This article seeks to derive some mathematical results for optimizing Attribute ratios. The derivations assume some familiarity with multivariable calculus and Lagrange multipliers.
Let r be the crit rate probability () and d be the crit damage () in decimal form (so 150% crit damage would be ).
Suppose the damage is x without crit. The expected damage is[Note 1]
The objective is to maximize this function, namely the product .
In general, choose the crit rate r and crit damage d that maximizes their product to maximize the expected damage. That is, ceteris paribus (holding all other attributes constant):
Maximize the product of CRIT Rate and CRIT Damage.
If one assumes as certain distribution between crit damage and crit rate, then it is possible to derive a special optimal crit ratio. For example, a 2:1 ratio between crit damage and crit rate is sometimes recommended. This 2:1 ratio is optimal if one assumes a special distribution between crit damage and crit rate.
Within Genshin Impact, there is a tendency for crit rate and crit damage to appear in a 1 to 2 ratio, whether for artifacts, weapons, ascensions, etc. That is, for every 1 point in crit rate we gain, we lose 2 points in crit damage. We can express this using the following constraint:
for some constant c.
Our new goal is to maximize the product rd with respect to the above constraint. We can do this using Lagrange multipliers.
Solving this system gives
so the optimal ratio between crit damage and crit rate assuming the given constraint is 2 to 1, as desired.
- Here damage is modeled as a random variable using a 2-point distribution. That is if is a bernoulli random variable that takes 1 with probability r and 0 otherwise, then in this model, damage can be written as the random variable . The expected value or first moment is then .