I am a relatively new player. I main Ashe, and enjoy the concept of an ADC carry. I found myself wondering how much damage Ashe could do if she was fed. What would be the best set of items on her?

To find out, I wrote a computer program that would examine every reasonable item combination for Ashe and calculate its damage per second at level 18. It would then output suggested item builds that maximize damage over time. I offer them now to you.

Please note, this is not a guide on "how to play Ashe". I do not qualify to write such a guide, being a relatively new player. If you want to know how the plat players use Ashe, look elsewhere. But if you want to inject some math into your game, read on...

If all you care about is just maximizing damage per second (which is probably a bad idea, as will be discussed below) then this item set is the best you can do:
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The above build makes some assumptions about what is considered "optimal", as follows:

• We assume our opponent has 150 armor.
  More info

  All champions have between 60 and 100 base armor, so this could represent a champion with a single high-armor item like a Randuin's Omen or several smaller items, like a Ninja Tabi and a Sunfire Aegis. This is less armor than a tank would have, but more than somebody not specifically focusing armor. (If you feel I should assume a higher base armor, make a case for it in the comments, and I'll re-run the numbers)
• We try to maximize single target damage.
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  This means when calculating damage per second, we ignore the passive ability of Runaan's Hurricane. (Yes, in the above build, you're buying it solely for the +70% attack speed! Because crit chance was already nearly maxed, the +70% attack speed was more valuable than any other alternatives.) We also don't worry about the damage that Volley is dealing. We don't consider that the Statikk Shiv can occasionally hit multiple targets.
• We try to maximize sustained damage per second, not burst damage.
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  This means damaging active effects are not incorporated into the formula. This is probably why the build above doesn't include Blade of the Ruined King; the active is nice, but if you ignore that, there are better ways to maximize damage output. (If this is a problem, go to the sections labelled "But I need my Blade of the Ruined King!") This also means we don't factor in damage dealt by volley, spellblade, and so forth. It also means the Statikk Shiv's burst effect is only counted 1/10th normal (because it occurs once every ten times you attack.)
• As an exception to the above, we assume Youmuu's Ghostblade is always on.
  More info

  After discussing the issue with my higher elo friends, we decided that because the active on the Ghostblade is six seconds long, with a short cooldown of 45 seconds, and most engagements last less than six seconds, I should run my numbers as though the Ghostblade is always on.
• If it ever matters, we assume our opponent has Magic Resistance equal to his Armor
  More info

  Yes, my program is also trying out gimmicky builds like a Nashor's Tooth, Muramana, and Seraph's Embrace build. They don't do nearly as well as a straight Attack Damage build.

With those assumptions made, you simply cannot get a higher damage per second then the above build. However...

Yes. Yes it is junk. Anybody more intelligent than an ox can tell you that you need more than simple DPS to make a good ADC. Let's go over some problems with maximizing sheer DPS and how to solve them.

Problem 1: The build does not have boots. Solution: Tell the program it must pick boots of some kind. (If you do this, it invariably picks the greaves, for obvious reasons.)

Problem 2: The build doesn't have any defensive items. Solution: Make the program leave a spot for any one defensive item (your choice). This means that (including the greaves) the computer only has four item slots to play with. That's fine. It'll still do the best it can.

Problem 3: The build doesn't include any lifesteal. Solution: It's more complicated than just setting aside a slot for Bloodthirster. Optimizing for multiple variables (such as damage per second and life stolen per second) is a tricky business. I can show you some solutions, but you'll still have to make some tough calls. To explain, let's explore a concept called the "Pareto-Optimal Frontier."

An image helps to explain this concept:

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After testing every possible combination of items, the computer gives the final DPS of each build, and plots it against the final Life Stolen per Second. Each diamond shows a build. In each of those builds, you cannot increase one stat without decreasing the other.

There were many builds that could be shown in the lower left corner section of the graph. Those builds were omitted (for clarity). They are not worth it, because by switching your itemset, you could either improve your Life Stolen per second, your DPS, or both.

There are no builds in the upper right segment of the map. Without changing your assumptions (for example, by selling your boots or defensive item) there are simply no itemsets that will get you past the blue line.

The blue line is called the "Pareto-Optimal frontier", and represents the tradeoff you, as an educated league-player, must make between lifesteal and DPS.

So, which build do you go for? The best builds "stick out" towards the upper-right corner. Besides that, choosing one is just a matter of preference.

For example, there's one location, at about 280 Damage per second and 40ish lifesteal per second, that seems to strike a good balance, labelled "Build 2":
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As mentioned earlier, there are only five items in this build because there is a slot "reserved" for a defensive item.

Also note that the calculations for the damage and lifesteal on this build rely on you using the ghostblade. Without the ghostblade on, you are better off buying a second phantom dancer. (In fact, a second phantom dancer instead of a ghostblade only reduces the DPS of this build by 10 and the Life Per Second by one. This build is just barely pareto-optimal!)

On the other end of the spectrum, there is another point (labelled "Build 3") that gives up some raw DPS for increased lifesteal per second:
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This build replaces the Youmuu's Ghostblade with another bloodthirster. The lack of crit chance and attack speed is only compensated for in part by the bloodthirster's additional attack damage. This lowers overall the overall DPS of the build, but you steal about 1.5 times more life overall.

The last build I want to mention on this graph (labelled "Build 4") is especially interesting because it includes a Blade of the Ruined King:
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This strikes a middle ground between the damage-heavy build and the life-steal build, but it also includes a Blade of the Ruined King, which has a great active. It's on the pareto-optimal frontier, which proves you cannot get both higher life stolen and DPS in a single package. In the end, if you run an Ashe with a defensive item plus greaves, this is probably the best build, assuming you are not a n00b who forgets to use your actives.

Even though most Ashe guides suggest a defensive item, I've seen that it's often popular (at least in lower elos) to run ADCs with nothing but greaves and attack items. I'm not pro enough to comment on whether or not this is a good idea; I only want to help those who care understand the tradeoffs.

It's easy to see why it would be tempting to go with five attack-enhancing items. Because attack damage, attack speed, and crit chance all stack in a multiplicative with each other, a single additional item can drastically increase your damage output. It could potentially work, if you're excellent at positioning and have teammates with heavy, hard crowd control.

It's equally easy to see how this could be a bad idea. No defensive items makes it way harder to survive! I don't care what your "life stolen per second" is, a bursty Brand or Akali can ruin your day so fast that the half-second of life you steal is essentially a non-issue. A Banshee's Veil can let you survive the burst so you can start healing again from lifestealing crit-arrows while pursuing the now-cooling-down burster (or fleeing, lest he burst you again). Also, if your opposing ADC is fed due to a poor support or jungler (or, you know, poor skill on your part) then a Warmog's Armor or other large armor can increase your survivability, increasing your dueling capability. And seriously, the revive on Guardian Angel is awesome.

If you do decide to go without an armor item, here is the new Pareto-Optimal Frontier:

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The old pareto-optimal frontier is still shown, still in blue, so you can get a sense of what has changed. The new one is shown in brown. If you look, the addition of a single item makes a great difference. With it, you can add 60% to your DPS, or triple your life stolen per second!

Again, I have highlighted some of the more interesting builds, which I will show below. The first is build five:
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This build deals over 50% more damage than the BotRK build above, and steals nearly twice as much life per second. It is my current favorite, surprising people at low Elo. It's especially effective against ADCs sporting Warmog's Armor, because they have low armor, meaning you can steal enough life to keep ahead of most four-item ADCs in a duel.

Again, it's just slightly better to have a Youmuu's Ghostblade than a second Phantom Dancer. If you are bad at activating abilities, or just want to never be caught on a cooldown, you can safely replace the ghostblade for a second phantom dancer for a paltry drop of about 15 DPS.

As with all the rest of the builds in this chapter, this build is weak against bursty casters, because you aren't reserving a slot for a Banshee's Veil or something.

If you want even more damage (but less lifesteal) there's another option, again shown above in red...
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This build has much less AD than the previous build, but much higher attack speed. Not only that, but you always crit. Those two facts together add up to a much higher overall DPS. Giving up that bloodthirster costs you life per second, though, and again, you are reliant on the active from the youmuu's ghostblade. Remember that you can't swap this ghostblade out for another phantom dancer like on some other builds. Doing so would put you over the crit chance cap, wasting the extra and lowering your DPS significantly.

But I need my Blade of the Ruined King!

Okay, okay, then I have an option for you:
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This build, when compared with build five, gives up fifteen life stolen per second for an extra eleven damage per second. This would probably be a poor trade were it not for the fact that you get the BotRK's active, which is nothing to sneeze at. As with Build 4, it's still Pareto-optimal; you can't get both better lifesteal and better DPS. If your playstyle centers around having a BotRK and you're willing to give up a defensive item, this is probably your build.

Some Ashe guides I have seen suggest that very late game, if you are well fed, you should sell your boots for another offensive item. In some ways, this makes sense. The +25% attack speed from Berserker's Greaves pales in comparison to another Phantom Dancer or something. The movement speed you lose you can get back in other ways, such as the active of Youmuu's Ghostblade or lots of move-speed increasing items such as Phantom Dancers. On the other hand, it hurts to give up the kiting potential of Enchantment: Furor or the cooldown reductions of Enchantment: Distortion. Again, I'm not going to tell you what call to make, I'm just going to show you what your options are should you choose to go this route.

If you do plan to sell your boots, here is the new Pareto-Optimal frontier, shown in green. (The blue and brown lines are still there for comparison)

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Not quite as good as if you have five items and greaves, but still a significant improvement. Again, there are three builds that stick further out than the rest. The funny thing about these three builds is that they are almost exactly like builds 5, 6, and 7, above. The only difference is that they omit the greaves. (And that build eight gives up the ghostblade for a phantom dancer, which just barely is better after losing the greaves.)

Even though builds 8, 9, and 10 are almost identical to builds 5, 6, and 7, respectively, I'll still list them here so you can see their math and final attributes:
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Again, the second phantom dancer just barely beats the ghostblade in this build. I would prefer to have my phantom dancer (which is always on) than worry about the active of the ghostblade, but if you want to get the move speed buff from the active, you can swap out the second phantom dancer for a ghostblade for only a small hit in DPS.
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Nothing much to say here. But again notice the 100% crit chance!
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As above, this build takes a middle road between damaging and lifesteal-ey, but has the benefit of a BotRK. As you can see from the Pareto-optimal frontier diagram above, it's actually not as bad as the similar five-item build. (It doesn't "poke in" as much.)

Then you're probably crazy.

But hey; what do I know. I'm low elo. At worst, this could be a cool thought experiment. At best, perhaps you could run some sort of slow-based "hyper carry" that can kill Udyr before he gets into Blazing Stampede range. Let's see how much damage we can tweak out of a no-defenses-whatsoever build...

Here's the new Pareto-Optimal frontier, this time in purple.

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There are two clear winners here, poking out much further than normal. I've labelled them as build 11 and build 12. For BotRK lovers, I've also pulled out a build 13, that trades out some lifesteal for a botRK. (Also, we see our first naive, sustainless build 1 down there at the "no lifesteal" end of the curve.)

With no further ado, I give you the builds with the most damage Ashe can deal:
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Hey! It's a Last Whisper! We haven't seen one of those pop up yet!

That's mostly because percentage based armor pen is almost (but not quite) a damage multiplier. Until you stack enough damage into a single build, it's better to get another phantom dancer than a last whisper. If you don't believe me, look at the section labelled "Why is there no Last Whisper?" for the math.

Anyway, this build is only about 37 damage away from the most Ashe can possibly do, but it also steals 90 life per second. Not a bad tradeoff.

The next build is on the high-lifesteal end.
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This is the most health you can steal per second without taking a major hit to your DPS. You get more than 150 life per second against an opponent with 150 armor. You get more if his armor is lower! The damage is also higher than any possible five-item build, and well more than twice as much as the four items + greaves build. You steal almost three times as much life as well.

To work a BotRK into this build is more costly than other builds, but if you need one, we've got a build for that...
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This build gives up 24 life stolen per second for a measly 4 DPS, and a decidedly not-measly Blade of the Ruined King active ability. It's up to you to decide whether or not to make that tradeoff. You could probably siphon away more health with the BotRK, but only if it happens to not be on cooldown at the time.

And that's it for the builds!

Or rather, questions I assume will be asked and I hope to head off at the pass.

Why do you hardly ever use the Last Whisper?

The current meta (from what I've seen in the Ashe guides) is to have one Attack Speed item and three Attack Damage items. Typically, people recommend you use the Last Whisper as your third "Attack Damage" item. Problem is, the Last Whisper is mathematically inferior to another attack speed item.

Let's say you have an Infinity Edge, Bloodthirster, and a Phantom Dancer. Should you get another phantom dancer, or a last whisper?

Attack Damage Crit Chance Avg. Damage / Att Dam. / Att (after armor) Attack Speed Total DPS

........ ........ ........ 294.8 0.55 537.9 258.3 0.97 249.9

........ ........ ........ 254.8 0.85 579.6 217.4 1.29 281.9

...and, of course, more DPS = more lifesteal. The phantom dancer is a clear winner.

But what if our opponents have 250 armor instead of 150?
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Attack Damage Crit Chance Avg. Damage / Att Dam. / Att (after armor) Attack Speed Total DPS

........ ........ ........ 294.8 0.55 537.9 196.8 0.97 190.4

........ ........ ........ 254.8 0.85 579.6 158.1 1.29 205.0

Even at 250 armor, the Phantom dancer still beats out the last whisper! Hopefully that'll be enough to convince the people who were confused by the choice.

Why do you never use the Statikk Shiv?

Aside from the passive (and cost, and a tiny bit of move speed), the Statikk Shiv is worse than the Phantom Dancer in every way. Thus, the algorithm always picks the phantom dancer.

The passive from the statikk shiv does combo well with Ashe's passive, Focus, because it can crit. The ability to deal an extra 250 magic damage to multiple targets gives Ashe some burstiness she didn't have before. The problem is that I am optimizing for Damage Per Second, not burst damage. Because of that, I ran my calculations as though the statikk shiv activated once every ten attacks, averaging an extra 10 damage per attack (before taking into account crit chance, the passive from Infinity Edge and so forth).

This is actually worse than a straight +10 Attack Damage! Because the shiv deals magic damage, you cannot lifesteal from it, it doesn't benefit from masteries that improve AD by a percent, and Armor pen doesn't apply! This makes it worse than physical damage.

If you feel you need a bit of extra burstiness to Ashe, then in any build above, you can substitute a statikk shiv for a phantom dancer. Doing so lowers the DPS by about 10% or so, but the shiv and the phantom dancer fill about the same role, so it's viable.

Thanks to JhoiJhoi for his guide on guides. Thanks to Dr. Goodrich at BYU for teaching me about Pareto-Optimality and search strategies for optimization functions. Thanks to my wife, who plays a mean jungle Udyr.

If you have any questions or comments, let me know in the discussion, and I'll try to add it to the guide. If you think some of my core assumptions are wrong, suggest better ones. (Especially with armor. I've never played top or mid, and I have no idea how much armor those guys really have.) If you have any other numbers you'd like me to run, let me know.

Thanks for reading!