D&D 5E Calculating Overkill Damage

FrogReaver said:

I think the thought is that he can remove overkill damage from DPR calculations to make them more accurate (it doesn't but I think that was his original belief / premise). It's probably best to let him answer for himself though...

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guachi said:

Yes. But to my knowledge no one has attempted to apply this to damage per attack for better comparisons between abilities. Is doing 120 damage in one attack better than doing 10 damage in each of 10 attacks? Maybe.

Can you tell me if it's better? I'll wait for an answer and your reasoning, with math, as to why.

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guachi said:

I used two examples (goblins and a random monster with 93 HP) that shows that DPR alone, like you assert, is not a better judge of actual effectiveness. My calculations match exactly with reality. I have yet to see you use any math of any kind to make your point.

I'll repeat this example again with a little more information:
40 damage per attack. 1 attack per round. Always hits.
16 damage per attack. 2 attacks per round. Always hits.
93 HP target.

You claim that because 40 > 16x2 that the first one is better.
My math shows that, if you account for overkill damage, they both have exactly the same DPR, 31.
Reality shows that they both kill the creature in 3 rounds for an average DPR of (drum roll) 31.

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It's almost as if my formula has some value and matches with reality. Almost. as. if.

I'm also still waiting for any support that DPR alone is all you need in telling me what form of attack is better at killing a goblin. You've made an assertion. Show some math to back it up. If DPR alone is better, surely you can show me how using -5/+10 gets the goblin dead faster. (it doesn't but I think that was your original belief / premise. It's probably best to let you answer for yourself though...)

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1. What is the calculation you are doing? I will take on faith that you have accurately calculated the probabilities for each possible damage value since I understand what those values mean, and, in principle, how to calculate them. So, starting from there, I'd like to understand the rest of the calculation to get to the quantities you call "Average Damage" and "(Average) Overkill Damage". Given those, I understand how to get "Modified Damage" and "Damage Reduction". Note: I would have thought "Average Damage" would be straightforward, but I cannot reproduce your value for the example [Sword & Board; DEX/STR 20; duelist style; d8 weapon; 65% hit chance]; possibly I am making a simple mistake, though (I get 7.48 instead of 7.70).
2. ​What exactly is your claim about how the values that you are calculating should be generally applied?

guachi said:

When we do our white room calculations for damage I've never seen any calculations done to account for overkill damage. That is, damage done over and above what is necessary to bring a foe to 0 HP.

Generally, we assume our foe has infinite HP. This is, of course, not the case but it certainly simplifies our calculations.

I have, however, attempted to calculate it and give myself a robust spreadsheet to if I need to change parameters.

I chose a GWF GWM greatsword wielder as my first victim. Because there's nothing like starting with the hardest one first. First up is finding the chances of doing each potential value in your damage range. This is harder for a greatsword as it deals 2d6 damage on a normal hit and, thus, isn't linear Further, and the real pain, is accounting for the rerolls. But I did it! Despite a range (before static bonuses) of 2-24 about 60% of your values will range from 7-10.

Assumptions:
STR 20
hit chance 65%
hit chance post -5/+10 usage 40%
crit chance 5%always use -5/+10
target HP: 100 (roughly CR 5)

All of these are relatively easy to change so if anyone wants different results, I can give them to you.

Foe is a target dummy with a randomly determined HP at time of swing ranging from 1 to its maximum value (in this case 100). This is an attempt to simulate that your target can have variable HP when you attack it and aren't intentionally targeting a foe with high or low HP.

Given the above, our GWM, GWF, STR 20 greatsword user will average 1.16 points of overkill damage per swing, accounting for misses (which, obviously, have no overkill damage). Our attacker averages 9.75 damage per swing so his average damage per swing is reduced by 11.9% to 8.59.

When I get a free minute at work I'll see see how things change simply by removing the usage of the -5/+10 portion of the feat.

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m00 said:

Count me in the group that thinks this is a useful white-room exercise.

Here is how I think about this problem: There is a combatant fighting an infinite supply of identical enemies, each with H hit points. The probability that the combatant hits an enemy with an attack is p; when the combatant hits, they cause XdN + b damage. We're not interested in how long the combatant will last before going down, so the enemies do not fight back. As soon as the combatant kills an enemy, their next attack can effectively be targeted at the next enemy (i.e., no wasted attacks).

Imagine the "state" of the system is the amount of damage on the current enemy (from 0 to H). Each attack moves the "state" up according to the combatant's probability of hitting and how much damage they cause. We can make a table where the rows correspond to the current state of the system, and the columns correspond to the next state of the system; the ith row and jth column is the probability that the system goes from state i to state j during the current attack. The probabilities of going between states can be encoded to include things like critical hits, whether the attacker chooses to use GWF, etc. As soon as the state reaches H, it starts over again from state 0 (no damage). In this context, "overkill damage" just means that the system can never move beyond state H, even if the damage we've caused indicated we should have (i.e., even we did enough damage to go from 5 to 10, but H = 8, then our new state only goes to 8, so we incurred 2 overkill damage).

The advantage of this setup is that we've actually defined a finite state-space Markov chain, which lets us exploit a lot of powerful math. For example, we can compute the expected amount of time that the system takes to go from state H (the enemy just died) back to state H (the next enemy died), which is the inverse of the rate at which the combatant kills enemies*: if we are considering two combatants with different weapons/fighting styles, we may compare the ratio of their expected amounts of damage (against an enemy with infinite HP) to the ratio of their "kill rate", to get a sense of how much damage is being wasted/how efficiently they are killing enemies.

So, here's how this plays out with some real numbers. Here's our enemy: a 10 HP sack-of-HP that the combatant hits 60% of the time (H = 10, p = 0.6). Our combatant (attacker 1) hits for 1d8 damage (expected damage 4.5). Here's what our damage table looks like (I've included critical hits, and rounded for clarity):

	0	1	2	3	4	5	6	7	8	9	10
0	0.400	0.069	0.070	0.070	0.071	0.072	0.073	0.073	0.074	0.006	0.022
1	0.000	0.400	0.069	0.070	0.070	0.071	0.072	0.073	0.073	0.074	0.028
2	0.000	0.000	0.400	0.069	0.070	0.070	0.071	0.072	0.073	0.073	0.102
3	0.000	0.000	0.000	0.400	0.069	0.070	0.070	0.071	0.072	0.073	0.176
4	0.000	0.000	0.000	0.000	0.400	0.069	0.070	0.070	0.071	0.072	0.248
5	0.000	0.000	0.000	0.000	0.000	0.400	0.069	0.070	0.070	0.071	0.320
6	0.000	0.000	0.000	0.000	0.000	0.000	0.400	0.069	0.070	0.070	0.391
7	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.400	0.069	0.070	0.462
8	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.400	0.069	0.531
9	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.400	0.600
10	0.400	0.069	0.070	0.070	0.071	0.072	0.073	0.073	0.074	0.006	0.022

So, for example, attacker 1 misses 40% of the time, and causes 1 damage 6.9% of the time (including critical hits). Attacker 1 kills their current enemy in one attack 2.2% of the time!

The "kill rate" for this matrix is 0.229 enemies per round (remember the expected time per kill is the inverse of this rate). If we consider a new attacker (attacker 2) that causes 2d6 damage, the kill rate becomes 0.323 enemies per round. The ratio of kill rates is 0.708 (attacker one kills opponents 71% times as fast as attacker two). However, the ratio of their expected amounts of damage is 0.643 (4.5 / 7), which is obviously less than 0.708; attacker 2 is less efficient than attacker 1! The difference of these ratios is a measure of efficiency.

There are some obvious special cases. For example, when H = 1, every attack that hits kills an enemy, so both attacker 1 and 2 have identical kill rates (attacker 2 is wasting a lot of damage!). As H goes to infinity, the amount of damage attacker 2 wastes decreases, so the ratio of kill rates converges to the ratio of the expected damage.

There are obviously other ways to approach this problem, but I think this one could be useful!

* these quantities can be computed by solving for the "stationary distribution" of the rate matrix

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guachi said:

Yes. But to my knowledge no one has attempted to apply this to damage per attack for better comparisons between abilities. Is doing 120 damage in one attack better than doing 10 damage in each of 10 attacks? Maybe.

Can you tell me if it's better? I'll wait for an answer and your reasoning, with math, as to why.

Click to expand...