[Satrina]

Let's talk about armour, mitigation, and diminishing returns. First, two definitions so we're all thinking the same thing:
Mitigation - This is the amount of physical damage that your armour reduces. For example, if you have 10000 armour and are fighting a level 70 opponent, your armour gives you 48.6% mitigation, or, you take 51.4% of the damage dealt to you. For example, if the opponent hits you for 200 damage, you would take 103 points of damage. You can calculate your mitigation with this formula (assuming your oppoent is level 60 or higher - the formula is slightly different for under level 60):

              AC
M = -----------------------
    AC + (467.5L - 22167.5)

Where level is the level of your opponent, not your own level.
Time to Live - Also called survivability. Your hitpoints and mitigation from armour combined will allow you to stay alive for a certain length of time under a given rate of damage incoming. If I have 10000 armour and 10000 hitpoints, and my opponent is level 70 and deals 200 damage per second (DPS) then:

- I have 48.6% mitigation, as above
- My mitigation means that my opponent actually is doing 102.7 damage per second to me
- My 10000 hitpoints will last for 97.4 seconds at that rate. This is my time to live

As we see, my armour and hitpoints combine together to determine how much damage I am taking, and how long I will live. We note that armour matters only for physical damage. Armour does not do anything against a fireball - that's why we have resist gear.

Diminishing Returns
What are diminishing returns, anyway? Here is a definition:

In economics, diminishing returns is the short form of diminishing marginal returns. In a production system, having fixed and variable inputs, keeping the fixed inputs constant, as more of a variable input is applied, each additional unit of input yields less and less additional output. This concept is also known as the law of increasing opportunity cost or the law of diminishing returns.

In short, you have diminishing returns if, every time you add the same amount of input, you get less output. It is plainly evident that mitigation is subject to diminishing returns as the mitigation function given above is asymptotic to 1. In any case, we will show that mitigation is subject to diminishing returns using an empirical example:

1) I have 5000 armour.  Against a level 70 opponent, this means I have 32.1% mitigation.  
   Now I will add 1000 armour, giving me 6000 armour.
   Against a level 70 opponent, this means I now have 36.2% mitigation.  
   Adding 1000 armour increased my mitigation by 4.1%
	 
2) I have 10000 armour.  Against a level 70 opponent, this means I have 48.6% mitigation.  
   Now I will add 1000 armour, giving me 11000 armour.
   Against a level 70 opponent, this means I now have 51.0% mitigation.  
   Adding 1000 armour increased my mitigation by 2.4%
	 
3) I have 12000 armour.  Against a level 70 opponent, this means I have 53.2% mitigation.  
   Now I will add 1000 armour, giving me 13000 armour.
   Against a level 70 opponent, this means I now have 55.2% mitigation.
   Adding 1000 armour increased my mitigation by 2.0%

We clearly see at different levels of armour, adding 1000 armour returns less and less mitigation. This is, by definition, diminishing returns. For the more mathematically inclined, here is the general proof of this. At this point I will note that if you still believe that mitigtion is not subject to diminishing returns, I highly suggest that you contact scientific journals immediately and get your proof published, because you have discovered something about calculus that is brand new and will essentially change the modern view of mathematics.

So Why Do People Say There Are No Diminishing Returns?
Mostly because they are talking about time to live, and are confusing labels and terminology. Look at this example:

1) I have 5000 armour and 10000 hitpoints.  
   Against a level 70 opponent that deals 200 damage per second I will live for 73.6 seconds
   Now I will add 1000 armour, giving me 6000 armour.
   Against a level 70 opponent that deals 200 damage per second I will live for 78.4 seconds
   Adding 1000 armour increased my time to live by 4.7 seconds.
	 
2) I have 10000 armour and 10000 hitpoints.
   Against a level 70 opponent that deals 200 damage per second I will live for 97.4 seconds
   Now I will add 1000 armour, giving me 11000 armour.
   Against a level 70 opponent that deals 200 damage per second I will live for 102.1 seconds
   Adding 1000 armour increased my time to live by 4.7 seconds.
	 
3) I have 12000 armour and 10000 hitpoints.
   Against a level 70 opponent that deals 200 damage per second I will live for 106.8 seconds
   Now I will add 1000 armour, giving me 13000 armour.
   Against a level 70 opponent that deals 200 damage per second I will live for 111.5 seconds
   Adding 1000 armour increased my time to live by 4.7 seconds.

The 4.7 second value is rounded off. If you carry all decimal places, you see that in all three cases the increase in time to live is always 4.7359696898... seconds, matching exactly to however many decimal points you carry. This difference holds true for any given pair (X, X+1000) along the mitigation function for a fixed health value. For the mathematically inclined, the general proof of this was also here.

Well wait then, what's this? Yes, for every X amount of armour that you add, your time to live increases by the same amount, no matter how much armour you had already. This increase in time to live is indeed not subject to diminishing returns. This is where the confusion comes in. Here we plainly see that adding armour does not have a diminishing effect. But, it is the time to live function that is not subject to diminishing returns, not the mitigation function. You are talking about a different system here. Mitigation is a function of your armour and your opponent's level. Time to live is a function of your mitigation, hitpoints, and damage per second. In fact, mitigation being subject to diminishing returns is the whole reason that time to live works this way. If mitigation was not on diminishing returns, then adding X armour would change your time to live by different amounts every time.

Summary
What it all boils down to is this phrase: Mitigation is subject to diminishing returns, but armour is not. We use this one sentence to summarise both of the effects explained above. I suppose a more correct sentence might be "Mitigation is subject to diminishing returns, but survivability is not", but most people express it in the first way generally. The distinction between mitigation and armour/time to live/survivability is this:

- Mitigation is a function of your armour and your opponent's level
- Time to live is a function of your mitigation, your hitpoints, and how much damage per second you are taking

These are two different systems that are related. The second one uses the first in its calculation, and works the way it does because mitigation is subject to diminishing returns. Again we stress that this is valid against physical damage only. Once fireballs and shadowbolts begin to fly, it goes to your resistance and hitpoints instead of armour and hitpoints.

[Joanadark]

We clearly see at different levels of armour, adding 1000 armour returns less and less mitigation. This is, by definition, diminishing returns. For the more mathematically inclined, ere is the general proof of this.

broken link

This difference holds true for any given pair (X, X+1000) along the mitigation function for a fixed health value. For the mathematically inclined, the general proof of this was also here.

broken link

[Satrina]

Fixed, thanks!

[dragonragoon]

just one question

if the target is doing a constant 200 DPS as shown in your math wouldnt they actually be doing more damage to overcome the migitation ( using the thought that migitation reduces damage... if the damage stays the same,when the migitation increases to maintain the damage stream the damage must be increased to the same degree) and wouldnt that change the overall outcome of the math

just wondering :P

[Karih]

The 200 DPS is DPS prior to mitigation by armor, as opposed to the incoming DPS the hypothetical player would see on their character at those armor levels.

[Snarfy]

This is really impressive research. The clarifcation between Mitigation and Survivability cleared up confusion for me on the matter.

I have a question. I realize now that Survivability is an additive constant ...thing... based on Armor, and that this is so because of the nature of Mitigation's diminishing returns. I'm a visual person, not inclined towards math, but I can picture in my head how the survivability math relies on the mitigation math... But I can't picture WHY.

Why does mitigation have to have diminishing returns in order for survivability to work the way it does? I try to imagine how survivability would work if mitigation wasn't subjected to diminishing returns, and I sort of draw a blank. I can imagine that survivability would have increasing returns, but only because of reciprocal logic, not because I understand why.... What's the correlation? Do I need to take college math to understand that? lol....

[Snarfy]

Source: Snarfy But I can't picture WHY.

AH! I think I get it. I couldn't stop thinking about it until I understood it. I drew up the following scenario and afterwards the why came to me.

Attacker @ 100dps
50% Mitgation
50% Damage Taken = 50dps
Player @ 10,000 Health = 200s Survivability

Attacker @ 100dps
60% Mitgation
40% Damage Taken = 40dps
Player @ 10,000 Health = 250s Survivability

Attacker @ 100dps
70% Mitgation
30% Damage Taken = 30dps
Player @ 10,000 Health = 333.33s Survivability

So a 10% increase in mitigation from 50% to 60% is a 25% increase (50s) in survivability.
But, a 10% increase in mitagation from 60% to 70% is a 33% increase (83.33s) in survivability.

So there is some sort of curve going on wherever. Blizzard, wanting to compensate in favor of steady survivability increases, decided to affect mitigation with diminishing returns (I'm guessing by the same rate that the above scenario illustrates the increasing returns?).

So the why in my above post is because survivability takes your mitigation and applies it over time. Trippy.

Yay. Maybe this will help someone else as slow as me =D

[keith]

That is all good until you are hit with an 8k pyro which your armor mitigates 0.

so you jump into zerk stance to intercept and you are sheeped. oops xtra 10% damage comming -- nice diminishing return there huh.

1 thing that upsets me the most about being a warrior is that we are the only class who has diminishing returns when we do more damage. All other classes as far as i know have 0 diminishing returns when it comes to doing more damage.

When blizzard created this game the initial intent was not to have it turn into a pvp based game. Mages, Locks, all ranged dps that is were put in this game to do millions of points of damage to bosses. They were not put here to do millions of points of damage in pve. The burst damage of a mage is way too over powered and a warrior has no way of mitigating it.

I guess i am going off on a tangent but since this is about diminishing returns where is the diminishing returns on the other classes. Especially when it comes to damage being done. I heard in 2.4 casters will now be able to stack spell haste that means more dps that we have to deal with. How do we mitigate it umm i guess resiliance. Kinda hard to hit them when they just nerfed hamstring and they are sitting at 400 resiliance to mitigate our mele damage.

Sorry if I screwed up the thread a bit but just thought that it may have some kind of relevence to the diminishing returns that a warrior gets.

keith

[Finelle]

I think you missed the point. Mitigation has diminishing returns from armor, because if it didn't, you hit exponential scaling. Kind of like what happens when you get over 70+% avoidance on dual wield bosses. Time to Live is the important metric for a tank, and the diminishing returns from armor to TTL is small enough to be insignificant.

Hence, Satrina's point is that armor has no diminishing returns for a tank, because for what's important, it scales linearly. PvP is a whole different ballgame, and if I may add, warriors have always been the highest represented class in Arenas.

[commacozzee]

i think the confusion between armor for survivability and armor for mitigation comes from the terminology and how it is applied when you're gearing up and fighting bosses, and i believe a slight clarification is necessary.

there is an in depth look into the math and application of armor mitigation vs. survivability by death and taxes' tank Alextrebek, in which he names them damage reduction and relative damage reduction. relative damage reduction (time to live, survivability from OP) refers to the fact that each point of armor linearly increases a tanks survivability (we're not factoring avoidance here, we're assuming we're hit 100% of the time) w/o incoming heals, where as damage reduction (or mitigation) increases at an exponentially decreasing rate (diminishing returns).

furthermore, the application of such knowledge as far as healing goes is that a healer's burden is decreased for each point of damage reduction, not relative damage reduction. therefore, gearing armor for survivability is negligibly inconsequential to a healer's burden, which i believe is a more practical look at how armor affects gameplay.

in response to Finelle, i believe that TTL is much less important in a raid than damage mitigation is. there are t6 bosses who are doing 9k dmg on the tank, increasing TTL by 0.5 sec doesn't mean anything when that 0.5 sec is a space where there is no damage, and is just waiting time before that cleave of death (of course we all know a tank needs to be able to survive for a small amount of time w/o heals, 2-3 seconds or so or 2-3 worst case hits/crushes).

i don't believe armor is the right stat to look for to increase TTL, i believe avoidance and good button pushing are the way to survive when heals aren't enough. mitigation and avoidance (in general) are crucial overall stats that work hand in hand to decrease incoming damage to your tank, and thus the amount of survivability is increased and amount of healers needed for your tank is reduced directly from these two stats. of course, this leads to an avoidance vs. EH debate, but that is reserved for another thread

this is not my own work, the link How to Armor - A (short) guide by alextrebek is taken from death and taxes' guild forum as a response to a 'How to Warrior' EJ thread.

[phaze]

We're both straying off topic here, but:

i don't believe armor is the right stat to look for to increase TTL, i believe avoidance and good button pushing are the way to survive when heals aren't enough.

The exponentional gains of increased avoidance are at odds with the logarithmic reductions to the worst-case. How these two balance out depends on your choice of points on each curve.

When modeling for the worst-case scenario, you assume worst-case avoidance, with respect to your current EDF. For those circumstances, it's unlikely that your gearing will be in a position to massively improve your EDF through large avoidance gains. You are unlikely to make a discrete improvement to the worst-case event (boosting EDF to go from expecting 4 hits in a row at the worst, to just 3 hits in a row, requires a lot of extra avoidance).

So for these situations, you'll want sufficient EH to take the hits. The other gearing option for mitigation is Block Value, and that's not a guaranteed occurrence either. For most encounters, the fastest way to adjust gear in order to survive, is to boost EH through increased armor and stamina.

Of course, once you're comfortable with surviving your worst-case scenario, you can modify your gearing decisions. You can either work on improving your worst-case scenario (with better EDF through more avoidance), or you can work on more readily surviving the current scenario (with better EH through more armor/stamina).

By this point though, you're already surviving, so most people turn to other concerns: improving threat, and boosting raid efficiency.

mitigation and avoidance (in general) are crucial overall stats that work hand in hand

That's the important part. =)

[commacozzee]

We're both straying off topic here

argh you're right :P my post was a look at how i view balancing armor with other stats and how it affects TTL and overall damage mitigation. it was mostly in response to the previous post about how TTL is the important metric to look at when gearing your tank:

Satrina's point is that armor has no diminishing returns for a tank, because for what's important, it scales linearly.

TTL is obviously crucial for the worst-case scenarios that will always happen, and armor directly contributes to your EH.

Of course, once you're comfortable with surviving your worst-case scenario, you can modify your gearing decisions.

that's the idea i'm looking for. gearing for armor and stamina increase the odds you'll survive the worst-case scenario, but once you get past your EH minimums, you'll want to look elsewhere for gear choices. and depending on your gear level, you may be looking for more survivability, more threat, or more avoidance. the beauty of a tank is that you are balancing many different factors to maximize your efficiency.

When modeling for the worst-case scenario, you assume worst-case avoidance, with respect to your current EDF. For those circumstances, it's unlikely that your gearing will be in a position to massively improve your EDF through large avoidance gains. You are unlikely to make a discrete improvement to the worst-case event (boosting EDF to go from expecting 4 hits in a row at the worst, to just 3 hits in a row, requires a lot of extra avoidance).

you are absolutely correct about assuming worst-case avoidance, and that should never change. my point is gearing for avoidance to decrease the chance of getting hit 4 times in a row (as opposed to lowering your EH minimum by overgearing for avoidance, which i don't endorse at all except for players better than myself :P) is more effective for raid stability than increasing your EH beyond minimums, but again;

We're both straying off topic here

i've brought an avoidance argument to a mitigation thread. please forgive me, i only want new tanks stumbling upon this thread to get a more diverse view of gearing choices as they read on, as i know through experience that focusing too much on one aspect of your character is easy and inferior to balancing all factors of your characters stats.