"Total" EH  incorporating different damage types into EH
Moderators: Fridmarr, Worldie, Aergis, theckhd
"Total" EH  incorporating different damage types into EH
This thread in the Gear forum got me thinking about the appropriate way to estimate Effective Health. This post will serve as a quick "Basics of EH" calculation, as well as a demonstration of how we can incorporate magical and bleed damage into the mix to get a more versatile and realistic definition of Effective health.
Note: This is not an "armor is bad" screed. Armor is exceptionally good, as covered in section III. However, it's also good to know where your limitations are. I also think that it's good to have a complete picture of what's going on, even if in practice you use a ruleofthumb rather than the full equations. The point of this post is simply that  to have the "full" derivation written down somewhere.
Note #2: I've updated the values for patch 4.0.1, and rerun the graph. However, I have not modified the entire derivation to include the new block mechanic. Keep this in mind when using this math for anything practical  shaving 30% off of every melee hit via block effectively adds 30% mitigation to all physical damage, in essence adding a factor of (1Mb)=0.7 anywhere you see (1Ma).
I. Basics of Effective Health Theory
EH theory isn't very complicated fundamentally. The basics are outlined here and here, but I'll try and present a slightly more accessible derivation:
Consider a boss that hits for purely physical damage. His hits deal raw damage D. If your physical mitigation is M, then the damage that shows up in the combat log is d:
In other words, if the boss hits for an unmitigated D=20k, and I have M=50%=0.5 mitigation (from all sources, but primarily armor), then I take d=10k damage.
We can rearrange this into a useful quantity I'll call E, for effectiveness (or efficiency if you like):
E is the ratio of raw damage taken to actual damage taken. Viewed another way, it represents the effectiveness of each point of health we have, because to take d=1 point of damage we'd need to be hit for a raw damage of D=E. Since we can arguably take as much damage as we have health, this means we have an "effective health" of
where H is our maximum health. This is the formula we all have come to know and love.
To go further, we need to know M more explicitly, so we need to know the player's Armor A. M can be calculated as:
where K is given by K = L*2167.5  158167.5 for an attacker of level L. There's no intuitive explanation for this, it's just how the game is coded.
Given these three equations, we can figure out how much armor is equal to one point of health for EH purposes. First, we plug equation (2) into eq. (3) to get
We then differentiate both sides, using differentials to represent small changes in quantities:
Let's say an amount of health dH increases our Effective Health by d(EH). To find the amount of armor dA that would give us the same EH increase, we set the two terms on the right hand side equal to one another:
which is exactly the form given in Satrina's derivation.
To convert this to stamina S, we have to recognize that H=f*S, where f is a different constant for each class. Since for paladins, talents, plate specialization, and BoK give us f=17.7502, then:
and
again, exactly the form that Satrina gets.
Thus the amount of armor dA we'd need to get the same EH contribution as dS = 1 point of stamina is dA = 17.7502*(K+A)/H.
II. Total Effective Health  Including other forms of damage
The major weakness with this theory is that it's focused on purely physical damage. It can cover purely magical damage by substituting a magical mitigation factor for M (which would be a complicated formula based on talents and resistance). However, it doesn't accurately reflect fights that contain both magical and physical damage, which means "nearly every fight in the game." I will now suggest a fairly simple way to extend EH to cover any fight. Taking our cue from Satrina, we'll call this "Total Effective Health" or TEH.
Proof:
To fully incorporate other types of damage, we need to be more careful. First of all, there are three basic types of damage we want to consider:
1) "Regular" damage  physical damage mitigated by armor
2) Bleeds  physical damage that is not mitigated by armor
3) Magical damage
In section I, we assumed that physical damage was just mitigated by armor. This isn't technically true, there's also talented mitigation to consider. This becomes especially important once we add other damage types into the mix, because the talented portions can apply differently to each damage type. Some talents add only magic mitigation, while others add to all mitigation.
Let's return to the beginning. Now, instead of taking damage D per hit, let's talk in aggregate. The boss puts out D raw damage in a period of time T, broken down into Dp "regular" physical damage, Db bleed damage, and Dg magical damage. T and D could represent the entire fight, but a more realistic estimate of tank death is a small window of 510 seconds where we take burst damage. For the moment, let's assume we're talking about a small burst window, so that D and the Di represent the broken down components of the burst (i.e. Dg from a magical attack followed by a melee for Dp and a bleed tick of Db).
To get a general form, we express the damage you take d by:
Where we've used the following mitigation factors:
Ma is the mitigation due to armor, defined as M is in section I.
Mt is the mitigation applied to physical damage due to talents
Mg is the mitigation applied to magical damage due to talents
Mr is the mitigation applied to magical damage due to resistances
In this form, we can account for everything, and can raise or lower physical and magical talented mitigation independently from armor and resistances if we want to.
At first, this looks like it will be hopelessly complex. We want to solve for D/d, but we don't have Dp, Db, or Dg, nor do we know what percentage of D any of those represent. Luckily, we won't need any of this information in the final result.
we can define values P, B, and G to represent the relative percentages of the boss's total raw damage output D as follows:
As we said, we don't actually have access to this information, but it turns out we won't need it, and it will help simplify the math in the meantime. This notation lets us rewrite equation (8) as:
and rearranging this to express TEH we have
Before we go any further, let's consider what we see on a WoL parse. The damage that we read off of the parse is postmitigation. In other words, we don't have Dp directly, we have Dp*(1Ma)(1Mt), and similarly for Dg and Db. We can represent what we see on a combat log parse as X and Y:
Here X is the percentage of our damage intake that's from bleed effects, Y is the amount of damage taken from magical sources, and 1XY is the "leftover" amount due to regular physical damage.
We want to be able to express EH in terms of X and Y; to do that we need to eliminate some variables. To do this, we rewrite equations (13) in a different form:
If we multiply (14b) by (1X)/Y and add it to (14a), we have
Similarly, (14b) multiplied by X/(1Y) and added to (14a) gives us
Using this, we can rewrite B(1Mt) + G(1Mg)(1Mr) as
and equation (12) becomes
This still has a P in it though. To eliminate that, we substitute eqns (15) into (10) and solve for P:
Where I'm using Z = (1Mt)(1Ma)/(1Mg)(1Mr) as an temporary variable to simplify the expression slightly. Plugging (17) into (16) gives us the final form of the expression for effective health:
We can write equation (18) in a slightly more intuitive form:
The first term is our EH against "regular" physical attacks (or PEH) multiplied by the percentage of our damage intake that those attacks represent. The second term is our EH against bleeds (BEH), multiplied by the percentage of our intake due to bleeds. And the third term is our EH against magical damage (MEH) multiplied by the percentage of our intake that's magical.
In other words, TEH is properly calculated as the weighted average of our EH against all damage sources, with the weight factors simply being the percentages of our intake that those sources represent (during the burst event we want to consider).
Just to check that this makes sense, let's consider some special cases:
Y=0, X=0: Second and Third terms disappear, and TEH simplifies to H/((1Ma)(1Mt)), exactly the form the simple version took once talented mitigation is included.
Y=1, X=0: The first and second terms disappear, and TEH simplifies to H/(1Mg)(1Mr). This is exactly what we'd expect for a purely magical fight  our only mitigation here is from talents.
Y0, X=1: The first and third terms vanish, leaving TEH = H/(1Mt). Again, exactly what we'd expect  the mitigation from armor disappears, and we're left with only physical mitigation from talents.
To find the armor:stamina EH relation, it's simpler to start from equation (12). Differentiating gives us
d(Ma) evaluates to:
Which substituted into our expression for d(EH) gives
Again, we equate the first and second terms, and solve for dA:
It should be clear by inspection that the second fraction in that expression is simply 1/(1XY), giving us the final forms of dA:
Here we finally have an equivalence that accurately relates armor to stamina for a multifaceted fight. We see that as Y>1 (P>0), the second factor blows up. The other way to understand this is that the second term in the differentiated equation goes to zero, removing dA from the equation. This is as expected, since no amount of armor will give you any TEH for Y=1.
Alternatively, solving for dS:
So to find out how much stamina an amount of armor is equivalent to, we simply divide by the conversion factor (K+A)/H and multiply by the percentage of damage the armor will help mitigate (1XY). What this means practically is that armor loses effectiveness linearly with the amount of physical damage in the fight. For a fight with 50% physical damage, armor will only be worth 50% as much EH.
Let's see what happens as Y varies from 0 to 1 for a tank with H=150k, A=40k. In other words, how the Armor:Stamina ratio changes as the amount of armormitigated physical damage decreases:
I've truncated the graph at 70% since it blows up as Y gets larger. However, here are a few representative points:
So for a purely physical fight, armor seems like a pretty good deal. But for a fight with 20% magical damage, Armor's EH contribution drops by 20%. This will generally be enough to make a Stamina trinket provide more EH than an armor trinket.
For an example, let's look at the Glyph of Indomitability, since that's what started this thread. It gives 1792 armor, which is equivalent to 209 stamina. However, for a fight with Y=20%, we get only (1Y)=80% of that, or 167 stamina. For a fight with 30% magical damage, it's only worth 146 stamina, and so forth.
III. A note on healing
Traditionally, healing is not included in the EH metric. This is by design, since healing isn't going to be consistent from encounter to encounter or even between attempts. In essence, healing is out of the scope of the question being asked by EH ("How much raw damage will it take to kill me in the worstcase scenario").
But what if we asked, "How much raw damage does it take to kill me given h points of healing?" While this wouldn't be a true EH metric, we can obtain some insight by considering the effect that this would have on the result.
The answer is surprisingly simple. Every point of healing acts exactly like every point of health, in that it is worth one point of damage after mitigation effects. To see how receiving h points of healing affects the formulas, one needs only to replace H with (H+h) in the formulas.
So adding healing is sort of like adding health. If you receive one full HP bar's worth of healing during a burst scenario, it's mathematically equivalent to having twice as much health as far as the formulas are concerned. It's clear from equation (23) that if you double your health, you also double the effectiveness of armor  if 10 armor was worth one point of stamina, it's worth two in that healing scenario.
While you're not guaranteed to receive a particular amount of healing during a burst scenario, you're fairly likely to receive some. In practice that means that armor is more valuable for survival than the EH equations predict. Unfortunately, the only way to generate something quantitative from this intuition is to examine a lot of data: look at your parses, and see what your deaths look like. If you're receiving healing and die a slow, "trickledown" death, then armor is going to be much stronger than the EH equations suggest. If you're dying in only a few seconds with almost no incoming heals, then the EH equations will be a fairly accurate representation of the two stats.
IV. Conclusion
It is quite simple to extend the Effective Health calculation to incorporate nonphysical sources of damage. The result is simply a weighted average of one's effective healths vs. "regular" physical (PEH), bleed physical (BEH), and magical damage (MEH). The weighting is determined by the percentages of one's damage intake from bleed and magical sources X and Y respectively. The formulas are:
where Mt and Mg represent talented mitigation factors for physical and magical damage respectively, Mr is the mitigation due to resistances, Ma is the mitigation due to armor (defined below), K is the armor decay factor (also defined below), H is the player's fully raidbuffed max health, and A is the players raidbuffed armor.
And here are the definitions of Mt and Mg for a paladin:
Mg = 0.1
Mt = 0.1
I think Mr should be 0.1 with an aura up, but I haven't looked up the exact resistance binning mechanisms at level 85. If it's anything like level 80, 0.1 is the guaranteed value, and ~0.2 is the average value.
The takehome message of these formulas is that Armor loses effectiveness linearly with the percentage of "regular" physical damage intake for a given fight. In other words, for a fight with only 50% nonbleed physical damage, armor is reduced in effectiveness by 50%. If an armor trinket is worth 100 stamina on a purely "regular" physical fight, it will only be worth 60 stamina on a fight with 15% bleed damage and 25% magic damage (60% "regular" physical).
However, Armor also interacts with healing in such a fashion that its survival benefit can be greatly increased. Incoming healing can (for example) double or triple the survival effectiveness of armor, but has no effect on stamina. Thus, using EH as a hardandfast rule for choosing between Armor and Stamina likely won't give an accurate representation of which is more likely to save your life.
Also note that the old Ardent Defender, the "Strength of Wrynn" buff in ICC, and other multiplicative stamina bonuses have no effect on the armorstamina relationship.
Note: This is not an "armor is bad" screed. Armor is exceptionally good, as covered in section III. However, it's also good to know where your limitations are. I also think that it's good to have a complete picture of what's going on, even if in practice you use a ruleofthumb rather than the full equations. The point of this post is simply that  to have the "full" derivation written down somewhere.
Note #2: I've updated the values for patch 4.0.1, and rerun the graph. However, I have not modified the entire derivation to include the new block mechanic. Keep this in mind when using this math for anything practical  shaving 30% off of every melee hit via block effectively adds 30% mitigation to all physical damage, in essence adding a factor of (1Mb)=0.7 anywhere you see (1Ma).
I. Basics of Effective Health Theory
EH theory isn't very complicated fundamentally. The basics are outlined here and here, but I'll try and present a slightly more accessible derivation:
Consider a boss that hits for purely physical damage. His hits deal raw damage D. If your physical mitigation is M, then the damage that shows up in the combat log is d:
 Code: Select all
d = D*(1M)
In other words, if the boss hits for an unmitigated D=20k, and I have M=50%=0.5 mitigation (from all sources, but primarily armor), then I take d=10k damage.
We can rearrange this into a useful quantity I'll call E, for effectiveness (or efficiency if you like):
 Code: Select all
E = D/d = 1/(1M) (1)
E is the ratio of raw damage taken to actual damage taken. Viewed another way, it represents the effectiveness of each point of health we have, because to take d=1 point of damage we'd need to be hit for a raw damage of D=E. Since we can arguably take as much damage as we have health, this means we have an "effective health" of
 Code: Select all
EH = E*H = H/(1M) (2)
where H is our maximum health. This is the formula we all have come to know and love.
To go further, we need to know M more explicitly, so we need to know the player's Armor A. M can be calculated as:
 Code: Select all
M = A / (A+K) (3)
where K is given by K = L*2167.5  158167.5 for an attacker of level L. There's no intuitive explanation for this, it's just how the game is coded.
Given these three equations, we can figure out how much armor is equal to one point of health for EH purposes. First, we plug equation (2) into eq. (3) to get
 Code: Select all
EH = H*(K+A)/K = H*(1+A/K) (4)
We then differentiate both sides, using differentials to represent small changes in quantities:
 Code: Select all
d(EH) = dH*(K+A)/K + H*dA/K
Let's say an amount of health dH increases our Effective Health by d(EH). To find the amount of armor dA that would give us the same EH increase, we set the two terms on the right hand side equal to one another:
 Code: Select all
dA = (K+A)/H * dH (5)
which is exactly the form given in Satrina's derivation.
To convert this to stamina S, we have to recognize that H=f*S, where f is a different constant for each class. Since for paladins, talents, plate specialization, and BoK give us f=17.7502, then:
 Code: Select all
dH = f*dS = 17.7502*dS (6)
and
 Code: Select all
dA = 17.7502*(K+A)/H * dS (7)
again, exactly the form that Satrina gets.
Thus the amount of armor dA we'd need to get the same EH contribution as dS = 1 point of stamina is dA = 17.7502*(K+A)/H.
II. Total Effective Health  Including other forms of damage
The major weakness with this theory is that it's focused on purely physical damage. It can cover purely magical damage by substituting a magical mitigation factor for M (which would be a complicated formula based on talents and resistance). However, it doesn't accurately reflect fights that contain both magical and physical damage, which means "nearly every fight in the game." I will now suggest a fairly simple way to extend EH to cover any fight. Taking our cue from Satrina, we'll call this "Total Effective Health" or TEH.
Proof:
To fully incorporate other types of damage, we need to be more careful. First of all, there are three basic types of damage we want to consider:
1) "Regular" damage  physical damage mitigated by armor
2) Bleeds  physical damage that is not mitigated by armor
3) Magical damage
In section I, we assumed that physical damage was just mitigated by armor. This isn't technically true, there's also talented mitigation to consider. This becomes especially important once we add other damage types into the mix, because the talented portions can apply differently to each damage type. Some talents add only magic mitigation, while others add to all mitigation.
Let's return to the beginning. Now, instead of taking damage D per hit, let's talk in aggregate. The boss puts out D raw damage in a period of time T, broken down into Dp "regular" physical damage, Db bleed damage, and Dg magical damage. T and D could represent the entire fight, but a more realistic estimate of tank death is a small window of 510 seconds where we take burst damage. For the moment, let's assume we're talking about a small burst window, so that D and the Di represent the broken down components of the burst (i.e. Dg from a magical attack followed by a melee for Dp and a bleed tick of Db).
To get a general form, we express the damage you take d by:
 Code: Select all
d = Dp*(1Ma)*(1Mt) + Db*(1Mt) + Dg*(1Mg)(1Mr) (8)
Where we've used the following mitigation factors:
Ma is the mitigation due to armor, defined as M is in section I.
Mt is the mitigation applied to physical damage due to talents
Mg is the mitigation applied to magical damage due to talents
Mr is the mitigation applied to magical damage due to resistances
In this form, we can account for everything, and can raise or lower physical and magical talented mitigation independently from armor and resistances if we want to.
At first, this looks like it will be hopelessly complex. We want to solve for D/d, but we don't have Dp, Db, or Dg, nor do we know what percentage of D any of those represent. Luckily, we won't need any of this information in the final result.
we can define values P, B, and G to represent the relative percentages of the boss's total raw damage output D as follows:
 Code: Select all
P = Dp/D (9a)
B = Db/D (9b)
G = Dg/D (9c)
P + B + G = 1 (10)
As we said, we don't actually have access to this information, but it turns out we won't need it, and it will help simplify the math in the meantime. This notation lets us rewrite equation (8) as:
 Code: Select all
d = D*[P(1Ma)(1Mt) + B(1Mt) + G(1Mg)(1Mr)] (11)
and rearranging this to express TEH we have
 Code: Select all
H
TEH = H*D/d =  (12)
P(1Ma)(1Mt) + B(1Mt) + G(1Mg)(1Mr)
Before we go any further, let's consider what we see on a WoL parse. The damage that we read off of the parse is postmitigation. In other words, we don't have Dp directly, we have Dp*(1Ma)(1Mt), and similarly for Dg and Db. We can represent what we see on a combat log parse as X and Y:
 Code: Select all
B(1Mt)
X =  (13a)
P(1Ma)(1Mt) + B(1Mt) + G(1Mg)(1Mr)
G(1Mg)
Y =  (13b)
P(1Ma)(1Mt) + B(1Mt) + G(1Mg)(1Mr)
Here X is the percentage of our damage intake that's from bleed effects, Y is the amount of damage taken from magical sources, and 1XY is the "leftover" amount due to regular physical damage.
We want to be able to express EH in terms of X and Y; to do that we need to eliminate some variables. To do this, we rewrite equations (13) in a different form:
 Code: Select all
X*P(1Ma)(1Mt)  (1X)*B(1Mt) + X*G(1Mg)(1Mr) = 0 (14a)
Y*P(1Ma)(1Mt) + Y*B(1Mt)  (1Y)*G(1Mg)(1Mr) = 0 (14b)
If we multiply (14b) by (1X)/Y and add it to (14a), we have
 Code: Select all
P(1Ma)(1Mt) = (1XY)*G(1Mg)(1Mr)/Y (15a)
Similarly, (14b) multiplied by X/(1Y) and added to (14a) gives us
 Code: Select all
P(1Ma)(1Mt) = (1XY)*B(1Mt)/X (15b)
Using this, we can rewrite B(1Mt) + G(1Mg)(1Mr) as
 Code: Select all
B(1Mt) + G(1Mg)(1Mr) = (X+Y)*P(1Ma)(1Mt)/(1XY)
and equation (12) becomes
 Code: Select all
H*(1XY)
TEH = H*D/d =  (16)
P(1Ma)(1Mt)
This still has a P in it though. To eliminate that, we substitute eqns (15) into (10) and solve for P:
 Code: Select all
X (1Ma)(1Mt) Y (1Ma)(1Mt)
1 = P + B + G = P + P** + P**
(1XY) (1Mt) (1XY) (1Mg)(1Mr)
(1Mt)(1Ma)
(1XY) = P*[(1XY) + X(1Ma) + Y*]
(1Mg)(1Mr)
(1XY) = P*[(1XY) + X(1Ma) + YZ]
(1XY)
 = (1XY) + X(1Ma) + YZ (17)
P
Where I'm using Z = (1Mt)(1Ma)/(1Mg)(1Mr) as an temporary variable to simplify the expression slightly. Plugging (17) into (16) gives us the final form of the expression for effective health:
 Code: Select all
H*[(1XY) + X(1Ma) + YZ]
TEH = H*D/d =  (18)
(1Ma)(1Mt)
We can write equation (18) in a slightly more intuitive form:
 Code: Select all
(1XY) X Y
TEH = H*D/d = H* + H* + H* (19)
(1Ma)(1Mt) (1Mt) (1Mg)(1Mr)
The first term is our EH against "regular" physical attacks (or PEH) multiplied by the percentage of our damage intake that those attacks represent. The second term is our EH against bleeds (BEH), multiplied by the percentage of our intake due to bleeds. And the third term is our EH against magical damage (MEH) multiplied by the percentage of our intake that's magical.
In other words, TEH is properly calculated as the weighted average of our EH against all damage sources, with the weight factors simply being the percentages of our intake that those sources represent (during the burst event we want to consider).
Just to check that this makes sense, let's consider some special cases:
Y=0, X=0: Second and Third terms disappear, and TEH simplifies to H/((1Ma)(1Mt)), exactly the form the simple version took once talented mitigation is included.
Y=1, X=0: The first and second terms disappear, and TEH simplifies to H/(1Mg)(1Mr). This is exactly what we'd expect for a purely magical fight  our only mitigation here is from talents.
Y0, X=1: The first and third terms vanish, leaving TEH = H/(1Mt). Again, exactly what we'd expect  the mitigation from armor disappears, and we're left with only physical mitigation from talents.
To find the armor:stamina EH relation, it's simpler to start from equation (12). Differentiating gives us
 Code: Select all
dH d(Ma)*H*P(1Mt)
d(TEH) =  + 
P(1Ma)(1Mt)+B(1Mt)+G(1Mg)(1Mr) [P(1Ma)(1Mt)+B(1Mt)+G(1Mg)(1Mr)]^2
d(Ma) evaluates to:
 Code: Select all
dA A*dA dA*K dA*(1Ma)
d(Ma) =    =  =  (20)
(K+A) (K+A)^2 (K+A)^2 (K+A)
Which substituted into our expression for d(EH) gives
 Code: Select all
dH dA*H*P(1Mt)(1Ma)
d(TEH) =  + 
P(1Ma)(1Mt)+B(1Mt)+G(1Mg)(1Mr) (K+A)*[P(1Ma)(1Mt)+B(1Mt)+G(1Mg)(1Mr)]^2
Again, we equate the first and second terms, and solve for dA:
 Code: Select all
(K+A) P(1Ma)(1Mt) + B(1Mt) + G(1Mg)(1Mr)
dA = **dH
H P(1Ma)(1Mt)
It should be clear by inspection that the second fraction in that expression is simply 1/(1XY), giving us the final forms of dA:
 Code: Select all
(K+A) 1
dA = **dH (21)
H (1XY)
17.7502*(K+A) 1
dA = **dS (22)
H (1XY)
Here we finally have an equivalence that accurately relates armor to stamina for a multifaceted fight. We see that as Y>1 (P>0), the second factor blows up. The other way to understand this is that the second term in the differentiated equation goes to zero, removing dA from the equation. This is as expected, since no amount of armor will give you any TEH for Y=1.
Alternatively, solving for dS:
 Code: Select all
H
dS = *(1XY)*dA (23)
17.7502*(K+A)
So to find out how much stamina an amount of armor is equivalent to, we simply divide by the conversion factor (K+A)/H and multiply by the percentage of damage the armor will help mitigate (1XY). What this means practically is that armor loses effectiveness linearly with the amount of physical damage in the fight. For a fight with 50% physical damage, armor will only be worth 50% as much EH.
Let's see what happens as Y varies from 0 to 1 for a tank with H=150k, A=40k. In other words, how the Armor:Stamina ratio changes as the amount of armormitigated physical damage decreases:
I've truncated the graph at 70% since it blows up as Y gets larger. However, here are a few representative points:
 Code: Select all
Y(%) Armor 1/(1Y)
0 8.59 1.0000
10 9.54 1.1111
20 10.73 1.2500
30 12.27 1.4286
40 14.31 1.6667
50 17.18 2.0000
60 21.47 2.5000
70 28.63 3.3333
So for a purely physical fight, armor seems like a pretty good deal. But for a fight with 20% magical damage, Armor's EH contribution drops by 20%. This will generally be enough to make a Stamina trinket provide more EH than an armor trinket.
For an example, let's look at the Glyph of Indomitability, since that's what started this thread. It gives 1792 armor, which is equivalent to 209 stamina. However, for a fight with Y=20%, we get only (1Y)=80% of that, or 167 stamina. For a fight with 30% magical damage, it's only worth 146 stamina, and so forth.
III. A note on healing
Traditionally, healing is not included in the EH metric. This is by design, since healing isn't going to be consistent from encounter to encounter or even between attempts. In essence, healing is out of the scope of the question being asked by EH ("How much raw damage will it take to kill me in the worstcase scenario").
But what if we asked, "How much raw damage does it take to kill me given h points of healing?" While this wouldn't be a true EH metric, we can obtain some insight by considering the effect that this would have on the result.
The answer is surprisingly simple. Every point of healing acts exactly like every point of health, in that it is worth one point of damage after mitigation effects. To see how receiving h points of healing affects the formulas, one needs only to replace H with (H+h) in the formulas.
So adding healing is sort of like adding health. If you receive one full HP bar's worth of healing during a burst scenario, it's mathematically equivalent to having twice as much health as far as the formulas are concerned. It's clear from equation (23) that if you double your health, you also double the effectiveness of armor  if 10 armor was worth one point of stamina, it's worth two in that healing scenario.
While you're not guaranteed to receive a particular amount of healing during a burst scenario, you're fairly likely to receive some. In practice that means that armor is more valuable for survival than the EH equations predict. Unfortunately, the only way to generate something quantitative from this intuition is to examine a lot of data: look at your parses, and see what your deaths look like. If you're receiving healing and die a slow, "trickledown" death, then armor is going to be much stronger than the EH equations suggest. If you're dying in only a few seconds with almost no incoming heals, then the EH equations will be a fairly accurate representation of the two stats.
IV. Conclusion
It is quite simple to extend the Effective Health calculation to incorporate nonphysical sources of damage. The result is simply a weighted average of one's effective healths vs. "regular" physical (PEH), bleed physical (BEH), and magical damage (MEH). The weighting is determined by the percentages of one's damage intake from bleed and magical sources X and Y respectively. The formulas are:
 Code: Select all
(1XY) X Y
TEH = H*D/d = H* + H* + H* (19)
(1Ma)(1Mt) (1Mt) (1Mg)(1Mr)
17.7502*(K+A) 1
dA = **dS (22)
H (1XY)
H
dS = *(1XY)*dA (23)
17.7502*(K+A)
where Mt and Mg represent talented mitigation factors for physical and magical damage respectively, Mr is the mitigation due to resistances, Ma is the mitigation due to armor (defined below), K is the armor decay factor (also defined below), H is the player's fully raidbuffed max health, and A is the players raidbuffed armor.
 Code: Select all
Ma = A/(A+K)
K = L*2167.5  158167.5 (for an attacker of level L)
And here are the definitions of Mt and Mg for a paladin:
Mg = 0.1
Mt = 0.1
I think Mr should be 0.1 with an aura up, but I haven't looked up the exact resistance binning mechanisms at level 85. If it's anything like level 80, 0.1 is the guaranteed value, and ~0.2 is the average value.
The takehome message of these formulas is that Armor loses effectiveness linearly with the percentage of "regular" physical damage intake for a given fight. In other words, for a fight with only 50% nonbleed physical damage, armor is reduced in effectiveness by 50%. If an armor trinket is worth 100 stamina on a purely "regular" physical fight, it will only be worth 60 stamina on a fight with 15% bleed damage and 25% magic damage (60% "regular" physical).
However, Armor also interacts with healing in such a fashion that its survival benefit can be greatly increased. Incoming healing can (for example) double or triple the survival effectiveness of armor, but has no effect on stamina. Thus, using EH as a hardandfast rule for choosing between Armor and Stamina likely won't give an accurate representation of which is more likely to save your life.
Also note that the old Ardent Defender, the "Strength of Wrynn" buff in ICC, and other multiplicative stamina bonuses have no effect on the armorstamina relationship.
Last edited by theckhd on Thu Jan 14, 2010 11:46 am, edited 11 times in total.
"Theck, Bringer of Numbers and Pounding Headaches," courtesy of GrehnSkipjack.
MATLAB 5.x, Simcraft 6.x, Call to Arms 6.0, Talent Spec & Glyph Guide 5.x, Blog: Sacred Duty
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theckhd  Moderator
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Re: "New EH"  incorporating different damage types into EH
ahh delicious theorycrafts
my preciousss...
Edit: magical damage makes me want to /wrists
my preciousss...
Edit: magical damage makes me want to /wrists
10 SIN
20 GOTO HELL
20 GOTO HELL

æ  Posts: 213
 Joined: Sun Oct 12, 2008 8:02 pm
 Location: Santa Barbara
Re: "New EH"  incorporating different damage types into EH
It's going to take a day off to have the time to read through all of this. I ready about 1/3rd of the way though, will have to continue later (mainly posting so that you know your hard work is appreciated rather than thinking nobody cares). It's a good point on the armor vs magical damage intake and the effect that has on gear choices.
 Strendarr
 Posts: 110
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Re: "New EH"  incorporating different damage types into EH
nothing to contribute here but..
i just made a screenshot for when my guildis
call me a nerd of numbers and theorys again.
i just made a screenshot for when my guildis
call me a nerd of numbers and theorys again.
 "take THIS, im not even close to that guy"

Lave  Posts: 187
 Joined: Fri Aug 10, 2007 6:50 am
 Location: Berlin, Germany
Re: "New EH"  incorporating different damage types into EH
Nice graphic.
Seems to show that, while armor is a nice thing on itself, it's practically never worth sacrificing stamina for in case of EH scenarios.
Seems to show that, while armor is a nice thing on itself, it's practically never worth sacrificing stamina for in case of EH scenarios.

Thels  Posts: 1236
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 Location: The Netherlands
Re: "New EH"  incorporating different damage types into EH
Another really nice post, Theck. Could you rerun the (Armor/Stam)/% magical damage graph with a value for N that would reflect having a resist aura up? Many fights are limited to one or two types of magic damage that we can reasonably assume a near100% uptime of a significant resist amount for. From the source post for your N value, 128+ resist will guarantee an additional 10% magical mitigation (first bullet point in the "Notes" section at the bottom of the post).
Kriskringle  80  Duskwood
Greatfather Winter is a sham.
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 Kriskringle
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Re: "New EH"  incorporating different damage types into EH
Damnit.. I started reading this, went crosseyed and am now mumbling incoherently to myself...
I'll have to go back and read more of it later when A.D.D isn't kicking in
I'll have to go back and read more of it later when A.D.D isn't kicking in
 Shathus
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Re: "New EH"  incorporating different damage types into EH
Lave wrote:nothing to contribute here but..
i just made a screenshot for when my guildis
call me a nerd of numbers and theorys again."take THIS, im not even close to that guy"
/petpeeve alert
/hurts you for posting that in jpg and not png
Anyway thanks for this Theck. Its explained well step by step, and pretty much comes down to the easily readable graphic.
Now th eonly thing left to do is start sorting boss encounters by how much inc magical damage on average vs unavoidable hits etc etc.
Or do we? Is stam just better overall? Should they buff AC numbers on items by 15%? Should AC start contributing to resistances in some way?
10 SIN
20 GOTO HELL
20 GOTO HELL

æ  Posts: 213
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 Location: Santa Barbara
Re: "New EH"  incorporating different damage types into EH
Nice work Theckd. I went through the first part and checked your math, but then I realized it was almost lunch time, so I skipped doing the same on the second.
Your assumptions all appear valid.
This just confirms what we all suspected: Armor is overvalued in itemization. Thus, trinkets like the Glyph of Indominability are not worth the equivalent stamina trinkets in terms of effective health. The only thing they have going for them is the fact that you take less damage so life could be easier on the healers, but if you don't have enough EH you'd die anyway.
Another thing I noticed is that the value of armor goes down the more armor you have, but up the more health you have. It could theoretically get pretty good at very high health totals, but it means that stacking armor will never be worth it because it will actually get less and less useful. Blizz designed armor this way, sadly.
Your assumptions all appear valid.
This just confirms what we all suspected: Armor is overvalued in itemization. Thus, trinkets like the Glyph of Indominability are not worth the equivalent stamina trinkets in terms of effective health. The only thing they have going for them is the fact that you take less damage so life could be easier on the healers, but if you don't have enough EH you'd die anyway.
Another thing I noticed is that the value of armor goes down the more armor you have, but up the more health you have. It could theoretically get pretty good at very high health totals, but it means that stacking armor will never be worth it because it will actually get less and less useful. Blizz designed armor this way, sadly.
 Spectrum
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Re: "New EH"  incorporating different damage types into EH
No. It can be better and generally is on our current encounters in raids, but it is not by default better. So don't go throwing away your Armor trinkets yet boys.æ wrote:Is stam just better overall?
What we have now is a ratio of the value of Armor vs. Stamina with a basic idea of where they vary. Thus you can get Armor trinkets and roughly compare their value for the bosses ahead.
Also, consider that sometimes magical damage can be avoided but isn't for some reason (doesn't do enough damage to really worry me or it doesn't get interrupted in time). In those situations Stamina takes an even stronger importance
As a side note: By my understanding, the idea of 11.7 Armor = 1 Stamina relies on the assumption that the healers will have limitless mana (as they seem to have in almost every fight). Though worth considering that if there is a fight where healers having limitless mana is not a true statement(General Vezax as a quasiexample) the situation would need reexamining because suddenly Armor decreases the amount of healing needed since white damage is almost a guaranteed in every encounter (obviously if it's not white damage as the melee attacks, it's entirely moot). With less healing needed that gives a bonus to the raid that doesn't take the form of something that we can attribute to ourselves.
Personal opinion: No. AC isn't worthless, it's just less effective than Stamina as a means of getting EH in magical fights. But it's always been that way, not to mention we have tons of it thanks to our plate. Armor doesn't need to be better than Staminaæ wrote:Should they buff AC numbers on items by 15%?
No. Absolutely not. The amount of changes this would put into the game are staggering. The obvious pro would be that magical damage would be avoided and mitigated more often by tanks. But current numbers are balanced around the assumption of only a token amount of magical resistance (130 from a buff of some sort unless it's a resistance fight at the most), so as a result magical damage from enemy spells would be buffed. We would end up taking just as much damage in the end, only the magical portion would be stronger.æ wrote:Should AC start contributing to resistances in some way?
Next, Blizzard would have to go back and alter the damage of every spell in the game to account for suddenly every class having magical resistance, many of them (such as mail and plate wearers) having a hefty amount. Of course this would be pretty painful for casters as they are already low on armor, but that's balanced by the fact that they can get at range from their targets and usually keep them there. So while this wouldn't mean death for all clothies, it would strongly increase their down time as now if they aren't selfhealers, they are getting solid beatdowns every fight.
Now, maybe Blizzard could take the route of making AC give a resist boost via a talent. Well this would work for not having to buff damage of openworld casters, but suddenly you run the risk of creating unstoppable PvP gods. The one balancing fact for PvP is that even though a plate wearer usually can take a strong beating from melees, they can be nuked with spells fairly easily. Again, at most they have 130 resistance to one or two elements. But with Prot Paladins and Warriors having such a talent, suddenly they are all the rage (warriors... rage... hur hur) in PvP and are virtually unstoppable. Of course such a talent would need to be avaiable to DKs and Feral Druids too, which is even worse since at least the Prot trees are pretty focused on just tanking. Meanwhile PvP DKs and Feral Druids are even stronger having all the pros of a tank while still putting out rediculous DPS.
Also, with the current mechanics of resistances, this becomes a risky buff. AC is good because it's a static reduction in incoming damage, meanwhile Resistances are not. Honestly, resistances are more akin to a combination of Block and Avoidance. There's a chance for a spell to totally miss and a chance for the spell to be partially resisted cutting out 2575% of the spell. We've already seen the issue with having too much avoidance and walking around all the time with tons of Resistance would just mean they would need to buff the damage done perspell. Then we're at the "twoshot if you don't avoid one of the attacks" situation all over again, only instead of "avoid" change the term to "fully resist". No thanks.
 Alixander
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Re: "New EH"  incorporating different damage types into EH
Spectrum wrote:Another thing I noticed is that the value of armor goes down the more armor you have, but up the more health you have. It could theoretically get pretty good at very high health totals, but it means that stacking armor will never be worth it because it will actually get less and less useful. Blizz designed armor this way, sadly.
Care to elaborate on this? Saying that "the value of armor goes down the more armor you have" implies that at higher levels of armor, additional armor contributes less to TTL. This, however, conflicts with Satrina's findings at Tankspot (which that conclude that TTL increases linearly with armor, pointforpoint).

cypher  Posts: 14
 Joined: Tue Dec 02, 2008 6:38 pm
Re: "New EH"  incorporating different damage types into EH
Firstly, cypher: this is Satrina's post which verifies what Spectrum says (and what Theck alludes to as well), namely that the equivalence between health and armour depends on current levels of both.
In response to Theck's post: it's great to see a more rigorous treatment of effective health that includes various damage sources. My understanding of this is it's essentially an interpolation between the mitigation offered on the one hand by armour and on the other by its magical equivalent (essentially resistance, although as noted this functions probabilistically rather more like avoidance than armour) with the coefficients of that interpolation given on a perfight basis as the proportion of damage dealt which is physical versus magical.
The only thing I'd mention, and I'm quite sure you're aware of it, is that lumping together all "magical" sources is somewhat unsatisfactory (and doubly so by including bleed effects and other sources of physical damage that aren't mitigated by armour) because one's level of resistance to these things varies. Even if one were to assume all magic resistance buffs, meaning the amount of damage which could be mitigated would be the same for all and thus their source could just be treated as "magic" rather than frost, fire, nature, etc., one still has to deal with Holy damage and the aforementioned physical damage that cannot be mitigated by armour.
While it would doubtless be unwieldy to derive, and would still gloss over the awkwardness of the nondeterministic mitigation provided by resistances, I wonder if the complete (read: anal) derivation wouldn't include a factor for each damage type, the mitigation available for it, and the proportion of overall damage represented by that particular source. Needless to say on most fights all but one magical source would have coefficient 0 anyway.
In response to Theck's post: it's great to see a more rigorous treatment of effective health that includes various damage sources. My understanding of this is it's essentially an interpolation between the mitigation offered on the one hand by armour and on the other by its magical equivalent (essentially resistance, although as noted this functions probabilistically rather more like avoidance than armour) with the coefficients of that interpolation given on a perfight basis as the proportion of damage dealt which is physical versus magical.
The only thing I'd mention, and I'm quite sure you're aware of it, is that lumping together all "magical" sources is somewhat unsatisfactory (and doubly so by including bleed effects and other sources of physical damage that aren't mitigated by armour) because one's level of resistance to these things varies. Even if one were to assume all magic resistance buffs, meaning the amount of damage which could be mitigated would be the same for all and thus their source could just be treated as "magic" rather than frost, fire, nature, etc., one still has to deal with Holy damage and the aforementioned physical damage that cannot be mitigated by armour.
While it would doubtless be unwieldy to derive, and would still gloss over the awkwardness of the nondeterministic mitigation provided by resistances, I wonder if the complete (read: anal) derivation wouldn't include a factor for each damage type, the mitigation available for it, and the proportion of overall damage represented by that particular source. Needless to say on most fights all but one magical source would have coefficient 0 anyway.
 bashef
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Re: "New EH"  incorporating different damage types into EH
Lave wrote:"take THIS, im not even close to that guy"
Yeah, I get that a lot.
Kriskringle wrote:Could you rerun the (Armor/Stam)/% magical damage graph with a value for N that would reflect having a resist aura up? Many fights are limited to one or two types of magic damage that we can reasonably assume a near100% uptime of a significant resist amount for. From the source post for your N value, 128+ resist will guarantee an additional 10% magical mitigation (first bullet point in the "Notes" section at the bottom of the post).
Sure, In fact I can go back and add curves for several values of N on that same plot (i.e. N+5, N+10, and so on for the various plateaus of resistance).
æ wrote:Now th eonly thing left to do is start sorting boss encounters by how much inc magical damage on average vs unavoidable hits etc etc.
This is exactly where I intended to head with this concept, in fact. In concept, we can come up with a value of Y for every relevant raid encounter currently available. Once we have that, we can use a simple rule of thumb to determine which bosses are worth swapping armor trinkets in for. For example, "If Y < 15%, swap in armor trinkets."
æ wrote:Or do we? Is stam just better overall? Should they buff AC numbers on items by 15%? Should AC start contributing to resistances in some way?
Alixander covered most of this already, but just for clarity:
 Stam is better in general, because it works everywhere. It's the VISA of EH.
 Armor is American Express. More exclusive, but very powerful in the few places it should be used.
 This wasn't intended to imply that Armor needs to be reworked. I think that Blizz is better off leaving armor asis. But we need to be able to make intelligent gearing decisions as tanks, and knowing when to use or not use armor trinkets helps us do that.
Alixander wrote:As a side note: By my understanding, the idea of 11.7 Armor = 1 Stamina relies on the assumption that the healers will have limitless mana (as they seem to have in almost every fight).
Correct. As you and Spectrum both noted, armor does have the benefit of reducing total damage taken. It's certainly something to be considered, but again, it's the exception rather than the rule in current content. So far we've seen one fight that really stresses healer mana in that way  Vezax.
bashef wrote:My understanding of this is it's essentially an interpolation between the mitigation offered on the one hand by armour and on the other by its magical equivalent (essentially resistance, although as noted this functions probabilistically rather more like avoidance than armour) with the coefficients of that interpolation given on a perfight basis as the proportion of damage dealt which is physical versus magical.
Actually, I haven't included resistance at all here. N is based on our innate resistance to magical damage from talents. It's certainly possible to include resistances now thanks to their new implementation with Wrath (with a certain threshold of resistance, you can guarantee you'll resist a certain percentage of the damage).
bashef wrote:The only thing I'd mention, and I'm quite sure you're aware of it, is that lumping together all "magical" sources is somewhat unsatisfactory (and doubly so by including bleed effects and other sources of physical damage that aren't mitigated by armour) because one's level of resistance to these things varies.
This is a good point, because in the current derivation, I have bleeds lumped in with "magical" damage. However, our innate damage reduction from talents is slightly different for magical and physical damage.
What that means functionally is that the value of N we need to use is slightly different than the number I plugged in. It should be the weighted average of our magical and physical mitigation from talents (excluding armor), based on how much of each damage type is present in the encounter. The example plot is thus inferred as a fight with no bleeds, for example. The net result is not going to be significantly different though.
It's also worth noting that we have physical mitigation from talents as well as armor. That's not included in the model, though it could be trivially added as a multiplicative factor where (1M) shows up. I plan on working through the problem with bleeds treated separately just for thoroughness, but in the meantime this derivation illustrates the main point  that armor goes from 100% to 0% effectiveness linearly with (1Y). That result will not change even if bleeds are treated separately from magical damage.
Thanks for the feedback by the way. The next steps are as follows:
 Clean up the derivation. I intend to post this as an article, but I want to pretty it up some first by making TeX images for the equations. I also want to clean up the prose some, and make sure it's organized logically. I may wait until I have the more thorough derivation finished for this.
 Start compiling a list of bosses with some representative Y values based on fight parses. We can start this project in this thread, if we want. I can update the OP as we go with the full list.
"Theck, Bringer of Numbers and Pounding Headaches," courtesy of GrehnSkipjack.
MATLAB 5.x, Simcraft 6.x, Call to Arms 6.0, Talent Spec & Glyph Guide 5.x, Blog: Sacred Duty
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theckhd  Moderator
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Re: "New EH"  incorporating different damage types into EH
theckhd wrote:EH theory isn't very complicated
theckhd wrote:
 Code: Select all
d = D*(1M)
 Code: Select all
E = D/d = 1/(1M) (1)
 Code: Select all
EH = E*H = H/(1M) (2)
 Code: Select all
M = A / (A+K) (3)
 Code: Select all
EH = H*(K+A)/K = H*(1+A/K) (4)
 Code: Select all
dA = (K+A)/H * dH (5)
 Code: Select all
dH = f*dS = 12.54*dS (6)
 Code: Select all
dA = 12.54*(K+A)/H * dS
 Code: Select all
d = D_p*(1M) + D_m*(1N) (7)
 Code: Select all
D_p = P*D (8a)
D_m = (1P)*D = Q*D (8b)
 Code: Select all
d = D*[P*(1M) + Q*(1N)] (9)
 Code: Select all
H*D H
EH =  =  (10)
d P*(1M) + Q*(1N)
 Code: Select all
D_m*(1N) Q*(1N)
Y =  =  (11)
D_p*(1M) + D_m*(1N) P*(1M) + Q*(1N)
 Code: Select all
Y
Q*(1N) =  * P*(1M) (12)
1Y
 Code: Select all
P*(1M) + Q*(1N) = P*(1M)/(1Y) (13)
 Code: Select all
H (1Y)
EH = * (14)
(1M) P
 Code: Select all
Z = (MN)/(1N) (15)
 Code: Select all
(1P)(1N)(1Y) = YP(1M)
(1N)(1Y) = P[Y(1M)+(1N)(1Y)] = P[(1N)  Y(MN)] = P(1N)[1Y(MN)/(1N)] P(1N)[1ZY]
P = (1Y)/(1ZY)
 Code: Select all
H*(1  YZ) H*(K+A)*(1YZ)
EH =  =  (16)
1  M K
 Code: Select all
1 H*P*dM
d(EH) = dH* + 
P(1M)+Q(1N) [P(1M)+Q(1N)]^2
 Code: Select all
dA A*dA dA*K dA*(1M)
dM =    =  =  (17)
(K+A) (K+A)^2 (K+A)^2 (K+A)
 Code: Select all
1 H*P(1M)
d(EH) = dH* + dA* (18)
P(1M)+Q(1N) (K+A)*[P(1M)+Q(1N)]^2
 Code: Select all
(K+A) [P(1M)+Q(1N)]
dA = **dH
H P(1M)
 Code: Select all
(K+A) 1 12.54*(K+A) 1
dA = **dH = **dS (19)
H 1Y H 1Y
 Code: Select all
H
dS = *(1Y)*dA (20)
(K+A)
 Code: Select all
Y(%) Armor 1/(1Y)
0 11.7 1
10 13.0 1.1111
20 14.62 1.25
30 16.71 1.4286
40 19.49 1.6667
50 23.39 2
60 29.24 2.5
70 38.99 3.3333
 Code: Select all
H*(1  YZ) H*(K+A)*(1YZ)
EH =  =  (16)
1  M K
(K+A) 1 12.54*(K+A) 1
dA = **dH = **dS (19)
H 1Y H 1Y
H
dS = *(1Y)*dA (20)
(K+A)
 Code: Select all
Z = (MN)/(1N)
M = A/(A+K)
K = 467.5*L  22167.5

Invisusira  Moderator
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Re: "New EH"  incorporating different damage types into EH
Thank you Invisusira, You cleared that one up.
As for the Parses and the Y values, Theck is Y strictly the overall percentage of magical damage taken? If that is the case I can start compiling some values for you in TOTGC at least. For now the problem is that we do not really raid anything else.
I would have to say that if this works out perfectly, this is a soggy wet dream come true for a gear swapper like myself. To empirically prove what gear set you would use for what fight would be extremely powerful. This thread has already inspired me to modify my gear sets post to include the use of armor trinkets, with a few citations to come with our Y values (as they are posted). I have waned to tailor my gear set post to include references to what set to use for what bosses for a while now, as the information is useless if you do not know how to apply it. I feel like my gear sets post is Bill Gates' ATM card but we dont have the PIN number yet!
Even more important, I just came to the realization that identifying critical TTL points in progression encounters would tip the balance of what set to use for what fight. The first that comes to mind is Anub and P3, or Gormok's Impale/Bleed and Icehowl's Ferocious Butt, but alas i digress (whats new)...
As for the Parses and the Y values, Theck is Y strictly the overall percentage of magical damage taken? If that is the case I can start compiling some values for you in TOTGC at least. For now the problem is that we do not really raid anything else.
I would have to say that if this works out perfectly, this is a soggy wet dream come true for a gear swapper like myself. To empirically prove what gear set you would use for what fight would be extremely powerful. This thread has already inspired me to modify my gear sets post to include the use of armor trinkets, with a few citations to come with our Y values (as they are posted). I have waned to tailor my gear set post to include references to what set to use for what bosses for a while now, as the information is useless if you do not know how to apply it. I feel like my gear sets post is Bill Gates' ATM card but we dont have the PIN number yet!
Even more important, I just came to the realization that identifying critical TTL points in progression encounters would tip the balance of what set to use for what fight. The first that comes to mind is Anub and P3, or Gormok's Impale/Bleed and Icehowl's Ferocious Butt, but alas i digress (whats new)...
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