Crit % vs. Crit damage runes : the math

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EnderTheXenocide

Junior Member

01-26-2010

I don't know if someone else has already done this or not (if so, i didn't show up easily through the search function). I may not be the best player our there, but one thing I do know is math and statistics so I wanted to answer the question: so far as pure average dps goes, is it better to stack all Crit chance runes (~20% at tier 3, I believe), or crit damage runes (~ 50% for tier 3)?

First, let's look at how much damage a single attack does. We're not comparing crit to attack speed or armor or anything like that, so I'm going to ignore all that for now.

Since I'm a math inclined person, I'm going to assign variables to represent some important stats:

C = crit chance
D = crit damage boost (note, this means a boost ABOVE the standard 2 times base damage, for a normal, non-boosted crit).
B = base damage.

In this case, the average damage you do per attack is given by adding your normal base damage times the chance of NOT critting, plus the crit boosted damage times your chance of critting.
OR: average damage per attack = B * (1-C) + B*C*(2+D)

Since both the normal attacks, and critical attacks are dependent on your base damage, I'm going to throw that out for now too, to compare the different crit statistics.

So the average damager per attack normalized by the base damage (let's call this X) is given by: X = (1-C) + C*(2+D) = 1 - C + 2*C + C*D = 1 + C + C*D


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Now you may be wondering, how the heck is this useful? I now have a multivariable function that shows how my damage (averaged, normalized by base damage) depends on crit chance and crit damage boost, but how does that help me?

Well, what we really want to know is the RATE at which this function increases. For some small increase (say due to a rune, or a mastery, etc.) is it more effective to increase C, or D? Well to answer this question, we need the rate of change of this function. That's right, we gotta do some (basic) calculus now.

the rate of change (with respect to C) is:
dX/dC = 1+D
the rate of change (w.r.t. D) is:
dX/dD = C

So the more crit damage boost you have, the higher D is, the more important it is for you to raise the CHANCE of crit. Conversely, the higher a chance of crit you have, the higher C is, the more important it is to raise your crit DAMAGE boost.


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But the real question is, so far as runes go anyway, which is better? And we have to remember that all is not equal. We can either raise C by 20% or D by 50% if we totally stack our tier 3 runes one way or the other.

That is, when is the damage increase dX the same in both cases:
dX (from chance) = (1+D) * dC = dX (from damage) = C * dD (where dC and dD are the CHANGE caused by runes)

Lets look at a few examples. In all cases, dC = 20% = 0.2 and dD = 50% = 0.5

Presuming we start with no help whatsoever, C = 0 and D = 0 :
dX (from chance runes) = (1+0)*.2 = .2
dX (from damage runes) = 0*0.5 = 0

Ok this was obvious. If your chance to crit is 0%, then no amount of crit damage boost is going to help. At all. Ever.

But lets say you got a few useful items / masteries, and your crit chance is 20% already, but still no damage boost yet:
dX (chance) = (1+0)*.2 = .2
dX (damage) = .2*.5 = .1

So even if you already have 20% crit. It is still more effective to get MORE CRIT % than a crit damage boost.


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So where is the tipping point? At some point, crit damage boost is going to be better than a higher crit chance. When does this happen? We'll lets see. Let's find the very point where the damages from increasing chance, or boost damage, are equal to each other. This will be the tipping point.

Case 1: no crit damage boost
dX = (1+0) * .2 = C*.5 --> C=.4 = 40%
So if you have no masteries/runes/items to enhance your crit boost, then you need a 40% crit chance before crit damage boost runes (50%) are better than extra crit chance runes (20%).

Case 2: 10% crit damage boost from masteries
dX = (1+.1)*.2 = C*.5 --> C = 44%
If you put points into crit damage boost in masteries, you need to have > 44% crit chance before crit damage boost runes are better.


But let's face it. If you're even considering getting crit damage boost runes, you're probably playing a champion that will focus on getting infinity's edge ASAP, right? And you're going to put those points into the crit boost damage mastery. So what about this scenario?

Case 3: 10% crit damage from masteries and 50% boost from Infinity's Edge
dX = (1+.6)*.2 = C*.5 --> C = 64%
That's right, if you have infinity's edge, then only when you have a whopping 64% crit chance do crit damage boost runes (+50%) become MORE effective than extra crit chance runes (+20%).


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Again, I'm a math expert, not a LoL expert. So you tell me. How late in the game / how often, do you actually get up to 64% crit chance (NOT counting your runes, because the runes are, after all, the variables in question here). Assuming you never hit a 44% crit chance BEFORE getting that infinity's edge, up until you reach that threshold of 64% crit chance (w/o runes, with IE).... you've just DECREASED your damage output by pumping crit damage runes instead of crit chance runes. And really, at which point in the game does an extra edge from your runes really count the most? In the first half of the game (or until you reach 64% crit chance)... or the first chunk of the game, building your way up to those uber items?

As for me I'll be pumping crit chance. Which, hey, they're cheaper anyway so... score!

Of course, some heroes are special cases. Ashe's passive can let her have 100% crit chance sometimes, etc. None of this is taken into account. I'm not here to tell you what to do. I didn't look in to base damage, or armor pen, attack speed, or any of that. I just wanted to show you the math on this one issue, setting aside all that jazz, and just looking at crit statistics, alone.

If you want to compare your own build, one versus the other, on an averaged, per-attack basis, just plug in your own numbers.

dX (dmg increase) due to crit chance increase = (1 + crit damage boost %) * (crit chance increase)
dX (dmg increase) due to crit damage boost increase = (crit chance) * (damage boost increase)

Please give me your thoughts. Or if I've made a math mistake anywhere, please let me know.


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DonkeyBalls

Senior Member

01-26-2010

well that sucks. then why the heck are crit damage double the price in the store. pisses me off!


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cycoconutz

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Senior Member

01-26-2010

if your post didnt have any numbers or capital letters in it; it would be exactly what i thought after buying 15 crit dmg runes. so i switched to chance instead....17% only if we could SELL BACK RUNES


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Klazzix

Senior Member

01-26-2010

Nice post, thanks for the math. We all appreciate. +1

I already crunched this numbers in my mind but I'm envious of Shaco's that can do burst dmg with Crit dmg runes.


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WhysoSully

Junior Member

01-26-2010

the only thing i think that is an issue is the fact that you can buy crit chance items for cheap in game. it takes 2 minutes (or first blood) to get 880 gold and buy a +18% chance for crit. compared to spending alot alot of IP for the runes when you could have maybe a little more mana for that extra siphon strike, parlayy, or gold card.


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EnderTheXenocide

Junior Member

01-26-2010

I thought the same thing at first WysoSully, as did some friends of mine. "Oh crit % is easy to get, so clearly it's better to spend my IP on crit damage runes. Only Infinity's Edge actually raises crit damage boost."

The thing is, yes you can just buy +18% chance for crit items. But the damage you get from a higher crit % doesn't really have diminishing returns. All the way to 100%, higher crit rate is always good. So yes you can buy a +18% crit chance item after 2 minutes or first blood. But even still, by the math it's STILL better to have crit % runes instead of crit damage. Cuz think about it, with that +18%, and +20% runes, that will give you 38% crit chance after two minutes (or first blood).

And you only have so many item slots. Until you stack THREE of those +18% crit items, it's still better to have crit % runes than crit damage boost runes, doing the math... And in a real game, you can't just focus 100% on crit chance. You need damage, armor pen, regen, etc. So using your favorite, most effective item builds, the real question is, when do you reach the "magic" 44% crit (no IE, or 64% crit with IE)?


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mege

Junior Member

01-26-2010

What would a distribution look like with a possibility of a 'sweet' spot in obtainable numbers (with and without runes)? Or is that 40% crit/50% crit damageconsidered a good sweet spot?

In which case... I can get 40% crit and 50% crit damage in game fairly easilly and use all ArPen runes Best of both worlds!


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EnderTheXenocide

Junior Member

01-26-2010

Unfortunately, the math won't make the decision for you, mege

There is not real "sweet spot" for these values, because they're just linearly proportional to each other.

At a fixed value of D (crit damage), every increase in your crit rate of 20% is going to increase your average damage output, per attack, by 20%. So there's no efficiency sweet spot. 40% crit rate is great. In fact, it's literally 40% more damage (on average) per attack (compared to 0% crit rate, with 0 crit damage boost). But 60% is well... 60% better. 80% is 80% better. etc.

I haven't done the math on armor pen yet, so I can't comment on that.


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Klazzix

Senior Member

01-26-2010

What sucks is that luck is involved. Even if you have ~50%, there's no guarantee you'll always be critting, so it's better to have more of those %s to make sure you do.


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Klazzix

Senior Member

01-26-2010

About ArPen, I had the chance to try it too, I had an IE + 1 BF Sword as my dmg items, ~30 ArPen will give you about 120 more dmg (double/2.xx when it crits) on armored heroes (everyone has base armor, right?).

EDIT: This was tested on YI, so if you plan to do that math, it was with Wuju Style on.