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TBC vs. LW: Sims keep putting TBC ahead

Creator: Pranzatelli June 8, 2012 1:30pm
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Pranzatelli
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My friend and I made a program on MatLab for DPS under ideal conditions. We've been using it to run through options for item sets and compare them.
One thing that struck me is LW almost NEVER wins. Compared to flat armor penetration (well, TBC is reduction, but we'll pretend the target has at least 45 armor), percentage penetration
a) stacks poorly with itself (multiplicatively) in comparison to flat pen/red, which becomes more effective as the value of pen/red gets higher and
b) is on an item that gives less stat (and in only one kind, AD, instead of two kinds that multiply against each other, AD and AS) outside of penetration than TBC or, oh God, Youmuu's (now THERE'S an undervalued item).
That's what I see. On every realistic dummy that isn't a Taric with Frozen Heart and a Thornmail (jokes, though my friend plays this/Malphite/Rammus with a similar build for the lols), TBC seems to come ahead.

(PM me if you want to see the formula.)

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I would really love to see the math you did on this. I have an idea as to what you did, and if I'm right I know a few reasons why it's not ideal, but I really can't comment without actually seeing your work.
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Here's the formula, simplified.
(AD)*(bAS*(1+AS))*(1+(CC*CD))*(100/(100+((%R*Armor)-RP)*(%P)))
AD is attack damage, base and from masteries/items/runes. Duh. I sometimes put a modifier in here (eg., Draven with Spinning Blades always up gets this value multiplied by 1.85). Most carries get a bonus 3 in here from masteries, too.
bAS is a character's base attacks/second, which comes as a decimal. For example, Maokai's bAS value is .694. AS is a character's percent attack speed, which it gets from leveling up (*18) as well as from items. This is also represented as a decimal, so a Phantom Dancer adds .55 to this value.
CC is a character's crit chance represented as a decimal. IEdge adds .25 to this value.
CD is bonus crit damage. This starts at 1 (because crits are 100% bonus damage) and goes up to 1.1 with masteries. Iedge can bring this to 1.6. Pretty straightforward.
%R is percent reduction. Applied first to armor.
RP is flat reduction and penetration. Put both as one value now that TBC is the only non-champion source of reduction (so it's unlikely that anything's going below zero).
%P is percent penetration. With masteries this is .9; with masteries and a last whisper this is .54.

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To be fair, at 112.5 enemy armor, LW > TBC. Anything less and TBC> LW.
This is assuming the champs you're using have the exact same AD, AS and Crit chance.

If we don't assume these things, I really don't think the argument holds water here.
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^THIS MATH is, in my opinion, faulty.
You have ignored your masteries as well as your champion's interaction with armor. These provide penetrations and reductions in differing amounts that will affect this magical value of 112.5. Many carries run with the 6+10% armor penetration from their masteries.
In addition to being done from a point of view ignorant of all other sources of penetration/reduction, your math also ignores the rest of the item. You are comparing the items as if they are the same price with no other DPS values on them. THIS is the importance of the rest of the equation. A character with TBC isn't going to get the same value from his item per gold spent as someone who picks up a LW. If the purpose of buying items which interact with armor is to increase damage, then why do we suddenly ignore damage and attack speed? The argument is: which one is the better buy against a number of target dummies with different values of armor, taking in all forms of damage increase? I mean, at the end of day, it's whichever item makes the biggest red number come off the target.

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I know that most AD carries are running 21/X/X masteries but, for any equation to be accurate, we have to eliminate variables.

Carry A on blue team might be running 21/X/X masteries while taking both Havoc and Weapon Expertise . AD carry on the purple team however is taking 21/X/X masteries with Vampirism instead of Havoc .

We're also assuming that all AD carries are running the same runes, right? Urgot Might be running more armor pen to make the best use of Noxian Corrosive Charge while Graves would be using more AD to get better usage from all his skills.

Also, I completely forgot about the 10% armor pen bringing the break even point of TBC v. LW a bit higher.

Like I said, I'm just trying to make thing equal.

If a lot of your team deals Phys damage, TBC is probably better because it increases everyone's output instead of just your own.
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Uhm. For the eight millionth time, it's not this simple at all.

If you're not talking about runes when you talk about these items, you're not thinking deeply enough.

If you aren't talking about the liklihood of the enemy team building armor (AKA are they a tanky team or are they likely to be squishy) then you aren't thinking hard enough.

Example; Ranked game starts.

Your enemy's comp is a double AP comp, where not much armor can be built to be effective. Zhonya's might be seen. Jungler might get some armor. AD carry could get wriggles and support can get some armor.

No full tanks are to be seen, MAYBE one in the jungle like a nautilus.

Bust out the 25 ArP page and buy a TBC. Perfect, we're gonna reduce 45 and pen 25, bringing those already shallow armor values down 70 points. Keep in mind most of these champs will have between 80-150 armor at the most, and reducing that by 70 will bring reduction values down to 9% - 44%, not even 50% at the highest value.

Yeah guys. That is how you use TBC. In conjunction with flat ArP runes vrs squishy targets. It's important to plan ahead what method you'll most likely be using instead of deciding on the fly.

If you are using AD runes, then you -should- use LW almost every time.
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Here's the formula, simplified.
(AD)*(bAS*(1+AS))*(1+(CC*CD))*(100/(100+((%R*Armor)-RP)*(%P)))
AD is attack damage, base and from masteries/items/runes. Duh. I sometimes put a modifier in here (eg., Draven with Spinning Blades always up gets this value multiplied by 1.85). Most carries get a bonus 3 in here from masteries, too.
bAS is a character's base attacks/second, which comes as a decimal. For example, Maokai's bAS value is .694. AS is a character's percent attack speed, which it gets from leveling up (*18) as well as from items. This is also represented as a decimal, so a Phantom Dancer adds .55 to this value.
CC is a character's crit chance represented as a decimal. IEdge adds .25 to this value.
CD is bonus crit damage. This starts at 1 (because crits are 100% bonus damage) and goes up to 1.1 with masteries. Iedge can bring this to 1.6. Pretty straightforward.
%R is percent reduction. Applied first to armor.
RP is flat reduction and penetration. Put both as one value now that TBC is the only non-champion source of reduction (so it's unlikely that anything's going below zero).
%P is percent penetration. With masteries this is .9; with masteries and a last whisper this is .54.

About what I expected. While this does give the average DPS (over an infinitely long period of time) on the two, it requires you to ignore one very key difference between LW and TBC. LW gives its full effect on every hit, while TBC has no effect on the first hit, -15 armor on the second, -30 on the third, and its full effect on the fourth and beyond (if you keep hitting). This isn't that much of an issue against tanks, who might be able to take a dozen or more hits, but could be very important against squishier characters who might die in three or fewer hits. This will also be very important in teamfight settings, where your team is going to be focusing down targets, since all the other damage coming in is going to lower the number of hits you'll get on any target before they die.

The way I've been wanting to model AD carry DPS, in order to compare these things, is to use a ceiling function to represent total damage done over time at a given armor value. Using the notation ]x,a[ as the ceiling function of x with steps of a, my function over time t for LW would then be
LW: ]AD*(1+(CC*CD)*(100/(100+((%R*Armor)-RP)*(%P)))*t,1/(bAS*(1+AS))[
Because TBC has differing values for the first three hits, but then remains constant from the fourth hit on, I have to use a piecewise function ending in the ceiling function. The function over time t for TBC is then
TBC: { ]AD*(1+(CC*CD)*(100/(100+((%R*Armor)-RP)*(%P)))*t,1/(bAS*(1+AS))[ for 0<=t<1/(bAS*(1+AS))
{ ]AD*(1+(CC*CD)*(100/(100+((%R*Armor)-RP-15)*(%P)))*t,1/(bAS*(1+AS))[ for 1/(bAS*(1+AS))<=t<2/(bAS*(1+AS))
{ ]AD*(1+(CC*CD)*(100/(100+((%R*Armor)-RP-30)*(%P)))*t,1/(bAS*(1+AS))[ for 2/(bAS*(1+AS))<=t<3/(bAS*(1+AS))
{ ]AD*(1+(CC*CD)*(100/(100+((%R*Armor)-RP-45)*(%P)))*t,1/(bAS*(1+AS))[ for 3/(bAS*(1+AS))<=t
For the TBC function, you don't want to include TBCs actual passive in the RP number, because it doesn't remain the same between the early hits. It's also worth noting that the time t is measured in seconds for these calculations. These functions allow you to much more easily see when certain damage thresholds on reached on certain armor values, which lets you know when, in seconds, targets die (in the absence of healing and such). They also clearly show the millisecond-by-millisecond differences in damage between the two items on a given build, in ways that linear functions cannot, which is extremely relevant when on of the items provides AS and the other does not.

Of course, there are still factors that the above model doesn't handle well, but I hope you'll find it useful. The game is never as simple as our models would like it to be.

EDIT: Fixed a couple of things in the model.

EDIT 2: For anyone with Mathematica, here's working versions of both functions. CrD is bonus only, and you'll want to use /. Armor -> to set armor to make the functions plotable.

LW[AD_, bAS_, AS_, CrC_, CrD_, FArP_, FArR_, PArP_, PArR_] := Ceiling[AD*(1 + (CrC*(1 + CrD)))*(100/(100 + (.6) (1 - PArP) ((1 - PArR) Armor - FArP - FArR)))*t, AD*(1 + (CrC*(1 + CrD)))*(100/(100 + (.6) (1 - PArP) ((1 - PArR) Armor - FArP - FArR)))/(bAS*(1 + AS))]

TBC[AD_, bAS_, AS_, CrC_, CrD_, FArP_, FArR_, PArP_, PArR_] :=
Piecewise[{{Ceiling[(100*AD*(1 + CrC*(1 + CrD))*t)/(100 + (1 - PArP)*(-FArP - FArR + Armor*(1 - PArR))),
(100*AD*(1 + CrC*(1 + CrD)))/((1 + AS)*bAS*(100 + (1 - PArP)*(-FArP - FArR + Armor*(1 - PArR))))], Inequality[0, LessEqual, t, Less, 1/((1 + AS)*bAS)]},
{Ceiling[(100*AD*(1 + CrC*(1 + CrD))*t)/(100 + (1 - PArP)*(-15 - FArP - FArR + Armor*(1 - PArR))),
(100*AD*(1 + CrC*(1 + CrD)))/((1 + AS)*bAS*(100 + (1 - PArP)*(-15 - FArP - FArR + Armor*(1 - PArR))))], Inequality[1/((1 + AS)*bAS), LessEqual, t, Less,
2/((1 + AS)*bAS)]}, {Ceiling[(100*AD*(1 + CrC*(1 + CrD))*t)/(100 + (1 - PArP)*(-30 - FArP - FArR + Armor*(1 - PArR))),
(100*AD*(1 + CrC*(1 + CrD)))/((1 + AS)*bAS*(100 + (1 - PArP)*(-30 - FArP - FArR + Armor*(1 - PArR))))], Inequality[2/((1 + AS)*bAS), LessEqual, t, Less,
3/((1 + AS)*bAS)]}, {Ceiling[(100*AD*(1 + CrC*(1 + CrD))*t)/(100 + (1 - PArP)*(-45 - FArP - FArR + Armor*(1 - PArR))),
(100*AD*(1 + CrC*(1 + CrD)))/((1 + AS)*bAS*(100 + (1 - PArP)*(-45 - FArP - FArR + Armor*(1 - PArR))))], 3/((1 + AS)*bAS) < t}}, 0]
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I don't like referencing guides on this forum, but ninjaginge's The Rat Reevaluated (Extensive PvP Twitch) [updates inc!] has a graph of Damage Per second of The Black Cleaver vs Last Whisper and The Bloodthirster.

RED: The Bloodthirster
BLUE: Last Whisper
Purple: The Black Cleaver
The DPS of black cleaver is better than DPS of last whisper until your given target has >650 armor
N.B. I cited this guide because i felt as if the proper citation was in order
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^ Garbage.
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