[Mathcraft, Game Theory] Magic Damage Runes Guide: Penetration or AP?

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WinterAyars

Senior Member

08-20-2009

Hello, forums!

So if you're building a page for a magic damage character you are maybe asking yourself: what should i do: AP runes or spell penetration runes? (You might also ask if you want cooldown runes, but that's something i'm not going to deal with here.) You could just do what someone else does, or try both and go with your gut.

Or you could actually find out which one is better and go with that.

I decided to do that last one, and as per my usual approach i'm not going to just keep all my knowledge secret and locked up on my hard drive. I'm going to share it. Of course, i can be confident that won't hurt me in any way since everyone will just ignore me or try to argue that math and logic don't prove anything.

But hey, maybe it'll help someone.

!!! WARNING WARNING WARNING !!!

This is a gigantic wall-o-text post. At the end of the post i've helpfully included some executive summary, as well as some basic rules of thumb to help you decide. The gigantic wall of text is in case you actually want some evidence or want to know how things work for real rather than just listening to me talk.

(End warning.)

So first of all, if we're going to compare damage we need to know how to calculate it. I'll diagram the formula i'm using, then explain it (note that i'm taking most of the formulas from this thread):

Code:
Base magic resist ->
Archaic Knowledge ->
Abyssal Scepter ->
Other flat reduction ->
Other percent reduction ->
Final magic resist number
Final spell damage will be, for our purposes:

Code:
Base spell damage -> AP bonus (AP * ratio) -> final spell damage
This is ignoring several things (like Havoc), but it'll do for now.

Take the final resist number with this formula:

Code:
100 / (100 + final magic resist) = percentage reduction

1 - precentage reduction = damage percent remaining

Damage percent remaining * final spell damage = damage dealt
We're concerned, here, with comparing the damage dealt.

Some notes:
  • "Base magic resist" is counting all your items and runes and stuff. That's all added together before any reduction is calculated.
  • Archaic Knowledge multiplies this by .85 if you have it, does not change anything if you don't. (Archaic Knowledge is going to make a pretty big difference, ultimately.)
  • Abyssal Scepter can reduce the target's magic resist below 0, though nothing else(?) can. (Anything else that could reduce resist below 0 goes here also, but that's only Scepter as far as i know.) Subtract 30 from the magic resist if the target is under the effects of Abyssal Scepter.
  • Spell penetration has no effect past (below) 0 magic resist. The rest of it is just chopped off. Subtract it from magic resist, otherwise.
  • Other reduction might be bugged, but the way it should work (and the way i'm going to assume it works) is that if the target's magic resist is below zero this doesn't change it. Otherwise, reduce the magic resist by the listed percent. I'm pretty much going to ignore this entirely, though.
  • I'm going to list AP ratios in terms of percents, rather than actual ratios (ie 1:1) just for the sake of convenience.
  • I'm not doing these calculations by hand. I wrote a spreadsheet for it instead, because i am lazy. It's an open office .ods though, so i'll zip it and upload it. (Sorry, Excel fans.)
Okay, let's talk about what we're doing here:

Spell Penetration works by reducing the target's magic resist and, thus, increasing the spell's damage. AP runes work by increasing the spell's damage directly. What we want to do is to figure out which one is better and when.

I could work up a formula that would do this for arbitrary inputs, but i don't think people would like that and it would (again) require me to be un-lazy. So instead i'm just going to do a series of examples.

Right at the moment there are five runes we're concerned with:
  1. Greater Glyph of Insight (+0.83 spell penetration each)
  2. Greater Quintessence of Insight (+1.67 spell penetration each)
  3. Greater Glyph of Potency (+0.87 AP)
  4. Greater Glyph of Force (+2.88 AP at level 18)
  5. Greater Quintessence of Potency (+4.09 AP)
In theory, we could also consider Greater Quintessence of Force (+7.38 AP at level 18), but as you have to be level 11 before those really beat Potency i'm going to ignore them. If you people really think those are worth taking over the Potency Quints i can do calculations with them, but i think in a real game the +4 straight is going to be a better way to go. (In comparison, Glyphs of Force are better than Potency at level 6. I think you'll agree that's a much more auspicious crossover point.)

I'm furthermore going to assume you're stacking a full page of either spell penetration glyphs/quints or a full page of AP glyphs/quints. The reason is because (as we will see), spell penetration has increasing returns out to magic resist 0 so if you go spell penetration runes you should go as many as you possibly can. We're just stacking AP runes because that's the way to get the biggest numbers out of straight AP, though ultimately you might pick a different setup. This isn't really intended to be "real world", but more a demonstration of the effects of the runes.

Let's start with some basics. I'm ripping some spell examples from Kira's guide, largely because i'm lazy.

Let's start with Fiddle's ult:

Quote:
Rank 1: 850 damage (170 dps)
200 mana (4.25 dpm)
150s cd (5.66 dpc)
2.5:1 per second ratio, or 0.5:1 total.

Rank 2: 1250 damage (250 dps)
300 mana (4.16 dpm)
150s cd (8.33 dpc)
2.5:1 per second ratio, or 0.5:1 total.

Rank 3: 1650 damage (330 dps)
400 mana (4.125 dpm)
150s cd (11 dpc)
2.5:1 per second ratio, or 0.5:1 total.
This is quite strong. 330 damage per second plus .4 AP per second. We'll work on damage per second for now, just multiply that by five for the total damage (if you manage to land the full duration, that is). I'm going to assume level 18 here.

So: 50 base MR, +0 AP except from runes, Archaic Knowledge is learned, Crowstorm is level 3:

Force runes: 242.3 dps (1211.5 total)
Potency runes: 237.09 dps (1185.45 total)
Insight runes: 253.81 dps (1269.05 total)

So here, spell penetration wins pretty handily. Let's make it a little bit crazier and throw an Abyssal Scepter into the mix. Same otherwise, but +70 AP and the Abyssal debuff now included:

Force runes: 331.8 dps
Potency runes: 325.21 dps
Insight runes: 357.93 dps

Spell penetration still wins... what has changed, though? In the first case, Insight (highest) dealt 7.05% more damage than Potency (lowest) and in the second case Insight (highest) dealt 10.06% damage more than Potency (lowest. Or to put it another way: getting Abyssal Scepter increased the advantage of Insight (spell penetration) runes by about 43%. Why is that?

Well, Spell Penetration has increasing returns: the more you get the better it is. Abyssal Scepter adds more, so the effect increases. Secondly, the more damage the spell already does (as compared to the effects of the AP runes) the less the AP runes will matter. On the other hand, spell penetration doesn't do anything on its own... every point of damage you gain is further enhanced by spell penetration.

Let's consider the effect of adding AP more closely. Imagine if we take the exact same scenario, but now add a Mejai's Soulstealer (0 charges) for +70 additional AP (+140 total):

Force runes: 356.69 dps
Potency runes: 350.1 dps
Insight runes: 385.92 dps

Insight wins again... becoming somewhat of a theme, so far. Don't count Force out yet, but let's learn something from this example:

This time the difference between the highest and lowest has gone up to 10.23% (meaning spell penetration pulled ahead another 1.7%). This is not a big difference, but even so: why is the increase not linear? It's because, as i said, each point of spell penetration enhances each point of AP you have. On the other hand, the AP runes can only add linearly to your damage. So the higher the damage compared to the AP runes the better off spell penetration is.

What if the Mejai goes up to full charges (+210 AP total):

Force runes: 381.58 dps
Potency runes: 374.99 dps
Insight runes: 413.92 dps

Insight wins again, which shouldn't be a surprise by now. And again, it pulls ahead of Potency runes as compared to the previous example by 1.5% extra damage (to 10.38% advantage). As compared to the initial example (7.05% difference) spell penetration is now doing 47% damage more when compared to Potency runes than it did.

What does that mean? Maybe increasing AP just improves damage in a more than linear fashion? Well, consider a comparison of Force runes to Potency runes: In the last example Force runes do 1.76% extra damage, whereas in the first example they do 2.2% extra damage! How does that happen?! AP runes are actually getting worse off the more AP you have--they're not even increasing linearly, they have diminishing returns!

Don't think it's all shiny happy spell penetration land, though. What happens if, tired of getting owned by Fiddle's ult, your enemy buys items and gains 100 additional magic resist (going up to 150)? Well, let's start with the basic numbers from the first example, but with 150 resist this time:

Force runes: 151.77 dps
Potency runes: 148.51 dps
Insight runes: 153.47 dps

Insight still wins, but Force runes are really close. The difference between Insight and Potency, here, is only 3.34%, as opposed to 7.05%! So, just as getting more damage helped spell penetration: the enemies getting more magic resist blunts (or eliminates) spell penetration's advantage.

In addition to that, Fiddle's ult is pretty big to begin with... what if we took a look at a different spell? Let's look at a hypothetical spell that deals 200 damage and has an AP ratio of 1:

(0 AP, Archaic learned, no items, 50 magic resist)
Force runes: 167.15 damage
Potency runes: 154.14 damage
Insight runes: 153.82 damage

Now Force has pulled ahead, and Insight isn't even beating Potency runes! Force runes, here, deal 8.67% damage more than Insight runes. What if we start adding more items though?

(Same, but with Abyssal as per above)
Force runes: 273.95 damage
Potency runes: 257.47 damage
Insight runes: 269.95 damage

Spell penetration runes have started to catch up--now beating Potency runes. Force runes are still ahead, though...

(Same, but with Mejai (0 charges) as per above)
Force runes: 336.17 damage
Potency runes: 319.69 damage
Insight runes: 339.93 damage

Insight has pulled ahead, again.

(Same, but with Mejai (7 charges) as per above)
Force runes: 398.39 damage
Potency runes: 381.91 damage
Insight runes: 409.92 damage

Insight is even farther ahead.

So we see the same sort of effect.

Earlier i said that if you have spell penetration it's better to get more of it. Let's take a look at that: first with an example similar to the ones we've already considered: 70 AP (a Mejai's Soulstealer with 0 charges), Archaic Knowledge, 50 magic resist:

Force runes: 216.27 damage
Potency runes: 203.26 damage
Insight runes: 207.66 damage

Here, Force runes win out. What if we swapped that Mejai for an Abyssal? We already know the damage that does (look up to the start of the 200 damage/1:1 ratio section). With Mejai's in this scenario, Force runes have a 4.1% damage advantage. With the Abyssal, however, the advantage is only 1.5%! The Force runes have exactly the same advantage over the Potency runes in either case (+6.4%), so again: spell penetration gets better the more of it you have--each point is worth more than the previous point.

If Abyssal makes a difference like that then what about Archaic Knowledge? What if you can't take it, for whatever reason? I'm again going with the 200 damage, 1:1 ratio, and 50 magic resist... just no Archaic Knowledge this time:

Force: 158.79 damage
Potency: 146.43 damage
Insight: 145.43 damage

Similar to the example with Archaic Knowledge, Force wins... this time, though: it wins by 9.19% instead of 8.67%! In other words, without Archaic Knowledge AP runes are better and spell penetration runes worse. What if we go the full Abyssal+full Mejai's route?

Force: 373.49 damage
Potency: 358.04 damage
Insight: 381.32 damage

Insight still wins, but not by as much as it did previously. (2.1% advantage as compared to a 2.9% advantage with Archaic Knowledge.) In other words: Spell Penetration and Archaic Knowledge work together.

HOLY EFF THAT WAS GIGANTIC!!!

It really was, wasn't it?

I think we did what we set out to do, however. What lessons can we learn from this?

AP is flexible (works reasonably well even when you don't use 9-12 runes of it), whereas spell penetration really works best when stacked exclusively.

Although Spell Penetration gets wrecked by increasing magic resist, so do AP runes. If your opponent goes crazy buying magic resist there's really not a whole lot you can do about it. In that case, it's up to your teammates to pick up the slack and so optimizing for that worst-case scenario is probably not the best way to go if it means sacrificing your damage in all other cases.

Conversely, AP is better when you are attacking with spells that don't start with much damage (especially if they have 1:1 AP ratios--think Jax) and before you start stacking AP items. In other words: early game. (If you have high damage, low ratio spells--think Fiddlestick--then consider Spell Penetration.)

If you're getting Archaic Knowledge you'll want to consider spell penetration, if you're skipping Archaic Knowledge you'll want to look more to AP runes, and if you're getting spell penetration you should get Archaic Knowledge.

If you already have magic resist reduction (either abilities or Abyssal Scepter) then think about getting spell penetration over AP.

Finally, if you have abilities that don't do damage but that use AP (think: Soraka) then you'll want to look at AP. Spell penetration does not help those spell, as it only reduces magic resistance of targets.

Whew.

Okay.

I have to stop somewhere, so this is it... for now. I'll touch up my spreadsheet and send it up onto the internet (both .ods and an exported .xls, which is probably going to turn into ****) soon here. Please let me know if this is helpful, what mistakes i've made, and what (if anything) is still confusing.

(Edit!)

Okay, there was a request for an executive summary of the executive summary. Here we go:

Riot can do math(!!) so it's not like one clearly beats the other, however: casters (particularly ones with Archaic Knowledge, Abyssal Scepter, magic resist reducing skills, huge damage spells, and full pages solely for spell damage) should stack as much spell penetration as possible. Other people (particularly those with low damage/high AP spells like Jax or spells that don't do damage but use AP like Soraka) should go AP runes.


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Dischord

Junior Member

08-20-2009

Solid post and information, thanks for working all of that out Winter


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WinterAyars

Senior Member

08-20-2009

Okay, here's the spreadsheet.

Some notes:

The first page contains the actual calculator, the second contains the numbers i used in this analysis.

On the first page, the italic fields (Archaic Knowledge, Abyssal Scepter, Spell Penetration, Misc. Reduction, Base Damage, Item AP, Rune AP, AP Ratio, and Magic Resist) are the ones you should fill in with your desired numbers. All the rest are automatically calculated and should not be touched.


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raidil142

Junior Member

08-20-2009

Very helpful. I suppose there's a similar scenario between armor penetration runes and attack damage runes.

Is there any way to know if the enemy is using magic pen. or AP runes? If one knew this, a magic pen. build could be easily countered.


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Albrecht

Junior Member

08-20-2009

Are Teemo's passive poisons always going to be a flat value, or are they affected my magic pen as well?


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WinterAyars

Senior Member

08-20-2009

Quote:
Originally Posted by Albrecht View Post
Are Teemo's passive poisons always going to be a flat value, or are they affected my magic pen as well?
From what i understand the poison damage over time will be affected by spell penetration. However, i'm not sure on that--debuffs might not get spell penetration.

Asking a developer is probably the way to go.


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The Senate

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Senior Member

08-20-2009

I've been using Eve with spell penetration lately, and have needed very little AP to make effective kills in early game. Does that sort of fall in line with your findings?


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BL4zD

Senior Member

08-20-2009

That was an awesome write-up! I already suspected much of what you wrote to be true, but now it's confirmed. Thank you very much.


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PiousFlea

The Council

08-20-2009

TL;DR version:

If you are a tank or healer get AP, if you are a damage caster get Spell Pen

It's a pretty intuitive conclusion, but it's always good to see that the obvious is backed up by hard math. Thanks OP for doing the math.


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Yuralz

Senior Member

08-20-2009

Great, you've now forced me to change my full AP stack page into penetration with AP in marks and seals. Thanks a lot, that's 12 runes I have to fish for with the dumb generator. /me shakes fist angrily!


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