[Derivation] Armor/Mastery/Avoidance Calculations

Warning: Theorycraft inside.

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[Derivation] Armor/Mastery/Avoidance Calculations

Postby theckhd » Sat Jan 01, 2011 11:59 am

This is a cleaned-up version of this derivation thread. The point is to present the general expression for damage intake, and use that expression to derive useful relationships between avoidance, mastery, and armor.

The types of questions we're trying to answer are, "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).

Each relationship will get its own post, to make it easier to link back to individual calculations. I'll reserve some posts for future use as well. If you have a question that you think we can answer with this framework, feel free to post it and I'll take a crack at it.


Table of Contents
  1. Damage Taken Formula
  2. Holy Shield Model
  3. Mastery & Armor
  4. Avoidance & Armor
  5. Avoidance & Mastery
  6. Meta Gems
  7. Reforging an Avoidance/Threat combination item
  8. Conclusions (TLDR summary)

In addition, I've written a MATLAB script that performs all of the calculations in this thread. It doesn't use anything too fancy, so it should work in the free MATLAB alternatives (FreeMat, Octave, etc.). You can download it here if you want to fool around with it or plug your own values in.
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Postby theckhd » Sat Jan 01, 2011 12:00 pm

I. Damage taken formula and problem set-up

For a boss melee swing of damage Do, the actual damage we take is
Code: Select all
D = Do*Fa*S*[0*Av + (1-Bv)*Bc + 1*(1.024-Av-Bc)] = Do*Fa*S*[1.024-Av-Bv*Bc]         (I.1)

Where Av is your decimal avoidance (i.e. 30%=0.3, the sum of parry and dodge from your character sheet), Bv is your decimal block value (the amount of an attack you block, or 30%=0.3 baseline in 4.2), Bc is your decimal block chance, S is the Sanctuary damage reduction factor (S=0.9), 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)            (I.2)
dFa = -dAr*Fa/(Ar+K)                              (I.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:

Code: Select all
dD/Do = dFa*S*[1.024-Av-Bc*Bv] + Fa*S*[-dAv-Bv*dBc-Bc*dBv]
      = -dAr*S*Fa/(Ar+K)*[1.024-Av-Bc*Bv] + Fa*S*[-dAv-Bc*dBv-Bv*dBc]
      = -dAr*S*Fa/(Ar+K)*[1.024-Av-Bc*Bv] - Fa*S*[dAv+Bc*dBv+Bv*dBc]           (I.4)

From this expression, we can start making comparisons between the different types of avoidance/mitigation.
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Postby theckhd » Sat Jan 01, 2011 12:01 pm

II. Holy Shield Model

Before we go any further, we need to briefly discuss Holy Shield. To get anything useful out of these equations, we'll need to assume a value for Bv, the time-averaged block value, which means we need to decide on a model for Holy Shield. There are a few easy models we can try, and together they should give us a fairly complete picture.

Model A: The first is to ignore it entirely, and assume a static value of Bv=0.3 (or 0.31 with the block meta gem). This will give us a set of "worst-case" values that tell us how things look when HS is on cooldown.

Model B: The second is to us a simple model where we use it on cooldown, and use the time-averaged block value Bv=0.3667. This serves as a model for a "sloppy" tank, but will also give us more reasonable values than ignoring it.

Model C: Another way is to use the 4.1 value of Bv=0.4. This is probably an optimistic estimate of the upper bound that a tank could reasonably achieve by precisely timing Holy Shield to maximize its effectiveness. It also gives us a reference point for comparing pre-patch to post-patch.

Model D: Finally, we could consider the case where Holy Shield is always on, or Bv=0.5. While this is definitely an over-estimate of a time-averaged block value, but it does give us values representing the period while the new HS is active. This might be useful for players trying to optimize the survivability boost of HS at the expense of average damage reduction.

In the subsequent derivations, I'll be evaluating expressions for all three of these block values and displaying the results in a table to show the variation with block value model.
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Postby theckhd » Sat Jan 01, 2011 12:01 pm

III. Mastery and Armor

To determine an equivalency between Armor and Mastery for damage taken, we want to set the two terms of equation (I.4) equal to one another and solve for either dBc or dAr. dBc is both easier and slightly more logical, so let's do that. We'll ignore dAv and dBv for now by setting it equal to zero.

Thus, we get
Code: Select all
Bv*dBc = dAr*[1.024-Av-Bv*Bc]/(Ar+K)            (III.1)


dBc 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, dBc = dRm*0.0225/Cm 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:

Code: Select all
dRm/dAr = (Cm/(0.0225*Bv))*[1.024-Av-Bv*Bc]/(Ar+K)    (III.2)


Now, let's plug in some simple numbers. Let Ar=40k, Av=35%=0.35, Bc=55%=0.55. With the values given above for Cm and K, this evaluates to

Code: Select all
Bv             0.3     0.3667        0.4        0.5
dRm/dAr     0.1863     0.1414     0.1246    0.08761
dAr_1r       5.368      7.071      8.025      11.41
dAr_1s       962.4       1268       1439       2046
160/dAr_1r   29.81      22.63      19.94      14.02


In other words, it takes 1/0.1863=5.37 armor to give you the same damage reduction as one point of mastery rating without Holy Shield, 7.1 armor with the "simple" model, 8.0 with the old 4.1 Holy Shield, and 11.4 with 50% block value. That works out to 961, 1268, 1439, or 2046 armor per one point of mastery skill, respectively. Note that the exact values will vary as we change armor, avoidance, or block.

In any event, given the simple model, 1 armor should be equivalent to ~1/7 a point of mastery, or about 14% as effective. In terms of itemization, we only get 4 armor for every ipoint (trinkets give 1285 armor, 321 mastery/agi/etc., or 482 stam), making it only about 57% as effective 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. Note that this is true even for the worst-case HS model, in which armor is only 74% as effective as mastery.

Considering the 160 armor shield enchant in this light, it's worth about 160/5.37 = 29.8 mastery, which is less than the 50 afforded by the Mastery enchant. The value drops even further in the models with higher Bv.
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Postby theckhd » Sat Jan 01, 2011 12:01 pm

IV. Avoidance and Armor:

You can do a similar calculation for avoidance instead of block. To do so, we need to employ the diminishing returns equation, which applies to dodge and parry separately. The DR equation is
Code: Select all
1/A = k/a + 1/C   (IV.1)

a and A are the pre-DR and post-DR avoidance percentages, respectively, and k and C are the avoidance constants found here (k=0.9560 and C=0.65631440 at level 85 in our notation). By convention, lowercase variables indicate a value before diminishing returns is applied, uppercase represents a value that's already had diminishing returns applied.

Thus if we're considering dodge, a is the pre-DR dodge percentage from dodge rating, which can be read off of the character sheet tooltip. Similarly, if we were working out diminishing returns for parry, a would be the pre-DR parry gained through all parry rating sources. If we wanted to specify which type of avoidance we're talking about, we could put subscripts on A and a - i.e. Ad and ad for dodge and Ap and ap for parry. To get Av from these values, we'd need to add the Ad value for dodge and the Ap value for parry to our base miss, dodge, and parry chances. In other words,
Code: Select all
Av = 5% + [5% + Ap] + [5% + Ad].

This also means that if we differentiate Av, we simply have dAv=dAp+dAd. Generally we'll only be varying one of the avoidance sources at a time, which means that we will usually drop the subscripts on A and a, and can equate dAv and dA directly in these calculations. For the rest of these derivations, we'll assume that A and a represent dodge for clarity.


We start our derivation by differentiating the diminishing returns equation and solving for dA in terms of da, A and a:
Code: Select all
1/A = k/a + 1/C
-dA/A^2 = -k*da/a^2
dA = k*da*(A/a)^2    (IV.2)


And if we solve the DR equation for A/a and plug in we can eliminate a:

Code: Select all
dA = (da/k)*(1-A/C)^2   (IV.3)

Now that we have the post-DR dodge percentage dA gained by adding da pre-DR dodge percent to our existing A post-DR dodge, we can plug dA in for dAv in (I.4) to get the equivalent to equation (III.1) for avoidance. The goal is to relate dAr to da here since we're going to use the pre-DR values to convert to rating:

Code: Select all
dAr*[1.024-Av-Bv*Bc]/(Ar+K) = dAv = dA
dAr*[1.024-Av-Bv*Bc]/(Ar+K) = (da/k)*(1-A/C)^2       (IV.4)


Reminder:
A is only our post-DR dodge, so it's (char_sheet_dodge_% - 5).
Av is our total avoidance, including base avoidance (char_sheet_dodge%+char_sheet_parry%+char_sheet_miss%).

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:

Code: Select all
dRv/dAr = (100*k*Ca)/(1-A/C)^2*[1.024-Av-Bv*Bc]/(Ar+K)     (IV.5)


This is the equivalent to equation (III.2), with all of the same definitions for Av, Bc, and Ar. Plugging in Av=0.35, Bc=0.55, Ar=40k, A=0.1 (i.e. 15% dodge minus the base 5%) and the constants k,C, K, and Ca, we get:

Code: Select all
Bv          0.3     0.3667        0.4        0.5
dRv/dAr  0.1649      0.153     0.1471     0.1293
dAr_1r    6.064      6.535      6.798      7.735
dAr_1%     1072       1155       1201       1367


Which is the avoidance equivalent to the Mastery/Armor table in section III. dAr_1r is the armor required to match the average damage reduction of one point of avoidance rating, which falls between 6 and 7. dAr_1% is how much armor it takes to match 1% avoidance.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby theckhd » Sat Jan 01, 2011 12:02 pm

V. Avoidance and Mastery:

This is easy, because since dAr=0, the first term in equation (I.4) is zero. We only need to solve:
Code: Select all
dAv = Bv*dBc
Using (IV.3) and the equations for dRm and dRv, we find:

Code: Select all
(0.01*dRv/Ca)/k*(1-A/C)^2 = Bv*0.0225/Cm*dRm
dRv = (100*Ca)*(0.0225*Bv/Cm)*k/(1-A/C)^2*dRm
dRv = (2.25*Bv*Ca*k/Cm)/(1-A/C)^2*dRm              (V.1)


Plugging in the same numbers as before, we find that dRv = 1.082*dRm for the simple HS model at this level of diminishing returns, indicating that we're above the crossover point where mastery becomes better than avoidance. We can explicitly calculate the crossover point A=Ax by letting dRv = dRm = 1 and solving for Ax:

Code: Select all
Ax = C*[1 - sqrt(2.25*Bv*Ca*k/Cm)]         (V.2)


Code: Select all
Bv          0.3     0.3667        0.4        0.5
dRv/dRm  0.8853      1.082       1.18      1.476
A_x       13.29       7.76       5.19     -1.945
A_x+5     18.29      12.76      10.19      3.055
R_x        2815       1487        952     -319.1


For the simple HS model, that's 12.76% on your character sheet, corresponding to 1487 rating. This increases to 18.29% if we don't use Holy Shield at all, and drops to 10.19% with the 4.1 Holy Shield model (952 rating). While Holy Sheld is active, however, the crossover point is below zero, and mastery is strictly superior.

Finally, it might be handy to have a comparison of avoidance and mastery for combat table coverage. This one is even easier, since we just have to solve

Code: Select all
dA=dBc           (V.3)


Again, using (IV.3) and the equations for dRm and dAv, we have,

Code: Select all
(da/k)(1-A/C)^2 = dBc
(0.01*dRv)/(k*Ca)*(1-A/C)^2 = (0.0225/Ca)*dRm
dRv = (2.25*k*Ca)/(Cm*(1-A/C)^2)*dRm


Plugging in our values, we get dRv = 2.95*dRm. In other words, one point of mastery rating is worth about 3 points of either avoidance rating for combat table coverage.
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Postby theckhd » Sat Jan 01, 2011 12:04 pm

VI. Meta Gem Comparison

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 (I.1). Da and Db are straightforward modifications:

Code: Select all
D  = Do*S*K/(Ar+K)*(1.024-Av-Bv*Bc)     
Da = Do*S*K/(1.02*Ar+K)*(1.024-Av-Bv*Bc) (VI.1)
Db = Do*S*K/(Ar+K)*(1.024-Av-(Bv+x)*Bc)  (VI.2)


Where I've plugged in Fa = K/(Ar+K) in each case, and let the block meta have value x (just in case they decide to toy with its value again). The damage reduction of each meta is 1-Di/D:
Code: Select all
1-Da/D = 1 - (Ar+K)/(1.02*Ar+K) = 0.02*Ar/(1.02*Ar+K)    (VI.3)
1-Db/D = 1 - (1.024-Av-(Bv+x)*Bc)/(1.024-Av-Bv*Bc)
       = x*Bc/(1.024-Av-Bv*Bc)                           (VI.4)


To compare the two meta gems, we'll calculate the Armor crossover point - in other words, the value of Ar does it take for the two gems to mitigate the same amount of damage. To do this we solve 1-Db/D = 1-Da/D for Ar and plug in x=0.01:

Code: Select all
0.01*Bc/(1.024-Av-Bv*Bc)=0.02*Ar/(1.02*Ar+K)
(1.02*Ar+K) = 2*Ar*(1.024-Av-Bc*Bv)/Bc
Ar*[2(1.024-Av-Bc*Bv)/Bc - 1.02] = K
Ar = K/[2(1.024-Av-Bc*Bv)/Bc - 1.02]                     (VI.5)


Something interesting happens once we reach block cap. In that circumstance, Bc=1.024-Av; in other words, your block chance Bc fills up the full remainder of the 102.4% CTC that Avoidance hasn't covered already. Plugging that in, we find that block chance and avoidance completely drop out of the equation:

Code: Select all
Ar = K/[2*(1.024-Av-(1.024-Av)*Bv)/Bc - 1.02]
   = K/[2*(1.024-Av)(1-Bv)/Bc - 1.02]
   = K/[2*(Bc)(1-Bv)/Bc - 1.02]
   = K/[2*(1-Bv) - 1.02]                                 (VI.6)


As usual, we'll evaluate these equations at several values of Bv for Av=0.35. For (VI.5), we'll also use two different values of Bc (0.45 and 0.55). These will give us the armor break points we're interested in:

Code: Select all
Bv          0.3     0.3667        0.4        0.5
AR(55%B)   39.2       46.7      51.63      75.59
Ar(45%B)  23.68      26.22      27.71      33.39
Ar(cap)=  85.72      132.1        181      -1629


These armor values are given in thousands, and the average raid-geared tank should have around 40k armor. So in other words, the 1% block meta is almost identical to the 2% armor meta in the absence of Holy Shield for a tank sitting at 55% block and 35% avoidance, a little over 10% below block cap. It also gives that tank more net damage reduction against blockable damage than the armor meta does with almost any model of Holy Shield.

However, the results are fairly sensitive to B and Av. Dropping Av or Bc (in this case, setting Bc=0.45) is enough to push the armor meta is back ahead of the block meta. This is shown in the third line of the table.

On the other hand, once you reach block cap the armor break point goes up dramatically. Even without using Holy Shield, a block-capped tank would need over 85k armor before the armor meta would end up mitigating more damage. Factoring in any model of Holy Shield usage inflates that above 130k; worse yet, during the period where Holy Shield is active, there's literally no amount of armor that will allow the armor meta to catch up.

Given this, your choice will probably depend on your gear level. For a tank that's far from block cap, the armor meta is the stronger choice. If you're only about 10% away, then the two are pretty close and it's a toss-up. But once you're homing in on block-cap (<5% away), the block meta becomes far and away the better choice, especially during your Holy Shield uptime.
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Postby theckhd » Sat Jan 01, 2011 12:04 pm

VII. Reforging a combination Threat/Avoidance item

If an item has Rv avoidance rating and Rt threat rating, which one is better to reforge into mastery? The answer depends on whether you want to minimize damage taken or maximize combat table coverage (i.e. minimizing the chance to take an unblocked hit).

To minimize damage taken, we need only consider the second term of equation (I.4), as dAr=0 in this comparison. We can also ignore Fa and S, giving us only:

Code: Select all
dAv + Bv*dBc            (VII.1)


We can express dAv, the post-DR avoidance gain, in terms of da and A using equation (IV.3). Once again, note that A is only our post-DR dodge, so it's (char_sheet_dodge_% - 5).

For this derivation, it's worth noting that we can treat avoidance diminishing returns as modifying Ca, the rating to avoidance conversion factor. If we define Ca' as the "effective" conversion factor post-DR, we can re-write equation (10) in terms of ratings and express Ca' in terms of Ca, k, C, and A:

Code: Select all
dA=dRv/Ca' = dRv/(k*Ca)*(1-A/C)^2     (VII.2)
Ca' = k*Ca/(1-A/C)^2                  (VII.3)


We will also define Cm' = Cm/2.25 because it simplifies later expressions. Cm' can be interpreted as the amount of rating required for 1% block. We can represent dBc and dAv as:
Code: Select all
dBc = dRm/Cm'                 (VII.4)
dAv = dRv/Ca'                 (VII.5)


Thus, we can rewrite expression (VII.1):

Code: Select all
dRv/Ca' + dRm*Bv/Cm'         (VII.6)


We want to compare two specific configurations; one with Rv avoidance and 0.4*Rt mastery (reforged from threat rating), and one with 0.6*Rv avoidance and 0.4*Rv mastery. The change in avoidance rating between the two situations is dRv=(Rv-0.6*Rv), while the change in mastery is dRm=0.4*(Rt-Rv):

Code: Select all
(Rv-0.6*Rv)/Ca' + (Rt-Rv)*0.4*Bv/Cm'        (VII.7)


We can set this expression equal to zero and solve for Rt to find the break-even points:

Code: Select all
(Rv-0.6*Rv)/Ca' + (Rt-Rv)*0.4*Bv/Cm' = 0
(Rt-Rv)= -Cm'/(Bv*Ca')*Rv
Rt    = [1-Cm'/(Bv*Ca')]*Rv                       
Rt/Rv = [1-Cm/(2.25*k*Bv*Ca)*(1-A/C)^2]       (VII.8)


Plugging in the value of A=0.1 that we used earlier, we can calculate the ratio Rt/Rv:

Code: Select all
Bv               0.3   0.3667     0.4     0.5 
Rt/Rv (TDR)  -0.1295  0.07591  0.1528  0.3223 


The negative sign in the first column means that it's always better to reforge the threat stat in the absence of HS. For the simple HS model, the break-even point occurs when the amount of threat rating on the item is 7.6% of the avoidance rating on the item. For an item with less threat rating, you would be better off reforging the avoidance rating to mastery. Since this should not be the case for nearly all (all?) items available right now, you will in general get a larger reduction in damage intake by reforging the threat rating to mastery. In the 4.1 implementation of HS the break-even point is 15.3%, and model D gives 32.23%.



To maximize combat table coverage, the relationship is a little different. The break-even point is found by minimizing

Code: Select all
dAv + dBc         (VII.9)


We again use equations (33)-(34) to express this in terms of rating (dRv,dRm) in a similar fashion to equation (36) and plug in our two configurations:
Code: Select all
(Rv-0.6*Rv)/Ca' + (Rt-Rv)*0.4/Cm' = 0  (VII.10)


Solving that equation, we find that

Code: Select all
Rt = (1-Cm'/Ca')*Rv    (VII.11)


If you ignore diminishing returns (A=0), this gives you (1-Cm/(2.25*k*Ca)) = 0.5284, or about 53%. Using a more realistic value of A=0.1, we get

Code: Select all
Ca'            235.1
Cm'            79.68
1-Cm'/Ca'     0.6611


or 66%.

So for most practical purposes, if the threat rating on the item is less than about 66% of the avoidance rating on the item, you get better combat table coverage by reforging the avoidance rating. This value will increase as our avoidance goes up due to diminishing returns on avoidance.
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Postby theckhd » Sat Jan 01, 2011 12:04 pm

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Postby theckhd » Sat Jan 01, 2011 12:05 pm

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Postby theckhd » Sat Jan 01, 2011 12:05 pm

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Postby theckhd » Sat Jan 01, 2011 12:05 pm

VII. Conclusions (aka TLDR)

  • Mastery & Armor
    • 1 mastery rating ~ 7 armor for damage reduction
    • The 160 armor shield enchant is only worth ~30 mastery at best, so the 50 mastery enchant is better for blockable damage.
  • Avoidance & Armor
    • 1 avoidance rating ~ 6-7 armor for damage reduction at projected avoidance levels for T12
  • Avoidance & Mastery
    • 1 mastery rating is better than 1 dodge or parry rating for total damage reduction at 1487 parry rating, or 12.76% on the character sheet, based on a simple model of HS that assumes 33% uptime. The crossover point is 18.29% (2815 rating) in the absence of HS.
  • Meta Gems
    • At 35% total avoidance and 55% block, the block meta and armor meta are roughly equal in the absence of HS. Increasing HS uptime pushes the block meta ahead, as would block-capping.
    • At lower avoidance/block (30%/50%), the armor meta is more effective than the block meta.
  • Reforging threat/avoidance items
    • For reducing damage taken, reforge the threat rating into mastery.
    • For maximizing combat table coverage, reforge the threat rating if it's at least 66% of the avoidance rating. Otherwise reforge the avoidance rating to mastery.
"Theck, Bringer of Numbers and Pounding Headaches," courtesy of Grehn|Skipjack.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby ck5uperman » Mon Jan 03, 2011 9:40 pm

Would the Effulgent Shadowspirit Diamond be up for consideration?
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby Digren » Wed Jan 05, 2011 12:11 pm

Most of the contents of this post were made obsolete with the change to agility with patch 4.2. Agility no longer provides dodge.

Strength provides parry rating at a rate of 27%, i.e. 27% of your strength will appear as parry rating in your parry rating total. When evaluating gear, remember that the printed strength will be increased by 5% with Blessing of Kings, so for gear evaluation use 28.35% of strength as parry rating.
Last edited by Digren on Sun Jun 19, 2011 10:48 pm, edited 3 times in total.
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Re: [Derivation] Armor/Mastery/Avoidance Calculations

Postby theckhd » Wed Jan 05, 2011 12:26 pm

ck5uperman wrote:Would the Effulgent Shadowspirit Diamond be up for consideration?


Not especially, for two reasons:
  1. Large magical attacks don't seem to be as frequent or large as they were in the first few tiers of Wrath. In most cases, we seem to be dying to a string of melee attacks, and the vast majority of damage taken is physical. Both of those make the Effulgent diamond less attractive.
  2. This derivation is only valid for blockable physical damage. Magical damage is neither of those, obviously. So to do a rigorous derivation that includes multiple sources of damage, we'd need to go back to the methodology used in the Total EH thread. I'm not sure it's worth going to that much trouble, given #1


Digren wrote:This seems to fit best in this thread.

I agree. I'll see if I can incorporate it in a logical manner. The take-home message seems to be, "1 STR gives 0.25 Parry Rating, 1 AGI gives 0.61 Dodge Rating. Converting an item with X STR to an item with X agility is roughly equivalent to adding 0.36*X avoidance rating, provided that your character sheet dodge and parry percentages are about the same."

It's tough to come up with anything more accurate than that, since it changes with dodge and parry DR.
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