Probability distributions associated with WS spam

(Correction: 06/15/2010. I thought Sekkanoki lasted one minute but after reading up on it, it lasts either for one minute or until the next weapon skill, whichever comes first. So, my discussion of the consequences of TP overflow elimination now refers to a hypothetical "Sekkanoki 2.0," which would reduce the TP cost of all weapon skills to 100 TP.)

Who cares about "TP overflow"?

"TP overflow" seems to be the de rigueur term referring to any landed hits that don't contribute to spamming weapons every time 100+ TP is accumulated. TP overflow is inevitable when more than one landed hit per attack round is possible, so it's not like anyone can do much about it except attempt to minimize it by spamming WS. This absolutely does not mean it is harder to "cope" with TP overflow using a multi-hit weapon (when weapon delay is the same as a non-multi-hit alternative). Rather, slack effort means squandering the benefit of the more rapid TP gain of the multi-hit weapon.

So why care about TP overflow? One argument is that it should be "accounted for" when doing item comparisons pertaining to damage efficiency, possibly to be more accurate.

Consider, for example, Soboro Sukehiro, which is considered to average 1.9 attacks per attack round, with the probability of two attacks being .5 and that for three, .2. Given 100% hit rate and 0% DA rate, it takes 3.46553 attack rounds, on average, to be able to execute a weapon skill in six hits, with the actual average number of hits being 6.584507 (note that 6.584507/3.46533 = 1.9 attacks per round), so almost 9% of the hits occur in excess of the target number of hits.

What if somehow there was a way to allocate the TP from those excess hits toward additional weapon skills? Well, Samurai has a level 60 job ability called Sekkanoki, which limits the cost of the next weapon skill to 100 TP. This seems analogous to job abilities like Elemental Seal or Divine Seal, which lasts for 1 minute or until a spell is used, whichever comes first. But what if Sekkanoki limited the cost of all weapon skills to 100 TP while active, say, one minute? This would effectively cause a re-allocation of TP toward future weapon skills. Let's call this "Sekkanoki 2.0."

If one were under the effect of "Sekkanoki 2.0" over a very long time interval, effectively all of the TP would go toward weapon skills, and so the average number of hits approaches 6. Since the average number of attacks per round is 1.9, then the average number of attack rounds approaches 3.157894737, which seems like a fairly significant reduction in average attack rounds until you realize that the concomitant "loss" of TP damage that results from TP overflow (which is eliminated under Sekkanoki 2.0 over an infinite period of time), along with the slight loss of WS damage, offsets the benefit of increased WS frequency. (Also, the proposed Sekkanoki 2.0 lasts for 1 minute out of 5, which means that some TP overflow is inevitable for finite time periods, so it's not like Sekkanoki 2.0 has this tremendous effect.) So, the argument about accounting for TP overflow is a bit overblown (not that you shouldn't, however).

So why care about TP overflow? Since there is no Sekkanoki 2.0, which itself would be a limited tool, you can't do anything about it, so why worry about it? Maybe it's more about players wanting to appear to be "clever" about a not-very-subtle consequence of multi-hit weapons, like asserting that the probability of TP overflow for a given WS is high. (One could easily retort that for Soboro, the fraction of excess hits over total hits would be around 9%.)

But, you know, I'm all about meaningless stuff, so let's finally get into how to define the probability distribution of excess hits (that contribute to TP overflow) associated with WS spam (this would be the same as the probability distribution of the number of hits you end up with under the condition that you spam weapon skills).

Excess hits contributing to TP overflow and the corresponding probability distribution

Let E denote the number of hits in excess of those that contribute to the 100+ TP (in six hits) required to spam a WS. Let's continue with the example of Soboro. For any given attack round, the probability of n landed hits is π_n, where n = 0, 1, 2, 3. These probabilities are straightforward to calculate. Not as straightforward to calculate is the probability mass function for E. An extremely tedious approach is to list all the possible combinations of attack rounds that result in 6 or more hits—the possibilities being 6, 7, or 8, which correspond to E = 0, 1, and 2, respectively. This approach requires knowing what to count (all the possible ways to get E = 0, 1, and 2), how to count (combinatorics), and knowing the closed-form expression for the sum of an infinite series, as the possibility of missing hits with non-100% hit rate means there are an infinite number of possible outcomes. (For a given combination of attack rounds leading to 100 TP, there could possibly be zero attack rounds that yield zero landed hits, one attack round that yields zero landed hits, two attack rounds that yield zero landed hits, and so on. These attack rounds are independent of those that yield hits.)

After spending more time than I care to admit, I obtained the p.m.f. of E, which is


This expression is quite unsightly, and rather useless. Not only is it useless merely because knowing the probability of TP overflow is useless, it also is useless because it refers only to the case where 6 hits are required to attain 100 TP. It requires no imagination to see that an expression for a dual-wield situation would be ghastly. It also is useless because you don't even need to knowledge of this p.m.f. to obtain the average number of hits in the process of getting to 100 TP (as I have shown repeatedly in the past). But there it is...

Again, using the Soboro example, P(E = 0) = 0.522579, P(E = 1) = 0.370335, and P(E = 2) = 0.107086, and thank goodness the probabilities sum to 1. The probability of "TP overflow" for a given WS with Soboro is almost 50%... not that you can really do anything about it. The correct response is, "who gives a shit?"

Even worse: the probability distribution of the number of attack rounds

Let R denote the (total) number of attack rounds that results in 100 TP. Again, with the Soboro example, R = 2, 3, 4, ..., and there is not much hope for an elegant formula for the probability distribution, because to obtain such a formula "by hand," one needs again to enumerate all the possible outcomes associated with each event. I only got as far as R =3 before I quit.


Again, using the Soboro example, P(R = 2) = .04, and P(R = 3) = .519. This is consistent with the average number of attack rounds being ~3, but if you already had the average number of attack rounds, why do you need the corresponding probability distribution. Useless!

A better approach for calculating these probability distributions: Markov chains

Perhaps I'll discuss this in a future entry. Aside from the fact that knowing the above probabiltiy distributions is quite useless—average weapon skill TP, average number of rounds, and average number of hits, among other things, are all easily obtained without any knowledge of these probability distributions—the Markov chain approach to obtaining these is much faster and far superior when no symbolic formulas are required. The interpretation of Markov chain output and manipulation is also much easier than it is with formulas for a specific case. It is also the only realistic way where dual-wielding is concerned, as you would have to be crazy even to consider deriving closed-form expressions for the probability distributions for that situation. It is so easy to make a mistake with a binomial or multinomial coefficient here or there, that I have to admit I didn't obtain the above expressions entirely "by hand," but with the help of Mathematica, which is quite handy for dealing with symbolic math.

A real great katana comparison

(Correction: 06/13/2010. Additional comments are in italicized red. Incorrect statements are crossed out.)

Earlier, I blabbed about the consequences of delay associated with the use of weapon skills in terms of modeling damage output mathematically, but did not justify how much delay should be specified because I didn't know how much the following attack round (after a WS) is delayed. Fortunately, I came across this presentation of results and discussion quantifying the amount of delay that is effectively added to the attack round following the use of a job ability (or weapon skill). The results of "stacking" job abilities aside (read for yourself), it is obvious that a two-second delay for the use of a weapon skill must be accounted for, at the minimum, when attempting to model theoretical damage output. (Using other job abilities while engaging an enemy would also have an effect on damage output, but the use of weapon skills, if spammed, is the dominant factor contributing to job ability delay. Consequently, many of my previous posts, which ignored this delay, likely have led to incorrect conclusions.)

For now, though, I think it would be instructive to show how much a weapon skill delay of two seconds obviously hampers the modeling of damage output. But I don't want to waste my time doing the "before" analysis, so I base my "after" analysis based on the conditions set forth in this comparison of great katanas (covering Hagun, Soboro Sukehiro, Kurodachi, and Radennotachi). There are some problems with it, especially with the implied use of /DRG (low DA rates but not accounting for the effect of Jumps, wut). Therefore, I do not merely reuse the computed figures given but provide my own in some cases. In any case, it may help to review that comparison and mine side by side as I wish not to waste my time rehashing said conditions.

Calculating WS frequency: Zanshin is relevant for main job SAM?

The effect of Zanshin on weapon skill frequency is something I had not considered in my previous posts, and I am kind of surprised the activation rate is apparently rather high for samurai as the main job. Recall that in the October, 19, 2006 version update, "the hit rate of the extra attack [was] increased." Moreover, there is very good evidence the Zanshin activation rate can be considered 45% for main job and 25% for subjob, with the hit rate bonus the result of +35 accuracy (source). Unfortunately, it is more difficult to furnish evidence as to how Zanshin interacts with double attack for auto-attack purposes, but it seems likely that Zanshin has a lower "priority" than double attack (if double attack processes, Zanshin doesn't, and if it doesn't, Zanshin can), so I'll just run with that. This means that accounting for Zanshin doesn't really matter all that much for multi-hit weapons, but since I do it for Hagun and Radennotachi, I might as well do it for the other two.

To start off with my "after" analysis (remember I want to show the effect of weapon skill delay not previously considered on an analysis that incorrectly ignores it), going back to the "before" analysis I cited previously, I should first point out that pDIF is apparently ignored in favor of a bogus assumption of a "baseline" 35:65 ratio of melee damage to WS damage for Hagun.

Since we are talking about theoretical damage output, it is nonsense to assume such a ratio. The baseline assumption is bogus, not that 35:65 may be observed in practice. If 35:65 is observed, surely average auto-attack damage and average WS damage are also observed (from parser output)! Use those values instead to back-calculate an "average" pDIF for both auto-attack and WS damage that should be fixed across all great katanas. The differences in WS frequency and weapon base damage will then account for the differences in the ratio of melee damage to WS damage, holding pDIF constant.

Anyway, I will return to the pDIF issue later. After accounting for the effect of Zanshin, I obtain the following averages for attack rounds from WS use to 100 TP, auto-attack hits in the process of getting to 100 TP after WS use, and the "effective" hit rate (landed hits per attack round), which encompasses the effects of accuracy, double attack, and Zanshin.

Weapon	Average no. of rounds	Average no. of hits	Effective hit rate
Hagun	5.05945	5.10742	1.00948
Soboro Sukehiro	3.11051	5.58467	1.79542
Kurodachi	3.99375	5.32642	1.33369
Radennotachi	5.05945	5.10742	1.00948

I am aware of the apparent absence of Brutal Earring (5% DA) for Soboro (but why use a Pole Grip then, implied with the stated 2% DA?), replaced by a mysterious source of accuracy +5, and accounted for those differences. I gave the benefit of the doubt, so to speak, with Soboro (94% hit rate after accuracy +5), even though it could easily be argued that, across all merit mobs encountered, the average hit rate could actually be closer to 93.5%.

My effective hit rate figures agree with the previous analysis more or less, but I do not compute effective hit rate directly. Instead, I compute it, as a kind of check on my calculations, after computing the average number of attack rounds and average number of hits (example: 5.10742/5.05945 = 1.00948) to make sure I didn't make any errors calculating the average number of attack rounds.

As always, the average number of rounds can be converted to the average time to accumulate 100 TP, but now the time between weapon skills must also account for the two-second weapon skill delay discussed previously. (This will be done at the end of the post.)

Accounting for average TP for the use of Tachi: Gekko

The previous analysis assumes maximum fSTR for each of the weapons (16, 12, 15, and 17 for Hagun, Soboro, Kurodachi, and Radennotachi, respectively), which would appear to be reasonable given the implied high STR modifier bonus used for Tachi: Gekko (152*.75*.83 = 94.62, which is close to the given 94). As mentioned previously, pDIF is completely ignored, but based on the attack bonus of Tachi: Gekko, it is reasonable to assume an average pDIF of 2.3 (based on a symmetric pDIF distribution between 1.9 and 2.7).

The only thing left is calculating the fTP bonus of the first hit for Tachi: Gekko, which requires calculation of average TP for each weapon when a one-hit weapon skill is used, accounting for double attack. This, in turn, requires knowledge of the probability distribution of TP return from a one-hit WS and the corresponding TP values, which is the same regardless of weapon.

This would seem straightforward except for the observation of 2-TP return with one-hit weapon skills (source), which would suggest that for weapon skills, Zanshin can occur on the first hit independent of the double attack (Zanshin still can't occur for the double attack hit, presumably). The presence of Zanshin effectively "reallocates" the probability of missing the first hit (and losing the full TP return of 16.7), which is 5% most likely, so ignoring the Zanshin effect for a one-hit weapon skill results in negligible error for TP return (but not necessarily WS damage).

Weapon	Average TP per WS (my calculation)	fTP bonus of 1st hit (with Gorget effect)
Hagun	101.27806	1.9829879
Soboro Sukehiro	109.24816	1.6914005
Kurodachi	104.93533	1.6779229
Radennotachi	101.27806	1.6664939

Note that average TP shouldn't be truncated because these averages are themselves based on the actual truncated TP figures to begin with (assumed 16.7 TP per main WS hit and auto-attack hits and 1.4 TP for off-hand WS hit).

Accounting for average Tachi: Gekko damage: ignore Zanshin?

Given 91% hit rate for any double attack hits (7% DA rate) for Tachi: Gekko (95% otherwise), the average number of hits per weapon skill is .95 + (.91)(.07) = 1.0137. Accounting for the 45% Zanshin rate, this average rises to 1.035075, of which .95 still corresponds to the first hit (which receives the fTP bonus), so 0.085075 of the hits in the average WS have an fTP = 1. The effect of Zanshin is, therefore, like adding 2.345% DA, which, for the purposes of Tachi: Gekko, constitutes approximately a 1.1-1.3% increase in average WS damage. (This is given the conditions stated in the "before" analysis). Whether or not this is accounted for (I will account for it), the effect of Zanshin very slightly "favors" weapons with worse WS "secondary" hit damage (compared to other factors), so it can be ignored for convenience.

A "fatal" flaw: consequences of the effect of haste with weapon skill delay

Because weapon skill delay, which is a fixed value (consider it two seconds), exists, the relative benefit of haste (or other forms of delay reduction) is higher for weapons with lower weapon-skill frequency compared to weapons with higher weapon-skill frequency. It follows that a weapon with higher weapon skill frequency CAN actually be "worse," on average, than a weapon with lower weapon skill frequency depending on the level of haste!

One way to think of this is to consider an arbitrary time frame during which weapon skills occur. The time associated with the WS frequency might be reduced with haste, but there is always an absolute weapon skill delay tacked on. Even if haste goes to 100% (meaning the time associated with WS frequency goes to 0) and you still decide to use WS for some reason, the sum of the absolute weapon skill delay for the weapon with higher WS frequency will be higher than equal to that for the weapon with lower WS frequency (WS frequency is rendered irrelevant if it takes zero time to build TP toward a WS), so the weapon with higher WS damage wins out in terms of WS damage output.

A "practical" consequence is that for "zerging" situations where maximum haste is involved, low-damage, multi-hit weapons (on average) can be worse than standard weapons. Similarly, multi-hit weapons may not be that good for meriting situations.

The "fatal flaw" with the "before" analysis is the unstated assumption that the haste level doesn't matter across weapons, so that the "pecking order" of great katanas always holds. Because weapon skill delay is not accounted for, the analysis does not hew to what is experienced in practice.

Repeat the analysis instead with ~65% haste (Hasso, Haste spell, double March, 20% equipment haste) along with the weapon skill delay of two seconds. The following figures are the result of a "per weapon skill" perspective, using average auto-attack pDIF 1.15 and average WS pDIF of 2.3. (Overwhelm 5/5 also used.)

Weapon	Avg. TP dmg	Avg. WS dmg	Time per WS	Dmg/sec	TP:WS dmg
Hagun	510.99763	910.7053192	15.281 s	93.04	36:64
Soboro Sukehiro	333.96349	619.4514917	10.165 s	93.79	35:65
Kurodachi	490.03068	744.1553438	12.810 s	96.35	397:603
Radennotachi	593.22713	835.6201688	15.281 s	93.50	415:585

Given 65% haste, relative to Hagun, Kurodachi is about (96.3474/93.0370 - 1)100% = 3.56% more efficient, and Soboro, about (93.7931/93.0370 - 1)100% = 0.81% more efficient. Radennotachi is about 0.5% more efficient. This jibes with the observation that Soboro is not really any better than Hagun in a typical merit situation.

Now, what happens given 80% haste?

Weapon	Avg. TP dmg	Avg. WS dmg	Time per WS	Dmg/sec	TP:WS dmg
Hagun	510.99763	910.7053192	9.589 s	148.26	36:64
Soboro Sukehiro	333.96349	619.4514917	6.666 s	143.03	35:65
Kurodachi	490.03068	744.1553438	8.177 s	150.93	397:603
Radennotachi	593.22713	835.6201688	9.589 s	149.01	415:585

Obviously, the damage figures (other than rate of damage) shouldn't change with haste. As they are fixed, changes in relative efficiency calculations (relative to 65% haste) involve only changes in time per WS (where applicable). The effect of 15% more haste benefits Hagun relatively more than it does Soboro because of the presence of the fixed two-second weapon skill delay. The result here shows that Hagun is more efficient than Soboro in a max-haste situation when spamming WS, and you should be. 910 damage, on average, in exchange for 2 seconds is better than 511 damage, on average, over 7.589 seconds.

(Correction: 06/13/2010) Incidentally, given 9% DA (the stated condition), Kurodachi is still better than Hagun even with maximum haste, so it is just better barring situations where WS damage is the predominant form of damage and WS frequency is an irrelevant consideration but as DA increases, Hagun eventually becomes better than Kurodachi. This should make sense (but even I overlooked this...) because the "full" benefit of a DA increase is not realized with multi-hit weapons such as Kurodachi, and definitely not with Soboro Sukehiro.

Conclusion

Weapon skill delay, which exists and can be considered to be two seconds, should be considered when doing a theoretical comparison of things related to doing damage.

A major consequence of weapon skill delay is that, as haste increases, weapons with lower WS frequency benefit relatively more than weapons with higher WS frequency. This affects the "correct" choice of weapon for situations where high levels of haste are achieved. For example, even though Soboro Sukehiro may be better than Hagun at low levels of haste, it is inferior at high levels of haste (on average, since there is some inherent variability of WS frequency associated with multi-hit weapons).

(Correction: 06/13/2010) However, it can be shown that Kurodachi is superior to Hagun when WS frequency is a relevant factor (e.g., not relying only on Meditate to generate TP). "Actually better" in theory, however, is contingent on how much base DA is present.

The effects of Zanshin on WS frequency, WS damage (fTP bonus and Zanshin hits), and TP return can be quantified. While the effects of Zanshin given low hit rate were not discussed, the effect of Zanshin can "safely" be ignored for relative comparisons given high hit rates.

A comparison of 5-hit Rindomaru with Hagun

The great katana Rindomaru is one of those new "fey" weapons that can be augmented through the quest "Succor to the Sidhe," and with the possibility that Rindomaru can be augmented with a heap of Store TP, a "5-hit Rindomaru" setup could theoretically rival the boilerplate "6-hit Hagun."

The main comparison here is whether the increased WS frequency from a 5-hit Rindomaru overcomes the Hagun's TP bonus, or whether a hypothetical 25% increase in weapon skill frequency overcomes the 20% increase in weapon skill damage with the TP bonus for the Yukikaze/Gekko/Kasha triumvirate.

A crude calculation of efficency with "most things being equal" could be something like [(88*4+700)/30/(86*5+800)*37.5 - 1]*100 = 6.91%, that is, Rindomaru is more efficient with really crude simplifications. But surely we can be more sophisticated than that.

To start, here is a description of a pretty good augmented Rindomaru (15 Store TP, +4% weapon skill damage, etc.) as well as some specific, full-Usukane equipment setups for both 5-hit Rindomaru and 6-hit Hagun with minor differences, such as Sword Strap for Hagun and White Tathlum for Rindomaru. I will be basing my calculations based on these setups... and the canonical Greater Colibri.

Calculating average time to 100 TP

Weapon	Average no. of rounds	Average no. of hits		Average time (s)
Rindomaru	3.774	4.123		28.310
Hagun	4.690	5.123		34.159

Here, I am assuming 15% double attack rate and 95% hit rate. For Hagun, the reduction of delay is from 450 to 437. Under these conditions, the increase in WS frequency is about 21%.

Calculating average damage to 100 TP (including weapon skill damage)

Weapon	AA "base" damage	WS "base" damage	Average AA damage	Average WS damage	Total damage
Rindomaru	88	167	580.640	709.992	1290.632
Hagun	86	169	705.051	818.463	1523.515

The assumption of using a weapon skill immediately upon getting 100 TP is not all that realistic, but then again all comparisons like this are based on "ideality" and the excuse that this is all supposed to be the case "in the long run." Take the comparison with a brick of salt as you consider the conditions under which it's made.

As another "ideal" assumption, I suppose that pDIF is maxed out for weapon skill damage, and is 1.6 on average in the auto-attack phase. The average number of hits per weapon skill is 1.0925.

How do I account for the "+4% weapon skill damage" on that hypothetical Rindomaru? It sounds like it could be incorporated into the fTP factor. Thus, the fTP factor for the first weapon-skill hit is 1.975 for Hagun and 1.7025 for Rindomaru. For this case, Hagun's average WS damage is about 15% higher.

I also accounted for Meditate but no Overwhelm, mainly because I don't know if Overwhelm affects all hits of a weapon skill.

Damage per second

Weapon	AA proportion of total damage	DPS	Relative efficiency
Rindomaru	.450	45.588	+2.21%
Hagun	.462	44.600	---

Here, you would want to compare the theoretical AA proportion of total damage to what you actually experience.

On paper, this idealized Rindomaru is actually better than Hagun on paper. At this point, you may wonder if using a weapon skill immediately at 100 TP is realistic, so consider also the fairly extreme case of waiting an additional round past 100 TP before WSing (equivalent to starting from 0 TP).

Another comparison of damage per second by "wasting" an attack round beyond 100 TP

Weapon	AA proportion of total damage	DPS	Relative efficiency
Rindomaru	.496	41.286	---
Hagun	.495	41.631	+0.83%

You can think of the last two tables as representing the ideal lower and upper bounds of how fast you WS after attaining 100 TP... in the long run. So if you hold TP, the additional TP will "benefit" Hagun more than Rindomaru and the relative benefit of having a 5-hit setup with Rindomaru is eroded.

Finally, note that this hypothetical Rindomaru has +4% weapon skill damage, which I assumed is an fTP bonus. I consider this the key attribute that allows Rindomaru to eke out a slight edge.

Of course, as far as Greater Colibri are concerned, polearm with attack buffs would be more "fun" as far as cranking out Penta Thrusts with an average easily exceeding 1k. I did manage to find a forum thread with a parse of a 5-hit Tomoe where the AA proportion was about .38.

Comparison of Love Halberd and Tomoe for samurai

I allocated way too much time this weekend to putzing around with spreadsheets, but let's just finish this off, shall we? Here's an example of doing a fairly simple comparison of a 7-hit Love Halberd with a 5-hit Tomoe, which is based on ideas presented in a prior comparison of weapons for warrior. In particular, I utilize the concepts of "expected number of rounds to clear 100 TP" and "expected number of hits to clear 100 TP" to make the arithmetic more tractable.

I didn't see any (good) hypothetical comparison of Tomoe 5-hit versus Love Halberd 7-hit for samurai (using Penta Thrust), so I thought I could do this really fast because I already set up the "black box" (this mess of a spreadsheet) to spit out an answer.

Calculating average time to 100 TP

Weapon	Average no. of rounds	Average no. of hits		Average time (s)
Love Halberd	3.959	6.488		26.131
Tomoe	3.774	4.123		30.197

This is the easiest step as the assumptions are reasonable if idealized, such as 95% hit rate, 15% double attack rate, and starting with some initial TP from the previous weapon skill.

Calculating average damage to 100 TP (including weapon skill damage)

Weapon	AA "base" damage	WS "base" damage	Average AA damage	Average WS damage	Total damage
Love Halberd	70	110	454.179	650.335	1104.515
Tomoe	96	136	395.890	822.614	1218.505

Again, there are more simple assumptions, like using Penta Thrust immediately after attaining 100 TP, using the same fSTR throughout, and assuming an average pDIF of 1. Using the expected values from the previous table, the average auto-attack and WS damage can be calculated.

Also, average WS damage is based on an average return of 5.035 hits.

Did I account for the effect of Meditate? Assuming Meditate recast is 150 seconds, we can assume all the TP goes to one WS and incorporate that damage into one cycle of AA and WS damage. For example, a "Meditate WS" is about 0.174 of a full WS in one cycle for Love Halberd, 0.201 for Tomoe, which makes sense as Meditate will benefit "slower-to-WS" weapons relatively more (Tomoe being slower).

Damage per second

Weapon	AA proportion of total damage	DPS	Relative efficiency
Love Halberd	.411	42.267	+4.75%
Tomoe	.324	40.351	---

Time for a reality check. Is it really possible for Tomoe auto-attack damage to account for only about 33% of total damage? I would have to see some parser output to validate these calculations. If you ignore Meditate, the proportions increase to .451 and .366. I will update this post when I can track down some parser output.

Even accounting for Meditate, Love Halberd comes out ahead on paper by almost 5%. Whether that 5% is worth expending virtue stones in a merit party is another issue altogether. You can't really argue differences in hit rate (if you want hit rate to drop below 95%) since the only real difference would be whatever is used in the ammo slot. As for attack differences, who knows how DEX +7 would compare to attack +5 and whatever's in the ammo slot.

Of course, the major issue, at least to me, is whether DA really stacks with virtue weapons. I've been assuming it does. Even if it doesn't though, Love Halberd is still slightly more efficient.