Blitzer's Roll

Blitzer's Roll (COR Lv.83)
Reduces melee attack delay for party members within area of effect. Lucky number: 4. Unlucky number: 9.

Seeing as this is the only interesting thing from the update as far as TP-burning for one-handed melee DD vs. Hasso (you self-identify as WAR/DRK/DRG/SAM? Are you a tool? It's either piercing Hasso or non-piercing Hasso) is concerned (although that supposes anyone still cares about FFXI), let's anticipate what the effect might actually be.

Thoughts:

• If this really reduces melee attack delay a la Sword Strap, maybe it will be exactly the same form of delay reduction as that from dual wield. (Why not?) If Blitzer's Roll actually stacks with dual wield and the bonuses are good, this would benefit dancer and ninja the most.
• One-handed melee jobs get more from Blitzer's Roll even if it reduces weapon delay separately from dual wield, but it doesn't mean it'd be bad for two-handed melee jobs (depends on the actual bonuses), just that Fighter's and Samurai would be better. n-hit builds are overrated.
• What if it's "job ability haste"? Wouldn't it be funny if the bonuses were large enough to obviate the use of Hasso and Haste Samba? Among other things that would follow: why are we using two-handed jobs for TP-burn then?

Samurai Roll and Fighter's Roll... for ninja

Having shown in the past that Fighter's Roll can be better than Samurai Roll for increasing warrior's damage output (despite the lack of consideration of WS delay, which actually would favor Fighter's Roll), I thought it might be nice to have a worked example for ninja (this time accounting for WS delay).

This image (Imageshack host) summarizes the computations, based on 95% hit rate, 15% DA rate, 55% haste, 40% dual wield, and a "cRatio" of 1.5, with a 10% critical hit rate, with Blade: Jin as the weapon skill. The specific katana combination is Hochomasamune/Uzura, which is something I would consider getting as a "high-value" option, meaning that it's a highly effective option that I don't have to waste as much time doing boring shit to get this as I would for other options.

First, let's start with Samurai Roll (augmented by the presence of a samurai). The following is a table summarizing the relative increase in the (theoretical maximum) rate of damage from each of several desirable Samurai Roll outcomes.

Roll total      Bonus     Damage/second     Relative efficiency
---------------------------------------------------------------
 -              -               113.607                       -
 2             42 STP           123.979                   9.13%
 7             26               119.672                   5.34%
 8             30               121.410                   6.87%
 9             32               121.414                   6.87%
10             34               121.458                   6.91%
11             50               126.684                  11.51%

The relative increase for each store TP bonus is decent, but not as high as you would expect for two-handed weapons since the proportion of total damage from Blade: Jin for katana is realistically never going to be as high as the proportion of total damage from the (generally) best weapon skills for two-handed weapons (e.g. Drakesbane, Raging Rush, Tachi: Gekko).

In contrast, Fighter's Roll (here augmented from the presence of a warrior) increases auto-attack damage, weapon skill frequency, and weapon skill damage, so it's not a surprise that Fighter's Roll is generally easily superior to Samurai Roll for increasing rates of damage:

Roll total      Bonus     Damage/second     Relative efficiency
---------------------------------------------------------------
 -               -              113.607                       -
 5             15% DA           129.567                  14.05%
 7             11%              125.291                  10.28%
 8             12%              126.358                  11.22%
10             13%              127.427                  12.16%
11             19%              133.857                  17.82%

Note that these are supposed to be the actual DA percent bonuses after the infamous August 2007 version update (source), you know, the one that resulted in ninja being disregarded as DD. If they were what they were before that update, assuming there was a change (higher than what they are now), Fighter's Roll would be even better.

So, if Fighter's Roll can be shown to be better for the job that gets the least relative benefit from Fighter's Roll (warrior), as well as for ninja (though this could be considered self-evident), are there even any legitimate reasons to use Samurai Roll? No, "TP overflow" is an incorrect and stupid answer that betrays a lack of conceptual understanding.

Now, that's not to say Samurai Roll has no application. Maybe it would be desirable to maximize the ratio of damage efficiency to TP "fed" to a mob rather than consider damage efficiency in isolation. I doubt Monk's Roll would do the job, but a case could be made for Samurai Roll.

Also, if it is desirable to maximize weapon skill frequency, Samurai Roll would generally be better for that purpose than Fighter's Roll. (Also related is maximizing TP per hit, which would be desirable for dancer.)

Also, Fighter's Roll can be said to be "contraindicated" for use with multi-hit weapons (that don't use virtue stones) such as Magian multi-hit weapons. (But this assumes Magian multi-hit weapons are actually good, which is not necessarily true.)

Aspir and Drain modeling: an incomplete picture

Earlier, I made some bold statements (at least by my standards) about the distribution of unresisted values of Drain and Aspir. I proposed a model from which I can make explicit, testable predictions.

Specifically, I wanted to see if the model holds at 114 dark magic skill, which can be attained by subbing /DRK on any job without any native dark magic skill (in my case NIN/DRK). Based on the model I described previously, the maximum Aspir without any potency-increasing equipment used is 114/3 + 20 = 58 MP, and the minimum is half that, or 29 MP.

I went out to cast Aspir on Stone Eaters (North Gustaberg (S)) and after the seventh cast I obtained an Aspir of 63 MP, which exceeds the stipulated maximum, so the model doesn't hold for 114 dark magic skill. The following are the observed data values (in order observed):

51 44 56 49 25 51 63 53 35 62 62 46 41 32 44

I then cast Drain on the same Stone Eaters with 246 dark magic skill (NIN75/SCH35 with Dark Arts) and obtained the following results (stem-and-leaf plot):

   6 | 49
   7 | 
   8 | 4
   9 | 
  10 | 
  11 | 
  12 | 
  13 | 4445
  14 | 68
  15 | 044
  16 | 46
  17 | 37
  18 | 9
  19 | 4
  20 | 23789
  21 | 79
  22 | 026
  23 | 55
  24 | 277
  25 | 36
  26 | 01346

(The reason I used Drain and not Aspir was that my scholar is only level 35, and Aspir is accessible at level 36. I would prefer to gather data for Aspir because Aspir values are obviously less variable than Drain values.)

Given 246 dark magic skill, the predicted maximum for Drain is 266 and the predicted unresisted minimum is 133 (both without any potency gear or day/weather effects), so the distribution of Drain (as represented by the sample) seems consistent with the model.

In conclusion, if one were to be technical in describing the scope of the model I proposed earlier, I would say the model is (likely) valid for Drain between 246 and 300 dark magic skill. It is valid for Aspir between 269 and 300 dark magic skill. And, finally, it is valid for Drain II between 285 and 300 dark magic skill. What happens in the 100s is just not that relevant. (In case you are wondering what data I am referring to, you'd have to check my old posts on Drain and Aspir).

The "new" melee pDIF

This post is definitely not for those who neither understand nor care about what melee "pDIF" is all about and why it can be of interest, so I find no point in making some sort of "for dummies" kind of introduction and will just jump into the results.

First, a reference to the "new" melee pDIF should be seen as a sarcastic gesture, as there likely have no been wholesale changes to pDIF after the August 2007 version update that brought the gameplay-altering "two-handed weapon adjustment." Therefore, the following results are assumed to reflect the actual changes to pDIF made in August 2007.

Data and results

The guy who plays Masamunai (currently of Cerberus) provided this spreadsheet of data, having tabulated the observed damage values for various ratios of attack to defense (without level correction), using both one-handed and two-handed weapons, on level 63-65 Lesser Colibri and then "standardizing" them to approximate observed pDIF values (acknowledging estimation error associated with in-game truncation of values). There are more details concerning the raw data and he provided his own analysis, but I prefer to do my own analysis so you don't necessarily have to review the spreadsheet yourself.

The following is an image attempting to plot 67,123 of the observed pDIF data values (almost of all the data) to show primarily how the minimum, maximum, and (most important to me) mean pDIF for both critical and non-critical ("normal") hits varies with the ratio of attack to defense:



It is somewhat difficult to plot 67,123 data values cleanly and elegantly with limited resolution, so I exploited transparency of data points, resulting in narrow "bands" that vary in opacity from top to bottom, an attempt to illustrate roughly the relative "density" of observed values. Each band represents the entirety of the data collected for a given attack/defense ratio. Another interpretation is that each band represents the observed conditional distribution of pDIF for a given attack/defense ratio.

The bands for critical pDIF are generally less "dense" or less opaque than those for normal pDIF, reflecting that fact that there are many more data points for normal pDIF (55,956 versus 11,127). Also, the bands are generally most translucent at the endpoints, reflecting the fact that the observed data at the extremes of each conditional pDIF distribution (for a given attack/defense ratio) occur relatively less frequently, which is consistent with the idea that pDIF is now a function of two uniform random variables (either the sum or the product), which follows a trapezoidal(-like) distribution. (But I will not be discussing probability distributions today.)

Aside from the plotting of the data values, regression lines for the mean pDIF (controlling for attack/defense ratio) were also plotted (lines based on ordinary least squares, which is justifiable as there are a lot of data points involved for each level of attack/defense considered). Regression was done in an informal piecewise fashion, as there are specific ranges of attack/defense ratio where the variance of pDIF is obviously not constant, specifically for three cases:

• where there is a critical pDIF upper limit imposed (3.15 when attack/defense is approximately greater than 1.65)
• where there is a normal pDIF lower limit imposed (1.00 when attack/defense is between 1.25 and 1.5), and
• where the mode of normal pDIF is 1.00 and the mode does not occur at the left endpoint of the pDIF distribution (when attack/defense is less than 1.25). It should be noticed that it is impossible to discern the mode of pDIF (conditional on a given attack/defense ratio) based on the above graph. One would have to consult the original source as cited above.
  

I hope that will suffice as an explanation for the elements of the graph.

Interpretations and conclusions

These are a few of the things one could take away from the graph above.

Aside from the maximum attack/defense ratio attainable, there appear to be no differences in pDIF between one-handed weapons and two-handed weapons. I have incorrectly thought otherwise in the past, but I assumed people who cared about this knew what they were talking about. Obviously not.

While there is no data for two-handed weapons below 1.398 attack/defense ratio, I would invoke model parsimony and assert there is no good reason to expect differences at lower values of attack/defense. Although it is not shown above (and cannot be shown above cleanly), 2.00 is the maximum attack/defense ratio for one-handed weapons, and 2.25 is the maximum attack/defense ratio for two-handed weapons. Support for the these maxima can be found in the spreadsheet.

The ceiling on critical hit pDIF first occurs near 1.65 attack/defense. Moreover, the value of the ceiling, 3.15, is the modal (most frequently occurring) pDIF for attack/defense ratios above 1.65.

Mean pDIF, as a function of attack/defense, does NOT increase at the same rate for critical hits as for normal hits for a given value of attack/defense. A consequence of this is there is no pat way to relate normal pDIF to critical pDIF, like critical pDIF = normal pDIF + 1. To see what I mean, refer to this blog entry (JP), particularly the first image, to get a sense of how pDIF was incorrectly perceived more than a year after the August 2007 version update (a mish-mash of the critical hit pDIF ceiling of 3.15, increased attack/defense ratio maximum, and the old pDIF model).

Irrelevant considerations

This is a matter of personal preference, but I consider the so-called "secondary randomizer" an irrelevant red herring. pDIF as a product of two uniform random variables or sum of two uniform random variables, so what? (But I will note that the slope of mean pDIF without the second random factor does not change if the factor is added and does change if the factor is multiplied.) I just know it's there and I can explain what can cause it, but it is not very important for estimating mean pDIF, which is why I even made this post in the first place.

I also do not care about exactness of any pDIF model. Approximately true is fine with me as far as modeling rates of damage is concerned. (There are other factors when completely ignored or incorrectly computed that cause much more error than mere sampling error based on 60,000+ samples).

Formulas for mean pDIF as a function of attack/defense ratio and whether the weapon is one-handed or two-handed

These formulas are based on the regression estimates. (You may have noticed discontinuities in the piecewise mean pDIF functions suggested in the graph, but I do not care that much about fudging the estimates to eliminate that.) For normal-hit pDIF, which I have denoted as M_Normal, the estimated mean of M_Normal, as a function of H, the number of hands required to wield a weapon (H = 1, 2), and R, the attack/defense ratio (level-corrected or otherwise), is



For critical-hit pDIF, the functional relationship between that and H and R is



The following is the output of the regression procedure. I only include this to show that there is no reason to expect that the coefficient of determination be high, mainly because there is inherent variability of pDIF.

***Regression for normal pDIF, ATK/DEF < 1.25****

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.225403   0.010107    22.3   <2e-16 ***
ratio       0.782699   0.009748    80.3   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.1621 on 12257 degrees of freedom
Multiple R-squared: 0.3447,     Adjusted R-squared: 0.3446 
F-statistic:  6447 on 1 and 12257 DF,  p-value: < 2.2e-16 



***Regression for normal pDIF, 1.25 < ATK/DEF < 1.5****

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.07274    0.04129   1.762   0.0781 .  
ratio        0.90232    0.02969  30.390   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.2369 on 14254 degrees of freedom
Multiple R-squared: 0.06085,    Adjusted R-squared: 0.06078 
F-statistic: 923.6 on 1 and 14254 DF,  p-value: < 2.2e-16 



***Regression for normal pDIF, ATK/DEF > 1.5****

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.294339   0.016866  -17.45   <2e-16 ***
ratio        1.162306   0.009566  121.50   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.2518 on 29479 degrees of freedom
Multiple R-squared: 0.3337,     Adjusted R-squared: 0.3336 
F-statistic: 1.476e+04 on 1 and 29479 DF,  p-value: < 2.2e-16



***Regression for critical pDIF, ATK/DEF < 1.65****

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.94139    0.01739   54.14   <2e-16 ***
ratio2       1.07335    0.01273   84.29   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.232 on 6745 degrees of freedom
Multiple R-squared: 0.513,      Adjusted R-squared: 0.5129 
F-statistic:  7106 on 1 and 6745 DF,  p-value: < 2.2e-16 



***Regression for critical pDIF, ATK/DEF > 1.65****

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.59673    0.04287   37.25   <2e-16 ***
highratio    0.68607    0.02344   29.27   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.1905 on 4377 degrees of freedom
Multiple R-squared: 0.1637,     Adjusted R-squared: 0.1635 
F-statistic: 856.8 on 1 and 4377 DF,  p-value: < 2.2e-16 

Drain and Aspir minimum and maximum

Figuring out things that were already figured out years ago

After reviewing all my posts on Drain and Aspir, I finally realized that I had enough information to specify the minimum and maximum unresisted values, as a function of dark magic skill, for both Drain and Aspir, and since the distribution of unresisted values can be considered approximately uniformly distributed (acknowledging flooring effects), the mean Drain and Aspir can also be specified with reasonable certainty.

Of course, this was something that was figured out years ago. This Drain and Aspir summary page (JP) has some interesting things to say about Vampiric Mitts/Boots, Diabolos's Pole, Y's Scythe, and factors other than equipment, day/weather, and magic burst affecting Drain that do not affect Aspir, but I will leave it to those interested to check out that source while I summarize the the minima and maxima for Drain, Aspir, and Drain II, and speculate on those for Aspir II.

All specified extrema are defined for dark magic skill between 0 and 300, and for clarity's sake, factors that increase Drain or Aspir potency are not included in the statements aside from dark magic skill. It should be understood that any of those other factors increase the maximum and minimum by some multiplicative constant, and these constants can be multiplied in succession (generally, the order of multiplication is first the constant for equipment, then that for day/weather, then that for magic burst, with some twists on Drain potency you can look up yourself) with flooring after each multiplication step to obtain the final values of the maximum and minimum.

Maximum and minimum of Drain

The maximum unresisted Drain is

skill + 20,

and the minimum unresisted Drain is

floor(0.5(skill + 20)).

It follows that the mean unresisted Drain is approximately

0.75(skill + 20).

Currently 320 HP is the highest possible Drain return before any other potency-increasing factors are considered.

Each 1-point increase in dark magic skill (up to 300 total skill) increases the maximum Drain by 1 HP. Other potency-enhancing factors can increase the magnitude of this marginal return.

Maximum and minimum of Aspir

The maximum unresisted Aspir is

floor(skill/3 + 20),

and the minimum unresisted Aspir is

floor(0.5floor(skill/3 + 20)).

It follows that the mean unresisted Aspir is approximately

0.75(skill/3 + 20).

Currently 120 MP is the highest possible Aspir return before any other potency-increasing factors are considered.

Each 3-point increase in dark magic skill (up to 300 total skill) increases the maximum Aspir by 1 MP. Other potency-enhancing factors can increase the magnitude of this marginal return.

Maximum and minimum of Drain II

This discussion of the Hirudinea Earring effect (Hirudinea Earring seems to increase the potency of Drain or Aspir by either 2.5% or 3%) also has Drain II data that is consistent with the following expressions for the minimum and maximum of Drain II.

One thing to note is that Drain II has less variability than Drain I. (A dot plot would help show this.) Another thing to note is that the observed minima seem as though they are based on the expression for the maximum of Drain I. Then the minimum unresisted Drain II is

skill + 20.

The maximum unresisted Drain II is

skill + 85.

The mean unresisted Drain II is

skill + 52.5.

Each 1-point increase in dark magic skill (up to 300 total skill) increases both the minimum and maximum Drain II by 1 HP. Other potency-enhancing factors can increase the magnitude of this marginal return.

Currently 385 HP is the highest possible Drain II return before any other potency-increasing factors are considered.

Note that unlike Drain and Aspir, the variability of Drain II values appears not to vary with dark magic skill. Instead, varying skill merely shifts the distribution of unresisted values to the left or the right.

Minimum and maximum of Aspir II?

While I have yet to see anything regarding Aspir II, I wouldn't be surprised to see the minimum be

floor(skill/3 + 20)

and the maximum be

floor(skill/3 + 85),

but this is subject to verification based on others' experience.

Summary

The following table (image) summarizes the maximum, minimum and (approximate) mean unresisted value of Drain, Aspir, and Aspir II as a function of dark magic skill ("capping" at 300 skill) when no other potency-increasing factors are present.



Other potency-increasing factors increase the minimum and maximum by their corresponding multiplicative constant (e.g. that for equipment), with flooring after each multiplication step.

Increasing Drain potency

This is the last of a series of posts describing my investigation of Drain mechanics, particularly what things increase Drain potency and by how much. The following findings are based yet again on casting Drain on Zvahl Fortalices, details of which (justification, limitations, and whatnot) I described in previous posts (but weren't that detailed).

The motivation: Excelsis Ring

Having received the Excelsis Ring from the Bastok sequence of Wings of the Goddess nation-oriented quests, I wondered whether this ring actually increased the potency of Drain, which should be understood as increasing the maximum Drain or the average (mean) Drain value.

Having previously shown that Drain potency could possibly "cap" at 300 dark magic skill holding all other relevant factors fixed, I felt that it would be useful to verify, in the process of determining whether Excelsis Ring actually increases Drain potency, that Drain potency doesn't increase at higher levels of skill (again), specifically 331 dark magic skill.

After establishing two baseline samples at 331 dark magic skill given no other potency-enhancing equipment (or day/weather effects), I then obtained a sample adding Excelsis Ring to the "baseline." After that, I figured it would be helpful also to quantify the effects of Pluto's Staff and Dark weather on Drain potency, both relative to the baseline.

Wait a second... aren't you ever going to describe an experimental procedure in more detail?

I could, but seriously, is it that hard to think of equipment swaps that guarantee your current HP is 350-400 HP lower than your maximum HP? Hint: HP-increasing equipment upon Drain casting counts. Also, it isn't that hard to think of ways 331 dark magic skill can be attained. I could take a screenshot of the equipment used, but that would mean logging into the game.

Results

The above image is yet another set of dot plots providing a visual summary of the five samples obtained. In the past, I made no statements about the actual distribution of Drain (controlling for resist level), but by now it should be pretty obvious that a uniform distribution is a good model for the data, so further insights will be based on this model.

Also, there were a few Drain resists observed, but as they are obviously resists they can be ignored for the purposes of potency estimation, and the vertical lines denote the suspected minimum (160 HP) and maximum HP (320 HP) drained for the control samples, to be discussed later. A table (ASCII, yes I am that lazy) summarizing the extrema and median of the unresisted Drain values for each sample follows:

                    ·--------·-------------·-------------·------------·
                    |    n   |   Minimum   |   Maximum   |   Median   |
·-------------------·--------·-------------·-------------·------------·
|   Baseline 1      |   48   |       162   |       305   |    223.0   |
|   Baseline 2      |   50   |       160   |       319   |    238.0   |
|   Excelsis Ring   |   51   |       171   |       336   |    255.0   |
|   Pluto's Staff   |   61   |       184   |       365   |    263.0   |
|   Dark weather    |   30   |       190   |       349   |    288.5   |
·-------------------·--------·-------------·-------------·------------·

Speculating on what the extrema could tell us, it looks as though the variability of the data (as indicated by the range of observed values) is highest for Pluto's Staff, which is consistent with the idea that Pluto's Staff increases potency by some multiplicative factor. Also of interest is the possibility that the minimum is merely half of the maximum, so the observed minima could provide some insight on what the maxima should be.

Another obvious thing to note is that even at 331 dark magic skill, there is still no evidence that Drain potency is higher compared to that at 300 skill, so it is reasonable to conclude, considering all the data to date, that the Drain "cap" (holding other potency-enhancing factors fixed) seems to be met somewhere near 300 dark magic skill (if not exactly at 300). Technically, I could invoke an equivalence test yet again (as I did in a previous post), but you would have to be a tool to demand one here.

Since we know that a Pluto's Staff and single Dark weather each increase direct-damage magic (as shown by changes to the initial damage of any of the Bio spells, allowing for truncation) by 15% and 10% respectively, it wouldn't be surprising if these factors increased Drain potency by the same amount. It stands to reason that Excelsis Ring could behave in the same way (but increasing potency by a lesser amount).

One way to estimate the mutiplicative factors is to divide the observed maximum Drain for each of the non-baseline samples by the observed maximum for the baseline samples combined.

Excelsis Ring appears to increase Drain potency by a percentage near (336/319 - 1)100% = 5.3%. But I suspect the maximum Drain for the baseline is 320 (I just wasn't lucky enough to observe it), which would mean the increase in Drain potency from Excelsis Ring could be 5%.

Keeping in mind that the maximum Drain for the baseline could be 320, then Pluto's Staff appears to increase Drain potency by 15%, and Dark weather appears to increase Drain potency by 10%, and these figures are consistent with their effects on direct-damage magic.

An alternative, more statistical method of potency estimation is to divide the mean Drain value for each non-baseline sample by the mean Drain value of the pooled baseline sample. The ratio is then an estimate of the multiplicative factor for the potency-increasing equipment of interest. From the basic bootstrap, a set of simultaneous 95% confidence intervals for the ratios of the means can be obtained.

                    ·----------------·-------------------------------·
                    |   Mean ratio   |   Confidence interval (95%)   |
·-------------------·----------------·-------------------------------·
|   Excelsis Ring   |     1.088056   |        (0.996482, 1.183355)   |
|   Pluto's Staff   |     1.144933   |        (1.057611, 1.237342)   |
|   Dark weather    |     1.193882   |        (1.085225, 1.303625)   |
·-------------------·----------------·-------------------------------·

(Edit: 07/24/2010. I realized the bootstrapped data also included Drain resists. Not correct to include them in the data. Estimates have been corrected.) Not surprisingly, statistical significance for the potency effect of Excelsis Ring isn't achieved, but this is merely a consequence of the sample size. The effects of Pluto's Staff and Dark weather are large enough to yield statistical significance.

Conclusion and open issues

Excelsis Ring appears to increase the potency of Drain by 5%. Pluto's Staff was verified to increase the potency of Drain by 15%, and single Dark weather was verified to increase the potency of Drain by 10%.

Now, this still leaves the accuracy effect of Excelsis Ring to be determined, but quantifying it is difficult as it likely does not impart a large accuracy bonus (if there is one at all). If I were to do so, I'd be interested in determining whether Dark weather or Darksday has an effect on Drain accuracy. If so, it wouldn't be unreasonable to generalize to other types of magic and conclude that the weather or day can affect magic accuracy in general.

Regarding the mechanics of Drain in particular, it seems that the distribution of Drain could be uniform (discrete or continuous with truncation, it doesn't really matter) and that it is parameterized only by the maximum, with the minimum possibly being exactly one-half the maximum. Provided that this holds, one obvious implication is that the variability of Drain increases with dark magic skill (up to a point), and that it also increases with other various potency factors present. Taken altogether, these factors can make it very difficult to draw any conclusions about Drain (and Aspir by analogy) based on eyeballing alone.

As far as the distribution of Drain values given some sort of resist, it still isn't clear how resists relate to non-resists, but resists aren't that interesting to me so I would never willingly investigate Drain resists.

Finally, why have I never made any explicit claims about what the actual value of the Drain cap should be? Considering that Zvahl Fortalices take increased magic damage, it's possible that they also take increased Drain damage, and 320 might not be the "true" cap given that no other potency-enhancing factors are in play. But I was interested in differences, not actual amounts.

When to use Snake Eye?

What is the appropriate use of Snake Eye?

Recently, I was challenged on the notion that "it is typical to use Snake Eye to get a lucky Phantom Roll total or avoid an unlucky one" (my words).

Instead, it is recommended to save Snake Eye to get an 11 (XI) or to avoid an unlucky roll, while not using Snake Eye to get a lucky total. (Note that there is no reason that Snake Eye couldn't be used on 10 if Snake Eye wasn't used to get there regardless of whether there is intent to use Snake Eye to get a lucky roll.) Can it can be shown probabilistically which Snake Eye tactic is better?

Example: Corsair's roll

Corsair's roll being the only relevant roll for experience parties, let's just use this as an example. For reference, here are the experience point bonuses for "desirable" outcomes based on a tactic of rolling based on the expected value of the bonus if you continue to Double Up, given your current total, exceeding the bonus for your current total, which I have called an "expected value on Double Up" criterion (EVDU). Of course, with Snake Eye available, the unlucky total can be avoided entirely. (See this spreadsheet if that doesn't make sense):

Roll total     EXP Bonus
------------------------
 5             20%
 7             15%
 8             16%
 9              8%
10             17%
11             24%

For all probability calculations, assume job abilities occur instantaneously without recast restrictions for the sake of clarity. Also assume for simplicity that Phantom Roll always has one of several desirable outcomes in effect (Phantom Roll never wearing off) and that Snake Eye can be used only once toward the final outcome. Also assume Phantom Roll lasts either 5 or 10 minutes because who merits Winning Streak? Finally, let us assume no bust-mitigation measures (for now).

First, let's consider the "recommended" course of action, which is to save Snake Eye to get an eleven or to avoid an unlucky roll. In other words, use Snake Eye on 9 or 10. This kind of makes sense to do, since eleven lasts longer and I don't really see a downside to Corsair's roll lasting 10 minutes.

The probability distribution (numbers rounded to ten digits) of getting one of the possible outcomes (shown previously) follows:

Roll total     Probability
--------------------------
   5           .3087705771
   7           .1935656731
   8           .1657878952
  10           .1333804876
  11           .1470336084
Bust           .0514617630

Obviously, with 11 lasting twice as long as each of the others, the relevant consideration is the "proportion of time spent under each effect," so the probabilities need to be adjusted:

Roll total     Probability
--------------------------
   5           .2691905221
   7           .1687532701
   8           .1445362135
  10           .1162829808
  11           .2563719263
Bust           .0448650871

Based on that, the expected (long run) EXP bonus is 18.357% if you use Snake Eye to get 11 and to avoid a 9.

Now, what if you take a more conservative tack and Snake Eye given three conditions: when you are on a 4 (one less than lucky, 5), when you are on a 9 (unlucky), or when you are on a 10? The probabilities (adjusting for time duration differences) shake out as follows:

Roll total     Probability
--------------------------
   5           .4864098317
   7           .1305837864
   8           .1050579060
  10           .0752777122
  11           .1621366109
Bust           .0405341527

Based on that, the expected (long run) EXP bonus is 18.538% if you use Snake Eye to get a 5 (lucky), 10 (avoid an unlucky), or 11. This approach is better probabilistically and you have a lower probability of busting!

For the sake of comparison, the expected EXP bonus if you take a timid approach and avoid busting completely (never getting an 11), but use Snake Eye where it makes sense, is 17.89%.

Um... why don't you account for bust mitigation?

Suppose hypothetically that you can re-roll (start over) indefinitely (and instantaneously) to avoid a bust. Then the asymptotic probabilities for both Snake Eye tactics are as follows:

Roll total     Snake Eye on 9 or 10     Snake Eye on 4, 9, or 10
----------------------------------------------------------------
 5             .2818350774              .5069589846
 7             .1766800353              .1361005051
 8             .1513254427              .1094962435
10             .1217450847              .0784579382
11             .2684143600              .1689863286

The (limiting) expected EXP bonus when using Snake Eye on 9 or 10 is 19.22%, while the (limiting) expected EXP bonus when using Snake Eye on 4, 9, or 10 is 19.322%.

Conclusion

Two Snake Eye tactics were considered for Corsair's roll. One tactic is to use Snake Eye on 9 or 10 (emphasizing getting an 11 at the expense of getting a 5), and the other tactic is to use Snake Eye on 4, 9, or 10 (no emphasis on getting an 11).

Using Snake Eye on 4, 9, or 10 is a superior tactic regardless of attempts at bust mitigation.

By "going for broke" (getting an 11), you give up a sure thing, and the trade-off is not worth it (even if the difference is slight), and this is before considering time spent under suboptimal EXP bonuses in the process of achieving a desirable total.