Armor penetration, flat attack damage, or attack damage per level; a case study.
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I will give a case study to show which runes are optimal for Tryndamere, armor penetration, flat attack damage, or attack damage per level, by comparing their qualities and the various ways in which they scale. Below are Tryndamere's innate statistics, from lolwiki
Hp 461 +98 Hp5 7.9 +0.9 Ad 56 +3.2 As 0.644 +2.9% Arm 14.9 +3.1 Mr 30 +1.25 Ms 320 Rng 125 The left hand column are his level 0 values, and the right hand column are his increases for each level. Now consider two possible runepages, one emphasizing armor penetration, the other attack damage. The mastery pages will be identical. RUNE PAGES Armor penetration rune page: +24 magic resist at level 18 +13 armor +25 armor penetration Attack damage rune page: +24 magic resist at level 18 +13 armor +15 attack damage MASTERY PAGE Summoner's wrath Executioner Demolitionist Tough skin (two ranks) +3 attack damage +0.5 attack damage per level +10% critical strike damage +4% attack speed +10% armor penetration +6 armor pentration +6 armor +6 magic resist The innate statistics for flat armor penetration Tryndamere (fapT), and flat attack damage Tryndamere (fadT) are below. I have excluded armor, magic resist, health, range, movespeed, hp5 and attack speed because they are identical. I have included flat armor penetration and percentage armor penetration as new base statistics. fapT Ad 59 +3.7 fap 31 pap 10% fadT Ad 74 +3.7 fap 6 pap 10% The equation for damage dealt with a single attack against a target with more armor than you posses in armor penetration is below. fapT DAMAGE DEALT = ATTACK DAMAGE * 100 / (100 + ARMOR  31) fadT DAMAGE DEALT = ATTACK DAMAGE * 100 / (100 + ARMOR) At the bottom of the thread is a 3d graph of the following equation which you may magnify in size by right clicking it and opening it in a new tab. z = x * 100 / (100 + y) with the following boundaries, and their reasons: x is greater than or equal to 59, the minimum attack damage for fapT x is less than or equal to 108, the maximum attack damage for fadT, full bloodlust passive with an attack damage item. y is greater than or equal to 31, the minimum armor fapT for which the equation is valid. y is less than or equal to 100, arbitrary value that i chose. z is greater than or equal to 29.5, computed. z is less than or equal to 103, computed. [ATTACH]attachmentid=494879&stc=1&d=1343250762[\ATTACH] The best case scenario for fapT at level 1 is for when all of the opponents armor is penetrated, and has an attack damage item and full bloodlust passive is the point (89, 0, 89) Therefore fapT will do no more than 89 damage on each normal auto attack at level 1. The same scenario for fadT at level 1 against the same opponent, with 31 armor, yields the point (108, 22.5, 88.16). 31 armor was penetrated by 6 armor penetration, then of the remaining 25, 10% of that was penetrated to produce effectively 22.5 armor. Therefore against the same opponent, fadT will do no more than 88.16 damage on each normal auto attack at level 1. This shows that fadT does less damage than fapT can do in its ideal case, to an opponent with 31 armor at level 1. Consider the same case but without maximum passive from bloodlust. fapT (74, 0, 74) fadT (93, 22.5, 75.91) This shows that a full health fadT does more damage than a full health fapT to an opponent with 31 armor at level 1. The point at which fapT does equal amount of damage to the same opponent that fadT does, can be figured by solving the following equation, where B stands for the passive attack damage boost from blood fury passive: 100*(93 + B ) / (100 + 22.5) = 100*(74 + B) / (100) .8163*(93+B) = 74 + B 75.92 + .8163*B = 74 + B 1.92 = .1837*B B = 10.45 The passive on bloodlust will only grant integer amounts of attack damage, so 11 points of bonus attack damage must be gotten before fapT does more damage than fadT at level 1 against an opponent with 31 armor. At least 11 points of attack damage will be granted to tryndamere from his bloodlust passive if he has less than 26% of his health remaining. We can make the following conclusion, for tryndamere hitting an opponent at level 1 who has 31 armor: If tryndamere is above 26% of his max hp at level 1, fadT does more damage. if tryndamere is below 26% of his max hp at level 1, fapT does more damage the difference in damage never exceeds 2 points. If the damage difference never exceeds 2 points, then why bother analyzing it? At levels above level 1, tryndamere will continue to earn attack damage from his items and from the flat bonus on bloodlust, but the passive scaling attack damage on bloodlust will be dwindled by those. This shows that even at level 1, fadT and fapT do comparable damage regardless of the bloodlust, and is a hint that we can ignore the scaling passive attack damage bonus, and simply account for the flat attack damage bonus in further analysis. Simply put, both fapT and fadT scale comparably with loss of health, so we can exclude that from further study. But how do they scale with levels? Recall the base statistics: fapT Ad 59 +3.7 fap 31 pap 10% fadT Ad 74 +3.7 fap 6 pap 10% We can use the previous equation for comparing fapT and fadT over how they scale with loss of health, but adapt it for use with levels. What we are really interested in is how they scale with increase in attack damage, which is granted at each level in the same amount. y = bonus attack damage for fadT granted by levels x = bonus attack damage for fapT granted by levels 100*(74 + y) / (100 + 22.5) = 100*(59 + x) / (100) .8163*(74 + y) = 59+x 60.41+.8163*y = 59+x 1.41 + .8163*y = x Now for example, if fadT had y = 66.6 for being level 18, then fapT would only need x = 1.41 + .8163*y = 55.78 attack damage from levels, implying that fapT would only need to be level 55.78 / 3.7 ~ 15 to do the same damage. Therefore without items or skilling bloodlust, a level 18 fadT does the same damage that a level 15 fapT does. So here we can see that fapT scales better with levels than fadT does. But does fapT scale better with levels than any other choice of runes? Consider attack damage per level runes, for attack damage per level Tryndamere (adlT). Attack damage per level rune page: +24 magic resist at level 18 +13 armor +35 attack damage at level 18 (+1.95 attack damage per level) adlT Ad 74 +5.65 fap 6 pap 10% now we use the equation from earlier 1.41 + .8163*y = x if fapT is level 18, then x = 66.6 = bonus damage from levels for fapT, and y = 79.86 = the required bonus damage from levels for adlT. Therefore adlT must be level 14 ~ 79.86 / 5.65 in order to do the same damage to a target with 31 armor that a level 18 fapT does. But how do they scale against armor? I will use the damage equations from earlier: x = bonus damage for fadT and fapT y = armor of opponent which is not to be less than 31 z = damage dealt 100*(74 + x ) / (100 + (y6)*.9) = z 100*(59 + x ) / (100 + (y31)*.9) = z In other words, we want to know the armor values for which the damage is the same for both fadT and fapT. Any value below that and fapT will provide more damage, and any value above that and fadT will provide more damage. We will always assume that the opponent has 31 armor or more. 100*(74 + x ) / (100 + (y6)*.9) = 100*(59 + x ) / (100 + (y31)*.9) (74 + x) * (100 + (y  31) * .9 ) = (59 + x) * (100 + (y  6) * .9 ) 7400 + 100*x + (74+x)*(y31)*.9 = 5900 + 100*x + (59+x)*(y6)*.9 1500 = (59+x)*(y6)*.9  (74+x)*(y31)*.9 1667 = 59*y + x*y  354  6*x  74*y  x*y + 2294 + 31*x 0 = 273 15*y +25*x y = 18.2 +1.67*x So for example, if x = bonus damage for fadT and fapT, including that gained by levels, items, and the blood fury passive = 100, then the required armor value for which fadT and fapT will do identical damage is y = 18.2 +1.67*100 = 185.2. In other words, fadT with 174 attack damage will do the same damage with his auto attacks to an opponent with 185 armor that fapT with 159 attack damage will do. Performing the same analysis but including last whisper gives us the following equation and result, where i have included the attack damage bonus of last whisper. The .55 multiplier comes from the 40% armor penetration of last whisper, and the 10% penetration of the armor left after that, .6 *.9 = .55 100*(114 + x ) / (100 + (y6)*.55) = 100*(99 + x ) / (100 + (y31)*.55) (114 + x) * (100 + (y  31)*.55) = (99 + x) * (100 + (y  6)*.55) 11400 + 100*x + (114+x)*(y31)*.55 = 9900 + 100*x + (99+x)*(y6)*.55 2727.3 = (99+x)*(y6)  (114+x)*(y31) 2727.3 = 99y  594 6x 114y + 31x + 3534 0 = 212.7 15*y +25*x y = 14.18 + 1.67*x So for example, if the bonus damage besides last whisper for fadT and fapT, including blood fury passive, items, and whatever is gained by levels = 60, then the required armor in order for fadT and fapT to have identical damage is y = 14.18 + 1.67*60 = 114.38. In other words, a fadT with last whisper and 60 ad from all sources except masteries and runes and the last whisper itself, will do the same amount of damage to an opponent with 114 armor that a fapT would do, given that fapT also has 60 ad from all sources except masteries and runes and the last whisper. But look what happens when you consider a Tryndamere with a very large amount of attack damage, say 250 on top of that given by runes and masteries and the last whisper. How much armor would be required in order for fapT and fadT to be equal? 14.18 + 1.67*250 = 431.68 The opponent would need more than a staggering 432 armor in order for the flat attack damage runes to beat out the flat armor penetration runes, and thats because armor penetration scales with attack damage very well. The more attack damage you have, the more flat damage increase you would get by penetrating armor, so the armor required to make a measily 15 flat attack damage become better than 25 armor penetration, when you already have a large amount of attack damage, is staggering, even with the last whisper. In conclusion
adlT attack damage per level Tryndamere fadT flat attack damage Tryndamere fapT flat armor penetration Tryndamere 
lots of fap. make a tl;dr with less acronyms that require you to read the whole post to understand. maybe when i take my 300 mg of vyvanse tomorrow i'll be able to focus on this freakin textbook.
in short, flat arpen is better up until +400 armor? 
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We want to know the armor values for which the damage is the same for both fadT and fapT. Any value below that and fapT will provide more damage, and any value above that and fadT will provide more damage. y = armor of opponent and is greater than or equal to 31 x = bonus damage (including only damage from items, and skills) y = 18.2 +1.67*x So if Tryndamere will have 100 bonus damage from items and skills, which build is better for him, fadT or fapT? It depends on the armor of the opponent he is attacking. To determine the critical point at which fadT becomes better than fapT use the equation. y = 18.2 + 1.67*x y = 18.2 + 1.67*100 = 185.2 armor .................................................. .................................................. .............................. There is a slightly different equation if Tryndamere will have a last whisper in his build. In this equation, include damage from all skills and items, excluding last whisper. y = 14.18 + 1.67*x The equation answers the same type of question though. .................................................. .................................................. .............................. The point is that as Tryndamere builds more and more damage, the value of armor penetration runes skyrockets past flat attack damage runes, even if you have a last whisper, except against targets with ridiculous amounts of armor. The more damage, the more armor required in order for flat attack damage runes to be worth more than flat penetration runes. At 250 damage bonus damage from items besides the last whisper, an opponent would need 432 armor in order for flat damage runes to be superior to armor penetration runes. 
Label your axes and show us at what point armor pen becomes better than flat AD. Generally flat AD gives enough of an advantage early game to snowball yourself into a better position than if you took armor pen runes.

Perhaps totally irrelevant, but doesn't flat AD do more damage to minions? So, last hitting more efficiently may be more important...

Math is hard.

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Wow. Hope you didn't use too much time on that math, since I could have told you the same with some quick napkin math and a dollop of common sense.
Runes aren't rocket science, and the math between ad/arpen has been shown time and time again. It's all about whether or not you feel you need the early game advantage, or you feel confident enough to aim for lategame. 
way to much wall of text for data that already has been done over and over in mutiple threads even with excel spreed sheets posted to do it ur self.
good effort for working out some stuff ur self :) hope u have Armour pen and Armour pen % and flat pen and flat pen % ordered correctly in ur data set top down order of application: flat % flat pen pen % 
I've always preferred flat AD as it gives better last hitting ability, and that can easily turn into quite a bit of gold. Considering that early on Trynd tends to run in, last hit, and E out, you want to make sure that you actually get that last hit off instead of having to wait around for a second hit. Any harass you put on the enemy must come with a W or else you'll lose the trade (except for ganks), so the ARP is about even with the AD until you start getting items.
The core of Trynd, though, is usually Berserker Greaves, Vamp Scepter, and Zeal, none of which give any AD and thus do not have any effect on the early ARP vs AD debate. By the time the enemies start getting armor items, you should have enough time to grab LW. Against some lane matchups, I actually get LW immediately after grabbing my core; it wrecks Glacial Shroud builds. 
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