Monday, 14 April 2014

Diablo 3 - Cooldown Reduction

Cooldown reduction is an interesting new thing they added to loot in the run-up to the expansion.  Some classes can reduce cooldowns with passive skills, but now everyone can have access to reduced cooldowns.

Unfortunately, reducing cooldowns stacks in a complete rational fashion.  If you have 10% reduced cooldowns and 20% reduced cooldowns you end up with 28% reduced cooldowns rather than 30%.  That is, it takes 10% off leaving you with 90% then take 20% of the 90% off, leaving you with 72%.

This is probably good because otherwise cooldown reduction would probably break the game when people stacked it to 100%.  On the other hand, it means that cooldown reduction gets more and more modest as you stack more of it.

Well, kind of.  It feels like it becomes more and more modest because reducing from 20 seconds to 18 seconds feels better than reducing from 10 seconds to 9 seconds.  But in reality it depends a lot on what exactly the cooldown is doing for you.

For cooldowns that have durations, cooldown reduction becomes more powerful the more of it you have.  Suppose you have a cooldown that makes you do 50% more damage for 20 seconds with a 120 second cooldown.  That ability is on one sixth of the time so in the long run it increases your damage by 8.33%.  Slap a 10% cooldown reduction on and now you can use it every 108 seconds so it increases your damage by 9.25%.  The next 10% cooldown reduction lets you use the ability every 97.2 seconds, meaning it is on 20.5% of the time and increases your damage by 10.25%.

Going from 108.33% of your damage to 109.25% increased your damage by 0.84% while going from 109.25% to 110.25% of your damage increased your damage by 0.91%.

Similarly, if you have an ability that lets you stun enemies for 4 seconds on a 15 second cooldown, then enemies are stunned 26.67% of the time - or, put another way, they get to act 73.33% of the time.  Reduce the cooldown by 10% and they get to act only 70.33% of the time.  Reduce it again and they get to act only 67.07% of the time.  The first 10% reduction reduced their acting time by 4.05%, the second one by 4.78%.

Of course these numbers look like they kind of suck.  Improving from one bad number to another bad number doesn't make things look good.  But if they get better the more you have, then the question isn't really whether 19% is better than 10%.  It is whether 55% is better than 50%.

Let's say you have a flawless royal diamond in your hat, a 20% cooldown reduction skill, 50 paragon points in cooldown reduction and two pieces of the Captain Crimson set on.  That would have you serving only 56.7% of your cooldowns.  How good is another 8% - the maximum amount you can roll on a non-weapon unique or set item?

I did some math for an Archon wizard build.  Right now on my wizard I fight with disintegrate and explosive blast when not in archon form and try to melee and mash arcane blast when in archon form.  One of the things about this build is that it converts cooldown reduction to damage both in and out of archon form.  Since the blast abilities can be used while making other attacks, lower cooldown means more spamming.

Quite a few things go into this calculation.  Slower weapons mean that cooldown instant cast abilities are more powerful.  Choice of archon rune makes a big difference.  I also did the math with and without an empowered shrine since wearing the shrine gloves is probably a very good idea for this build.

First of all, here is the archon uptime that you enjoy with and without the pure power rune.  The x-axis shows how many 8% cooldown reduction items you are wearing.  It is fairly easy to see how each is better than the last at improving archon uptime.

This chart shows how much damage you do while attacking and mashing that arcane blast button.  Not only do you do more with a slower weapon, but you also accelerate faster with a slower weapon.

So what happens when you have the four piece bonus from Vyr's set to get all archon rune, you use a 0.9 speed two-hander, and you have an empowered shrine active?  It turns out that the best 8% cooldown reduction gives you nearly 10% more overall damage.  For more realistic situations 8% cooldown reduction grants more like 3.5-5.0% overall damage.  Unfortunately, that makes it worse than a maximum roll on pretty much any other stat an item could have.  So cooldown reduction only manages to outweigh other stats if it is doing more for you than adding to damage.

Tomorrow, cooldown reduction and teleporting.

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