Update on Ezreal; Brutalizer

*outdated* Go here

The Brutalizer
I've been criticized on my insistence on going Brutalizer first on Ezreal. I've also been infuriated with Ezreals going Zerker Greaves. I've finally gotten my act together and made a spreadsheet to compute it, see if I was right. And I was.

The stats
In the sheet I'm assuming R>Q>E, that you're 21 in offense and that you have AD marks.
The enemy hit is assumed to have average armor plus flat seals plus the +5 mastery. I've added a Sunfire Cape at higher levels. R and Shiv only hits 2 targets, E hits a target half the time. All this can be modified in the sheet.

Brutalizer first item
The most cost efficient to go from the start is Brutalizer. Here's a comparison of Brutalizer vs some other typical items, showing loss of gold efficiency:
12%  BF sword
14%  Pickaxe
31%  Longsword
47%  Spirit of the Elder Lizard
67%  Manamune (stack at 250)
102% Sheen
120% Phage

The Core
Brutalizer -> Sheen -> CDR boots
People have then told me that it pays off when you get to Trinity Force. But that isn't true either. Going Brutalizer, Sheen and CDR boots is the most effective build at this price level (3-4k gold). Another comparison:
11% BT + CDR boots
15% BT
16% BT+ Zerk
19% Brutalizer + Phage + CDR boots
27% Brutalizer + sheen + Zerk
32% Trinity Force

Boots
Here we can also see that CDR boots are better than Zerker Greaves. And it gets even worse at higher levels. Don't buy Zerker Greaves on Ezreal please.

Conclusion about early build paths:
The only other option, trading damage for lifesteal, is going BT. BF sword is only a little weaker than Brutalizer, and same goes for BT compared to Brutalizer+Sheen+CDR. It also has more burst.

The optimal path for the BT build is:
BT -> Brutalizer/CDR Boots -> Remaining -> LW -> Tri -> IE -> BC
If you're maxed and want a defensive item, BT is the weakest if the enemy has just a little bonus armor, followed by BC if they have a little more bonus armor, followed by LW if they have a lot of bonus armor.

Typical Build:
Brutalizer -> Sheen -> CDR boots -> Trinity -> LW -> BoRK/IE/BT -> BC

If they have any bonus armor, LW is more cost efficient than Trinity
If they don't have much relevant bonus armor, while LW is still cost effective, other options provide more slot efficiency and more utility.

Going BT/BoRK means a loss in damage, but it's not terribly big.

About Tear: 
If you're buying Tear anyway (I do), upgrading it after Sheen but before Trinity is more damage. Trinity is 27% less efficient than upgrading Tear, and that's without it transformed.

Maximum Output

The single most damaging build with boots is BC+Tri+MM+LW+IE with 1000 DPS on an averagely armored target without items, and it includes both anti-armor items.
If you want a defensive item in there, the MM is the least slot effective unless the opponent has little armor, in which case LW is the least effective.
If you don't want boots, Static Shiv is the best output(1165). BoRK(1124) is close if you want the active and lifesteal.

Excel Sheet

The Unstoppable Force vs The Immoveable Object; Musing (Not Useful)

So, now that we've pointed out that DPS has diminishing returns and that Armor and HP does not, does that not beg the question: Do defensive stats scale exponentially?

We have a champ D and O with 1000hp and 0 armor, 1 APS and 100 AD. O is attacking D. It takes O 10 seconds to kill D.

If we give D a Warmogs and O a Bloodthirster, D doubles his EHP and O doubles his DPS. O attacks D. It still takes 10 seconds.

If we only give D a Warmogs, it takes O 20 seconds to kill D. A 10 seconds increase.
If we only give O a BT, it takes O 5 seconds to kill D. A 5 second decrease.
O only wins half as much time as D by having his item against someone without items.

So when the defensive champ has stat superiority, he gains more over those he has superiority over than the offensive champ does.

However, this is an extremely gross oversimplification, as there are obviously eight other champions and countless more factors. As you have probably noticed we're a bit past the 5 bruiser meta. I just thought it was an odd effect of the system.

Multiplier Theory: Stat Synergy explained with Polytopes

(Would like to start out apologizing for the somewhat awkward terminology used, I wanted to be clear, and the day-to-day terms for these things are very ambiguous in English)

The Concept
Imagine a quadrate. The length of one side is your AD, the length of another is your ASPD. The area of the quadrate is your DPS. The best ratio between circumference and area for a quadrate is found when all sides are equally long, i.e. when it's a square. So when you have a lot of AD, more ASPD gives the quadrate more area than AD does. Now, make sure you understand this metaphor, as I'll be using it a lot.

The two lengths of the sides of the quadrate is what I call a "multiplier" or an "axis" in mechanics lingo. And they're very important for understanding scaling and stat interacting. When there are two multipliers involved, it's a quadrate. Now, if I add Crit to it, what happens? It turns into a hexahedron, going from a 2-dimensional shape to a 3-dimensional shape. Now, the DPS is the volume. Once again, the longer the other sides, the better the shorter sides get.

The problem with this metaphor is that it gets fairly abstract once we add flat percentage damage increase like those from masteries or various abilities. Now it's the 4-dimensional equivalent to the 3D hexahedron and the 2D square. So it gets harder to visualize. I'll be sticking to 3D in the examples of this post though.

Concrete Examples
Lets look at the hexahedron with AD, ASPD and Crit as the length of its sides. How does stat extensions interact with this? Like Malphite's Shield scaling off a percentage of your health. I've been met with a lot of skepticism when I say it scales just as well off armor and magic resistance. It actually scales off EHP - effective health points. The defensive equivalent of DPS.

What about Infinity Edge? That multiplies Crit damage much like Malphite's shield multiplies HP. Well, the thing is, Malphite's shield always applies, it directly multiplies EHP. Crit damage is only applied to the Crit part of your DPS. So if you had 100% Crit, then yes, the Crit damage part of IE could be considered a full multiplier. Otherwise, it's worth 50% of your Crit Chance.

What about an item like Deathcap, that multiplies the main damage stat of AP which many considers the direct equivalent of AD. But it's not, because AD carries auto attack damage comes entirely off one polytope, where all the stats interact with eachother. AP only interacts with the scaling of abilities, not the base. So while a 10% increase in AD is a total auto attack DPS increase of 10%, an increase of 30% in AP is not 30% more ability damage dealt.

And that is the mechanical explanation of why the AD carry has the highest damage scaling. Almost all of the stats interact with each other in one polytope with many effective dimensions.

This is also why on hit effects, such as the one on BoRK, are less good than they seem. Especially when it first came out, BoRK's on hit damage was often compared directly to the AD of Bloodthirster. But that's not the case. BoRK does not scale with Crit or abilities, but AD does. It's the same with other on-hit effects and abilities.

There's a reason the timeless Zerker Greaves, IE, PD, LW build is so damn good on ADCs. It's two ASPD items, two AD items, two Crit items, an ARP item and a Crit Damage item.

Multiplier Theory: Stats stacking with themselves

Non-synergistic self-stacking

I stumbled upon two quotes from the Wiki.

"Liandry's Torment is probably best for ability power champions that can deal percent health magic damage like"

"Blade of the Ruined King is a great item choice for practically any autoattacking champion that does percent health damage like"

This is incorrect. Not that it cannot be true for Liandri's or BoRK to be good on one such champion, but that is never on merit of those two items dealing percent health damage as well as that champion having percent health damage. These two stats do not have increasing returns, they do not synergize with themselves. If anything it opens you more up to counterbuilding by focusing on resistances and offensive/utility items. 6 percent plus 4 percent is 10 percent. It's not like CDR or Penetration, which have increasing returns. It's like AD, ASPD, crit, AP. The first 10 points does exactly the same as the last 10 points.

Remember back in season 1 when everyone was really bad, building tons of HP on Cho'gath because "he already has a lot of HP"? I do. Maybe my friends were bad. Maybe I was bad and in the bronze of normals. But it annoyed me to no end. But I see some of the same a lot today.

It's not that having a lot of something makes more of that stat worth less - it just makes other stats that interact with it worth more.

Synergistic Self-stacking

There are however stats that synergize with themselves. That have increasing returns. One is Penetration. Flat penetration and % penetration works well together, flat penetration and flat reduction works good together as well. Penetration builds are particularly good on champions with great base damages and on champions that generally go for low-MR/Armor targets.

Y=Damage; X=ARP

Another is CDR. Up to the cap of 40% naturally.

y=percent more casts; x=CDR

Then there's two other stats with some technicalities.
One is movement speed, which has 3 different stats that scale multiplicatively with each other. Flat movement speed, percentage movement speed, and the other percentage movement speed from certain abilities. Not all movement speed increases from abilities fall in the "other" group though.
However, movement speed is affected by two softcaps - 20% reduction after 420 ms and 50% reduction after 490. So movement speed increases act a bit odd, gaining efficiency until 420ms, then losing a bit before regaining even more efficiency, then losing a lot at 490 before gaining more and more and more efficiency. Reference the vids of Hecarims and Rammuses that go halfway across the map in a few seconds.

Y=Post caps, X=pre caps

The other technicality is healing, through any means. Net change in health is damage taken subtracted by healing taken. Once healing taken rises above damage taken, you're immortal. The reason why I state this as a technical but not practical self synergy is because of burst damage and, in case of lifesteal and spellcasts, crowd control. It is, however, still a phenomenon that can be abused in some situation. Particularly low CC and low damage scenarios, like one-on-ones and small skirmishes. (And why Aatrox is busted in 3v3. Aatrox Spreadsheet)

Armor Penetration and the Journey to the Depths of Damage

When it comes to flat Armor Penetration, there are four categories of people: The ones that say ARP is better against low armor, the ones that say that armor doesn't matter, the ones that say that ARP is better against high armor, and the ones that have no clue. Most of the people among these groups, except the last, either don't know why or they're spouting incomplete nonsense that has nothing to do with mechanics.
I was part of the first group, and I'll show you why I was.

I took a random damage number and a random armor and made the following formula from the armor formula on the wiki.


I tried playing around a bit with damage and armor just to make sure they're not relevant to the proportions, and they weren't. It changes the location of the graph, but not the proportions.

Setting dmg and armor to constant and reducing the formula reveals how it's not linear, but exponential.

I'll show you a graph of how this works. I made a function with dmg and armor being constant, leaving only ARP as a variable, then simply took the function of ARP and subtracted the function of ARP-1, to see how much the flat increase in damage was compared to the last point of ARP. Below graph shows the result.

Damage on the Y axis, ARP on the X axis.

We clearly see that it's not linear at least, and that it's an increasing non-linear function.
This is why I figured that ARP was better on low armored targets. It was the same back in WoW (they used a similar armor mechanic), and mostly everyone discussing the subject thought it was true.

However, this is not the bottom level. This is not the relevant graph.
Damage is good. What is damage good for? Well, killing guys. So why is more damage good? Because it kills guys faster. The function I was looking for had something to do with time.

So I made a simple function that tells me how many seconds it takes to die to someone.

Like last time, we can derive the function with a little math, but I prefer showing this graphically. Again the dps, hp and armor values are irrelevant to the shape of the graph, I checked. So I just chose some random values.

Y axis is Seconds To Death, X axis is Armor Penetration.

See. See! It's linear. The effectiveness of Armor Penetration is not dependent on the target's armor. The first 10 will decrease the time it takes to kill your target by exactly the same amount of seconds as the last 10. Obviously it doesn't do anything past 0 armor.
Just to double check, I set DPS, hp and armor to constants and reduced the formula to 5-0.2*arp - a linear function.

Armor Penetration has neither increasing or decreasing returns.

So I was now very embarrassed about being in the first group for so long. I planted myself firmly in the second group with about as much conviction as when I was in the first. For a minute or two.

I looked at DPS. I thought, what if i attack once per second for 100 damage, and I add 10 AD, it increases DPS linearly. But back when I calculated ARP, its effect on DPS was exponential.
Which means all the other stats have Diminishing Returns, as they increase DPS linearly.

Seconds to Death on the Y axis, DPS on the X axis.

So, in comparison to its competition, ARP has increasing returns, meaning it's better against low armored targets. Putting me right back into the first category. I hope you will all join me.