pDIF distributions

Anyone with at least a passing interest in how the game calculates physical damage is likely aware of the so-called pDIF factor, which is a function of the ratio of one's attack to the opponent's defense (ATK/DEF). A given value of ATK/DEF corresponds to a specific range, or distribution, of possible damage values constrained by a minimum and a maximum, and one can treat the pDIF graph as a concise summary of the possible distributions from 0 to 2 ATK/DEF.

But what is the underlying probability distribution for all possible ranges? A uniform distribution with the parameters of pDIF minimum and pDIF maximum has strong intuitive appeal because random numbers from a uniform distribution are simple to compute. It would seem impractical for the programmers to mess with normal distributions, and the apparent reliability of pDIF max and min in predicting a range of damage values (at least for one-handed weapons these days) basically precludes the use of standard normal. (It makes no sense to parameterize a normal distribution with pDIF min and max, anyway.)

Moreover, assuming a uniform distribution makes it easier to calculate damage with an expected value of pDIF, which would be just the midpoint between the endpoints of a given distribution if it and all others were really uniform.

But is it really the case? To get a sense of it I considered what would be the easiest, least riskiest, least costly and least time-consuming way to collect data without actually paying attention to the game, which basically meant poking at Campaign fortifications with dual-wield katanas I already had (Mamushito +1).

I acknowledge that my original goal in doing so was not really to gather evidence for a uniform distribution but rather to see to what extent the distribution of damage values might change with an increase in attack from a meat mithkabob (told you I was going on the cheap, and I was thinking maybe the distributions aren't uniform). I also ended up concluding that fortifications are a nice target for testing this, in a way; because of the extremely limited range of actual damage values due to their damage-reduction properties, I had no need to trouble myself with appropriate histogram binning.

A rank promotion later, I put together some "composite histograms" in Excel to summarize my peculiar results:

"Lower attack"

"Higher attack"

While the "higher attack" case didn't yield any surprises, the "lower attack" case was definitely not uniform in the least, but why the bias toward 6 damage? What's up with that?

Nonplussed, I attempted to find any snippets of comments regarding pDIF using Google, and I came across an interesting statement about pDIF, which is paraphrased as follows:

"For a given pDIF distribution, if pDIF 1.0 is within the range of possible pDIF values, pDIF 1.0 has a probability of 1/3, with the other possible values being uniformly distributed otherwise."

This statement, if true, would apply to cases of ATK/DEF between 0.5 and 1.5, which pretty much encompasses everyday conditions when fighting. It seems plausible enough in light of the data I collected, but why would anyone go to the trouble of making it so?

At this point, I thought it might help to try some simulation with random uniform numbers to see if I could obtain similar results to what I showed in the graphs above, and by doing so illustrate a possible method for creating a bias toward pDIF 1.0. The biggest problem was making an educated guess about the fortification's attributes, especially the damage reduction property, but I had to run with something.

For the "higher attack" case I managed to get a similar result to my obtained data with a ATK/DEF ratio of about 1.521:


For the "lower attack" case I was unsure how to simulate a result similar to what I obtained from data collection and I looked for further clarification. One idea held that pDIF 1.0 at the endpoint of a distribution is the result of random values below 1.0 (or above) being rounded up (or down) to 1.0. But this doesn't jibe with a large data set I collected while poking at a fortification (when I regrettably neglected to record STR and attack) where 6 damage (the mode) seems to correspond to 1.0, yet 5 damage was recorded also:


But wait! Ignoring the 6, don't the data suggest a long right-hand tail? A uniform distribution doesn't have tails! And why does the range of damage go from 5 to 14? At 395 attack, maximum damage was shown to be 11. I probably wasn't using a meat mithkabob, and I try to maximize attack speed so I don't bother with attack equipment. But, it might be useful for reference later.

So ultimately, I have no conclusion that I'd rely on. I did perform another simulation to demonstrate how the "lower attack" case described a long time ago might come to pass. Let's say about 25% of all pDIF random values on the interval [1,1.65] (ATK/DEF ratio 1.375) end up being converted to pDIF 1.0, ensuring that 1.0 is the mode of any pDIF distribution that contains it. Otherwise, the data are random uniform numbers. Then, this criterion works in my simulation (rather, I ran the simulation a bunch of times until I found a result that looks similar to the one above):


It's too bad getting a feel for the underlying distribution from a random sample is quite annoying in the case of pDIF. Maybe I'll try again with attack lower than 344 next time.

Data collection was made possible with the "offense detail" feature in kparser. Otherwise I wouldn't even bother.