#### Ovinomancer

##### No flips for you!

Overkill Damage is often brought up in DPR discussions as a theoretical offset to the damage that characters with fewer harder hitting attacks are doing. The basic concept seems reasonable on the surface but what never gets accounted for is that a character with a single attack can kill enemies faster on average than a character with 2 attacks even when they do the same DPR.

Proof:

Enemy has 5 hp

PC1: 60% chance to hit. 8 Damage (no variation). 1 Attack

PC2: 60% chance to hit. 4 Damage (no variation). 2 Attacks

PC 1:

View attachment 106907

PC 2

View attachment 106908

As you can clearly see above, the Character that makes a single attack actually kills the 5 hp enemy faster on average than the PC making 2 attacks. (In the example provided both PC's have the same overkill damage whenever this particular enemy dies). So why is PC 1 killing this enemy faster? Because the chance to kill on rounds X distribution.

Why do we consider PC 1's overkill damage to be higher than PC 2's (given that it was the same in the example provided)? Because the other half of enemies PC 1 fights will leave him doing more overkill damage. Thus, on average PC 1 does more overkill damage than PC 2

On the enemies where PC 1's overkill damage is higher, what causes that? Basically it happens because the hp value fell is such a place that PC 1 needed X hits to kill it on average but PC 2 only needed 2X-1 attacks. This occurs any time an enemies hp divided by 4 is an even number. When the enemies hp divided by 4 is an odd number PC 2 needs 2X attacks.

Why do I discount overkill damage? Because, the most important factor is how fast the enemy dies. If the enemy dies faster then you get to start applying your damage to the next enemy that much faster. Since equal DPR characters kill different enemies faster or slower on average (and their chance to kill an enemy on round X distributions are never the same) then looking at overkill ignores the most important factor. In fact, that's why I call overkill damage a fallacy. It's a nearly meaningless stat in the grand scheme of things that some individuals regard as providing a significant insight in analysis. It can't do what they want it to do because equal DPR characters don't kill enemies at the same rates. Oftentimes the fewer attacking higher damage character will on average kill enemies faster.

What more work needs done in this area?

Adjust the results for different chances to hit. Different numbers of attacks. Probably most importantly would be to adjust for variable damage dice as opposed to a specific damage value for every hit. It would also be helpful to compare a few enemies that PC 1 kills faster and some PC 2 kills faster and see if the magnitudes and relative values of such faster average kills are much different.

It took me a bit to figure out what you did to get it, but I did recreate your numbers. First, let me explain what I understand you did so we're on the same page.

The above charts look only at the chance that the PC kills the target in that round. For the single big attack, it's straighforward enough. You kill it or you don't. The cases are round 1 hit, round 1 miss round 2 hit, round 1&2 miss round 3 hit, and so on.

The second case has more cases to consider, and these expand rapidly per round. You still kept the analysis at the single point where PC 2 kills the creature in that round. So, on round 2, you consider the case for round 1 - 2 misses and round 2 - 2 hits, round 1 - 1 hit and round 2 - 1 hit and 1 hit round 2, and, finally, round 1 - 1 hit and round 2 - 2 hits. And so on.

However, in both cases, I think you've made an error in analysis. Each round should have the cumulative chance that the creature is killed in that round or in any previous round, not just the chance that the target is killed in that round. You're essentially ignoring all the cases where the killing is done already. But, let me drop some illustrations first.

Here is 3 rounds of expanded hit cases and associated odds for each for PC 1:

And here is the same for PC 2:

You can see that the column marked % Kill the round shows the numbers you presented. The column marked % Kill Overall is the total odds at the end of each round that the PC is standing over a dead body. As you can see, PC 1 is ahead in rounds 1 and 2, but starts to lag behind in round 3. This continues, although it remains close, because the number of successful kill cases for the 2 attack character grows faster. This is why there is a larger delta in round 1, where PC 2 has 1 kill case out of 4 and PC 1 has 1 kill case out of 2, and almost none in 3 and four due to the similarity in kill cases. Past 3 rounds, the % Kill Overall is essentially 1 for both characters.

So, overall, your average rounds to first kill seems a bit wonky, as it's ignoring the odds of a previous round resulting in a kill already. I'm not sure that your number tells us anything, and, honestly, I'm not sure you can just add up your percentages to get to an average time to first kill. The extended results above, though, do suggest that the single big hitter will kill first, and possible more often. I crunched some more numbers to look at this in a case where there's always at least one more foe to attack. So, over 3 rounds:

PC 1 kills none 6% of the time, 1 29% of the time, 2 43% of the time, and 3 22% of the time. This is an 'at least 1' of 94% and an at least 2 of 65%.

PC 2 kills none 4% of the time, 1 41% of the time, 2 50% of the time, and 3 5% of the time. This is an 'at least 1' of 96% and an at least 2 of 55%

It's very interesting, I'd say. The single big attack spends more time at the ends than the multiple attacks and less time in the middle. What's also interesting is that the average damage output per round for both PCs remains constant and equal -- as you might expect. This was one of my primary checks for my maths - did I maintain expected average damage per round?

And, all of this is using your numbers. Some have advanced this use case is maximized for the big hitter based on target hp, but the analysis is stable for all x less than 9 and greater than 4. Until you get into the hp that PC 1 can't kill with 1 hit or into hp that PC 2 can kill with one hit, the numbers don't change. However, if you use different numbers, the analysis will change. This assumes that big hit precises doubles little hit, and that's not a valid assumption, nor is it valid that big hit will always kill and little will not. Different cases may yield very different outcomes. It should be possible to build an expanded case chart that could handle very different numbers, but it would take more work than I've done so far (a good bit) so unless there's a lot of interest, I'm not going to try.