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.

Warrior's Charge for TP generation

A typical way to use Warrior's Charge

When I had the one perfunctory merit for Warrior's Charge—not like there was anything compelling in Group 2—I reserved it exclusively for weapon skills instead of TP gain. For an ability that can be used on demand when available, this is a typical application especially for "zerging," when you can't really ensure that the potential TP gain from the guaranteed double attack will let you squeeze out another WS in the 45 seconds of Mighty Strikes. But to be honest, I liked the big numbers for Steel Cyclone.

A more efficient way to use Warrior's Charge?

Of course, if you're doing some long-term activity, like meriting, it is theoretically more efficient (that word again...) in the "long run" to use Warrior's Charge in the auto-attack phase than for weapon skills. By "long run," I'm basically referring to TP-burning, where the benefit of the average increase in TP gain from Warrior's Charge actually and most ideally manifests in higher average WS frequency per unit time.

Would you actually want to waste 22 merits to reduce the recast time to 5 minutes, though? Perhaps it would help to quantify the effective increase in double attack rate from using Warrior's Charge every five minutes.

Expressing the effect of Warrior's Charge as a rate of double attack

First, a preliminary observation. While Warrior's Charge confers an absolute increase of one double attack per use, its relative contribution to long-run rate of damage decreases with increasing delay reduction.

For example, if there are few attack rounds in the time period between uses of Warrior's Charge, then the contribution of Warrior's Charge is relatively large. But if there are many attack rounds in that window, then the contribution of Warrior's Charge is relatively small.

With that in mind, it is necessary to make an explicit statement about the amount of delay reduction present before determining the effective double attack rate with ideal use of Warrior's Charge. I acknowledge that the effect of Warrior's Charge is "discrete," and not continually present, but the rate of double attack is just an expected value (average) anyway, a mathematical conceit, so it's natural to account for the effect of Warrior's Charge in a weighted average, which is the "effective" double attack rate in the long run (to be shown later).

As an example, suppose that you have 5/5 Warrior's Charge, for a five-minute recast. The average DA rate for Warrior's Charge, one double attack every five minutes, is equivalent to 2.8% DA given 504 delay, and 2.296% DA given 504 delay and 18% haste. These values are calculated as follows:

\[\frac{1\ \mbox{DA}}{5\ \cancel{\mbox{min}}}\cdot\frac{1\ \mbox{min}}{60\ \mbox{s}}\cdot\frac{1\ \mbox{s}}{60\ \mbox{delay}}\cdot\frac{504\ \mbox{delay}}{1\ \mbox{round}}\cdot100\% =2.8\%\ \frac{\mbox{DA}}{\mbox{round}}\]
\[\frac{1\ \mbox{DA}}{5\ \mbox{min}}\cdot\frac{1\ \mbox{min}}{60\ \mbox{s}}\cdot\frac{1\ \mbox{s}}{60\ \mbox{delay}}\cdot\frac{504(.82)\ \mbox{delay}}{1\ \mbox{round}}\cdot100\% =2.296\%\ \frac{\mbox{DA}}{\mbox{round}}\]
Again, the length of the interval between attacks, which is affected by delay reduction, determines the relative contribution of Warrior's Charge on a per-round basis.

That makes sense, but how does Warrior's Charge actually affect the effective double attack rate?

The effective double attack rate, which takes into account the contribution of Warrior's Charge, is not the sum of the base DA rate and the contrived DA rate from Warrior's Charge. It's a weighted average based on how often WC takes effect, ideally as often as possible, or once every five minutes. That may be a confusing statement, so I provide further corny explanation.

Approaching this question from a probabilistic point of view, the above rates can be treated as the unconditional "probabilities" that Warrior's Charge takes effect in one attack round. There is nothing probabilistic about how often Warrior's Charge is used, but I am using probability language for the sake of explaining how the effective double attack rate is calculated.

If you are familiar with the phrase "percent of the time" appended to a number, perhaps this explanation will actually make some sense. Probability statements are often colloquially expressed in terms of "percent of the time." Given 0% delay reduction, Warrior's Charge "takes effect 2.8 percent of the time." Given 18% delay reduction, Warrior's Charge "takes effect 2.296 percent of the time," and so on.

Given that Warrior's Charge has just been used, the "probability" of a double attack in the subsequent attack round is 1; otherwise, it is whatever your DA rate normally is. Therefore, the "effective" double attack rate is just a weighted average, an application of the law of total probability treating the DA rates as probabilities. As a probability statement, the effective double attack rate is

\[P(\mbox{DA}) = P(\mbox{DA} \mid \mbox{WC})P(\mbox{WC}) + P(\mbox{DA} \mid \overline{\mbox{WC}})P(\overline{\mbox{WC}})\]
Note that this expression is valid in extreme hypothetical cases. If your delay reduction approaches 100%, the relative effect of Warrior's Charge tends to 0. Therefore, your effective DA rate (which is a semi-probability, so to speak) cannot exceed 1. If you never use Warrior's Charge, then your effective DA rate is just your base DA rate.

To give an explicit example, suppose my DA rate from Warrior's Charge is 2.296% (shown above) and my base DA rate (before Warrior's Charge) is 19%. Then, my effective DA rate is

\[P(\mbox{DA}) = 1(.02296) + (.19)(1-.02296) = .2085976\]
This effective double attack rate, which accounts for the discrete contributions of Warrior's Charge, can then be used to estimate the average number of rounds to 100 TP or the average number of hits in a weapon skill.

Showing that Warrior's Charge is better (in the "long run") for TP gain than for weapon skills

Certainly, if you don't find occasion to use Warrior's Charge for meaningful TP gain, you might as well use it for weapon skills. No one ever said anything about not using one's discretion and judgment.

Still, it is easy to argue that you get more out of Warrior's Charge for TP spamming without doing arithmetic. In the auto-attack phase, on average the extra DA increases TP gain, leading to higher WS frequency, and also contributes to auto-attack damage. For weapon skills, the extra DA merely gives a slightly higher average TP return and slightly higher WS damage.

Numbers provide a nice summary, however, so I present the results of some number-crunching for TP spamming with

• sufficient Store TP for a "6-hit" setup (5 hits to 100 TP given sufficient TP return from the previous weapon skill)
  
• a 3-hit weapon skill (like Raging Rush or King's Justice)
  
• 18% delay reduction
• 5/5 Warrior's Charge (so 2.296% DA rate from WC)
  
• 19% base double attack rate (so effective DA rate of 20.86%)
  
• 95% hit rate
• 106 "base" damage for TP (average pDIF of 1)
• 159 "base" damage for WS (average pDIF of 1)
  

(All numbers in the following table are averages.)

Comparison of average damage per second for a "cycle" of auto-attack and WS damage (5 hits to 100 TP)

Use of Warrior's Charge	Rounds	TP Hits	WS Hits	Time (s)	DPS
5/5 Warrior's Charge only for TP gain	4.497	5.164	3.211	30.979	34.149
5/5 Warrior's Charge only for WS ("e-penis")	4.557	5.151	3.228	31.388	33.752
No Warrior's Charge	4.557	5.151	3.211	31.388	33.662

As expected, the e-penis approach is slightly better than nothing, but worse than the "optimal" approach, provided you actually have opportunities to use Warrior's Charge for TP generation. 22 merit points into Warrior's Charge only gets you up to a 1.446% improvement in theoretical, long-run rate of damage, which itself is an inefficient use of merit points compared to other warrior-specific options such as Group 1 double attack rate.

Moreover, as shown earlier, Warrior's Charge becomes relatively less effective for increasing WS frequency the more delay reduction you have, as it provides only a static increase in TP gain while being unaffected by delay reduction.

By this point, it should be easy to accept using Warrior's Charge for increasing WS frequency where applicable, but let's return to the 45-second Mighty Strikes "zerg." Warrior's Charge is relatively less effective with increasing haste (which is desirable for zerging), but the key word is "relatively." In that small time frame, you should still be better off using Warrior's Charge for the extra TP to get another WS off than to tack on another hit to a weapon skill.