Help with Maths?

[Mark Chance]

Let's say a D&D-esque monster attacks at +6/+6 and does 1d8 points of damage per attack, but that I want to have the monster attack only once, end up with same average damage per round, and then just describe the results as a flurry of claws, bites, whatever. What's the easiest way to convert multiple attack routines into a single attack bonus and damage roll? Or is there an easiest way?

 

[Janx]

Let's say a D&D-esque monster attacks at +6/+6 and does 1d8 points of damage per attack, but that I want to have the monster attack only once, end up with same average damage per round, and then just describe the results as a flurry of claws, bites, whatever. What's the easiest way to convert multiple attack routines into a single attack bonus and damage roll? Or is there an easiest way?

Interesting problem. The basic approach is to calculate the outer bounds (1-16 damage possible on hit(s)), and calculate the probable damage. Then compute a single attack that matches that signature.

1d8 is 4.5 damage per round (take die size, divide in half, add .5 to get average die result, then multiply by quantity).

Like calculating projected lottery winnings, you take the % chance of success times the "winnings". Each plus one is 5%, so +6 is 30%.

30% of 4.5 damage is 1.35

Since there's 2 attacks, that's 2x1.35=2.7 average damage per round

It's kinda tricky matching 1-16 damage (the 1 because only one attack might hit). If we accept some imprecision, 2d8 is 2-16 and is close enough for government work.

2d8 average result is 9 (i.e. 4.5x2), now we just need to figure out what to-hit gets us to 2.7 average per round. For that, we just do the reverse math we did for % lottery winnings (2.7/9=30%

Which is the same +6 to hit as a single attack for 2d8 damage.

I can't say mathologically that my logic is perfect, but that's my basic approach to get a good enough answer.

 

[Mark Chance]

I don't think that makes sense to me, which means I might not be phrasing my question clearly. I was with you up until here:

Since there's 2 attacks, that's 2x1.35=2.7 average damage per round

I mean, I get that 2x1.35=2.7, but would doesn't the math have to take into account the odds of both attacks hitting in the same round?

 

[Janx]

I don't think that makes sense to me, which means I might not be phrasing my question clearly. I was with you up until here:



I mean, I get that 2x1.35=2.7, but would doesn't the math have to take into account the odds of both attacks hitting in the same round?

bear in mind, somebody with greater math skills will point out some more correct calculation method. But here's my logic.

It's like the lottery caclulation, you pay $1 for a ticket and you have a 1% chance of winning the $500 lottery (I'm using easy numbers for math here).

The effective value of that ticket for the decision to pay $1 for a chance to win $500 is equal to $500/1% which equals $5. Since $5 is > $1, buying the ticket is a good investment. That formula is sound, and if you plugged in real lottery numbers, you'd know when it's a waste of your money.

Now, if you buy 2 tickets (kindof like having 2 attacks), you add them together, because you've "statistically won" $5 each.

So, with each single attack being worth 1.35 statistically, we add them together for their total worth.

I'm using the wrong term, but it's statistical worth is not the same as it's actual value (which won't be known until the dice are rolled).

Let's use another common example, claw/claw/bite for 1d6/1d6/1d8 at +6/+6/+8 attack. This would demonstrate how to apply the method to an assymetrical attack pattern (non-matching quantities).

The average damage is 3.5/3.5/4.5

each claw at 30% chance is worth 1.05
the bite at 40% is worth 1.8

Add them all up:
1.05 + 1.05 + 1.8 = 3.9

The damage range on hit is 1-20 (6+6+8). however, rolling a 1d20 is a bit swingy, compared to what would actually happen if all 3 attacks hit. I would argue, like before, that we always just roll all the dice for these "cumulated" attacks as 2d6+1d8 on a hit.

With that, all I need to do is calculate what % to hit will give me a 3.9 damage from the average roll generated by 2d6+1d8 (which is 11.5 for the raw die average)).

3.9/11.5 = 0.3391304347826087

So 33% which in D&D needs to be 30% or 35%. Rounding to the higher value seems fair, so 35% is +7 to hit for 2d6+1d8 damage.

As I disclaimed before, the math is solid enough, but I can't say the algorithm is perfect. If you had +0 to hit, my formula would fail (0% of any die roll would imply you always miss), as it's missing something to account for that (namely some non-zero baseline).

But for non-zero multi-attack coaggulations, it calculates a correct enough result based on my very tiny testing size.

 

[Storminator]

If you keep exactly the same attack bonus, and just add the damage dice together, you will get exactly the same average damage. Your round by round damage will be much more swingy tho. If that matters you need to move away from just expected damage. In that case, increase the attack bonus a bit, and reduce the damage from the pure sum to something less than that.

PS

 

[Nagol]

Let's say a D&D-esque monster attacks at +6/+6 and does 1d8 points of damage per attack, but that I want to have the monster attack only once, end up with same average damage per round, and then just describe the results as a flurry of claws, bites, whatever. What's the easiest way to convert multiple attack routines into a single attack bonus and damage roll? Or is there an easiest way?

The easiest way to keep the same damage when combining attacks with the same to-hit factor is to roll once and if you hit, roll all the damage dice.


So for a +6/+6 1d8/1d8 aattack sequence would become +6 / 2d8.

Note this makes the critter more swingy as the chance for substantial damage (both of the original atacks hit) and the chance for no damage (chance neither original attack hits) is greater when rolling once than rolling twice.

 

[Mark Chance]

Okay, that makes sense now. I think. I don't think the result is quite what I'm looking for. Thanks, everyone! Back to the drawing board.

 

[Janx]

Okay, that makes sense now. I think. I don't think the result is quite what I'm looking for. Thanks, everyone! Back to the drawing board.

Since you gave a mirrored example (both attacks are the same), Nagol and Storminator's method is faster than mine to the same result, which is roll all the damage at once, use the +6 modifier.

My example simply goes through the statistical hoops to come to the same conclusion, demonstrating how to do it for dissimilar multiple attacks.

But I be thinking we grasped your question:

You have a monster with +6/+6 attack for 1d8/1d8 damage. You don't want to have to roll twice, you just want to roll once to-hit, and if you hit, roll some dice.

You wanted the to-hit/damage to be statistically equivalent to the normal monster design, such that the monster isn't stronger or weaker than by the RAW.

I believe we have calculated that for you. I'm not sure what you thought the numbers would look like (+8 for 1d4 damage?). Mathematically, there are probably other BAB/damage combinations that equal what what the monster has as multiple attacks, but that would be non-trivial to identify them for any given starting set.

 

[Janx]

As an extension of Mark Chance's base idea, imagine applying this to 3e's multiple attacks for a fighter (where subsequent attacks are -5 less than the previous.

a figher with a BAB of +16 gets +16/+11/+6/+1 if I recall correctly (or whatever the table says, it doesn't matter). I'd be curious if we can systemetize this such that we can give the fighter 1 attack, that does more damage, the higher level he is. If you calculate enough "levels" a pattern would likely emerge, enabling quantifying this to something where fighters might get better than +1 per level and a bonus to damage as they increase in level instead of getting multiple attacks. This would speed up high level combat by reducing dice rolls.

 

[Nagol]

It's hard, actually. The expected damage is pD where p is the probability of landing the blow and D is the damage range. p is dependent on the defender's AC as much as it is the attacker's bonus. Worse, the attack probabilities are bounded between 5% and 95% and the range of variation is up to 75% for the bonuses.

Dealing with same bonus attacks is simple. When trying to extrapolate to different bonuses, you have to try to consider the effect the bounded probabiliites have. Does a +16 attack hit x + 75% more frequently than a +1 or has one of the attacks hit the bounds of the game?

A set of extrapolated adjustments can be made for the attack sequence against any specific AC, but each AC will have a different expected damage range for the single attack.

Dropping to a single attack also makes the critter much more swingy on a round-by-round basis as well and that is often not a desireable feature.