@DeputyChuck This relation above I’ve come to showed that if initial cdr c is 0% and the animation time is also 0, having
d% cooldown
d% chance of 100% recharge
should give the same DPS so I thought it would be good to have some simpler proof without complicated calculations:
Let’s say, for simplicity’s sake, that we use our skill every second.
If we then add d% cdr to it, we attack
1 / (1-d)
more frequently / have that many times more attacks.
(For example for 50% cdr we get 1 / (1-50%) = 2 times more attacks)
But from high school we know that
1 / (1-d) = 1 + d + d^2 + d^3 + d^4 …
And the right side is basically the average hits in the d% chance scenario
One can see it from expected value definition or see that:
probability of 1 hit - 100% = 1
probability of additional hit - d
probability of yet another hit - d^2
etc.
…
so the average number of hits is this sum of 1’s (of hits weighted by chances they happen) which is 1 / (1-d)
which is like having d% cdr due to this identity of formulas
Now to more general case:
if we add animation, it only negatively impact the chance scenario so it’s always worse then
(since it’s equal with no animation)
if we add some initial cooldown c instead,
it’s also quite clear that it benefit standard case more because
d% chance benefits the dps to the same degree (1+d+d^2… = 1/(1-d) times more hits)
no matter if we have initial cooldown or not
yet intuitively simply adding cooldown gets better and better the more you have it
precisely it would be (1 - c) / (1 - c - d) > 1 / (1 - d) which is always true
(can be written like that 1 + d / (1 - c - d) > 1 + d / (1 - d) to see it better for example)
[or simplify completely to c > 0 ]
Well that seems to put my question to rest then
Thanks for indulging me
That raises another interesting (but unrelated) question: those skills with a modifier that either add
a cooldown and increase damage or that remove cooldown while reducing damage should have a definite “best” version for a given scenario too. (Same goes for items that add 1 sec cooldown to said skills, like the Vanquisher set does for Callidor’s tempest)
Since you also have to consider animation time only in the spammable version.
I’d be curious to see what DPS gains can be made and how good or impactfull those modifiers/items really are.
But I’ve derailed that thread enough, so I’ll post my question separately