Optimizing Effective Health Math with Spirit of the Ancient Golem Tanks

GundayMonday·Jul 16, 2014·5 votes·888 views

UPDATE: Edited based on forum feedback. Now using the correct new health costs, and took a closer look at the optimal health/resistance ratios. Thanks for pointing that out Pushover!

Wrote a post walking through the math behind the new Spirit of the Ancient Golem and how to optimize a jungle tank build: http://graceunderfiero.blogspot.com/2014/07/league-of-legends-spirit-of-ancient.html

TLDR; Summary:

Spirit of the Ancient Golem decreases the cost of health from ~2.67g to ~1.9g, which changes when it's most effective to buy health versus resistances from an effective health per gold spent standpoint. With Golem % health bonus and % hp runes, you want your ratio of health to resistance to be ~14 to optimize effective health as opposed to ~9.5 without the bonus.

The item adds a lot of utility for additional health regeneration from tank items/masteries making champions like Dr. Mundo very scary during the lategame.

The item offers a considerable boost to abilities that scale with health like Sejuani's Flail of the Northern Winds.

Riot did a great job balancing this item since Ancient Golem tanks can now focus more on buying health instead of resistances, which allows for better survivability against mixed damage team compositions.

18 Comments

PushoverJul 17, 20143 votes

I would like to point out that % health regeneration doesn't favor health stacking over resistances. 1% of your health every second is 1% of your EHP every second, regardless of if you are a 4000 HP 100 armor tank (40 health is 80 EHP), or a 2000 HP, 300 armor tank (where 20 health is 80 EHP). Flat health regen favors resistances too, as each point of health is worth more if you have higher resistances.

You have a mistake in your math when determining the price of health. If 1 point of health costs 2.67 gold, after running your setup you get 2.67/1.4 = 1.9 gold per HP, not 0.5 gold/HP as you mention. And that skews your numbers severely, and changes your findings significantly.

EHP is a maximization problem of spending gold, something like maximize: (base HP + (gold spent on HP/gold per HP)) * (1 + (base resistance + (total gold spent - gold spent on hp)/100))

If you know calculus, you can find the maximum point (derive and find where the zero is), or you can simply graph it where gold spent on HP is X.

Still, aside from the big change in your numbers, a good post.

KSHarrisonJul 17, 20141 votes

Yeah I was wondering about that 40% increase in value making an 80% decrease in the cost.

The optimal ratio of health to resist is 1 resist per 1% of your HP. So for a champion with 2000 HP at level 18, they want 20 HP per 1 resist. If you factor in 40% armor pen, it comes out to 33 health per resist. Since most champions achieve around 70 - 80 base resist anyways, ideally you go for around 500 bonus HP before touching resists.

In practice, health is not as available as resists and you end up just going mixed resist items like Randuin's and BV for maximum efficiency. I'm not sure the SotAG makes a difference in this rote build order; a key item to begin with would be Warmogs. The cost analysis would need to be redone with the corrections you suggested.

PushoverJul 17, 20141 votes

The optimal ratio of health to resist is 1 resist per 1% of your HP. So for a champion with 2000 HP at level 18, they want 20 HP per 1 resist. If you factor in 40% armor pen, it comes out to 33 health per resist. Since most champions achieve around 70 - 80 base resist anyways, ideally you go for around 500 bonus HP before touching resists.

This isn't really true, supposing armor for some reason now cost 500 per 1 armor, and health was 1 gold per HP, then health would easily be more effective than armor, and you would get a significant amount of health before wanting armor. You wouldn't spend 500 gold on 1 armor and 20 gold on health if this were the case. It probably works as a rough guideline, though.

What you do see is a maximum point for HP to armor, and then spending gold equally on resistances and armor maximizes EHP against physical damage. Penetrations move the maximum point (by making resistances more expensive for % penetration, or by moving it mostly in the direction of more resistances with flat penetration)

KSHarrisonJul 17, 20141 votes

Yes, I was speaking on purely the mathematical relationship between the two initially, and then I went on to state that in practice it doesn't function this way, because of the difference in availability which can be interpreted similar to relative costs differences.

And I wonder if flat resistances move it in the direction of stacking more resists. It would depend on the scenario, but assuming you don't sell anything back, you would have to buy armor planning for an eventual late game scenario where you face 40% resist pen. Then it may be that health is still more desirable, all the more so because the flat pen of a Brutalizer is so often built into a Black Cleaver for increased % Pen.

This might be relevant because flat pen is primarily early game, but it might be that health is more desirable early game, where it has a larger relative impact on your base health? 180 health from a Ruby Crystal when you only have 500-900 HP might be worth more than, say, 15 armor.

PushoverJul 17, 20141 votes

Flat penetration should be handled by building whatever resistance gets penetrated. If you are a tank with 2000 HP and 100 armor, and get attacked by someone with 20 armor penetration, you have EHP of 3600. If you have 1900 HP and 110 armor, despite having EHP of 3990 (versus the 4000) before penetration, you have EHP of 3610.

Lategame against 40% penetration, you need to recalculate the maximum, accounting for the fact that each point of armor is only worth 0.6 times as much. You can't neglect armor, but you can account for the penetration, and you will end up finding a maximum that involves building less armor.

This goes back to maximizing the problem I wrote above, if you want to account for someone having penetrations, you can do that. There's a point where it's more efficient to build armor than health, and then there's also a point where it is more efficient to build health than armor. Adding flat penetrations makes buying armor more effective for longer, but just like normal, there's a point where buying health is better than buying resistances. Health Potions earlygame also favor buying resistances, although Ignite makes health better.

KSHarrisonJul 17, 20141 votes

There's a point where it's more efficient to build armor than health

My question was where that point would be. I was just musing aloud I guess.

GundayMondayJul 17, 20141 votes

I just took a look at this based on your conversation with Pushover. If you take all of your health, and all of your armor, then divide one in the other like this:

Health / Armor

You generally want that ratio to be ~14 for the most effective health. So if you're under 14, buy more health. Over, buy more armor.

Without the Golem passive health gain, that same ratio is ~9.5. Kind of hard to do that math on the fly, so I would recommend looking at common build paths for your favorite tanks, and doing a little homework before the game to figure out when the best times to buy different items are.

I cheated and just made an excel model that does this for me based on how much gold I have :-)

KSHarrisonJul 17, 20141 votes

That's strange, my math always came out to 1% effective health per resist, with no optimal fixed ratio for health to armor.

GundayMondayJul 18, 20141 votes

It's definitely not fixed, you're right. Might be ~13.5 early, ~14.5 late, but generally, ~14 seems to be a local equilibrium with golem and %hp runes. Trying to keep it general since it's rough doing math during a game.

PushoverJul 18, 20141 votes

Given a champion that is optimally tanked (say 2000 HP, 167 armor if we take 400 g = 150 HP and 300 g = 15 armor), you want to spend money equally on armor and health, leading to a ratio of 150 HP per 20 armor, or 7.5. The only reason you get different numbers when you divide health/armor is that armor starts at effectively 100% of your health, at 0 armor you get 100% of your health as EHP. With 1 armor your ratio looks absurd (with 0 it breaks). To get a consistent number you want health/(100 + armor). For a 1 armor tank, the optimal HP is 755, leading to a ratio of 755.

As for finding the optimal values of health and armor you want to maximize (x/gold per HP) * (1 + ((C-x)/gold per resist)/100) x = the gold spent on HP C = total 'gold' spent, gold is in quotation marks because this does not account for base stats. Also due to how negative armor works, this will not work for low values of health and armor.

To maximize this you can use algebra (you get a downward opening parabola, find the vertex) or calculus (derive to find the vertex). In either case, Wolfram Alpha is your friend.

Alternatively you can use an optimally tanked set of numbers, such as the 2000 HP 167 armor above, and just scale it up or down. But keep in mind this only deals with 1 source of damage and doesn't take shields or heals into account, so at some point simply doing math isn't going to solve your tank problems.

GundayMondayJul 17, 20141 votes

Hey thanks for reading. I think you're right about the cost of health. Let me take another look and repost.

GundayMondayJul 17, 20141 votes

Changed the new health cost and took a look at effective health per gold spent with and without the golem bonus. Appreciate you pointing that out!

I left the health regen section alone, because all I was really trying to say is that 1% of a bigger number is a bigger number. I still think that's valid :-)

Any other feedback?

PryotraJul 17, 20142 votes

You look like you enjoy math. If you don't mind, I'd like to reference your post if someone were to ask me in the future about health and armor values.

Also, I'd like to invite you to math with me over in my math thread. I'll link it incase you are interested.

Link

GundayMondayJul 17, 20141 votes

Great idea. Maybe Riot could build a Theory Crafting section of the boards where this could live?

PryotraJul 17, 20141 votes

As much As I want that to be a reality, I figure the best we can hope for is a sub board to an up and coming "Improvement Corner" type board.

Mr KrypJul 16, 20141 votes

Very very nice.

logiqueJul 22, 20141 votes

Hi, I got in touch with you in twitter.

So basically even % seals are outright better and scale more than hp/lvl.

(hp/lvl) Would be - 216 *1.25 = 270

(% seals) 1.35 *200 = 270

I was wrong.

All right. I don't want to invest in %quints yet - please how would i reinterpret your table to determine new balance between hp / armour?

omg I can't believe how expensive they are.

GundayMondayJul 22, 20141 votes

Yeah they're pretty pricey. I just finished grinding ip to buy all of them last week.

The chart from the article is converting the stats you would get for buying health + armor items into gold, then showing your total effective health per gold spent on each stat. Just trying to show how you should spend your gold to be as tanky as possible with and without the golem passive. The new golem item + % hp runes shifts the sweet spot of buying health/armor to favor more health since you get more effective health for your gold. More bang for your buck.

This still changes as you get more levels + health, and is champ specific since champs can have stat modifiers like Rammus's shield, so take that advice with a grain of salt.

Generally the %hp runes are better on pure tanks that are going to be building a lot of hp. This is because you can get the late game benefits earlier in the game. A champ like Sejuani loves these.

Generally HP/level runes are better on champs that will go off tank or bruiser builds with a few damage items thrown in. A champ like Wukong would use these.

If it's interesting, I could compare hp/level runes to the %hp runes and try to get some general guidelines for when to pick what.