How to choose: Ability power or magic penetration
This one is for the mages. LeagueMath.com loves mages, and wants to help them make great mid-game item decisions. So, can math tell us which is best between mass ap, flat penetration, and Void Staff? Yup.
First, let's look at how damage from abilities is calculated in League of Legends. Here's the official LeagueMath.com reckoning:
The damage done by an ability is a function of six variables; the base spell damage, champion ability power, the spell's ap ratio, the target's magic resist, and two variables for percent magic penetration and flat magic penetration. The decision of what items to buy comes down to maximizing the output of this function.
Let's begin by looking closely at three common item options.
The first option is to buy tons of ability power. For 2460 gold you can buy both a Needlessly Large Rod and a Blasting Wand. This gives you a total of 120 bonus ap, and noting else. There's a vague sense that this might be good, especially if your abilities have high ap ratios, but what if your opponent has lots of magic resist?
The second option is to opt for flat magic penetration. For 2585 gold you can buy Sorcerers Shoes and Haunting Guise. This gives you 25 ap, 200 health, some movement speed, but most importantly 30 flat magic penetration. Is this better than the similarly priced Deathcap components above? The answer is, unsurprisingly, sometimes. We'll discuss when below.
The third and final option is to choose percent magic penetration. For 2295 gold you can buy Void Staff. This gives you 70 bonus ap and 35% magic penetration. The bonus ability power here is nothing to scoff at, but it's the percent magic penetration that makes this choice unique.
Ok, but when do we choose which?
Let's look at some plots (LeagueMath.com loves plots).
Here I've plotted the extra damage you'll do after buying a Needlessly Large Rod and a Blasting Wand, as a function of the ap ratio of the spell you're casting and your target's magic resist.
(I wrote a summary of how to read these contour plots in my post on armor penetration, so if you're confused you may want to review that.)
The first observation here is that the less mr your target has and the higher the ap ratio of your spell, the more damage you're going to do buying flat ap. Nothing too surprising there. The fact that the contour lines are less steep as you go to the right on the plot means that when your opponent is going flat ap there is more benefit to buying magic resist, the higher their ap ratios are. Maybe we already knew that, but it's good to see it quantified here so clearly.
Let's compare this to the option of buying flat magic penetration.
Mmmm, that orange stuff in the bottom right is juicy. That's where we want to be. For an ability with a good ap ratio and against targets with low magic resist, buying flat magic penetration is by far the most efficient thing you can do to increase damage. However, as usual, there are tradeoffs. The fact that the contours are flat in this plot means that the benefits of this build fall off extremely quickly as your opponent buys magic resist. Conversely, if you see an opponent with lots of flat magic penetration (ahem, Rumble top) buying a Negatron cloak will go a long way to mitigating their damage gains.
Another thing to think about with this plot is minions and farming. I've always wondered by buying Sorcerers Shoes seems to make such a big difference in my ability to farm mid with mages. This plot shows it very clearly. Since minions start with zero magic resist, and gain it slowly (~1/min). There is great benefit to having some flat magic penetration when casting spells against minions.
Finally, let's look at the plot for percent magic penetration.
Boom! Like percent armor penetration, percent magic penetration (Void Staff) puts your damage on literally a different curve. Though the Void Staff itself doesn't confer directly the damage benefits of the other two builds, the nearly vertical contours on this plot mean that it matters much less how much magic resist your target has.
These benefits extend to every aspect of a mage's scaling, too. Any additional ability power or spell levels that increase damage all cut through better with Void Staff. It's simply the only way to go against targets in excess of 100 mr.
Note that the benefit of Void Staff is pretty minimal against low magic resist targets (think creeps). That means it's not going to help you farm, at least not as much as the other options explored in this post. But for late-game shredding front-line as a mage, there's only one way to fly.
That's it for this look at mid-game item decisions for mages (magic penetration vs ap). I'm happy, and I hope you are too. /all Worth. I'm again impressed at how 'real' these decisions are. These calls are not at all obvious, and they depend heavily on the state of the game and the moment-to-moment decisions of your teammates and opponents. Well done Riot.
Everybody stay cool, and hang around; I'll be back shortly with more league fun and math fun here at LeagueMath.com.
Tibbers was not harmed in the making of this article. A big shout out to calculus, Mathematica, and the sunshine here in Boulder, CO.
That's enough math for one day.