Direct Hit; how much is "enough"?

[nand]

That being said, the real question in practice is more complicated to answer (and depends strongly on the game), because the constant factor on each stat plays an important role in determining the stat weights within a bounded region (say, from 0 to 3000). The previous image illustrates that perfectly - the only reason direct hit rating outscales critical hit rating for low stat amounts is because the constant factor on it is better. So in some games it's still best to ignore all stats except the best one, if that stat is so good that it's better than the rest even after the "opportunity cost"; and it seems like this might be the case for FFXIV.

I wanted to think some more about this problem. To use a simple model, say we have two linear multipliers with relative weights A and B, depending on the amount of stat (a) or (b) invested. Also, assume that we have constant offsets +1, i.e. the multiplier for stat A looks like 1 + A·a where a represents the amount of stat we have. Finally, to use the same general assumption as in the previous post, let's assume we have a bounded amount T of total stat available to us, and the only thing we control is the distribution x, i.e. a = x·T and b = (1-x)·T.

We now arrive at a result that looks like (1 + A·x·T)(1 + B·(1-x)·T). This expands (via WolframAlpha) into this second-order polynomial: -ABT²x² + ABT²x + (AT - BT)x + BT + 1

In order to simplify this formula somewhat so we can squint at it from a distance, let's combine T and x by rewriting it in terms of a = Tx again. We can also kill off the constant term because we're only interested in the local maximums of this polynomial, resulting in -ABa² + ABTa + (A-B)a. If we furthermore observe that T = a+b we can simplify this into AB·ab + (A-B)·a. Some observations:

1. For small T, the (A-B)·a term is what rewards us for having more of the better stat.
2. For large T, the AB·ab term is what punishes us for having either stat close to zero.

To figure out the best long-term ratio, we need only look at the limiting growth behavior, which depends only on the quadratic AB·ab term. Since AB is constant, we need only maximize a·b = T² · x · (1-x), which (by symmetry) is obviously maximal exactly when a = b, i.e. x = 0.5 - same as if the stats had been equally good.

To figure out the best ratio near T = 0, we need only look at the linear term, which is (A-B)·a = (A-B)·x·T (as a function of T), therefore the best slope is achieved by maximizing (A-B)·x, i.e. putting everything into the best stat.

(1/2)

[Phoenicia]

I will first admit that I haven't gone through most of the thread (I read the last 3 posts because graphs are beautiful!). But I will attempt to answer the OP:

At our current gear levels, critical hit finally passed the point where "DH scaled better" because of its constant linear scaling. But we cannot reach any point where "enough x stat is enough" unless it is skill/spell speed but that is due to the stat's nature, not scaling (not affecting oGCDs, throwing GCDs off alignment with our own or raid buffs, etc).

The short answer is: If a piece of equipment can take it: Go for critical, next is direct hit unless you can break a SkS or determination threshold. Since you are a dragoon, here are a few "Best in slot" examples (credit goes to the Balance discord):

2.43 New and improved BiS (Eureka included)
http://ffxiv.ariyala.com/18LZD


2.45 (old BiS)
http://ffxiv.ariyala.com/175G9 (High Crit; 2.45 GCD)
*Crafted pieces hit stat tiers cleanly and have premium stat spreads.


2.17 high sks set. (Eureka included)
http://ffxiv.ariyala.com/18LZE

If you do not like fast dragoon, the 2.42 set is the 3rd best option. Is is not considerably worse than the 2.19, but the gcd tier is slightly awkward.
http://ffxiv.ariyala.com/177ZA (2.42 GCD)

[Cetonis]

There's no magic number at which DH's value starts degrading; it's degrading the whole time. Every 39.5-ish points of DH increases your Direct Hit Rate by 1%. So your first 395 points increase your damage by 2.5%, since you do 25% more 10% of the time. But since you're already doing 102.5% of your pre-DHit damage, increasing that to 105% is only a net gain of +2.44%. The next 395 points gain you +2.38%, and so on.

By comparison, the first 395 points of Det get you +2.37% of your pre-Det damage, the next +2.315%, the next +2.26%, and so on. So you can imagine that, if you have very low Det and very high DH, Det will actually be a bigger damage benefit.

However, it's not quite so straightforward to find, because if you're comparing DH and Det, you also need to consider that your pre-DH damage and your pre-Det damage are two different values. For instance, if you have much more DH, than your pre-Det damage is a higher number than your pre-DH damage, so a seemingly smaller boost from Det could actually be the larger damage increase.

This is why the numbers folk have avoided stat weights - the substats are too close and there's too much nuance for a simple overview like that to be sufficient. Thankfully the math itself, at least for Det and DH, is super simple, so you can always work it out for yourself pretty quickly. And if you put a little elbow grease in, depending on your job it can be reasonably straightforward to get numbers on Crit as well.

[nand]

So we know some of the curve's behavior in general:

1. Initially, it's best to put everything into the best (linear) stat available to you.
2. Long-term, it's best to have an exact 50%/50% ratio of both stats.
3. Somewhere in between, there is a transition region from one to the other. We know that this must be smooth and continuous everywhere (because we are only dealing with polynomials).

Since this example was only based on two stats, we haven't necessarily learned anything about actual games here, but I conjecture that these two facts apply to all systems of linear multipliers, i.e. the best initial behavior is to maximize the best stat and the best long-term behavior is to evenly distribute them. The details are left as an exercise to the reader.

It also doesn't directly apply to systems involving quadratic scaling stats like critical hit, but we can at least project the "long-term behavior" result by considering a quadratic scaling stat as two independent linear stats that have the restriction that they must always be exactly equal. But since we already know (under my conjecture) that the optimal long-term behavior is to make them exactly equal, this means that long-term, you want to have twice as much of a quadratic scaling stat as other any linear scaling stat. In general: in the limit, balance each stat weighted by its polynomial degree.

(2/2)

[Phoenicia]

Math is hard!

Very nicely explanations to stats.

Wanted to let you know that you can bypass the character limit by posting, then editing your post with more wordssss!

In Stormblood, Direct Hit was the best stats at low item levels. The only stat that actually surpassed it is the quadratic crit stat.

Speed should have a curve graph instead of a line, but I don't think we can have enough of it to matter and it doesn't work well for most jobs anyways.