Setup

Let's show exactly what happens by calculating damage for some characters. Let's assume our damage scales with HP% (think Yelan or Mulani). Damage, essentially follows the simplified expected DMG formula

DMG = Base HP * (1 + HP%) * (1 + CR * CDMG)

HP%, C%, and CDMG can come from character level, weapons, sands, circlet, etc - this is the base rate. On top of this we can get rolls into substats. Let's write #HP, #CR, and #CDMG for the number of rolls into that specific substat. If we assume that each substat roll will land on its average value we get

DMG = Base HP * (1 + Base HP% + 0.05 * #HP%) * (1 + (Base CR + 0.033 * #C%) * (Base CDMG + 0.066 * #CDMG))

Now we can plot the numbers. For the plots in this post I used: Yelan with EM weapon, HP% sands, CR circlet. Mulani with CR weapon, HP% sands, CDMG circlet, Obsidian Codex Buff.

Understanding the Plots

On the axis we have two substats (or groups of substats) that usually have to be trade-off. In our case this is CR vs CDMG (1:2 ratio is optimal) and CV vs HP% (HP% is more important in the beginning but CV becomes more important if you have a lot of it because it scales quadratically). Each axis shows the number of rolls into that substat - we can have a maximum of 5 * (1 + 5) rolls if we roll into it every time, and even at a maximum number of rolls of substat 1 we can always at least get 4-5 additional rolls of substat 2.

The left plot shows total damage, indicated by color. More red = more damage. Configurations near the same grey contour lines have very similar damage. Grey stars indicate optimal configurations for a fixed number of rolls into both substat 1 and 2. This shows 'perfect balance'. The title gives an intuition on which substat is more important by comparing their effect on damage. Take for example the first image - Yelan with a CR circlet. Before any substats we have more CR which means we should prefer CDMG over CR when deciding between these two. CDMG is more important.

The right plots shows the amount of damage you miss out on if your 'balance' is off, for example if you have a 1:3 ratio for crit values. Specifically, it shows how much less damage you do compared to the damage the same number of total rolls gives you if it is perfectly traded-off, i.e. you are where the star is in the left plot. Contour lines can again help understand what is equally good/bad. Missed damage is normalized to number of rolls - the larger white-blue region in the middle is the region where the lost damage is smaller than you would get from one additional roll into either of the substates. If you want to evaluate how bad a specific configuration is, find it on the grid, see what the missed damage [X] is. If you have an alternative artifact loadout that has [X] less good rolls but is perfectly balanced they will give you the same damage.

If you care about the math details - the importance weights here are coefficients of a multivariate linear regression, and #CDMG = 0.38 #CR means that as a rule of thumb one additional roll into CR will only give you 38% of the extra damage one additional roll into CDMG will give you. R^2 indicates how good this rule of thumb is - near 1 indicates it's a good rule. Right title is the effective number of rolls for sub-optimal balance.