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.