I thought this would be a fun exercise using properties of expectation (probability theory) to compare the performance of Fortitude Axe, Perdu Voulge, and Engetsuto on a level 82 Greater Colibri (67 VIT, 327 defense) in terms of (approximate) average ideal damage. My idea of "average ideal damage" is based on a so-called "cycle" of TP generation to reach 100 TP followed immediately by a weapon skill.
Why compare the expected damage resulting from a "cycle" of auto-attack damage
and WS damage? While this implies an unattainable ideal (along with generally uncontrollable factors "in the field," such as Pecking Flurry and the like, it is practically impossible to WS as soon as possible and still get the maximum "realized" damage from that WS because of overkill), it is a concise representation of efficiency
in the long run that includes damage from both auto-attack and WS "phases." Talking about the long run means considering the variability of auto-attack and weapon skill damage.
Some motivation: arguments based on a wall of arithmetic tend to be annoying. Those who present these walls of arithmetic seem to labor under the pretense of precision (why calculate anything if you don't want to be as precise you possibly can?) yet their lack of clarity obscures "typical" mistakes like using cRatio as a stand-in for pDIF. This obviously underestimates the relative effect of attack where percent changes are involved. Another example is adding or subtracting percent changes. While the propagation of error may be slight because percent changes tend to be based on values near 1, it's still error.
On the other hand, this kind of junk tends to be done in an off-hand manner for quick and dirty comparison of
what's better (or worse), which is understandable. Yet where putatively non-trivial comparisons are concerned, why not answer the question of
how much better (or worse)? This is what I attempt to do with a comparison of Fortitude Axe and Perdu Voulge by outlining the steps I take to make the necessary calculations.
As for the specific example, I wish to mount a counter-argument against the "hype" that Fortitude Axe and a polearm (Engetsuto being a typical polearm for warrior to use) are meaningfully superior to Perdu Voulge. To me it's not at all obvious that the increased WS frequency associated with Fortitude Axe should overcome the base damage discrepancy despite facile intuition-based arguments that are biased toward Fortitude.
But before starting, why not make some preliminary assumptions?
Some general assumptions
- 95% hit rate.
- Meat mithkabobs for great axes and crab sushi for Engetsuto (the latter to ensure 95% hit rate especially without merits and high-end accuracy equipment).
- 19% double attack rate. It is known that the explicitly named Double Attack trait "stacks" with the virtue weapon property. Therefore, it is possible to attack up to four times in a single round (supposedly).
- Sufficient TP to achieve 6 hits from 0 to 100 TP for all weapons (also considering the 7-hit case for the polearm).
- Assume sufficient TP return from a previous WS to be able to reach 100 TP in 5 hits (or 6 hits in the case of the polearm) for the single "cycle" described earlier. In the "long run," which is what expectation is all about in a sense, the extra hit needed to attain 100 TP to begin with makes a slight contribution to the expected value.
- Berserk active only. This is merely for my own convenience.
- fSTR 6 for TP, 11 for WS.
- Average pDIF based on assumption of symmetry of pDIF distribution, which seems to hold for cRatio above 1.5. This assumption ignores behavior of pDIF at the extremes.
- 18% critical hit rate. This is completely arbitrary and corresponds to some level of DEX that I choose not to specify at this time.
- WS damage always assumed to be at 100 TP. The expected TP per WS for Fortitude Axe would obviously be higher than that for Perdu Voulge (where damage or crit rate varies with TP). This may be worth keeping in mind.
Some of the jargon and terminology (pDIF, cRatio) is explained in this
FFXIclopedia article and immediate links to other articles.
I will also explain briefly how I derive all the computations that follow and state any further assumptions, if not justify them due to lack of verification through data or resources/comments "officially" sanctioned by SE.
1. Expected number of attack rounds to 100 TP
Given 100% hit rate and 0% double attack trait, the expected number of attack rounds to 100 TP for Fortitude Axe is 3.5625 given sufficient TP return from the previous WS, compared to 5 for Perdu Voulge (obviously) and 6 for Engetsuto. The 3.5625 value can be computed using conditional expectation and can be verified through simulation.
Of course, I assumed 95% hit rate and 19% DA rate to begin with, so the expected values of the number of rounds to 100 TP are different. Here is a table for the expected number of attack rounds for some values of hit rate to give some sense of the trend. As one should expect, the rate of change of the number of attack rounds
increases as hit rate
decreases for any weapon.
Table 1. Expected number of attack rounds to 100 TP for various values of hit rate.
| Hit rate
|
| Weapon | 95% | 90% | 85% | 80% |
Fortitude Axe
| 3.259 | 3.423 | 3.606 | 3.812 |
Perdu Voulge/
Engetsuto (6 hit)
| 4.557 | 4.803 | 5.077 | 5.386 |
Engetsuto (7 hit)
| 5.442 | 5.736 | 6.066 | 6.437 |
2. Expected pDIF for auto-attacking and WS (including the effect of critical hits)
I am basing pDIF and fSTR values (fSTR values of 6 and 11 stated earlier) loosely on my own equipment but do not assert complete fidelity to that.
With Berserk, a Meat Mithkabob, and 18% critical hit rate, the attack ratings, cRatios, and expected pDIF are as follows for both auto-attack and WS. The pDIF without critical hit contributions is necessary for the WS damage calculations done in a later step.
Table 2. Computed values for obtaining expected pDIF
| Weapon | Attack | cRatio | pDIF
(no crit)
| pDIF
(w/ crit)
|
Fortitude Axe
(auto-attack)
| 576 | 1.411 | 1.346 | 1.526 |
Fortitude Axe
(WS)
| 595 | 1.469 | 1.381 | 1.561 |
Perdu Voulge
(auto-attack)
| 596 | 1.472 | 1.383 | 1.563 |
Perdu Voulge
(WS)
| 607 | 1.506 | 1.407 | 1.587 |
Engetsuto
(auto-attack)
| 445 | 1.010 | 0.963 | 1.143 |
Engetsuto
(WS)
| 468 | 1.081 | 1.047 | 1.227 |
3. Expected number of hits in a 3-hit and 5-hit WS
Double attack is known to process twice on multi-hit weapon skills for two-handed weapons, but is it possible to double attack three times (or more) where applicable? While this may be an unsatisfying explanation, given the only data set (for Penta Thrust TP return) I have actually seen that attempts to answer this question, it seems very unlikely that DA can proc three times or more. Therefore, based on common experience, DA can proc only two times on a WS.
(This issue could merit its own post at a later date. For now, you could review my comments on DA and multi-hit weapon skills under the "double attack" tag, one of which addresses the Penta Thrust data set just mentioned.)
Based on the above, it is possible to obtain the probability distribution for the number of hits in the 3-hit and 5-hit WS cases. Since the first hit of a WS can have a different fTP (gorget effects or intrinsic property) and therefore different damage than the successive hits, these probabilities must be broken out by whether the first hit occurs in order to computed the expected damage. The total number of hits for a
n-hit weapon skill is described using the notation "
x+
y," where
x = 0, 1 are for the first hit and
y = 0, 1, ...,
n + 1 are for the rest.
Table 3. Probabilities for the number of hits in a 3-hit WS with 19% DA
No. hits
| Probability |
0+0
| .0000919133 |
0+1
| .0035232531 |
0+2
| .0343440735 |
0+3
| .0111244407 |
0+4
| .0009163196 |
1+0
| .0017463527 |
1+1
| .0669418089 |
1+2
| .6525373956 |
1+3
| .2113643724 |
1+4
| .0174100715 |
Table 4. Probabilities for the number of hits in a 5-hit WS with 19% DA
No. hits
| Probability |
0+0
| .0000002099 |
0+1
| .0000160425 |
0+2
| .0004616123 |
0+3
| .0059589813 |
0+4
| .0299088222 |
0+5
| .0123274910 |
0+6
| .0013268409 |
1+0
| .0000039875 |
1+1
| .0003048071 |
1+2
| .0087706328 |
1+3
| .1132206449 |
1+4
| .5682676222 |
1+5
| .2342223291 |
1+6
| .0252099764 |
4. Expected WS damage for King's Justice and Penta Thrust
I chose these weapon skills because neither can crit and I'm not sure what the critical hit rate for Raging Rush is at 100 TP. The properties of the alleged "damage spike" of King's Justice are not well known, so I do not account for this in the calculations.
Again, it is assumed that weapon skill damage is based on 100 TP. While this is an unrealistic assumption given the presence of double attack, it may be worth keeping in mind.
For the "effective" base damage for King's Justice, I used D + 11 + 42 = D + 53, where D is the base damage of the great axe. Here, the first hit is under the effect of an appropriate "sea" gorget (+0.1 fTP). For Penta Thrust, I used D + 11 + 32 = D + 43 and did not incorporate a gorget effect, but multiplied the damage by 1.25 to account for the piercing bonus on Colibri.
Table 5. WS damage (King's Justice for great axe, Penta Thrust for polearm) by hit
| Before pDIF | After pDIF |
Weapon
| 1st hit
| others
| 1st hit
| others
|
Fortitude Axe
| 128 | 117 | 176.863 | 161.664 |
Perdu Voulge
| 163 | 149 | 229.426 | 209.721 |
Engetsuto
| 160 | 160 | 167.588 | 167.588 |
Finally, we can calculate the expected WS damage based on the probability distributions in part 3:
Table 6. Expected WS damage
Weapon
| Damage
|
Fortitude Axe
| 521.256 |
Perdu Voulge
| 676.195 |
Engetsuto
| 856.547 |
5. Expected number of auto-attack hits per "cycle"
The expected number of rounds to 100 TP (with TP return from the previous WS) is related to the frequency of WS while the expected number of auto-attack hits to 100 TP (also with TP return from the previous WS) corresponds to the damage dealt during the auto-attack phase. Using conditional expectation, the number of hits to 100 TP is summarized as follows:
Table 7. Expected number of auto-attack hits to 100 TP for various values of hit rate
| Hit rate
|
| Weapon | 95% | 90% | 85% | 80% |
Fortitude Axe
| 5.527 | 5.499 | 5.472 | 5.444 |
Perdu Voulge/
Engetsuto (6 hit)
| 5.151 | 5.143 | 5.135 | 5.127 |
Engetsuto (7 hit)
| 6.151 | 6.143 | 6.135 | 6.127 |
Unlike the expected number of
rounds to 100 TP, which approaches infinity as hit rate tends to zero, the expected number of
hits to 100 TP approaches 5 for a 6-hit setup and 6 for a 7-hit setup as hit rate tends to zero. It is easy to see that TP for WS often exceeds 100 for Fortitude Axe.
6. Expected total damage per "cycle" (auto-attack and weapon skill damage)
Finally, by combining the results of steps 1-5, I obtain the expected total damage per "cycle" of auto-attack damage that concludes with WS use immediately upon attaining 100 TP.
Table 8. Expected total damage per "cycle"
Weapon
| Auto-attack
| Weapon skill
| Total in cycle
|
Fortitude Axe
| 590.767 | 521.256 | 1117.073 |
Perdu Voulge
| 821.619 | 676.195 | 1497.815 |
Engetsuto (6-hit)
| 669.821 | 856.547 | 1526.368 |
Engetsuto (7-hit)
| 799.837 | 856.547 | 1656.384 |
Total damage per cycle is a rate of damage. However, the duration of the cycle is different for each of these cases. To compare the efficiency of each of these, it is easier to convert damage per cycle into a damage per unit time.
At this point it I should see if these calculations do not flagrantly disagree with parsing of merit parties with both Fortitude Axe and Engetsuto. Informally speaking:
"REALITY" CHECK with Fortitude Axe:- Auto-attack damage: The expected damage in this exercise is 106.8817 and accounts for critical hits. From a short merit party (27 mobs), I have an average damage of 126.069 without Dia II (174 hits). However, this party had 2x Minuet for what it's worth.
- Weapon skill damage: The sample mean of King's Justice damage (12 attempts) was 662.50. The expected value is 521.256 and the standard deviation is just under 102. Since the sample mean is approximately normal for sample size 12 (I checked this with simulation), it is obvious that 662.50 is not at all consistent with the estimated mean. Then again, a mere ~100-attack increase from 2x Minuet would raise the expected value to 652.810. This doesn't account for the alleged spikes in KJ damage.
- Auto-attack damage as a proportion of total damage: The expected proportion for Fortitude Axe is .5289, which compares favorably with the observed proportion of .5326.
- CONCLUSION: The calculations so far seem reasonable. There is no compelling reason to question the validity of the calculations.
"REALITY" CHECK with Engetsuto:- Auto-attack damage: The expected damage in this exercise is 130.019 and accounts for critical hits. From a short merit party (128 mobs), I have an average damage of 178.6634 without Dia II (621 hits). Adding 100 more attack (approximating the effect of 2x Minuet) brings the expected damage up to 167.212, which is still lower than the observed. (Remember that I've been assuming Berserk always active.)
- Weapon skill damage: The sample mean of Penta Thrust damage (84 attempts) was 984.61. The expected value is 856.547 and the standard deviation is about 122.39. An additional 100 attack increases the expected value to 1046.136 and standard deviation to 153.
- Auto-attack damage as a proportion of total damage: The expected proportion for Engetsuto is .4829, which is a bit lower than the observed .5327.
- CONCLUSION: There don't appear to be any egregious discrepancies...
(I note that the Perdu Voulge latent bonuses are inactive above 100 TP but do not account for this in the calculations.)
7. How long is a "cycle"?
The answer is based on the expected number of attack rounds to 100 TP, which was computed in step 1.
Without any delay reduction whatsoever, and letting the first attack round occur at
x delay after 0 delay, where
x is the delay of the weapon of interest, one can approximate the time duration (in seconds) of the cycle by using the conversion 60 delay = 1 second.
Table 9. Cycle duration in terms of delay and seconds
| Weapon | Delay | Time (s)
|
Fortitude Axe
| 1643 | 27.38
|
Perdu Voulge
| 2297
| 38.28 |
Engetsuto (6 hit)
| 2187 | 36.46
|
Engetsuto (7 hit)
| 2612
| 43.53 |
Finally, damage per second for a single cycle can be computed from the results of step 6 and the current step.
Table 10. Damage per second in a single cycle
| Weapon | DPS |
Fortitude Axe
| 40.614
|
Perdu Voulge
| 39.129
|
Engetsuto (6 hit)
| 41.868
|
Engetsuto (7 hit)
| 38.049
|
After all that work, it turns out that Fortitude Axe does edge Perdu Voulge. It is worth mentioning that I tried to minimize any rounding except where called for by game mechanics. Intuitively speaking, the effect of excess TP above 100 will have a very minor effect on these DPS values, but for the sake of comprehensiveness, excess TP can be considered in section 8.
No combat skill merits are considered for the polearm cases. With that in mind, it seems futile to use polearm on Colibri without spamming crab sushi to achieve capped hit rate (or getting a Madrigal). Without meat mithkabobs and the high hit rate, the piercing bonus can't overcome the attack deficit.
8. Can we account for the excess TP beyond 100 that contributes to WS damage?
This is a bit troublesome because I have been assuming "sufficient TP return from the previous WS to get to 100 TP in
x number of hits." Actually, I can calculate expected TP return from the results of step 3, assuming the minimum Store TP for a "true"
n-hit build.
Table 11. Expected TP return from a weapon skill (5-hit polearm WS, 3-hit great axe WS)
Enough Store TP
for a ...
| TP return
|
Great axe 6-hit
(504 delay)
| 18.487
|
Polearm 6-hit
(480 delay)
| 20.8582
|
Polearm 7-hit
(480 delay)
| 18.1621
|
From here on, excess TP is relevant only for King's Justice.
One way to think about this problem is conditioning the expected number of hits
beyond 100 TP on the previous number of hits attained (before 100 TP). Unfortunately, I do not see an easy way to calculate, say, the probability that the previous number of hits is 3. It is just easier to estimate the expected TP per WS by simulation of the required probabilities. Note that there
will be excess TP even in the case of only 5 hits to 100, based on the expected TP return.
Table 12. Estimated probabilities of obtaining excess TP over 100 TP
| Weapon | +1.9 TP
| +18.6 TP
| +35.3 TP
| +52.0 TP
|
Fortitude Axe
| .5795
| .3239
| .0883
| .0083
|
Perdu Voulge
| .848
| .152
| N/A
| N/A
|
The expected King's Justice TP, fTP values, and WS damage (applicable only to the first hit) are as follows. (I assumed linearity of the fTP function with TP between 1.0 and 1.25.)
Table 13. Estimated expected TP and fTP for King's Justice (with sea gorget effect)
| Weapon | Estimated
expected TP
| Est. fTP
(1st hit)
| Est. damage
(1st hit)
|
Fortitude Axe
| 110.7612
| 1.126903
| 182.1795846
|
Perdu Voulge
| 104.5254
| 1.1113135
| 233.0656868
|
Recall that the expected damage on the first hit at exactly 100 TP was 176.863 for Fortitude Axe and 229.426 with the Perdu Voulge.
After all the intermediate calculations (results not shown), the updated DPS values for the great axes are obtained.
Table 14. Damage per second in a single cycle accounting for excess TP above 100
| Weapon | DPS |
Fortitude Axe
| 40.800
|
Perdu Voulge
| 39.219
|
The difference still favors Fortitude Axe, but again, it is slight. People seem to like percent changes, so it's about
3.194% better.
9. Why exactly are polearm and Fortitude Axe played up for Greater Colibri?
Good question, considering the inconvenience of farming virtue stones (even though soloing xzomits is simple). Using a weapon with inferior combat skill is also pretty inconvenient when you have a A+ weapon with 8/8 combat merits. In my own experience, I never felt that Fortitude Axe was that much better than Perdu Voulge, but "all things being equal" rarely applies when experience is discussed.
As for an informal argument as to why Fortitude Axe might be superior, here seem to be the main points, with rudimentary supporting calculations that someone might perform for a forum post:
- "Fortitude Axe provides 50% more damage, going from 0% DA to 50% DA, compared to a featureless 64-damage, 504-delay great axe." Of course, no warrior has 0% DA. In the presence of 19% DA, Fortitude Axe's WS frequency is expected to be about 1.40 times that of a generic great axe (based on time per cycle).
- "The double attack property of Fortitude Axe compensates for the low base damage." This is not really a self-evident statement. A 32-point difference in base damage from 64 to 96, given 11 fSTR, is approximately a 43% increase in damage. Yet from the standpoint of damage in the auto-attack phase, Fortitude Axe is still better roughly by about [(70*5.527/27.38)/(102*5.151/38.28) - 1]*100 = 2.941%, ignoring attack differences.
- From the standpoint of WS damage per unit time, where virtue weapon DA does not apply, Fortitude Axe is better by about [117/27.38/(149/38.28) - 1]*100 = 9.781%, again ignoring attack differences.
- A naive person would incorrectly add these percent changes, but in reality the calculation of difference is more like [(70*5.527+117*3.135)/27.38/(102*5.151+149*3.135)*38.28 - 1]*100 = 6.160% in favor of Fortitude Axe, yet again ignoring attack differences. (3.135 is the expected number of hits of a 3-hit WS.) This should not be counter-intuitive because TP damage is slightly more influential than WS damage.
Practically speaking, the percent difference is closer to 3.194% in favor of Fortitude because attack differences still exist, even for Greater Colibri.
As for polearms, I can see where it can be pretty effective with sufficient Store TP for a 6-hit, appropriate support, and meat mithkabobs. I wouldn't say it's particularly efficient or easy to implement though.
10. Conclusion
Fortitude Axe is better than Perdu Blade "on paper" against a Greater Colibri by about 1.04 damage/second or, in terms of percent difference, about 3.194%.
If you had polearm merits and enough accuracy (whether from equipment or a Madrigal), a combination of polearm and meat mithkabob spamming could be more "efficient" (more so for 6-hit) if your idea of efficiency is wasting a bard song on Madrigal or spending millions on the scarce accuracy equipment (basically Toreador's Rings or Sniper Rings +1 and Cuchulain's Mantle) to eke out more accuracy. Without sufficient accuracy and support though, meriting with polearm seems to be a waste of time.