Armor Penetration and the Journey to the Depths of Damage

When it comes to flat Armor Penetration, there are four categories of people: The ones that say ARP is better against low armor, the ones that say that armor doesn't matter, the ones that say that ARP is better against high armor, and the ones that have no clue. Most of the people among these groups, except the last, either don't know why or they're spouting incomplete nonsense that has nothing to do with mechanics.
I was part of the first group, and I'll show you why I was.

I took a random damage number and a random armor and made the following formula from the armor formula on the wiki.


I tried playing around a bit with damage and armor just to make sure they're not relevant to the proportions, and they weren't. It changes the location of the graph, but not the proportions.

Setting dmg and armor to constant and reducing the formula reveals how it's not linear, but exponential.

I'll show you a graph of how this works. I made a function with dmg and armor being constant, leaving only ARP as a variable, then simply took the function of ARP and subtracted the function of ARP-1, to see how much the flat increase in damage was compared to the last point of ARP. Below graph shows the result.

Damage on the Y axis, ARP on the X axis.

We clearly see that it's not linear at least, and that it's an increasing non-linear function.
This is why I figured that ARP was better on low armored targets. It was the same back in WoW (they used a similar armor mechanic), and mostly everyone discussing the subject thought it was true.

However, this is not the bottom level. This is not the relevant graph.
Damage is good. What is damage good for? Well, killing guys. So why is more damage good? Because it kills guys faster. The function I was looking for had something to do with time.

So I made a simple function that tells me how many seconds it takes to die to someone.

Like last time, we can derive the function with a little math, but I prefer showing this graphically. Again the dps, hp and armor values are irrelevant to the shape of the graph, I checked. So I just chose some random values.

Y axis is Seconds To Death, X axis is Armor Penetration.

See. See! It's linear. The effectiveness of Armor Penetration is not dependent on the target's armor. The first 10 will decrease the time it takes to kill your target by exactly the same amount of seconds as the last 10. Obviously it doesn't do anything past 0 armor.
Just to double check, I set DPS, hp and armor to constants and reduced the formula to 5-0.2*arp - a linear function.

Armor Penetration has neither increasing or decreasing returns.

So I was now very embarrassed about being in the first group for so long. I planted myself firmly in the second group with about as much conviction as when I was in the first. For a minute or two.

I looked at DPS. I thought, what if i attack once per second for 100 damage, and I add 10 AD, it increases DPS linearly. But back when I calculated ARP, its effect on DPS was exponential.
Which means all the other stats have Diminishing Returns, as they increase DPS linearly.

Seconds to Death on the Y axis, DPS on the X axis.

So, in comparison to its competition, ARP has increasing returns, meaning it's better against low armored targets. Putting me right back into the first category. I hope you will all join me.