Diminishing Returns - Math!
Here we deal with mitigation due to armour using some Calculus to show its properties.
We begin with the basic formula for physical damage mitigation from armour for opponents level 60 and higher:
For the purposes here (467.5*level - 22167.5) is a constant (C). The variable x will be used to represent armour (simply because I'm too lazy to re-do all the graphic calculus bits with an a.) Using these definitions, the basic formula for mitigation simply looks like:
M = -------------------------------
Armour + (467.5*level - 22167.5)
It's important to realize that there are two significant effects of damage mitigation:
Both are significant to the healer in different ways. The former affects how much health/sec must be restored to keep you alive, and the latter affects the amount of mana that can be regenerated between burst healing.
1) The effect on the rate of health loss
2) The effect on the amount of time it takes to go from full health to dead.
Effect of Armour on Rate of Health Loss
If h is the health at any instant in time, H is the maximum health, D is the pre-mitigation damage per second incoming, t is time, and P(t) represents healing as a function of time, then the formula for health at a given time looks like:
Then, this term represents post-mitigation damage:
The rate of change of health would be the derivative of health with respect to t:
This makes sense: dh/dt increases as x increases, and goes to (0 + dP/dt) as x goes to infinity. Now we can examine the effect change in armour has on the rate of health loss. The change in the rate of health loss with the change of armour (that is x) can be found by taking the derivative of the above equation with respect to x. Note that as healing is not a function of armour, the healing term P(t) drops out as zero.
(Remember back to the quotient rule if you don't see how to get here (d/dx)(u/v)=[v(du/dx)-u(dv/dx)]/v^2). This equation represents the change in health/second of damage taken with change in armour. We see the slope of the armour v. mitigation curve is on the order of 1/(x+C)^2. The extent of diminishing returns (i.e. the change in the slope of the relationship between armour and damage mitigation) can be found by taking the second derivative with respect to x:
(If you've forgotten, the rule to get here is (d/dx)(u^n)=nu^(n-1)(du/dx)). The negative second derivative signifies a flattening of the slope as armour increases, meaning a lessening of returns for each quantity of armour added. We see that there are diminishing returns on armour with respect to health/sec mitigated. It diminishes as -2DC/((x+C)^3).
We can also take the second derivative with respect to armour of the mitigation function above and see that it is also negative, and thus there are diminishing returns on mitigation in the absence of time. As expected, mitigation alone diminishes as -2C/((x+C)^3) - the same as the health/sec mitigation, less the damage per second component.
Effect of armour on lifespan
It may seem contradictory to say the effect of armour on the time it takes to go from full health to dead is different than the effect on health/sec mitigated, but it is. For the purposes of this section, we refer to the amount of time it takes to go from full health to dead as "lifespan" which we'll call T going forward here. The difference between the analysis of lifespan and the analysis of damage mitigated stems from the nature of mitigation. As mitigation approaches 100%, each 1% increase has a greater effect on lifespan than the last. (e.g. going from 50% mitigation to 51% mitigation will extend your lifespan approximately 2%; whereas going from 98% mitigation to 99% mitigation will extend your lifespan by 100%). Using the basic formula for health (h) from the previous section, we see that t=T when h=0. Therefore, solving for T with h=0 yields:
Now with this equation, we can examine the effect of armour on lifespan by taking the derivative of T with respect to x (uses the same division rule noted above).
This is a very interesting equation, and can be rearranged as:
After a bunch of factoring:
The term x^2+Cx factors into x(x+C), which cancels the denominator and results in simply x, giving:
Which leads simply to:
It is interesting (and significant) that dT/dx is not a function of x. This means that the slope of armour vs. lifespan is a constant (i.e. d^2T/dx^2=0) so there are no diminishing returns on lifespan as armour increases. As the saying goes, mitigation is subject to diminishing returns, but armour is not. More properly, mitigation is subject to diminishing returns, but lifespan is not.
1) As armour increases, the damage mitigated per second is subject to diminishing returns. The effects diminish as:
Where D is the unmitigated damage per second, x is armour, and C = 467.5*level - 22167.5.
2) As armour increases, the effect of increasing the amount of time required to go from full health to dead is not subject to diminishing returns. Each increase of some quantity of armour will increase a tank's lifespan equally (that is, if going from 4000 to 6000 armour increases your lifespan by 1 minute, then going from 12000 to 14000 armour will also increase your lifespan by 1 minute).