Phantom Roll and support for discretion to roll however you want

Before I discuss the optimality of several approaches to using Phantom Roll, I want to talk glibly about whether Phantom Roll even involves the use of a fair die in practice, which is the major assumption underlying Phantom Roll "strategies." I will use Pearson's chi-square test to check badness of fit of the following count data.

This is the only discussion I've seen so far (2006) that entertains the possibility that the outcome of Phantom Roll (I through VI inclusive) is not uniformly distributed. The point of the data collection was to find some evidence that the die becomes weighted in the presence of the optimal job associated with the specific roll (Bard with Choral Roll, etc.). You can read the thread for details.

The first example, with 700 uses of Corsair's Roll, did not yield compelling evidence against unbiasedness (p-value .0811).

The other eight examples involved sampling 100 times under varying conditions. At this point it bears reminding that the distribution of p-values under the null hypothesis of a fair die is (asymptotically) uniformly distributed (keeping in mind bin specification for the sake of generating histograms), as illustrated below with a bunch of simulated p-values sorted into histogram bins, given a sample size of 100.


I bring this up only as a reminder of what the possible p-values can be under the null hypothesis.

For the remaining eight data sets, tests for unbiasedness yield p-values of .4614, .2739, .3601, .007439, .7974, .3101, .09696, and .2763. Based on this crude analysis, only the data set for Healer's Roll with WHM present showed statistically significant evidence of biasedness (specifically 30/100 for a roll of I), but compared to the other non-significant results, it seems difficult to attribute this to something other than Type I error.

Of course, the primary question of interest was not whether Phantom Roll gives unbiased rolls regardless of situation, but whether the presence of the optimal job changes the "weight" of the roll. Tests for homogeneity for each specific roll (multiple testing duly noted) give p-values of .5402 (bard), .1077 (white mage), .6433 (ranger), and .1099 (thief).

Generally speaking, chi-square tests have pretty low power, and one tends not to "invert" these to (sets of) confidence intervals to get a good sense of how (in)adequate the sample sizes are. But considering this data as a whole there isn't a particularly good reason to think that the Phantom Roll die is biased.

Now, optimality of two Phantom Roll approaches

The following could basically be summarized as comparing the pros and cons of busting more versus busting less depending on how you go about doubling up.

There is a spreadsheet that provides a convenient summary of whether to Double-Up for various roll types, based on a criterion of conditional expectation (actual buff value), given the current roll total. Basically, consideration of (conditional) expected value is a formal way to make a decision that can be mostly carried out using common sense--you will never double up with a total of 11, as the expected value of the buff after Double-Up must be 0--but addressing borderline cases where it may not be obvious whether one should double-up, for example if your current roll is 6. I awkwardly call this the "expected value on double-up" (EVDU) approach.

The spreadsheet also gives an unconditional expected value of the roll after doubling up based on the expected value criterion, which could be useful for comparing different types of rolls on a "long-run" basis.

For wannabe nerds who can't even calculate the conditional expectations or understand probability, that is one way to go about it. Not unexpectedly, these min/maxing wannabe nerds frown upon conservative approaches that seek to minimize the probability of a Bust, with the implication that people who refuse to Double-Up on a 6 are "suboptimal." For the remainder of this post, I will call categorical refusal to Double-Up on 6 (unless 6 is unlucky), yet still Doubling-Up if one gets an "unlucky" total (therefore risking a Bust), as the "conservative" approach (and the only one I will consider in this post).

I am willing to bet that most of the people who advocate EVDU (implicitly or not) never actually bothered to compare EVDU with more conservative approaches quantitatively, especially for specific types of rolls. By quantitatively, I mean comparing (unconditional) expected values under each approach to see how much better in the long-run EVDU is, and also comparing the busting proportions under each approach to see how much riskier in the long-run EVDU is.

Personally, I don't really give a shit what rolling strategy a corsair actually uses, since to me it mostly falls under the purview of individual playing style.

Consider Corsair's Roll, for example. Under EVDU, the expected percentage increase in EXP is 15.66% while the conservative approach gives an expected increase of 15.55%, which to me is a really trivial difference. Moreover, the probability of busting under EVDU is .051 while the probability of busting "conservatively" is 0. If you are willing to assume an actual 5% (non-zero) chance of busting for a theoretical 0.11% long-run increase in EXP, fine. But here, the tradeoff between risk and reward is not all that good.

I also estimated the probabilities for the Corsair's Roll bonuses under each strategy (since I didn't want to waste even more time thinking about how to hand-calculate them) to make it easier to compare the strategies in probabilistic terms. (Relative frequencies may not add up to 1 due to rounding.)

COR Roll	EVDU	Conservative
Bust	.051	.000
8%	.134	.082
13%	.000	.309
15%	.193	.142
16%	.165	.114
17%	.095	.044
20%	.309	.309
24%	.052	.000

I colored the relevant probabilities one "side" might use to make a case against the other. Note that the probability of obtaining the "lucky" result (20% EXP increase) is the same regardless of approach. I also did the same for Hunter's Roll (melee and ranged accuracy) and Chaos Roll (melee and ranged attack) without the presence of the optimal job.

Again, the tradeoff between risk and reward is not so great. Whether you, as a corsair, want to make that tradeoff should be up to you and not to dumbasses who need to rely on mindless rules of thumb because they don't know any better. Personally, I would rather allocate all of my busting risk to another roll rather than to Corsair's Roll if the increased risk is actually worth it on another roll. But when is it worth it? I repeat the above exercise with both Hunter's Roll and Chaos Roll, rolls that are available early on.

For Hunter's Roll, the expected value under EVDU is 29.63 accuracy, and 28.09 taking the more conservative tack. Clearly, a 1.54-point difference in average accuracy is such a profound increase as to assume a greater risk of busting. The estimated probabilities are given below.

RNG Roll	EVDU	Conservative
Bust	.135	.057
20	.000	.3o9
25	.194	.142
27	.161	.101
30	.124	.063
40	.264	.264
50	.122	.064

For Chaos Roll, the expected value under EVDU is 18.6% attack increase (47.5/256), and 17.6% attack increase (45.0/256) playing it conservatively. Again, a 0.98% average difference in attack obviously warrants the increased risk of busting. The estimated probabilities are given below.

DRK Roll (xx/256)	EVDU	Conservative
Bust	.134	.058
32	.000	.3o8
40	.193	.142
44	.163	.101
48	.124	.063
64	.265	.265
80	.124	.063

Sure, a 1-point or 1% difference may be important enough to you, but 0.11%?

I spent time constructing this post while considering whether to level corsair to 75. (I won't but not based on what I found in this post. Ultimately I'd rather buy an account with a ready-to-play COR75 than waste time leveling another job to 75.) Take-home message: do whatever the hell you want as long as you can support it logically.