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.

37

Comments

  • #27 SoresuMakashi

    Hah, I've done the same equations on GD before, except I also factored in starting health and resistances, and calculated the gold that should be spent on each stat based on existing tankiness. Honestly though, the equations are so impractical to use that ultimately a simple case-by-case calculation is the best, I find.

    SHP x (δAR + δMR) / 2= δEHP, assuming 50% damage from each source
    δHP x (1+(SAR+SMR)/200) = δEHP

    , where SHP, SAR and SMR denote your current stats. Efficiency is then calculated by (δEHP/Gold cost). The question then becomes simple: Which choice gives more?

    Last edited by SoresuMakashi: 8/2/2012 1:19:37 AM
  • #33 MerryLane

    Quote from SoresuMakashi »

    Hah, I've done the same equations on GD before, except I also factored in starting health and resistances, and calculated the gold that should be spent on each stat based on existing tankiness. Honestly though, the equations are so impractical to use that ultimately a simple case-by-case calculation is the best, I find.

    SHP x (δAR + δMR / 2)= δEHP, assuming 50% damage from each source
    δHP x (1+(SAR+SMR)/200) = δEHP

    , where SHP, SAR and SMR denote your current stats. Efficiency is then calculated by (δEHP/Gold cost). The question then becomes simple: Which choice gives more?


    Please replace SHP x (δAR + δMR / 2) => SHP x (δAR + δMR) / 2

    And the choice is simple :
    -get shitloads of HP if you have to favorise both Damages AND "Tankability" (like, 1 or 2 items being purely defensives, items such as atma, wits end and hexdrinker not being considered like defensives)
    -get a good mix between health, armor and magical resists if you want to be a "real" tank, but prefer health>armor>mag res, tho still trying to be in phase with the clues this article gave, unless you have an urge to counter some specific situation (such as really huge magical damage or %hp damages)

    I would like to add a subject to discuss, "What to build on AP carries".
    According to the graphs on top, best for them is to get 2 health items (2 Roa, or 1 Roa + rylay) and then get a zhonya and an abyssal, to be better affected by ennemies damages. Of course, discussing about the use of zhonya (, wota) and abyssal isn't my point, I only want to talk about the "real" tankability of AP carries in front of both damages in a real teamfight in which they started with full hp.

    Last edited by MerryLane: 8/1/2012 1:38:36 PM
  • #25 Railzar

    I just want to point out one thing that people always forget to mention on this debate.; Lifesteal.  I'm not referring to the Lifesteal you have which is essentially seen as mini-heals.  I'm referring to the Health your opponent gets off you.

    First, at your proposed values you say you increase your durability by 116 "Health", which in effect is 1.6% more (assuming all your damage came at "max" %penetration).  I won't say that's not a lot because any increase is good.  But let's just remember, that's currently the plus in your tradeoff.

    Continuing from your post, starting at 191 Armor.  The carry effectively is fighting you at 103 Armor, which roughly cuts his Lifesteal down to 49.2%.  At the low end of 148 Armor, the carry's Lifesteal is only brought down to 55.6%.  That's a difference of 6.4%, or you can say that the carry has 13% more benefit from his Lifesteal when you choose to go lower Health.  So a carry going in at 1k DPS (which is fairly low for a full build carry) will Lifesteal 16.5 more Health a second when choosing a high Health build.  Two seconds of autoattacks and the carry recovers 1.7% more of their total Health back when compared to a resistance build (one second if Zeke's Herald is in effect).

    And this is only considering the carry; other champs are leeching off you as well, stealing about 16-17% more as opposed to the carry stealing the 13%.

     

    Despite me saying all this, Health builds are better to use.  First, Health has a bigger impact while leveling due to smaller Health pools.  Second, there's no awkward stage when going Health where you're extremely vulnerable to either type of damage.  Third, there's no real build that nets a respectable amount of Health without Warmog's.

     

    TL;DR - Due to Lifesteal, building Health to be more durable makes the carry even more durable.

  • #29 MerryLane

    The thing is, there are really really few times in which you will go for a "real tank build".

    This simply means that out of the 5 item slots, you will only have one or two of them for "defensive" stats. Long story short, you don't have enough the possibility of maximising your resistances, your armor, and your health, while you are focusing for damages mostly.

    So you have to get these items to boost a maximum your EPH, and I'm pretty sorry, but the most efficient ones when you have a limited amount of item slots, is big health items like warmog and a frozen mallet, rather than getting a FH and a FoN (just exagerating a bit so you get my point)
    We can then discuss on which items should be the most cost efficients, those that would give you the best tankiness, those that would give the ennemy AD carry less hp lifestolen, ... But you get the point.

  • #12 koach23

    hey, what about poppy, her passive is pretty good, but i dont know what to buy, more resist or hp, to make it even better, there are a lot of theories about her passive but i really dont know which one is true

  • #17 GentlemanGustaf

    It would seem to me that since it's based on the damage received (applied after MR/Armor) that it doesn't matter which you get. DR should stack equally well with health or resistances, and the condition of her DR is also based off of EHP, so I think you can ignore her passive for determining what items to get (although doesn't poppy normally want hybrid damaging items anyway?)

    EDIT: Also keep in mind that Poppy can choose who damages her, so your choices are even more flexible.

    Last edited by GentlemanGustaf: 7/31/2012 5:47:07 PM
  • #19 koach23

    ok, i got this, thanks
    about items, i like getting FH for cdr which is really strong stat on her, and AS debuff , when I am facing double AP or even 3 AP i always buy FoN, movement speed and mr are awesome, GA=OP as fuck
    even 2 of these 3 items give a lot of resists but no HP so i was wondering, am i doing right to not invest in HP

     

  • #21 GentlemanGustaf

    Assuming your health is about 2000 (average for a champion), and your armor is about 95 (average for a tank with masteries/runes), and your MR is about 60, if you get FH (100 Armor), GA (68 armor + 38 MR) and FoN (76 MR), you will have about 2000 health, 265 armor, 175 MR. This gives you about 6770 Effective HP vs physical damage (assuming your opponent has 9 points in offense for % armor pen), and 5150 Effective HP vs Magical damage (same assumption)

    If you cut GA for warmog's, your numbers would be:
    Phys - 9151 (greater than 6770 by 35%)
    Mag - 7369 (greater than 5150 by 43%)

    With LW/Void Staff, Warmog's wins even more:
    Phys - 6810 (greater than 4862 by 40%)
    Mag - 5741 (greater than 3890 by 47.5%)

    Essentially, by cutting one resistance item for a health item, you'd give yourself 35-45% more tankiness.

  • #9 BooleanLobster

    Sir - 

    Well done. The only wrinkle in your analysis is that you used the cost-per-stat for basic items. That's fine if my choice is between Chain Vest and Giant's Belt, but not if my choice is between Warmog's and Randuin's, or Wriggle's and Triple Doran's. This is still quite useful when building rune/mastery pages, assuming you know what finished items you intend to build.

    One interesting computation that I'd like to see your take on is the following: Suppose I build a gp10 item and use the proceeds to buy health/mana pots. How does that compare to the typical sustain items? Consider that hp5 is wasted while you are at full health, but gp10 keeps ticking. How much of each stage of the game is spent at full health/mana (TEST THIS)? Also remember that the gp10 won't really start sustaining you until you buy the first two health pots or so, so HoG effectively costs 895. What about gp10 runes? What about lifesteal/spellvamp?

     

  • #11 GentlemanGustaf

    My instincts say this won't work, but I'll look into it, and if there's something there, you'll see an article in 3 weeks on it (my next two topics are mostly set).

    Maybe in the future I'll post an item by item analysis, as opposed to going just by stats. As mentioned, I could use better sleep habits, and I just moved; so once I'm settled, maybe I'll revisit this topic.


    EDIT: Also, I really like your name.

    Last edited by GentlemanGustaf: 7/31/2012 5:41:09 PM
  • #24 GentlemanGustaf

    After some thought, I'm pretty sure this wouldn't work. Getting gold items means you are relying on basic strength to not lose lane, and waiting for late game for the returns to pay off. But gold items will lose in trades vs super items like glacial shroud or hexdrinker or brutalizer or Doran's or Hextech Revolver.

    A) Your potion sustain won't actually end up being sustain, but will instead just make up for the trades you're losing.
    B) Because of the potion sustain cost, you won't actually end up with much more gold late game, defeating the purpose of the gold items.
    C) Potion sustain won't help you in later team fights; spell vamp and life steal give you pseudo-tankiness from sustain.

  • #26 BooleanLobster

    You can't compare HoG-pots to Brutalizer. One is a sustain item, one is a damage item. Your A) objection makes no sense because you assume your opponent has managed to buy both strength AND sustain, whereas you have only bought gp10-sustain. If he hasn't bought sustain, you don't care about losing trades because you'll out-sustain him (as long as he doesn't just splat you).

    B) Stop thinking that the gold item is an income source. It isn't. It's a sustain item, like Regrowth Pendant. There might be potential upside in gold when you no longer need sustain, but for now just pretend you are forced to spend the money on potions.

    C) Not all champions actually want burst-heal, or tankiness at all. Champions that do might be better served by just buying health + lane sustain. Depends on cost.

    5gp10 comes out to a potion every 70s. That's 10.4 hp5 or 7.1 mp5. Factor in that a good third of the game is spent at full, and it looks more like 15 hp5 or 10 mp5. All of a sudden, that Kage's Pick looks like 500g of AP + 585g of mp5, all for 765g + 2x 35g investment. That seems pretty efficient! Anyway, no lifesteal/spellvamp items give mana regeneration.

    Last edited by BooleanLobster: 7/31/2012 10:12:29 PM
  • #28 corallein

    Comparing sustain to damage is never safe because they're situationally built. Sometimes you want sustain. Sometimes you want damage. Sometimes you want tankiness. And sometimes you just want to farm for a big item or two.

    Sustain doesn't help you much if your enemy can constantly harass you out of CS range because they trade so much better than you. Then they shove the lane to your tower and recall while you desperately try to get all the CS under your tower. And then they come back to lane with more items and kill you straight up, because you couldn't recall. Even if you don't die, every time your opponent backs and you can't because your lane is shoved, he gets further and further ahead. 

    Gold per 10 items CAN be good, but you shouldn't be looking them as a sustain item (except for Philo Stone, which is a sustain item). You also shouldn't build them unless you actually intend to upgrade them later, because it takes a good 15 minutes for them to begin to pay off and they aren't cost efficient for the early-game. Yes, the extra gold you get from them can be used to purchase pots; but that's true of ALL gold, and sometimes buying gold per 10 items instead of items that are more immediately gold-efficient can cost you far more in lost CS than you gain in gold over time.

  • #31 BooleanLobster

    Perhaps I am not being clear. Obviously you don't buy sustain items if you don't need sustain items. We agree on that much.

    But if you ARE buying a sustain item, consider HoGPots, which costs 895g and gives 200 hp and 15 hp5. Or KagePots, which costs 835g and gives 25 ap and 10 mp5. 

  • #32 GentlemanGustaf

    Quote from BooleanLobster»

    5gp10 comes out to a potion every 70s. That's 10.4 hp5 or 7.1 mp5. Factor in that a good third of the game is spent at full, and it looks more like 15 hp5 or 10 mp5. All of a sudden, that Kage's Pick looks like 500g of AP + 585g of mp5, all for 765g + 2x 35g investment. That seems pretty efficient! Anyway, no lifesteal/spellvamp items give mana regeneration.

    I'm not sure I see where your numbers are coming from. Each potion gives 150 health/100 mana over 15 seconds. During that time, that's 10 hp5 or 6.66 mp5. But since it's only about 25% of the time, that's only 2 hp5 or 1.66 mp5. Seems like a philo outclasses that AND gets you gold.

  • #34 BooleanLobster
    No. 150 health divided by 15 seconds is 10 health PER SECOND or 50 hp5.

    Since I get 150 health every 70 seconds, that's 2.1 health PER SECOND or 10.7 hp5. Since the regen from Regrowth Pendant (15 hp5) is wasted a third of the time, while the gp10 keeps ticking for consumption when you are actually low, it's fair to discount normally-purchased hp5. Hence 15 hp5 from Regrowth Pendant is only as good as 10 hp5 from gp10-pots.
    Last edited by BooleanLobster: 8/1/2012 2:05:56 PM
  • #35 corallein
    Or you could just buy the pots and an item that you'll actually build into something later. 895g is a Vamp Scepter and 12 health pots.
  • #36 GentlemanGustaf

    Oops, thanks for pointing that out; but even with your assumption, regrowth gives as many stats as both sets of potions combined. Why not just get regrowth, and get health (HoG) and AP (Kage's) elsewhere, from more efficient items?

  • #37 BooleanLobster
    "Or you could just buy the pots and an item that you'll actually build into something later."
    Like Randuin's Omen? If you aren't building that item on 75% of your bruisers/tanks, you are doing something terribly wrong.

    "Why not just get regrowth, and get health (HoG) and AP (Kage's) elsewhere, from more efficient items?"
    Because there aren't more efficient items if you have to carry the deadweight of a 435g Regrowth Pendant or 390g Meki Pendant.
  • #8 klvkboom

    Very nice article. This whole time I was stuck into this mindset where I had to get a respectable amount of armor to survive the late game damage of AD carries, but now it looks like I only have to opt for more HP as it's more cost efficient to do so. Also it's good since it also gives effective HP in both spectrum of magic and physical damage.

    I want to know how this fits more into the current meta though. Currently, you see a a good amount of anti-ADs stack a considerable amount of armor, like Malphite, who on many occasions would stack Omen/FH/Tabi depending on the team, nearing 300 armor. Other champs easily get nearly ~200. Based on your article, it would be better if those champions opted for maybe a Warmogs over Omen or FH. Why don't they? Is it because they want to shut down champions before LS, or is the unique effects of the items too good to pass?

    Last edited by klvkboom: 7/31/2012 5:08:32 PM
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