When is Infinity Edge the best option in a build, with respect to early game success and ranged carry types?
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While this question is short and to the point, there are a number of facets to it that make it a very interesting, if not difficult, question to answer.
First, we will identify what Infinity Edge is. Infinity Edge gives +80 Attack damage, 25% Critical Strike Chance and +50% Critical Strike Damage (Unique) at a total cost of 3830 gold.
The flat damage increase (+80) is the highest bonus to damage you can find in a single item (with the exception of stacking items such as Sword of the Occult and Bloodthirster). Source. Infinity Edge is also the only item to offer an increase to the Critical Strike Damage that a champion deals though there are Runes and Masteries can be used to increase this "stat".
Critical Strike Damage is the bonus damage that is dealt when a champion scores a critical strike. Critical Strike Damage has a base of 100% which means that a normal attack with no modifiers that would normally deal 100 damage deals 200 damage instead. (100 base + 100 base * 100% critical strike damage). The same attack with an Infinity Edge would have dealt 250 damage (100 base + 100 base * 150% critical strike damage). Furthermore, according to the Lethality Wikia entry,
Critical Strike Damage increase is additive, meaning a crit with Lethality and an Infinity Edge equipped would do 200 + 50 + 10 = 260% crit damage.
As a sort of bonus, Infinity Edge also increases the chances of scoring a critical strike by 25%.
Now that we have an understanding of the item in question lets look at the different types of R-DPS champions (Cr Type Sort and find R-DPS). For your convenience the same champions are listed here: Ashe, Caitlyn, Corki, Ezreal, Graves, Kog'Maw, Miss Fortune, Sivir, Teemo, Tristana, Twisted Fate, Twitch, Urgot and Vayne.
While Infinity Edge can be bought by any of the above champions as carries, it shines on some a little brighter than others. We will first dismiss Infinity Edge on the following Champions from the above list (brief description to follow):
INFINITY EDGE NOT RECOMMENDED ON
Please keep in mind that there are a huge number of champion builds out there with different focuses and an ever changing metagame.
Ezreal - Ezreal is not as much of an Attack Damage (AD) Carry as an AD Caster. That isn't to say he can't do a lot of damage or he shouldn't build AD but his damage comes primarily from his spells, not his auto attacks. This means that you wont be getting the full benefit from Infinity Edge as you are mostly going to end up wasting the Critical Strike Chance and Damage Increase it offers (his Q applies on-hit effects but it does not crit). Ezreal tends to do more damage with items like Bloodthirster, Black Cleaver, Sheen/Trinity Force that offer additional effects on top of flat damage.
Kog'Maw - Kow'Maw's abilities, with the exception of his Living Artillery (Ultimate), scale exclusively with Ability Power (AP). This alone makes Infinity Edge a questionable item on Kog. Furthermore, his Bio-Arcane Barrage already deals damage equal to a percentage (6% at ability level 5) of the target's maximum health. This makes attack speed items significantly more useful on Kog'Maw than flat damage. Items that offer attack speed can also offer some useful proc effects. Three such items are Malady, Wit's End and Madred's Bloodrazor. Malady and Wit's End will be more useful than an Infinity Edge to Kog'Maw early game simply because they are cheaper and Madred's Bloodrazor will help him melt takes and high hp enemies late game with an effect similar to his Bio-Arcane Barrage.
Teemo - Teemo's Toxic Shot is easily his best "DPS" ability and it scales off of ability power and attack speed. With an argument similar to the one given for Kog'Maw above, Teemo just doesn't have the need for an item like Infinity Edge. Another option for Teemo, including those items listed under Kog'Maw, is Frozen Mallet. It gives him some needed durability and the slow on-hit really aids his chasing/finishing power.
Twisted Fate - Twisted Fate's Wild Cards ability is arguably his best source of damage in the early stages of the game. It has a very long range and wide area of effect, a short cooldown and decent damge that scales with...Ability Power. There are Twisted Fate builds that focus attack damage, critical strikes and attack speed but those builds tend to leave his Wild Cards ability under leveled and underused. It basically removes one of his abilities since the flat damage it offers late game is lacking if you haven't purchased AP.
Summary of why Infinity Edge is Not recommended on the above champions
With all of the above champions it becomes a game of efficiency where Infinity Edge just isn't really the BEST item for the money you are spending on it since other items will give you more punch for their cost through the various stages of the game.
That leaves us with a list of champions that...
INFINITY EDGE IS RECOMMENDED ON
Ashe - Of the typical R-DPS (carry) champions Ashe is actually on the lower end of the damage scale. She offsets this with her utility and farming abilities with her ultimate (stun) and Hawkshot (free CV and bonus gold). That being said, Ashe needs to build Attack Damage early in the game to stay viable and to stay a threat. Since Ashe naturally gains critical chance through her Innate ability (Focus), her first attack in a team-fight will often score a critical hit. Since Ashe needs the bonus attack damage and crits naturally it makes Infinity Edge a natural item choice for her. It is recommended to build it very early in the game (first major purchase) since she is lacking a real sustained form of an attack steroid (Attack Speed, Sustained Critical Chance, Bonus Damage, etc.). Berserker Greaves/+3 boots, zeal and Last Whisper are common mid game items for Ashe to pick up AFTER Infinity Edge.
Caitlyn - Caitlyn's innate ability Headshot described below
(Innate): Every 8th / 7th / 6th autoattack is enhanced to be a headshot, dealing 150% damage to a champion or 250% damage to a minion or monster. Attacks from brush increase the attack counter by two instead of one.
cause her attacks to deal 150% of their normal damage when the ability procs. As Infinity Edge offers the highest bonus to attack damage in a single item (source above) it is an obvious choice. This attack can also score a critical stike (but the bonus damage is not increased by the critical strike damage increase stat). With an attack damage stat of 100 and an Infinity Edge in hand Caitlyn's Headshot attack would deal 300 damage (100 base * 150% headshot + 100 base * 150% critical strike damage). Caitlyn also has the benefit of having one of the longest natural attack ranges in the game which allows her to auto-attack enemy champions from beyond their maximum range. Source Since Caitlyn's attacks are her main source of damage she really benefits from having critical chance and critical damage increase, which are stats that abilities (except for Gangplank's Parrley) do not benefit from.
Corki - Corki is one of the R-DPS champions that likes Infinity Edge but tends to build it later in the game rather than rush it from the start. This is because Corki tends to get more of his damage from his abilities during the early phases of the game. Sheen, specifically, is a good early game item for Corki since it fixes his mana issues and gives him a little more burst in damage. Wriggle's Lantern is also a good item to pick up early in the game. Corki's innate ability (Hextech Shrapnel Shells) deals an additional 10% of his attack damage as true damage when he attacks. This is a big part of why he likes flat attack damage items like Infinity Edge, Black Cleaver and Bloodthirster over items like Phantom Dancer which is an item commonly found on other R-DPS champions.
Graves - Graves enjoys carrying around an Infinity Edge because of his high attack damage ratios on Buckshot and Collateral Damage and he gets a free attack speed buff from Quick Draw. These factors combined with defense bonus from True Grit means that there isn't much Graves needs to be a threat. He doesnt have to spend gold on a lot of health or defenses thanks to true grit and quick draw lets him skip an early attack speed purchase. Infinity Edge gives graves everything he doesn't already have except for life-steal which becomes more important later in the game. Infinity Edge as the first major purchase on Graves is almost always the right move.
Miss Fortune - Miss Fortune has a 50% attack speed buff through her Impure Shots ability and good attack damage ratios on her Double-Up and Bullet Time abilities. These abilities combined with the movespeed bonus from Stut and her aoe slow (Make it Rain) Miss Fortune makes good use of Infinity Edge when chasing enemies and when well protected in a team-fight.
Tristana's high attack speed and powerful attack speed steroid make Infinity Edge especially cost efficient on her.
There isn't really much more to say. Her Rapid Fire ability increases her attack speed by 90% and her Innate: Draw a Bead gives her the longest (passive) attack range in the game. Infinity Edge is a strong purchase for Tristana early in the game because it gives her some real attack damage, which she is lacking from base stats alone.
Twitch - After de-stealthing from his Ambush ability, Twitch has a nice attack speed buff. This, combined with the critical strike chance and increased critical strike damage from Infinity Edge, makes Twitch dangerous. While Infinity Edge is usually purchased after boots and sometimes a survivability item like Wriggle's Lantern, it should be incorporated into most Twitch builds.
Urgot - While most or all of the champions that benefit heavily from Infinity Edge it tends to be more of a "nice addition" to Urgot. Early-game, Urgot needs additional mana and cooldown to significantly increase his damage and lane presence. Brutalizer, Sheen and Tear of the Goddess are strong items for Urgot to start with. Once in the mid to late-game Infinity Edge allows Urgot continue being a threat. After Urgot's mana issues are solved and he has some Cooldown Reduction, damage will come from using Acid Hunter as a sort of second (or primary) basic attack. If you are able to auto-attack your targets between uses of Acid Hunter (with the added damage from Infinity Edge), Urgot is capable of doing a lot of damage very quickly.
Vayne - Vayne can pick up an Infinity Edge as one of her first major purchases but it does not necessarily need to be rushed. Vayne's Silver Bolts deal damage based on the maximum health of the target and her innate ability (Night Hunter) gives her a move speed buff so long as she is moving toward an enemy she can see. What Vayne is lacking early in the game is an escape. Picking up a Zeal and boots early on will allow you to close, get additional procs of you Silver Bolts ability and escape with the added movement speed if necessary. Tumble can dodge some abilities and Condemn can stun enemies if they are foolish enough to stand next to walls but it is better to assume that your enemies will do what will cause you the most harm and account for your ability use.
R-DPS champions that have attack steroids (bonuses to attack speed, damage, critical strikes, etc.) and tend to rely on their auto-attacks for damage gain a significant DPS boost from having an Infinity Edge. The time at which you purchase the item depends on what stats you need to scale into the mid and late game phases. Item purchases in League of Legends is about spending your gold on what is most efficient. This means the most benefit for your champion given the gold cost. Infinity Edge, while being one of the most expensive items in the game, is also easily the single most damage enhancing item for champions that rely on their auto-attacks.
IE is like rabadons for ad carries. It is the best damage item, so as an ad carry, you should get it during the game. It depends what you head for. Basically, if you own your lane and you have a really good farm, you can get it as the first item. However, if you want more of a lane sustain, you will probably go for some lifesteal item first: vampiric scepter is the basic, you should probably have it before getting IE. If you re getting pushed and you want to get lane sustain, with better farming ability and also some durability, get lantern first, as it gives you lifesteal, damage and armor which is very good for an ad carry. Very viable as first item is also blood thirster. I usually get blood thirster when I feel that I slightly dominate the lane or I have a character with attack speed ability, then you get a nice advantage by having a lot of damage and lifesteal. Now I like the last option even more, when you lose only half stacks when dying.
The important goal is to win the lane. One consideration with respect to IE, beyond who it is a good fit for (fine analysis, there), is when to get it.
The goal is to win the game. The immediate goal in service of that is to win your lane. Sometimes an AD Carry - even one who favors IE - is not going to get IE first. Sometimes you need the sustain so you can survive trades. 3x Doran's Blade + Life Steal mastery can take care of that, certainly. But note that's 1400 and change gold spent.
Some people like to go for Blood Thirster first, and given the recent change on how death affects it stacks I suspect that option could become more popular. Doran's items add safety in addition to their other benefits, and are cost effective, but don't build into anything. Plus, you have to go back to get anything. Sometimes it works out that you can go back and get a BF sword your first or second time back. If you need the sustain, BT might make sense.
OTOH, you frequently see people shoot for IE but pick up a Vampiric scepter (to make up up for swapping out Doran's blades for other items, at times) quickly to help with sustain. That's valid too.
If you are winning your lane, and aren't frequently getting bailed out of engagements, you probably want to shoot for IE on characters where it makes sense. If you are having trouble, you might instead need to consider more defenses/sustain.
I usually go blood thirster on my AD carries first for sustain in lane. However, I find myself getting IE on Ashe as her passive allows her to crit more often than other carries
For AD carries you should really rush IE. That crit chance and damage is just beastly in lane. Usually I try to get it before the 20 minute mark.
Depending on which carry you are, I like getting the IE after my sustain item (Bloodthirster, Wriggle's Lantern etc.) because it gives me more ability to stay in my lane and continue the ever so needed farming. Sometimes, I don't even get it at all. (I find Madred's Bloodrazors to kill quicker with Kog'Maw.) Also, if you do choose to get IE then get zeal first, the pair are really good, especially with zeal's upgrade Phantom Dancer.
IE should really be your first item on all carries. It's just too good.
IE should usually be the first item. There are exceptions, however. For instance, Vayne might be better off getting a Phantom Dancer first. PD complements Night Hunter and Silver Bolts quite well.
With carries that already have crit, like ashe, build it relatively early. But with carries that don't, build it after a Phantom Dancer or two.
Ranged carries generally have IE or Trinity Force as their major damage dealing item. This depends on the champion. Some champs like Ezreal and Corki that spam spells a lot go trinity force. Others like Ashe and Caitlyn should get IE as their main item. They mostly auto-attack for their damage in teamfights.
For AD carries IE is a must buy item. You should rush it as soon as possible. After that buy items depending on the enemy team, and your champ/play style.