Update on Ezreal; Brutalizer

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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.