Resistances vs Health

 

Resistances vs Health, an in-depth cost analysis

Gentleman Gustaf, back again and this time I’m here to talk about health and resistances. Guides have been written ad nauseam about the optimal amount of health and resistances to buy, but all of them are missing crucial pieces of information: resistance penetration and reduction. Items like Last Whisper and Void Staff (taken on almost every carry) and masteries like Weapon Knowledge and Arcane Expertise (taken on almost every damage dealer) guarantee that there will be % Penetration in almost every game. On top of that, Magic Penetration Marks, Armor Penetration Marks and Quints, the Sunder mastery, and items like Abyssal Scepter can add magic penetration and reduction. I will show how Penetration should affect your choices about health and resistances

 

Back to the Basics

First, how do you know when to get resistances vs health?

Well, the equation for Effective Health, a measure of how much damage of a given type it takes to kill you, is as follows:

EHP = Health * (100 + Armor)/100

To see how much EHP is affected by any given point of health or armor, we take the derivative of EHP with respect to health and with respect to armor:

δEHP/δHealth = ((100+Armor)/100)
δEHP/δArmor = Health/100

Essentially, every point of health increases your effective health by 1 + Armor/100. As well, every point of armor you buy increases your effective health by Health/100. So the more armor you have, the more health benefits you, and the more health you have, the more armor benefits you. This is exactly what makes it seem that armor has diminishing returns, a myth debunked by DiffTheEnder here. It’s not that armor has diminishing returns, it’s that health and armor have increasing returns with each other, so getting more armor makes health more valuable, and as such, makes armor less valuable by comparison.

So when should you get Armor over Health? Set the two derivatives above equal to each other, divide each by the costs of the relevant stat, and solve for one variable:

δEHP/δHealth/2.58 =  δEHP/δArmor/15.56
((100+Armor)/100)/2.58 = Health/100/15.56

Health = ((100+Armor)*15.56/2.58
Health = ((100+Armor)*6.03
Health = 603 + 6.03*Armor
Armor = Health/6.03 – 100

When we graph this, we get the following:

As you can see above, from both the equation and the graph, you don’t want armor below about 600 health, and from there, you want about 6 points of health per point of armor.

Of course, this is highly inaccurate. Given that the average champion health at level 18 is about 1950 (and average armor is about 77), and the corresponding armor value is 223, this would mean that everybody who cares about tankiness would start their build with a Frozen heart or Thornmail plus a Randuin’s Omen. But why are these numbers inaccurate?

For starters, there are two types of damage: magical and physical. The above numbers assume that all of the damage coming in is of the same type. You can replicate the math easily yourself using the method I did above for armor, but substituting the cost of MR For the cost of armor, and you will get (basically) the same thing. However, this makes our equations get a fair deal more complicated. We now have Physical Effective Hit Points and Magical Effective Hit Points, and our Effective Hit Points are just the average of the two.


PhysicalEHP: PEHP = Health * ((100 + Armor)/100)
MagicalEHP: MEHP = Health * ((100 + MR)/100)
EHP = (PEHP + MEHP)/2

What we want to know is how much EHP is affected by any given point of health or a resistance (and for now, we assume equal points in each). To get this number, we want to know how much you get per point of health, so we take the derivative of EHP with respect to health, armor, and MR, and get the following:

δEHP/δHealth = ((100+Armor)/100+(100+MR)/100)/2
δEHP/δArmor = Health/200
δEHP/δMR = Health/200

Assuming we spend equally on Armor and MR, we want to know when 1 point of (both) resistances are equal to one point of Health:

δEHP/δArmor + δEHP/δMR = δEHP/δHealth

Health = 1203 + 12.03*(Resistances)

'Resistances' in the above equation refers not to total, but average resistances. This gives us more reasonable numbers:

You will notice two peculiarities. First, the draft dips into negative resistances. This is because (due to reduction), you could (in theory) have a choice between more health and enough resistances to cancel out the reduction below 0, and those numbers are also determined by this equation.


Second, I took a brief shortcut in merging armor and MR into 'Resistances' This is safe for two reasons. First, the cost of armor and MR are so similar that it will hardly affect our numbers. Second, each point of any resistanc increases your EHP by Health/200, as shown above, so the distribution of our resistances does not matter. We can get a 50/50 split or or 25/75 split and it will help our EHP the same amount. As such, we can leave decisions about WHAT resistances we need aside, and instead focus on how many points of resistances we want to buy (for now).

However, something that has largely been left out of discussion of health and resistances is penetration. Because basically every damage dealer will have at least 10% Penetration (from going to 9 points in the offensive tree), and almost every carry will have 40% Penetration (from Void Staff or Last Whisper), should we expect these efficient numbers to change? Intuitively, the answer is obviously less. Every point of resistances you buy will be less effective. But what is the mathematical effect of these penetrations and reductions?

Let's start by assuming 10% reduction (masteries). We can simplify to get the following equation:

Health = 1337 + 12.03*(Resistances)

Adding an additional 40% reduction to the mix (Void Staff/Last Whisper) gives us:

Health = 2228 + 12.03*(Resistances)

Summary

But Gentleman Gustaf, it looks like you just threw a bunch of numbers above, but I don't really know what they mean or why they matter.

So here's a basic summary: first, here are the equations comparing health and resistance, and how much you should get depending on how much penetration the other team has.

No Penetration:
Health = 1203 + 12.03*(Resistances)
Resistances = Health/12.03 - 100

10% Penetration:
Health = 1337 + 12.03*(Resistances)
Resistances = Health/12.03 - 111.139

46% Penetration:
Health = 2228 + 12.03*(Resistances)
Resistances = Health/12.03 - 185.2

Assuming your health is around 3500, this means you should get about 191 Armor/MR, assuming no penetration. However, this number falls to 180 assuming 10% penetration, and 106 assuming 46% penetration. Cutting that armor will give you a little bit of extra money, allowing you to put more into both health and resistances. This gives us the following, equal gold-cost options

0% Penetration: 3500 Health, 191 resistances
10% Penetration: 3566 Health, 185.3 resistances
46% Penetration: 4010 Health, 148.1 resistances

How much better is this?

if we stayed at 3500 Health and 191 resistances, our effective health would be:

0% Penetration: 10185
10% Penetration: 9516.5
46% Penetration: 7109.9

By going more health heavy (as above), we change our effective health to:

0% Penetration: 10185
10% Penetration: 9519
46% Penetration: 7225

By going to a health-heavier build, we increase our tankiness by a small amount. 116 damage may not seem like a lot, but everybody has been in fights where that would have made the difference.

All too often, people have set numbers in their head: they want 3500 health and 200 resistances. Then when somebody gets a Last Whisper or a Void Staff, they try to get back up to those numbers; they think since I'm losing armor/MR, I need more to make up for it. The appropriate solution, however, is to get less armor/MR, not more, and to replace it with health. Since you can assume that at least the AD carry will be getting Last Whisper, and the AP Carry will PROBABLY be getting Void Staff, it's best to begin erring on the side of too much health to begin with. Obviously there are mitigating factors (% Health damage, health scaling abilities, armor scaling abilities, Galio's passive), but for the most part, you won't want to go too overboard on resistances, even if they are heavily concentrated with one type of damage, because of % Penetration.

Exceptions

The exceptions and special cases may seem obvious, but there are so many it's worth pointing out.

  1. Anybody with a shield, especially a spammable one, should get more resistances. Essentially, they should multiply the number of shields they expect to get in a combat by the strength of their shield, and add that value to their health, when calculating how many points of resistances to get. Why? Shield spam is essentially temporary health. Heals on your team have a similar effect.
  2. If you have abilities or stats scaling off of armor (I'm looking at you, Malphite/Rammus), more armor should be purchased than predicted. With the recent commonality of Malphite, this is important. Same goes for MR, Galio.
  3. Anybody with a resistances buff obviously wants more health (Shyvana used to be scary with just Frozen Mallet, Wit's End and Warmog's, with an Atma's coming up).
  4. %Health damage is all magical or true, so if you're against one of those champions, get some MR to go with your massive amounts of health, and if you're against Vayne, cry because she doesn't really care what you build.

TL;DR

Normally, I try not to do TL;DRs, but the amount of math this post requires already made it very dense, and that's with some of the math glossed over to avoid making the article impossible to understand. As compensation, this post may be for the math nerds like me, but the next one will be much simpler, and with a nice snazzy conclusion you can try out pretty easily.

The takeaway message is this; since you can pretty much assume that 3 or 4 people on each team will have 10% Pen, and another 1 or 2 will get 40% pen at some point, cut a little bit of the resistances from your build unless their damage is all one type.

 

EDIT:

After a number of comments, I thought some clarification was in order. I am not advocating stacking nothing but health, and I am especially not advocating such with champions with % health damage. What I am saying is that given that every team will have an AD and AP carry, and LW and Void Staff are pretty much givens in most games, tanks should get 20-40 less armor than would typically be predicted by a health/resistances comparison.

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