D&D 5E Calculating Overkill Damage

[guachi]

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...

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.

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...)

 

[FrogReaver]

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.

All I can tell you is that it depends on what you are fighting and numerous other factors.

I can say that DPR accurately predicts that over the range of all possible situations that the 120 damage single attack character will on average perform better than the 10 damage, 10 attack character. Will that be true in every situation? Nope. Just like it isn't true in every situation that a 13 damage, 10 attack character would be better than the 120 damage single attack character.

 

[FrogReaver]

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.

If we simply change the enemies hp to 110 then your theory falls flat on it's face.
3 round kill by the 40 damage character. (10 overkill damage)
4 round kill by the 16 damage 2 attack character. (2 overkill damage)

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...)

Claim was that DPR is better and more useful without trying to factor overkill damage from it than it is with trying to factor overkill out of it. The goblin example and other 15 and 17 hp critter examples are best handled by probabilities as even you reverted to using those instead of your overkill damage calculations when actually trying to determine what was better.

 

[m00]

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

 

[Sadras]

Mathturbation Thread

Personally like @FrogReaver I do not see the value in calculating overkill damage. It does not provide meaningful information to the user, and the information this exercise does provide is already apparent to the user.

 

[Harzel]

[MENTION=6785802]guachi[/MENTION] Before having an opinion, I'd like to be sure (as much as possible) what it is I'm having an opinion on. So even after reading the thread so far, I still have two basic questions.

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?

 

[CapnZapp]

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.

My only contribution is that GWM/SS is voluntary, and that this quality is often overlooked.

For instance, if you are about to attack a goblin or wounded orc, you should probably not activate -5/+10.

This is a not-insignificant quality of the feats that I definitely consider to contribute to their OP-ness.

Z

Sent from my C6603 using EN World mobile app

 

[jasper]

Rule 37 There is no OverKill . There is open fire and time to reload.

 

[FrogReaver]

Your math is off. There is no chance to kill a 10 hp enemy in 1 attack. You listed it as a value above 0. Never mind you are factoring in cross.

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

 

[Aenorgreen]

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.

It depends entirely on the situation. Is the target a single creature with 100 hp, or are there 10 creatures with 10 hp each. If the single creature than one attack is better as you have your hit percentage chance to kill it. It is nearly impossible with the 10 attacks. Conversely with the multiple creatures it would be better to have the more attacks. It is like trying to compare a fireball spell to a disintegrate. They both have their uses. The entire question's premise of one being "better" doesn't make sense to me. It depends on many conditions such as target hp, AC, ad/dis, the target's expected damage to you, etc. There is no single best.