Master Yi just got a rework, and that means it's time for all of the bad Master Yi players with bad Master Yi builds to come out of the woodwork! He has so any steroids that it can be hard to tell what stats he wants most! To ease the transition, here's some math on Master Yi's abilities!
First, let's get basic assumptions out of the way. For now, I'm using AD Marks/Quints, as well as 21/9/0 Masteries. I am assuming this will be the strongest route, simply because some of your damage is true damage, making the option of Armor Pen weaker. AS runes also prove strong, but don't particularly affect the relative strengths of items.
Now, let's look through each of his abilities:
Alpha Strike scales with three stats: AD, AS, and Crit %
Each point of AD increases its damage by 1.
Each point of Crit% increases its (expected) damage by .6*AD*.01
This means that at 166 AD, Alpha Strike will be doing 331 damage. Getting a 1% crit means that your effective damage will go up to 167, as will 1 AD. Beyond that point, Alpha Strike scales harder with Crit. How easy is it to get 166 AD? Turns out, not that hard. At level 18, your AD will naturally be 109, with another 30 from runes and masteries. Wuju Style's passive will give you another 21 AD, putting you to 160! Even at level 1, you should have about 75 AD, so a few levels (plus some Wuju Style) and a BF Sword should put you at that sweet spot. Basically, Crit always outshines AD, once you've left the laning phase. Unfortunately, Alpha Strike is the only skill that preferences Crit on Master Yi, so it won't be where we focus our efforts.
Each point of AS decreases its cooldown by a small amount. At 1.0 AS (level 18 without highlander), the cooldown is about 6 seconds. Highlander puts that to about 1.5, which brings the cooldown to 4.5 seconds. At this point, your Attack Speed is already pretty huge. Adding 50% Attack Speed, like you'd get from a PD brings you to 1.75, or a 4.2 second cooldown. The more attack speed you get, the more will be wasted from kiting, and the less you'll reduce the cooldown, anyway. With 2.5 AS, you'll end up with a 3.2 second cooldown. That's a lot of attack speed to invest in for 23% CDR. Essentially, it's the same reason you don't need to build Attack Speed on Skarner; the base passive is strong enough. But on top of that, you already get a ton of AS.
Wuju Style is a tricky spell. It benefits from both AD and AS. Assuming you're level 18 buying items (not an in-game scenario, but a good mathematical ceteris paribus comparison), you can find out that Attack Speed is SLIGHTLY better than attack damage for the active of Wuju Style. 1550 gold worth of AD (45) boosts your DPS from Wuju Style from 95 to 111. 1550 gold of Attack Speed brings it to 117! Not a huge difference. What about the passive effect? It varies, much like auto attacks in general do. You want a sweet spot of AD, AS, and Crit for this bad boy, but rather than calculate it separately, I've just included it in my calculations for DPS for every other ability.
If it isn't obvious, while Double Strike scales with AD, AS, and Crit, it's going to scale BEST with on-hit effects, because they won't do half damage. This ability alone makes me consider BotRK as the frontrunner for your first item.
This ability gives you a ton of AS, making AD and Crit the better purchases, while also boosting the value of on-hit effects
Overall, we've got Crit>AD>AS for Alpha Strike, AS>AD for Wuju Active, everything dependent on everything else for Wuju Style Passive, on-hit > everything else for Double Strike, and everything but AS for Highlander. It's a bit hard to make heads or tails of anything without going into the math.
First, let's look at how our 1550 gold dumps of stats affect DPS.
At level 18, vs 0 armor, we do 453 DPS, assuming our steroids stay up and assuming Alpha Strike is used right away, and not saved to dodge damage.
If we get 1550 gold of AD, we up our dps to 573.
If we get 1550 gold of AS, we up our dps to 551
If we get 1550 gold of Crit, we up our dps to 560.
It would seem that Yi wants AD>Crit>AS, but this is not true. The higher their armor gets, the more valueable your true damage steroid becomes, and the less valuable crit becomes.
At 50 Armor, our base dps is 334, AD gets us 419, AS gets us 407, and Crit gets us 405; AS has taken the lead over Crit! And at about 200 Armor, AS surpasses AD. You can generally assume that average team armor won't be 200, so it looks like AD is Yi's 'core' stat. You should be able to see this in the charts that follow.
Since the items cost different amounts, I have divided the gold cost of the item by the dps increase (from the no-item dps), to give the effective gold cost of each point of dps increase.
On top of that, BotRk gives you a 4.1s Alpha Strike CD, compared with 4.59 on IE/BT and 4.0 on PD.
At low levels of armor, BotRK is obviously the right choice for your first item, and this continues until above 200 armor. At some point of armor, BT becomes the strongest first item (it surpasses IE at around 150 armor).
We should not be surprised by BotRK's strength on Yi. Every steroid he has benefits from it or synergizes with it, either from AS (all but Highlander), AD (all), Crit (all but Wuju Active), or on-hit effects (all but Wuju).
Nor should we overestimate the value of IE. It may have slightly more damage efficiency than BT, but once you include Vamp for Lifesteal, that amount goes down (and you have significantly less vamp). However, this debate is largely unimportant, because of the option of BotRK.
Once you have BotRK, what should you be working on?
If we're looking just at damage, IE is the clear second choice. If you need more ability to chase (after Highlander and BotRK), PD is not a bad second choice, but against what team are you playing where they're outrunning/kiting a Highlander Yi? Probably one with a lot of hard CC. At that point, you're probably better served with Merc Treads and/or Zephyr and/or Spirit of the Ancient Golem than with a bit more Movement Speed. You should probably just get Merc Treads every game, in fact, because of your already ridiculously high Movement and Attack Speeds.
As such, I feel that BotRK + IE is the best Yi build, and from there, you can tank up, or get a LW right away if you feel gutsy. Your True damage is not a crazily significant portion of your DPS (it amounts to about a third of your auto attack damage), so you will still benefit from Last Whisper.
Potential Assumption Flaws
Alpha Strike is best when saved to dodge a CC or damage, so we won't be using it as often, so crit won't help us as much. I think this should mostly bias us towards BotRK over IE.
Steroids won't always be up. But they probably will be up long enough to have either won or lost the teamfight/skirmish. I'm not too concerned here.
I calculated Shiv without as much movement as I probably could have, and it came up weaker than PD. It is probably better than PD because of how much running Yi does, but given that AS and Crit don't seem as exciting on Yi as they could, I made the assumption that LW would be a better third item. This could prove wrong, but I think I'd have to assume a LOT more kiting than is reasonable for the math to put SS above IE or LW. Part of this is because BotRK + Highlander is already a TON of Attack Speed.
You could start IE instead of BotRK, and then get PD instead. This would give you 8.59 gold/dps even with a Vamp Scepter. But I felt uncomfortable holding onto just that Vamp Scepter. Would it become a BotRK? Then why not just build BotRK to start? When are you getting your Last Whisper? Are you building 4 offensive items and only one defensive? I made the (I think safe) assumption that you'd want 3 offensive items, boots, and 2 defensive items. It could be that higher level Yi players abuse Alpha Strike well enough to only need one defensive item, in which case the optimal build is PD + IE + (BT or BotRK) + LW.
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