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"Total" EH - incorporating different damage types into EH

Warning: Theorycraft inside.

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Re: "New EH" - incorporating different damage types into EH

Postby Andris » Wed Nov 25, 2009 10:19 am

Not to heap even more on your plate, Theck, but I'm wondering if we could also figure out what the crossover points in terms of magic damage between Flask of Stoneblood (1300 hp), Lesser Flask of Resistance (+50 resist), and Flask of Chromatic Wonder (+35 resist, +18 stam). Of course, in some cases you also get resistance from auras or MotW.

Note that you'll actually need to split damage three ways for armor vs. hp vs. resistance comparisons
  • D_p (physical damage mitigated by armor)
  • D_u (non-resistable, non-armor mitigated damage, e.g. bleeds)
  • D_r (resistable magical damage).

For example, Gormok's impale would be D_u damage, for which the only way to increase EH is to increase H (or take more damage reduction talents), but Anub's aura is D_r; Mimiron's spellfire is D_r, but you only get the MotW resistance rather than fire resist aura for that damage. I'm sure you've already thought about this a bit, but it would be interesting to see how things stack up now. (I remember finding that the Resistance flask was a lot better in 3.0 for Sarth+3, but they boosted Stoneblood since then. Still, with the current levels of health, I wouldn't be surprised if some fights like Jaraxxus still benefitted from the resist flask.
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Re: "New EH" - incorporating different damage types into EH

Postby Alixander » Wed Nov 25, 2009 10:22 am

Wrathy wrote:For now the problem is that we do not really raid anything else.
I can help with regular Ulduar10 along with possibly Naxx10 (seeing if I can get my guild to run it either this week or the next since we will be missing quite a few regulars due to either holiday or just travel). While obviously a single fight for each of theses bosses is not enough

Also, Theck, I'm guessing we're going to want the unmitigated values of the attacks coming at us (that is, we either list what our mitigation numbers are or we figure out what it is before we post it).
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Re: "New EH" - incorporating different damage types into EH

Postby Florisia » Wed Nov 25, 2009 10:35 am

Stellar post. I was debating just today whether or not I should go and snag the Glyph, and this at least allows me to make a better educated decision regarding it.

I'll admit, it took me two or three readings to understand the logic behind it, but great none-the-less.

Edit: If there's anything we can do to assist with data collection, just let me know what we can/need to do.
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Re: "New EH" - incorporating different damage types into EH

Postby theckhd » Wed Nov 25, 2009 12:32 pm

Wrathy wrote: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.

Yes, it would be strictly the overall percentage of your damage intake from magical sources.

I posted a quick breakdown of a single parse for Uld25 and ToGC25 over in the Glyph of Indomitability thread. What we'd ideally want is to compile data from a number of parses to find reasonable averages, since I expect the amount will vary some from parse to parse.

Though as Andris pointed out, we should really break it down into three values.
Y for strictly magical damage
X for bleed damage (physical damage not mitigated by armor)
and the rest (1-X-Y) for regular old physical damage.

Alixander wrote:Also, Theck, I'm guessing we're going to want the unmitigated values of the attacks coming at us (that is, we either list what our mitigation numbers are or we figure out what it is before we post it).

Actually, it will suffice to just figure out the relative percentage of intake. The way I've developed the theory, we don't really need to know what the un-mitigated damages D_i are. I derived it that way on purpose, to make it easier to actually use the theory rather than having it just sit there and look pretty.

In other words, if the parse says that 50% of the damage you took was physical, 20% was physical bleeds, and 30% was magical, those are the values we need (Y=30%, X=20%).
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Re: "New EH" - incorporating different damage types into EH

Postby theckhd » Wed Nov 25, 2009 12:37 pm

Florisia wrote:Edit: If there's anything we can do to assist with data collection, just let me know what we can/need to do.

The data collection is actually pretty easy, you don't even need to do anything in-game.

  1. Find a World of Logs parse of an encounter being tanked by a paladin. This could be one of your own parses or someone else's.
  2. Look at the "damage taken" summary for that player, and add up all of the %'s due to magical or bleed damage to find Y and X
  3. Post the results here

It's also worth noting that there are several tanking jobs to consider for some fights. Tanking Freya, for example, will have a different breakdown than tanking the Stormlasher and Water spirit. So we'd want to know which job that player was doing (though this should also be discernible by the information in the parse).
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Re: "New EH" - incorporating different damage types into EH

Postby Alixander » Wed Nov 25, 2009 1:12 pm

Should we even bother with the overtly magical fights? I'm specifically thinking of single-tank Hodir (with or without resist gear), but I'm sure there are others if I stopped to think of it. In those type fights, you're taking so much magical damage that I can't imagine that Armor is the better option, but we might want the numbers just for knowledge's sake (who knows when it will be useful later).
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Re: "New EH" - incorporating different damage types into EH

Postby Wrathy » Wed Nov 25, 2009 1:26 pm

CRAP...

This is going to be a lot of work. I'll try to create a rough spreadsheet for ToTGC and post it here so that I can verify all of the information with everyone. Basically for ToTC/ToTGC, I am going to break it up as follows:

Northrend Beasts - Three part fight, Three different values for total damage, X, and Y
Gormok D X and Y
Worms D X and Y
(assmumptions will have to be estabilished for the numbers to align e.g. entire duration of the phase and how your guild tanks it)
Icehowl D X and Y

Jaraxxus Boss vs add tank - My thoughts are that the Infernal damage taken by the MT should be included.

FC - Overall D X and Y

Twins - Overall D X and Y (this fight will have a very high variable in magic damage based on the strat and skill of soakers)

Anub - MT vs OT (It can be argued that OT doesn't need data, as we have established what is needed for that gear set)
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Re: "New EH" - incorporating different damage types into EH

Postby Hammerjudge » Wed Nov 25, 2009 7:02 pm

Hi Theck,
I'm actually a DK tank nowadays, and wanted to use the calculation you've done to compare two sets of talents, (frost) gives 3% avoidance and 2% flat damage reduction, the other (unholy) 6% magic damage reduction, if all other factors remained the same.

If I followed you to conclusion, below is the formula I would use, and just insert values. Is that correct?

Code: Select all
      H*(1 - YZ)      H*(K+A)*(1-YZ)
EH = ------------  = ----------------      (16)
        1 - M               K

- Gravity
btw, you rock.
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Re: "New EH" - incorporating different damage types into EH

Postby theckhd » Wed Nov 25, 2009 7:44 pm

Hammerjudge wrote:Hi Theck,
I'm actually a DK tank nowadays, and wanted to use the calculation you've done to compare two sets of talents, (frost) gives 3% avoidance and 2% flat damage reduction, the other (unholy) 6% magic damage reduction, if all other factors remained the same.

If I followed you to conclusion, below is the formula I would use, and just insert values. Is that correct?

Code: Select all
      H*(1 - YZ)      H*(K+A)*(1-YZ)
EH = ------------  = ----------------      (16)
        1 - M               K

- Gravity
btw, you rock.

You may want to wait until I post the newest version of the theory (which is now finished). I'm going to begin typing it up now.

The reason is that the new version accounts for armor and talent mitigation separately, which more accurately models the system you're interested in. For example, the physical damage you take is actually:

P*(1-Ma)*(1-Mt)

Where Ma = A/(A+K) is your mitigation from armor, and Mt is the mitigation provided by talents.

In your case, a 2% flat damage reduction would increase both Mt and Mg (magical mitigation, which I used N for in the previous calculation), whereas a 6% magic damage reduction would just increase Mg.

After I post the theory, I'll make more specific comments about how you'd use it to solve your dilemma.
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Re: "New EH" - incorporating different damage types into EH

Postby theckhd » Wed Nov 25, 2009 8:43 pm

I'll pretty this up later. For now, I just want to get the math parts typed up for review before I go to bed.

Question: Should I just go back and revise the OP, or just leave the revised version of section II here? In the long run, I'm going to re-write the whole thing to make it a useful article anyway.


II. 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.

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 over the course of a fight, broken down into Dp "regular" physical damage, Db bleed damage, and Dg magical damage.

To get a general form, we express the damage you take d by:

Code: Select all
d = Dp*(1-Ma)*(1-Mt) + Db*(1-Mt) + Dg*(1-Mg)(1-Mr)         (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 re-write equation (8) as:
Code: Select all
d = D*[P(1-Ma)(1-Mt) + B(1-Mt) + G(1-Mg)(1-Mr)]         (11)

and re-arranging this to express EH we have
Code: Select all
                                 H
EH = H*D/d = -----------------------------------------            (12)
              P(1-Ma)(1-Mt) + B(1-Mt) + G(1-Mg)(1-Mr)


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 post-mitigation. In other words, we don't have Dp directly, we have Dp*(1-Ma)(1-Mt), 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(1-Mt)
X = -----------------------------------------          (13a)
     P(1-Ma)(1-Mt) + B(1-Mt) + G(1-Mg)(1-Mr)

                   G(1-Mg)
Y = -----------------------------------------          (13b)
     P(1-Ma)(1-Mt) + B(1-Mt) + G(1-Mg)(1-Mr)


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 1-X-Y 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 re-write equations (13) in a different form:

Code: Select all
X*P(1-Ma)(1-Mt) - (1-X)*B(1-Mt) +     X*G(1-Mg)(1-Mr) = 0         (14a)
Y*P(1-Ma)(1-Mt) +     Y*B(1-Mt) - (1-Y)*G(1-Mg)(1-Mr) = 0         (14a)


If we multiply (14b) by (1-X)/Y and subtract it from (14a), we have
Code: Select all
P(1-Ma)(1-Mt) = (1-X-Y)*G(1-Mg)(1-Mr)/Y       (15a)


Similarly, (14b) multiplied by X/(1-Y) and subtracted from (14a) gives us
Code: Select all
P(1-Ma)(1-Mt) = (1-X-Y)*B(1-Mt)/X       (15b)


Using this, we can re-write B(1-Mt) + G(1-Mg)(1-Mr) as
Code: Select all
B(1-Mt) + G(1-Mg)(1-Mr) = (X+Y)*P(1-Ma)(1-Mt)/(1-X-Y)


and equation (12) becomes
Code: Select all
                H*(1-X-Y)
EH = H*D/d = ---------------            (16)
              P(1-Ma)(1-Mt)


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      (1-Ma)(1-Mt)          Y      (1-Ma)(1-Mt)
1 = P + B + G = P + P*---------*-------------- + P*---------*--------------
                       (1-X-Y)      (1-Mt)          (1-X-Y)   (1-Mg)(1-Mr)

                                    (1-Mt)(1-Ma)
(1-X-Y) = P*[(1-X-Y) + X(1-Ma) + Y*--------------]
                                    (1-Mg)(1-Mr)

(1-X-Y) = P*[(1-X-Y) + X(1-Ma) + YZ]

(1-X-Y)
------- = (1-X-Y) + X(1-Ma) + YZ            (17)
   P

Where I'm using Z = (1-Mt)(1-Ma)/(1-Mg)(1-Mr) 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*[(1-X-Y) + X(1-Ma) + YZ]
EH = H*D/d = ----------------------------             (18)
                  (1-Ma)(1-Mt)


We can write equation (18) in a slightly more intuitive form:
Code: Select all
                   (1-X-Y)           X               Y
EH = H*D/d = H*-------------- + H*-------- + H*--------------             (19)
                (1-Ma)(1-Mt)       (1-Mt)       (1-Mg)(1-Mr)       

The first term is our EH against "regular" physical attacks multiplied by the percentage of our damage intake that those attacks represent. The second term is our EH against bleeds, multiplied by the percentage of our intake due to bleeds. And the third term is our EH against magical damage multiplied by the percentage of our intake that's magical.

In other words, EH 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.

Just to check that this makes sense, let's consider some special cases:
Y=0, X=0: Second and Third terms disappear, and EH simplifies to H/((1-Ma)(1-Mt)), exactly the form the simple version took once talented mitigation is included.
Y=1, X=0: The first and second terms disappear, and EH simplifies to H/(1-Mg)(1-Mr). This is exactly what we'd expect for a purely magical fight - our only mitigation here is from talents.
Y-0, X=1: The first and third terms vanish, leaving EH = H/(1-Mt). 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(1-Mt)
d(EH) = ------------------------------------- + -----------------------------------------
         P(1-Ma)(1-Mt)+B(1-Mt)+G(1-Mg)(1-Mr)     [P(1-Ma)(1-Mt)+B(1-Mt)+G(1-Mg)(1-Mr)]^2


d(Ma) evaluates to:
Code: Select all
           dA         A*dA       dA*K       dA*(1-M)
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(1-Mt)(1-Ma)
d(EH) = ------------------------------------- + ---------------------------------------------
         P(1-Ma)(1-Mt)+B(1-Mt)+G(1-Mg)(1-Mr)     (K+A)*[P(1-Ma)(1-Mt)+B(1-Mt)+G(1-Mg)(1-Mr)]^2


Again, we equate the first and second terms, and solve for dA:
Code: Select all
      (K+A)   P(1-Ma)(1-Mt) + B(1-Mt) + G(1-Mg)(1-Mr)       
dA = -------*-----------------------------------------*dH
        H                  P(1-Ma)(1-Mt)


It should be clear by inspection that the second fraction in that expression is simply 1/(1-X-Y), giving us the final forms of dA:
Code: Select all
      (K+A)      1
dA = -------*---------*dH                       (21)
        H     (1-X-Y)


      12.54*(K+A)      1
dA = -------------*---------*dS                 (22)
           H        (1-X-Y)

Here we finally have an equivalence that accurately relates armor to stamina for a multi-faceted 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 EH for Y=1.

Alternatively, solving for dS:
Code: Select all
           H
dS = -------------*(1-X-Y)*dA                   (23)
      12.54*(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 (1-X-Y). 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 check some numbers. For a tank with H=50k, A=30k, Ma=0.6433. For paladins, Mt=0.1686 and Mg=0.1156. Let's see what happens as Y varies from 0 to 1:
Image
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/(1-Y)
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

So for a purely physical fight, armor seems like a pretty good deal. But for even a fight with 20% magical damage, Armor becomes devalued 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 153 stamina. However, for a fight with Y=20%, we get only (1-Y)=80% of that, or 123 stamina. For a fight with 30% magical damage, it's only worth 107 stamina, and so forth.



III. Conclusion

It is quite simple to extend the Effective Health calculation to incorporate non-physical sources of damage. The result is simply a weighted average of one's effective healths vs. "regular" physical, bleed physical, and magical damage. 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
                   (1-X-Y)           X               Y
EH = H*D/d = H*-------------- + H*-------- + H*--------------             (19)
                (1-Ma)(1-Mt)       (1-Mt)       (1-Mg)(1-Mr)       

      12.54*(K+A)      1
dA = -------------*---------*dS                 (22)
           H        (1-X-Y)

           H
dS = -------------*(1-X-Y)*dA                   (23)
      12.54*(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 raid-buffed max health, and A is the players raid-buffed armor.

Code: Select all
Ma = A/(A+K)
K = 467.5*L - 22167.5


The take-home 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% non-bleed 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).
Last edited by theckhd on Fri Nov 27, 2009 10:39 am, edited 6 times in total.
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Re: "New EH" - incorporating different damage types into EH

Postby Joanadark » Wed Nov 25, 2009 11:33 pm

I tend to hesitate before subscribing to any sort of defined stam-armor equivalency ratio.

The problem for me is that I don't think your equation models correctly the part of damage mitigation which I've always considered the most important; the way incoming damage is clustered.
While you can create formulas such as yours if you are basing your assessments of value purely on Damage In, I find that in real situations the relationships between the stats do not remain static depending on damage clustering trends.

Ultimately, what EH theory represents is the answer to the big question "How big a burst can I survive before my next heal lands?". The problem with stam has always been that an extra couple stam gems, or a single or even multiple stam pieces, or even a stam trinket are generally not enough to make a difference in being able to take an additional hit unhealed, and the difference between stam stacking and not is generally lost within the normal variations of damage range.
Where stam really starts to have a definite effect on tank survivability is when hits are relatively small, but frequent. A good example would be Algalon.

Armor, on the other hand, contributes in increasing effectiveness as the size of hits go up. The higher the hit, the more damage you are removing from the burst for the same absolute quantity of armor.
Thus, the real stam-equivalency of a given amount of armor is directly connected to hit size and the way damage is clustered, and thus cannot be modeled quite as simply as you've done.
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Re: "New EH" - incorporating different damage types into EH

Postby jathitimus » Thu Nov 26, 2009 1:27 am

Joanadark wrote:I tend to hesitate before subscribing to any sort of defined stam-armor equivalency ratio.
.......
Thus, the real stam-equivalency of a given amount of armor is directly connected to hit size and the way damage is clustered, and thus cannot be modeled quite as simply as you've done.


This is my original post:

I would agree that the EH boost of armor fluctuates based off the size of the in coming hit.
.1% worth of armor would be an EH gain of 1 on a 1,000 attack, it would also be an EH gain of 100 on a 100,000 attack.


This is what I found when I tried to prove myself correct:


at 26500 armor you have a 61.43% reduction from a lvl83 attacker
at 27000 armor you have a 61.88% reduction from a lvl83 attacker

500 armor in this case would yeild .45% additional reduction.
500 armor would be 455 armor pre-talents, 455 armor is roughly the same as 32.5 stamina in itempoints. 32.5 stamina after talents and buffs would be 406 health.

being hit for 100,000 the .45% reduction would reduce damage by 450
being hit for 20,000 the .45% reduction would reduce damage by 90


Considering that armor reduces the size of EACH hit, the EH boost of armor would be also tied to how many hits one could take before death.

If one could take 2 hits for 100,000 in the above case, then it would be effectively a reduction of 900, 4 hits would be 1800, etc.
With everything else being the same, someone that could take 2 hits for 100,000 could also take 10 hits for 20,000 this would be a reduction of 900......


It would seem that if my logic and math is correct, EH gain of armor would not be tied to the size of the hit as much as one would think, if at all.
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Re: "New EH" - incorporating different damage types into EH

Postby Candiru » Thu Nov 26, 2009 2:57 am

Armour is only tied to hit size in so much as it affects the effectiveness of block value.

Hit size is irrelevant for armour contribution without Block Value.

(EG you either take 10* as many hits at 1/10 each, or 1 big hit, either way its the same total damage and same total armour contribution. If you are blocking these attacks, however, then if your armour lowers the damage taken to be equal to your BV, you take no damage. This makes the 1/10*10 htis do 0 damage while the one big hit does 9/10 damage.)
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Re: "New EH" - incorporating different damage types into EH

Postby Harlequina » Thu Nov 26, 2009 3:47 am

Thanks alot Theck
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Re: "New EH" - incorporating different damage types into EH

Postby Ragingsoul » Thu Nov 26, 2009 4:38 am

Joanadark, Theck also made a thread about hit size vs HP, when more stamina is worth less because of the hit size, if you can't find it, I'll link it when I get home, or maybe someone else can link it.
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