Reign of Gaming

Gentleman Gustaf here to re-analyze Health and Resistances, now that Pre-Season 3 has started to settle down. A few things have changed:

1. Health's cost is down (5%)
2. Resistances' costs are up (15%)
3. %Pen stacks with Flat Pen without wasting any
4. %Armor Reduction is readily available

How has this affected the balance you should maintain between health and resistances?

Before I dig deep into the numbers, I want to point a number of things out: Resistances don't just scale your burst 100-0 tankiness: they also provide multiplicative benefits to lifesteal, shields, heals, and regen. So if you are a champion who has such abilities, you should bias yourself more towards resistances. The way I would do so would be as follows: calculate how much you expect to lifesteal/shield/heal/regen during an 8-10 second interval, and add that amount to your health.

Calculating Tankiness

With that out of the way, let's jump right in. First, let's recap the important equations.

We have the equations for calculating Effective Health vs Physical Damage (PEHP) and Effective Health vs Magical Damage (MEHP), as well as their average (EHP)

From there, we can calculate the increase each point of Health or Armor (sub for MR as needed) gives you.

Let's say that the enemy team has nothing but physical damage. Do we want nothing but Armor? Turns out, no, for a pretty intuitive reason. The value of each point of Health, as shown above, is (1+Armor/100), and the value of each point of Armor is Health/100. So given any combination of Health and Armor, if we stack armor infinitely, the value of Health will go up and up (as your amount of Armor goes up and up). Eventually, Health will just be worth more. By factoring in the relative costs of Health and Armor, we can calculate how much of each we want:

So if you're against an entirely Physical team, you want a lot of armor, but not nothing but Armor. A good sweet spot is about 3000 Health and 300 Armor. But wait! Aren't they going to at least going to have 8 %Pen from masteries? That should mean you get less armor, right? Yes. However, how much may come as a surprise. Even though I knew what numbers to expect, having done these calculations before, I feel a bit weird seeing them. The amount less is a flat amount. That is, while %Pen may go through a percentage of your armor, you shouldn't buy a percentage less. This is because there are two factors at work. The first, the %Pen of the Armor you ALREADY have, decreases the value of health. The second, the % Pen on Armor you would buy in the future, decreases the value of Armor. These two cancel out, leaving you with a flat amount. How much? 8.7 Armor, or (1/(1-.08)-1). Now, let's add LW into the mix, for another 35% Pen! The results are the same as above, you should get (1/((1-.08)*(1-.35))-1) less armor, or 67. Add on BC's % Pen, and you're up to 139! Here the graphs are together:

But Gentleman Gustaf, what about the flat pen on Black Cleaver?

Up until now, I've just been looking at % Pen. But let's account for the 10 Pen from Cleaver, and the 5 from masteries. If there were no %Pen involved, the answer would be easy: Get 15 Armor to make up for the 15 Pen and get back on the optimizing line. However, since % Pen is involved, things get a little bit more complicated. Why? Because 15 Armor isn't worth 15 Armor anymore. To actually get 15 Armor, you need 15/.4186, or 15/(1-%Pen). In this case, that's 35.83 Armor.

And this is exactly the problem with Black Cleaver, conceptually. Brutalizer says 'get 10 Armor to make up for this Pen!'. Then you upgrade it to Black Cleaver, and get %Pen, which says 'get 4 more Armor to make up for the Flat Pen being harder to make up!', but follows it up with 'also, get 43 less armor, because your armor is worthless!'

More on the past of Black Cleaver, if you're interested:

Of course, most games, the team probably won't be all physical or all magical damage, and you'll end up having to weight your choices accordingly.

So what should you do if you are faced with an even split?

The close observer may note that this graph is the same as just Armor, but with 100 points fewer spent on resistances. So if you have 3k health against only physical damage, you want about 300 Armor. But if you have 3k health against split damage, you want about 200 resistances (summed). From there, it's a matter of figuring how much of their damage is physical, and how much is magical, and finding an appropriate middle ground between a perfect Resistances split and a 100% Armor page.

TL;DR

Don't get resistances until you have about 1500 HP.

Then get 0-100 points in the resistance of your choice, based on physical/magical balance of their team; 100 being all one type, 0 being an even split.

Then get 1 point in the resistance of your choice for every 75 health you get

Flat Pen - Get extra armor equal to their flat pen divided by the complement of their %Pen.

%Pen - Get less armor equal to 100/(%Pen)-100

Obviously, items don't come in continuous quantities, so just aim to be somewhere near this rubric.

Also, different champions are different, so if you have shields or heals, get some more resistances, and if they have % Health damage, get more resistances.

 For more of my work:

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