Optimal Ratio for Crit Chance/Dmg

I am wondering what is the optimal crit chance/damage ratio? After plugging in numbers into the equation below, I noticed that the magic value appears to be around 1:3. Is my math flawed?

Equation: z=1+xy where z = damage output, x = crit chance, y = crit damage.

Solution 1:

TL;DR: simplified: 1% chance = 10% damage, keeping them even gives double the bonus.

Your equation is correct. You start off with a base of 5% crit chance and +50% damage giving you effectively an average "effective damage" of 102.5% damage.

Adding 10% crit chance will always give you the same bonus to effective damage as +100% to crit damage disclaimer: this is only true when your critical chance and critical damage are proportionately equivalent (e.g. 5% chance of +50% damage, 10% chance of +100% damage, +20% chance of +200% damage).

However, I will show below how, from balanced values, it is twice as beneficial to get +10% crit chance and +100% to crit damage rather than just getting +20% in crit chance or +200% increased critical damage.

Example:

From base, lets say you add either 10% chance or 100% damage:

1 + ( 5% * 50% ) = 102.5%

1 + ( 15% * 50% ) = 107.5%

1 + ( 5% * 150% ) = 107.5%

Taking that example, if you add another 10% chance or 100% damage to what you already did... (adding +20% chance or +200% damage to base):

1 + ( 5% * 50% ) = 102.5%

1 + ( 25% * 50% ) = 112.5%

1 + ( 5% * 250% ) = 112.5%

But what if you added 10% to base and 100% to damage?

1 + ( 5% * 50% ) = 102.5%

1 + ( 15% * 150% ) = 122.5%

Conclusion:

Going for a mixed balance of critical chance and critical damage will provide higher damage output than maxing one or the other. Going from base values and adding 20% crit chance is equivalent to adding 200% crit damage, but adding 10% chance and 100% damage yields double the bonus to your "effective damage". From the example above, going from a base of 102.5% effective damage up to 112.5% or 122.5% is an increase of 10% or 20%.

Sidenote:

Be aware of what items can give you because you can get more value out of some items. For example... Gloves can give up to +10% chance and up to +50% to crit damage.

The best crit damage will come from weapons (up to +100% on weapon iteself with a socket of up to 100%). Theoretically with dual wielding you can get up to +650% damage on critical hits (not including unique bonuses from legendary items)...

If you have +0% crit damage on weapons, the highest you can get is only +250%.

Guide to Crit from items

Solution 2:

A more mathematical approach

• First, take the hyperbolic function f(x,y) = 1+xy and look at 2 things,

  1. Contours: Set f(x,y) to some constant C. C = 1+xy behaves basically like y = 1/x i.e. the two are inversely related. Scaling x by a factor of k and y by a factor of 1/k is just moving along this level curve.

  2. Directional derivative and direction of steepest ascent: The gradient vector of f(x,y) is obviously <1,1>, meaning that the fastest way to increase f(x,y) is to go along the line x = y.

  This means THE FASTEST WAY to increase your damage is to increase your crit chance and crit damage concurrently with some scaling factor. This obviously may not be possible, but the math here only tells you the fastest way to increase damage, not the most cost-efficient way - leading to the next point

• Having looked at z = 1+xy, devise a cost function associated with each point in R^2, i.e.

  h(x,y) = "cost of items associated with x crit chance and y crit damage"

  Generate this using some estimation/modelling techniques and data from the AH (if you have that much time)

  Gven the damage function f(x,y) and the cost function f(x,y), finding the optimal path for "efficient damage increase" becomes a constrained optimization problem which is not something I can do off the top of my head on the get-go - lagrange multipliers, etc.

  Mind you, the cost of items factor in more than just crit cost / crit damage, so a more comprehensive approach might be to use a different equation for damage/cost all together.

Going back to the previous answer by DiabloMonkey

• Take 5% crit chance (cc) and 50% crit dmg (cd) to start, then scaling cc by 1.2 (1% cc increase) is equivalent to scaling cd by 1.2 (10% cd increase). The damage increase is the same.
• Now, again with 5% cc & 50% cd, say I scale cc up by 1.2 (1% cc increase), and then scale cd down by 1/1.2 (roughly 8% or 9% of cd decrease I think). Damage output stays the same.

• So wait, did I just show that 1%cc = 10%cd AND 1%cc = 8.x%cd at the same time? No. BOTH of those statements are WRONG. There are no such equalities. They both only work under certain assumptions and objectives: "I want to throw away cc/cd, but keep same damage, what should I do?" (traverse along the same level curve given constant C) or "I want more damage as fast as possible, what should I do?" (moving towards the next level curve C* along the best path aka direction of the gradient vector). Also keep in mind that different given starting points of (x,y) produce different results when scaled.

  Does that mean DiabloMonkey is wrong?

• If your base chance/damage is 5%/50%, then no. If you have other passive skills/runes that affect your base value, then his method will be most certainly sub-optimal.

• He is basically using the equation f(x,y) = 1+(0.05x)(0.5y) where x and y now represent your % scaling bonus of cc/cd from items as opposed to actual values of cc and cd. Note that the gradient vector is <0.05,0.5>, a scalar multiple of <1,10>, hence the idea that 10x damage is equivalent to 1 chance (but again, this holds true only under a specific set of assumptions).

What does this mean? (TL;DR)

• If you are looking for a optimal ratio between crit chance and damage (as in how much of each you should have), then I'm sorry to tell you that it does not exist since f(x,y)=1+xy has no global maximum. This is not to say that it might not exist at least locally for other functions.
• If you are looking for an optimal way of increasing your crit chance and damage, then the FASTEST way is to increase both numbers at a constant rate, i.e. increase both your chance and damage by xx% if possible (i.e. if your base cc/cd happens to be 5%/50%, then going up by (1%,10%) as prescribed by DiabloMonkey is a good place to start.)

• Alternatively, use,

  f(s,i,w,c,d) = (1+s)*(1+i-w)*(1+c*(1+i)*d)

  where,

  s = damage_stat/100 (str/dex/int),

  i = total_ias/100 total increase attack speed %,

  w = weapon_ias/100 need this because weapon ias already factored into weapon dps

  c = total_cc/100, total crit chance %

  d = total_cd/100, total crit damage %

  Helps you get a nice idea about how much more dex/ias/cc/cd you want (I play DH, so this is somewhat useful for me), but still doesn't tell you everything (most notably, the effect of +min/max damage bonuses)

Solution 3:

If you have a choice between increasing crit chance vs. increasing crit damage, you are best off increasing the lower one (as seen on the character "details" window) first, if possible.

If you don't have any items, your character should have 5% crit chance and 50% crit damage, so you are best off adding crit chance at first. This will be true until you have equal crit chance and crit damage. At that point, you should increase both evenly if possible.

I say "if possible" because at some point, you won't be able to add any more critical chance (especially never beyond 100%), so you have no choice but to increase damage. The most important thing is that increasing either one always helps some. It's just that increasing the lower one helps more than increasing the higher one.

If you are a demon hunter, and you are using the "sharp shooter" passive skill (I think that's the one), then your crit chance will go up to 100% after you've been walking around looking for monsters to kill. With this passive skill, you are probably best off investing in gear that gives you as much crit damage as possible.

Other classes besides DH have to live with much less than 100% crit chance at any given time, so they need gear to increase the crit chance.