Minmaxing damage with equipments: a theoretical approach

Hello there good sir/lady/object/animal (I identify myself as a racist against attack helicopters). This thread was made to help you understand how to choose the right equipments for your DPS build. Now you may ask yourself: "Why did you let this buffoon make this thread?! He's not BK/Cyph/[insertnamehere]!". True. But first of all, I'm a math freak so you can expect me to throw you (semi-)solid proofs. Second, I love this game and I'm really invested to help people with it. Third, Y U NO DO EET URSELF DEN? =P
Note that:

• I do NOT claim any of the findings I publish.
• This is aimed at people who want to play competitively but do not understand how each stat mechanic works, and more spefically help those people find the best equipments. So if you're a pro that has no intention of improving this guide or a casual player, while you're still welcome to read this guide, any non-constructive critique will be ignored (same treatment for the rest).
• A lot of what I'm writing is going to sound obvious to many of you. Again, this is aimed at people who don't understand much about how damage calculcation works.
  
• I'm a human, therefore flawed. If you could help me improve this guide, it would be much appreciated!

So, with further ado, let's start with the bases for my calculations!

Premiss​

In many games, the damage calculation of a skill has a formula (without criticals or whatever) of this shape:

(attack_value + skill_attack_value)*skill_damage_multiplier​

(For simplicity, we will call the attack value ATK and the skill's attack value B)
When we include criticals, the formula becomes this:

(ATK + B)*skill_damage_multiplier*critical_multiplier​

(Again, for simplicity, we will call the critical multiplier CDMG)
Now, you will notice that we have two variables (ATK and CDMG) and two constants (B and the skill damage multiplier) in this formula. However, only B will matter in our damage calculations, therefore we can get rid of the skill damage multiplier, resulting in a simpler formula:

(ATK + B)*CDMG
​

(You can skip proofs if you believe me from the get-go)

Spoiler: Proof

This proof will be more intuitive than mathematical.
Suppose we want to compare two builds with different ATK and CDMG. The difference between the two will be a constant we will call C.

(ATK_1 + B)*skill_damage_multiplier*CDMG_1 - (ATK_2 + B)*skill_damage_multiplier*CDMG_2 = C
ATK_1, ATK_2, CDMG_1, CDMG_2, B, C ⊆ ℝ
B ≥ 0 (by definition; most (if not all) values of B in games are this way)
⇒ ((ATK_1 + B)*CDMG_1 - (ATK_2 + B)*CDMG_2)*skill_damage_multiplier = C
​

If we divide C (a constant) by the skill damage multiplier (another constant), we will obtain a constant that we will call D. We also have to divide it in the left side, effectively getting rid of it.

⇒ (ATK_1 + B)*CDMG_1 - (ATK_2 + B)*CDMG_2 = D
D ⊆ ℝ
​

If you continue further, you'll see that there is no real way to remove B from the left side.

Comparing the highest possible crit damage​

With this in mind, we can figure out a function that will allow us to compare the equipment effectiveness of equipments in a build for crit damage:

f(x) = (ATK_X + B)*CDMG_X
ATK_X = Base_ATK_X*ATK_multipler_X + Bonus_ATK_X
CDMG_X = Base_CDMG_X* CDMG_multiplier_X+ Bonus_CDMG_X
​

Note: ATK_multiplier_X and CDMG_multiplier_X are equivalent
How do you calculate it? The easy way is to use an already established calculator and replace ATK_X and CDMG_X with the attack values and crit damage values it returns, then find the skill's damage value (NOT THE MULTIPLIER!) AKA B (Coryn Club's calculator and dopr.net's skill descriptions come to mind). The hard way is to calculate these values yourself.
To compare two different builds, just calculate the values for both of them. Highest value is better.

What can we conclude from this? Higher critical damage has a much bigger impact than higher attack. (The proof is pretty intuitive, since CDMG_X multiplies everything, including Bonus_ATK_X and B (unlike ATK_multiplier_X). This means you need a much higher ATK_multiplier_X or Bonus_ATK_X to produce similar results.)
Note :Bonus_ATK_X includes both the Weapon Atk (Wep ATK% stats are fairly useless so I'm including it here) and the ATK up from equips such as Don Yeti

Comparing the average damage with a maxed stability​

The above fomula is flawed however: it assumes that you will always crit. To determine whether you will crit or not, we have the critical rate (CRATE). The formula to determine the average damage you will deal is the following:

g(x) = f(x)*CRATE_X/100 + (ATK_X + B)*(1 - CRATE_X/100)​

Spoiler: Proof

The event of executing a crit is a discrete random variable. The expectation of such a variable is Σ(x_i*p_i). The rest is easily deduced.

So, what can we conclude from this? Well, not much really. Because of the crit rate being a thing, many situations where crit damage would beat attack in terms of impact are negated or even completely flipped over. The best way to find out which is better for your build is by trial and error. However, thanks to the existence of calculators and damage tables, all you need to do is to use these tools and find out which is best.

I've run a couple analyses with fluctuating ATK and CDMG stats and fixating the starting crit rate at various key values (25, 50, 75, 100). The results were the following:

Crit rate equips trump over getting offensive stats.
Now you may be wondering: "Why is that?". Well the reason is simple: getting access more often to the crit damage multiplier is worth much more in average than boosting your attack in all cases. Yes, your maximum damage is higher if you use attack boosts, but we're not talking about lucky situations here. We're talking about getting a consistent damage output. Let's compare two typical examples. Suppose we want either an ATK7% or Crate+12 with Crate12%. Normally you would think: "ATK7% boosts my attack quite significantly regardless of crits so it wins!", but then you would be forgetting about one small detail: it doesn't increase Weapon ATK, extra ATK from equips and the B value. The crit damage multiplier, however, does increase all of it, so having a higher chance to crit makes for a pretty consistent increase in overall damage. Crate+12 with Crate12% wins over CDMG7% by a landslide: having a higher chance to crit gives a much more consistent increase than a higher damage if you crit.
Now, is crit rate the "slam dunk" winner? For equips, most likely. For stats, it's much more debatable as most of the damage comes from a pretty good stat build. My advice is only grab CRT once you've maximized your damage through your main stats. (And before people ask, yes survivability is better in many cases, but read the title again please.) The only exception to that rule are builds that can reach 100% crit rate naturally (only Sword-Fu with Triple Kick are able to do this), but that's a pretty obvious exception.

There is no clear winner between CDMG and ATK
It all comes down to this question: how much crit rate and crit damage do you have? Low crit rate and crit damage means ATK is better; high crit rate and crit damage means CDMG is better. In-between, it's very difficult to say.

Comparing the average variable with fluctuating stabilty​

Again, the formula above is flawed (yaaaaaay...): it assumes that you have maxed stability. The stability (STAB) value determines how low your damage will go. Here's the formula for the average damage I came up with:

h(x) = g(x)*(1 + STAB_X/100)/2​

Spoiler: Proof

Normally, the random event concerning stability also follows a discrete random distribution. More specifically, it follows an uniform distribution U[STAB_X/100, 1]. While we don't know the exact number of points possible, it's not a big issue since the expectation of a uniform does not depend of that number (I've had a long issue with this, but I'm not going to mention the details): (1 + STAB_X/100)/2. Now, first of all, both the event described here and in the previous point are linked by a multiplication (the damage is calculated considering the crit rate, then multiplied by the stability). Second, these events are independent (the crit rate and stability calculations are separate). There's a nifty theorem that says the expectation of the product of two independent events are the product of their expectations, and g(x) is the expectation of the first event, so we simply need to multiply the two to get the average damage when taking stability into account.

What can we conclude from this? Again, as above it's just trial and error, except there's even more variables! Again, just use a calculator for convenience's sake.
(More info later)

 

RESERVED

 

NOT GONNA LET YOU TAKE THAT RESERVED POST STARFOX

 

WHY IS IT WEIRD THAT I UNDERSTAND THIS JUST BY SKIMMING THROUGH IT?!?!?!?!?

 

Wow. An einstein playing mmorpg.

 

Not sure if I like it or not ewe ...good luck with this anyway

 

Because you're 2mart4me =P

Lol I'm no Einstein. He would probably be ashamed if he saw all my mistakes.

 

Interesting...
So in these formula,
Bonus_ATK_X <--- is this supposed to be ATK+ flat stat ? Or some other bonus?
Also theres something that has been bugging me for quite a while.
How do find out our base ATK?
Is it supposed to be like this:-
TotalATK - (WeaponATK*RefineMultiplier) - SkillMultipliers ?
For a char equipped with nothing but a stat less weapon and all stat points distributed to stats that give the highest DPS growth.
Because we can not consider a bare handed char ATK as baseATK as it has completely different stat growths than a weapon.

 

No hablo numero

 

Also i believe it should be "+" in there instead of a "-"
If it was a "-" that would mean that non crt hits would make the boss recover dmg :v
Poor bg snipers.

 

Both weapon attack and flat ATK boosts.
And yep that's pretty much how you do it. You also need to divide that by all the base attack multipliers (like Atk+7% or X Mastery skills).

And yep, my bad (when you write a guide that would make a light novelist and a mathematician squeal, you kinda do these mistakes).

 

All the math; my head is tolling

 

wow. someone is actually putting some high school math to use.

 

I skipped the "proofs" parts. :")

 

i saw some calculus here *flies away*