[Advanced POE Math] Lightning Coil and the Armour Assumption

Hi everyone,

For a tl;dr, skip to "The spreadsheet!" section.

Introduction

I couldn't sleep, so I followed my spreadsheeting instincts and examined one of the basic assumptions about the Lightning Coil Desert Brigandine. For those who are new:



Lightning Coil (LC for short) is a famous Unique Armour in Path of Exile that is very useful for effectively reducing the amount of Physical Damage that you take, thereby boosting survivability.

However, it dawned on me that phys->light conversion shouldn't result in a strictly better situation in all scenarios - clearly, if you have 100% physical reduction before equipping Lightning Coil, but only 75% lightning resist, then you go from taking 0 damage to taking some damage, which is bad, mmkay? At this point, I realized that the standard logic of Lightning Coil was missing something. Let's take the legacy LC linked above as an example; 40% phys->light along with 75% light reduction means that 75% of 40%, or 30%, of your phys damage taken is reduced. This much is indisputable fact - out of every 100 physical damage points you would have taken, 30 of them are removed. However, most analysis of Lightning Coil stops there - you get 30% phys damage reduction from LC and that is that. It almost implies that you will take 30% less physical damage.

However, that would only be true assuming Armour were 0 - in truth, 40% of the physical hit gets partially reduced, but Armour is still there doing the work, and sometimes might even reduce it more than the lightning resistance does (as the lightning resistance still allows 10% of the physical hit to get through). So, a more accurate model was possible to determine exactly how much the benefit is - and I found out that sometimes, even with legacy LC, there are very real situations where having LC on causes you to take more damage than before equipping LC! (Overall, of course, LC still provides a huge benefit.)

Walkthrough of reasoning

As is usual with game mechanics, the math itself is fairly simple, but identifying the moving parts and how they interact can prove tricky.

Before anything else, I want to define a funky symbol I use. "Reduced Damage Taken Multiplier" is a mouthful and doesn't actually mean much, nor is it an official term - in normal speech I would call it "what you times by." The idea recurs throughout the formula, so I use )<' to represent it. It is a crude ASCII version of a model of a particle bouncing off of a shield:

• )<' = damage taken multiplier for reductions

The reason for this type of variable is because doing calculations with things like "20% reduced" is not possible - it must be changed into "times 80%" before calculations can be done. If you like reading aloud, think "multi" instead of ")<'" - it will make reading the rest of this section a lot easier.

First of all, the basic definition of a less/more relationship is simply the ratio of the new to the old, so we can start with that.

Final)<' = New)<' / Old)<'

• The Old)<' is simple. It is simply an application of the armour formula for whatever chest you were wearing before Lightning Coil. New)<' is a little more complicated because of the conversion.

New)<' = NewPhysical)<' + Converted)<'

• In order to find New)<', we split it into the physical and converted components. We will still assume legacy LC (40% lightning conversion) for the following line, which might make it clear why it is split:

New)<' = 0.6 * NewArmour)<' + 0.4 * Resistance)<'

• Basically, 60% of the damage is going to physical, which will go through the Armour formula, and 40% of the damage is going to lightning, which will go through the (thankfully simple) Resistance formula.

New)<' = 0.6 * [1 - (A/(A+10*Draw1))] + 0.4 (1 - Lightning Resist)

• This is a bit of a mess. Check the Armour page on the wiki if you want to try to understand the A and Draw stuff. But while we're at it, we can define Old)<' pretty easily from the same formula:

Old)<' = [1 - (A/(A+10*Draw0))]

• Note that different Draws are used, because armour reduction is calculated after conversion, which means a lower hit damage is registered in the armour formula for New)<' vs Old)<':

Draw1 = 0.6 * Draw0

• For clarity and completeness.

At this point, the variables Draw0, A, and Lightning Resist can be fairly easily pulled from the game and substituted into the formulas to yield our actual comparative damage multiplier (i.e. how much less damage we actually take by switching to LC). One adjustment I have not explicitly spelled out is the difference in Armour; it is assumed that you have the exact same Armour rating before and after equipping the LC. This is not true, of course, but I make the appropriate adjustments in...

The spreadsheet! (Ta da!)

Obviously, the calculations in the Walkthrough of reasoning are stupidly complicated and no sane human being would perform each step by hand. Instead, we have this spreadsheet: https://docs.google.com/spreadsheets/d/1nsFKJQsEZAFGtnM4sR5o61r96qbuOLPkoUte0sIyR6E/edit?usp=sharing

EDIT: Click here for a spreadsheet you can edit directly, assuming no one goes in and vandalizes the whole thing.

The blue italic numbers represent the ones you should change around in order to make the bold orange Output dynamically tell you the difference. The way to read the orange number is that switching from one chest armour to Lightning Coil causes you to take x% less damage - so if the orange number reads 15%, you're taking 85% as much damage wearing LC as you did before wearing LC.

The really interesting part is when the orange number goes negative. This means that you actually take more damage wearing LC than you did before, taking into account just the physical portion of a hit! This happens when the physical damage reduction provided by Armour exceeds your Lightning Resistance by a significant amount, and will only happen for "small hits." Alternatively, it also means that if you have a massively, massively high Armour, Lightning Coil does more harm than good.

Example calcs

• At 500 non-chest Armour, 1000 Phys Hit Damage, 75% light res, and 560/560 Chest Armour-ratings, Lightning Coil provides 32.5% less damage taken. This is actually higher than 30% because of the dual effects; there is the conversion effect where 30% is mitigated, and there is the increase in Armour's efficiency because of the smaller hit from conversion. In this case, Armour's efficiency is greatly increased by the smaller Hit (Draw1), so that plus the less-than-30% conversion effect exceeds 30%.

• At 1000 non-chest Armour, 1000 Phys Hit Damage, 75% light res, and 560/560 Chest Armour-ratings, we get 33.4% less damage taken. The mitigation actually increases, because the smaller-hit effect is increasing faster than the conversion effect is decreasing. If you look at the Armour wiki page graph you can probably identify this at the lower Armour ratings, where a small change in Raw Physical Damage has a large effect on the phys reduction %.

• At 26000 non-chest Armour, 1000 Phys Hit Damage, 75% light res, and 560/560 Chest Armour-ratings, we get 23.02% less damage taken. This means LC has less of a relative benefit when you have 26k Armour than 1k Armour. As Armour gets very high, the conversion effect is dominant; you can see that pre-Coil damage reduction exceeds 72% in the black box, meaning Armour is beginning to be competitive with Lightning Resist.

• At 60000 non-chest Armour, 1000 Phys Hit Damage, 75% light res, and 560/560 Chest Armour-ratings, we get 8.7% more damage taken. This is the "massively high armour" I mentioned. At this point, for 1000 raw damage, 60k Armour means that wearing LC actually causes you to take more damage than before! Of course, the actual damage taken is less than 150 per Hit, as original damage reduction is over 85%, so the point is "moot" as they say.

• At 3000 non-chest Armour, 40 Phys Hit Damage, 75% light res, and 560/560 Chest Armour-ratings, we get 36.5% more damage taken. This is the "very small hit" I mentioned. Once more, the reduction % from Armour exceeds 75%, which causes you to take more damage. However, the difference is 1 or 2 points of damage per Hit, so once again, the point is "moot."

Conclusions?
The calculated effectiveness of LC ultimately boils down to the ratio of Hit damage vs your Armour Rating, which highlights LC's core purpose: avoiding one-shots by mitigating huge Physical Hits. However, this does open up LC wearers to taking more damage from very rapid small hits. On the other hand, if Armour was already very high and/or Hits were already very low, the implication is that these situations where you take "more damage" by wearing LC are nothing to worry about. This makes LC still an extremely valuable Armour for mitigating Physical Damage.

very interesting post, thanks for sharing it with us.

When I was messing around with the ehp calculator I saw LC actually made you take more damage vs tiny hits, as in 10-100 damage hits which are harmless anyways.

From a practical stance, LC is not going to be combined with massive armor over 12,000 because the armor itself does not have great armor, the cases where someone has 60,000 armor AND LC AND still maintain offensive power is importable at best.

You need to make assumptions about the base build in which:
-Armor final will not exceed ~12,000 because scaling it much higher is not optimal unless youre a zero dps tank build
-Small hits are irrelevant an you can practically out life regen them


With these 2 things in mind LC is by far the most powerful armor in the game at this current patch, mandatory for surviving past level 92 ish. It's even stronger on evasion builds with no armor builds who use arctic armor + taste of hate. Frankly it is far worse than path of life nodes because at least passives are free. Mandating everything to use a coil so their not constantly at the edge of their seat wondering when the next one shot will come.

Great post.

Also it makes you shocked if you take a critical hit :)
This is a very real danger for reflect with builds like spectral throw or physical bow builds. As a phys bow player I always stick to evasion rare chest pieces and avoid LC :)

u never really get shocked in reality, the amount of lightning is too low for the shock to stick and you cant crit yourself on reflect, its just a return on damage dealt so you cant shock yourself without chance to shock i believe. Youll be so much safer vs phys reflect with a coil.

"

Nephalim wrote:

From a practical stance, LC is not going to be combined with massive armor over 12,000 because the armor itself does not have great armor, the cases where someone has 60,000 armor AND LC AND still maintain offensive power is importable at best.

You need to make assumptions about the base build in which:
-Armor final will not exceed ~12,000 because scaling it much higher is not optimal unless youre a zero dps tank build

I have heard legends of 100k+ armour with Iron Reflexes, ironskin-Granite, and reflexes-Jade. If I'm looking at this right, even if you had 0 armour and no nodes, those two flasks together would grant about 15,000 Armour, so the rumours of 100k+ don't seem too far fetched. With just ironskin-Granite (no Jade) and no Iron Reflexes, I get 11.2k Armour rating with Granite up. But maybe the rumours were greatly exaggerated.

"

-Small hits are irrelevant an you can practically out life regen them

With these 2 things in mind LC is by far the most powerful armor in the game at this current patch, mandatory for surviving past level 92 ish.

Glad you agree with my conclusion!

"

It's even stronger on evasion builds with no armor builds who use arctic armor + taste of hate. Frankly it is far worse than path of life nodes because at least passives are free. Mandating everything to use a coil so their not constantly at the edge of their seat wondering when the next one shot will come.

LC better for Evasion or Armour builds is arguable. Looking purely at <theoretical differential numerical reduced damage taken from physical Hits> (i.e. "how the numbers are different for one Hit after putting on LC"), Armour probably has more synergy than Evasion. Evasion benefits from LC very consistently - because you have practically 0 Armour, the differential will be 30% less Physical hit damage no matter the Hit size. However, Armour also benefits from the reduced size of the hit, which adds to physical damage reduction % from the Armour formula. Armour builds have more of a "sweet spot" where you get more than 30% less Physical hit damage [than before]. This is when you have medium Hit damage and medium Armour, or a large hit with large Armour. Right now on the sheet I have 1000 Ar/250Hit/75% res/40% conv/560/560 and it shows 35.995% less hit damage, which I believe is the maximum for 1000 Ar w/o chest. It seems to slowly decay asymptotically to 30% as Hit damage increases, and of course is less strong for insignificantly lower Hits (50 raw -> 2 more damage points taken!)

Interestingly, I thought the potential synergy would go up with higher Armour rating, but it seems that the maximum potential for less damage is capped at 36%. So best-case Armour seems to beat out best-case Evasion by 1-0.54/0.6 = 10% Additional Less Damage (is this the 2nd derivative or something?). As Armour goes up, the weakness of the conversion effect actually takes over, making you have less than 30% less damage more frequently. So it's hard to say.

This is, of course, ignoring things like the versatility of combining LCL's phys reduction with Evasion's Attack dps reduction.

EDIT: Also, if it's worth any salt, I've never gotten Shocked wearing LC :P

If I can contribute in any way I can say while having 5,3k hp life the "problem" with being shocked while taking phys hits its not a problem. LC is really great while coupled with good rolled granite flask and having 5+ Endurance charges. I'm playing dw cleave and phys dmg mitigation is great while wearing legacy LC + granite of Iron Skin + 6 EC. I can easily shrug off most of phys dmg I take.
Legacy LC is really worth it if you're not some kind of build that utilizes high rolled chest piece and IR + Grace and Determination auras. There's isn't such thing as "not enought phys mitigation" in my book. And I think that LC is even better for chars that utilizes evasion and/or Acrobatics.