Saturday 14 September 2013

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.

Friday 13 September 2013

Effective HP, Armor and Diminishing Returns

A lot of people don't believe me when I say that armor does not have diminishing returns, using various arguments and, regretfully, metaphors. I've also been asked to deal with the subject. So I hope this post can clear up the workings of mitigation and penetration.
(in this post I'll be talking about armor; MR acts exactly the same. Note that I'm ignoring negative armor)

First I'd like to introduce a term called Effective HP (EHP). It's the amount of premitigation damage (e.g. the damage the enemy has in his UI on his ability buttons) you can sustain before dying. It is derived by dividing your hp with your percent damage reduction.
Each point of armor always adds 1% of your HP to your EHP.
The formula on the wiki under damage reduction is a damage multiplier, so it's the number to multiply damage with to find out how much damage you deal. We can find our EHP by dividing our HP with this number.

HP/(100/(100+armor))=EHP

Lets take an example where you have 1000 hp and 0 armor. This will put you on 1000 EHP.
1000/(100/(100+0))=1000    
Now lets add a Thornmail
1000/(100/(100+100))=2000   (1/2 mitigation)
Putting you have 2000 EHP, an increase of 1000EHP.
Now, if armor has diminishing returns, the next Thornmail added will increase your EHP by less than 1000. Lets add another Thornmail.
1000/(100/(100+200))=3000 (1/3 mitigation)
It added exactly as much EHP as the first Thornmail.

Lets have a look at how a Last Whisper and Weapon Expertise (total 40% armor penetration) affects these numbers. We have the same starting point, 1000hp with 0 armor for 1000EHP. Lets add a Thornmail
1000/(100/(100+100*0.6))=1600
So the Thornmail effectively added 60 armor after penetration and thus adds 60% of your hp to EHP.
Lets add another thornmail
1000/(100/(100+200*0.6))=2200
 And again the second thornmail adds the same amount of EHP as the first.

Lets look at it from a less abstract angle. There's a champion attacking you. He has 100 AD and attacks once per second. You have 1000 HP and 0 armor. You'll live for 10 seconds.
Adding a thornmail, he'll deal half damage. You'll live for 20 seconds; 10 seconds longer.
Adding a second thornmail, he'll deal a third normal damage (your UI will show you 67% damage reduction), and you'll live for 30 seconds. Linear increase, no diminishing return.

Armor does not have diminishing returns.

Aatrox Itemization

I crunched a few builds for Aatrox. It was less work, as I've avoided most builds without lifesteal. It just seems like anyone would prefer building lifesteal on Aatrox, and I agree with that notion. Especially after looking at the numbers.

9/21/0 masteries are assumed. The sheet assumes 50% uptime on ult.

Runes
ASPD and AD are exactly even for BoRK, but ASPD pulls ahead with more hp
ASPD is better for BT
ASPD is better lategame
ASPD is better with dmgW at level 1
ASPD is better without ult
AD is better with healW at level 1
AD is better for burst
AD is better with ult on

According to the sheet, ARP is very good, especially later on. It means smaller heals and less damage taken early, but more healing later when you have a big LS item and less damage taken. But the E of Aatrox deals physical damage in my model, so it might be skewed too heavily.
ARP is even better lategame
ARP is better with BoRK
ARP is better against low armored targets
ARP is worse against heavily armored targets

For junglers I suggest taking the usual 3 ARP marks regardless of whether you go with AD or ASPD. I recommend against going full ARP for junglers, as so much of it is wasted against some of the jungle camps.

Personally I decided on ASPD runes for jungle Aatrox, but none of these 3 offensive runes are bad - they're all very close. It depends on where you want your strengths.


Slot 1
If you only have one damage item, at level 11, against an opponent with the usual average hp, average armor + flat seals + 5 from mastery and a Sunfire Cape, using dmgW, Last whisper is actually the most cost efficient choice. Even without having any other items. That was a bit of a surprise. IE comes in second, followed by BT and then BoRK. BT and IE are nearly equivalent at 9% and 11% cost effeciency lost compared to LW, BoRK losing 21%.

Slot 2
If you don't have a LW, now is the time. BoRK is best paired with it, second comes IE with 2% loss, lastly BT with 13% loss.

Slot 3
Of course, at this point, the good old IE+LW+PD is the best, followed by IE+LW+BoRK at 10% loss, then BT+LW+IE/PD both at 15% loss, followed by BT+BoRK+LW at 21% loss.


Lifesteal
Now that is amazing, the double LS build is actually very competitive damage wise. You gain a ton of extra resilience from having as much LS as that build offers. Let me show you the numbers.
It reads "HPS with dmgW" / "HPS with healW" / "HPS with healW under 50%"

-45/51/152    IE+LW+PD   
78/177/283   IE+LW+BoRK
32/133/246   IE+LW+BT
121/227/343 BT+BoRK+LW

Look at that amazing healing. You're literally unkillable 1v1 against low damage targets. Hell, you could probably take two. So I very strongly endorse builds with lifesteal. In fact, I very strongly advocate builds with BoRK specifically, as it shines much brighter than BT lategame since you're not getting much crit. BoRK is after all a premium bruiser item.


Suggested builds
If you really want your early midgame damage to shine, get BT->LW->BoRK. If you value BoRK active and/or midgame and lategame, get BoRK->LW->IE/BT. If you're only getting two offensive items, get BoRK->LW almost no matter what.

Trinity Force
As for Trinity Force, I'm afraid it's not good on Aatrox. It's at 64% loss for a first item, 39% as second item (with BoRK), and 28% as third item with BoRK and LW.

Hybrid Defensive Items
When it comes to the hybrid defensive items, both Atmas and Maw are decent. At Slot 3, Atmas loses you 27%(at level 18 with +400 hp), Maw loses you 34% (at 75 assumed average AD from Maw)


Slot 4
I haven't crunched for slot 4 with Aatrox, because with that LS, getting resistances and making sure you can't be bursted is simply too good overall. I heavily recommend against going glass cannon. I can however almost guarantee you that BT/BoRK+LW+IE+PD is the best build.

Spreadsheet Link
Aatrox Spreadsheet

Sunday 1 September 2013

Itemizing the new Master Yi

The new Yi, even after the nerfs, is a DPS monster. He scales so well internally despite the shitty base stats, and adding items just makes it better. As the only stat he doesn't gain innately is crit it turns out to be quite important for a high damage build.
Be warned, this is a long and detailed post. Look for interesting headlines or go to the bottom for the TLDR if you're not all that interested.

A 6 slot Yi at level 18 can kill a squishy a second with his sustained damage. He can kill a tank with Warmogs, Sunfire, Frozen Heart and Randuins in less than 4 seconds. At 4th offensive item, you'll deal over 1000 damage against a Sunfire Cape opponent if you choose any of the top 20 builds.

I've made a spreadsheet including more than 100 item combinations. I've mainly worked with it at 1 Q target with ult on and E up 60% of the time against an average opponent with armor seals, armor masteries and Sunfire Cape.  That's the default setting of the sheet, but it's easy to change. The link is at the bottom.

Firstly, Infinity Edge is as mandatory as it's always been for auto attackers, and Last Whisper is as cost effective for physical damage as we remember it. The Life Steal items are not very good for damage, it's actually a significant sacrifice to get that delicious Life Steal early.
Something that surprised me somewhat was the effectiveness of GhostBlade, even assuming only 50% uptime on the active. It was very often among the top builds. Lastly, the new Trinity Force is actually extremely good midgame when paired with Infinity Edge

I'll try to give an overview of the best build paths.

Path of Highest Damage

Slot 1
The first item should be IE or GB. GB can be gotten earlier, IE has good parts to build from and is more cost effective once finished.

Slot 2
If you went GB, you go IE now.
If you went IE, adding Tri is the highest damage midgame, but IE+PD transitions better into lategame and is quicker to assemble. Tri also has a fairly shitty build path. But the field is open with IE, it adds just what you need alongside your ulti.

Slot 3
Add a PD if you don't have one, add a LW if you do.

Slot 4
Add a LW if you don't have one, add another PD if you do.

It's worth noting that these builds are the top contenders even when your ult is down.

Ghostblade

The GB build is almost even with the IE PD build against a Sunfire worth of armor, but ahead against no armor opponents. The PD LW build pulls ahead of the GB build at Slot 3, and is almost even again at Slot 4. This is assuming 50% uptime on the active. It is worse off against armor stacking though, as it doesn't gain LW until 4th item. In my experience, it has less relevant uptime, but it does really wreck face when it's up in combination with E. I personally favour the PD build, but I think it's personal preference at this point.

Triforce

If you went IE with Tri in Slot 2 you gain 10% over the PD build, but you lose 22% at Slot 3 against PD+LW and 12% at Slot 4 against LW+double PD.

Lifesteal

If you want Lifesteal, BT is generally at higher damage output, but only significantly at Slot 1 and Slot 3. BoRK has the advantage of the active, and being better when ult is down. BT does more Q damage and damage to squishies. Note that unlike on most other champions, BoRK does not really give more healing than BT. BT is significantly better against towers.
If you want to rush lifesteal, go Lifesteal->IE->PD->LW. If you went BoRK and they stack several armor items, switching IE and LW is better.
If you're willing to wait for the most opportune time, getting Lifesteal Slot 3 or 4 is where you lose the least.
Hydra is only defensible if the AoE is going to be very very relevant
Personally, I prefer BoRK over BT, but both are entirely defensible.

With Lifesteal, it'll be about 30HPS at Slot 1, 50HPS at Slot 2, 120HPS at slot 3 and 140 at Slot 4.

You'll reach 200HPS at 3 items with IE+BT+BoRK, dealing 30% less damage than top build. At 4th item, adding a PD, you'll reach 250HPS and deal 25% less damage than top build.

Triforce and Lifesteal

With BT and Tri, you'll lose 9% at Slot 2. You'll lose 34% cost effectivness at Slot 3, and 30% at Slot 4 though.

Last Whisper

If the average armor of your relevant targets surpass 125ish, it's time for LW almost no matter what.

Defensive Items

If you want an offense/MR hybrid, go Hexdrinker/Maw over over Wits End. Its damage scales with E and can crit, and it's better against burst damage than WE. If you want offense/armor hyrbid, Atma's is actually not all that bad, even without HP items. If you've stacked a lot of HP (say, your two defensive items are Randuins/Spirit and Mallet), it's actually the strongest Slot 3 item against <150 armor opponents sorting by cost effectiveness.
At Slot 3, Maw loses you 11% compared to strongest build, and Atma's loses you 7% at base HP.

And just to note it, I do not endorse glass cannon Yi in almost any situation. It's simply too volatile and, well, bad against competent opponents. You don't have range, you simply can't make up for squishyness entirely with positioning.

Tenacity

I'm not much fan of Zephyr for tenacity. As a second item rather than PD, you lose 43% effectiveness. As a third item, you lose 18% effectiveness. The CDR is not worth much on Yi, neither is the movement speed. In my opinion, if you want tenacity, Merc Treads or Spirit is the way to go, and then just return to your regular build path.

Runes

I use attack speed runes. And I know that's weird, considering the massive attack speed boost you get from your ult. But you don't get to start with your ult. ASPD runes gives you the most stat per rune compared to gold, and it facilitates the IE rush. AD runes are also fine for last hitting in lane, you only lose a few percentage of effectiveness.
But, if we pretend there is no early game, crit damage runes are absolutely amazingly unfair. At 4 items, it gets you 150 more DPS. But it's a rather silly idea.

TLDR Best Builds

Best build for damage: IE->PD->LW->PD
Best build with Tri: IE->Tri->PD->LW
Best build with Lifesteal: IE->PD->BT->LW
Best build with rushed lifesteal: BT->IE->PD->LW
Best build with both Tri and Lifesteal: IE->Tri->BoRK->LW

Spreadsheet Link

Master Yi Spreadsheet