crit and crit dmg scaling

Hi all, wonder if someone who's done some theory could point out my mistake.

In an attempt to determine the relative values of various DPS nodes on the passive skill tree, I've made a rudimentary DPS calculator. To keep things simple I'm using a very low level character for now. When I tried to even think about it in terms of my higher level character, my brain about exploded (and he's only level 20-ish!) so this will have to do for now. Iterative process and all.

There is either a rounding error or a problem with the way I'm applying local weapon APS mods. Anyway, it's minor and I'll figure it out later. Shouldn't make a huge difference for my purpose now.

Goal: Fast forward to an endgame build/gearing maxing crit and crit dmg and sacrificing all else. Basically, I put together a build with 111 points taking every feasible crit/critD node.
Now add to this the maximum amount of available crit chance possible on gear. End result is around ~70% crit overall.
Also add to this a high quality +critD gem and assume you have one of such applied to everyone of your main skills. End result is about 600% critical dmg.

%attack speed and %physical damage mods have a very slight diminishing return - going from 50% to 55% on either should be worth more than going from 55% to 60%. While crit and critD have this feedback effect - so that rather than diminishing in value the two each become considerably more powerful as you stack the other.

Reality check 1: For my low level character, a 4% attack speed skillpoint, or a 6% physi skillpoint is worth several times more than a 10% critD or a 15% crit skillpoint. Okay. That is as it should be.

Reality check 2: When I get to the maxed out crit-critD build/itemization, attack speed, +phys... +ele... skillpoints should all be worth LESS per skillpoint than both crit and critical damage. By a pretty significant amount. But that's not where I arrive. I get the following ratio (dps shift per skillpoint):

+4 APS: 13
+6 phys: 6
+15 crit: 4.5
+10 critD: 9

conclusion: because you have to spend skillpoints and gear budget/gems on crit+critD just to get them to a point of meaningful value, many of your points are worth much less than the above ratio implies. A crit / crit damage build shouldn't be able to compete with a haste / +dmg build with these rules in place.

Of course the haste build consumes more mana. But the way flasks are, and the way this game is itemized, makes it pretty easy to add more regen to your kit (or your build) to compensate for that. It should be a non-issue.

My limited DPS analysis tool is here. Google docs isn't a filehosting service, save a copy to your account or computer and then close the tab out. This way others can get a look too.

I'm mostly interested in hearing from people who (1) know what I did wrong (2) read my whole post (3) examined my formulas a bit. Please no anecdotal replies. I know this is a long-winded post.

A few ideas where my logic may be off. First, the current value of crit/critD feed back into the value of all PREVIOUS points of haste/+phys. Will have to make another pure phys/haste build and compare them closely in the calc.

Hi,
you might not be the most interested in my post, but
1) I might know your mistake
2) I have read the entire post
3) I looked at your formula, got a little headache and made my own.

So, my dps formula is the following:
DPS = DPH*(1+ID) * APS*(1+IAS) * (1+CC*CM)
where
DPS: damage per second
DPH: damage per hit
ID: inceased damage percentage
APS: attacks per second
IAS: increased attack speed
CC: critical chance
CM: crit damage multiplier

If this is wrong, than I just wasted everybody's time.

So, here's my anecdote:
If not, that there's no point in comparing ID and IAS to either CC or CM, because the last two get multiplied. You could only compare them the the product CC*CM, but that doesn't help you in dermining the actual damage increase for each passive node.


Edit: of course I made a mistake. Let's redefine CM to 100%-'crit damage multiplier', then the formula is correct. Details, god damn details

"

mokschi wrote:

So, my dps formula is the following:
DPS = DPH*(1+ID) * APS*(1+IAS) * (1+CC*CM)
If this is wrong, than I just wasted everybody's time.

First, thanks for your input. A couple issues with your formula, though:
- %physical damage only applies to the physical component of damage (as you have it written, it applies to the whole thing).
- crit size = (critical damage multiplier)*[(non-crit size) + (4/3)*fire component of a non-crit]
- average hit size = (crit chance)*(crit size) + (1-crit chance)*(non-crit size)
- DPS = attacks per second * average hit size

I've compared this process to my character tooltips, and it seems to be accurate. Cooking down the above into a 1-line formula like yours gives:

DPS = CC*CM*DPH + (1-CC)*DPH
= [1 + CC*CM - CC]*DPH*[1+AS]
... but this still doesn't account for +PD or the Burning effect on crit.

"

If not, that there's no point in comparing ID and IAS to either CC or CM, because the last two get multiplied.

I understand where you're coming from, but the individual value of each is very meaningful. We've expressed DPS as a function of four variables: DPS(%AS, %PD, %CR, %CD). Now I'm extracting the "directional derivative" of this function with respect to one skillpoint.

dDPS/dS where dS can be: 4 %AS, 6 %PD, 15 %CR, 10 %CD, or any other combination of stats on the skilltree. Any way you phrase this derivative already accounts for how much damage all the other stats are contributing.

Your approach of looking at crit chance * crit dmg shows how much all your existing crit chance and crit dmg are getting you.... My approach shows how much the next increment will give me. Then compares that to how much the next increment of %AS and %PD will give me. See?


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Now, I've figured it out somewhat. It's all about context. I don't understand the context quite well enough to be 100% correct on all this, even if my numbers are right.

For instance, I still have to go through the same mental exercise with haste/physi: to make a build and put it through the same process. Reason being that you can't use a directional derivative to max a non-linear function (and dps is non-linear against crit/critD).

Second, when looking at a hybrid build like mine (1h/shield crit build shadow) that uses a lot of physical attacks AND spells, it's pretty clear the crit build will win regardless of how the numbers turn out. That's because investing heavily in +%physical damage and +%attack speed gimps your spellcasting. Most crit nodes buff attacks AND spells.

Okay, I was only considering PD and did not account for burn damage from fire-crits, but in my defense, I did correct myself for the missing (1-CC). see:

"

mokschi wrote:

Edit: of course I made a mistake. Let's redefine CM to 100%-'crit damage multiplier', then the formula is correct. Details, god damn details

Anyways, I'm still a bit confused. So you're only interested in the next best step for your buid? Something along the line:
- I have PD=x1, IAS=x2, CC=x3, CM=x4
- I can take x1+6%, x2+4%, x3+15%, x4+10%
- calculate 4 different damages
- choose best
- repeat from top

In that case, you'd never take CC or CM although taking both 10 times might be better than taking 20 time PD.
Isn't the real problem that for high x3 and x4, CC and CM become stonger and stronger?

Edit: Just wanted to add this for fun: it's not a Markov Process ;-)

I'm actually interested in finding out whether max DPS lies along the enhanced damage - haste route, or the crit - crit damage route. In math this is referred to as a "boundary value problem" but in this case the "boundary" is very difficult to express.

Expressing dDPS is one step towards solving the problem, but not end-all. You need to check against boundary conditions (a "boundary condition" would be optimal gearing for the build). You see, I didn't choose skillpoints that were currently the best along the whole route, or I would have picked physical dmg - haste all along the way. My build is focused on crit/critD.

My point: crit/critD eventually need to have a larger influence on total DPS than haste/%dmg. After you've stacked them sufficiently. But they don't. At every gearpoint I checked, even gearpoints with crit and critD stacked very high (60% crit 400% crit dmg on char tooltip) the next crit/critD skillpoint is still worth less to your DPS tooltip than the next haste skillpoint. This is the thing that made me think I had a problem. Please tell me if you understand. I couldn't make crit or crit dmg worth more than haste (per skillpoint on the passive tree), ever. In the context of the problem I explained, maxing DPS would require you to shift back into the haste/phys build.

What I'd forgotten to consider at the time was that different passives effect different subsets of our active skills. A +critD boost will benefit EVERY active skill. A +%PD boost or a +%haste boost will only benefit attacks.

So I think that I've found my answer: crit/critD CAN have a bigger impact on total dps, depending on your playstyle. If you're a linear hack n'slash melee type with very little spell damage, the haste/physi build will win. If you're a hybrid spellcaster the crit build will win. If you're a linear spellcaster again the +ele/haste skills would be worth far more than a crit build.

Arguably, the devs' intention with crits is not to be superior DPS: if they were, straight haste/physical damage as you describe would be strictly inferior endgame, because of the auxiliary benefits crit damage provides (higher chance of outright one-shots, and the elemental status ailments, especially shock).

Here's the calculation for what the game lists your DPS as:

=(((MinDamage+MaxDamage)/2*AttackSpeed)*(1-CritChance)+((MinDamage+MaxDamage)/2*AttackSpeed)*(CritChance)*(CritDamage))*(Accuracy)

Note that your Accuracy is a floating point number, and can be several digits longer than what's displayed on your character screen. So, if you're seeing a slight difference between your calculated value, and your in-game value, you didn't calculate accuracy right.

To calculate your Min/Max Damage:

=(WeaponDamage + AddedDamage)*(1 + IncreasedDamage)*(1 + MorePhysicalDamage)*BaseDamage

Then add that to your elemental damages.

If any of the above is unclear I can provide more information.

You forgot Ice Spear's Crit chancce bonus. Also, you're wrongly assuming that a crit build will max out the 10~15% crit/crit damage nodes. A crit build might visit the Templar, Shadow, and Witch starting areas and pick up only the big crit nodes.

You will never get far in dps if you are only stacking one stat, whether it's just attack speed, damage, or critical.

Damage and attack speed are both linear increase, meaning you will get less total damage boost from each nodes as you stack them higher and higher. Critical chance is not linear, but stacking high into it meaning huge sacrifice in other aspects. Keep in mind tho, it is possible to reach 95% critical chance cap.



Anyway, the easiest way is to have your own stats/numbers ready, and every time you are forced into deciding which choice is better, plug in the numbers into each choice and then divide the outcome by your current one; the one with highest ratio wins. (You can do this for long term planning too, but you also need to consider what sort of gear you will be getting)