[PSA] "Damage is Lucky" effect on DPS.

So, we're getting more and more mechanics in this game that rely on Lucky, a.k.a. take two rolls and keep the better result. And we know that Damage is Lucky is a Good Thing. But how Good is it? That requires some probability, which is not my forte. Thankfully u/nightcracker on reddit has done all the heavy lifting for us. There is some math wizardry revolving around integrals that essentially say the probabilistic benefit is something like 2/3 over 1/2 for the difference between the midpoint and the maximum, something like having 2 equally spaced midpoints instead of 1 midpoint as your expected value.

Basically, the wider your damage range, the greater the multiplier to your DPS, ranging from 0 to 33.33%.

Formula: (2*min + 4*max)/(3*min + 3*max)

Alternatively: where x = min and y = max

Spreadsheet to toy around with values:
Locked version (copy to your own Drive to use)
Unlocked version (others may edit at the same time as you)

Basic stuff at the top and looking at proportional stuff on the bottom. See text below in this post for more details. You can experiment with different values with the "Input" cells highlighted in brown.

Derivation:


An intuitive understanding? Sadly, the relationship between proportion of the damage that's uncertain (max minus min) does not scale the multiplier between 0 and 33.33% linearly as I was hoping (so if your damage range was 50-100, aka half random half guaranteed, you DON'T get a 33.3%/2 = 16.6% more DPS multiplier). So it's a bit rough to do mental math. If we sub in 1 for maximum (aka 100%) and then conceptualize minimum as a decimal portion of 1 we can graph it as follows:


showing basically the 1/x relationship but slightly modified. As min approaches max, the DPS multiplier approaches 1 (for 0% more damage). However starting at 0 min dmg you have 33.33% more damage and it decreases in concave fashion as one might expect of a rational function.

You may also try the 3d version if you are interested in visualising how min and max damage can change at the same time:
https://www.geogebra.org/3d?lang=en
Set z equal to this: If[0<=x<=1&&0<=y<=2,(2x+4y)/(3x+3y)-1] and use the "Move Graphics View" tool to drag the axis scales to where you want to see them. The formula has minus one (hundred percent) so you can easily see the plane (converts 1.33 multiplier to "33% more" basically).


A bad approximate heuristic I got is multiplying the proportion complement twice, so for example if your damage range is 333-3330 you get 9/10 of 9/10 of 1/3 for .9*.9*.3333 is about 0.27, which approximates the true 0.2727 repeating you get from the formula above. The heuristic gets worse and worse as min increases, but may be a nice thing to try if you like doing mental math like me.

Alternative intuitive understanding: u/Divinicus1st also points out you can simply look at added base DPS being (max - min)/6.

Again, credit goes to u/nightcracker on reddit to for his version of this PSA and inspiration.