Advanced Runeology: Attack vs Crit Damage

It's a common question; how do I build my nukers? Most people will ostensibly advise Rage/Blade and similar combinations with 2 and 6 slots being attack, and the 4 slot Crit Damage, but while these well-intentioned summoners are not wrong, per se, there's a lot more to the old adage.

So let's have a look at how Crit Damage and Attack stack up against each other assuming the same base attack (whether its your true Base or Base plus multipliers doesn't matter as long as your starting point is fixed). Here I've assumed a Base Attack of 100.

At first glance, Crit Damage looks like a sure winner. You start off with an added 50% and given Crit Damage multiplies your base attack damage even more, it's a sure win. Right?

Wrong. Because that nifty little chart there doesn't take into consideration that you have to actually land a critical hit in order to do Crit Damage.

To get a better sense of whether or not more Attack or Crit Damage will make your big, scary attacker even more deadly, you need to consider your probability to deal a critical attack, or in other words, your Critical Rate.

So how can we determine what our real damage will be on average? Why, with a little math of course!

Let's start by defining the variables, assuming:

Total Attack = (Attack x (1 +  % Crit Damage))
Probability that You Don't Crit = P(A)
Probability that You Do Crit = P (C)

Then your average damage will be:

Total Damage= [(Attack x P(A)) + (Total Attack x P(C))]

Or in layman's terms:

Total Damage = [(Attack x   Chance to NOT Crit)+ (((Total Attack  x Crit Rate )]

Example:

Attack: 100
Crit Damage: 150% or 1.5
Crit Rate: 40% or 0.4

Total Damage =  [(100 x 0.6) + (100 x (1+ 1.5)) x  0.4)]

OR if you prefer seeing percentages as values instead of decimals...

Total Damage= [(100 x  60) + (100* (1+1.5)) x 40)]/100

Attack vs Crit Damage with Crit Rate 

So let's see how that stacks up now that we are considering how likely we are to strike a critical hit at all.

Holy numerals, Batman! What just happened here?

Following that long red line (Attack) we can see that while Crit Damage wins out almost nearly every time when multipliers are under 150%, , Attack steadily begins winning out over Crit Damage above it from every Crit Rate below 100% (technically it's closet to around 97% but that's close enough to 100% for Summoner's War to not quibble). The key finding?

The same increase in Crit Damage as Attack will always win out----but only when the increase is below 150% and you're running over 60% Crit Rate. 

To clarify more, let's use a real monster example with everyone's favorite, homicidal clown, Lushen!

Assuming you're running 2 6 Star Attack Runes and have no other runes on you, the following damage can be expected.

As you can see from the red highlights in the chart, while initially even abysmally low Crit Rates can deal more damage than pure Attack, as multipliers increase, you need higher and higher Crit Rates to get the same damage output when using Crit Damage vs Attack. At 150% (understandably when Crit Damage looses its initial 50% head start), only 90 and 100% Crit Rates win.

Fatal vs Rage Sets

The implications of this mean that at the 40% increase (the increase you can get from choosing Fatal or Rage) any Crit Rate above 45 will have Crit Damage win all else equal. That means if  can get 45 CR or more and none of your subs are Attack or Crit Damage, you should probably build whatever it is Rage as your 40% set increase combined with the 80% slot 4 increase will give you exponentially more damage than Attack slot 4 with Fatal which will give you that 40% set increase plus 63% from slot 4.