Double attack and weapon skills, part 2

So many "known" things about random mechanics in FFXI seem poorly substantiated due to a lack of data, bad methodology when data are collected, and poor or non-existent analysis and interpretation after the data collection. Then again, it's not as though you really need to know, say, how many hits per attack round you can expect from a Kraken Club. Even if you have one, such considerations are beside the point.

That said, it's almost delightful to see some real data (not some useless parse), and even better when there are some easily tested hypotheses that follow from the purpose of the data collection. This thread on double attack during WS generated some interesting speculation about how many times double attack can process based on the data gathered but ventured no further, and no one really provided and tested a model of how DA interacts with weapon skills, the closest being a proposal that Penta Thrust may receive up to 3 DA "checks" per WS.

This proposal followed from data collection on TP return (a measure of number of hits in a WS) for Penta Thrust, which is summarized as follows:

10% DA rate (warrior subjob)
95% hit rate (lv 73 dragoon vs. lv 47-54 diatryma)

196 total WS

3 hits: 3 (.015)
4 hits: 42 (.214)
5 hits: 120 (.612)
6 hits: 30 (.153)
7 hits: 1 (.005)

However, I am not interested in seeing whether a "3 DA check" model is a good fit to the data since it is "known" that double attack cannot proc more than twice on a WS. (I hope this is a correct assumption. Besides, it doesn't seem likely that people who love to jack off to WS damage, and make their obnoxious asses known on popular FFXI forums, wouldn't run their mouths about a 8-hit Penta Thrust. Sometimes the persistent absence of evidence is strong evidence--NOT PROOF--of absence.) Rather, I'm looking to clarify how exactly double attack can proc twice at most based on my previous post.

As a reminder, I proposed the following models for how DA might work with WS: (1) double attack can proc twice on specific hits of the WS (thought to be the first two hits per FFXIclopedia), and (2) double attack may proc a maximum of two times on a WS (not restricted to specific hits). Is it even possible to tell the difference between these two models for Penta Thrust, given only 10% DA rate?

Fortunately, the probability distributions under each model are fairly easy to calculate (the calculations for Penta Thrust are similar to those for 3-hit WS last time) and are summarized in the following graph:



The difference between the two is fairly stark, so it wouldn't take that much data to support one over the other, assuming either one is true. In particular, the difference between the two models is most pronounced for the 6-hit and 7-hit cases. A sample proportion of .153 for 6-hits is very unlikely for the "2 DA maximum" model, where the theoretical proportion is .262. The "DA 2 hits only" seems a decent fit, so run with that.

The FFXIclopedia article on DA was changed February 10 of this year to state that DA can activate on the first two hits only (instead of being able to proc twice at most and on any of the hits). Aside from the fact that one cannot distinguish between the 4 ways DA can proc consecutively on two hits in Penta Thrust (saying it procs on the first two hits is nothing more than a guess if you don't know how it's programmed), I wonder if that change was motivated by the evidence of sample data or if it was just a shot in the dark. At least I found some evidence for that.

If you are interested in playing around with the probabilities of the number of hits for your favorite multi-hit weapon skill, the following is some R code I wrote to generate them. You can change p1 (hit rate), p2 (double attack rate), and y (number of normal hits in the WS) to suit your particular situation. Some slight modification would have to be made to isolate the probability of the first hit occurring for the purposes of calculating average WS damage (where fTP isn't 1.0).

# p1 - hit rate
# p2 - double attack rate
#  y - number of normal hits in the weapon skill

p1 = .95
p2 = .15
y = 2

# double attack can process on only two hits

p_2x = rep(0,(y+2))
for (i in 0:(y+2)) {
  p_2x[i+1] = sum(dbinom(max(i-2,0):min(i,y),y,p1)*dbinom(i-max(i-2,0):min(i,y),2,p1*p2))
}

# double attack may process a maximum of two times

p_max = rep(0,(y+2))
for (i in 0:(y+2)) {
  if (i < 2) {
    p_max[i+1] = sum(dbinom(max(i-2,0):min(i,y),y,p1)*dbinom(i-max(i-2,0):min(i,y),y,p1*p2))
    next
  }

  p_max[i+1] = dbinom(i-2,y,p1)*sum(dnbinom(0:(y-2),2,p1*p2))

  if (i != (y+2)) {
    p_max[i+1] = p_max[i+1] + sum(dbinom((i-1):i,y,p1)*dbinom(i-(i-1):i,y,p1*p2))
  }
}

# probability mass functions

round(p_2x,10)
round(p_max,10)

# expected number of hits

hit = seq(0,(y+2))
exp_hit_2x = sum(hit*p_2x)
exp_hit_max = sum(hit*p_max)

exp_hit_2x
exp_hit_max

Some checks: for a 2-hit WS, the two models are indistinguishable. As DA tends to 0%, the two models are indistinguishable in the limit. (The negative binomial distribution is degenerate when p2 = 0.) When hit rate is 100%, there are no number of hits less than the number of normal hits.