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    Diminuishing Returns - Armor

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    Vepar
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    Diminuishing Returns - Armor

    Post  Vepar on Wed Feb 18, 2009 5:07 pm

    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):
    Code:

    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:
    Code:

    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:
    Code:

    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 armor and hitpoints.

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