to 3 times the single target damage to have the same impact.

I think the only difference in the case where average damage from the AoE is enough to kill enemies outright is that single target damage from a single spell or attack is capped at one enemy's worth of HP; so it's not that the AoE gets more efficient, it's that single target damage gets less efficient.

If, say, the rest of the party collectively would kill one enemy per round, and your AoE averages one enemy worth of damage to each enemy, then we have P = K = 1, and PM(M+1)/2 = M(M+1)/2, which is to say, on average you prevent all of the enemy's turns. Meanwhile, an equivalent amount of single target damage would mostly be overkill, so we wind up having to reduce MPD to just D, and we just reduce the number of enemy turns by ... MP = M. Which is correct, since we effectively deny each monster one turn by offing the first one before it can act.

 

[Esker]

Which means for most situations where you would use an AoE it would require 2 to 3 times the single target damage to have the same impact.

This isn't usually too hard to achieve, though. If a fireball hits two creatures (which most people would consider a poor use of a fireball), then its single-target-equivalent damage value is essentially 12d6. A Paladin using a 3rd level smite gets 4d8 out of that slot, on top of their regular complement of attack damage, which is maybe 4d6+8 at the level fireball is starting to come up.

Since the Paladin can smite on either attack, they are getting 22*(hit %)+18*(1 - (1 - hit%)^2)+18*0.05+18*(1-hit%)*0.05 with their action and slot (assuming they smite on the first hit, which could be a crit). The crappy use of a fireball, meanwhile, is doing the equivalent of 42 * (fail % + 0.5(1 - fail%)) = 21 * (1 + fail%). If we set fail% = hit% = 0.60, then the paladin is doing 29.6 damage with their action and 3rd level slot, whereas the crappy fireball is doing an adjusted 33.6. Now to be fair to the paladin, they might have missed both attacks and not wind up using the slot, in which case they can use it next round. But it will have lost value by then.

 

[NotAYakk]

Or, secondary target damage is worth 50% of primary.

Actually if you aren't doing a full encounter AOE the srcondary target damage can be worth more. Primary target damage is worth X times as much as the last monster to die damage; when you add that up the average is 0.5 for 2nd through Xth.

But if there are 4 monsters and your AOE hits 2, and the 2nd target is who you drop next, that secondary damage is worth more than 50%.

In a more general situation, often the secondary target is a target of opportunity, and not whom you'd pick as your next target. Even if you do end up dropping them next, it would often be better to have dropped someone else.

Suppose you hit an ideal 50%. Well the "tall" half of a triangle has 3/4 of the area of the triangle; so you are 75% as useful as hitting all the monsters!

In general, hitting the top Z% reduces the area of 1-(1-Z)^2 of the triangle (the area is total damage taken), or 1-(1-2Z+Z^2) or 2Z-Z^2, or a factor of Z(2-Z), compared to an even AOE.

For .5 this is 3/4. For .25 this is 5/16. For .1 this is 19%.

If you had 100 monster and attacked 10, this is 19% as useful as AOEing all of them. AOEing all of them is 50% (well, 50.5%) as good as single target focused damage (of the same total). 50%*19% is 9.5%; so an AOE on 10/100 is 95% as efficient as single target damage would be (assuming you win fight).

In the limit, AOE damage is worth half single target. Halving 2Z-Z^2 then dividing by Z gives us 1-Z/2 as the efficiency of an AOE that hits Z% of monsters compared to focused single target damage.

.1 is 95%
.2 is 90%
.5 is 75%
.8 is 60%
1.0 is 50%

This neglected the already main target AOE target; we can use this for secondary targets.

I'll have to check if we want to use "fraction of available secondary targets" or "fraction of targets" as our weight.

 

[seebs]

Most of the time you aren't fighting horde after horde of 10 hp enemies - and if you are then you already have a good idea of how impactful fireball will be.

The non-trivial question about fireball starts to take place in the range where fireball doesn't outright kill the enemies. That's really the area we have been examining - and coincidentally what you will typically find to be the most common situations you face.

You're missing the point. Most of the time you aren't fighting horde after horde of 10hp enemies. But sometimes you fight a horde of them, or you fight some 10hp enemies next to other things. The spell is situational; if you carefully define the situation you're looking at to be the one it's not designed for, you are not getting any meaningful view of its power. Real situations will also vary in lots of other ways. Maybe the other party members can't focus fire on a single target right now; in that case, assuming that they would gives wrong answers again.

Trying to come up with a generic answer to a tactical question, in the absence of the variety of situations you will actually encounter, produces nonsense results.

 

[FrogReaver]

You're missing the point. Most of the time you aren't fighting horde after horde of 10hp enemies. But sometimes you fight a horde of them, or you fight some 10hp enemies next to other things. The spell is situational; if you carefully define the situation you're looking at to be the one it's not designed for, you are not getting any meaningful view of its power. Real situations will also vary in lots of other ways. Maybe the other party members can't focus fire on a single target right now; in that case, assuming that they would gives wrong answers again.

Trying to come up with a generic answer to a tactical question, in the absence of the variety of situations you will actually encounter, produces nonsense results.

seems we have it defined and solved The question of tactical impact pretty well.

piecewise function
where dmg is over enemy hp and their are many enemies then very high Impact

where dmg is below enemy hp then the equations in this thread are great estimators of impact.

so what exactly are you arguing about?