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.)

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.

Cutting corners with Store TP and weapon skills

Edit: Another table appended.

Last week I referred to "minimum store TP" to achieve so-called n-hit builds from the standpoint of going from 0 to 100 TP in n hits or reaching 100 TP in n - 1 hits starting with sufficient TP return from the previous weapon skill, but practically speaking I should have called it "worst-case scenario store TP if you're using a multi-hit WS." With all the TP you'll get after the first hit (and when was the last time you saw only the first hit land when your WS didn't kill your target?), there aren't too many compelling reasons to maintain "true" store TP totals if it means using equipment you wouldn't touch otherwise. The question is how much store TP to drop while still maintaining a "virtual" n-hit.

As you might have guessed, you can turn to probability to answer this. Consider first the case of a 5-hit polearm with a 5-hit weapon skill. After calculating the probabilities for obtaining sufficient TP returns from a single WS (no need to present such clutter, but I hope I didn't screw up), we can see the relationship between dropping store TP and the lowered probability that you will be able to get 100 TP in n - 1 hits of the next TP-generating "cycle." Of course, these probability calculations are based on the assumption that DA can proc only twice on a multi-hit weapon skill (fewer than seven hits).

I am assuming 95% hit rate for the first WS hit. Since (lack of) accuracy does affect TP return, I thought it would be useful to show the effect of a lower hit rate.

Table 1. Probability of getting 100 TP in 4 hits (after a WS) for a 5-hit polearm (480 delay, 17% double attack rate)

Minimum hits after 1st WS hit	Store TP	95% hit rate		80% hit rate
0	54	.95		.95
1	53	.949996		.948865
1	52	.949996		.948865
2	51	.949677		.930353
2	50	.949677		.930353
3	49	.940513		.815681
3	48	.940513		.815681
4	47	.822485		.490462
4	46	.822485		.490462
5	45	.233998		.105841

You can see there is not much of a drop by shedding up to 6 store TP and still being pretty close to a true 5-hit. Remember that the first hit of a WS can still miss.

The probabilities shown are cumulative probabilities in the sense that, given some amount of store TP, what is the probability that I will be able to get 100 TP in 4 hits after a weapon skill? More specifically, given some amount of store TP, what is the least amount of hits I need to land to be able to get 100 TP in 4 hits with an acceptable probability? Remember that .95 is pretty much as good as it gets.

If you have 95% hit rate, 48 store TP gives you a 94% chance of generating 100 TP in 4 hits, requiring at least a 4-hit return from the previous WS (1st hit TP and TP from at least 3 other hits). If you have a "true" 5-hit build, shedding 6 store TP may be a good trade-off. For example, I've seen 5-hit polearm builds with 49 store TP (including merits), suggesting awareness that 54 store TP is rather superfluous.

If you have 80% hit rate, 48 store TP gives you a 82% chance of generating 100 TP in 4 hits, so you might want at least 50 store TP if being around 80% hit rate is more realistic for whatever you are doing.

This exercise can be repeated for both 6-hit polearm and 6-hit great axe.

Table 2. Probability of getting 100 TP in 5 hits (after a WS) for a 6-hit polearm (480 delay, 21% double attack rate)

Minimum hits after 1st WS hit	Store TP	95% hit rate		80% hit rate
0	29	.95		.95
1	28	.949996		.948948
1	27	.949996		.948948
2	26	.949705		.931688
3	25	.941320		.823837
3	24	.941320		.823837
4	23	.832807		.513000
5	22	.284425		.130744

29 store TP not all that easy to obtain as a warrior (maybe you want to use Aurum Cuirass), but 24 is possible with a bunch of ticky-tack pieces. 15 from /SAM, 5 from Rajas, 1 from Brutal, 1 from Chivalrous Chain, 1 from Ecphoria Ring, and 1 from Engetsuto gives 24 total. Then again, if you're spamming crab sushi, some of these may not be very optimal for Penta Thrust.

Table 3. Probability of getting 100 TP in 5 hits (after a WS) for a 6-hit great axe (504 delay, 21% double attack rate)

Minimum hits after 1st WS hit	Store TP	95% hit rate		80% hit rate
0	22	.95		.95
1	21	.948478		.923695
2	20	.889887		.702636
2	19	.889887		.702636
3	18	.034124		.017160

I have only 6 store TP on equipment for warrior anyway. I can live with 21 store TP if I actually am using /SAM for some reason. What about the likes of 6-hit scythe and 6-hit polearm, both with four-hit weapon skills (like Guillotine and Drakesbane)? The following table compares the minimum TP for a "true" 6-hit build to the minimum TP for a "virtual" 6-hit build.

Table 4. Minimum Store TP requirements for 6-hit builds with 4-hit weapon skills

		Minimum Store TP
Delay	Base TP	True	Virtual
528	14.4	16	14
513	13.9	21	18
501	13.6	23	20
492	13.3	26	23
480	13.0	29	26

With "virtual" store TP builds, the corresponding probability is .9449 given 95% hit rate. (Of course, lower hit rate will lower this probability.) If that .0051-difference in probability really troubles you and is unacceptable, by all means be hyper-conservative.

Dumb thread(s) of the day

Here's a new feature where I talk briefly about crappy replies to decent questions. It would be a lot easier just to take pot-shots all day at shitty FFXI forum threads, which I might just do rather than play with numbers all the time.

Apparently, there is a thread on BG discussing why Allakhazam is so maligned, which usually is done by repeatedly knocking down the straw man that anyone actively endorses TPing in STR or DEX rings. When talking about a signal-to-noise ratio, the noise component is rather substantial on Allakhazam but the signal is pretty small in absolute terms for any FFXI forum, really. Even BG has threads like this, where bald-faced assertions are made without referencing sources and people can say they get 8-hit Drakesbanes with a straight face.

As another example, if you were talking about the relative efficiency of a 6-hit polearm build, you would pretty much get the same content-free, inane answers whether you posited this question on Allakhazam or Blue Gartr. Apparently, it is so difficult to use an average auto-attack damage, use an average WS damage, estimate the time between weapon skills, and use all this information to estimate roughly the relative efficiency of a 6-hit build. (Hint: a 6-hit is not even close to being 20% more efficient than a 7-hit). Instead, you have a reasonable OP followed mostly by dumb-fuck snark and drivel.