## An Optimal Buildpath

Fair warning: if you don’t play League of Legends this post may not be interesting to you. Or it still might. Who knows. Oh, and for people who actually want to use relevant versions of these statistics, this post was made in the middle of Patch 6.2.

Armor is a statistic that reduces incoming physical damage taken. It’s similar to magic resist, which is another defensive stat that reduces incoming magic damage taken. The Wikia article I linked does a good job of explaining how armor is calculated, but i’ll restate it here: all incoming physical damage to your champion is multiplied by a factor of $\frac{100}{100+A}$ where A stands for armor. For example, if you have 100 armor, you’ll receive $\frac{100}{100+100}$ or 0.5 times all incoming physical damage.

It sounds like you can theoretically stack armor forever and reduce your incoming physical damage to a multiplier so low that you’d take close to zero damage at all times. This graph of the multiplier function shows you why that’s not possible:

The horizontal axis is armor, while the vertical axis is the multiplier. Even at 400 armor, which is an absurdly high amount, you would still receive 20% of incoming physical damage which is relatively high compared to what you invested.

Another reason why stacking solely on armor is a bad idea is because of the notion of effective health. Effective health is the amount of damage required to kill you, taking resistances into account. If you build resistances, your EH will always be higher than your HP. If you have a lot of armor but not that much base HP, it won’t matter how much armor you have since it doesn’t take a lot to kill you. In addition, physical damage isn’t the only source of damage: magic damage and true damage (damage that is dealt directly to your HP without exception) is also prevalent. Here’s a simple example: if you have 3000 HP and 100 armor, you have an EH of 6000, which is equivalent to having 1500 HP and 300 armor. The downside to having 300 armor is when the opposing team has a good source of magic damage as well, making your armor useless.

Since every point of armor requires you to take 1% more of your maximum health in physical damage to be killed, armor doesn’t have diminishing returns per se, but it’s a much better idea to find an optimal balance between HP and armor to have the optimal EH from a frontline tank’s perspective.

It’s apparent that investing in straight health is useful because it raises EH regardless of magic or physical damage. In contrast, armor will only raise EH if physical damage is concerned, and magic resist will only raise EH is magic damage is concerned. Health is a universal defense for physical, magic, and true damage. Of course, there is also a point where simply buying HP gives diminishing returns with respect to EH. Simply put, the more well-rounded the enemy team composition is in terms of physical/magic damage, the more highly you should invest in health. If the enemy composition is almost entirely AD (physical) or AP (magical), then they made a giant mistake as long as you can abuse it: stacking the appropriate resistances here are much more helpful than raw health due to the percentage nature of how armor and magic resist works.

There’s another advantage to armor: it makes healing EH easier. Most healing abilities on champions heal a flat amount of HP. If you only buy health, it makes heals weaker in terms of effective health (it’s just a 1:1 healing to effective health increase ratio), but investing in resistances forces enemies to go through your HP bar more slowly, making the healing:EH ratio greater than one.

So the question that remains is: how do you know what to invest in at a given stage of the game? Should you prioritize flat health or resistances? To answer that, we need to optimize our EH function given a set amount of money c. How do we do that?

We first need some unit of account to gauge how much armor, magic resist, or HP is really worth. Fortunately, there are three “basic” items that grant solely armor, MR, or health. These items are Cloth Armor (300g, +15 armor), Null-Magic Mantle (450g, +25 magic resist), and Ruby Crystal (400g, +150 HP) respectively. Dividing through, we get unit costs of 20g/+1 armor, 18g/+1 MR, and 2.67g/+1 HP. Note that armor actually costs a bit higher than MR.

The tricky thing is that given constant resistance r, investing in +1 hp raises effective health by $\frac{100+r}{100}$. For example, if I have 200 armor, each point of health I invest in raises EH by 3. Taking it the other way around, every point of armor I invest in given constant health h raises EH by $\frac{h}{100}$. For example, investing in +1 magic resist with 2000 health gives 20 additional EH. (This is all assuming single-type damage, and not split.)

Since 2.67g gives +1 HP and 20g gives +1 armor, 1g of HP will give you $\frac{100+r}{2.67(100)}=\frac{100+r}{267}$ EH, and 1g of armor will give you $\frac{h}{20(100)}=\frac{h}{2000}$ EH. We now know the value (in terms of effective health) of both armor and health: the key here is to recognize that setting these values equal to each other means that our buildpath is optimized. Doing so and solving for h, we get roughly $h=750+7.5r$. This equation is what we’ve been looking for.

The horizontal axis is armor, the vertical axis is HP. If you find yourself above the line (aka to the left), invest in armor to shift right and get yourself back on the equilibrium line. If you’re below the line, invest in health to shift up.

Magic resist is pretty much the same thing. Repeating the calculations done above gives a graph of $h=670+6.7r$.

I won’t get into specifics, but the equilibrium line will shift upwards if the incoming damage is hybrid instead of just one type. This is because mixed damage favors building flat health. You should still build resistances, but not as much as if the enemy composition is mostly one type of damage.

The exact defensive items that you should buy I won’t get into, as this post is already getting a bit long. But if you can multiply your resistances by seven and add 700 to that total in your head, you should be in good shape to make some better item choices on the Rift.