Magical Spreadsheet Singalong of Penetration

Blue Katarina Is Best Katarina

The Obsession with True Damage

I can't be the only one who's noticed everyone's love of true damage. I hear about it everywhere, from posts recommending armor penetration runes on AD carries to "deal true damage in lane!" to casters seeing a triple mpen build and saying "Katarina will be doing true damage to the carries now!" or streamers swooning over Olaf's new potential to deal near true damage by using the reworked Black Cleaver. But the discussion point that's never mentioned is that by building all these penetration items they're choosing to not build other damage items. We know triple pen Kat is a monster, but have you wondered if she'd be as effective going with an AP-centric build?

Considering magic damage specifically, the main idea is that skill damage comes from both the base damage and a ratio of your ability power. Opponents' magic resist attenuates both parts of the damage (hurr durr that's obvious). Building more ability power increases the amount of damage getting through MR from the scaling component only, but penetration items help both the scaling component and the base damage get through magic resist better. For champions with high base damages and/or low ability power scaling, better results can frequently be gotten through building magic penetration to get more of those base damages through MR. 

Well this sort of things interests me, even if you're already dozing off. Cutoff points intrigue me. There has to be a quantitative way to know when building a Haunting Guise gives you optimal damage or a way to choose between going for ability power or magic penetration runes. Dodging the questions with "build pen if they have high base damage" doesn't cut it. For now I'm only going to look at the magic damage side of things, simply because there are fewer complicating variables - there are no items that give magic resist reduction, but Black Cleaver is real (and seemingly awesome). Once I struggle through the math I'd like to tackle AD and revisit this topic.

Starting Equations

One of the complicating factors in this analysis is just how many damned variables go into determining the amount of damage you deal. You have the ability's base damage, ability power ratio, your current damage stats (ability power, percentage magic pen, and flat magic pen all factor in), and even your opponents magic resist values play a role - that's six variables (seven if the spell does %hp damage)! Combined into the equation, the damage dealt by a single ability is:

(Base + AP*Ratio) * 100

100 + (1-%Mpen)*MR - Mpen

The sheer number of variables poses a bit of a problem. We can't graph it (not even with one of the fabled/hated rainbow plots!). My first approach was to find the partial derivative with respect to each variable to get the change in damage output per change in each stat. Unfortunately that's (a) not easy math to explain (b) extremely ugly equations and (c) not even useful since they were all cross-dependent equations. tldr; yuck.

So instead of going that approach, what about a method to simply compare builds? That sounds cleaner. A quick way to say "does this move do more or less damage with build A or build B?". Since the whole point of this writeup is to compare how mpen builds stack up against stacking ability power, I settled on a concept of "effective AP". A Katarina that builds Haunting Guise does the same amount of damage on this target as she would if she build x amount of AP - therefore, Haunting Guise is equivalent to x "effective AP". 

Honestly, since I can't figure out how to write the full equation on Blogger without it looking like utter poopoo, you'll just have to trust me that it's complicated yet correct. The diligent can fact check me based on the spreadsheet formula.

The effective AP value of each penetration item will change from champ to champ, ability to ability, and target to target, but if you crunch the numbers for a variety of realistic scenarios you'll be able to eyeball it and see that in most cases either mpen or AP is better. Think of this as an interactive walkthrough - I'll be providing copy/pastable spreadsheet equations that you can plug in your own values and get your own results. 

If you're very impatient or simply not interested, I suggest you skip the following two sections and go to "Haunting Guise Results". But really, what's the fun in that? I'm trying to teach a man to fish or something.

It's Like a Sing-Along! 

Everyone should have easy access to a spreadsheet program, either Microsoft Excel, OpenOffice Calc, or even Google Spreadsheet. Any of the above should work great for stepping through this process for your favorite AP. Create a new spreadsheet and copy and paste the cells below into it. They should take up cells A1 to K2. 

	Ability Information		Shared Stats			Magic Penetration Build			Target Resistance	Required AP to Match
Name	Base	Ratio	Flat Mpen	Percent Mpen	Ability Power	New Flat Mpen	New Percent Mpen	New Ability Power	--	--

As an example, mine looks like this with a little bit of column resizing so the text doesn't cut off: Don't get hung up on what the columns mean yet or what values go where - we'll cover each soon enough.

Reasonable Assumptions for MPen Values

Because of how the damage equation works, we cannot assess how much extra damage we get out of increasing penetration or ability power by a given amount without knowing how much penetration and ability power we already have. So let's start by figuring what you'll get from runes and masteries at the start of the game. 

If you're looking to optimize spell damage and are planning on building significant amount of mpen or ability power, 9-x-x or 21-x-x are probably your two options for mastery configurations. The 9-x-x build gives you 8% magic penetration and 1 AP/lvl, while going a full 21-x-x gives 8% mpen, 1 AP/lvl, and 6 flat AP. There are a few other masteries here and there that would marginally affect your damage output but I'm going to ignore them. Doing a special case for whether or not you have Executioner or Archmage would be significantly more trouble than it's worth.

For runes, the marks are the slots designed around offense. However, most of the primary stats for marks help AD champs far more than AP. When you deal magic damage, the only mark that provides decent stats per slot is magic penetration (which most people already run, so this is old news). With the Season 3 adjustments to magic penetration values, you now get a total of 7.83 mpen instead of the old 8.55.

For the quintessences, both ability power and magic penetration are reasonably good value per slot. However, it works out that the 14.85AP gives more spell damage than the 6.03 mpen for pretty much any spell. I cannot really recommend running mpen quints if you're looking for damage. 

Overall, I'm going to assume 20 starting ability power, 8% magic penetration, and 22.83 flat magic penetration. This corresponds to a 21-x-x setup with magic penetration reds and AP quintessences but does not use offensive seals or glyphs. If you prefer a different rune page simply adjust my values while following along. It also assumes that Sorcerer's Shoes are purchased for the additional 15 magic penetration since these are bought nearly all the time in midlane and seen on AP toplaners as well.

Detailed Haunting Guise Walkthrough

Let's start putting some values into our spreadsheet. As I mentioned, I'm going to run through this for Katarina's abilities and you're free to mimic it for your champion. What I'm interested in here is how much "effective AP" is Haunting Guise worth on Kat? Does it really get her more damage than 1485g worth of real AP would? While not realistic, I'm going to just assume level 18 and use the values for max rank skills.

• Ability Information - Name. Column A is not involved in the calculations at all, it's merely an identifier to distinguish between rows. Since the effective AP will vary from skill to skill, I'm adding one row for each move.
• Ability Information - Base. Column B will have the base damages for the abilities you're assessing. I simply fill these in with the information from the League of Legends wiki for B3 to B6.
• Ability Information - Ratio. Column C will have the AP ratios for each ability. Again, I pull these numbers from the wiki and simply fill them in from C3 to D6.
• Common Stats - Flat Mpen. Column D is the amount of flat magic penetration both builds have. Since both a Haunting Guise build and an AP build will have 7.83 mpen from runes and 15 mpen from Sorc Shoes, I put 22.83 in each of the cells D3 through D6.
• Common Stats - Percent Mpen. Column E is similar, it has the amount of percentage magic penetration both builds would have. Unless both builds have a Void Staff, Column E will almost always be 8% for the mpen mastery.
• Common Stats - Ability Power. Predictably, Column F is how much AP both builds will have. From runes and masteries I'm approximating 40 AP and put that value in F3 to F6.
• Magic Penetration Build - New Flat Mpen. Column G holds the new flat mpen values you'll have with the mpen oriented build. Since I already have 22.83 mpen from Column D and am adding 15 mpen with Haunting Guise, I put 37.83 in each cell from G3 to G6.
• Magic Penetration Build - New Percent Mpen. Since I'm not building a Void Staff in this sample build, my percent mpen is unchanged and I put 8% in each cell from H3 to H6. If you need to account for an added Void Staff, the new value would be 40% because of the multiplicative stacking.
• Magic Penetration Build - New Ability Power. How much extra AP do you get from your penetration items? With my example Haunting Guise, Column I will be 25 more than Column F, or 65 from I3 to I6.
• Target Resistance. Who are you hitting? Since my Katarina will target squishy champions with 60 magic resist, I put 60 in each cell from J3 to J6. To simulate hitting a tank you might increase this to 150.

That wasn't so bad was it? With values included, my spreadsheet now looks like this: 

The last column is a little gross... It contains all the heavy lifting calculations to figure out how much "effective AP" your magic penetration build is giving you. Copy and paste the equation below into K3 (make sure you include the equals sign!) Once it's pasted, you need to grab the lower right hand corner of K3 and pull the formula into the other rows you have (if any). 

=(((B3+C3*I3)/(100+(1-H3)*J3-G3))*(100+(1-E3)*J3-D3)-B3)/C3-F3

Haunting Guise Results!

You now have values! As you may see faintly in my image below, Haunting Guise on Katarina will do the same amount of damage as 77 to 125 flat AP, depending on the ability. W scales the hardest on magic penetration and her ultimate (surprisingly?) benefits more from flat AP over magic penetration than her other moves. 

Keep in mind that this is for this specific scenario; if I target the tank with 150 MR instead, the Guise is only worth about 55-80 AP (meaning the mpen is giving me about 30 to 55 effective AP since Guise does have 25 AP on it as a stat). Does these numbers make Haunting Guise a good purchase choice? I would say yes. There's no way to buy 77 to 125 AP with 1485g so your damage against carries should be higher with Haunting Guise than any other mid game item. Even against tanks you get a Needlessly Large Rod's worth of damage out of your W. Considering that Haunting Guise also gives you 200 HP to help survivability it sounds like a worthy choice over a glass cannon AP build. 

One caution while using this spreadsheet setup! Make sure the target resistance is set high enough that you're not reducing MR past 0. I'm generally not setting target resistance below 60 for builds with two penetration items or below 70 for builds with three. Don't go lower!

Interesting Penetration Builds

The same analysis can be done to play with any mpen build. For instance, I was curious how building Void Staff instead of Abyssal Scepter affected her damage output. Void Staff is less expensive and offers percentage penetration which should increase damage against tanky foes. In situations where the magic resist on Abyssal Scepter isn't that helpful, how would a Void Staff function? 

Interestingly, Void Staff offers better slightly better damage (about a 10% improvement) against squishy targets (the top group of 8 rows) than Abyssal Scepter. With the changes in penetration application order in Season 3, it's actually impossible for the three flat penetration items to give less penetration than two and a Void Staff even against low MR targets. And as seen by the second group of 8 rows, Void Staff is much better against high MR targets. W does a whopping 106% more damage against a 150 MR enemy by switching from an Abyssal Scepter to a Void Staff. 

Isolating Katarina's Sinister Steel, the plot of the effective AP of Sorcs/HG/Void compared to Sorcs/HG/Abyssal shows just how dramatic the difference becomes. While close in value for low MR targets (the point where they'd meet would be true damage at 63MR), Void is equivalent to huge AP values at high magic resists and significantly outdamages Abyssal. Cool!

There may be cases where the magic resist on Abyssal or the aura is worthwhile enough to you to still go Abyssal instead of Void staff, but Void gives an absurd amount of extra damage against high MR champions. The goal here is not for me to tell you exactly what builds are good on which characters, but rather how to use a plug-and-chug spreadsheet to educate your own item choices.

As a second teaser tidbit, a few of my friends have been enjoying Elise's ability to build a little bit of magic penetration top lane and then go tanky while still having great damage. The equations change a bit when you start playing with %hp damage and I won't be going into there here, but she gets a insane amount of damage out of Void Staff with the Season 3 penetration changes. If you thought the Katarina graph was shocking, Elise is on a whole different level when you switch from flat penetration to a Void Staff... Perhaps if there's interest %hp moves could be explored in the future. 

Enjoy exploring your item options in Season 3! There's a whole slew of new things to play with. An AD-oriented version of this may be upcoming soonish that incorporates Black Cleaver's armor reduction (applied completely separately from penetration) or maybe on something entirely different. There are a lot of ideas bouncing around in my head.