[Derivation] Armor, Mastery, and Avoidance

This came about because of something Wrathblood posted in the EJ prot pally thread a few days ago. Something about the armor weighting he came up with seemed fishy to me, so I wanted to sit down and work it out analytically.

Now that I've sat down and worked it out, I'm fairly certain there's an error somewhere in his math, probably nestled deep in Zarko's spreadsheet (so not Wrathblood's fault). In any event, I don't want to equationspam the EJ thread, but I did want to put the full version of the derivation somewhere easy to find and link to. So you get another [Derivation] thread to chew on.


I. Damage taken formula and problem set-up
The question we're trying to answer is, "How much armor does it take to reduce damage intake by the same amount as 1 mastery rating?" We're only going to consider blockable damage in this derivation. Obviously for unblockable (and unavoidable) damage, block and avoidance are useless and armor is the strongest of the three. Since none of them help against magical damage, we can ignore it entirely (since we're not trying to relate mastery to stamina).

For a boss melee swing of damage Do, the actual damage we take is

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D = Do*Fa*[0*Av + 0.6*B + 1*(1-Av-B)] = Do*Fa*[1-Av-0.4*B]         (1)

Where 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:

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Fa = 1 - Ma = 1 - Ar/(Ar+K) = K/(Ar+K)            (2)
dFa = -dAr*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.

Differentiating the expression for D, we get:

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dD/Do = dFa*[1-Av-0.4*B] + Fa*[-dAv-0.4*dB]
      = -dAr*Fa/(Ar+K)*[1-Av-0.4*B] + Fa*[-dAv-0.4*dB]
      = -dAr*Fa/(Ar+K)*[1-Av-0.4*B] - Fa*[dAv+0.4*dB]           (4)

II. Mastery and Armor

To determine an equivalency between Armor and Mastery for damage taken, we want to set these two terms equal to one another and solve for either dB or dAr. dB is both easier and slightly more logical, so let's do that. We'll ignore dAv for now by setting it equal to zero.

Thus, we get

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0.4*dB = dAr*[1-Av-0.4*B]/(Ar+K)            (5)

dB is linear in mastery, at 2.25 percent per point of mastery, or 0.0225/Cm percent per point of mastery rating, with Cm being the mastery rating conversion factor (Cm=179.28 @ level 85). In other words, dB = 0.0225/Cm*dRm for mastery rating dRm. So we can write an exact expression for how much mastery rating it takes to see an equal amount of damage reduction as dAr points of armor:

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dRm = Cm/(0.4*0.0225)*[1-Av-0.4*B]/(Ar+K)*dAr = (Cm/0.009)*[1-Av-0.4*B]/(Ar+K)*dAr          (6)

Now, let's plug in some simple numbers. Let Ar=40k, Av=30%=0.30, B=45%=0.45. With the values given above for Cm and K, this evaluates to

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dRm=0.14273*dAr      (7)

or in other words, it takes 7 armor to give you the same damage reduction as one point of mastery rating (or 1255 armor to match 1 point of mastery skill). You can double-check the numbers; plug 40k armor, 30% avoidance, and 45% block into the first two equations in this post. Then do it again for 40k armor but 47.25% block, and again for 41278 armor and 45% block. the last two should come out the same (0.2294), indicating that 1278 armor is the same as 1 mastery or 179.28 rating. Note that the exact values will vary as we change armor, avoidance, or block.

In any event, given this, 1 armor should be equivalent to ~1/7 a point of mastery, or 0.143 (14.3% as effective), rather than the 0.35 that Wrathblood found. On the other hand, in terms of itemization, it seems that you get 4 armor for every ipoint (trinkets give 1285 armor, 321 mastery/agi/etc., or 482 stam), making it 57.14% as good as mastery in terms of raw itemization. Thus, a mastery trinket should be better than an armor trinket in most cases (i.e. for blockable damage), ignoring on-use effects.

Considering the 160 armor enchant in this light, it's worth about 160/7 = 23 mastery, which is less than the 36 afforded by the Blocking enchant.

III. Avoidance and Armor:
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:

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1/A' = k/A + 1/C   (8)
-dA'/A'^2 = -k*dA/A^2
dA' = -k*dA*(A'/A)^2    (9)

And if we solve the DR equation for A'/A and plug in, we get dA' in terms of dA and A':

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dA' = (dA/k)*(1-A'/C)^2   (10)

Now that we have the post-DR avoidance gained by adding dA pre-DR avoidance to our existing A' post-DR (i.e. character sheet) avoidance, we can plug dA' in for dAv in (4) to get the equivalent to equation (5) for avoidance. I'll use dA here to indicate that it's pre-DR, since we're going to use that to convert to rating:

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dAv = dA' = (dA/k)*(1-A'/C)^2 = dAr*[1-Av-0.4*B]/(Ar+K)    (11)

Note: A' is only our post-DR dodge or parry (depending on which one you're considering for dA'), so it's either (char_sheet_dodge_% - 3.9705) or (char_sheet_parry_% - 5):

dA is simply equal to 0.01*dRv/Ca, the added avoidance rating divided by the avoidance rating conversion factor (Ca=176.7189) times 0.01 to put it in decimal notation. So plugging in for dA, we get:

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dRv = dAr*(100*k*Ca)/(1-A'/C)^2*[1-Av-0.4*B]/(Ar+K)     (12)

This is the equivalent to equation 6, with all of the same definitions for Av, B, and Ar. Plugging in Av=0.3, B=0.45, Ar=40k, A'=0.075 (i.e. 12.5% parry minus the base 5%) and the constants k,C, K, and [/b]Ca[/b], we get:

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dRv = 0.1543*dAr     (13)

Which is the avoidance equivalent to equation (7).


IV. Avoidance and Mastery:

This is easy, because since dAr=0, the first term in equation (4) is zero. We only need to solve:

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dAv = 0.4*dB

. Using (10) and the equations for dRm and dRv, we find:

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(0.01*dRv/Ca)/k*(1-A'/C)^2 = 0.4*0.0225/Cm*dRm
dRv = (Ca/0.01)*(0.009/Cm)*k/(1-A'/C)^2*dRm
dRv = (0.9*Ca*k/Cm)/(1-A'/C)^2*dRm              (14)

Plugging in the same numbers as before, we find that dRv = 1.0811*dRm at this level of diminishing returns, which is expected (mastery should eclipse parry at around 10% parry due to DR, we're at 12.5% parry). We can double-check that this gives us the right value of A' by letting dRv = dRm = 1 and solving for A':

A' = C*[1- sqrt(0.9*Ca*k/Cm)] = 5.1896%

Or 10.1896% on your character sheet, corresponding to 952 rating. Note that this is in excellent agreement with the numerical solution found here (the difference of 10 points is due to discretization; we were finding x parry rating such that x+10 parry rating made mastery and parry equivalent for damage reduction; the exact crossover point is thus 952 rating).

Interesting. Not surprisingly, I used a relatively similar methodology, though instead of solving as a problem I just put together some sample numbers, adjusted them, and then went back and measured the differences, which I then compared with the results I got for Mastery to get the relative weight.

I'm still reading through what you did, but while I puzzle it through, I though I'd note a couple things for clarification. First is that the value I got for Armor was for 1 itemization point worth of Armor not for 1 point of armor, so my result is low rather than high compared to yours. Second, its not off by quite as much as it seems because I was calculating based on getting 3.5 Armor per itemization point, and bringing it up to 4.0 moves Armor up to ~0.45 (though it jumping all the way up to 0.45 is a little worrying). So, still off, but closer. I'll comment further once I think it through a bit more.

Edit - Ah, figured out why the value moved so much when I went from 3.5 to 4.0 armor per itemization point. In the original case I wasn't giving the bonus armor credit for any armor multipliers (toughness, touched by the light, etc) and I fixed that when I went to 4.0, hence the size of the shift.

Ok, I think I've followed your math all the way through, and I haven't found a mistake with it, though I've also pondered my math a bit and haven't found an error with it (yet) either. At first, this made me think that the problem was me overstating the value of Mastery, making Armor look worse by comparison. However, your general avoidance number appears pretty close to my Parry number (~0.91 vs your ~0.925) which is calculated at 12.84% parry (close to your 12.5% avoidance; my dodge number is significantly different but since its at 16.35% you'd expect it to be) suggesting that the avoidance/mastery ratio appears valid.

Hmm, for comparison, here's exactly what I did (its pretty simple) with armor:

Buffed Armor of 40,477 which gives damage reduction of 0.554104

I then added 10 itemization points of armor (40 armor) which, after buffs, increased armor to 40,522 which gives damage reduction of 0.554378.

Comparing the two numbers gave a 0.04952% reduction in damage taken thanks to the additional armor. Since 10 Itemization points of Mastery gave a 0.11061% reduction in damage taken, the weighting ends up around 0.45.

It occurs to me that I'm basing my math off itemization points while you're working off the % and then working backwards to the itemization points. Could some buff be accounted for differently in the different approaches and thus skew the value of one stat or the other?

wrathblood wrote:Edit - Ah, figured out why the value moved so much when I went from 3.5 to 4.0 armor per itemization point. In the original case I wasn't giving the bonus armor credit for any armor multipliers (toughness, touched by the light, etc) and I fixed that when I went to 4.0, hence the size of the shift.

You have to be careful there, actually. Only the base armor on items gets the toughness and meta gem modifiers. Bonus armor (i.e. "green" armor on an item), armor from enchants, trinkets, consumables (elixir), and Devotion Aura does not get modified by toughness or the meta gem.

I probably should have specified this in the derivation, but that's why I didn't include any of those modifiers on dAr. We're usually trying to compare one of the "special" types of armor, like trinkets or enchants, that doesn't get the extra benefit.

wrathblood wrote:It occurs to me that I'm basing my math off itemization points while you're working off the % and then working backwards to the itemization points. Could some buff be accounted for differently in the different approaches and thus skew the value of one stat or the other?

I don't think so. Looking at your process, it seems like you're calculating the relative change in damage taken and I'm calculating absolute change. In other words, I'm calculating dD/Do, which is the change in damage taken relative to raw boss hit size. You're calculating dD/D, change in damage taken relative to the size of the mitigated hit before the additional armor. That will probably skew the numbers enough to explain the difference. It's questionable which is the more relevant parameter; I could see arguments for both.

Would that really explain it? If Mastery and Armor have a certain ratio of damage prevented in absolute terms, shouldn't it be roughly the same in relative/mitigated terms?

If it is indeed the difference, then the appropriate methodology would presumably be driven by the intended usage. For me, I want a quick and dirty "How can I tell which piece of gear is better? (usually trinkets, but not always)". So the decision is never made in a vacuum, its always made in the context of a certain level of gear and comparing one or more pieces against each other. Thus, relative damage saved would be more useful.

If I was constructing a ground-up damage taken model, I'd want the absolute measurement, because that's where I'd be starting.

wrathblood wrote:Would that really explain it?

No, actually, What I said was completely wrong, the ratio should be exactly the same for dD/D. I was thinking of the absolute values of dD/D and dD/Do, which are obviously different. But the ratio of armor to avoidance or mastery that we calculate from those two expressions is exactly the same.

Which means there's some other explanation for the discrepancy. I'll try and fiddle with your numbers after raid.

Thanks Theck, I came here today just for this sort of info.

Ive thought for a while Blizzard has been stingy with AC on items.

wrathblood wrote:Hmm, for comparison, here's exactly what I did (its pretty simple) with armor:

Buffed Armor of 40,477 which gives damage reduction of 0.554104

I then added 10 itemization points of armor (40 armor) which, after buffs, increased armor to 40,522 which gives damage reduction of 0.554378.

Comparing the two numbers gave a 0.04952% reduction in damage taken thanks to the additional armor. Since 10 Itemization points of Mastery gave a 0.11061% reduction in damage taken, the weighting ends up around 0.45.

It occurs to me that I'm basing my math off itemization points while you're working off the % and then working backwards to the itemization points. Could some buff be accounted for differently in the different approaches and thus skew the value of one stat or the other?

I think I found the error. You're not comparing like quantities. I had to make some guesses here about your mastery process, so feel free to correct me if those assumptions are wrong.

Assumptions based on your earlier posts:
Ar=40477,
dAr=45 (incorrect for most types of armor, but we'll keep this to make sure we get your values),
Av=12.84%+16.35%+5%=34.19%=0.3419,
dB=10*0.0225/179.28=0.001344980.
From those values I'm inferring a block chance of B=51.06%=0.5106 . I started with 45% here and got answers close enough to your mastery reduction percent to feel certain this was the right avenue (0.105006% @ B=45%), so I back-calculated from your exact value to figure out the most likely Block % that the spreadsheet gave you.

Mastery
(assumption): You added 10 mastery to the spreadsheet and wrote down the DTPS for before and after (D and D'). You calculated the reduction as 1-D'/D. When I do that explicitly with my expression for D (eqs 1 & 2), I get:
D = 0.2023744837
D' = 0.2021506403
1-D'/D = 0.0011060854 = 0.11061% (0.105% @ 45% block) This is exactly equivalent to calculating the damage reduction of 10 mastery by calculating the amount of block dB that mastery adds, and plugging into dD/D=0.4*dB/[1-Av-0.4*B]:

dD/D = 0.0011060854

It should be obvious why, because plugging D'=D+dD into 1-D'/D and simplifying gives you my expression for dD/D.

In other words, the mastery reduction you're calculating is relative damage reduction (because it's based on dD/D). If you were taking 10k DPS before adding the gem, you'd take 9.9889k DPS afterwards (99.889%, or 0.99889=1-0.11061).

Note that this is different than what I've calculated; dD/Do is absolute damage reduction, which gives dD/Do = -Fa*0.4*dB. However, this doesn't effect the relative worth of each stat at all, as we'll see later.


Armor
Here, you calculated the armor mitigation factors Ma and Ma' for the before and after cases, and evaluated 1-Ma/Ma'. Using the assumed values above:
Ma = 0.5541037242
Ma' = 0.5543782364
1-Ma/Ma' = 0.0004951713 = 0.04952%

However, it's worth noting that this is not equivalent to 1-D'/D. It's clear from equation (1) that D'/D in this case is:
D'/D = Fa'/Fa = (1-Ma')/(1-Ma),
and
1-D'/D = 1-(1-Ma')/(1-Ma) = (Ma'-Ma)/(1-Ma)

In other words, 1-Ma/Ma' isn't a relative damage reduction metric, and shouldn't be compared to your earlier value of mastery. The proper value to compare would be 1-(1-Ma')/(1-Ma) or (Ma'-Ma)/(1-Ma). When I evaluate that, I get:
1-D'/D = (Ma'-Ma)/(1-Ma) = 0.0006156414 = 0.06156414%

Note that this is the same as what we get analytically from dD/D when dAv=dB=0:

dD/D = -dAr/(Ar+dAr+K) = -0.0006156414 = 0.06156414

(I've cheated a bit here; technically this is the finite difference rather than the derivative dD/D = -dAr/(Ar+K). The two converge when dAr is truly infinitesimal).

And comparing that to the value for mastery, we get armor clocking in at 0.5566, or 55.66%. I should note here that using dAr=40 armor instead gives 0.4948, lower than the ~0.57 value I got because of the increased armor, avoidance, and block used in your calculations. If I use Ar=40k, Av=30%, and B=45%, I get the value I posted in the OP.


What exactly is the 1-Ma/Ma' value you calculated? Well, if you use Ma=1-Fa and Ma'=1-Fa', you can re-write it as:
(Fa-Fa')/Ma' = (D-D')/(D*-D)
where D* = D/Fa = Do*[1-Av-0.4*B]. D* is the damage you would take if you didn't have any armor at all. In other words, the value you calculated represents the reduction in damage taken due to the extra armor relative to the reduction in damage taken due to all of the armor that preceded it. I'm not sure whether this value has any importance; it could be interesting, but in any event it's definitely not going to give you a fair comparison to the relative damage reduction calculated for mastery.


A note on relations between dAr, dAv, and dB:
This brings us to the comment I made in my last post - I was thinking about the fact that dD/Do and dD/D have different values (as seen in this post's derivation). Without thinking too hard about it (because I was in a hurry), I suggested that this could give us different relative values, which is incorrect. If you solve for one of (dAr, dAv, dB) in terms of another with either the dD/Do expression or the dD/D expression, you'll get exactly the same answer. It should be obvious why; all three differential factors only appear in dD, and relations can be found by setting dD=0. You can divide dD by anything you want (provided it's not 0 or infinity), and it won't change those relations. In other words, if solving F(x,y)=0 gives me x=1+y, then solving F(x,y)/C=0 will still give me x=(1+y).

The problem in your calculation occurred because you were comparing between different metrics. An analogous error would be to find dD/D due to 10 mastery and compare that to dD/Do due to 40 armor. You'd be comparing apples to oranges, and thus get a fruit basket instead of a reasonable result.

Just noticed that you can use this formulation to very easily compare our two meta gem choices.

Let D be our damage intake without either meta, Da be our damage intake with the Armor meta, and Db be our damage intake with the block meta.

D is given by equation (1). Da and Db are straightforward modifications:

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D  = K/(Ar+K)*(1-Av-0.4*B)      (20a)
Da = K/(1.02*Ar+K)*(1-Av-0.4*B) (20b)
Db = K/(Ar+K)*(1-Av-0.4*B-x*B)  (20c)

Where I've plugged in Fa = K/(Ar+K) in each case, and let the block meta have value x so we can test both 5% (tooltip, bugged) and 1% (current) versions. The damage reduction of each meta is 1-Di/D:

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1-Da/D = 1 - (Ar+K)/(1.02*Ar+K) = 0.02*Ar/(1.02*Ar+K)           (21a)
1-Db/D = 1 - (1-Av-0.4*B-x*B)/(1-Av-0.4*B) = x*B/(1-Av-0.4*B)   (21b)


We can figure out what x needs to be to match the Armor meta by solving 1-Db/D = 1-Da/D:

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x*B/(1-Av-0.4*B)=0.02*Ar/(1.02*Ar+K)
x=(0.02*Ar/B)*(1-Av-0.4*B)/(1.02*Ar+K)    (22)


Plugging in Ar=40k, B=0.5, and Av=0.35, we get x=0.0098, or 0.98%. So in other words, even the "bugged" 1% block meta gives us more net damage reduction against blockable damage than the 2% armor meta does. It's a close call though, if you divide (21a) by (21b) for x=0.01, you find that the armor meta is 98.13% as effective as the block meta. However, for x=0.05, the block meta gets 5x more powerful, and the effectiveness of the armor meta drops by a factor of 5 (as expected) to 19.63%.

It's worth noting that this is also fairly sensitive to B and Av. Dropping Av to 0.3 and B to 0.45 is enough to cause a massive swing, such that the armor meta is 26% more effective than the block meta.

Also note that this is only looking at blockable damage. Given that the armor meta is applicable to all physical damage, it's the better choice as long as the efficiencies are close.

This might explain why the block meta is still "bugged," by the way. At 5% it is phenomenally good, at 1% it's right on par with the Armor meta. It may not be "bugged" at all - this might be an intentional balancing change they decided to make during beta, and they just never updated the tooltip. If that's the case, I'd argue they erred, because the block meta needs to be slightly better against blockable damage to be balanced against an Armor meta that applies to everything. If they made it 2% block value, it would be just about right (~2x as good against blockable damage, but useless while stunned/attacked from behind/etc.).

You're absolutely right about what I was doing with armor. In a nutshell, I was calculating X/Y when I should have been calculating X/(1-Y). Using actual numbers I was calculating 1/0.55 when I should have been calculating 1/0.45, resulting in a solution ~25% higher. Recrunching my numbers, my answer agrees with yours.

Its funny, while reading your post for the 2nd time, I was struck by an insolvable problem. Was the amount of time it was taking me to parse exactly what you were saying the reason I went for the MBA instead of graduate work in Economics, or would I have been able to read it faster had I gone for the grad work in economics instead of the MBA. Chicken and egg.

wrathblood wrote:Its funny, while reading your post for the 2nd time, I was struck by an insolvable problem. Was the amount of time it was taking me to parse exactly what you were saying the reason I went for the MBA instead of graduate work in Economics, or would I have been able to read it faster had I gone for the grad work in economics instead of the MBA. Chicken and egg.

If it helps any, I have the same thoughts from the other side of the coin. When reading scientific papers, I sometimes question whether I should have just gotten a job in engineering with my bachelor's.

"Wait, you want me to do what with a rank 4 tensor??"

And I thought my math teacher was trying to church his profession up when he said You'll use math everywhere.

Not to say TLDR - I love this thread - but what have we arrived at from this from a gearing perspective?

Let's see. Firstly, we have this gem:

Theck wrote:Considering the 160 armor enchant in this light, it's worth about 160/7 = 23 mastery, which is less than the 36 afforded by the Blocking enchant.

So the blocking enchant is worth about one and a half times the armor enchant, assuming the damage can be blocked.

Secondly, we can answer that damn Meta question that's been plaguing some of us (me at least):

Theck wrote:Plugging in Ar=40k, B=0.5, and Av=0.35, we get x=0.0098, or 0.98%. So in other words, even the "bugged" 1% block meta gives us more net damage reduction against blockable damage than the 2% armor meta does. It's a close call though, if you divide (21a) by (21b) for x=0.01, you find that the armor meta is 98.13% as effective as the block meta.

The block meta scales better. Given that armor scaling is mostly nominal compared to avoidance scaling, tanks with less than 50% block and 35% avoidance should get the Austere gem. Tanks who reach that avoidance level should consider getting the Eternal gem. Around that level it's kind of a wash, so I'd only consider changing gems whenever you get a helmet upgrade unless you have metas to burn for some reason. (As an addendum, this math is within a few avoidance percent points of accurate for warriors, too, unless I missed a variable.)


Are these accurate conclusions to draw?

theothersteve7 wrote:Not to say TLDR - I love this thread - but what have we arrived at from this from a gearing perspective?

....

Are these accurate conclusions to draw?

Yep. Keep in mind that the "point" of this thread was to have the derivation all written out somewhere convenient, so that when I need to use it to draw conclusions in other threads, I have something to point to.

That said, it seems that I'm getting enough mileage out of this framework that it might be worth pretty-ing it up and putting together a list of conclusions at the end. Maybe including the comparison of Stamina to Mastery in terms of average TTL. I'll add that to the list of possible "holiday break projects."