There are a lot of notes if you want to spend some time hovering over the Notes indicators above.

Each jungle champ has a unique strategy to win that will vary based on team composition in each game. Yi happens to be quite straightforward and specific so we can work on refining the nuances of that strategy.

Yi's singular goal is to do heaping piles of damage and whack things very fast. With Lethal Tempo and his kit focusing around the concept of attacking faster all the time the ideal items to maximize output are pretty clear: Attack speed items making you hit many times + On-Hit perks + Guinsoo's Rageblade to make them hit even more.

There's other builds that are fun to try (Crit, Brawl, Tanky), but in the end it is going to end up being On-Hit winning in the vast majority of situations. You'll need to diversify when the enemy comp is really just hard countering you but focusing committing to On-Hit is the best for Yi.

Yi wants damage. He attacks fast and Lethal Tempo makes him attack even faster, so all On-Hit perks are high value to him. In general we have to balance AD and AS for good base damage versus attack speed/On-Hit frequency. Later in the game we balance in Crit, Armor, MR, Utility, Armor Pen, etc as needed but only one or two items max - we prioritize AD/AS/OH if we can.

Early game

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Balanced stat growth

(+On-Hit) Kraken is a great first item because it's throwing down a little bit of everything for you right out of the gate with the exception of Lifesteal. It also offers a huge amount of On-Hit ramping damage which is great for burst ganks on squishies and sustained fights/ganks with tankier champs.

You'll get Magical Footwear boots at some point before 12:00 depending on how effectively you've ganked/invaded. These offer you a significant amount of attack speed and it's hard to find a good alternative to sell them for in the late game so you'll basically plan on getting these soon, and keeping them forever. If you buy these first then you had a bad start.

Mid game

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Amp up our On-Hit perks

(+On-Hit) BORK gets us more AD + AS and some lifesteal to help us sustain. You can take BORK first instead of Kraken Slayer if you prefer as they will have similar results overall but I find Kraken's power spike to perform better in general unless you're going to be stuck in long fights early and need the extra sustain or need the slow on-hit perk for some very specific target.

(+On-Hit) These stats are kinda meh and AP is useless but who cares because the passive ramps your attack speed, which means more attacks, which means your On-Hit stuff will proc more often, then when fully stacked it can proc twice instead of once! In terms of damage, this is the best interaction for Yi to maximize output when building On-Hit. You don't need this second because the stats aren't good enough and you wouldn't have any proc's benefitting from it so it is either 3rd or 4th depending on the game.

Mid/Late game

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Get more specific

(+On-Hit) If your problems are AP based or CC then Wit's End is always Item 5 because it gives you what you already want with some MR for free. You'll get huge attack speed out of this and be throwing down On-Hit procs like crazy. The insane attack speed will be trimming down your Q cooldown to a point that lets you stagger takedowns very consistently in sustained fights.

If your problems are AD based then Death's Dance is a good Item 5. This mitigates a lot of damage but requires you to close the takedown. It makes your strategy depend a little more on high commitment dives in order to clear the mitigated damage if you're playing on the razors edge, but overall the stats are good. Wit's End is better, but this is good for some AD based sustain and high commitment play styles.

If your problems are getting nuked then Sterak's Gage may be a good Item 5. The lifeline shield is pretty handy, but you have to track it's cooldown or risk overplaying yourself. The extra health is preferred when you are up against a mixed AD/AP comp with no clear carry damage type. When there is a clear carry damage type you should build Armor/MR instead unless they have obscene amounts of penetration built out (such as Veigar).

If your problems are sustained fights then Experimental Hexplate may be a good Item 5. It offers some flat HP, some much needed Attack Speed and a fairly significant boost to your R mode if you need to make a lot happen in a short time.

End game

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Solve situational problems

Despite not having on On-Hit perk, LDR is actually the best item 6 to synergize with Kraken Slayer and Blade of the Ruined King. Tanks with big health pools will completely melt and you'll still do great damage versus squishies. If you need more sustain then go for Titanic Hydra instead.

(+On-Hit) If you have good presence in teamfights then consider running TH last in order to spew damage to everyone all at the same time while gaining some extra HP to help your sustain.

(+On-Hit) Do you like to Q? Do you already have Wit's End? Do you want to Q even more? Get Navori Quickblades today! Yes, NQ is handy because it drops your cooldowns to a point where as long as you can keep hitting the enemy, you can whiff abilities. Note that if they can CC you and stop the attacks then NQ is not the pick so this item requires an adjustment to your engagement strategy.

Is the support repeatedly CC'ing you when you approach? Maybe Edge of Night can simply negate it! You can get perks like this from Morgana, but in some rare cases EoN can be the end-all solution to a highly specific problem of CC in one instance - plus the stats are already favorable so you're basically just paying for the perk. It's slightly wasteful if no one (or none of your targets) are running armor.

(+On-Hit)I've experimented with this quite a bit and it's not bad. Yi basically only wants this if there is excessive armor present and fights are longer than you want so you would benefit from scaling de-buffs and scaling armor during fights. If most of the team is running armor then it can be a good tool to trim it, but less ideal if there isn't an excess of armor in the game. Prefer Lord Dominik's Regards or some other utility in that case.

Avoid buying this if you can. Yi doesn't see huge benefit to this in his playstyle because in most cases when you die then you'll be in a terrible position and unfit to do anything when you respawn - unlike brawlers, who can make better use of GA. It's nice to avoid the death timer after 18 but rarely does it pay off with more value than other items.

Do not buy this. If you need it then your team composition or strategy has a structural problem and it's not going to save you. The stats are a complete wash, so you're better off doing almost anything else. You could consider a Seeker's Armguard if you're desperate, but your money is better spent elsewhere.

When you are full build (all slots full) then spend extra money on this elixer. It's a substantial boost, and better suited than the other elixers in every situation.

Invade

Track the enemy jungler. If you know he will be at a particular camp and you can get there around the same time without being seen then do it, let them use their kit on the camp, be patient and blow them up. Repeatedly doing this guarantees the enemy to use wards in their own jungle, delays the enemy jungler progress, gets you free takedowns, and tilts them (which is important).

Track

If you don't know where the enemy jungler is then you can't make the best descisions. You should know what camp they start on and where they want to go next. When you know where the enemy wants to go then you can make better choices about where you want to go.

Prefer Grubs

Grubs offer a lot of XP early and the first Grub in each group will yield more xp than the others, so stealing at least one is almost as valuable as taking the other two if it is contested. If you were to prefer Drake then it would be all or nothing! Because of this, Grubs are currently the preferred first camp. In fact, starting topside is preferred too because you can back after a full clear and have the first two camps spawned back when you're on the way to Grubs - which is the most effective gold pathing when there are no other effective options for your pathing besides farming. I'll update this if Grubs change value in later patches.

Choose to contest or avoid

The time you spend taking your own camps is time your team does not benefit from you being in the game. Your preference is to not be on your own camps. When the game starts you should try to figure out how to harass everyone, including the enemy jungler. If you can, then you need to plan to be at every major objective to either get it first or contest it from the enemy jungler. If you make deliberate choices to give up an objective then you need to make sure you have good trade value in whatever else you decided to do.

Even when you don't have a favorable matchup, try stealing the enemy's camps whenever possible in order to force them into unfavorable situations and tilt their mental. By removing their options from the map, you force them into bad options. When you know they're headed for Red Buff and you just took it, you know what their next options are - like probably a mid-gank - and you can be prepared to counter them.

Counter Ganks

Yi isn't the best at counter ganks because he can't prevent his laner from dying in any other way than simply demolishing the enemies. Therefore you need to be prepared to accept that you might lose allies due to Yi's lack of utility in his kit throughout the entire game but in return you are very likely to be able to secure multiple takedowns with ease, it just comes with a small up front cost to ensure that the enemies are prepped for demolishing. With a little sacrifice and if you can leverage that to snowball then you will inevitably carry the team. And, of course, they will totally understand. (not really, you'll get flamed for not playing a support jg as you cut the nexus into swiss cheese, get used to it.)

Always make a plan before doing anything. Yi has zero support utility for his team. Your only tools are DPS, surpises, and mental fear. Fear is a notable point because somewhere in the game after you start snowballing, enemies will start running away from you into terrible situations because it simply means less death than the alternative of being anywhere near Yi - which would be instant death. You need to eventually start planning for this and use this fear to your advantage.

In general these are your goals:

Give your laners an advantage
Secure all shared resources (Crabs, Grubs, Herald, Drakes)
Make useful ward placements
Invade enemy jungler's camps to force him into unfavorable situations

Applying pressure takes time but if you can get enemy laners to use their Summoner Spells then you have tons of time to now apply pressure and harass that lane with repeated attacks. So don't see ganks as a loss just because they don't end in a takedown. Giving your laner an advantage is going to pay off.

Wards last 120 seconds. Don't waste them getting information you already have or don't need. Put wards in places that are valuable and will confirm that the plan you've already made is going as expected, or in places that confirm your plan is about to have a problem.

Stay ahead of the enemy jungler. Think of their plan first, and figure out where they're going or what they want to do. Are your lanes shoved? Well, you can't gank so just ignore it right? Not really, the enemy sees that as an opportunity and might show up there soon so you could be counter ganking with a little patience. It's better to know what the enemy is doing first, then establish your own path to ruin their plans.

Yi requires this kind of planning because his kit is limited. If you can't stick to enemies you can't get Q cooldown bonuses and this leaves you very vulnerable. Your plan going in needs to include how you're going to get resets and keep going either in or out.

Yi needs gold. Enemies are made of gold. Your own jungle is simply a fallback when nothing else is available. You can't farm your own camps all the way to victory, get in lanes, invade, claim tempo advantage, contest.

Don't linger around while losing objectives unless you will attempt to steal.
Don't waste time taking your own camps if something else is available.
Always get crabs before your own camps to deny enemy jg vision, gold, xp.
Try not to engage with Q if you can avoid it. You can use it to catch Dashes.
Prefer AD/AS items instead of Armor.
No game plays the same. Find a strategy to counter specific players.
Prioritize helping allies over everything else.