Crit Chance vs Crit Damage -- the MATH

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EasymodeX

Senior Member

03-05-2010

Quote:
So, in conclusion, I would say that, regardless of the math, critical damage and attack damage are actually the only two runes you should ever be socketing.
1. For raw damage, last hitting, etc, armor pen is better than attack damage. It serves the same purpose of consistent damage amplification, but does it much better per rune slot.

2. Early game (and early-mid), champs have a very low crit chance. To have a vaguely reliable "spike" damage, you would need a moderate composition of crit chance runes. When your crit chance is somewhat reliable, then your argument for increase spike via crit damage becomes meaningful.

3. Past early game, you lose the opportunity to "disengage and re-engage" at your leisure. This means that you have much less room for a failed spike. The result would lean the item strategy towards scaling damage via heavier static damage (base damage, ArPen), because the moderate chance to spike was already acquired in the early game (see #2).

4. Lastly, note that you can take the "extreme spike" strategy to an extreme, in the early game. If you utilize ArPen runes, and you bank on your low (~5-8%) chance to crit, you can crit for "mega" damage in the early game, but it's not something you can base a strategy around.

In summary: ArPen owns both for consistent damage and for rare spike damage (but is capped in effectiveness at the enemy armor level). Crit damage is only a little bit good for rare spike damage. Crit% is good for frequent spike damage. Base damage runes suck in all aspects.


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Phi Trigger

Senior Member

03-07-2010

Quote:
Originally Posted by EasymodeX View Post
1. For raw damage, last hitting, etc, armor pen is better than attack damage. It serves the same purpose of consistent damage amplification, but does it much better per rune slot.

2. Early game (and early-mid), champs have a very low crit chance. To have a vaguely reliable "spike" damage, you would need a moderate composition of crit chance runes. When your crit chance is somewhat reliable, then your argument for increase spike via crit damage becomes meaningful.

3. Past early game, you lose the opportunity to "disengage and re-engage" at your leisure. This means that you have much less room for a failed spike. The result would lean the item strategy towards scaling damage via heavier static damage (base damage, ArPen), because the moderate chance to spike was already acquired in the early game (see #2).

4. Lastly, note that you can take the "extreme spike" strategy to an extreme, in the early game. If you utilize ArPen runes, and you bank on your low (~5-8%) chance to crit, you can crit for "mega" damage in the early game, but it's not something you can base a strategy around.

In summary: ArPen owns both for consistent damage and for rare spike damage (but is capped in effectiveness at the enemy armor level). Crit damage is only a little bit good for rare spike damage. Crit% is good for frequent spike damage. Base damage runes suck in all aspects.

Curious - do creeps have armor? I never paid attention, so I perhaps made a faulty assumption that armor penetration would have no effect on farming. If this is incorrect, what armor do the various creeps have and is it consistent? Let's play out some math just to see it, I think it's a good analysis.

And yes, you need 15-20% crit chance first, for at least semi reliability.... I think I mentioned that?

I agree that late game disengaging isn't possible any longer, but I feel the midgame superiority will grant you enough of a monetary advantage to more than make up for the difference in damage output, whatever it is.


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seanzies

Senior Member

03-07-2010

Quote:
Originally Posted by Biri Biri View Post
nearly 25% starting CC is far better then anything else. It will win you a lot of early ganks.

armor pen > crit chance / crit dmg especially for ashe/shaco

im sure there are some exceptions like pirate perhaps.


edit: armor pen can also make certain physical abilities much better "volley being a huge example"

again pirate comes to mind with that example as a possible exception to that rule.


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Phi Trigger

Senior Member

03-08-2010

Quote:
Originally Posted by ceoofdarkness View Post
Think of the armor an opponent has as a static reduction. No matter which build you choose, he has the same amount of damage reduction from armor. Thus no matter what you choose he reduces the total by a static amount. With this information in hand, you should choose to increase whichever stat increases your dps by the greatest marginal amount.



You need to know the current levels of your statistics before you can answer this question. Use the same logic as in my above post to compare marginal damage.
(1-CC)+(CC*CD)>Atk(CD-1)?
Plug in the values of interest and you should receive the results of which you should increase. Remind yourself that this is on margin. If the left side is greater, it's better to stack Atk. Using just the approx values of rune levels it seems that:
(1-.2)+(.2*2.5)>25(2.5-1)?
1.3>37.5
It is far better to stack crit chance marginally before you reach the 100% cap. What I would advise you, from having played 300 games though, is that you need to use your judgment as to which stat to increase. Certain games you will be stomping the opposition and can wait to increase your stats in the most effective ways. Other times you will be having difficulty, the opponents are winning, and it may give you the best results to compromise and increase less efficient stats to try and "even" the game up.




All I can tell you is that I definitely notice 20% crit at the start of the game. It is empowering and you crit 1 more out of 5 times. As you begin accumulating more crit chance, you will notice less and less, but in the scheme of things, you will do 20% more damage assuming you only did physical attacks. With a larger sample size, your mean number of crits will average out to the expected value of crits (aka your crit chance). If your crit chance is 20% lower, then as your sample size increases, the probability that you will crit will be 1/5 lower than if you chose the crit chance rune page. This is the central limit theorem.
I am not sure how you derivated your marginal formula, but upon further investigation it is just incorrect. Rather than try to figure out how you went about finding it, I will simply post a counter example. According to your marginal formula, an attack damage of 10 would certainly mean critical hit % would be the preferred rune to use.

I'll start by using a very very basic formula that assumes no critical damage or armor penetration bonuses, and assumes 0 armor on the target.

2 * Attack * CC + Attack * ( 1 - CC ) = DAMAGE OUTPUT

Very basic, but I will explain each part. The portion to the left of the addition is the damage when a critical occurs, and to the right the damage when a critical does not occur, taken at their relative percentage chance of outcome then added to produce the total weighed value. Again, very basic, and I apologize if this is too simple for any readers, but I know not everyone on these boards is a mathematician.

If we take a generic red damage rune, we have +.32 damage. A generic red critical rune is +.93 CC. Let's first calculate what the damage output average is with 10 base attack, then calculate how much the damage output increases with the addition of each type of rune. Since the math here is discrete, there is no reason to differentiate as it's just more trouble than it's worth.

Base 10 attack:

2 * 10 * .2 + 10 * .8 = 12

+ .32 damage rune:

2 * ( Attack + .32 ) * CC + ( Attack + .32 ) ( 1 - CC ) = DAMAGE OUTPUT

=> 2 * Attack * CC + 2 * .32 * CC + Attack ( 1 - CC ) + .32 ( 1 - CC ) = DO
=> 12 + 2 * .32 * .2 + .32 * .8 = DO
=> 12 + .128 + .256 = 12.384 = DO

+ .93 CC rune:

2 * Attack * (.2 + .0093) + Attack ( 1 - ( .2 + .0093 ) ) = DO
=> 2 * Attack * .2 + 2 * Attack * .0093 + Attack ( 1 - .2 ) - Attack ( .0093) = DO
=> 12 + 2 * Attack * .0093 - Attack * .0093 = 12.093

So clearly attack damage has a much more dramatic impact on damage, with critical damage and armor neglected. The question is, at what point is attack damage high enough that a marginal increase of 1 rune, .32 damage, is surpassed by the marginal CC rune chance, .0093. I don't have time to solve this equation at the moment but just for a quick reference point:

100 attack damage = 120 DO

+.32 damage rune w/100 Attack = 120.384 DO
+.0093 CC w/100 Attack = 120.93 DO

So at 100 damage, CC rune is stronger.

50 AD = 60 DO

+.32 w/50 AD = 60.384 DO
+.0093 w/50 AD = 60.465 DO

So at 50 damage, CC rune is stronger but very close.

50 is probably the most relevant point, as most base damages are somewhere around there. So CC is more damage at that point, but as I have pointed out in previous posts, AD serves more than one purpose, as it's also used to more effectively farm. The negligible increase in DO is far outweighed by enhanced farming, in my estimation.


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EnderA

Senior Member

03-08-2010

Quote:
Originally Posted by Phi Trigger View Post
50 is probably the most relevant point, as most base damages are somewhere around there. So CC is more damage at that point, but as I have pointed out in previous posts, AD serves more than one purpose, as it's also used to more effectively farm. The negligible increase in DO is far outweighed by enhanced farming, in my estimation.
However, many people have no trouble getting last hits. In my opinion it's more important to be able to deal more damage to champions than to creeps.

As for creep stats, the melee creeps get approximately 2 armor every 3 minutes, the ranged creeps get 2 MR every 3 minutes, starting with 0 of each. They also get a minor amount of the other type. If I recall correctly, the siege creeps get a bit more armor than the melee creeps, and a small amount of MR.


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Phi Trigger

Senior Member

03-08-2010

Quote:
Originally Posted by EnderA View Post
However, many people have no trouble getting last hits. In my opinion it's more important to be able to deal more damage to champions than to creeps.

As for creep stats, the melee creeps get approximately 2 armor every 3 minutes, the ranged creeps get 2 MR every 3 minutes, starting with 0 of each. They also get a minor amount of the other type. If I recall correctly, the siege creeps get a bit more armor than the melee creeps, and a small amount of MR.
So if a player begins with the armor penetration mastery maxed, their damage is not going to see any benefit from armor penetration for creeping for quite some time - long enough that they could have a damage boosting item and not worry anymore.

It's possible to last hit with low damage, but when creep waves become larger and during times where creeps fire synchronously by variation, it will be difficult to land last hits with lower damage, and in fact in some instances it could well be impossible or highly improbable depending again on how damage intervals line up.


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EnderA

Senior Member

03-08-2010

Quote:
Originally Posted by Phi Trigger View Post
So if a player begins with the armor penetration mastery maxed, their damage is not going to see any benefit from armor penetration for creeping for quite some time - long enough that they could have a damage boosting item and not worry anymore.

It's possible to last hit with low damage, but when creep waves become larger and during times where creeps fire synchronously by variation, it will be difficult to land last hits with lower damage, and in fact in some instances it could well be impossible or highly improbable depending again on how damage intervals line up.
I've essentially only had trouble last-hitting as a mage. For example, Anivia is pretty bad, because her attack animation is horrible and her damage is horrible. In that case, the comparison isn't between attack damage and armor penetration, but rather between attack damage and magic penetration, which is an entirely different choice.

Ultimately, I think the massive bonus that armor penetration gives versus champions far outweighs the small generic damage bonus that damage runes give. For Marks, I would much rather do around 12% extra damage to champions than have approximately 3 extra damage, or have 1 damage per level.


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shanthia

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Junior Member

03-09-2010

In case you only wanted to run your red runes for crit damage/chance:


damage = base + CC*(1+CD)*base

d(damage)/(dcc) = base*(1+CD)
d(damage)/(dcd) = base*(CC)
d(damage) = base*(1+CD)*dcc
d(damage) = base*(CC)*dcd
(1+CD)*dcc = (CC)*dcd
(1+CD)*dcc/dcd = CC
CC = (1+CD)*dcc/dcd

dcc = 0.0093
dcd = 0.0223

CC = (1+0)*0.0093/0.0223 = 0.417 = 41.7%
(with IE CD = 0.5)
CC = (1+0.5)*0.0093/0.0223 = 0.626 = 62.6%

I'll neglect proof that above the threshold CD is more important than CC (but it should be logical).

You could change the values of CC, CD, dcc, and dcd above the find the values for any other rune slot or combination of already allotted runes.