[Derivation] Armor/Mastery/Avoidance Calculations

Warning: Theorycraft inside.

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Re:

Postby Aerron » Fri Jan 07, 2011 1:14 pm

theckhd wrote:Meta Gems
  • At 30% total avoidance and 50% block, the block meta and armor meta are roughly equal.
  • At lower avoidance/block, the armor meta is roughly 25% more effective than the block meta.


For purposes of knowing when to make the switch, does the 30% total avoidance above include the normal 5% miss chance? Or should we just go with what's showing up on the Character sheet?
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby theckhd » Fri Jan 07, 2011 3:15 pm

That includes 5% miss, yes. Or more accurately, 4.4% against a level 88 boss. I've neglected the boss's inherent level-based avoidance reduction in most of these calculations.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby Fetzie » Fri Jan 07, 2011 5:02 pm

Does the block meta gem's effectiveness increase in the same fashion that the armor one's does when avoidance and block chance are set below 30 and 50% respectively?
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby wrathblood » Fri Jan 07, 2011 7:40 pm

Armor always works, whereas the block meta only works when you, naturally, block an attack. Anything that increases the % of incoming non-avoided swings that are blocked, increases the value of the block meta, and vice versa.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby theckhd » Sat Jan 08, 2011 9:40 am

Pyrea wrote:Does the block meta gem's effectiveness increase in the same fashion that the armor one's does when avoidance and block chance are set below 30 and 50% respectively?


I think you misread that section. The armor meta does not increase in absolute effectiveness as avoidance and block go down. In fact, it doesn't change in effectiveness with avoidance or block at all. The block meta gem gets weaker as your avoidance and block go down, so the armor one becomes more effective relative to the block meta.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby Durability » Sat Jan 08, 2011 7:37 pm

theckhd wrote:
I think you misread that section. The armor meta does not increase in absolute effectiveness as avoidance and block go down. In fact, it doesn't change in effectiveness with avoidance or block at all.


Shouldn't it? It would have no effect on an avoided attack, so I would think its relative power would go down if avoidance went up and block stayed the same (since that doesn't affect the block meta's effectiveness with attacks on a single roll). Admittedly I did tl;dr the full theory on the previous page, but yeah.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby theckhd » Sat Jan 08, 2011 9:14 pm

Look at equations 20 and 21. The definition of damage reduction we're using is

1-(damage taken with meta)/(damage taken without)

You can see from those expressions that the avoidance and block term cancels out. Conceptually, this makes sense. If you avoid an attack, you'd be taking no damage regardless of meta choice, so it has no effect on the damage reduction of the armor meta. If you avoid more, you'll take less damage overall, of course. But the armor meta will still reduce damage taken by a fixed percentage.

The block meta does change in value with your avoidance and block, which is why it becomes more or less effective than the armor meta, changing the relative worth of the two gems.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby Button » Sun Jan 09, 2011 9:48 pm

Do I understand correctly from step 12 that you're using scientific form (100% = 1.0) for your diminishing returns calculations? I've run into problems with that before - it seems like Whitemane's DR formula assumes you're using colloquial-form percentages instead.

For example, my warrior's paperdoll right now shows 747 parry rating (I logged right after Tol Barad, so I'm mostly in DPS gear) and 9.14% total parry chance. Given that 5% of that is base parry, that means the 747 parry rating is providing 4.14% parry chance. Undiminished, it should give 4.22705% parry chance. Solving the DR equation by expressing that as .0422705 gives me

Code: Select all
1/x' = 0.01523660 + 0.9560/.0422705 = 22.63148197


for an x' of .0441862, or 4.42% - more than the undiminished number! Using the colloquial notation of 4.22705 however gives

Code: Select all
1/x' = 0.01523660 + 0.9560/4.22705 = 0.241399054


for an x' of 4.1425183, the observed 4.14%.

Naturally this discrepancy between notations would throw your avoidance vs X calculations way off.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby theckhd » Sun Jan 09, 2011 10:47 pm

It's no wonder you've run into problems with that, because you've used the formula incorrectly. :P

The DR formula is given in equation (8):
Theck wrote:You can do a similar calculation for avoidance instead of block. For the moment, let A be pre-DR avoidance, A' post-DR avoidance, and k and C be the avoidance constants found here (k=0.9560 and C=0.65631440 at level 85 in our notation). If one differentiates the diminishing returns equation (1/A' = k/A + 1/C) and solves for dA' in terms of dA and A, they get:
Code: Select all
1/A' = k/A + 1/C   (8)


Note that you can use either scientific form (1.0) or percentage form (100%) as long as you use the appropriate form for C. The value of C you're using is 65.63 (from which you got 1/65.63=0.0152), which locks you into using percentage form. If you plug in A in scientific form, you'll get incorrect A' values, as you demonstrated.

However, since I've consistently used scientific form for avoidance throughout the derivation, I've also used scientific form for C (1/C=1/0.6563=1.52). You can check for yourself that this works properly by using 1.52 instead of 0.0152 in your first equation.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby Button » Sun Jan 09, 2011 10:52 pm

Ah, I see. Yeah, I just took the value of 1/C from Whitemane's table of constants underneath his table of combat ratings, rather than calculating it myself.

I guess that means it's my spreadsheet that contains an error in valuing armor vs avoidance. And I was so hopeful after our armor vs mastery lined up so well, too.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby Awyndel » Sun Jan 23, 2011 11:34 am

I have been trying to come up with some numbers to put on mastery/avoidance for the purpose of making a ( theoretical ) block capped set. So basicly the relationship between the 2 for combat table coverage.

I take str as 0.25 parry and agility as 0.61 dodge. But I haven't been able to solve a relation between mastery and parry/dodge. For the level of DR we can take an average, I'm not looking for one number to fix it all, just for an estimate to compare gear with.
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Re:

Postby The_Marsh » Tue Jan 25, 2011 8:10 am

theckhd wrote:[size=130][u]
Where [b]Av
is your decimal avoidance (i.e. 30%=0.3), B is your decimal block chance, and Fa is your armor mitigation factor. The armor mitigation factor is defined as follows:
Code: Select all
Fa = 1 - Ma = 1 - Ar/(Ar+K) = K/(Ar+K)            (2)
dFa = -dAr*K*Fa/(Ar+K)                            (3)

where Ar is your armor, K is the armor coefficient for a level 88 boss (K(88)=32573), and I've evaluated the derivative of Fa with respect to armor for future use.


Hey Theck, got bored while the servers are down so I decided I'll kill some time while I've got nothing to do. I looked at your working here and I cannot work out how you got formula 3. By my justification:

If:
Code: Select all
Fa = 1 - Ma = 1 - Ar/(Ar+K) = K/(Ar+K)            (2)

Then using the quotient rule we should obtain:
Code: Select all
(dFa/dAr) = -(dAr*K)/(Ar+K)^2
=> dFa = -[K*(dAr)^2]/(Ar+K)^2
Taking out K/(Ar+K) for Fa:
dFa = [Fa*(dAr)^2]/(Ar+K)


Please correct me if I'm wrong, it's university holidays and I haven't done proper maths for a few months now!
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Re: Re:

Postby theckhd » Tue Jan 25, 2011 9:30 am

The_Marsh wrote:If:
Code: Select all
Fa = 1 - Ma = 1 - Ar/(Ar+K) = K/(Ar+K)            (2)

Then using the quotient rule we should obtain:
Code: Select all
(dFa/dAr) = -(dAr*K)/(Ar+K)^2
=> dFa = -[K*(dAr)^2]/(Ar+K)^2
Taking out K/(Ar+K) for Fa:
dFa = [Fa*(dAr)^2]/(Ar+K)


Please correct me if I'm wrong, it's university holidays and I haven't done proper maths for a few months now!


Well, we both made mistakes. Let's do yours first. :P

That's not how partial differentiation works. If you're evaluating dFa/dAr, then the differential dAr in the numerator on the right-hand side shouldn't be there. The differential should only be in one of those two places. So you either have dFa/dAr=-K/(Ar+K)^2 or dFa=-K*dAr/(Ar+K)^2. See, for example, Chapter 4 of Boas, which covers partial differentiation.

As for my mistake, you're right that it should have been dFa=dAr*Fa/(Ar+K). My equation (3) has an extra K in the numerator. However, it should be clear from equation (4) that when I plugged in for dFa, I didn't include the erroneous K. So it looks like I just made a typo in equation (3).

Thanks for catching that.
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Re:

Postby Digren » Sun May 08, 2011 5:38 pm

theckhd wrote:[*]Avoidance & Mastery
  • 1 mastery rating is better than 1 parry rating for total damage reduction at 952 parry rating, or 10.1896% on the character sheet.
  • For dodge, this happens at 8.9705% on the character sheet. A rating equivalence is unreliable due to agility raid buffs.

Doesn't giving an exact rating for parry ignore the contribution from strength?
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Re:

Postby Digren » Sun May 08, 2011 6:15 pm

In equation (1), you use Av to represent post-DR avoidance. You continue this convention through equation (6).

In equation (8), however, you use A to represent pre-DR avoidance and A' to represent post-DR avoidance. There are a few places later when you substitute back and forth, such as in equation (11) where you point out that dAv equals dA' (as it should, since Av means the same thing as A').

I've read through these formulas many times, trying to extract little bits of information to gain insight into a gearing question. I've just realized that, for me, the most complicated part is this change of convention for expressing pre- and post-DR avoidance. It messes up my mind when I try to jump back and forth from the earlier to the later expressions.

If you had a day with nothing better to do, could you standardize the equations on a single set of variable names?
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