That’s the one, yes (linked in the OP)
Thanks, I was curious too
The real question though: what is better: a 24% chance of 100% cooldown… or a 24% cooldown reduction ?
I’m inclined to think they are equivalent, but I’m not sure…
In Crucible when fishing for the best run it might actually be the first one lol
The hardest wave and then you get i.e. Aegis chain
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On the other hand, I generally see that most Crucible runners prefer less randomness and are more in favour of consistency.
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Some very quick (and questionable) maths tell me that a chance at 100% cooldown is almost 1.5 times as good as a cooldown reduction of the same value.
If it was about a fixed number of instances, they would be equivalent… but they both deal with time, so a “free” use frees up some time to try and get more…
…Anyone here good with maths ?
Luck based so we all know how it will end 
Concept of an unstable build sounds interesting though. You get as many cdr chance sources as possible and you are a walking RNG machine! Getting lucky and your Mirror is permament or something like that haha!
the same bonus value should be about 41% better in practice than a flat cooldown rate
(in other words, a 20% chance of 100% cooldown, is (on average) as good as a 28.2% coodlown bonus)
Anyone actually good with maths, please correct me
Edit: fixed bad maths with more bad maths
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I’m trying to calculate it - give me some time because I’m rusty AF.
BTW you are taking additional animation time of additional procs into account 
?
This animation doesn’t matter in standard case of course - because it doesn’t delay further hits then.
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No, I’m thinking straight theoretical for now. (this is already complex enough for me 
)
The %chance gets more out of higher attack/casting speed though, so if we get to that point, it would be worth checking.
Mine if of course super theoretical too, not taking gameplay into account, assuming perfect execution.
I think if you have d% of 100% cdr then the average chain length (how many hits in a row)
is 1 / (1-d)
A - animation time
Now all of these bring more animation time: A * [1 / (1-d) - 1] = Ad / (1-d)
So after let’s say K normal recharges (taking T time, from your standard cooldown), you will do
K / (1 - d) hits but it will take you not K * T time, but K * [T + A*d / (1-d)]
so in case of having d% of 100% cdr our DPS is (number of hits per time internal):
K / (1 - d) / (K * [T + A*d / (1-d)]) = 1 / [(1-d)(T + Ad / (1-d))] = 1 / [T*(1-d) + A*d] = [T*(1-d) + A*d]^{-1}
We can see that the formula above makes good sense because precisely if A > T (animation takes longer than recharge)
- we actually get worse DPS than if we were hitting simply every T time without this 100% CDR chance
Now in the standard case d is boosting our base cooldown, so we need to calculate benefit over 1/T here (K hits in K*T time without any bonus).
c - base cooldown
R - base recharge
T = R (1 - c)
T’ = R (1 - c - d)
DPS = 1/T’ = 1 / T * [T / T’] = (1 / T) * (1 - c) / (1 - c -d)
Ok, let’s take some sample numbers.
A = 0.2 (200ms) <- is it ok animation time???
C = 0.2 (20% cooldown)
D = 0.2 (20% cooldown)
T = 3 seconds = 3
With chance of 100% reduction we get 1 / (3 * 0.8 + 0.2 * 0.2) ~ 0.41
And when d gets added to c: 1 / 3 * 0.8 / 0.6 = 0.4444…
Unboosted DPS would be: 0.33333
so this example and if I haven’t make mistake the standard cooldown is better.
[edit] Not sure, maybe in case of other sample numbers it’s different or is it more general (i.e. with Grim Dawn range of numbers)?
You can try some other numbers in these 2 formulas, assuming it’s correct.
DPS in chance case: 1 / ( T * (1-d) + A*d )
DPS in standard case: (1 / T) * (1 - c) / (1 - c - d) = 1 / ( R * (1-c-d) )
T - is recharge ( R) after reduction by base cooldown c
c - base cooldown (the formula linking these is T = R * (1-c))
d - additional cooldown / chance
A - animation time
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I’m in no position to critique your maths up there, so sorry if my way of calculating it is a bit basic, but I come up with roughly the same result you have now.
I think an easier way to look at this problem is counting the number of hits you get on average when you activate the skill.
With %chance cooldown, you get a number of hits (between 1 and… technically infinity), then go into cooldown and the cycle repeats. This is a variant (with different probabilities) of the “getting head or tails X times in a row”
Let’s take some sample numbers (that are conveniently easy to divide)
Skill has 4 second cooldown
we have 25% cooldown / 25% chance of 100% cooldown
Normal cooldown gets 1 hit every 3 seconds, flat
Chance cooldown gets 1.33 hit every 4 seconds
(75% chance of getting 1 hit, 18.75% chance of getting 2, 4.69% of getting 3, 1.17% chance of getting 4 and so on… comes up on average to 1.33 hits)
After 100 seconds, you will have
33.3 hits with the normal cooldown
33.25% hits with the chance cooldown
This is pretty much the same thing.
I don’t think including the animation time is relevant though, since it will have to play the same number of times for each, so come up to the same time (that is, unless cooldown starts immediately once the skill is activated and not once the animation is complete… in which case, every time you get 2 or more hits, you lose the animation time in unreduced cooldown, but this only happens 18.75% of the time.)
My previous calculation was based on a different problem that looked similar on the surface. I just imported my calculation from there assuming it would work the same. Sorry if that was misleading.
Cooldown starts immediately imo. I mean I haven’t looked closely but come on 
I’ll bet my head on it.
Btw I’ll now make a different example because I think the result might be different:
T = 3
c = 0
d = 0.2
a = 0.2
DPS in chance = 1 / (3 * 0.8 + 0.2 * 0.2) ~ 0.41 as before
DPS in standard = 1/3 / 0.8 = 0.416…
Standard won this time too 
But I’m still not sure if it’s always the case, 1 more example
Or maybe I’ll try comparing these two in general sense.
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Ok, I think I’ve proven it. Standard is always better than chance. I compared two formulas and got to
Tc + A(1-c) > 0
where standard was on the left. d even disappeared. Even with 0 animation time it should be always better.
[edit] wait no! there’s actually the case when it’s the other way around!
I divided by (1 - c - d) once
No, no, I’m stupid 1 - c - d is always > 0 in our examples, we cannot reduce cooldown lower than 0 
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In any case, The difference is marginal though, and in practice it’s probably a wash.
The difference however might be more meaningful in gameplay:
(What follows is not maths, just opinions)
Assuming the total DPS is (roughly) the same, I can see a case for chaining abilities but using them less often versus being on a shorter timer.
If the ability has lingering effects (like Heart of wrath on Judgment or the debuff on war cry) that are slightly shorter than the cooldown, you obviously want that cooldown consistently faster to have a longer uptime on that debuff. So those don’t benefit from chance cooldown.
But on skills like doom bolt, getting to cast it twice from time to time can be better from a gameplay standpoint, as you don’t need to progress through your piano as fast while getting roughly the same dps. It makes kiting easier, and getting 3 straight doom bolts to the face puts a serious dent in every boss’s health bar.
In short: You can compensate on gameplay for longer cooldowns easier than enemies can compensate for getting nuked twice as hard.
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Another example of usability:
Would you rather have a free use of Mirror ready, or a cooldown that’s 4-5 seconds shorter ?
When I use mirror, I’m usually in trouble… Getting a second use right away might just save me, while getting a cooldown of 14 seconds instead of 19 seconds likely won’t do me any good in the immediate (If I just survived 14 more seconds, I wasn’t THAT much in trouble). I can always briefly pause between scraps to compensate anyway.
Same goes for the healing skills, or things like overguard, Ascension, Nullification.
Ok, so I can give counter example.
Let’s say you’re kiting stutter stepping with doom-bolt (imaginary, not sure I’d play like that)
In practice, to ensure double Doom Bolt you’d have to hold your button after every cast (because reset is not reactable)
Which in some cases could mean your death, i.e. when fighting Ravager / or not be feasible.
Perfect piloting with some bosses could be harder with that style.
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Fair enough, but it’s an easy fix to train yourself to double-click.
I guess the examples of very high cooldown skills is better to illustrate the benefit. And whatever you gain from instant cooldown is out the window if you need to pilot perfectly (like when fighting a super boss) If you need perfect piloting to begin with, you can’t do unpredictable. (and thus, are better off with regular cooldown)
For most players/cases however, I think there is a small gain
My internet died from all of this, I’m tethering from phone 
I don’t think it will work with double click even. You’d have to delay the second click I think
We are theorycrafting yet here’s opinion from one of the best players (DMT played TSS to death)
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Which is precisely the kind of players that gains more from predictability than from bursts of luck.
I’m not saying it’s better overall, just that I can see cases where it could be more interesting than regular cooldown.
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@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 ]
-
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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
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