I've been reading some of the Comparison threads that Turin the Mad has been posting, and have become quite interested in the math.
Something Turin said caught my attention:
Now, to be realistic, one should probably factor in chances of missing, criticalling, yadda yadda ... but I'm not a statistician.
Now, I'm not specifically a "statistician", however I became adept at creating damage modals back in my Everquest days when trying to compare weapons in that game (before they had handy numbers all figured out).
So, here's how I'd figure out damage against a given AC, taking into account all factors possible.
(I'm making this in a way so that you can fill in the blanks and then pop numbers into a calculator... maybe I'll make an XML file to do all this.)
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Let's take Lenny's stats as our example:
Attacks: +41/+41/+36/+31/+26
Damage: 2d6 + 27
Critial: 17-20/x2
Step 1: Average damages
Regular 34 (2d6+27)
Critical 68 (4d6+54)
Step 2: Determine Critical Chances
17-20 = 4 point range = 20%
Step 3: Factor critical odds with average damage
80% of 34 = 27.2
20% of 68 = 13.6
Step 4: Determine chances to hit
35 - 41 = -6 (only miss on a 1), so 95% chance
35 - 36 = -1 (only miss on a 1), so 95% chance
35 - 31 = 4 (miss on 3 or lower), so 85% chance
Iteratives are at 5 point intervals, so 25% lower chance on each iterative once you get a real number below 100%.
26 = 60% chance
Step 5: Determine Critical Confirmation chances
Since you roll a second time at the same bonus, simply multiply the chance to hit, with a second chance to hit (the confirmation).
This is where having confirmation bonuses or automatic confirmations change the odds.
+41 has 95% hit, (.95 x .95), 90.25%
+36 has 95% hit, 90.25%
+31 has 85% hit, (.85 x .85), 72.25%
+26 has 60% hit, (.6 x .6), 36%
Step 6: Apply average damage to the odds
+41 = (95% of 27.2) and (90.25% of 13.6) = 38.114
+36 = same as +41, 38.114
+31 = (85% of 27.2) and (72.25% of 13.6) = 32.946
+26 = (60% of 27.2) and (36% of 13.6) = 21.216
Step 7: Total the damage for one full round
38.114 x 3
32.946
21.216
--------
168.504
or
~168.5 avg dmg
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This gets more complicated when adding in damage dice effects (flaming weapon) and even more complicated when you add things that affect criticals (flaming burst weapon + critical confirmation bonuses), however it is all much more manageable once the base rules have been set down.
My Office Excel is on my other computer, but I think I might actually sit down and make a "punch in the numbers and get your final results" xml sheet for this.
I'm interested enough to write it out for myself.
If anyone finds fault in my statistical rundown, let me know and I'll update this accordingly.
Hope this helps folks do some "run the averages" comparisons.
If anyone finds fault in my statistical rundown, let me know and I'll update this accordingly.
Hope this helps folks do some "run the averages" comparisons.
No fault, I can suggest a slightly easier way to calculate the damage from crits though. Helps a bit if you write it all out as an equation, so starting with normal damage (ignoring crits at this point)...
chance to hit * normal damage
There's no need to pull out the chance to threaten at this point. If you're attack hits you're dealing at least normal damage, since a crit that fails to confirm still does base damage.
So all you really need to know is the amount of extra damage that you'll do on a crit...
normal damage * (crit multiplier-1) the -1 is because this is the additional damage from a crit, not the total.
And your odds of doing that additional damage, which would be...
chance to threaten * chance to hit (multiplied by chance to hit for the confirmation roll)
If you have damage that is not multiplied by a crit you need to pull it out, and I'll get to how to handle that in the last step.
Ok, now putting it all together is going to be a little hard to read if I keep writing everything out, so I'm going to use some shorthand.
toHit: chance to hit
toThr: chance to threaten
AvDmg: average damage on a normal attack
Multi: crit multiplier
So the full equation for damage is...
(AvDmg*toHit) + (AvDmg*(Multi-1)*toThr*toHit)
The second term also includes AvgDmg multiplied by ToHit, so we can factor them out like this...
(AvgDmg*ToHit) * (1+(Multi-1)*ToThr)
Almost everything is going to be either a x2 or x3 multiplier, so in the end, you only have to calculate your average damage times your chance to hit, then multiply it by 1 + your threat percentage (for x2), or 1 + double your threat percentage (for x3). And, since the chance to threaten and crit multiplier never change, you don't even have to calculate it per attack, you can find the average damage for a full round, multiple rounds, whatever, as long as the attacks are all using the same weapon and threat range, then multiply the total at the end.
Ok, so an example, to show how easy this is to use, because the equation by itself may be a bit hard to follow.
A character with a +4 to hit, attacking an AC 15, dealing 1d8+4 on a normal attack, threatening on a natural 20, and doing x3 damage on a successful crit.
Normal Average Damage is 8.5 (1d8=+4.5, +4 flat damage)
Chance to hit is 50% (11 or better)
Chance to threaten is 5% (natural 20)
and Crit multiplier is x3
So starting with (AvgDmg*ToHit) you've got 8.5 times 50%, or 4.25.
The threat range is 5%, and it's a x3, so that's 1+(.05x2), or 1.1
4.25 x 1.1 is 4.675, so an average attack, including crits, will do 4.675 damage.
Now the one problem, I mentioned earlier that you have to seperate out damage that doesn't multiply on a crit, but this actually isn't that bad. Since all you care about on something like fire damage is your chance to hit and your average damage, you can just add it seperately after you multiply in crits.
So if in the previous example they were doing 1d8+4+1d6 sneak attack, you'd just need to multiply 3.5 (average on 1d6) times .5 (chance to hit) and add it at the end, giving you 6.425
The second term also includes AvgDmg multiplied by ToHit, so we can factor them out like this...(AvgDmg*ToHit) * (1+(Multi-1)*ToThr)
Aha! See, I knew there was a way to factor down the equation.
This actually helps quite a bit for the xml file I'm creating for this. Since I need it to be flexible enough to toss in multiple weapons and such, I can't simplify as much as you mentioned. However, this above bit of math will help keep the formulas easier to input.
Thanks a bunch!
The second term also includes AvgDmg multiplied by ToHit, so we can factor them out like this...(AvgDmg*ToHit) * (1+(Multi-1)*ToThr)
Aha! See, I knew there was a way to factor down the equation.
This actually helps quite a bit for the xml file I'm creating for this. Since I need it to be flexible enough to toss in multiple weapons and such, I can't simplify as much as you mentioned. However, this above bit of math will help keep the formulas easier to input.
Thanks a bunch!
Glad I could help.
Oooh.. only problem I ran into. How do you accomodate such things as "crit confirmation bonus".
So basically your toHit to confirm the threaten is different from the regular toHit. In other words, if your example above had a +2 to confirm critical hits, you'd have a 60%, not 50% on confirmations.
Also, I have to take into account the Fighter's end cap "Auto confirm" criticals.
I may have to write this out the long way in order to keep the xml more flexible.
*Edit*
It's still better than my OP formula...
Basically, I'm taking your unfactored equation and modifying one part:
(AvDmg*toHit) + (AvDmg*(Multi-1)*toThr*ConfHit)
Oooh.. only problem I ran into. How do you accomodate such things as "crit confirmation bonus".
So basically your toHit to confirm the threaten is different from the regular toHit. In other words, if your example above had a +2 to confirm critical hits, you'd have a 60%, not 50% on confirmations.
Also, I have to take into account the Fighter's end cap "Auto confirm" criticals.
I may have to write this out the long way in order to keep the xml more flexible.
Yeah, when I originally did the math on all this there weren't any abilities that had a bonus to confirmation, so that does sort of put a kink in things. I'll see if I can't figure out a simplified way of handling it, but it might take a bit.
Impressive work gentlemen and ladies (as necessary), impressive work. I had no idea excellent material like this would spring forth... :)
KUTGW!
Here's the xml file, if anyone is interested in using this as an automated thing.
It's not exactly pretty math involved, but it gets the job done. I left a crapload of spaces for Attacks... when doing the numbers on a Monk/Fighter for the TWF + Flurry thread, I ran into around 8 or more attacks on a primary weapon, and additional attacks on top of that... so hopefully 3 weapons with 8 attack slots each will be flexible enough for nearly any PC build.
Yours is very, very close to how things work in DnD, it's just missing edge cases. You need to deal with auto fails, auto succeeds, and the cases where your threat range is bigger than your hit range (the latter is actually somewhat likely with big threat weapons, keen, and imp crit).
Yeah, I realized the edge cases were there when I started thinking about bonuses to the confirmation roll, I'm hoping I can find a simpler way to express it all that will take into account those edge cases without having to know the individual toHit for each attack.
The nice thing about the simplified equation is that since Average Damage and the specific to Hit were separated from the other terms, you could make meaningful comparisons about crit multipliers/ranges, that would apply to any character with that multiplier/range. For instance, you could say "x2 crit, with an 18-20 range, is going to give the character 110% base damage. (-5% for automatic misses, +15% for the crits)"
Especially important is that you didn't have to know what the character was fighting, you just had to know their stats, and with the exception of some extreme edge cases, it would be true for every fight. Those edge cases just didn't come up that often, since it's fairly uncommon for a character to have a crit range larger than the range in which they would successfully hit.
Once you include bonuses to the confirmation though, you start butting up against the other end of the range, and getting values where it matters that you can't (normally) go above a 95% chance to confirm, and that is a problem, because high level melee characters frequently have attacks that hit on anything but a 1.
I hope I can find some way to factor out the To Hit Percentage and Average Damage, so making comparisons between various setups doesn't have to be so complicated.
The excel file takes into account autosuccess and failure. Basically if the ToHit ends up lower than 5% or higher than 95%, it will cap it.
Except when you select Autoconfirm as Y, then it makes confirmations 100%.
The excel file takes into account autosuccess and failure. Basically if the ToHit ends up lower than 5% or higher than 95%, it will cap it.
Except when you select Autoconfirm as Y, then it makes confirmations 100%.
Right, I was just saying that needing to calculate the ToHit means you also have to know the AC of what you're attacking and the specific attack bonus for the attack. Where if the ToHit and AvDmg can be factored out, you can solve each side independently, and calculate the relative percentage of damage for various crit/multipliers/etc independently from the specific attack bonus and foe.
Not sure there's going to be a way to do that that also factors in confirmation bonuses and edge cases, sadly, I'm just hoping something comes to me, since it simplifies comparisons so much.
Oops! Sorry, I was responding to Tarkisflux, not your post.
I was thinking.. in terms of fixing the excel file, I might just write a further nested IF/THEN function to check if the Threaten % range exceeds the ToHit % range.. and if it does, the equation will use the ToHit % instead.
Meaning, if you only land a hit on a 20 (5%), and your crit range is 17-20 (20%), then instead of using 20%, it'll use 5% to get closer to the actual damage you'd do.
So..
IF "Threaten > ToHit" THEN <equation using ToHit> ELSE <equation using Threaten>
Now to fit that into 24 attack space equations.. ugh.
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If you care about these things, here ya go:
AdjHit% = <Minimum(95%, <Maximum(Hit%, 5%)> )>
AvgTotalDmg = (AvgStdDmg*AdjHit%) + ((AvgStdDmg+BaseBurstDmg) * (Mult-1) * <Minimum( AdjHit%, ThreatRng/20)> * <Minimum( AdjHit% + ConfirmBonus/20, 95%)> ) + PrecisionDmg + OtherNotMultipliedOnCritDmg + OtherOnCritDmg
This is functionally identical to the method I used to generate those graphs for Turin's Lenny the Barbarian.
Umm... At the risk of sounding like a broken record, have any of you guys actually found a character that has better damage output by using Improved Vital Strike?
Sam
That's a good question.
When I finish fixing the threaten/tohit part of the xml I'll put Vital Strike and Improved Vital Strike through it's paces and see what we get.
Here's the latest .XML file for download.
Here's a rundown of what it does:
- It will take the Attack bonus in a given slot and give the odds of successfully hitting the target AC, with caps for going over 95% or lower than 5%.
- It will find the odds of successfully landing the crit, taking into account modifiers to confirmation rolls (or autoconfirmation).
- It will calculate the damage by adding up the Base damage, the critical damage (and take into account when the Threaten range is greater than the Hit chance), as well as adding in one dice type of critical bonus damage and two dice types of standard damage.
- It can do this with up to 3 different weapons in one attack routine, and up to 8 attacks per weapon.
If you need more than 8 attacks on a single weapon, you can just fill in the same weapon information on the secondary or tertiary fields and get another 8 attacks with that weapon, although it reduces the number of weapons useable in a given attack routine to two.
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So basically you can run the numbers on a +1 Flaming Burst weapon, with a +2 crit confirmation bonus feat applied, and get the proper damage modal.
Now to run some numbers regarding the Vital Strike line. I'm also curious to see if Power Attack numbers stand up when you get exceedingly high Strength/BAB scores.
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Now to run some numbers regarding the Vital Strike line. I'm also curious to see if Power Attack numbers stand up when you get exceedingly high Strength/BAB scores.
As an example, here's a graph of the numbers I ran for Turin's Lenny the Barbarian at 17th Level:
Lenny the Barbarian, 17th Level, Full AttacksWith the greater heroism and haste buffs backing it up, the Power Attack does pretty well. Vital strike also does well, but Improved Vital Strike is a trap for a heavy damage bonus character like this.
Sam
Okay, ran ton of numbers with regards to Vital Strike and Imp Vital Strike.
It seems that there isn't really ever a situation where you'd really want to use Imp Vital strike, unless you are up against something with MUCH higher AC than you should be fighting. And even then, it's a paltry bonus.
EXCEPT in ONE unique situation.
A Fighter built with Twohand weapons would benefit from Improved Vital Strike if he also got TWF and ITWF and made those offhand attacks with unarmed strike (say, a kick). At that point, trading the 1d4 20/x2 damage on two lower bonus attacks would be worth the added 2d6 damage per mainhand attack.
Essentially, it'd take two extra feats, but you could essentially give up -2 attack to get +4d6 damage. That's with keeping all your twohander attacks.
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Honestly, since I don't think the point behind Improved Vital Strike was to only be the most beneficial when specifically getting a crappy offhand weapon to dump attacks with.
I think Improved Vital Strike should get a rewrite, as it stands..
Okay, ran ton of numbers with regards to Vital Strike and Imp Vital Strike.
It seems that there isn't really ever a situation where you'd really want to use Imp Vital strike, unless you are up against something with MUCH higher AC than you should be fighting. And even then, it's a paltry bonus.
EXCEPT in ONE unique situation.
A Fighter built with Twohand weapons would benefit from Improved Vital Strike if he also got TWF and ITWF and made those offhand attacks with unarmed strike (say, a kick). At that point, trading the 1d4 20/x2 damage on two lower bonus attacks would be worth the added 2d6 damage per mainhand attack.
Essentially, it'd take two extra feats, but you could essentially give up -2 attack to get +4d6 damage. That's with keeping all your twohander attacks.
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Honestly, since I don't think the point behind Improved Vital Strike was to only be the most beneficial when specifically getting a crappy offhand weapon to dump attacks with.
I think Improved Vital Strike should get a rewrite, as it stands..
Raises a mug of hot-buttered rum in salute to thread readers after a hearty, albiet somewhat furry, meal.
It is my belief that the intent behind Vital Strike and Improved Vital Strike is to function as an "acceptable substitute" for trimming down the one and two lowest attack-bonus swings a high-BAB full-attack provides. Obviously, over the course of many threads - 'ours' especially - it is alarmingly clear that they do so rather poorly. Both feats probably need to be re-written, at least enough so that they are effectively an irresitable temptation.
I'd prefer of course to just trim down the iterative attacks in an integrated revision of the combat rules... but I suppose that will have to wait for the pertinent sections of the rules set to be opened up for in-depth discussion. ^_^
Until then, enjoy a tasty beverage of your choice gentlecritters!
Yeah, the feat shouldn't even just "break even" with it's change from lots to less attacks, it should be giving something as well. There's no "inherent" benefit to doing less attacks in a round other than maybe spending less time rolling. And you are spending a Feat slot to do this.
Plus, it's the second feat in a chain, that by virtue of the nature of the feats includes the bonus from the first one. It's not like say Power Attack and Cleave, where you can use and benefit from both at the same exact time. The second feat in the chain is "adding on" to the first feat, like Two Weapon Fighting feats.
It *needs* to be a good bonus to be worth spending essentially two feats on.
And right now it's doing the opposite...
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Maybe I can use this tool to try various alternatives to the current feat and see what would work better in it's place.