[Calculus Inside] Finding the Perfect Balance for Health and Armor
Hello everyone! This is my first [Math Inside] thread. Except, this is a [Calculus Inside] thread. Basically the same, but it scares away small children. I was wondering how to optimize effective HP. As an example, if you had 3000 health and 20 armor, it is more costefficient to buy Cloth Armor instead of a 4th Ruby Crystal. Likewise, with 1000 health and 400 armor, buying health will be more costefficient to raising Effective HP(hereafter called EHP). I was thinking I could set up some equation with health and armor in it, and solve for an answer like H/A = 6, which would mean that it is optimal to have 6x more health than armor.
I set up an equation. I looked at it for a while. I grabbed my calc textbook and turned to the chapter entitled "Vector Calculus." The equation I got was EHP = H + HA/100. You guys can test it; it works. The first thing I noticed is that this equation can't be solved as H/A = X. The third variable prohibits that. Then a random memory of a question in my textbook asking "In which proportions do we increase the independent variables to most rapidly increase the dependent variable?" The first step was to take the gradient of the function. To do this, we take the partial derivative with respect to H and add unit vector h, then do the same for A, but with unit vector a. gradEHP = (1 + A/100)h + (H/100)a To use this, you would enter your current A and H amounts, and it would tell you the optimal proportion of each to build from there. However, it seems to really highly prize armor over health. This can be remedied by adding a few constants equal to the gold value of one point of armor/HP. 1 gold will get 0.0643 armor or 0.3874 health. Replacing H and A with 0.3874H and 0.0643A will inform the equation of how expensive armor and health are. (1 + 0.0643A/100)h + (0.3874H/100)a or (1 + 0.000643A)h + (0.003784H)a Enjoy the fruits of my labor! Mathcheckerpeople, feel free to look over what I did and inform me if I messed anything up! Edit: For more practical decisions, you should also include MR in the equations. I'll do that sometime tomorrow(Oct 11), with weights you can add depending on the opposing team composition. 
MR is also a factor in EHP, although I suppose that would significantly complicate the equation.

The thing is, the calculation is done with minimal information. Champions can do %health damage, shred armor/mr in flat or percent amounts, and health regeneration based on time not at full health.
When I did the math a year back, I got something like 3000 health = 250 armor/mr and 1800 health = 67 armor/mr. So early on, health is better to get, but once you get near 2k health, you should start grabbing armor/mr. The exception is for high sustain champs or lifesteal/vamp—resists give you more EHP than health since they regenerate a lot of health. 
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