Reign of Gaming

For almost the entirety of League of Legends, Rabadon's has been a considered rush item. There have certainly been times when it hasn't been the dominant item, for sure, like when Deathfire Grasp was buffed out of obscurity, or when Abyssal was discovered to be a bit on the OP side because of its aura range, or whenever health is in vogue and people rush Rod of Ages. But those items have rarely fully displaced Rabadon's Deathcap. at best, they were its strong replacements on some champions, while the 'standard' champions still picked up Rabadon's, and at worst, it was the item to which other items were compared to determine value. Today, I'm here to challenge the notion that Rabadon's Deathcap should be a rush item in two ways. On the mathematical hand, I shall dispute the notion that it should be a necessary rush item on the basis of its stats and will argue that several items that can replace it. On the theoretical hand, I shall argue that its existence as a rush item consistently is actually a design flaw.

Why Rabadon's?

The whole point of Rabadon's is to add some pseudo-multiplicativity to the AP role, without breaking it. Why is this important? AP Carries are a role that are VERY strong in the early-mid game, but without creative help from itemization, will inevitable fall off. Why? Because it is typically the nature of spells that they have much higher cooldowns than auto attacks. Because of these higher cooldowns, they typically have much higher base damage than attacks. (It may seem strange to consider an attack as having 'base' damage, but just consider the 'base' damage of an attack to be that damage which Sheen applies and you'll see that it's a stat which the game itself uses). While base damage on an auto-attack is only about 100, base damage on a non-ultimate spell can range much higher. Take even the low cooldown Qs of Karthus and Ryze; they have base damages of 240 and 208 (minimum mana you can have on Ryze by level 9). This is because spells will never be able to approach the low 'cooldowns' an auto attack can have. While attacks can have a cooldown of as low as .4 seconds, spells can only have cooldowns as low as 60% of their base cooldowns.

As a result of this, spells tend to have higher ratios than auto-attacks, whether they are AD spells or AP spells, although AP spells tend to have even higher scaling than AD spells, given that AD will also scale your auto-attacks, making it more valuable. For example, while auto attacks scale 1:1 with bonus AD, Pantheon's Q scales with 1.4*bonus AD, Riven's Q with 2.1*bonus AD, Annie's Q scales with .7*AP, and Gragas' Q scales with .9 AP.

But Gentleman Gustaf, I thought you said AP spells had higher scaling! It seems to me like the AD spells have higher scaling!

1. AD is much more expensive than AP; a BF sword and a NLR cost about the same, but you get 80 AP to 45 AD. So you have to consider the difference in scaling.
2. Riven's Q is 3 separate casts, and on a much longer cooldown.

As such, it seems we want to consider a slightly different way of measuring scaling. Riven doesn't have a 2.1 AD ratio on her Q, she has a 2.1 AD/13 seconds on her Q. Compare this to Annie who has a .7 AP/4 seconds on her Q, and you'll see that the real scalings are .16AD/second and .175 AP/second. Factoring in the relative costs of AD and AP, you can see that Annie is getting about double the returns Riven is in damage from her Q as Annie is. Pantheon and Karthus both come close to having the highest scaling Qs in the game, with Pantheon getting .35AD/second, and Karthus getting .6AP/second. When you adjust for the relative cost of AP, however, you will see that Karthus is WELL ahead of Pantheon.

While comparing just champions' Qs won't give us full knowledge of scaling, it can deliver some insight into the world of scaling.

So why the higher scaling? For AD Spells, it's obvious. If you're buying a lot of AD, if your spell ratios were less than 1:1, at some point, you'd have to say 'why am I not auto-attacking instead?' This exact plight is what faces AD Carries like Caitlyn; their spell-casting becomes useless in a teamfight because they'd rather just auto-attack a few times.

So what's the point of Rabadon's?

So now that you have higher scaling and base damage, how do you compete with auto-attacks? Both auto-attacks and spells have damage (both base and scaling) and frequency (Attack Speed vs CDR) stats for scaling. But auto-attacks have a chance of critting, something spells obviously can't consistently have (Panth and GP have it, but they're fringe cases) for balance purposes. Can you imagine the sad, terrifying world we'd live in if Darius could spell-crit his ult? Or if Cho could spell-crit Feast? Or if Annie could spell-crit Tibbers? Karthus would become terrifying beyond belief when he replaced his last AP item with Rabadon's Spell-Crit Stick, which gave you 100 AP and a 25% chance of any given spell doing double damage. The higher base damage of spells is always going to make building tankiness feel viable, and having Rabadon's as an option is basically the game telling you 'no really, please buy AP! It's even worth MORE now!'

And this is where I have my theoretical beef with Rabadon's. It occupies the same conceptual ground as an item like Atma's, which says 'hey, I'm going to reward you for building tanky by giving you some free stats based off of your tankiness, so you don't end up feeling useless'. Rabadon's says 'I know your scaling starts to suck, so here's some MORE SCALING. In this way, it reminds me of Aghanim's Scepter, in that it essentially buffs your abilities. That is, Rabadon's Deathcap could just as easily read: 120 AP + UNIQUE PASSIVE: your spell's AP ratios are all increased by 30%. However, it is nothing like Atma's. When do you buy Atma's? When you already have a decent amount of health. What does it give you? Armor and AD. So while Atma's helps to round you out, Rabadon's just says on the gratuitousness. If you're looking for an item that's just a stat dump of AP, well, you're going for Rabadon's Deathcap, which gives you 156 AP. But if you're looking to buff your late-game scaling so future AP is better for you? Well, you also buy Rabadon's Deathcap. This confluence of power means most champions look at Rabadon's Deathcap with a hungry greed in their eyes.

This leads to Rabadon's Deathcap being THE item to which people look, and Riot has understood this balance concern for awhile. Back when Rabadon's Deathcap was Zhonya's Ring, not only was it your super AP item AND your extra scaling item, but it was also your 2-second stasis defensive item. Since it was everything to everybody, it became increasingly hard to balance. People who loved the stasis (Kennen) made it hard to balance the scaling front. People who didn't love the stasis could find it lackluster if it got nerfed. This made it a huge balance concern. Splitting it into two items was a good solution. But it still puts the giant stat-stick (Deathcap) in an awkward position where it tends to force other (pure damage) items out. Abyssal and RoA had their time in the limelight, sure, but not because of damage, but demands for tankiness.

So what has Riot's response been? Well, there are other items which increase your scaling. Rabadon's may increase all of your scaling by 30%, but Lich Bane just gives you an additional .75 AP ratio on every spell (essentially...there are some champions who rely too heavily on combos to at all use Lich Bane). And Deathfire Grasp? It adds some % Health damage, while also scaling all of your abilities; both their scaling AND their base values. But the catch? It's only on one target. So we have all of these scaling items competing, with Riot buffing or nerfing them to make them all competitive options. So what does that mean? Well, it's a scary time to fight against a mage, when every mage has an option better for them.

The Pseudo-Multiplicative Triad

I argue that every mage concerned with damage (and only damage) should get one of these three items (Rabadon's/Lich Bane/DFG) first. Why? Because they give huge amount of AP AND increase your scaling, making more AP more valuable. I myself am a bit theoretically disturbed that these stats come together on the same item. I'd much prefer to see stronger Unique passives on these items, but lower amounts of AP. This would put them in a position similar to Atma's; you'd get one to round out your build/define the tone of your mid-late game.

But which one? Well, it depends on a lot of factors, but I argue that Rabadon's really only ought be bought by long-ranged casters (who can't use Lich Bane's proc or DFG's active that well) and AoE casters. This runs strictly counter to the text listed in the strategy section of the wiki of League of Legends, which reads "Rabadon's Deathcap is essential for any champion with ability power ratios, tanky or carry, wishing to deal significant magic damage that will carry into late game." As such, I'll be bringing some math into play to prove my point.

Why I don't always rush Rabadon's Deathcap

Let us take, as our example, one of the quintessential Lich Bane casters: Twisted Fate.

Twisted Fate has two low-cooldown damage spells: his Q and his W. His W naturally procs Lich Bane, and turns his auto-attack damage into magic damage as well, while his Q is a long-ranged AoE spell. I am assuming AP from Quints, as well as masteries.

As we can see, his spells do significantly more damage with Rabadon's. With Rabadon's, he reaches 215 AP, compared to 125 with LB and 167 with DFG. This gives TF's spell-combo 69 more damage with Rabadon's than with DFG, and 131 more damage with Rabadon's than with DFG. So it's a clear win for Rabadon's, right? Not so fast. Even if you only get one LB proc off in your combo, Lich Bane gains 144 damage, putting it 13 ahead of Rabadon's (and if you get a second Lich Bane proc after your Q, you will gain even more)! And if you use DFG's active, DFG gains 225 damage (assuming a 1500 health target, since we only have 1 item). That puts it 156 damage ahead! Not to mention, if we manage to LEAD with DFG's active, our Q/W/E damage will be amplified by 20%, gaining another 170 damage for DFG. So if the active is up, DFG is MUCH stronger than Rabadon's, and the same strength as Lich Bane (again, unless you get to lead with the active). In reality, you should at least be able to W-stun -> DFG -> Q -> E, gaining you some damage. Important note: this holds true even if you hit multiple targets, as you can see from the right-most columns; DFG will still be well ahead, and even Lich Bane doesn't fall behind Rabadon's until 3 targets hit. This is because Rabadon's, as discussed above, is a weird rush item. It amplifies the power of future AP you get, when you don't yet have much AP. This is especially clear when you compare Rab+LB to LB + DFG. Despite having the same base AP, you end up with 72 AP more with Rab than with DFG. On the other hand, when comparing JUST Rabadon's to JUST DFG, you are only 48 AP ahead.

Once you're to your second item, we see that LB + DFG is the strongest combo counting the active, although without the active, it loses to Rabadon's Lich Bane. But both of the 'strong' combos include Lich Bane. Rabadon's Lich Bane almost beats Rabadon's + DFG even leading with the active.

So we see an interesting set of data. DFG is the best first item on Twisted Fate, especially with how bursty his ganks are (and how linked to his ultimate they are, guarantee that DFG will be up in time), but Lich Bane takes a close second. However, once you have two items, you want to have a Lich Bane for sure, whether it be Lich Bane + Rabadon's or Lich Bane + DFG.

Finally, as a side-note, as the number of targets you hit goes up, Rabadon's > DFG > Lich Bane (more AP is better).

I'm not going to tell you which item to rush in which circumstances; I'll let you make that choice, but now you have the numbers for it to be an informed choice. But I know that I personally would probably go for LB + DFG first with these numbers in mind, because why bother going for the AP-multiplier when you don't even have that much AP yet? Lich Bane and DFG are both effectively AP multipliers, either through Lich Bane procs or through DFG's active, but Lich Bane comes with 50 base damage on its proc, and DFG has a chunk of % health base damage.

But here's a general rule. Lich Bane gives you 50 + .75 AP for each proc you can reasonably expect. Rabadon's gives you .3*the sum of your AP ratios. DFG gives you .2*your AP ratios + .2 times your base damage.

So compare those three values (taking into account that Lich Bane gives 40 AP less) should tell you which of the three items will benefit you most. If you have super high AP ratios (lots of AoE helps, too), like Karthus, go for Rabadon's. If you have low-cd spells, auto-attack a lot, and don't have the highest AP ratios or base-values, go for Lich Bane. If you have really high base damage, regardless of your AP ratios, and are primarily single-target, go for DFG.

tl;dr

It sure is too bad this article was too long for you to read; you may have learned something!

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