Tank Builds, Math, and You!

The forums have made it pretty clear that tanks feel bad to play right now. With the usual forum echo chamber comes increasingly dubious claims: "ADC kills me in 3 seconds lategame", "ADC with lifesteal outsoaks tanks", "I got oneshot as a tank", and so on... Taking even a few of these seriously, I've decided some of you need help with tanking/soaking builds. Tl;dr at the end if you don't want to see the math.

1) EHP = HP x (100+defense)/100 x other mitigation This is your survivability, each point of defense (armor or MR) provides a linear increase to your EHP, not a diminishing return. Other mitigation includes any %reduction like Annie's E, Garen's W, Ninja Tabi, etc. (Note the formula listed is EHP against only one type of damage)

1.1) ΔEHP/gold: This is the stat we want to optimize, not total EHP (only correct at 6 items), not total EHP/gold (health pot wins), not gold efficiency (gold value doesn't account for ingame effect). To create a rule of thumb, compare ΔEHP/gold of pure armor and pure HP purchases and see where they break even (this will be the optimal point because, anywhere else, you would continue to purchase the stat with the higher ΔEHP/gold)

First, figure out how much each point of armor and HP increase EHP: ΔEHP for armor = HP x Δarmor/100, ΔEHP for HP = ΔHP x (100 +armor)/100. Now divide each by Δgold: ΔEHP/gold for armor = HP/100 x Δarmor/Δgold, and ΔEHP/gold for HP = ΔHP/Δgold x (100+armor)/100. Setting the two equal and plugging in ΔX/Δgold from cloth armor and giant's belt, we get our optimization: HP = 7.5 x (100+armor) (Obviously we can't control our stats to perfectly hit this point, and this also doesn't account for the unique combos of stats on finished items, but it's a decent guide to knowing which stat you want next)

1.2) Mixed damage and %pen: I won't show the full math on these, but both of these effects shift the optimal stat split toward needing more health. In the case of 50% physical/50% magic, you'd need twice as much health: HP = 15 x (100+defense), with equal amounts of each defense (armor = MR = defense). Other damage splits need less health, down to the minimum at 100% physical/magic (covered above). Percent pen increases the amount of health needed too, by devaluing the defense you've purchased.

1.3) Mitigation factors: Indicated in the formula earlier, these are multiplicative with other mitigation factors, and hence won't ever lead to 100% damage reduction. As such, usually these cancel out (X% damage reduction makes you take X% less damage, regardless of what you build). The only time they might not is when facing mixed damage, if the mitigation only works against one type (Ninja Tabi's %autoattack reduction).

**2) Remaining Armor = Base Armor (1 - %pen) + Bonus Armor (1 - %pen) (1 - %bonus pen) - flat reduction/pen. ** In this formula, %pen and flat pen are synergistic, which is one reason the ghostblade + cleaver combo is so strong. You should be expecting these penetration stacks when planning your builds. Note that Last Whisper's %bonus pen only operates on bonus armor, while Void Staff and Cleaver's %pen operate on total defense.

Alright, enough of the lectures and onto some context-based recommendations:

Optimization in lane: Assuming 30 defense just naturally, against a pure-AD or pure-AP opponent, the first optimization curve says you aim for 975 health before purchasing any additional armor. Against a mixed-damage laner, this health goal can go up to 1950 HP. Yes, potions and kit sustain can count against this, but these will do far less in all-ins.

Optimization lategame: You should be expecting more mixed damage and more %pen at this point in the game, which means the optimization curve is shifting toward health. Against a perfect 50/50 split of damage, with 150 of each defense, you should have 3750 HP, plus extra HP to adjust for %pen. In this way, health is a tank's best friend both earlygame and lategame.

The tank's natural enemy: ADCs specialize in outscaling tank defensive stats with crit, so extra consideration should be given to items that specifically counter their damage, like: (no, I didn't forget to list Thornmail... don't buy it). Don't underestimate these items lategame, as they all effectively scale with your total armor! Consider Ninja Tabi: the 12% reduction of autoattack damage is a multiplier to the usual reduction from armor. If you have 100 armor, you would cut the damage in half, then Ninja's reduce the remaining damage by 12%, all told taking 0.5*0.88=0.44 of the original damage (44%). In order to get that from armor alone, you would have needed 127 armor, making Ninja Tabi's passive worth ~27 armor. If you redo this scenario for 200 armor, Ninja Tabi's passive is now worth ~45 armor. Attack speed debuff and crit reduction from the other items shown function similarly.

PS: This mitigation factor to only physical damage means you should shift your defense split further toward MR than you might otherwise expect.

Build to succeed Just hear me out - this isn't as braindead as it sounds! While I've just told you to build Ninja's, that's with the goal in mind of soaking the most damage. If you need to wade through piles of CC to get your spells off, Mercs might be the better choice. Same for Thornmail - while it sucks at soaking, its purpose is additionally to return damage. Build how you need to build to succeed, BUT... don't complain that you can't soak when you're not building to soak. For example, Randuin's is the best armor+HP item for soaking, and building or are building suboptimally for that purpose (not to mention )

TLDR: Build health early, build health late [HP = (7.5 to 15) x (100+defense)].
Last Whisper can't pen the armor you don't build.
Build to tank ADCs, build more MR than normal to compensate. Don't build , the rest of your team can do better damage for cheaper.
Yea, this pen stack ADC build is stronger early, but it doesn't scale as hard as crit did late.
Feel better about building health.