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.)

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.

Fighter's Roll versus Samurai Roll

Time for another unnecessary applied probability exercise. This time, I compare the efficacy of Fighter's Roll to that of Samurai Roll in terms of improving rate of damage. There are several factors to consider when making this assessment, including how to apply roll tactics and the attendant theoretical distributions of possible results that follow from combinations of various tactics.

Note! Fighter's Roll may not be as potent as reported on FFXIclopedia. It seems that after the August 2007 major version update, the effect of Fighter's Roll was "nerfed" per "All or Nothing." This is worth noting that even using the "nerfed" Fighter's Roll values (which themselves appear to be point estimates), it can be shown that Fighter's Roll is better than Samurai Roll for a particular case, which is the whole point of this post aside from having a terrible excuse to waste time doing probability exercises.

What kind of decision rules will you use for Phantom Roll?

It is typical to use Snake Eye to force a lucky outcome and to "escape" unlucky ones, so I apply this rule. Some time ago, I discussed an approach to rolling using knowledge of conditional expectation to optimize the expected outcome (average) of any roll and that is what I also use here.

In other words, the decision to Double Up on your current total depends on whether the conditional expected bonus of the final total (given your current total) exceeds the actual bonus of your current total. If it does, you continue rolling, and if it doesn't, you stop.

For example, suppose 9 is an unlucky total. From the standpoint of conditional expectation, you may be better off Doubling Up on average despite your conditional busting rate of 2/3. (It will depend on how good the 11 bonus is.)

Using properties of conditional expectation and applying the above rules, it is much simpler to compute the expected outcome of each roll (both with job bonuses and without) without knowledge of the underlying probability distribution of possible outcomes (given the above rules), but with some effort I managed to compute these probabilities, which are presented as follows.

Probability distributions for Phantom Roll outcomes

Final roll total	Fighter's Roll (no WAR)	Fighter's Roll (w/ WAR)	DA rate bonus (+5% w/ WAR)	Samurai Roll (no SAM)	Samurai Roll (w/ SAM)	Store TP bonus (+10 w/ SAM)
2	-	-	-	.333333333	.333333333	32
5	.529320988	.529320988	10%	-	-	-
7	-	.142103909	6%	-	.142103909	16
8	.138010117	.114326132	7%	.165787894	.142103909	20
9	-	-	-	.165787894	.142103909	22
10	.173396776	.126028807	8%	.138010117	.114326132	24
11	.067794067	.044110082	14%	.105602709	.081918724	40
Bust	.091478052	.044110082	-	.091478052	.044110082	-

It is easy (for me) to verify that the expected outcomes are the same regardless of using explicit probability distributions or using properties of conditional expectation. I also give the Snake Eye "usage rate," which is the probability that Snake Eye is used to obtain the final outcome.

Expected (average) Phantom Roll values (outcomes)

Roll	EV (no job bonus)	Snake Eye rate	EV (w/ job bonus)	Snake Eye rate
Fighter's Roll (double attack)	8.595571845	.370263203	13.35133745	.346579218
Samurai Roll (store TP)	25.1661094	.166666667	34.48816872	.166666667

You may wonder what's the point of doing some cumbersome probability calculations when you can just manipulate the expected values to obtain percentage increases in rate of damage, which itself is an average anyway.

Consider the Samurai Roll average store TP without a samurai in the party, which is about 25. For great axes (504 delay), 25 more store TP is more than sufficient to go from a 6-hit scenario to a 5-hit scenario (assuming 22 store TP initially, necessary for "true" n-hit setups). However, it's not like you will get a 5-hit setup all the time just because the average store TP bonus is 25.

Stopping on an 8 (store TP 20) probably doesn't get you there, not to mention the rate of busting, which is about 9%. If you don't have enough store TP from Samurai Roll to get a "true" 5-hit setup about 26% of the time given the described rolling criteria, you may want to account for that in your analysis. This requires knowing the associated probabilities of obtaining the roll totals 2, 8, 9, 10, and 11 (along with the bust probability) to obtain a weighted average.

Establishing a baseline for application of Fighter's Roll and Samurai Roll bonuses

Since I am attempting to compare the efficacy of Fighter's Roll with Samurai Roll in terms of increasing rate of damage, there must be an explicit baseline rate of damage, which requires some statements about weapon damage, hit rate, number of hits in a WS, etc., if only to obtain the implicit "TP damage to WS damage" ratio that is necessary to account for the full benefit of using either Fighter's Roll or Samurai Roll.

I'll admit that since I went to the trouble of calculating the above probabilities, I'm going to use them. I personally would not be content just arguing that, "oh, when I'm on WAR, the 'DoT' increase from Fighter's Roll is about 11.2% on average starting with 19% DA." This facile conclusion may or may not be justified by a more thorough analysis, which I'm about to describe.

• Warrior job, so Fighter's Roll additional bonus of +5% DA applies
• 106 "base" damage for TP, 159 for WS (average pDIF 1)
• 95% hit rate, 19% double attack rate
• 3-hit weapon skill with no pDIF(-like) bonus property
• Assume sufficient TP from the previous WS to maintain a "n-hit setup" always
• No delay reduction

What's the effect of Fighter's Roll on rate of damage?

In the past, I have defined "rate of damage" to be the "ideal" average damage from a "cycle" of TP-phase damage along with the damage from a weapon skill used immediately after attaining > 100 TP. The rate of damage will be calculated under each Fighter's Roll effect and a weighted average taken to obtain the long-run average rate of damage under the effect of Fighter's Roll.

DA rate	Average no. of rounds	Average no. of TP hits	Average no. of WS hits	Average time (s)
+15%	4.117693416	5.241823719	3.496	34.59
+11%	4.226407623	5.219613414	3.420	35.50
+12%	4.198690485	5.225270309	3.439	35.27
+13%	4.171336013	5.230855361	3.458	35.04
+19%	4.014533945	5.263054002	3.572	33.72
+0% (Bust)	4.557017596	5.151708392	3.211	38.28

Now that we have the "frequency" calculations, we need the "potency" calculations next. For the sake of simplicity let there not be an fTP bonus (or other bonus) on the first hit.

Calculating average damage to 100 TP

DA rate	No. hits to 100 TP	AA dmg	No. WS hits	WS dmg	Total dmg
+15%	5.241823719	555.633	3.496	555.864	1111.497
+11%	5.219613414	553.279	3.420	543.78	1097.059
+12%	5.225270309	553.878	3.439	546.801	1100.679
+13%	5.230855361	554.470	3.458	549.822	1104.292
+19%	5.263054002	557.883	3.572	567.948	1125.831
+0% (Bust)	5.151708392	546.081	3.211	510.549	1056.630

Finally, the rate of damage can be obtained for each DA rate bonus. The auto-attack proportion of total damage is consistent with "empirical" observation that it's around 50%.

Damage per second

DA rate	AA prop. total dmg	DPS
+15%	.499	32.134
+11%	.504	30.901
+12%	.503	31.208
+13%	.502	31.515
+19%	.495	33.385
+0% (Bust)	.516	27.603

After computing the weighted average, the rate of damage in the presence of Fighter's Roll is 31.631 DMG/s, which is about 14.6% higher than the DPS without any Fighter's Roll effect (27.603). Recall that the naive estimate of percent increase of "damage over time," which doesn't even account for the increased WS frequency from higher DA rates along with increased average number of hits for the WS proper, is only (1.323513/1.19 - 1)100% = 11.2%.

What's the effect of Samurai Roll on rate of damage?

For the sake of convenience, I will just consider the case where the full effect of Samurai Roll is attained (when a samurai is present as Samurai Roll is applied). This makes the analysis much easier since, aside from busting, any roll equal to 2 or above 6 ensures a 5-hit setup for a 504-delay great axe.

n-hit?	Average no. of rounds	Average no. of TP hits	Average no. of WS hits	Average time (s)
5-hit	3.67229159	4.151525643	3.211	30.85
6-hit (Bust)	4.557017596	5.151708392	3.211	38.28

Again, with the frequency figures taken care of, we turn next to the potency figures and then the final rates of damage.

Calculating average damage to 100 TP

n-hit?	No. hits to 100 TP	AA dmg	No. WS hits	WS dmg	Total dmg
5-hit	4.151525643	440.061	3.211	510.549	950.610
6-hit (Bust)	5.151708392	546.081	3.211	510.549	1056.630

Damage per second

n-hit?	AA prop. total dmg	DPS
5-hit	.462	30.816
6-hit	.516	27.603

After computing the weighted average, the rate of damage in the presence of Samurai Roll (5-hit always except for busting) is 30.816 DMG/s, which is about 11.1% higher than the DPS without any roll (27.603).

How can we reconcile this percent change with the typical arguments for Store TP? Well, one can argue that going from a 6-hit to a 5-hit is a 25% increase in weapon skill frequency, but that doesn't really say anything about the increase in rate of damage (my definition). Without bothering with a detailed analysis, assume a 50:50 split in TP:WS damage (noting that this usually varies with the required number of hits to 100 TP!), so that an estimate of percent increase is actually 12.5%, which overestimates the "actual" value of 11.1%.

Perhaps I completely botched that silly argument, and you can correct me in the comments section.

Conclusion

In the case of a warrior spamming a 3-hit weapon skill, it can be shown that Fighter's Roll is more effective than Samurai Roll from the standpoint of increasing damage starting with a 6-hit setup (and other conditions). While Fighter's Roll does not reduce the average time to 100 TP as much as Samurai Roll, it increases both weapon skill damage and auto-attack damage. The combination of increased potency (damage) and WS frequency surpasses increased WS frequency alone.

Accommodating Samurai Roll with two-handed weapons

Regardless of what you think about Samurai Roll compared to other types of offensive rolls, I was kind of interested in whether it actually removes a hit from a "n-hit setup" for various delay values. A "n-hit setup" is usually understood to be an equipment/food configuration for a two-handed weapon (because no one seems to care too much about Store TP for one-handed weapons) with enough Store TP to achieve at least 100 TP in n (landed) hits starting from 0 TP. Of course, in practice one starts from 0 TP only after zoning or upon TP reset because weapon skills do give TP if they don't miss completely, but I am keeping this general to avoid accounting for whether a WS is multi-hit or not.

To satisfy my curiosity, I looked up some typical delay values for two-handed weapons and computed the minimum Store TP to achieve a 7-hit, 6-hit, and 5-hit setup for each of the delay values. Assuming that I have attained a n-hit setup for some arbitrary delay, if someone is going to use Samurai Roll effectively (this means not stopping at 6 for the most part), at least I will know if I am taking advantage of the Store TP bonus.

Since the effect of Store TP is generally considered "discrete" in the sense that you want just enough TP to attain a n-hit setup (any excess Store TP trait having no effect and thus superfluous), the Store TP bonus from Samurai Roll will usually overshoot or fall short of the minimum requirement for a (n - 1)-hit setup.

Starting with a 7-hit setup, the following table summarizes the TP "surplus" or "deficit" with Samurai Roll in effect, without the +10 Store TP bonus from having a SAM in the party. Why should a SAM necessarily be present?

TP surplus/deficit with Samurai Roll going from 7 to 6 hits (5 hits with 11)

	Minimum	Roll total
Delay	Store TP	II	VII	VIII	IX	X	XI
528	0	+16	0	+4	+6	+8	+1
513	3	+14	-2	+2	+4	+6	-1
504	5	+15	-1	+3	+5	+7	-1
501	6	+15	-1	+3	+5	+7	-2
492	8	+14	-2	+2	+4	+6	-3
480	10	+13	-3	+1	+3	+5	-4
450	25	+11	-5	-1	+1	+3	-9

I omitted roll totals 1, 3, 4, 5, and 6 because those are generally not desirable. In general, Samurai Roll does shave a hit off 7-hit setup except when the roll total is 7. If the roll total is 11, the +40 Store TP bonus falls short of shaving two hits off a 7-hit setup. Of course, if a SAM is present the +10 Store TP bonus overcomes these deficits. Suppose you happen to be subbing /NIN and have sufficient TP for a 7-hit setup. In this situation, Samurai Roll would "work."

TP surplus/deficit with Samurai Roll going from 6 to 5 hits (4 hits with 11)

	Minimum	Roll total
Delay	Store TP	II	VII	VIII	IX	X	XI
528	16	+9	-7	-3	-1	+1	-18
513	21	+9	-7	-3	-1	+1	-19
504	22	+8	-8	-4	-2	0	-21
501	23	+7	-9	-5	-3	-1	-21
492	26	+7	-9	-5	-3	-1	-22
480	29	+7	-9	-5	-3	-1	-24
450	46	+4	-12	-8	-6	-4	-32

In the 6-hit scenario, however, Samurai Roll generally does not provide enough TP to achieve a 5-hit setup unless the roll total is 2 or 11. If a SAM is present, however, Samurai Roll generally does remove a hit. In either case, the 11 roll is not even close to providing enough TP to achieve a 4-hit setup.

TP surplus/deficit with Samurai Roll going from 5 to 4 hits

	Minimum	Roll total
Delay	Store TP	II	VII	VIII	IX	X	XI
528	39	-3	-19	-15	-13	-11	+5
513	44	-4	-20	-16	-14	-12	+4
504	46	-5	-21	-17	-15	-13	+3
501	48	-4	-20	-16	-14	-12	+4
492	51	-5	-21	-17	-15	-13	+3
480	54	-7	-23	-19	-17	-15	+1
450	74	-12	-28	-24	-22	-20	-4

Let us now turn to the fanciful situation of having a 5-hit setup before Samurai Roll, which is, practically speaking, reserved only for polearm-using samurai. A 4-hit setup with Samurai Roll is possible only by rolling a 11 (without SAM) or by rolling a 2 with a SAM present.

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.