Mathematically evaluating the assassin rework
First off, this is going to be a nerdy post, but the TL;DR is at the bottom for those who don't like math. For other nerds, feel free to check my math to make sure this is legit.
The 9.17 patch notes on assassins reads:
(3) Trait Bonus: 125% Crit Damage ⇒ 75% Crit Damage & 10% Crit Chance (6) Trait Bonus: 350% Crit Damage ⇒ 150% Crit Damage & 25% Crit Chance
After some quick calculations, the expected damage output looks numerically identical before and after the change. However, there are a couple other considerations (reliability and items) to think about when evaluating the change. Let's break it down.
1. Identical expected damage. By "expected" damage, I mean the average output over a long period of time. RNG will make your actual damage output either more or less in an actual fight due to the limited number of autos per round. Currently, crit damage increase is additive instead of multiplicative. There's a base crit of 25% and crit damage of 150%. This means your total expected damage output for standard attack, the old assassin synergy, and the new synergy is expected to be:
Example = total expected damage multiplier = (chance of base damage X base damage) + (chance of crit X crit damage multiplier)
Normal auto attacks = 1.125 = (.75 x 1) + (.25 x 1.5)
Old (3) = 1.4375 = (.75 x 1) + (.25 x 2.75) Old (6) = 2 = (.75 x 1) + (.25 x 5)
New (3) = 1.4375 = (.65 x 1) + (.35 x 2.25) New (6) = 2 = (.5 x 1) = (.5 x 3)
This means when considering your base chance and strength of crits, your normal autos are expected to have a 12.5% increase in damage compared to no crit chance at all. Every champion in the game (currently) has this as their base. Adjusting for this, an Assassin 3 synergy bonus is a 28% increase in damage compared to normal champs, and an Assassin 6 bonus has a 78% increase. The total expected damage output for the old and new synergy is identical.
2. Consistency. This one is hard to put a numerical advantage to, because it depends on RNG and the specific situation. However, the easiest way to think about it is this: you don't want to overkill your target. Any damage dealt more than their health is "wasted" damage that you "could" have been dealing to the next victim. The change from lower damage per crit, but more crits, makes it so that you are much less likely to overkill, and you overkill by less. This is a slight buff.
3. Items. There are currently three items that affect crit directly: 
, and 
, and 
.
is better for the new synergy, because it's already more likely that the old crits overkill the target. From a consistency standpoint, it's more likely that the new assassins will be able to utilize but not waste the extra damage. More than that, though, a higher crit chance means this item applies its bonus damage more often. This is a reasonable buff.
changes the calculations of the damage output. The new damage calculations after the item kicks in are simply the crit damages, and we compare them to the 1.5 natural crit damage for a base champion:
Normal crit attacks: 1.5 Old (3) = 83% increase, or (2.75/1.5) - 1 Old (6) = 233% increase, or (5.00/1.5) - 1
New (3) = 50% increase, or (2.25/1.5) - 1
New (6) = 100% increase, or (3.00/1.5) - 1
The numbers are super clear: A new assassin with is shockingly worse than an old assassin. Comparing these numbers, got a 27% nerf with assassin (3), and a 57% nerf with assassin (6).
was already a shutdown against assassins, but it's even more so now. Again the calculations change, but the exact opposite way, where we're left with only the base auto chance multiplied by 1. This give us:
Normal Auto attacks = 33% decrease in total expected damage, or 1 / (.75/1.125)
Old (3) = 48% decrease, or 1 - (1 / (.75/1.4375)) Old (6) = 63% decrease, or 1 - (1 / (.75/2))
New (3) = 65% decrease, or 1 - (1 / (.5/1.4375)) New (6) = 75% decrease, or 1 - (1 / (.5/2))
In short, it used to be that cut your damage either ~1/2 or ~2/3, but now it's either ~2/3 or ~3/4. Comparing these comparisons, a got a 36% buff against assassin (3), and got a 20% buff against assassin (6). However much of a shutdown item it was before, it's even more so now.
TL;DR: Total expected damage is identical, but reliability is up (slight buff). is likewise better, but got a 27% nerf with (3) and 57% nerf with (6), and got a 36% buff against (3) and 20% buff against (6). In short, the damage may be identical, but assassins are harder to build, and easier to build against.
