D&D 5E Quantifying AOE impact

[FrogReaver]

And interact with initiative bonuses and so on. Probably not worth trying to model.

I think it would be fair to make a rule of thumb - something like if an enemy could be killed in half a round it doesn't get it's attack, if it will take over half a round to kill the enemy then it get's it's attack. Not perfect but get's a bit closer to reality. It seems like a good estimate would be that you are actually saving an additional M/2 attacks - both in the AOE case and in the non-AOE case.

 

[Esker]

Using the same notation, if the party does D damage per round and there are M monsters with KD hit points each, then we expect the combat to last about MK rounds. A monster who dies in round R gets R turns. As a reference point, if all monsters are worn down evenly and all die in the last round, they will collectively have gotten M^2*K turns in total.

In general, doing enough extra damage to kill one monster a round earlier than the party would have otherwise has a follow-on effect; since everyone can then move on to the next monster a round earlier than they would have otherwise, and so if you can do a burst of extra damage in the first round equal to PD, then that should have roughly a PD/KD = P/K chance of moving up any given monster's expiration time by a round, so on average you should save about MP/K monster turns.

This one is more back-of-the-envelope, and it's late so I'm not sure I'm reasoning super clearly any more, but does that strike you as a reasonable approximation?

I'm doubting it at the moment, since it seems to entail some backwards-seeming results, so I'm wondering if I screwed up something obvious, or have a variable misplaced somewhere.

 

[FrogReaver]

Using the same notation, if the party does D damage per round and there are M monsters with KD hit points each, then we expect the combat to last about MK rounds. A monster who dies in round R gets R turns. As a reference point, if all monsters are worn down evenly and all die in the last round, they will collectively have gotten M^2*K turns in total.

In general, doing enough extra damage to kill one monster a round earlier than the party would have otherwise has a follow-on effect; since everyone can then move on to the next monster a round earlier than they would have otherwise, and so if you can do a burst of extra damage in the first round equal to PD, then that should have roughly a PD/KD = P/K chance of moving up any given monster's expiration time by a round, so on average you should save about MP/K monster turns.

This one is more back-of-the-envelope, and it's late so I'm not sure I'm reasoning super clearly any more, but does that strike you as a reasonable approximation?

I'm doubting it at the moment, since it seems to entail some backwards-seeming results, so I'm wondering if I screwed up something obvious, or have a variable misplaced somewhere.

P/K represents the number of rounds you are initially skipping ahead. This get's multiplied by M because each subsequent enemy you kill also gets moved up by P/K rounds. So yes, MP/K monster turns saved in this non-aoe extra damage case. Or rather M' = monsters remaining. So M'P/K monster turns saved. This should generalize it to extra damage on any turn as opposed to just the first.

 

[FrogReaver]

P/K represents the number of rounds you are initially skipping ahead. This get's multiplied by M because each subsequent enemy you kill also gets moved up by P/K rounds. So yes, MP/K monster turns saved in this non-aoe extra damage case. Or rather M' = monsters remaining. So M'P/K monster turns saved. This should generalize it to extra damage on any turn as opposed to just the first.

@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?

P(M^2)/K

And possibly to have this make more sense

Let D be the extra damage we are doing. Let H be the HP of an enemy you are facing. Let M' be the number of enemies remaining. Then:

D/H * (M^2)

is the number of enemy turns saved by dealing D extra damage to each enemy on the current turn.

 

[FrogReaver]

I suppose the next interesting question is:

If there are M enemies and you do D extra single target damage then how many turns will that save?

Once we have this (assuming our work is correct) then we can compute what values of single target damage are comparable to AOE damage.

 

[DND_Reborn]

An Impact of AoE spells formula (taking a different approach of course ):

first draft/version

A = AoE area of spell (in sq. ft.)
D = AoE spell average damage
p = density of enemy population (1 = no empty space between enemies)
n = number of enemies present
S = AoE spell save DC
b = enemy saving throw bonus
e = space enemy occupies (in sq. ft.)
h = average enemy hit points

Briefly, it makes sense Impact increases as

• AoE area increases
• average damage increases
• enemy population or space density increases
• the number of potential enemies increases
• save DC increases

Impact decreases when

• enemy save bonus increases
• enemy space/size increases
• enemy hit points increase

It works well IMO for the different scenarios I've tried. This is more meant to gauge the effectiveness or "impact" of an AoE spell on a group as a whole. I am still working on what results would quantify a "good, great, poor, etc." rating.

Examples:
A Burning Hands (DC14) against 3 orcs with p = 1 results in I = 11.827
A Fireball (DC 14) in the same situation results in I = 224

Looking at those scenarios, BH would damage all three orcs, but unless some were already injured it won't kill any. Given their hp 15 and avg dmg is 10.5, it is a fair to good use of the spell, but not super effective.

FB on the other hand, with even saved damage of 14, is incredibly effective, to the point of overkill. You will mostly like (about 80%) kill 1 or more, and even those that save are severely injured.

The same FB against a single hill giant would roughly have an I = 1.259, damaging but hardly lethal or effective given the hill giants high hit points. The FB would more likely be better used on a group of lower hp foes, as you would expect.

The formula could be adapted to include spell level, spell slots available, and other factors if desired. For instance, if you consider BH is level 1 and FB is level three, using BH three times on the same three orcs would kill them all in a more thorough manner as even average damage saved each time (total 15.75) would kill each orc. The downside is, of course, you would require 3 rounds to cast all 3 BH spells instead of 1 round to cast 1 FB.

I suppose that is it for now. @FrogReaver wants to examine the impact relative to different factors. While @FrogReaver and @Esker are following a different thought process, I figured I would offer an alternative way to look at the problem.

 

[NotAYakk]

Sniff test: Feet radius doesn't matter. It matters how many things you damage. If you are fighting 5 things, a 5 million foot radius isn't not a thousand times better than a 5000 foot radius.

I believe the OP is talking about the question "should I fireball right now, dealing X damage to Y targets, or not", where you assume you may have a better or worse opportunity later in the day.

---

Taking a page from physics, we'll normalize things so our units are better.

Monsters deal 1 monster of damage per period of time. It takes 1 period of time to kill a monster using single target damage.

Your AOE damage is measured in fractions of monsters killed in one period of time.

The monsters in consideration are the monsters damaged by the AOE.

So if you have 10 monsters, it takes 10 rounds to kill them with single target damage.

A 0.1 AOE reduces this to 9 rounds.

The damage X monsters do is then (X)(X+1)/2 -- the Xth triangular number, aka 10+9+...+2+1.

Dealing 0.1 damage to all monsters then saves 0.1 of this; so 0.1 * X(X+1)/2.

Dealing 0.5 extra single target damage saves 0.5 rounds, but it shaves it off the left of the pyramid. So it saves 0.5 * X damage.

Now, the 0.1 damage actually did a total of 0.1 * X damage. The first 0.1 did it to the "primary" target, the rest of the 0.1*(X-1) damage to "secondary" targets.

If we deal 0.1*X damage to one target (or focused), the incoming damage is reduced by 0.1 * X * X rounds. A square!

Now, if our AOE does K damage (as a fraction of monster HP), and there are X monsters, how much single target damage do we need to reduce player taken damage the same amount?

K(X)(X+1)/2 = Y * X
K(X+1)/2 = Y
K + K*(X-1)/2 = Y

This means "AOE damage is worth the damage it does to the main target (K) plus 1/2 of the damage it does to secondary targets (K*(X-1)/2))".

Neat.

 

[FrogReaver]

An Impact of AoE spells formula (taking a different approach of course ):

View attachment 118006
first draft/version

A = AoE area of spell (in sq. ft.)
D = AoE spell average damage
p = density of enemy population (1 = no empty space between enemies)
n = number of enemies present
S = AoE spell save DC
b = enemy saving throw bonus
e = space enemy occupies (in sq. ft.)
h = average enemy hit points

Briefly, it makes sense Impact increases as

• AoE area increases
• average damage increases
• enemy population or space density increases
• the number of potential enemies increases
• save DC increases

Impact decreases when

• enemy save bonus increases
• enemy space/size increases
• enemy hit points increase

It works well IMO for the different scenarios I've tried. This is more meant to gauge the effectiveness or "impact" of an AoE spell on a group as a whole. I am still working on what results would quantify a "good, great, poor, etc." rating.

Examples:
A Burning Hands (DC14) against 3 orcs with p = 1 results in I = 11.827
A Fireball (DC 14) in the same situation results in I = 224

Looking at those scenarios, BH would damage all three orcs, but unless some were already injured it won't kill any. Given their hp 15 and avg dmg is 10.5, it is a fair to good use of the spell, but not super effective.

FB on the other hand, with even saved damage of 14, is incredibly effective, to the point of overkill. You will mostly like (about 80%) kill 1 or more, and even those that save are severely injured.

The same FB against a single hill giant would roughly have an I = 1.259, damaging but hardly lethal or effective given the hill giants high hit points. The FB would more likely be better used on a group of lower hp foes, as you would expect.

The formula could be adapted to include spell level, spell slots available, and other factors if desired. For instance, if you consider BH is level 1 and FB is level three, using BH three times on the same three orcs would kill them all in a more thorough manner as even average damage saved each time (total 15.75) would kill each orc. The downside is, of course, you would require 3 rounds to cast all 3 BH spells instead of 1 round to cast 1 FB.

I suppose that is it for now. @FrogReaver wants to examine the impact relative to different factors. While @FrogReaver and @Esker are following a different thought process, I figured I would offer an alternative way to look at the problem.

I appreciate seeing other directions. My first concern is that I can't replicate your numbers.

For fireball I get:
A = 1256 sq ft
D = 28
p = 1
n = 3
S = 14
b = 1
e = 5
h = 15

That computes to 1125.376. What am I doing differently than you?

 

[DND_Reborn]

I appreciate seeing other directions. My first concern is that I can't replicate your numbers.

For fireball I get:
A = 1256 sq ft
D = 28
p = 1
n = 3
S = 14
b = 1
e = 5
h = 15

That computes to 1125.376. What am I doing differently than you?

Well, like I said I know it wasn't the direction you and @Esker were discussing (which is also an interesting approach) so I am glad you can appreciate this view.

Any way, sorry, I rounded A to 1250 from when I first started and never updated it (my bad). This would produce a slight difference, from my original estimate. (225.0752 vs my posted 224).

You have e = 5, but it is e = 25 (5 x 5) because it is square feet. If you divide your computation by 5, you will get the 225.0752.

Again, unless I look at dozens of scenarios, spells, etc. I wouldn't know how to qualify the values (what is "good"? and so on). But to examine the relationship between variables and AoE impact, I think this shows some of them.

Also, does it hold up that everything contributes equally? Should the equation have something like n-squared instead of just n? Should I use A or sqrt(A)? and so forth.

Well, I will be interested in continue to read the thread.

 

[FrogReaver]

@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?