Content of the article: "A Mathematicians Take on Ability Haste vs. CDR"
I am a Master's student in mathematics who has played league for many years but never posted anything here. In this post I want to give my take on the new "Ability Haste" stat which is going to replace "Cooldown Reduction (CDR)". I am not a native speaker so excuse any (language) mistakes. For mathematical incorrectness I will of course take the blame. If you really hate math and don't want anything to do with it you might wanna skip to the tldr or even ignore the post. Although the math is very simple and I always try to give concrete examples rather than formulas or abstract explanations.
At first glance the concept of CDR is simple. Let's assume we have an ability that has a 10 second cooldown and you then buy items that give 20% CDR. Then this reduces the downtime of your ability by 20%, meaning your new cooldown is 10s*80%=8s. Nothing new so far. But something that I'm sure not everybody knows is that CDR gets "more and more valuable". What do I mean by that? Let's stick with the example of our ability that has a 10 second cooldown. For illustration, we assume our ability is a simple damaging ability. Of course each 10% CDR gives 1 less second on your cooldown. So with 10% CDR your cooldown is 9s, with 20% CDR your cooldown is 8s and so on. In that sense CDR scales linearly. But let's look now at how much damage you could do in a set timeframe, say 100 seconds. Without CDR you could cast the ability 10 times obviously. With 10% CDR ( meaning your cooldown is now 9s) you could already cast the ability 100/9≈11.11 times. This means you got an increase in your damage during this timeframe (or more simply an increase in your dps) of around 11.1% but you only bought 10% CDR. With 40%CDR you would even be able to cast the ability 100/6≈16.67 times which is a damage increase of 66.7%. So definitely no linearity here. This is why one could say that "CDR scales well" since from this dps perspective you have increasing returns.
In some sense Ability Haste is mirrored to what we discussed above. Again we have our damaging ability with 10s cooldown. If we buy 10 Ability Haste this means that we will be able to cast our ability 10% more often, if we buy 20 Ability Haste this means that we will be able to cast our ability 20% more often. What this means is that where originally you would be able to cast your ability 10 times in 100 seconds after buying 20 Ability Haste you can now cast your ability 12 times. So the damage you can do in those 100 seconds (or more generally your dps) increases by 20% with 20 Ability Haste. And in general you get 10% more dps with this ability for every 10 Ability Haste. So 10 Haste means 10% more dps, 20 Haste means 20% more dps, and so on. So unlike CDR, Ability Haste scales linearly with your ability dps. So one could argue that Ability Haste is more "balanced" if you will. But if you look at the downtime of your ability in seconds, Ability Haste only offers diminishing returns here. In our example 10 Ability Haste means you can cast the spell 11 times in 100 seconds. So our new cooldown is 100s/11≈9.09s. With 20 Ability Haste you would have a cooldown of 100s/12≈8,33s. In other words the first 10 Ability Haste give you approximately 9.1% CDR, the next 10 Haste give you only an additional 7.6% CDR and the ones after that give you only an extra 6.4% CDR. You can see the trend.
Personal take on what is "better"
From the point of game balance/allowing more variety in builds and things like that I can definitely see Ability Haste being the preferred option since it seems hard to balance around CDR (as suggested by the increasing returns discussion above). However, from a player point of view definitely think Ability Haste is more difficult to understand than CDR because in my opinion the way it functions does not align with the way we typically think. Let me give you an example: Imagine in your local supermarket your favourite product is on sale. Let's say instead of 30$ it costs 30% less now. What would you think? To be honest you would probably buy it without thinking anything. But if you would, the first thing anyone would do is probably calculate how much money you saved. Here you saved 30$*30%=9$. Not too bad. And this is basically how I assume most of us think about cooldowns when buying CDR. But thinking of it the "haste way" would be the same as saying something like "Since my favourite product costs only 21$ now I actually get 1.43 times as much value as before (because 30$/21$=1.43)" or "For the old price I can now buy not only 1 times my product but even 1.43 times." I'm pretty sure there is hardly anyone thinking this way. And this is why I feel Ability Haste might come across a bit unnatural to some people.
Ability Haste is like "Attack speed for abilities". 10 Ability Haste lets you cast your ability 10% more often, 20 Ability Haste lets you cast your ability 20% more often and so on. If you are bad at math and want to convert from CDR to Ability Haste or vice versa here are the two formulas.
- Spell Haste and You! Why CDR was secretly broken and why 2021’s Spell Haste is a mechanical upgrade.
- I made a chart to convert CDR to Ability Haste
- Wei as independent hero Hero Concept
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