Talk:Sivir/@comment-9048637-20140103214844

So, I've been playing around with some maths (yes, I'm one of those deviants), and I've come to the conclusion that Sivir actually gets less from building attack speed than perhaps any other Marksman in the game. Let me try and explain why.

Just a cursory glance of Sivir's kit reveals that she's far from an AS-reliant Marksman. She has phenomenally good AD scalings just from Boomerang Blade alone, and she gets a whopping AS steroid from On The Hunt. So far, so uncontroversial. In this respect, Sivir is hardly unique. Plenty of other 'casty' Marksmen get more out of AD and Armor-pen than AS; Graves is the quintissential example. But Sivir actually takes this a step further: her bonus AS is actually less than the notional amount. This is a distinction that I'm not sure any other champion actually holds, let alone any other Marksman.

At this point, you're probably wondering what the hell I'm on about. Okay, notional amounts, then, as opposed to real amounts. This is, simply, the difference between the amount of a stat you're advertized as getting versus the amount you actually get. Usually, this is something that's only relevant when discussing abilities, so it's an ability that I'll use as an example. Take Tristana's Q, Rapid Fire. As you're probably all aware, Rapid Fire gives Tristana bonus AS, 90% at max rank, for a period of 7 seconds. So, the notional AS Tristana gets from max rank Rapid Fire is 90%. However, when calculating Rapid Fire's effect on Tristana's overall DPS, one has to take into account the fact that Rapid Fire has down-time, the difference between its duration and its cooldown. Assuming you don't have any CDR, Rapid Fire's cooldown is 20 seconds. Therefore, to calculate the actual AS Tristana gets from Rapid Fire, that is the average over the period from activating Rapid Fire to its coming back off cooldown, one must 'spread' the notional AS out over that full 20s; that is, multiply it by the ratio of up-time to total cooldown, which comes to 90% x 7/20 = 31.5%.

Like I say, usually this is only relevant when it comes to abilities, because only abilities tend to have things like cooldowns which create a disparity between notional and actual stat increases; items, for the most part, just give you their stats constantly and without any modifications. When there is a disparity, indeed, it's usually positive: for instance, Rabadon's Deathcap will give you far more AP than the notional 120 thanks to its passive, which scales with your total AP (including, of course, the 120 from Rabadon's itself). Other examples tend to arise from the reaction between items and champion abilities: for example, Garen gets more actual bonus resistances than the notional amount due the passive effect of Courage.

Sivir fits into this latter category, but unlike nearly all other examples of such interaction, the effect in her case is negative: she actually gets less bonus AS than the notional amount, not more. The reason lies in how her innate AS steroid's own actual bonus is calculated. Unlike most steroids, Sivir's bonus from On The Hunt does not have a fixed duration: it lasts not a set period of time, but for however long it takes for her to perform three auto-attacks (subject to a cap after which it expires anyway, but for the purposes of calculating potential DPS, we assume that Sivir is constantly auto-attacking, as well as activating her steroid whenever it comes off cooldown). What further complicates this is the fact that, unlike with most abilities, Ricochet, the ability that actually triggers the steroid from On The Hunt, doesn't go on cooldown upon activation; it goes on cooldown once its three procs have been consumed.

What this means is that to calculate the up-time of Ricochet and the period between Sivir activating Ricochet and it coming back off cooldown, and thus calculate the actual AS bonus she gets from her steroid, one must take into account Sivir's total AS during Ricochet - including the steroid, but also including everything else. And this is where the devil in the detail emerges. Giving Sivir more attack speed will decrease Ricochet's up-time; while this will also make it come back off cooldown faster, when it comes to determining the ratio of up-time to total duration, since this reduction will reduce the numerator and the denominator of that ratio by an equal amount, it will, overall, reduce the ratio itself (as some of you may remember from your maths lessons). This means that the actual AS Sivir gets from her steroid is also reduced... and this reduction must be taken away from any AS Sivir buys! So, for instance, if you're buying a set of Berserker's Greeves on Sivir, you're not actually getting 15%; you're getting 15% minus however much effective AS you're losing from her steroid! Yikes!

I apologize if I didn't explain that particularly well; unfortunately, it's pretty darn complicated, and I'm not sure how to make it any simpler. But hopefully you managed to follow it. So, ladies and gentlemen, what does this mean in practical terms? Does it mean that one should avoid building AS on Sivir like the plauge? Well, no. Bonus AS will still improve her DPS; the amount you're losing from her steroid will never be enough to cancel out the permanent AS you're getting from the item/mastery/whatever in question. On top of that, the usual sources of bonus AS on a Marksman are Zeal and its upgrades, and those are still invaluable for the crit-chance and mobility they give (both of which Sivir obviously really likes). But it does mean that if you're thinking of building AS on her, then you may want to consider if there are alternatives. In particular, I no longer actually run the Fury mastery and Berserker's Greeves on her; I run Sorcery and Ionian Boots of Lucidity instead. Again because of the way her steroid works, these actually improve her DPS a lot more than they do for most carries, moreso (I'll spare you the numbers this time) than their AS cousins.

Plus, come on, it's kind of cool, right? To think that there's a champion out there with such weird mechanics! The only other I can think of who has a similer interaction with a stat is Kassadin, due to the way his passive works. Besides him, though, I think Sivir may be the only example. Anyway, I hope you found this as interesting as I did, and I hope I didn't fry anyone's brains with all the maths. Finally, if I did make a mistake anywhere (I'm pretty sure I haven't, but I'm only human), whoever finds it, please, let me know.