# The Overkill Damage Fallacy

#### Ovinomancer

##### Explorer
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.
No, weighted averages take into account what you've decided to measure and how you've decided to weight it. If I take the weighted average of the volume of a cat from tip to tail, I haven't said much about the dietary requirements of the cat, although some information towards that may be gleaned. Being able to take a weighted average does not, at all, mean you've successfully measured everything. This is reification -- you've done math and confused the concrete outcome of the math as applying to your assumptions. Nothing in stats will correct your assumptions -- tools will gleefully let you lie to yourself with the absolute certainty of a math equation. Don't confuse "I did math!" for "I did it right!"

For example, if you look at your round two, the information contained is that the creature wasn't killed in round one and here are the odds it is killed in round two. In mine, all of the information is present -- both the odds it was killed in round one and the odds it was killed in round two. These are answering different questions, so missing information isn't necessarily a bad thing, but you need to really look at what you've decided to measure to see if it suits your assumptions. Doing math doesn't fix faulty assumptions.

And, so far, I'm unclear as to what your assumptions are -- you seem to think that because you've shown something that this means that another thing is less valid (the overkill fallacy, as you've termed it). This is like taking the weighted average of a the volume of a cat and declaring that you know it's dietary needs, though -- these things may correlate on some level, but they aren't causative in the way you've presented them. You haven't shown how your weighted average says anything at all about overkill.

#### FrogReaver

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

View attachment 106992
Found the error. I needed to account for half rounds.

#### FrogReaver

##### Explorer
No, weighted averages take into account what you've decided to measure and how you've decided to weight it. If I take the weighted average of the volume of a cat from tip to tail, I haven't said much about the dietary requirements of the cat, although some information towards that may be gleaned. Being able to take a weighted average does not, at all, mean you've successfully measured everything. This is reification -- you've done math and confused the concrete outcome of the math as applying to your assumptions. Nothing in stats will correct your assumptions -- tools will gleefully let you lie to yourself with the absolute certainty of a math equation. Don't confuse "I did math!" for "I did it right!"
You keep saying things like that but it's simply not true. Calculating a weighted average for chance to kill on round (X) results in a value that is the number of rounds to kill. There's no misinterpreting what that means. It means exactly what I'm claiming it means.

For example, if you look at your round two, the information contained is that the creature wasn't killed in round one and here are the odds it is killed in round two. In mine, all of the information is present -- both the odds it was killed in round one and the odds it was killed in round two. These are answering different questions, so missing information isn't necessarily a bad thing, but you need to really look at what you've decided to measure to see if it suits your assumptions. Doing math doesn't fix faulty assumptions.
For comparing 2 different characters chances to kill. The rounds to kill an enemy is a much better metric than presenting someone with 2 probability distributions.

#### Ovinomancer

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

View attachment 106992
Sigh. You can't declare your answer to be the right one by fiat. Explain why you think this is so. Following my own advice, it's because the assumptions are slightly different. I'm assuming at least one more enemy, so the case on a round where there was 1 previous hit and two hits on that round carries the second hit into the new target, This reduces the overall average because I'm not stopping at one enemy so that the difference in distributions matters -- by assuming at least one more bad guy, I've removed the artifact of the different distributions of hit probability. I've applied mine to the continuum, not just the one specific situation. I've also accounted for that third wheel of a third hit.

To go back to an earlier point, have you investigated the region where target hp is from 9-12 in your construction? PC2 is ahead of the game, there. This means that bigger stick isn't always faster to the kill. What do you think this says (not snark)?

#### FrogReaver

##### Explorer
Sigh. You can't declare your answer to be the right one by fiat. Explain why you think this is so. Following my own advice, it's because the assumptions are slightly different. I'm assuming at least one more enemy, so the case on a round where there was 1 previous hit and two hits on that round carries the second hit into the new target, This reduces the overall average because I'm not stopping at one enemy so that the difference in distributions matters -- by assuming at least one more bad guy, I've removed the artifact of the different distributions of hit probability. I've applied mine to the continuum, not just the one specific situation. I've also accounted for that third wheel of a third hit.

To go back to an earlier point, have you investigated the region where target hp is from 9-12 in your construction? PC2 is ahead of the game, there. This means that bigger stick isn't always faster to the kill. What do you think this says (not snark)?
I didn't. I showed my calculations. There was an error in them though. I needed .5 rounds accounted for. That was throwing it off. Now as expected the 1 attack characters and the 2 attack character are killing 5-8 hp enemies at the same rate.

So basically disregard the premise of this post as the math around it was incorrect (stupid .5 rounds).

But I think it's given me some ideas on how to calculate the effect of overkill damage.

#### Ovinomancer

##### Explorer
You keep saying things like that but it's simply not true. Calculating a weighted average for chance to kill on round (X) results in a value that is the number of rounds to kill. There's no misinterpreting what that means. It means exactly what I'm claiming it means.
No, this is the weighted average of the "chance to kill on round X", not the weighted average of the" number of rounds to kill". Your metric shows that a kill is most likely to occur sometime in round 2, because that's the weighted average of the chance to kill across all rounds (assuming the later rounds are essentially zero). You can't shift what you've measured into something new with a weighted average, so this isn't the weighted average of the "number of rounds to kill."

There is a different between 'chance to kill ON round X' and 'chance to kill BY round X'. You show the former, not the latter. Your average is the average chance to kill ON round X. My method shows chance to kill BY round X. Different things.

This is, admittedly, a narrow point, but if you're going to lamblast me for lack of understanding, I feel it's a vital one.

For comparing 2 different characters chances to kill. The rounds to kill an enemy is a much better metric than presenting someone with 2 probability distributions.
Again, not quite what you're showing. So, the argument that yours is more useful without being able to say what mine is useful for shows that your assuming instead of discussing. You show kill ON, I show kill BY. Both seem pretty useful information.

Example: As PC1, what are the odds I will kill the monster ON round 3? Your method is the proper one. Those odds are ~10%.

Also as PC1, what are the odds I will kill the monster BY round 3? My method answers. Those odds are ~94%.

Different things. You don't show which round the monster will be killed on or by, and, honestly, neither do I. We both answer different questions. Playing the game is the only way to answer which round the monster will be killed on.a Don't fall into the trap of reification and assuming that your math says more than it does.

#### Ovinomancer

##### Explorer
I didn't. I showed my calculations. There was an error in them though. I needed .5 rounds accounted for. That was throwing it off. Now as expected the 1 attack characters and the 2 attack character are killing 5-8 hp enemies at the same rate.

So basically disregard the premise of this post as the math around it was incorrect (stupid .5 rounds).

But I think it's given me some ideas on how to calculate the effect of overkill damage.
According to what I've put up, it shows that certain DPRs are equivalent against a particular foe. PC1, for instance, has the same kill ratio for 8 damage as for 5 damage against the 5hp foe. Thus, overkill doesn't hasten kill rate. This will be very specific to the foe, though.

#### FrogReaver

##### Explorer
[MENTION=16814]Ovinomancer[/MENTION], by the way I can account for multiple enemies etc in my formulas. The only thing I can't implement yet is variable damage dice.

My formula is surisingly easy to use. Simply list rounds out. Find first round enemy can be killed and then copy paste my formula in every cell.

#### jgsugden

##### Explorer
You guys don't see the irony here, do you? The posts in this thread absolutely - and without doubt - prove that overkill is highly devastating. The lenth of posts in this thread was enough to kill any joy in topic five pages ago.

#### Ovinomancer

##### Explorer
You guys don't see the irony here, do you? The posts in this thread absolutely - and without doubt - prove that overkill is highly devastating. The lenth of posts in this thread was enough to kill any joy in topic five pages ago.

#### Ovinomancer

##### Explorer
[MENTION=16814]Ovinomancer[/MENTION], by the way I can account for multiple enemies etc in my formulas. The only thing I can't implement yet is variable damage dice.
Cool. Of course you can. My point was that you didn't.

My formula is surisingly easy to use. Simply list rounds out. Find first round enemy can be killed and then copy paste my formula in every cell.
Can you make changes to target hp, PC damage, PC hit chance, PC number of attacks, and also adapt to changes from those to the number of rounds needed to kill? The PC1/PC2 sheets are pretty easy to do if you're limiting the range of possible inputs so that you can hard code things. It's when you have to create the probabilities density functions for variable inputs that it becomes a challenge. For example, for a given round on PC 2 with variable inputs, I needed to figure out:

Number of hits needed to kill (HTK), then,

Chance no hits have yet occurred times chance HTK occurs this round (not a given) PLUS Chance 1 hit has yet occured times chance at least HTK-1 occurs this round PLUS ... PLUS chance HTK-1 hits have occurred yet times chance at least 1 hit occurs this round.

A number of these end up as zeros in my equations depending on the PC numbers for a given scenario, so error correction is also needed.

The formula for determining chance to kill for no previous hits is:

1. iferror(binom.dist(0,#att*(N-1),hitchance,false),0) x iferror(1-(binom.dist(,@att, hitchance, true),0)

This gives the probability mass function (ie, chance that in N trials you have exactly r results) for no successes in all previous rounds (accounting for attacks per round) at the given hit chance, and returns 0 if binom,dist fails due to improper numbers (if it's round 1, then you have 0 trials, and this fails). This is then multiplied by the formula to determine the chance to hit with all attacks, up to the number of attacks in this round. This uses the cumulative probability function to find the cumulative probability for all hit combos up to HTK and subtract from 1 to find the probability for HTK hits. Error correction catches if HTK exceeds current attacks per round and returns 0.

This then happens for each scenarios from 1 hit to HTK-1 hits. I'm currently using 6 iterations (meaning up to 6 hits needed to kill) but can easily C&P to more (the joys of named cells making formula easier and of relative references).

I know, I know, I'm weird. But, I can generate the weighted average of the probability to kill on round X for up to six hits needed to kill for Y attacks per round at P chance to hit. (Turns out damage dealt and target hp are feed in variables to the one that matters -- how many hits to kill.)

#### Garthanos

Focus fire is just so effective under D&D style hp rules, though...
Heck even ignoring hp oddities, attacks against an enemy tend to disrupt their attacks against you... ie you might have better effective armor class against any enemy you are attacking. So someone making broad sweeping attacks with a chance of hitting multiple enemies would be better defended from those enemies too.

Basically enemies not threatened have a significant advantage. So you want to threaten everyone even if your multistrike is itself at a penalty to hit.
Note this does not necessarily take complex mechanics either... if you weren't attacked last round you gain a bonus this round (could be bonus damage if that is easier and you are playing to the bounded accuracy gods - note this would reward both surprise and initiative situations too)

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#### Tony Vargas

Note this does not necessarily take complex mechanics either... if you weren't attacked last round you gain a bonus this round (could be bonus damage if that is easier and you are playing to the bounded accuracy gods - note this would reward both surprise and initiative situations too)
It's a tangent, but, sure: it'd be fairly simple to give a bonus (in 5e, say advantage) to a character who has not been attacked since the end of his last turn and was not threatened at the start of his turn. It'd be a mild counter-incentive against somewhat unrealistic focus fire, and make 'suppressive fire' a thing in D&D's Fantasy Vietnam - heck, we already have recon by fire(ball). ;P

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#### Garthanos

It's a tangent, but, sure: it'd be fairly simple to give a bonus (in 5e, say advantage) to a character who has not been attacked since the end of his last turn and was not threatened at the start of his turn.
That is a wicked amount of bonus might undermine almost completely focus fire temptations

#### FrogReaver

##### Explorer
It's a tangent, but, sure: it'd be fairly simple to give a bonus (in 5e, say advantage) to a character who has not been attacked since the end of his last turn and was not threatened at the start of his turn. It'd be a mild counter-incentive against somewhat unrealistic focus fire, and make 'suppressive fire' a thing in D&D's Fantasy Vietnam - heck, we already have recon by fire(ball). ;P
I rather like this idea.

#### 5ekyu

##### Explorer
I rather like this idea.
Iirc there was a version (perhaps more) of Traveller which had a pretty serious penalty for "took damage since last round."

It changed play quite a bit and frankly in many good ways.

Honestly if one wanted to do do, one could setup feats that would give bigger bonuses *if* not hurt since last turn, or bonus action maneuvers that stretched from turn to turn with big bonuses if you made it.

Heck, maybe even a growing turn by turn focus - that worked like concentration. Damage can cause you to lose focus.

Lots of potential there.