Why do ADCs always go for damage over armor?

Considering bot lane, it is customary for the ADC to build ONLY damage or armor pen while the support builds his gold income and sightstone. The idea behind this is for the ADC to be able to dish out as much damage as possible while depending on his support to keep him alive. I get this - it sounds reasonable.

What if you beleive you can last-hit just fine, you want to be able to trade well against the enemy, perhaps even win a 2v2 or 1v1 engage? Obviously as ADC you also would like to stay alive during a full engage, making hp or armor/resist a wish. What it comes down to is prioritizing, i get that. I also get that everyone seem to prioritize building only damage. I just don't get WHY. Say you go back as adc with 875 gold, so does the other adc. One of them buys a pickaxe the other one buys a chainmail and health pots. Once back in the lane, the fighting starts to break out. Let's examine the following scenarios:

1. Only the two adcs are present (both supports still at shop or for some reason just recalled). 1v1
2. All four bot laners are back in lane. 2v2
3. Disengage and continue to farm and harass

The defense adc can pop a potion just before the engage (chain mail being cheaper than pickaxe), effectly increasing hp as well. Let's also assume both teams target the enemy adc. The pickaxe adc will dish out more damage (+25) per hit, only to get it reduced by the armor (+40).

The way i understand armor is, that each 1 point in armor is increasing (effectively) hp by 1%. Making the armor a +40% hp. I'm aware this only works for attack damage, meaning any magic or true damage will go though as normal.

I'm aware the calculations are getting pretty hard now, so i'm gonna try and simplify it by assuming it's only a 1v1 adc/adc match. Also i'm assuming all damage from both sides is going to be pure physical since after all we are comparing a physical boost (+25 attack damage) to a physical defense (+40 armor).

Let's also assume both adcs are caitlyn (honestly it's easier and also my lol assistant listed her as free this week, making it easy to one-click and read stats). Let's assume they are both level 5.

(Effective) hp for damage adc: (390hp + (4 lvl * 80hp)) * (1 + (17armor + (4lvl * 3.5armor)) / 100) = 710hp * 1.31 = 930hp (Effective) hp for armor adc: (390hp + (4 lvl * 80hp)) * (1 + (17armor + (4lvl * 3.5armor) + 40 armor) / 100) = 710hp * 1.71 = 1214hp

Giving the armor adc an extra (excluding the healing pots) 1214hp-930hp = 284hp (it's equivalent to 710hp*0.4armor - just wanted to make it clear that i was factoring it all in)

Let's finish it up. 284hp/25 damage = 11,36

Without calculating the healing pots, the damage adc will win the engange if it lasts beyond 11 (atleast 12) auto attacks. That is a lot of auto attacks. Consider a normal fight - it should last less than that if you factor in the skills again (QWER) and also if you calculate the supports. Speaking of which. If supports are pure AD based, the favor obviously still swings towards the defense adc. If the supports are pure AP (kind of unrealistic since auto attacks should still be ad) then the formular still holds. The factor damage dealt/received by the two adcs would still favor the armor untill the 12 adc auto attack. Which will be even less likely since both supports are also focusing the adcs - making the engage last fewer turns.

I'm fully aware all this will chance once you factor in ganks, bursts, etc. I just wanted to review a pretty common scenario.

My calculations (if correct) shows that the early game would favor the armor adc - but i do understand the importance of building the Infinity Edge first. I just think getting the early kill(s) - if the armor adc succeed in capitalize on his advantage - probably would put you in a situation where you probably would still get there first, and this time also wearing a chain mail?

Am i missing something or are adc just blinded by a lust for more damage?

BTW, i main support, which is why i probably don't get the adc mindset...

There are two main reasons why you don't build armor as an ADC.

1st: The enemy Support as a target

Often you don't want to focus the ADC but also the enemy supporter (Let's just assume that it isn't a Leona or Braum but a Sona or Janna instead). Remember that about 70% of the Botlane work is done by the supporter. If you manage to kill the supporter (which is a lot easier if you already have bonus AD) the enemy ADC usually won't be able to do anything by himself vs 2. And like this you can snowball really hard, which is basically the thing to do as ADC.

Now of course you don't always want to focus the supporter and if the supporter is something rather tanky then don't even think about focussing him. So another thing you should take into consideration:

2nd: Your role in Teamfights

You are the Attack Damage Carry. The last word is the most important one. You bring the highest damage to the teamfight. If you waste your money on armor early on just to gain a small advantage for a small amount of time during the game, you won't be able to do your job right. Your jobs are Kiting, Positioning and Killing Whatever gets in range. If you build armor you just won't be a thread to anyone since ADCs heavily scale with AD and not with armor.

3rd: Farming

Well this isn't as severe as the other two points but... Farming is also a lot easier with more AD early on. You are able to lasthit better under the turret and with more Attackspeed you won't miss creeps that easily.

One thing you're not taking into account is that the ADCs aren't going to stand still, trading blows until one is dead. The C part of ADC is "carry", which means that the team is going to rely on them later in the game to carry the rest of the team.

The ADC, therefore, must make it its goal to deal as much damage as possible in as short a time as possible. If he can't deal as much damage as the opposition's ADC, he's putting his whole team at a disadvantage in team fights.

As a result of this, as you've noticed, the ADC doesn't have much survivability. This is why ADCs are classically known as "selfish" characters. Since they are so important to the team fight dynamic, but also so fragile, they often have to save themselves at the expense of other teammates either by not joining the fight or running early. Positioning is crucial for everyone, but especially so for the ADC; one step out of line could result in a team-fight-altering death.

So while the math may say that overall fighting ability goes up with armor in some cases, that's not actually taking into account the full spectrum of what the ADC needs to do. Hit and run tactics and focusing down targets in the mid-to-late game win out strategically over being able to tank the opposing ADC while you trade blows.

All the answers above seem to be valid, but everyone is forgetting another important point: scaling! Champions in League of Legends have these nifty abilities that often do damage, which is increased by a percentage of a certain stat. For ADCs, this stat is often Bonus AD (as in, AD that only comes from items, runes, masteries and buffs).

Lets look at Graves, first. His Q (buckshot) does damage in a cone, and scales off Bonus AD by 80-144%, in the case of a pickaxe that's an extra 25(.8) - 25(1.44) -> 20 - 36 damage every 10ish seconds. You probably won't get a second chance to back before level 6, so let's tack on his ultimate too: 120-150% bonus AD -> 30 - 37.5 damage. Put these in a combo, and you've got an extra 50 - 73.5 area of effect(!) damage. Hit 2 targets with each (or just one), which is not unreasonable, and all of a sudden you've gotten a lot out damage out of your pickaxe. This damage does get reduced by armor, but your opponent is only auto attacking you for ~50 damage at a time, while you're hitting him for ~40. The auto attack speed steroid that graves gets isn't even factored in here, and that gives even more benefit when you've purchased AD.

For a more common ADC, with no AD scaling, let's look at Tristana: She has a steroid called rapid fire, that at level 1 gives +30% attack speed for 7 seconds. With her q on, she'll get off about 5.9 attacks in the 7 seconds. Without it, she'll get off 4.6 attacks. If we use the same calculations of they attack for 50, you attack for 40, you do 5.9 * 40 = 240, and they do 4.6 * 50 = 230. Chain vest Tristana lost out on 10 damage! Keep in mind, though, that these "attacks per 7 seconds" calculations are done with BASE attack speed, and by the time you have a chain vest / pickaxe, the difference in damage dealt during the trade will continue to swing in the favor of the pickaxing ADC, as they'll have more higher damage attacks.

Armor doesn't work with any of the other stats that ADCs buy: life steal, crit chance (crit damage, in the case of IE), attack speed, Attack Damage works beneficially with all 3!

If the math hasn't convinced you, I offer you this: Do you really want a chain vest, pickaxe and bf sword while the other AD Carry has an Infinity edge? They're melting enemies with their 250% damage crits, while you're... dying slower and doing significantly less damage.