Store TP and dual-wielding

Store TP is a trait that is considered to have an all-or-nothing effect on weapon skill frequency and, therefore, damage over time. In general, there is a minimum amount of store TP required to attain 100 (or more) TP in n number of hits. In turn, it is desirable to minimize n without incurring counter-productive opportunity costs. Finally, since n is an integer, there will be discrete "tiers" of store TP, meaning there are specific ranges of store TP that correspond to each n, with the left endpoint of each "tier" as the minimum store TP to achieve an "n-hit setup."

In the case of dual-wielding, because n is generally high (well over 10), the store TP "tiers" are not as long. But what about the variability of TP return as a result of the off-hand hit and any other extra hits after the first? Doesn't that affect n?

Who cares? Even if it's not 95% of the time given 95% hit rate and depending on your double attack rate, the vast majority of the time n will be what you expect it to be. If you are aiming for a 5.0 TP/hit setup, most of the time you will require only 18 hits after a WS to reach 100 TP. Just because you might miss the off-hand hit doesn't mean you throw up your hands and dismiss the effect of the store TP or any additional quantities of store TP that might actually have a significant effect.

It is true that the number of required hits to attain 100 TP can take on three or more values depending on what the TP return of the previous WS actually is. As a consequence, even relatively small changes in store TP (for a given weapon combo and level of dual wield), will lower the average number of hits to reach 100 TP.

With that in mind, how does one illustrate the effect of changes in store TP on the rate of damage? First, let's examine the effect of changes in TP per hit on the average number of hits to reach 100 TP.

Because actual TP return from a WS is a random variable, the number of required hits to attain 100 TP, given that TP return, is also a random variable with an associated probability distribution. This was illustrated briefly in an earlier post analyzing the viability of the Tsukumo/Perdu Blade katana combination. For the case of 4.7 TP per hit, there is one probability distribution, and for the case of 4.5 TP per hit, there is another.

Given a three-hit weapon skill (like Blade: Jin), it is fairly straightforward to show how the probability distribution of the number of required hits to attain 100 TP changes with the amount of TP per hit, as shown in the following table. Since the double attack rate, hit rate, and dual wield delay reduction affect these, it is necessary to state them. I will use 15% double attack rate, 95% hit rate, and 40% dual wield delay reduction.

Probability distributions of the required number of hits to 100 TP (three-hit weapon skill)

TP/hit	24	23	22	21	20	19	18	17	16	15	14	13	Avg. no. hits
4.0	.0933	.8877	.0165										23.69
4.1	.0092	.1534	.8372										22.84
4.2	.0025	.0933	.8877	.0165									22.75
4.3	.0002	.0092	.1534	.8372									21.84
4.4			.0933	.8877	.0165								21.75
4.5			.0027	.0965	.9008								20.77
4.6			.0025	.0933	.8877	.0165							20.75
4.7				.0092	.1534	.8372							19.84
4.8				.0025	.0933	.8877	.0165						19.75
4.9				.0002	.0092	.1534	.8372						18.84
5.0					.0025	.0950	.9025						18.77
5.1					.0025	.0933	.8877	.0165					18.75
5.2					.0002	.0092	.1534	.8372					17.84
5.3						.0025	.0950	.9025					17.77
5.4						.0019	.0727	.7080	.2173				17.53
5.5							.0027	.0965	.9008				16.77
5.6							.0025	.0950	.9025				16.77
5.7							.0025	.0933	.8877	.0165			16.75
5.8							.0002	.0092	.1534	.8372			15.84
5.9								.0025	.0950	.9025			15.77
6.0								.0025	.0933	.8877	.0165		15.75
6.1								.0019	.0727	.7080	.2173		15.53
6.2									.0027	.0965	.9008		14.77
6.3									.0025	.0950	.9025		14.77
6.4									.0025	.0933	.8877	.0165	14.75
6.5									.0018	.0727	.7080	.2173	14.53

Here, I emphasized the probability of the most common outcome (required number of hits to attain 100 TP). Generally, 0.2-0.4 increases in TP per hit are "likely" to reduce by about 1 the average number of hits to 100, which will improve your rate of damage ever so slightly (being that WS damage is a low proportion of total damage). The question is how much store TP is sufficient to attain such increases.

To give an example, without any store TP the Senjuinrikio/Perdu Blade combination corresponds to 4.5 TP per hit before any store TP. The average number of hits (not average number of required hits) to 100 TP is 20.77, with 20 the most typical number of hits to 100 TP after Blade: Jin. With Rajas Ring, the TP per hit rises to 4.7 with a modest decrease in the average number of hits to 100 TP (19.84).

Of course, it is the average number of rounds to 100 TP that is needed to estimate the increase in rate of damage from increases in TP per hit from store TP. But I needed the above probability calculations to obtain a weighted average of required number of rounds because hits in excess of 100 TP do not contribute to increasing weapon skill frequency. These results are shown below, along with the average time to 100 TP (given 250 delay with 40% dual wield reduction) based on the average number of rounds to attain 100 TP.

Average time to attain 100 TP in the long run

TP per hit	Average no. of rounds	Average no. of hits	Average time (s)
4.0	10.84	23.69	45.17
4.1	10.45	22.84	43.55
4.2	10.41	22.75	43.39
4.3	10.00	21.84	41.65
4.4	9.96	21.75	41.48
4.5	9.51	20.77	39.61
4.6	9.50	20.75	39.57
4.7	9.08	19.84	37.84
4.8	9.04	19.75	37.67
4.9	8.62	18.84	35.93
5.0	8.59	18.77	35.79
5.1	8.58	18.75	35.76
5.2	8.17	17.84	34.02
5.3	8.13	17.77	33.89
5.4	8.02	17.53	33.43
5.5	7.68	16.77	31.98
5.6	7.68	16.77	31.98
5.7	7.67	16.75	31.94
5.8	7.25	15.84	30.21
5.9	7.22	15.77	30.07
6.0	7.21	15.75	30.04
6.1	7.11	15.53	29.61
6.2	6.76	14.77	28.17
6.3	6.76	14.77	28.17
6.4	6.75	14.75	28.13
6.5	6.65	14.53	27.71

To map store TP to TP per hit, I will use the example of Senjuinrikio/Perdu Blade, which has 4.5 TP/hit without store TP. Since I already specified some damage conditions in the Tsukumo post, I will use those to illustrate the relationship between increasing store TP and increasing damage per second.

In general, there are major "tiers" of rates of damage, and jumping from a lower tier to a higher up represents a decrease in the average time to 100 TP. The second lowest tier, from 5 to 8 TP, could correspond to having only Rajas Ring (or Usukane Sune-Ate) equipped. The next highest, from 9 to 15, could correspond to having both Rajas and Usukane Sune-Ate equipped.

While it is necessary to know the proportion of auto-attack to WS damage (which itself is determined by a variety of factors, some not very well understood) to determine how efficient jumping from a lower tier to a higher one is, for this specific example, equipping a Rajas Ring without any other store TP is about a 1.6% percent increase in damage per second not accounting for the other bonuses. Store TP is actually doing something, it's just tedious to quantify how much.

Of course, there aren't many store TP options for ninja, and those that are available are generally good all-around options. Perhaps someday there will be a way to reach 16 store TP without assuming high opportunity costs, such as accuracy food with store TP or other good all-round pieces of equipment comparable to Usukane Sune-Ate.

We can also see that, far from doing "nothing," Samurai Roll can provide a substantial increase to damage over time for dual-wielders, assuming the weapon skills aren't feeble. As store TP increases, obviously the ratio of auto-attack to WS damage approaches parity (albeit slowly), so it is inappropriate just to assume that auto-attack damage will always be something like 66% of your total damage in a WS-spamming situation regardless of store TP.

Notwithstanding possible factors such as time to execute a weapon skill (both human reaction time and any possible in-game delay), the good Samurai Roll totals (2, 8, 9, 10, 11) can increase damage per second up to 7-12% given the above conditions, and that is already accounting for the effect of double attack. Compare this to a warrior with a 6-hit setup. To get to 5 hits, a samurai has to be present (store TP +10) and while the increase in WS frequency is theoretically 25% without any double attack, roughly speaking the percent increase in damage per second will be less than 12% with non-trivial amounts of DA.

The two-fold effect of double attack

When you speak of damage over time, what do you actually mean? The answer may reveal whether you hold a minor, yet "fundamental" misunderstanding of damage "mechanics" in this world of OCD fuck-headed douchebaggery.

First, I hope you understand that the only valid view of damage over time accounts for weapon skills, regardless of when and how you use them (spamming them or whatever). Like, weapon skills contribute to damage, and you're doing damage over some time interval. Duh! Since when do the ideal and the actual have to coincide, anyway?

Anyway, I also hope we can all agree there are factors such as accuracy, double attack, and haste that affect the frequency of attacks (that land) and, therefore, damage per unit time (an average), while factors such as attack (rating) and strength affect the potency of damage per hit.

But as I've mentioned in passing here and there, while accuracy and double attack can affect the frequency of auto-attacks and weapon skills, it should be obvious that they also affect the average damage of weapon skills, which haste does not affect. But this fact is often elided for the sake of convenience without any apparent recognition.

For example, given 85% hit rate, 4 accuracy will increase auto-attack and weapon skill frequency by 2.35% ideally. This doesn't mean a 2.35% increase in overall damage per unit time, which again should account for the contribution from weapon skills, because that figure doesn't account for the effect of accuracy on average WS damage (per use). Therefore, it is a slight underestimate of the "true" percent change.

Does it matter? Practically, not really, mainly because it's pretty inconvenient to account for the two-fold effect of accuracy and double attack. But there are some "interesting," counterintuitive (indulging the conceit that FFXI players have any intuitions about how anything actually works) consequences that I demonstrate in the following example.

Does double attack ever "beat" haste?

Sure, a specific amount X of double attack, given some initial level of double attack, can be more efficient than a specific amount Y of haste, given some initial level of haste. Why does this comparison ever come up, anyway? I can't think of any situation where haste and double attack are in direct competition. Whatever increases your "efficiency" without incurring ridiculous opportunity costs should be good enough.

A possible explanation is that players often are deluded into thinking they have "capped" accuracy and rapid TP gain is "sexy," which both haste and double attack affect. However, on a per-point basis, haste is plainly more efficient than double attack at increasing the rate of TP gain because haste directly lowers the time between attack rounds, which is fundamentally more efficient than tacking on an occasional extra attack per attack round.

Of course, I didn't mention the effect of double attack on average weapon skill damage, which, when actually considered, is actually enough for double attack to be more efficient than haste for increasing overall damage over time (not just for TP gain) for specific situations.

How is this even possible? Fundamentally speaking, haste increases damage over time at an instantaneous rate of 100/(100 - H)², where H is the amount of haste (as an integer percentage), so the effect of total haste is relatively slow at the beginning but eventually ramps up rapidly as the amount of haste increases. This is why haste is the "gold standard" for increasing rate of damage.

But, since the rate of increase is relatively slow at the low end, there is the only "opportunity" for double attack and accuracy to be ever so slightly more efficient than haste if all you really cared about is damage efficiency.

A counter-intuitive example

The most convenient way to compare the efficiency of two competing options in the game, whether it be pieces of equipment, two types of "buffs," etc., is to determine the percent difference in "output" (damage over time, gil over time, whatever) since we are generally interested only in the relative difference and any factors that are fixed between any two options can be factored out and don't need to be "given" as a matter of convenience.

Where damage is concerned, this requires knowledge of the functional relationship between rate of damage and the factors of interest (holding all else fixed). Unfortunately, I don't see any way to derive an exact relationship between damage over time (including the weapon skill contribution) and double attack, so it is just easier for me to give a specific example illustrating where double attack is ideally more efficient than haste.

The example I give is based on the following conditions:

• Suppose 106 "base" damage per auto-attack hit and 159 "base" damage per weapon skill hit with average pDIF of 1 in both phases
• Six required hits to achieve 100 TP (6-hit setup), so five hits to 100 TP given sufficient TP return from the previous weapon skill
• A 3-hit weapon skill used instantaneously after achieving 100 TP
• 95% hit rate. Therefore, the average number of landed hits (giving TP) per weapon skill is 3(.95) = 2.85
  
• Starting from 0% double attack and 0% haste, it will be shown how damage per second varies with DA and haste, respectively

These conditions are sufficient to give an estimate of damage per second, ignoring the slight effect of having insufficient TP return from the previous WS to get to 100 TP in 5 hits thereafter, and possible variability in weapon skill damage based on TP, among other factors. But it's not the estimate of damage per second that is of interest, but how damage per second changes with either DA or haste.

After some boring spreadsheet calculations, which are boring and unnecessary to show, a plot illustrating the efficiency of double attack and haste is presented.


As expected, it takes a "while" for the effect of total haste to ramp up, but fundamentally damage per second must tend to infinity as total haste approaches 100%. In comparison, double attack is actually more efficient than haste initially because of the two-fold effect of double attack (on the rate of TP gain and the average WS damage) but the instantaneous rate of change increases very slowly. This makes sense because the increase in the average number of hits for any multi-hit weapon weapon skill is 0.020 for every point of double attack. At the same time, the time to WS is decreased.

When determining percent changes with double attack, it should now be obvious that the relative efficacy of double attack depends on the damage per hit of the weapon skill as well as the number of required hits to 100 TP. The relative efficacy of haste is independent of damage per hit or expected number of hits in a weapon skill, though.

It follows that the efficiency of double attack is blunted if TP isn't spammed. As a check on my calculations, I dropped the weapon skill component of damage per unit time (which is wrong to do) and generated a different plot illustrating auto-attack rate of damage.


This should be a familiar result. All is right in the world. Going from 0% to 25% haste results in a 33% increase in overall damage per second, as illustrated here and in the previous figure. Going from 0% to 25% double attack results in a 25% increase in auto-attack damage per second, as illustrated in the last figure alone.

If for some reason you never use a weapon skill, double attack can never be more efficient than haste at increasing the overall rate of damage, which must account for average weapon skill damage lest you look like a dipshit. But when does that actually happen if you are OCD'ing about damage to begin with?

What do I take away from this?

Double attack can proc on multi-hit weapon skills. Therefore, double attack not only affects weapon skill frequency and the rate of auto-attack damage, but also average WS damage. Don't forget average weapon skill damage matters somewhat when evaluating the effect of additional double attack. (Doing the actual calculations is another issue, though.) The same goes for accuracy, too, although I didn't provide a specific example.

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.

Stupid posts

Sadly, I wasted a bit of time combing English-langugage forums this weekend looking for anything interesting, but it's not as wasteful as actually posting on forums. If only I knew Japanese, then I'd be able to make this blog a lot more informative instead of cluttering it with arithmetic junk. Here are some threads amusing enough to waste 20 seconds commenting on each.

(FFXIclopedia) Playing the logic card in a video game when considering the possibility of cutting up Marinara Pizza into slices for convenience. "How do you end up with more pizza than you started out with?" How about, how the fuck do you fuck up cutting a pizza into slices? (Recall that you can HQ curry buns.)

(Allakhazam) How do you get a Kikoku when you're this ignorant? Never mind, you actually upgraded Kikoku as opposed to something useful and parasitized your Dynamis LS or RMT'ed it in the process or sat on a fat pile of gil before general deflation.

(BG) Dick-riders actually defend the Quicksand Caves map requirement for "Moogle Kupo d'Etat," like you dipshits didn't look up the mission info beforehand to save time (the smart thing to do) and actually finished this particular mission through trial-and-error. This is from someone who has all the maps and didn't leech some of them (like the useless Promyvion maps) like you probably did. No, most maps are completely useless.

(BG) Typical Fenrir "player" contributing nothing of value (asking about possible ANNM solos), except the valid point that vBulletin's search function is complete shit, and the premise that we should even search the forums for highly fragmented information is complete shit, too.

(Allakhazam) What is it with bards and Alkalurops anyway? Not that you should actually read this, but take my word for it that I just cannot have a legitimate conversation about magic accuracy with anyone in that thread except two people. There is still a lot to learn and confirm in theory (how about the "developer" team of monkeys not being so fucking opaque about everything?), but it's not getting done with some stupid back-and-forth.

(FFXIclopedia) Again, why does anyone care about beating StarOnion/InvincibleLeg/MeteorBrian?

(Allakhazam) Are players really that invested in becoming a great [insert job here]? Considering all the inane hero worship that surrounds so-called "celebrity" players in FFXI, no, no one who's played long enough should be all that surprised. Still, being a knowledgeable, even "skilled" (to the extent that skill is involved) player involves a bit more than micromanaging a single job. Not that leveling only one job is a bad thing (why throw even more good time after bad in a MMORPG?), but I guarantee that one-job only types who talk a game about "perfecting their craft" had to leech at some point or another. Especially thieves.

Tsukumo

The nature of recent posts hardly belies the fact that they are not at all interesting to construct. While they could easily be construed as a form of showing off, they were also motivated by my interest in analyzing the game in a more simpler way than just hand-waving about shit like everyone else. To me, it juts makes a lot more sense to deal with averages, which are a nice way to ignore variability in both damage and attack frequency and really simplify quantitative comparisons. Sometimes you just have to do the boring shit yourself since few others are capable and even fewer willing.

Aside from the posts concerning samurai weapons, everything else I had some degree of "personal" interest in, including this one about Tsukumo. Should I even bother trying to get a Tsukumo? I attempted to assess the viability of Tsukumo as a replacement for Senjuinrikio in the main hand.

Could Tsukumo with DMG +5 alone supplant Senjuinrikio in the main hand? The way I'm currently (not) using ninja, 190 delay and a decent damage rating on the main hand would be enough to let Senju collect dust, but all the hand-wringers can't seem to let go of the 6% crit rate and 38 base damage, never mind that you'd be better off with Fudo if you want higher average damage in the fantasy land of high pDIF.

Whatever your idea of high pDIF is for ninja, there is a way to show quantitatively whether Tsukumo/Perdu Blade is relatively less (or more) efficient (I'm starting to hate this term) than Senjuinrikio/Perdu in terms of spamming Blade: Jin. The case of dual wielding requires just a little more work than single weapons.

From the standpoint of spamming weapon skills, TP return from the previous WS looms large in terms of determining how many hits, on average, it will take to get to 100 TP. The fact that the off-hand weapon contributes a full-TP hit to Blade: Jin's TP return makes it more difficult to determine the "true" average number of hits to 100 TP after a Blade: Jin, but there is a way to calculate it.

Given the following...

• 95% hit rate
  
• 15% double attack rate
  
• 5 Store TP (Rajas Ring)
  
• 40% delay reduction from Dual Wield

... what is the probability distribution of the required number of hits to 100 after a Blade: Jin (with TP return being a random variable)? This information is necessary to calculate exactly the average number of hits to 100 TP along with the average number of rounds to 100 TP.

So far, I have gotten away with assuming there is always sufficient TP return from the previous WS to get to 100 TP in n - 1 hits for a n-hit setup. Ignoring the 5% of the time that there isn't sufficient TP return is not really necessary, but it is convenient.

While this simplification may be tolerated for two-handed weapons, this is not acceptable for the dual-wield situation. Nominally speaking, both Tsukumo/Perdu (4.6 TP) and Senjuinrikio/Perdu (4.7 TP) require 22 hits to 100 TP starting from 0 TP. (I hope these TP values are correct as they are the linchpin of this "analysis.") After a Blade: Jin, which can take on any of a finite set of TP values, the required number of hits to 100 TP is broken out as follows:

Comparison of probability distributions for required number of hits to 100 TP after Blade: Jin

Weapon	22 hits	21 hits	20 hits	19 hits
Senjuinrikio/Perdu (4.7 TP)	.0002	.0092	.1533	.8373
Tsukumo/Perdu (4.6 TP)	.0025	.0933	.8877	.0165

It shouldn't be surprising that even a 0.1-TP difference results in dissimilar distributions. Most of the time, the Senjuinrikio/Perdu combination requires 19 hits to reach 100 TP. In contrast, the Tsukumo/Perdu combination requires 20 hits to reach 100 TP most of the time. We can use these probabilities to obtain a weighted average of the average number of hits to 100 TP after Blade: Jin.

Average time to 100 TP in the long run

Weapons	Average no. of rounds	Average no. of hits	Average time (s)
Senjuinrikio/Perdu	9.081	19.842	37.84
Tsukumo/Perdu	9.497	20.752	36.09

These figures are based on spamming weapon skills continuously, showing that Tsukumo/Perdu doesn't have much of an advantage. Note that this a different viewpoint than just looking at the rate of TP gain (Tsukumo/Perdu being the obvious "winner"), which is not very meaningful for assessing spamming efficiency because, from this point of view, TP return from the previous WS is meaningless (like in Campaign).

If you've read this far, congratulations! I posted a spreadsheet that calculates the average number of rounds to 100 TP and average number of hits to 100 TP based on hit rate, DA rate, and whether you're dual wielding or not. While making a comprehensive, user-friendly spreadsheet for melee damage calculations is not very interesting to me, I can see where these calculations can be incorporated into one.

(Note: simulation was used to validate these averages.)

Calculating average damage to 100 TP

Weapons	No. hits to 100 TP	AA dmg	Hits/WS (main/sub)	WS dmg	Total dmg
Senjuinrikio/Perdu	19.842	691.708 + 571.463	2.9925/1.0925	701.261	1964.433
Tsukumo/Perdu	20.752	1195.305	4.085	652.783	1809.689
Tsukumo/Perdu (5% crit)	20.752	616.329 + 597.652	2.9925/1.0925	666.847	1880.829

Now that the average time to 100 TP has been taken care of, all that's left is to calculate average damage per hit and damage per WS. Hey, why not consider Tsukumo with 5% critical hit rate, too?

Here, I am assuming an average of 1.6 pDIF given 20% base critical hit rate across the board (still delving in fantasy, obviously) except where modified by weapon properties, 4 fSTR for auto-attack, and 9 fSTR for weapon skills. Given Blade: Jin, I used a WSC value of 53.

As I understand it, during a weapon skill double attack procs once for the main weapon and one for the sub, so that there are up to 6 hits possible for Blade: Jin. Therefore, it is necessary to split the average number of hits for Blade: Jin to account for any base damage (or expected damage) differences.

Another assumption I made with absolutely no concrete evidence is that that there is a critical hit rate bonus of +10% at 100 TP (analogous to Evisceration) on top of the 20% base rate and weapon crit rates, and that it applies to all hits. Assuming there is even a bonus, does it even apply to the off-hand hit, or does the off-hand hit crit at the normal rate? Is the second potential double attack from the off-hand swing? Obviously, I never closely examined these subtleties. Anyway, these details aren't terribly important.

Damage per second

Weapons	AA prop. total dmg	DPS
Senjuinrikio/Perdu	.643	51.916
Tsukumo/Perdu	.647	51.207
Tsukumo/Perdu (5% crit)	.646	52.114

The auto-attack proportion of total damage is just under 2/3 for WS spamming, which seems rather low. Maybe the assumption of a critical hit bonus at 100 TP is wrong. Tsukumo/Perdu with 5% crit rate barely edges Senju/Perdu, which is similar to Fudo barely edging Senju given high pDIF.

Final thoughts

Since Tsukumo is really only acceptable in the main hand if at all, I would consider Tsukumo with DMG +5 (any other augments being gravy) for both TP-holding (Campaign) and weapon-skill spamming. Frankly, if ninja users actually cared about optimality, why weren't they using Fudo for high pDIF?

Second to one?

Fay weapon augmentation seems so popular these days that players will even organize pick-up parties to spam the Grauberg (S) fight, which apparently is one of the easier ones. As embarrassing as it is to admit, I myself did a couple of pick-up runs to try to get a good Tsukumo for ninja, which I hardly use because I don't solo much on it anymore (I dusted it off to solo one of the latest Wings of the Goddess missions, though), when I could probably get a lot more mileage from a properly augmented Erlking's Kheten for warrior. But is it worth it when warriors already have access to Perdu Voulge as an all-purpose weapon? I attempt to provide some insight on this by playing with numbers.

Looking at possible Erlking's Kheten augments, we see that base damage can range from 3-6, STR from 3-5, attack from 5-7, and even double attack is possible (1-2). Some combination of base damage, STR, attack, and double attack (say, three out of four of these attributes) seems pretty competitive with Perdu Voulge. But would a strong Erlking's Kheten actually surpass Perdu Voulge, which has 5 accuracy (below 100 TP) and therefore might be better to use when hit rate isn't already at 95%?

Before doing the number-crunching, first recognize that is easy to calculate the percent difference between the two on a per-hit basis alone. It should be easy to see that Perdu is still better. A corny calculation could be (.925*108)/(.95*106) - 1 = -0.0079 in favor of Perdu. But let's see how we can account for the decrease in rate of TP gain relative to Perdu Voulge.

To put it another way, does the higher "potency" of a properly augmented Erlking's Kheten overcome the accuracy advantage of Perdu Voulge?

Using this example of a competitive Erlking's Kheten, which has total base damage 97 (DMG +6), STR +5 (effective base damage at least 98 given high STR relative to mob VIT), and attack +5 (either +8 or +9 effective attack rating), we can compare this particular Kheten to Perdu Voulge at various hit rates to get a sense of how fast these great axes get to 100 TP on average and how different average weapon skill damage could be.

This specific comparison follows the same template as a previous comparison among Fortitude Axe, Perdu Voulge, and Engetsuto, which was meant to serve as a proof of concept of how to simplify the calculations by first considering the average frequency of attacks and then considering the average "potency" of individual attacks to arrive at some answers--the proportion of total damage in weapon skills in particular--that can theoretically be compared against reality through parser output.

First, start off with calculating average frequency of attacks. Assume 19% DA rate and no delay reduction. Also assume sufficient TP from the previous WS to get to 100 TP in five hits.

Calculating average time to 100 TP for a "six-hit setup" - Perdu Voulge

Hit rate	Average no. of rounds	Average no. of hits		Average time (s)
70%	6.136	5.111		51.547
80%	5.386	5.127		45.244
90%	4.802	5.143		40.342
95%	4.557	5.151		38.278

I chose 70% overall hit rate as the lowest acceptable hit rate for meleeing anything. Notice that the average number of rounds and average time to 100 TP appears to decrease with increasing hit rate at a decreasing rate. As hit rate approaches 100%, the average number of rounds approaches 4 about 4.336. (Remember that double attack is present.) Also notice that the average number of hits increases with hit rate at a constant rate. Observing these trends is just to help verify that I didn't screw up the calculations. (Of course, I'm doing this all with spreadsheets.)

Calculating average time to 100 TP for a "six-hit setup" - Erlking's Kheten (DMG +6, STR +5, attack +5)

Hit rate	Average no. of rounds	Average no. of hits		Average time (s)
67.5%	6.358	5.107		53.414
77.5%	5.555	5.123		46.667
87.5%	4.936	5.139		41.463
92.5%	4.676	5.147		39.283

In this exercise, I am assuming that Perdu Voulge has received the full benefit of accuracy +5 to attain 70%, 80%, 90% and 95% hit rate, so that the hit rate difference for Erlking's Kheten is relative to Perdu Voulge. I am also assuming that changes in hit rate is not discretized by units of 1% hit rate as calculated by the game.

Now, it's time to compare the "potency" of Erlking's Kheten to Perdu Voulge, keeping in mind the whole point here is to see whether the augmented Kheten overcomes its accuracy deficit relative to Perdu Voulge.

Calculating average damage to 100 TP - Perdu Voulge

Hit rate	Avg. no. hits to 100 TP	Average AA damage	Avg. no. hits per WS	Average WS damage	Total damage
70%	5.111	812.771	2.616	592.524	1405.295
80%	5.127	815.310	2.854	646.431	1461.741
90%	5.143	817.850	3.092	700.338	1518.188
95%	5.151	819.121	3.211	727.291	1546.413

Let's suppose fSTR is 10 and average pDIF is 1.500. By considering some arbitrary average, I don't have to care about critical hit rate or the distribution of pDIF or anything like that. Let's also suppose a WSC value of 45 for King's Justice. (Of course, if you are attacking some mob for which you have poor hit rate, it is doubtful your average pDIF against that mob will be as high as 1.500.) I am also assuming you maintain exactly the same hit rate and average pDIF for WS'ing as you do for TP'ing and 95% hit rate for the first hit of KJ, along with no fTP bonus from a gorget or TP in excess of 100.

For the average number of hits per WS, notice that the average increases with hit rate at a constant rate. Also, the percent difference in number of WS hits between any two levels of hit rate is going to be the less than the percent difference in hit rate because the first WS hit is assumed to have 95% hit rate.

Calculating average damage to 100 TP - Erlking's Kheten

Hit rate	Avg. no. hits to 100 TP	Average AA damage	Avg. no. hits per WS	Average WS damage	Total damage
67.5%	5.107	825.804	2.5565	593.197	1419.002
77.5%	5.123	828.386	2.7945	648.421	1476.808
87.5%	5.139	830.969	3.0325	703.646	1534.615
92.5%	5.147	832.261	3.1515	731.258	1563.519

For this great axe, suppose fSTR is 11 and average pDIF is 1.497 relative to Perdu Voulge (-1 attack; for -2 attack, use 1.493). Unfortunately, this average must depend on how you think it changes with attack, which in turn depends on your target's defense rating as well as your critical hit rate, which also depends on your target's AGI. Still, I think it's inappropriate, since I have come this far, to avoid the nuisance of calculating the change in pDIF. Also assume WSC of 47 (+2 higher than with Perdu Voulge).

Damage per second - Perdu Voulge

Hit rate	AA prop. total dmg	DPS
70%	.578	27.26
80%	.558	32.31
90%	.539	37.63
95%	.530	40.40

Counter to the silly notion that there is "diminishing return" when it comes to accuracy, the computed damage per second here (average amount of damage output in a "cycle" of 5 hits to 100 TP plus damage from the executed weapon skill) appears to increase with increasing hit rate at an increasing rate. (Funny phrase.) This should make sense when you consider that accuracy not only increases the average damage of King's Justice, it also increases the frequency of WS (kind of like a delay reduction). The two-fold effect of accuracy could be considered "synergistic."

Damage per second - Erlking's Kheten

Hit rate	AA prop. total dmg	DPS
70%	.582	26.56
80%	.561	31.64
90%	.541	37.01
95%	.532	39.80

On paper, alas the "potency" advantages of the augmented Erlking's Kheten (DMG +6, STR +5, attack +5) do not overcome the accuracy bonus (+5) on Perdu Voulge when you receive the full bonus of Perdu Voulge. There is no "critical value" of hit rate at which the two weapons are equivalent. If Perdu Voulge is at 95% hit rate and receives the full effect of accuracy 5, on paper it is (40.40/39.80 - 1)*100 = 1.507% more efficient. Based on the previous comments about the effect of accuracy both on average WS frequency and average WS damage, this value should be higher than the rudimentary calculation [(.95*106)/(.925*108) - 1]*100 = .801% for auto-attack damage alone.

So, even though the Kheten is more potent, it must be slower to WS when hit rate is not capped and that is the decisive factor to the extent that accuracy matters for you.

Maybe if I got DA +2% on Erlking's Kheten (along with those other augments, though? Yeah, right) I could redo the calculations.

Conclusion

Erlking's Kheten: sorry, try again! If accuracy matters with "most other things being equal," Perdu should always be better. If you're always at 95% hit rate (not unlikely given the practicality of marinara pizza)... then why did you read this post?