Restraint: tentative findings

(Edit 07/09/2010: edited some figures to account for a Critical Attack bonus of 5%.)

This time I'm going to start with the current legitimate claims about the effect of Restraint, a level 78 warrior job ability that "enhances your weapon skill power with each normal attack you land, but prevents you from dealing critical hits" per the help description. Then I will go over the support for those claims, one at a time, and then discuss some implications for the effectiveness of Restraint.

Claims

• Restraint's enhancement seems to manifest as a damage multiplier distinct from other factors such as TP bonus, pDIF, and "TP modifier" (fTP).
  
• Restraint's enhancement is not exact but actually has some variability, controlling for the number of attacks landed. (The damage multiplier is effectively a random variable.)
• The damage multiplier of Restraint appears to have a maximum of 1.5 (+50% bonus).
  
• Restraint's enhancement is dependent on weapon delay. Generally speaking, the higher the weapon delay, the higher the damage increase per normal attack landed.
• The damage multiplier appears to increase linearly with the number of landed attacks up to a maximum of 1.5 (controlling for weapon delay).
  

Is Restraint's effect really a simple damage multiplier?

Using the weapon skill Spirits Within, whose damage function I described when discussing how I determined Fencer's TP bonus, it is straightforward for anyone to show that Restraint doesn't provide a TP bonus like Fencer does.

But first, why use Spirits Within? Its damage is completely deterministic and can be calculated exactly given your current HP and current TP. If there is then any deviation from the predicted value, that deviation can be attributed to whatever factor you had changed. Of course, this doesn't say anything about weapon skills one would actually want to use (for reasons to be discussed later), but getting a general idea of how Restraint appears to work should help focus further investigation (in theory because no one really gives a shit about doing it).

Anyway, ruling out a TP bonus is easy enough as soon as you observe a damage return from Spirits Within under Restraint that is impossible were it the result of a TP bonus. To go over briefly how I determined this, I whacked a Zvahl Fortalice with a Trainee Sword/Trainee's Needle combination (5.1 TP/hit with Dual Wield II) until I got 107.1 TP, then used Spirits Within. (Actually this basically is the general experimental procedure performed to reach some of the conclusions about Restraint I described earlier.) Given my current HP of 1148, the predicted damage, given 107.1 TP, is 147, but the observed damage was 164. It is impossible to obtain 164 damage from a TP bonus (the damage equation I provided is exact and has yet to fail), so a TP bonus can be ruled out.

As for pDIF, obviously pDIF doesn't enter into Spirits Within damage, so one cannot really speak of any kind of pDIF bonus, whether additive or multiplicative.

Ruling out an fTP bonus like that from an elemental gorget is not as straightforward. A conceptually simple method is to determine, using the same weapon(s) (holding weapon delay and therefore TP/hit constant), whether the damage of one weapon skill scales by approximately (accounting for flooring) the same factor as the damage of another weapon skill with a different fTP "profile" given the same TP and the same number of landed attacks under Restraint.

If the scaling factors are dramatically different, this could be considered evidence of an additive fTP bonus and one can rule out the idea of a damage multiplier. However, I chose not to do this for the following reason.

The damage increase from Restraint has some variability...

Continuing to whack on a Fortalice, I made the unpleasant discovery that, even though Spirits Within damage is supposed to be completely deterministic (knowing only two pieces of information, TP and current HP, means being able to calculate the damage exactly), I observed some variability of damage return holding TP constant under the effect of Restraint. The predicted damage values (based on 1148 current HP) and the observed damage values (which have a relationship to the number of attacks landed with Restraint active) are given in the following text table as I was too lazy to use my inelegantly constructed table markup:

Attacks landed     TP        Predicted damage     Observed damage (Restraint)
------------------------------------------------------------------------------
19                 102       143                  160, 158, 160, 160, 158, 158
20                 107.1     147                  164, 163, 160, 166, 163, 164
21                 112.2     147                  167
22                 117.3     152                  174, 170, 179
23                 122.4     156                  177, 179
24                 127.5     161                  183, 180
25                 132.6     165                  184, 189
26                 137.7     170                  192, 197
27                 142.8     170                  195, 202
28                 147.9     174                  201, 201
29                 153       179                  209, 209, 209
30                 158.1     183                  214, 219
31                 163.2     188                  218, 223
32                 168.3     188                  225, 229, 218, 218
33                 173.4     192
34                 178.5     197                  232, 236

To me, the fact that the "designers" apparently decided to make the Restraint enhancement a random variable is extremely obnoxious. (What the fuck is the point? Or was this unintentional?) If this is not merely an "anomaly" specific to Spirits Within, it makes it that much more annoying to pin down the effect of Restraint using weapon skills whose damage is normally variable. But at least now people should be aware of this.

Perhaps this is a glitch specific to dual wielding? I also checked for single wield and also observed variability of Spirits Within damage. I also attempted to check for damage variability with a magical WS whose damage is also "deterministic" (controlling for resist), but after getting a 87-damage quarter-resist and then a 171-damage half-resist with Seraph Blade (given 117.3 TP), I got very annoyed and switched back to Spirits Within for the purposes of exploring Restraint's effect further. Note that 171 is less than twice that of 87 (174), which can be considered evidence of variability fundamental to Restraint.

Restraint's maximum damage increase appears to be 50% (1.5 damage multiplier

For one iteration of Restraint, I wanted to see how much of a damage increase to Spirits Within I could get by accumulating as many landed attacks as possible within the 5-minute duration. I managed to get 110 hits in before using Spirits Within, which gave 807 damage, which happens to be exactly 1.5 times 538, the usual damage at 300 TP given 1148 HP.

Studio Gobli's version update notes also suggest that 50% might be the upper bound for the weapon skill damage bonus. But more important, the update notes indicates that Restraint's effect appears to be dependent on weapon delay.

The effect of Restraint depends on weapon delay

Reiterating Studio Gobli's notes, given 20 landed attacks, the weapon skill bonus (the weapon skill used is not stated) is highest (+21%) for the weapon with the highest delay (444), lower (+17%) for the weapon with the second-highest delay (264), and lowest (+13%) for the weapon with the lowest delay (218). Again, I used a Trainee Sword/Trainee's Needle combination (187 delay per weapon given Dual Wield II), and referring back to my text table, the damage increase given 20 hits was observed to vary from +8.84% (160/147) to +12.9% (166/147), so my results are consistent with Studio Gobli's claims (seemingly unsourced by the way). Moreover, my use of dual wield suggests that Restraint is affected by effective weapon delay.

Therefore, the effect of Restraint on Spirits Within cannot be generalized to other weapon skills because of this dependence on weapon delay. But findings from my Spirits Within investigation can be considered a kind of lower bound on Restraint's WS damage bonus (not that I would actually check for anything below 187 weapon delay).

For a given weapon delay, Restraint's (apparent) damage multiplier may increase linearly with the number of landed attacks

Given the above data in the text table, as well as some other observations toward the "extremes" (based on the number of landed attacks below 19, the minimum number to get to 100 TP after Spirits Within, which never misses, and above 34), I just performed OLS regression mainly to see visually if it is "safe" to assume that the Restraint damage bonus scales linearly with the number of landed attacks:


It appears that linearity is a valid enough assumption, and it could be said that the Restraint damage bonus increases by about 0.00588515 (0.058%) for every additional landed attack, up to a maximum bonus of 1.5 (50%). Also note that my 110-attack observation is well off the trend line. Extrapolation here is not fatal; I predict that either 85 or 86 is the minimum number of landed attacks to reach the maximum bonus.

Implications for Restraint use

Here it may be useful to reflect on snap judgments about Restraint's potential utility (or lack thereof).

First, it has been shown that Mighty Strikes is unaffected by Restraint (but is Restraint unaffected by Mighty Strikes?), so there shouldn't be any disadvantage to using Restraint for the purposes of zerging, whatever the bonus is (or isn't).

Second, suppose that it is even desirable to achieve the maximum Restraint bonus to start a zerg off (say, for a 300 TP Steel Cyclone). There aren't that many situations where this is feasible, due to unavailability of mobs to "power up" Restraint and/or time limitations. (The "stored" WS damage potential disappears when Restraint wears off.)

Other than that, let's look at the use of Restraint from the long-run, "optimal" perspective of dealing damage, which means WS spamming and whatnot. Losing critical hit damage for increased weapon skill damage may not seem like a good trade-off, but whether the trade-off is acceptable is determined primarily by what the actual bonus is, which no one has yet to determine for the "usual" range of landed attacks before using a great axe weapon skill. (I would say the range is between 5 and 9.)

Of course, this doesn't mean one can't estimate how much of a WS damage increase there needs to be to offset the loss of critical hit damage during Restraint. To do this, consider that for two-handed weapons, the loss of critical hits in the auto-attack phase is relatively more "severe" for a low attack/defense ratio than a high one. Also consider that the loss of critical hit damage is relatively more severe if your critical hit rate is high than when it is low. Yet another factor to keep in mind that the more hits that end up being landed in the process of getting to 100 TP (think multi-hit weapons and being lazy), the greater the Restraint bonus must be to offset the loss of critical hits. Finally, since Restraint has often been mentioned with King's Justice (because it is thought that Raging Rush is adversely affected by Restraint), I will base my estimation on the basis of improving KJ damage.

With that in mind, I estimate that, for a high attack/defense ratio (such that the maximum average pDIF is attained without any level correction involved), the Restraint damage bonus (as a percent increase) needed to offset the loss of critical hit damage is between 3.1% and 4.6% given a 9% critical hit rate and between 8.2% and 12.3% given a 24% critical hit rate (on average). It may be that the actual WS damage bonus (for 5, 6, ... landed attacks) exceeds the above estimates given 504 delay, although it remains to be determined.

For a relatively more modest attack/defense ratio (corresponding to an average non-critical pDIF of 1.5 and average critical pDIF of 2.6; these are rough estimates based on someone's empirical observations, which I will not go over at this time), the Restraint damage bonus needed to offset the loss of critical hit damage is between 6.0% and 9.0% given a 9% critical hit rate and between 16.0% and 24.0% given a 24% critical hit rate. (I give ranges based on the some of the current "final upgrade" Magian great axes. But the low estimates are based on the horrible "occasionally attacks 2-3 times" great axe.)

The above suggests a place for Restraint where attack is high or where critical hit rate is low (or a combination of both, which might be experienced in a merit party), but more work needs to be done to justify that contention.