Elitist Jerks (Relative) Damage Reduction from Armor, explained.

 05/19/08, 7:43 PM #1 ALEXTREBEK Glass Joe     alextrebek Tauren Warrior   Blackrock (Relative) Damage Reduction from Armor, explained. This post has been (re)moved, as its usefulness as a whole thread on these boards simply came to an end. If you are interested in reading the entire post, it can be found at www.dtguilds.com located in our Warrior Class forums. The gist/significant segment of the post is as follows: In all matters of correctly assessing the situation: The amount of physical damage reduced by Armor and the amount of HPS necessary to keep a Tank alive in relation to this amount is, and has always been, directly proportional to the amount of Damage Reduction from Armor you have, and not Relative Damage Reduction. In addition: Armor does not suffer from "basically negligible" diminishing returns, it suffers from serious and profound diminishing returns as a function of reducing incoming physical damage. However: Armor more or less does not suffer dimishing returns in its ability to increase your survival time without heals, which is the only accurate case in which Armor functions relatively. Last edited by ALEXTREBEK : 05/22/08 at 10:58 PM.
05/19/08, 9:13 PM   #2
Muphrid
Don Flamenco

Gnome Mage

Stormrage
To me, the key problem we're dealing with when we talk about relative increases, instead of absolute increases, in effectiveness is that people naturally assume that relative increases don't matter, or that absolute increases do matter.

The usual argument that armor is "fair" goes along the lines of "Well, if I add +X armor to both a mage and a warrior, they both live 5 seconds longer against someone attacking them. They both got the same 5 seconds, so this is fair." We can make this a bit more concrete:

Say a mage lives 15 seconds against a rogue attacking him.
Say a warrior lives 30 seconds against a rogue attacking him.

If we add some armor, an equal amount of armor to both the mage and the warrior, we could end up with...

The mage lives 20 seconds
The warrior lives 35 seconds

The problem is that this is inherently not the same balance that we had before--you can't say that adding armor was fair. Let's say I Fraps the original fights where the mage lived 15 seconds and the warrior 30 seconds and play these fights at a slower speed, such that...

The mage lives 20 seconds
The warrior lives 40 seconds

This is a simple transformation that is necessarily fair, because slowing down the speed of time is, well, obviously not going to change the outcome of anything. But for this to be fair, we can see the warrior should not get +5 seconds of time to live, but 10 seconds.

In short, the way armor works now, the mage's lifespan increases by 33.3%, but the warrior's increases by only 16.7%. This is why we say armor has relative diminishing returns: the more armor you have, the smaller the % increase in your lifespan.

To be fair, most stats have relative diminishing returns. Nearly all DPS stats do. Curiously, avoidance stats do not. Neither does the antithesis of armor--armor penetration.

Now, I've chimed in with my 2 cents on the subject of armor in general. Let me address the post itself.

 So, in fact, the healer's burden has not actually changed at all. Percent-wise (or relatively/whatever you want to coin it), yes, at 51% DR, the healer's burden is "2% less" than it would be at 1% DR, but actually speaking, the healer's burden of 20 HPS has not changed at all and is empirically, directly proportional to the amount of Damage Reduction from Armor you have, is theoretically proportional to the amount of Avoidance you have (a whole other story), and (obviously) differs depending on the type of damage the Tank is taking and the amount the Tank is taking. In all matters of correctly assessing the situation: The amount of physical damage reduced by Armor and the amount of HPS necessary to keep a Tank alive in relation to this amount is, and has always been, directly proportional to the amount of Damage Reduction from Armor you have, and not Relative Damage Reduction. In addition: Armor does not suffer from "basically negligible" diminishing returns, it suffers from serious and profound diminishing returns.
To be honest, I think the post you were responding to was saying something that didn't make sense in the first place. It is without meaning to say that the healing burden is related to the relative damage reduction. The relative damage reduction always has to be in reference to some baseline gear level or point. You can look at, say, how the healing burden has changed when you take a tier 4 healer and a tier 4 tank and upgrade them to tier 6, but to assess what the burden is is quite another story.

Now, that said, I think what they were trying to get at is that it's easier to increase one's HPS by 20 when you're already dealing 2000 HPS than it is to increase your HPS by 20 when you do only 1000 HPS. To some extent, this is true (because all healing stats scale with each other: +healing's effect is increased by +crit and +haste, for instance), but it's hard to say definitively without looking at the itemization budget and actually doing the math.

Also (and I'm not sure of the context that your post and the post you quote are coming from), it seems that you've addressed the diminishing returns of armor without mentioning that it takes progressively more armor per 1% mitigation. Indeed, if it took only +X armor for +1% mitigation, armor would have increasing returns. What gives armor diminishing returns is that the additional mitigation from +X armor is decreases more quickly than 1-mitigation.

 05/19/08, 9:25 PM #3 • Aldriana Mike Tyson   Aldriana Night Elf Rogue   Doomhammer So, you're absolutely correct, insofar as your answers extend; however, I'm forced to ask whether you're asking the right questions. I would argue that the question you're fundamentally trying to answer is how the value of the armor changes relative to the value of other stats as one's gear improves. Thus, the answer is not, in absolute terms, how does armor scale - but, rather, how it scales relative to other stats. So lets take a look at this question for a moment. There's two fundamental ways in which to measure the value of mitigation stats, namely: how do they affect my ability to withstand worst-case burst damage scenarios, and how do they affect the overall amount of damage I take. Lets look at these one at a time. First, surviving burst damage. The case under consideration here is: the boss, due to a parry or crushing blow or other fluke of damage, generates some huge spike of damage, which I utterly fail to dodge; hence, the relevant measure is how much damage I can absorb without healing assuming I totally fail to dodge everything. So if there's some incoming about of damage D, and I have H health and M mitigation, I want to make H as large as possible relative to D*M; in other words, I want to maximize the value H/M; the relevant measure for doing so is to figure out how much H I need to equal a given quantity of M. So if M is, say, 1 - that is, I take full damage from every attack - and I increase M by 1% so I now take M/.99 damage, or 1.01% more damage to kill. In order to become 1.01% harder to kill via stamina alone, I'd need to have 1.01% more health. Hence, 1% mitigation, at this level, is worth however much stamina is required to gain 1.01% of my current HP. Meanwhile, if M is, say, .5 - so I take half damage from everything - and I increase M by 1%, I now take M/.49 damage instead of M/.5; Hence, the increase in health needed to match is .5/.49 = 2.04%. That is, at 50% mitigation, it requires 2.04% more HP to equal the same benefit as 1% mitigation. Hence, in terms of it's equivalence to stamina, 1% mitigation is worth twice as much at 50% mitigation as it is as 0% mitigation. When one actually works out the numbers, it so happens that the stamina equivalence of one point of armor value does not depend at all on one's current armor value. So there's no increasing *or* decreasing returns in this case. Well, what then about overall damage taken? Given some input amount of damage D with avoidance A and Mitigation M, the net damage we take is DAM. If we want to reduce this by, say, 1% with avoidance, we have to change our avoidance to some new quantity A', which has no dependence on M. Meanwhile, if we want to decrease the amount damage we actually take by 1% with no mitigation (aka M = 1), we need to increase our mitigation by 1%, to .99; but if we want to do so with M = .5, we only need to reduce M to .495 - that is, to match a given gain of avoidance at 0 mitigation takes twice as much mitigation - percentagewise - as it does at 50%; hence, the value of every point of armor, measured in avoidance, is the same - again, no diminishing returns. Hence, regardless of whether you think mitigation should be measured in absolute numbers of percentagewise, the fact is that relative to any and every other stat, mitigation's marginal value relative to the other stats is completely invariant relative to how much mitigation you currently have; hence saying that it suffers diminishing returns, severe or otherwise, is somewhat misleading.
05/19/08, 9:31 PM   #4
Daedalix
Piston Honda

Draenei Shaman

Kilrogg
While interesting, is the Armor debate even worth analyzing at this point? When MC was the paradigm, warriors generally had 3 choices: lots of armor, lots of stam, and lots of avoidance. It was hard to stack any of them, given the gear, but you definitely had a choice. The amount of itemization "choices" on each piece went up drastically in TBC and the armor varied little from piece to piece and there's not much way to stack armor anymore (assuming a certain level of progression), nor are there consumable choices that pit armor versus anything worthwhile. People chug Ironshield pots, and there's really nothing else to do. Armor isn't much of a choice these days... you can't really gem for armor.

Now we deal mainly with avoidance, effective health, and threat. Armor is such a small part of the equation, but I guess that's your point in the first place.

This argument would seem even less practical as we step into Wrath. Though it may be worthwhile for Blizzard to explore itemization/encounter design around the fact that every tank may very well become armor capped without effort.

 Originally Posted by ALEXTREBEK In addition: Armor does not suffer from "basically negligible" diminishing returns, it suffers from serious and profound diminishing returns.
The math proves this out, depending on the relevant range, which is where I think a lot of the misconceptions in this argument arise. What tanks are generally dealing with (these days and not in the days of Conquest) is at the extremely high range of armor, where damage reduction is closer to the 100% mark than it is away from it. If people kept this argument to a relevant range, it'd be easier to come to a consensus. At that point the diminishing returns are clear. In the middle of the range (MC days which was the genesis of this argument) the diminishing returns are less clear and the returns less marginal. In order to not make players completely immune via Damage reduction, the graph never reaches 100% DR (or 75% or 0%, depending on what you are graphing) so each point of armor close to the limit can't "go anywhere." In order for there to be decreasing returns at the high end of the spectrum, there must be increasing returns at the low end of the spectrum.

05/19/08, 9:37 PM   #5
Muphrid
Don Flamenco

Gnome Mage

Stormrage
 Originally Posted by Aldriana Hence, regardless of whether you think mitigation should be measured in absolute numbers of percentagewise, the fact is that relative to any and every other stat, mitigation's marginal value relative to the other stats is completely invariant relative to how much mitigation you currently have; hence saying that it suffers diminishing returns, severe or otherwise, is somewhat misleading.
With respect, that's not true at all. It takes increasingly more +armor per 1% mitigation as your mitigation rises, while it takes the same +dodge rating, for instance, for 1% dodge. The value of armor relative to dodge is a function of both pre-existing mitigation and pre-existing avoidance. As mitigation goes up, the value of armor compared to dodge goes down. As avoidance goes up, the value of armor compared to dodge goes down, as well.

 05/19/08, 9:46 PM #6 • Aldriana Mike Tyson   Aldriana Night Elf Rogue   Doomhammer The math in my post says otherwise. It's true that armor requires more armor to get the same percent benefit - but if you'll note, that's exactly why it has no returns - positive or negative. Because dodge is fixed cost for 1%, it has *increasing* returns with itself; so it's fair to say that dodge scales with itself better than armor does. But then, armor scales better with itself better than stamina does. And it's scaling relative to the other stats is, regardless, quite independent of the point I was trying to make, which is: at a given amount of sta and avoidance, there is no diminishing returns whatsoever on the value of armor.
05/19/08, 9:56 PM   #7
Daedalix
Piston Honda

Draenei Shaman

Kilrogg
 Originally Posted by Aldriana at a given amount of sta and avoidance, there is no diminishing returns whatsoever on the value of armor.
Then how do you explain the fact that tanks cannot reach 100% damage reduction? At some point, an extra point of armor is worthless. Has to be. Otherwise, a tank could walk up and take no damage for 40 minutes.

05/19/08, 9:59 PM   #8
Happy October 19th!

Night Elf Druid

Dragonblight
 Originally Posted by Daedalix Then how do you explain the fact that tanks cannot reach 100% damage reduction? At some point, an extra point of armor is worthless. Has to be. Otherwise, a tank could walk up and take no damage for 40 minutes.
Does it have to be spelled out that, once you hit the cap, it has 0 value? That's not decreasing, that's zero.

05/19/08, 10:12 PM   #9
Muphrid
Don Flamenco

Gnome Mage

Stormrage
 Originally Posted by Daedalix Then how do you explain the fact that tanks cannot reach 100% damage reduction? At some point, an extra point of armor is worthless. Has to be. Otherwise, a tank could walk up and take no damage for 40 minutes.
Well, it's known that armor is capped at 75% reduction...

 The math in my post says otherwise. It's true that armor requires more armor to get the same percent benefit - but if you'll note, that's exactly why it has no returns - positive or negative. Because dodge is fixed cost for 1%, it has *increasing* returns with itself; so it's fair to say that dodge scales with itself better than armor does. But then, armor scales better with itself better than stamina does. And it's scaling relative to the other stats is, regardless, quite independent of the point I was trying to make, which is: at a given amount of sta and avoidance, there is no diminishing returns whatsoever on the value of armor.
Do you know the formulas involved?

$W= \frac{1}{1+\frac{w}{v}}$

Where W is the fraction of incoming damage (or DPS) taken (not mitigated), w is your actual armor, and v is a constant based on level (10557.5 for a level 70 target, for instance).

We can derive that, for a change in armor ∆w, we can figure out the new damage taken , W+∆W.

$W + \Delta W= \frac{1}{1+\frac{w}{v}+\frac{\Delta w}{v}}$

Divide by W, subtract 1, and you've solved for ∆W/W, the relative change in damage taken.

$\frac{\Delta W}{W}= \frac{1}{1+\frac{\Delta w}{w+v}} - 1$

Thus, the relative value of additional armor is, in fact, a function of w and thus a function the armor you already have (and thus the mitigation you already have).

05/19/08, 11:06 PM   #10
Oaken
Don Flamenco

Night Elf Druid

Uldum
 Originally Posted by ALEXTREBEK In all matters of correctly assessing the situation: The amount of physical damage reduced by Armor and the amount of HPS necessary to keep a Tank alive in relation to this amount is, and has always been, directly proportional to the amount of Damage Reduction from Armor you have, and not Relative Damage Reduction. In addition: Armor does not suffer from "basically negligible" diminishing returns, it suffers from serious and profound diminishing returns.
As Aldriana has pointed out, you are asking the wrong question. Another way of looking at it is If I measure 'diminishing returns' as total damage mitigated (or, what amounts to an equivalent measure, HPS), I can show armor has diminishing returns. Nobody is going to argue that because its true.

However, that isn't the best measure. If you go back to the original Conquest post that demonstrated armor has no diminishing returns, it was all about spike damage. Tanks rarely die because healers run out of mana (which is what overall mitigation measures - your first example). They also rarely die because average HPS is too low (your second measure). What kills them is spike damage - the tank dieing just before the next heal lands. So a "better" measure to look at when trying to assess how effective armor is would be "Time to Live" (TTL) - i.e., how long can I go without getting heals before I die?

In your terms, let's take the situation involving you having 50% DR from Armor but lets not worry about HPS, let's talk in absolute terms about HP and TTL.

Assume 20,000 HP, and your raid boss hits once per second for 2,000 unmitigated damage. So at 50% DR, you get:
TTL = 20,000 / (2,000*(100%-50%)) = 20s
You now increase your DR by 1% (in an effort to save your cock you may recall), putting you at 51% DR from Armor. From this, we're seeing these results:
TTL = 20,000 / (2,000*(100%-51%)) = 20.4s
After that girlfriend sunder (did she laugh at the size?), leaving you at 0% DR from Armor, we can measure
TTL = 20,000 / (2,000*(100%-0%)) = 10s
When miraculously you grow an extra percent of Armor, you get:
TTL = 20,000 / (2,000*(100%-1%)) = 10.1s
What's this then guv'nor? That extra 1% DR when I'm at 50% gives me another 0.3s to live over what it gave me at 0DR. That's an extra 0.3s for that next heal to land - which may mean the difference between walking away with a healthy libido or living the rest of your life as a eunuch.

I could show you the extra bit of math but basically that extra bit of TTL between 0% base and 50% base is close to the reciprocal of what you lose in absolute terms as armor goes up. In other words, armor gives the same TTL increase at 0% DR as it does at 50% DR - hence, no diminishing returns (if I recall, given the exact nature of the armor equation there actually is a small difference between the two but its pretty negligible in the grand scheme of things).

That's the problem with applied math - if I ask the wrong question I get a meaningless answer. I don't care how much damage you take, I care if you live long enough to get healed.

edited for clarity

Last edited by Oaken : 05/19/08 at 11:22 PM.

05/19/08, 11:19 PM   #11
• Aldriana
Mike Tyson

Night Elf Rogue

Doomhammer
 Originally Posted by Muphrid Well, it's known that armor is capped at 75% reduction... Do you know the formulas involved? $W= \frac{1}{1+\frac{w}{v}}$ Where W is the fraction of incoming damage (or DPS) taken (not mitigated), w is your actual armor, and v is a constant based on level (10557.5 for a level 70 target, for instance). We can derive that, for a change in armor ∆w, we can figure out the new damage taken , W+∆W. $W + \Delta W= \frac{1}{1+\frac{w}{v}+\frac{\Delta w}{v}}$ Divide by W, subtract 1, and you've solved for ∆W/W, the relative change in damage taken. $\frac{\Delta W}{W}= \frac{1}{1+\frac{\Delta w}{w+v}} - 1$ Thus, the relative value of additional armor is, in fact, a function of w and thus a function the armor you already have (and thus the mitigation you already have).
As a matter of fact I do. But you're not calculating what I suggested calculating.

So, if you'll note the quantity H/M I suggested calculating, where H = HP, M is the mitigation term and A is Armor Value, we have

$\frac{H}{M}= \frac{H}{\frac{1}{1+\frac{A}{k}}}=H\left(1+\frac{A}{K}\right)$

So, the amount of this quantity - which, roughly, effective hit points for resisting burst attacks - that you gain from changing your armor from A to A' is:

$H\left(1+\frac{A}{k}\right)-H\left(1+\frac{A'}{k}\right)=H\frac{A-A'}{k}$

So if $A' = A + \Delta A$, then $A-A' = \Delta A$, so the change in EHP is simply

$\frac{H \Delta A}{k}$

Note, if you will, that there's no dependancy on A; only on $\Delta A$.

05/19/08, 11:25 PM   #12
Muphrid
Don Flamenco

Gnome Mage

Stormrage
 Originally Posted by Oaken I could show you the extra bit of math but basically that extra bit of TTL between 0% base and 50% base is close to the reciprocal of what you lose in absolute terms as armor goes up. In other words, armor gives the same TTL increase at 0% DR as it does at 50% DR - hence, no diminishing returns (if I recall, given the exact nature of the armor equation there actually is a small difference between the two but its pretty negligible in the grand scheme of things). That's the problem with applied math - if I ask the wrong question I get a meaningless answer.
I'm curious as to what you mean by this. I mean, if you're referring to how much more armor it takes to see that 1% increase in mitigation, then you're correct--in fact, the whole armor system is clearly designed to make it so that, again, +X armor = +Y seconds of time to live, depending on the incoming DPS, but regardless of how much armor you already have.

But saying that armor guarantees an equal increase in time to live is, well, a point of mere mathematical interest. To me, it makes sense to evaluate armor as a stat and compare it to other stats using some metric or formula. The obvious metrics are, of course, effective health or expected time to live. Under these metrics, the math is unambiguous: 1 point of armor decreases in value compared to 1 point of, say, dodge rating, as the armor of the player is increased.

The problem we often get when talking about diminishing or increasing returns is that we don't agree on the definition of "neither increasing nor decreasing returns." I personally will always choose to define "neither increasing nor decreasing benefit" as "constant relative benefit," not "constant absolute benefit."

05/19/08, 11:38 PM   #13
Muphrid
Don Flamenco

Gnome Mage

Stormrage
 Originally Posted by Aldriana As a matter of fact I do. But you're not calculating what I suggested calculating. So, if you'll note the quantity H/M I suggested calculating, where H = HP, M is the mitigation term and A is Armor Value, we have $\frac{H}{M}= \frac{H}{\frac{1}{1+\frac{A}{k}}}=H\left(1+\frac{A}{K}\right)$ So, the amount of this quantity - which, roughly, effective hit points for resisting burst attacks - that you gain from changing your armor from A to A' is: $H\left(1+\frac{A}{k}\right)-H\left(1+\frac{A'}{k}\right)=H\frac{A-A'}{k}$ So if $A' = A + \Delta A$, then $A-A' = \Delta A$, so the change in EHP is simply $\frac{H \Delta A}{k}$ Note, if you will, that there's no dependancy on A; only on $\Delta A$.
Yes, you calculated the absolute change in your metric. You didn't calculate the relative change in your metric, which will naturally introduce A (the amount of armor one has) into the formula.

Since your metric only involves either max HP or armor, you only come to the conclusion that armor and max HP scale in the same "way." The value of 1 HP compared to 1 armor doesn't change under this metric, and since you have no other stats involved, the only question is how you define diminishing returns of a stat not relative to another stat--in terms of relative differences or absolute differences.

Since the thread was talking about relative differences, I'm confused as to why you posted a calculation that only shows an absolute difference.

05/19/08, 11:56 PM   #14
Oaken
Don Flamenco

Night Elf Druid

Uldum
 Originally Posted by Muphrid But saying that armor guarantees an equal increase in time to live is, well, a point of mere mathematical interest. To me, it makes sense to evaluate armor as a stat and compare it to other stats using some metric or formula. The obvious metrics are, of course, effective health or expected time to live. Under these metrics, the math is unambiguous: 1 point of armor decreases in value compared to 1 point of, say, dodge rating, as the armor of the player is increased. The problem we often get when talking about diminishing or increasing returns is that we don't agree on the definition of "neither increasing nor decreasing returns." I personally will always choose to define "neither increasing nor decreasing benefit" as "constant relative benefit," not "constant absolute benefit."
My response is to the OP's post and so its in the context of the OP's claim - as an absolute measure and not with respect to other stats.

Having said that, I still hold by the point that Armor has "constant relative benefit" in relation to other stats as well. It all depends on your frame of reference though (I'm having flashbacks to my special relativity courses) - i.e., which stats you choose. In a frame in which Stamina is considered to be of "constant relative benefit" you will find that Armor is also of "constant relative benefit" and, interestingly, Dodge is of "increasing relative benefit". By using Dodge as the constant which you seem to want to do, in your frame of reference both Stamina and Armor have diminishing returns. I suspect for most people that's an unnatural frame to be in though - if you look around these forums most people have come to the conclusion that Dodge has increasing returns which pretty clear shows which frame of mind they are in.

05/20/08, 12:03 AM   #15
ALEXTREBEK
Glass Joe

Tauren Warrior

Blackrock
 Originally Posted by Oaken My response is to the OP's post and so its in the context of the OP's claim - as an absolute measure and not with respect to other stats.
And this post has been made in response to the OP of The Protection Warrior thread here in these forums that claims that it is your RDR from Armor and not your DR from Armor that affects the amount of damage your Armor is reducing and the amount of healing needed to keep you alive. Anyways, I agree with what you have posted in this thread and am going to rephrase it just for a clarified, personal reference.
• Armor, as a function of Survivability without Heals more or less does not experience Diminishing Returns - and, in this case Relative Damage Reduction from Armor is what is proportional to your 'TTL'.
• Armor, as a function of Damage Reduction, does indeed experience Diminishing Returns, in the obvious sense that as your Armor increases, so does the amount of it needed to increase your Damage Reduction from it.
EX. (For the Timmys who manage to double-click to this website) 60% Armor --> 61% Armor will decrease incoming damage by 1%, as a function of survival (without heals) however, this 1% increase in Armor is worth 2.5 times its worth in Survival time without heals.

 Elitist Jerks (Relative) Damage Reduction from Armor, explained.