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Question Repeated for Convenience

When should Infinity Edge be prioritized on a carry? When is IE the best option in a build, with respect to early game success and ranged carry types?

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.

Twitch, Urgot, Vayne and a summary to come.Summary

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.

Question Repeated for Convenience

When should Infinity Edge be prioritized on a carry? When is IE the best option in a build, with respect to early game success and ranged carry types?

Twitch, Urgot, Vayne and a summary to come.

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.

Summary

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.

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Question Repeated for Convenience

When should Infinity Edge be prioritized on a carry? When is IE the best option in a build, with respect to early game success and ranged carry types?

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.

Infinity Edge

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%.

R-DPS Champions

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 - Right from her own Wikia Strategy Page:

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, Urgot, Vayne and a summary to come.