Legendary Dreadnought - Advancing Blows (Math Intensive)

[Kinerd]

Advancing Blows: Passive Bonus: You gain On Melee Critical Hit +1 to hit and damage for 6 seconds. Stacks 5 times.
(courtesy of ddowiki)

On average, what will our value for this buff be? Intuitively we recognize that it is possible to not get any crits in 6 seconds if we have less than 100% chance to crit, so it follows that our average value will be less than 5, but how to quantify this? Let us consider a toy problem where we attack once every 5 seconds with a crit chance of C. (Note: any attack time between 3 seconds and 6 attacks will result in the same average value, the key is that we are getting two attacks in our 6 seconds because getting the buff sets duration at 6 and losing the buff loses it entirely rather than losing one stack a la Blitz.) After 6 attacks, we will have an expectation value for each of...

0 buff
1 buff with 1 second remaining
1 buff with 6 seconds remaining
2 buff with 1 second remaining
2 buff with 6 seconds remaining
3 buff with 1 second remaining
3 buff with 6 seconds remaining
4 buff with 1 second remaining
4 buff with 6 seconds remaining
5 buff with 1 second remaining
5 buff with 6 seconds remaining

...and we can calculate to the next attack by doing...

0 buff at t7 = 0 buff at t6 * (1 - C) + [1 buff, 2 buff, 3 buff, 4 buff, 5 buff] with 1 second remaining at t6
1 buff at t7 with 1 second remaining = 1 buff at t6 with 6 seconds remaining * (1 - C)
1 buff at t7 with 6 seconds remaining = 0 buff at t6 * (C)
2 buff at t7 with 1 second remaining = 2 buff at t6 with 6 seconds remaining * (1 - C)
2 buff at t7 with 6 seconds remaining = 1 buff at t6 with 6 seconds remaining * (C)
3 buff at t7 with 1 second remaining = 3 buff at t6 with 6 seconds remaining * (1 - C)
3 buff at t7 with 6 seconds remaining = 2 buff at t6 with 6 seconds remaining * (C)
4 buff at t7 with 1 second remaining = 4 buff at t6 with 6 seconds remaining * (1 - C)
4 buff at t7 with 6 seconds remaining = 3 buff at t6 with 6 seconds remaining * (C)
5 buff at t7 with 1 second remaining = 5 buff at t6 with 6 seconds remaining * (1 - C)
5 buff at t7 with 6 seconds remaining = [4 buff, 5 buff] at t6 with 6 seconds remaining * (C)

...and it turns out that if we look at this for sufficiently high t we reach (or strictly speaking asymptotically approach) an equilibrium, where the losses and gains of every state are perfectly balanced. If we multiply out each state by its buff value and declare a set of C as 1, 1/2, ... , 1/9, 1/10, we get values for average buff of...

1	5
2	1.1625
3	0.679012346
4	0.490748355
5	0.387806897
6	0.321909816
7	0.275741168
8	0.241441324
9	0.214884001
10	0.193677982

...which looks pretty intense, but as the 1/2 indicates has a hidden pattern, which for C = 1/x is...

(2x-1) * (x^4+x^3+x^2+x+1)
(x^2+x-1) * x^4

...which is beautiful if still a little intense. It also hints that we can find an alegrabic expression for any number of attacks per our 6 seconds, but if we look at said expression for 3 attacks per our 6 seconds we get...

(3x^2-3x+1) * (x^8+2x^7+3x^6+4x^5+5x^4-26*x^3+23*x^2-8*x+1)
(x^3+(x-1)^2) * x^8

...which while still beautiful implies a 48th order polynomial factor if we get to 13 attacks per 6 seconds. Why is 13 important? Because by Vanshilar's Attack Index the attack rate for TWF at 45% (Haste + Boost) is 133.3 attacks per minute, or 13.33 attacks per our 6 seconds. A 48th order polynomial is a little too intense for me, so I will admit I just brute forced it out for 13 attacks per our 6 seconds and waited for the change in average buff to be less than 1E-15, good enough for jazz.

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The next question is what C we will use? I so happen to have a pure fighter SD khopesh user, so mine is 4/20 [crit rate of khopesh] * 1.9 [GTWF and capstone] = .38, which results in 4.95. In my anecdotal experience, when I can sustain continual combat I seem to keep Advancing Blows up pretty reliably. I use Improved Sunder to cut down my boost inactivity timer, but there is still a slight loss there, so that explains (or could explain) my empirical experience of seeming to lose the buff more frequently than the 0.003222316 probability of 0 buff suggests. It is a good question to ask just how long it takes to reach this (near enough) equilibrium, and for C = .38 it works out to 35.55 seconds, which to be honest is faster than I expected. If we take the average average buff over those 35.55 seconds, it still works out to 4.52. We get very close to the asymptote very quickly, so for long (even relative to 35 crummy seconds) fights it dominates and may as well be taken as the true value.

But what if we were to use the C of a THF greataxe user? Well, then we would have 2/20 * 1.1 = .11 because glancing blows never crit. This only reaches an average buff of 2.33, and only at 62.1 seconds, although the average average is still 2.13 so not much deflection. (This is also probably being slightly charitable because THF is slightly slower than TWF.) Obviously something like a falchion or eSoS would fare far better, but I at least cannot escape the reality of the eAGA, so I compare to it.

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mtl,dr: Advancing Blows favors khopesh over greataxe to the tune of 2.6 damage.

[scoobmx]

while it was definitely intuitive that advancing blows favored weapons with better threat range and faster attack rates, it's nice to put some numbers on it. thanks.