The Overkill Damage Fallacy

Jonathan Tweet · Jun 11, 2019

Another factor to consider is that reliability and consistency matter a lot to whichever side has the advantage. The underdog in a fight benefits from increased randomness, while the favored group benefits from a lack of surprises. In RPGs, the PCs are almost always the favored side. A disadvantage of a single large attack compared to two small attacks is that the single attacks are more likely to produce an unlikely result. When balancing 3E, we tried to take this factor into account, but it's a slippery one.

Ovinomancer · Jun 11, 2019

FrogReaver said:
Weighted average of the round you are killing the enemy on.

If im looking for that then I literally can’t use cumulative percentages as I need to know the exact chance I have of killing an enemy on exactly round X so I can use that value as my weight for round X

on a phone so lengthy math discussion is hard. Hopefully that helps

Dammit! My thumb keeps hitting the laugh button. Honestly, this is embarrasing.

Yeesss... but is this a useful question? I mean, further to the point that I'm pretty sure you can't sum the odds of "kill this round" and have it turn into "average rounds to kill." Only pretty sure because I haven't worked out if this is one of thise things where it happens to simplify out that way. I don't see it, but I could be missing something.

But, back to "average rounds to first kill" being a useful question to illuminate overkill. I don't think it is, because it doesn't address overkill but instead shows that single big hit is swingier (ie flatter) than 2 little hits. This is somewhat obvious if you think about it as PC1 has a much flatter arc over the rounds due to only one variable per round while PC2 is curvier due to doubling the number of tests. This is like showing that a d12 is likelier to roll a 10 or higher than 2d6. You've just shown that, over time, you're more likely to roll a 10 or higher on d12 than on 2d6 (1/4 vs 1/12).

I think that an overkill analysis might be better approached as an effective average damage. Say you do damage x, with probability p of hitting. Your target has hp y. The number of hits needed to kill is the ceiling of (y/x). Your effective maximum damage average is then y/(ceil(y/x)), which I'll call z. Note that if x=y, this is just x. Then, your average effective damage is p(z).

Example. You do 10 damage (x). You hit 60% of the time against this foe (p). The foe has 8 hp

. Ceil(y/x) = ceil(8/10) = 1. Z is then 8/1, or 8. Ave effective damage is .6(8) or 4.8 and not .6(10) = 6. 8/4.8 is average 1.67 rounds to kill.

Going the other way, if y is 12, ceil(y/x)=2. Z is now 6, and average effective damage is 3.6. 12/3.6 is 3.33 average rounds to kill.

FrogReaver · Jun 11, 2019

[MENTION=16814]Ovinomancer[/MENTION]

im still on phone

the method to obtain a weighted average is simple

Calculate
(chance X happens) * X
sum each value

whala you have a weighted average that tells you the average oh whatever X is.

If x is damage the. It’s average damage. If x is kill round then x is average kill round.

I Shouldn’t have to spend 3+ posts wxplainjng how weighted average works and that what I calculated was a weighted average.

FrogReaver · Jun 12, 2019

Ovinomancer said:
Sorry, was meant to be XP.

Um, not sure where you're going here. Weighted average of what? Depends on what question you're asking. I don't think "chance to kill this round only" does a good job of illuminating your premise. Actually, I don't think this entire exercise illuminates your title premise at all, although it is interesting and different from previous looks at damage per round.

The OP stated:
1. Mechanic X exists (where mechanic X is that one attack PC's with equal DPR kill some enemies faster on average than those with multiple attacks)
2. There is the proof mechanic X exists (it was provided in the OP)
3. Because Mechanic X exists, overkill damage isn't the only thing to consider.
4. The fallacy is not that overkill damage exists or doesn't. It most certainly does. The fallacy is the focus on overkill damage.

FrogReaver · Jun 12, 2019

Ovinomancer said:
Dammit! My thumb keeps hitting the laugh button. Honestly, this is embarrasing.

Yeesss... but is this a useful question? I mean, further to the point that I'm pretty sure you can't sum the odds of "kill this round" and have it turn into "average rounds to kill." Only pretty sure because I haven't worked out if this is one of thise things where it happens to simplify out that way. I don't see it, but I could be missing something.

I've already commented on this part. Please let me know if something with the calculation still doesn't make sense to you.

But, back to "average rounds to first kill" being a useful question to illuminate overkill. I don't think it is, because it doesn't address overkill but instead shows that single big hit is swingier (ie flatter) than 2 little hits.

Of course an example showing overkill wasn't being cared about in my OP. My argument was that a mechanic other than overkill exists and so there shouldn't be so much focus on overkill (at least until it can reasonably be quantified along with quantifying this other mechanic).

This is somewhat obvious if you think about it as PC1 has a much flatter arc over the rounds due to only one variable per round while PC2 is curvier due to doubling the number of tests. This is like showing that a d12 is likelier to roll a 10 or higher than 2d6. You've just shown that, over time, you're more likely to roll a 10 or higher on d12 than on 2d6 (1/4 vs 1/12).

The important part of my finding was not obvious. I had expected, along with just about everyone else, that the average rounds it would take to kill an enemy would be the same for a 1 attack PC and a 2 attack PC when their overkill damage was the same. That wasn't the case. That such a mechanism exists is why I'm disputing the relevance of overkill damage.

I think that an overkill analysis might be better approached as an effective average damage. Say you do damage x, with probability p of hitting. Your target has hp y. The number of hits needed to kill is the ceiling of (y/x). Your effective maximum damage average is then y/(ceil(y/x)), which I'll call z. Note that if x=y, this is just x. Then, your average effective damage is p(z).

Example. You do 10 damage (x). You hit 60% of the time against this foe (p). The foe has 8 hp . Ceil(y/x) = ceil(8/10) = 1. Z is then 8/1, or 8. Ave effective damage is .6(8) or 4.8 and not .6(10) = 6. 8/4.8 is average 1.67 rounds to kill.

Going the other way, if y is 12, ceil(y/x)=2. Z is now 6, and average effective damage is 3.6. 12/3.6 is 3.33 average rounds to kill.

This is interesting and is sufficient for a 1 attack doing exactly D damage. The method fails for a multi attack PC.

FrogReaver · Jun 12, 2019

Ovinomancer said:
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.

I did not. As already mentioned, chance to kill by round X doesn't help with computing average rounds to kill. This figure is the more important one for evaluating the 2 PC's.

I account for the killing being done on round 1 in my probability for round 1. I account for the killing being done on round 2 in my probability for round 2. I am simply not interested in the chance the enemy is dead by round X. I'm not sure why you think looking at cumulative probabilities instead of exact value probabilities is better.

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.

No problem with your math. It's the application. Mine shows what I claim it shows. Yours does as well. Neither of us did our calculations wrong but what I calculated was more useful than what you calculated. Average rounds to kill is a much more useful stat than a big cumulative probability distribution. It provides a summarization of the information that is easy for everyone to compare.

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%

Tell me the enemy hp, chance to hit and damage and I'll compute the average rounds to win the fight. Then we can actually compare those 2 characters in those 2 scenarios to see who is killing faster.

By the way, I check myself by ensuring all my by round percentages add to 1. It's an easy check to perform.

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?

That's to be expected. It still doesn't quantify who wins the fight faster.

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.

No issues there. I'm hoping to find a rather simple formula I can repeatedly use to answer the question overall. It's a work in progress but i'm making good strides.

Ovinomancer · Jun 12, 2019

FrogReaver said:
[MENTION=16814]Ovinomancer[/MENTION]

im still on phone

the method to obtain a weighted average is simple

Calculate
(chance X happens) * X
sum each value

whala you have a weighted average that tells you the average oh whatever X is.

If x is damage the. It’s average damage. If x is kill round then x is average kill round.

I Shouldn’t have to spend 3+ posts wxplainjng how weighted average works and that what I calculated was a weighted average.

Yes, I know what a weighted average is, I'm questioning whether or not what you did actually says what you think it does. I'm still not sure. And, no, I expect that you spending 1 post explaining your methods and intent is not spending 3. It's just the one.

Another thing that's interesting is just how much your analysis is dependent on the numbers chosen. For instance, if you go over 8 hps with your numbers, PC2 steps ahead of PC1 across the board. At least until you get to 13 hitpoints, where PC 1 regains the lead. It swaps again at 17. Other numbers cause very different patterns. This is, of course, all using your method of determining average rounds to first kill. I still don't think this is a useful metric, but I'm willing to be persuaded. I think it leaves a lot of information out to look at a very narrow probability curve that isn't very informative. I prefer that 'chance to have killed by x round'. This counts all cases that result in killing and not just those that occur on the given round.

I did build an extended calculator for your calculation (the weighted average). It was a bit of a challenge, but I was bored for a bit at work and fiddled with a crude way to represent the probability trees in excel. It makes heavy use of the binom.dist function to both find the odds of r hits in N attacks per round and also to determine the current state of a round (number of previous hits that didn't result in death earlier). It recreates your distributions from the OP and I verified it at another point (9 hp, to be precise). It allows for PC1 to need to make up to 3 attacks to kill and PC2 up to 5. I can expand it further, but don't see the need. It allows you to alter the target hp, damage dealt, and chance to hit. The number of attacks per round are fixed at 1 and 2, respectively, because the needed alterations for accounting for variable attacks is much larger that I wanted to deal with at the time. Although, now that I think about it, it might not be that hard? Hmm.

I also have a sheet for my preferred calculation, which shows the same patterns but has different numbers. It's also less cumbersome that your method. Not a lot, but less.

Ovinomancer · Jun 12, 2019

FrogReaver said:
The OP stated:
1. Mechanic X exists (where mechanic X is that one attack PC's with equal DPR kill some enemies faster on average than those with multiple attacks)
2. There is the proof mechanic X exists (it was provided in the OP)
3. Because Mechanic X exists, overkill damage isn't the only thing to consider.
4. The fallacy is not that overkill damage exists or doesn't. It most certainly does. The fallacy is the focus on overkill damage.

Okay, but you didn't show this. You shows something that wasn't about overkill damage and then claimed that since this not-overkill thing exists, asking about overkill (focusing on overkill?) is a fallacy. That's not how that works.

Your thing can be true AND overkill can still be an issue worth investigating. Your effort, while interesting in a technical sense, don't illuminate overkill at all. And, all your method really does it look at the difference between a flat probability curve and a less flat probability curve. You say above in one of your examples that your method is more useful than mine because it contains more information (I assume you mean the weighted average) but I challenge that proposition. You're excluding information, namely the chance that a kill event occurred earlier, to focus on a specific round's chances. This isn't more useful unless I'm betting on whether or not I kill something on round 3. If you're betting on whether or not you kill something before round 3, my method contains more information.

But this really isn't an who has more information thing -- I don't care outside of pointing out the math. I find your quantification interesting enough to have explored it more fully. However, in the end, if you're interested in who's more likely to kill first, there are other methods that are less cumbersome that provide that information.

As for overkill, you addressed my suggestion above and said it fails for multiattacking characters. It does not. Switch "rounds to kill" to "attacks to kill" then divide the result by number of attacks per round to determine "rounds to kill".

FrogReaver · Jun 12, 2019

Ovinomancer said:
You're excluding information, namely the chance that a kill event occurred earlier, to focus on a specific round's chances.

It's statements like the one above that make me question whether you really understand what a weighted average is.

Weighted averages by definition does take into account EVERYTHING. That's why I'm very puzzled when you make statements like these. My average rounds to kill takes into account rounds 1 to infinity. Your chance to kill by round X only takes into account rounds 1 to X.

FrogReaver · Jun 12, 2019

Ovinomancer said:
As for overkill, you addressed my suggestion above and said it fails for multiattacking characters. It does not. Switch "rounds to kill" to "attacks to kill" then divide the result by number of attacks per round to determine "rounds to kill".

I tried that before I posted. It gives an incorrect value of 1.667 rounds. The actual number of rounds is below: