Breakdown: New Ashe and Critical Strikes

Hello folks, a friend of mine and me (both mathematicians) have wondered whether the current implementation of Ashe's passive reduces how she benefits from critical strikes compared to "normal" ADCs damagewise. We have come to the conclusion: yes it does. Maybe you are puzzled about this, since "guaranteed critical strikes" seems like a super nice buff, even with the modified critical strike damage. But here is why we have figured that Ashe doesn't benefit as much from critical strikes as other ADCs. We are presenting our results, because we thought maybe some of you might be interested in such stats but have no time to ponder about them. ---- At first, let me start with clarifying the passive again: Ashe's basic attacks apply the "Frost" debuff to enemies. Let's disregard the slow utility, because our thoughts only concern the damage output. Basic attacks, aka auto attacks aka AAs, against frosted enemies a guaranteed to critically strike for 110 + (Crit Chance × (1 + Bonus Crit Damage))% AD. Otherwise Ashe cannot perform critical strikes at all. Let us get rid of the Bonus Crit Damage part and assume no crit damage modifications through items or runes. This simplifies the formula for the percentual bonus damage of critical strikes to "110 + Crit Chance". These are our assumptions for the first step: * 100 AD * 8% crit chance * 100 AAs We chose the stat values so that the origin of certain numbers in our formulas is more obvious, and the large number of AAs so that the calculations will be closer to the actual outcome for the normal ADCs where crits are actually a matter of randomness (law of large numbers). Here are the calculated and expected AD outputs for the setup: * **Ashe:** The first attack cannot crit and the other 99 attacks will definitely crit. A critical strike will deal (110 + 8)% of her 100 AD, which equals 118 AD. Thus she will deal 1 × 100 + 99 × 118 = 11782 AD. * **ADC:** Out of 100 AAs, 8 are expected to be critical strikes dealing 100% of the 100 AD as bonus damage. That means the expected output is 100 × 100 + 8 × 100 = 10800 AD. So far so good - Ashe deals almost 1000 more AD. Actually she deals exactly 1000-10-8 more AD, which is 100 times her natural **10**% applied to the 100 AD, minus the one shot that does not crit and thus neither deals this 10%, nor the 8% crit chance of her 100 AD. ---- If you take a closer look, you'll find that the common ADC equation applies for 100 AAs applied at any time, while Ashe's equation only renders true for 100 consecutive AAs, where "consecutive" means within the 2 seconds of Frost applied from the previous attack. If you "only" get 25 streaks of 4 consecutive AAs, the equation is 25 × 100 + 75 × 118 = 11350 AD, which is obviously a lot less than 11782 AD. Conclusion: to get any benefit from the passive regarding crits, Ashe is reliant on consecutive AAs. With that said, the only question is: where is the break even? How many auto attacks have to be follow-ups of previous AAs to have Ashe deal more damage than a different ADC with equal AD and crit chance would? The equation to answer this for the given assumpions is quite simple: (100-n) × 100 + n × 118 = 10800, where n is the number of AAs that have to be consecutive ones, so that the damage equals the expected output of 10800 of 100 common ADCs' AAs. We can now solve this equation (nice and slowly so that everyone can see it): * (100-n) × 100 + n × 118 = 10800 * 100 × 100 - n × 100 + n × 118 = 10800 * 10000 - 100n + 118n = 10800 * 18n = 800 * n = 44,44444... So after all, 45 of your 100 AAs have to be consecutive ones to have a greater damage output than common ADCs. That is not too bad. If you always get streaks of 2 auto attacks, you'll have 50 crits and benefit more from critical strikes than other ADCs would. ---- **BUT:** Now comes the point where the crit chance comes into play: so far, we assumed 8%. Let's re-calculate the last equation with assumed 10% crit chance. Obviously, the base formula will be (100-n) × 100 + n × 120 = 11000. The steps to solve the equation are the same: * (100-n) × 100 + n × 120 = 11000 * ... * 20n = 1000 * n = 50 As you can see, with increasing crit chance, the necessity to have consecutive auto attacks increases as well. Now let's go ham and look at what happens with a 100% crit chance build, as it is required for the maximum crit damage output: * (100-n) × 100 + n × 210 = 20000 * ... * 110n = 10000 * n = 90,90909... That is: 91 of your 100 AAs have to be follow-ups to deal more crit damage than any other ADC would. 90 means you already deal less damage. So 10 streaks of 10 auto attacks each will still **not** deal as much damage as normal ADCs do with any compilation of 100 AAs, be it consecutive ones or not. ---- So does this mean you should rather build AD instead of crit chance, because Ashe will benefit from AD more than other ADCs? No, actually the AD doesn't matter. Let us use our formula again with 10% crit chance, but instead of 100 AD, let's assume 200 AD. The damage dealt by critical strikes is now 240 for Ashe: (110+10)% of 200 AD. The common ADCs expected AD output is 100 × 200 + 10 × 200 = 22000 AD. So after all, the equation is * (100-n) × 200 + n × 240 = 22000 * (100-n) × 100 + n × 120 = 11000 which is the first line when we assumed 100 AD and 10% crit chance. As you see, the number of required consecutive AAs is independant from the AD you have. So what we have learned from this excursion so far is that the way Ashe's crits work are an early to mid game buff. The longer the game lasts and the more critical strike chance items she gathers, the harder it gets to benefit from that stat the way other ADCs do. ---- For the final step of this essay, let us analyze the last stat about critical strikes: the bonus crit damage we ruled out in the beginning to simplify the equations. Does Ashe benefit from items like Infinity Edge more than other ADCs? The answer is no. Let us assume 150 AD, 20% crit chance and a crit damage bonus of 50%, which equals a level 9 Ashe without runes or masteries and Infinity Edge as only item. An ADC with usual crit behaviour would have 100 × 150 + 20 × 150 × 1.5 = 19500 AD. Ashe's critical strikes deal (110 + (20 × 1.5))% × 150 = 210 AD. * (100-n) × 150 + n × 210 = 19500 * 100 × 150 - n × 150 + n × 210 = 19500 * 15000 - 150n + 210n = 19500 * 60n = 4500 * n = 75 Without the bonus crit damage, the equation would be * (100-n) × 150 + n × 195 = 18000 * 100 × 150 - n × 150 + n × 195 = 18000 * 15000 - 150n + 195n = 18000 * 45n = 3000 * n = 66,66666... As you can see, Ashe doesn't benefit **more** from an item like Infinity Edge than other ADCs; to do the same damage like other ADCs, she needs to get streaks of 4 auto attacks with IE, and just streaks of 3 without it. So is Infinity Edge a bad item on Ashe? The answer is again no. Let us look at the damage output: with Infinity Edge, 100 auto attacks with 75% crit rate deal 19500 damage. Without the bonus, it would only be 25 × 150 + 75 × 195 = 18375 AD, so you would miss out on 1125 damage for the same attainment of 75% consecutive auto attacks. Although other ADCs gain a larger damage increase from the bonus crit damage (for example 1500 damage vs. Ashe's 1125 damage), it would still be a waste of potential to not buy this item. ----- I hope this posting has granted you a deeper insight into how Ashe behaves when you buy items that alter AD, crit chance or bonus crit damage stats. If you have additional equations to highlight other relationships, share them in the comments below. Thanks for reading and have fun playing the new Ashe. And always remember: the trade-off in critical strikes is a balance to the slow and the empowered auto attacks from her stacked Q which gets stronger per spell level and stacks faster with attack speed.
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