Health to Armor ratios

vivimos·Oct 14, 2014·14 votes·6,393 views

I know there are diminishing returns on armor after 100 armor. I decided to investigate these returns to determine when I've purchased enough armor and it becomes more beneficial to begin buying straight health.

First I did some averaging across many items and determined that the average gold value of 1 health point is 2.7g and the average value of 1 armor is 19g. With these two metrics we can determine where the breaking point between armor and health is:

Health ------ Armor 1,500 --------- 100 2,000 --------- 175 2,500 --------- 250 3,000 --------- 325 3,500 --------- 375 4,000 --------- 450

So, if you have 1,500 health you should have no more than 100 armor, if you have 2,000 health you should have no more than 175 armor, and so on.

Here is a detailed diagram of the calculations: http://willow-tech.com/health_ratio.jpg

32 Comments

CupcakeTrapOct 14, 201420 votes

tl;dr: OP is right about "sweet spots", but it's worth bearing in mind that Armor does not give absolutely diminishing returns, but merely diminishing returns relative to Health.

Also, my Capitalization is completely out of Control here, because I'm not sure which words merit Capital Letters in this context. I'm too lazy to fix it.

Armor does not give diminishing returns in an absolute sense. Each point of Armor increases your effective HP by 1%, whether you're going from 5 Armor to 6 Armor or from 300 Armor to 301 Armor. This took me a while to understand, because the damage reduction formula is obviously non-linear.

Explanation: Effective Health

"Effective Health" means "incoming damage you can take before you die". This is the most practical definition of durability in this context.

Effective Health (vs. physical damage) = (Nominal Health) * (100 + Armor)/(100)

For example, let's say you have 1000 Nominal Health. (The Health meter says "1000".)

At 0 Armor, you have 1000 Effective Health versus Physical damage.
At 5 Armor, you have 1050. (+5%)
At 100 Armor, you have 2000 (+100%)
At 200 Armor, you have 3000 (+200%)

So the returns aren't diminishing in an absolute sense: every point of armor gives +1% Effective Health.

However, every point of Armor (or MR) you buy makes Health more effective. This means that as you buy more and more Armor, Health starts looking more and more attractive. This means that there are "sweet spots".

To explain further: each point of Nominal Health gives an amount of Effective Health proportional to your resistances. If you have 200 Armor, for instance, then you're basically getting three Ruby Crystals of Health for the price of one, at least with respect to Physical damage.

So really, it's less that Armor gives diminishing returns, and more that Health gives increasing returns (relative to Armor).

Why This Matters

This might sound like an academic distinction, but it's important when considering (e.g.) Armor Pen. I used to think of cutting someone down from 20 Armor to 10 Armor as being more important than dropping them from 200 to 190, because of the larger change in percentage reduction. However, the change to effective HP is the same. In other words, Sorc Shoes are just as valuable whether your target has 30 MR or 500 MR. (Well, I mean, except in the sense that I don't know why you're throwing Magic damage at someone with 500 MR; surely there must be a better target.)

Although I haven't fully error-checked this intuition, it occurs to me that this also means that cutting a high-Health Champion's resistances down is more powerful than cutting a low-Health Champion's resistances, because you're reducing a Health multiplier. In other words, let's say you're a shredding Champion and you're trying to help your team destroy the enemy Champions. Your allies will deal 5,000 points of damage. (Let's say it's all Physical.) You'll contribute more to the cause of killing the enemy team if you shred 20 points of Armor off of the Champion with 5,000 Nominal Health (result: enemy team's total Effective Health drops by 1,000) than if you were to shred 20 points of Armor off the Champion with 1,000 Nominal Health (result: enemy team's total Effective Health drops by 200). That said, there are many tactical reasons why you might not want to target the 5,000 Health Champion first.

vivimosOct 14, 20142 votes

Good point. If you reduce the incoming damage by 50% you have effectively doubled the damage you will receive.

JustMyBassCannonOct 14, 20143 votes

I was going to make this point, but you already did. I just have one minor nitpick.

This might sound like an academic distinction, but it's important when considering (e.g.) Armor Pen. I used to think of cutting someone down from 20 Armor to 10 armor as being more important than dropping them from 200 to 190, because of the larger chance in percentage reduction. However, the change to effective HP is the same. In other words, Sorc Shoes are just as valuable whether your target has 30 MR or 500 MR. (Well, I mean, except in the sense that I don't know why you're throwing Magic damage at someone with 500 MR; surely there must be a better target.)

Actually, flat MPen becomes less valuable based on how well your champion scales on AP if your opponent has high magic resistance.

Let's make up two spells for comparison; one of them deals 150+90% AP damage, while the other deals 250+50% AP damage. Let's also assume that you have Liandry's and Sorcs.

If your target has 30 MR--the amount most common--then you ignore 100% of their MR. The first spell deals 195 damage instead of 150 (~30% damage increase), and the second deals 275 instead of 211 (again, ~30% increase). In order to provide the same damage increase with AP instead of MPen, you need 65 AP on the first spell, and 165 AP on the second (on top of the 50 AP from Liandry's).

If your target has 200 MR, then you ignore 15% of their MR. The first spell deals 72 instead of 65 (~10% damage increase), while the second deals 102 instead of 92 (again, ~10% increase). In order to gain that much damage in AP instead, you need 24 AP on the first spell, and 61 AP on the second spell.

CupcakeTrapOct 14, 20144 votes

Indeed; I was simply remarking on the value of MPen relative to the target's MR. The balance between MPen and AP is a different question that brings in different factors.

JedenVojakOct 14, 201412 votes

It doesn't matter how much armor you have, it's not enough until you only take 1 dmg from every auto attack.

Crimson5pheonixOct 15, 20145 votes

Someone else already said it, but armor doesn't really have diminishing returns. While the percentage reduction in damage decreases, it's effect on your EHP remains linear so long as your HP holds static. You could make this same table with EHP as a function of health at given armor values as well as armor at given health values.

In short, defenses scale linearly by themselves, but geometrically with each other.

ZyorhistOct 14, 20143 votes

woah. the evaluations that JustMyBassCannon and UPfreely, and CupcakeTrap just threw out there really made me think of something to do when I am in match. I knew I needed to watch everyone's armor and MR but this just made everything click as far as what I am looking at. So while I am waiting on death timers I can run a quick evaluation on who should be my target and what I need to build to deal with people...thank you guys...

CupcakeTrapOct 14, 20145 votes

Veigar Happy to help! Just know that if the tables were turned, I would show you no mercy.

JustMyBassCannonOct 15, 20141 votes

Always glad to set the numbers straight and help a player join the glorious evolution

VentiraOct 14, 20142 votes

I could've sworn the diminishing returns occured at a much higher amount of armor...

Then again I suck at math.

vivimosOct 14, 20141 votes

The calculation for damage reduction is 1 - (100 / (100+armor)). This means that diminishing returns actually start at 2 armor and grow exponentially the more armor you get. But really we don't see large curves downward on diminishing returns until we get to 100 armor which is 50% damage reduction.

Champion SkinOct 15, 20141 votes

Well imagine if we didn't see a diminishing return... 200 armor 100% damage reduction... That would truly be a never ending battle as nobody would die... though maybe people would actually die because penetration and true damage (and this would so make Darius pretty good with % armor penetration and true damage)

ceaselessphantomOct 15, 20141 votes

Well imagine if we didn't see a diminishing return... 200 armor 100% damage reduction...

That is not how it works. You are not thinking in terms of formulas.

ZyorhistOct 14, 20142 votes

The thing here is that, while your calculations look correct, you aren't considering champions that scale on armor (and mr for that note) where even when you spend more on the dimished returns, you are getting more out of it because of abilities. Leona and Braum come to mind right off the bat.

vivimosOct 14, 20141 votes

Correct. those are all things to take into account. But this is just some numbers to give us a nice general idea of what is actually happening with all that armor. This is not a hard and fast rule. Just a guideline.

Duke AnaxOct 14, 20142 votes

I think you forgot the inherent armor of the champion here, as you don't need to buy that. If I see that right, you should need even less armor before getting more health.

vivimosOct 14, 20141 votes

Correct. I think tanks often buy too much armor and not enough health. These numbers are not specific to the items you buy. This is supposed to consider the total armor your champ has.

7ha7guy777Oct 14, 20142 votes

so let's say i decide to go full tank Zac and end up with 5567 hp (for calculations it might help to round up or down) and about 200 of both mr and armor. what would the effective health be? If i did the math right, i believe it's around 16500 hp against non-true damage but i'm not sure

CupcakeTrapOct 14, 20141 votes

so let's say i decide to go full tank Zac and end up with 5567 hp (for calculations it might help to round up or down) and about 200 mr and armor. what would the effective health be? If i did the math right, i believe it's around 16500 hp against non-true damage but i'm not sure

Sounds about right: it should take about 15,000 points of non-True damage to kill that Zac.

Note that if (say) you're a mage with a Void Staff, it only takes about 12,000.

Further note: each point of flat MPen costs about 20g, about the same as a point of AP.

ZyorhistOct 14, 20141 votes

actually, this is why I love items like item 3068 , gives you both

still though there is nothing like

Leona + item 3068 item 3110 item 3075 plus her w and watching to figure out if you can actually see the damage turrets deal to you...

17inchcorkscrewOct 15, 20141 votes

Randuin's is better for Leona because she wants to jump in and CC people, and she generally doesn't get much gold. Sunfire is generally for waveclear (which Leona doesn't do) and for sitting on the enemy's face (which Leona does when nobody's around).

UPfreelyOct 14, 20141 votes

There's easy methods of determining your effective health... health x (1 + .01 x armor) = effective health ie 100 armor = +100% effective health. without diminishing returns, armor would scale exponentially.

so say you had 2000 health and 100 armor 2000 x (1 + .01 x 100) = 4000 effective health so a simple way to calculate gains is like this, lets say we wanna buy a chain vest .4(bonus armor) x 2000(current health) = 800 bonus EH for buying that vest

say we could afford a giants belt 2(current armor) x 380(bonus health) = 760 bonus EH for buying the belt

Things to consider. Sustain makes armor/mr substantially stronger armor/mr is effective against one type of damage subtract your enemies flat penetration from your armor/mr before calculations. voidstaff/LW + masteries make your defensive stats about 60% as effective. so reevaluate your EH if your enemy purchases these.

honestly if you want a graph to help you I'd suggest making 2 line graphs. one just showing effective health and another at 60% defensive stat potency. for flat pen you can just shift that much up.

Also, black cleaver is kind of a nightmare to theorize about so i won't even touch base on that.

DaejinTyeOct 14, 20141 votes

This is why I health+Armor stack when I need so much armor.

Sunfire+Randuin's with a Thorn if I need mass armor. (As a bruiser/Tank, of course.) It kind of takes care of both avenues at the same time and helps minimize the DR of stacking one or the other.

I do the same with MR, Banshee's+Spirit Visage for example. On Mundo and Zac, even often on Volibear that's pretty much my full build: Sunfire+Spirit Visage+Banshee+Randuin's+Merc Treads+Warmogs/Thornmail.

Crimson5pheonixOct 15, 20141 votes

In fact, here's a graph of EHP with respect to HP and resistance. Health on the graph scales from 500 to 5000 while resistance scales 0 to 500.

http://www.wolframalpha.com/input/?i=3d+plot&a=*C.3d+plot-_*Calculator.dflt-&f2=x%2F(100%2F(y%2B100))&f=3DPlot.3dplotfunction%5Cu005fx%2F(100%2F(y%2B100))&f3=x&f=3DPlot.3dplotvariable1%5Cu005fx&f4=500&f=3DPlot.3dplotlowerrange1_500&f5=5000&f=3DPlot.3dplotupperrange1_5000&f6=y&f=3DPlot.3dplotvariable2%5Cu005fy&f7=0&f=3DPlot.3dplotlowerrange2_0&f8=500&f=3DPlot.3dplotupperrange2_500

hgrvaOct 15, 20141 votes

can you make the calculations counting last whisper? it would be more practical.

TenSlashTenOct 15, 20141 votes

and then they buy LS/VW and your resistances don't matter because 40% is worth more gold than all your defensive stats

CoolSpot2Oct 15, 20141 votes

Keep in mind that there are many ways to regenerate hp both overtime and instantly making armor even more efficient as sustain .

Sweet spot doesn't mean much because your items should be in reactions to big AD items like infinity edge or last whisper .

Also many items like BORK make hp even less efficient than armor . When good ADC carry gets Last Whisper as 4th item its a sign that you should 300 armor no matter how much hp you got . ( As far as pure tanks concerned )

SammanderOct 15, 20141 votes

there are no diminishing returns on armor. control your bronze logic

ParinoidPandaOct 15, 20141 votes

Some feedback on your chart design, I'd like to introduce you to this nifty little tool that 95% of chart designers can't do without:

Margins and the proper use of white space.

You have the obvious margin around your chart, but reading the individual columns is painful and confusing. I recommend putting a blank column in between each column section for readability.

Minsk and BooOct 15, 20141 votes

Fitting a function to this data gives: Desired armor = 0.1386 * Health - 102

What this means is that for a typical tank taking into account level HP and armor: Ideal armor against AD = (Level - 8) * 10 + Bonus HP / 7

There are limited armor item options to get to the theoretical 'ideal' endgame armor. Sunfire and Randiuns are both too heavy on HP to get to the HP vs armor optimum in the late game even though they're great against typical damage profiles.

If you're going for them, SotAG -> Warden's Mail -> Giant's belt is the correct order assuming a typical gold and xp income to lvl 8-9. Then go for more armor like Frozen Heart or Thornmail (crazy, I know) before upgrading the belt, or at least go for a HP/armor blend item like Randuin's or Sunfire. All of this assumes there is minimal AP damage on the enemy team...