Direct Hit; how much is "enough"?

To elaborate on the question, direct hit right now reminds me of an old problem with critical hit that dates back to world of warcraft years ago, where beyond a certain point; such as 33%, or maybe even 50%, getting additional critical hit rate didn't benefit the player as much because of how consistent you would critically hit.

FF14's current version of critical hit rate avoids this specific problem, ensuring it only gets stronger the more you get, and even benefits attacks guaranteed to crit already.

Direct hit on the other hand, might run into that 'softcap' problem eventually I described in that wow example. Apparently, with ~1700 DH on my RDM through deltascape savage + the occasional battlevoice from BRD, my DH rate was ~40% through fights.

I haven't taken any metric yet of how often I'm getting direct hits on my DRG with 2110 direct hit rate at this moment, but the core of my question is at what point should I not meld for direct hit, and just stick to crit/det melds to increase max possible damage instead?

Now, I don't actually know WoW theorycraft, but I know enough about numbers systems to confidently say that the only real time you stop investing in the most powerful stat is when you hit diminishing returns. IE, if X points represents 5% chance at 0% and at 70%, it's the same DPS increase. But when X points represents only 2% at 70%, it's diminishing returns and you'd instead invest in another stat.

An extreme example of this concept was accuracy in this game before 4.0. Before the accuracy cap, accuracy was hands-down the best stat you could invest in on any class. But at the accuracy cap (when you reach a 100% hit rate), it's a useless stat and you want to invest in another stat.

To answer your specific question... Direct hit has no diminishing returns. Each point in direct hit represents the same increase in frequency of direct hits. This does mean, of course, the direct hit has a theoretical maximum at 100% direct hit rate, but that maximum is higher than we can hope to reach in stats this expansion.

In fact, FFXIV doesn't really have any dps stats with dimishing returns. They are either flat multipliers (Det, direct hit, tenacity) or they have something of an exponential scaling (crit, spell speed). And the way each class interacts with those stats changes on a class-by-class basis. The only real times you'll see a "get to X amount in a stat" is in specific examples where, for example, a warrior might want to hit enough skill speed to fit another fell cleave into his inner release.

As a result of this, most classes are listed as having stat priorities, based upon how their kits benefit from said stats. Red mage, for example, is Crit > DH > Det > SpS.

To add to what has already been said, damage formulas seem to have interactions between stats so you don't, strictly speaking, really have stat priorities anymore.

Anyways, on the topic at hand you might find the following interesting : https://docs.google.com/spreadsheets...#gid=782468133

Direct Hit does theoretically have a cap, but it is currently unobtainable. The math would put it somewhere around 4,300 DH at which point you would have a 100% direct hit rate considering every approximately 40 points over the DPS base gives you a 1% increased chance to perform a direct hit which is always a 25% increase over your base attack. So, Direct Hit, as a stat scales linearly. Critical Hit, however, scales exponentially and theoretically has no maximum. Critical hit, as a stat increases your chances of landing a critical hit, and the higher your critical hit stat, the larger the damage increase for the landed critical.

Now all that out of the way, I don't know the value, but there is a point at which the bonuses from critical hit will outweigh the bonuses gained from direct hit, most likely around the point that critical hit will give you more than a 25% average chance to perform a critical. As a monk right now, I have about 2000 crit, and am performing critical hits (based on auto attacks in recent recorded parses) at a rate around 20%. So for the time being I would estimate that for most DPS jobs it's still best to just add in as much direct hit as possible and supplement the rest of your stats with critical and skill speed to comfort.

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Originally Posted by wereotter

So for the time being I would estimate that for most DPS jobs it's still best to just add in as much direct hit as possible and supplement the rest of your stats with critical and skill speed to comfort.

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At least Mnks, Brds and (since 4.2) even smns should focus crit over dh meanwhile : /

Ive heard various people say that DH does have diminishing returns at 1,800 (last patch) though Ive never been given a source to back it up.

Since Ive got no real way to test it, Ill generally aim for 1.8k, and just go crit for the rest anyways, not too big a loss if im wrong.

I honestly would focus more on DH for Red Mage, DH>Crit>Det
But all of what was said above makes sense.
Your weapon you will want 2 DH and other pieces were you cannot add DH you may add Crit.
Now i am not however suggesting you add Dh on everything possible but it also does not really hurt.
I have 1800 DH atm with 1629 Crit and have seen crits up to 25K

There are two scenarios where DH has 'diminishing returns'

The more common sense is that DH after a point gives less % per point. (So for example, after 1600, it'd take 60 points for 1%, after 2000 it takes 100, etc).

I think it's more along the lines that DH is a linear stat, like Det and Tenacity. Each point of DH is worth exactly the same, technically, as the previous, but the amount it increases is less due to the linear nature.

Using simplistic numbers, if DH 0 is 100% damage efficiency, then DH 160 is 101% damage efficiency. You go up by 1%. DH 320 is 102% damage efficiency, you go up by 1% again, but from 101% to 102% is less of an increase (.99%).

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Originally Posted by Kabooa

I think it's more along the lines that DH is a linear stat, like Det and Tenacity. Each point of DH is worth exactly the same, technically, as the previous, but the amount it increases is less due to the linear nature.

Using simplistic numbers, if DH 0 is 100% damage efficiency, then DH 160 is 101% damage efficiency. You go up by 1%. DH 320 is 102% damage efficiency, you go up by 1% again, but from 101% to 102% is less of an increase (.99%).

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It's the same multiplicative increase from the base. So, taking theoretical numbers, if 0 is 100%, 160 is 101% and adds 100 dps, and 320 is 102% and adds 200 dps, then every 160 is worth the same flat amount; 1% or 100 dps. Diminishing returns would be if later on another 160 points resulted in anything less than that same 1% or 100 dps.

So it's definitely a linear stat. There's nothing diminishing about this, just a weird way of arranging the numbers.

I'm always curious about these sorts of things and since I'm horrid with math I have to rely on the intelligence of others and personal experience about what works. Since I'm most proficient with smn, but I'm still causal for the most part I'm not sure what I think will weigh heavily but I'll take a top tier smn as a sample. Super summoner is running approximately and rounded around 2K crit, 1800 direct, 1200 det.

I'm guessing crit starts becoming more important than DH at some point, the question is where is the tipping point. (Also that doesnt take in account det). Here is one article I found a while ago by Ver Haad. (Maybe someone who's got a better brain for calculation can give more than me).
https://docs.google.com/spreadsheets...gid=1061959067

The thing with crit is you need such a high amount to see anything because basically it is a % chance of dmg happening were as other stats are more of a guarantee in dmg.
Now in the case of smn the better stat i guess is crit over DH because dots are involved but jobs like Red Mage DH is better

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Originally Posted by Maero

Now in the case of smn the better stat i guess is crit over DH because dots are involved but jobs like Red Mage DH is better

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its not just the dots what makes crit interesting/needful for smns the most important reason is ruin iv since we need those procs more than ever since ruin iii and ww got nerfed imo : o

In a way, flat multipliers are a diminishing returns by nature. Going from x1 to x1.1 is a gain of 10%, while going from x2 to x2.1 is a gain of 5% of your former dps. Even if you gained 0.1 in both case for the same amount of points.

Skill Speed and Spell speed are more efficient the more you put points in, in a way. Reducing 0.01 GCD from 2.50 is proportionnaly less powerful than reducing 0.01 from 2.00 GCD.
To take some obvious numbers, reducing 0.01 GCD from 0.02 GCD is an upgrade of +100% attack speed. While reducing 0.01 from 0.10 GCD is an increase of 10% attack speed.

None have (in themselves; see the post directly above) diminishing returns, but two double up on their gains (Crit and Speed, as the first also enhances the value of the extra chance given and the latter gives the same flat GCD reduction despite that becoming a greater and greater proportion of the remaining GCD).

As such, unless dependent on Crit or Speed, stats like Det and (as it may be overtuned for most anyways) DHit tend to be strongest at the very beginning of an expansion, but are quickly surpassed by Crit or Speed if you can get either high enough (either one a vast majority of your secondary stat total).

Edit: In general, unless you need a specific GCD (DRG, SAM, etc.) or you get procs from crits (Bard), you tend to be better off either avoiding or stacking Crit/Speed. The secondary stat options of your tier will largely make that decision for you, however. By next, we may well all be pushing for Crit and/or Speed.

We are kinda already at the place where crit is overtaking dh last time I've heard

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Originally Posted by Neela

its not just the dots what makes crit interesting/needful for smns the most important reason is ruin iv since we need those procs more than ever since ruin iii and ww got nerfed imo : o

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Except crit has nothing to do with ruin IV procs, it's not like the SS buff from back then.

The trait reads : "Grants a 15% chance that a pet action will trigger Further Ruin, upgrading Ruin II to Ruin IV." Nothing in here involves crit, it's just 15% each time an action is used by you pet.

No the reason Crit is better is because it suits best the spiky damage profile jobs, which summoner is, (especially when the crits hit the spikes), as opposed to more linear damage profile like DRK, NIN or RDM, who benefit more from DH. Well at least when we were under 1500, now with the new caps, and with next cap in 6 months able to reach 2000 up to 2500-2800 k easily, I think crit will return as the sovereign secondary stat for most jobs.

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Originally Posted by Neela

its not just the dots what makes crit interesting/needful for smns the most important reason is ruin iv since we need those procs more than ever since ruin iii and ww got nerfed imo : o

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Ruin 4 is triggered by pet actions, it has nothing to do with Crit.

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Originally Posted by Karshan

No, the reason Crit is better is because it suits best the spiky damage profile jobs, which summoner is

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That's also not the reason. Crit is preferred by a lot of classes because of its quadratic growth and its availability on gear, party compositions involving DRG/SCH/BRD/AST are also a factor in favor because of raid buffs that boost crit rate. Damage is compared using rotational simulators and the damage formula to find the best performing gear set. Crit scaling being quadratic in nature means that if enough of it is available on gear, it gets more powerful than the other stats with linear gains after a point. The point at which that happens depends entirely on the stat spread on the gear. If crit is paired with detrimental stats, like sks is in some cases, or if there's simply not enough crit on the gear, the gear set might not end up focusing on crit. The only jobs that have inherently crit skewed focus are the ones with class mechanics centered around crit, such as BRD or MNK.

Hey,

I'm new to FFXIV and found this thread while searching for resources on how damage formulas work in this game. I thought it was an interesing enough question to try and answer conclusively via the power of mathematics. I used the formulas from the following website to calculate the effective DPS gain from crit / direct hit rating, respectively:

https://docs.google.com/spreadsheets...F38/edit#gid=0

The way I decided to approach the problem was by making the assumption that we have a "fixed" pool of stats to allocate from (say, 3000 rating in total) and it's up to us to decide whether we want to lean that pool more towards crit rating or more towards direct hit rating. That is, I assume that we're presented with a series of choices such as "would I rather have 40 crit rating or 40 direct hit rating?". In the context of this framework, the question is: how much should I lean towards one or the other?

One thing that we can see quite clearly is that for the upper end of what we can realistically achieve with current gear, critical hit rating alone completely dominates direct hit rating. But this is not always the case - there's a phase reversal somewhere at around 3000 shared total stat between the two. More fascinatingly, in a result that I didn't expect at all, in this "transition" region, it's worse to have a mixture (say 50% crit and 50% direct hit) than it is to stack 100% of either, even though they multiply each other! (This is a direct consequence of the fact that crit rating is essentially useless if you don't stack enough of it)

To best visualize these results, I decided that we should use "50% of each" as the baseline - under the assumption that on average, we'll always be able to balance the two stats as a baseline, and the question is how much we want to "lean" towards one or the other as a function of how much stat we have available. That formula looks like this:

https://www.wolframalpha.com/input/?...000,+0%3Cy%3C1

https://0x0.st/zz6D.png

The short summary is that for <3000 total stat, you want to stack direct hit rating as much as possible. For >3000 total stat, you want to stack critical hit rating as much as possible. For ~3000 total stat, it doesn't really matter which one you go for - but always try to get as much as possible of the same stat, never mix the two.

The formula was generated using this haskell program: https://0x0.st/zz6d.hs

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Originally Posted by nand

The formula was generated using this haskell program: https://0x0.st/zz6d.hs

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Reproduced here (too long for the original post):

Code:

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import Data.Number.Symbolic

type Rate = Sym Double
type Chance = Sym Double
type Multiplier = Sym Double
type Distribution = Sym Double

baseRate :: Rate
baseRate = 364

critMultiplier :: Rate -> Multiplier
critMultiplier r = 1 + chance * bonusDamage
    where chance = (((r - baseRate) / 108.5) + 4.9511233) / 100
          bonusDamage = 0.4 + ((r - baseRate) / 108.5) / 100

directMultiplier :: Rate -> Multiplier
directMultiplier r = 1 + chance * bonusDamage
    where chance = ((r - baseRate) / 39.1) / 100
          bonusDamage = 0.25

statGain :: Rate -> Rate -> (Rate -> Multiplier) -> Multiplier
statGain cur inc f = f (cur + inc) / f cur

distGain :: Rate -> Distribution -> Multiplier
distGain pool dist = critMultiplier critRate * directMultiplier directRate
    where critRate = baseRate + pool * dist
          directRate = baseRate + pool * (1 - dist)

relGain :: Rate -> Distribution -> Multiplier
relGain pool dist = distGain pool dist / distGain pool 0.5

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Also, sorry for the triple post, but I wanted to dispel some of the claims made in this thread about the existence or nonexistence of "diminishing returns". While it's true that (assuming you haven't hit the 100% hard cap) Direct Hit rating is a "linear" stat, in a system of linear multipliers (which is pretty typical for games like these), it's generally better to balance your stats than it is to stack one. For example, if each stat "weighs" about the same on average, then instead of a 10% increase to one multiplier (=10% more damage), I could instead split them up to get two 5% increases (=10.25% more damage) instead. Mathematically, if stacking every stat into one linear multiplier results in a linear curve, then evenly distributing them across n linear multipliers results in an nth-order polynomial curve. Eventually, this will outscale the linear increase - it's just a question of how big your stat "pool" is. This phenomenon is what people often mean when they use the term "diminishing returns", but I think a better term for it is "opportunity cost". Or really, what's happening under the hood is sort of the "opposite" of diminishing returns - stats grow more powerful the less of a stat you have. So after a certain point, a badly scaling stat B will outvalue the well scaling stat A if the two become too imbalanced.

Incidentally, this can be generalized from linear stat systems to all systems of polynomials, including critical and direct hit rate. Indeed, once we go to absurdly high stat values (>50k and above) in the above graph, it again becomes better to mix your stats - because now you have a cubic scaling curve (quadratic scaling * linear scaling). So even though locally, critical hit rating seems strictly better than direct hit rating, we can still observe "opportunity cost" once going high enough. Of course, at that point, we hit the 100% hard caps anyway so this argument is moot - but it demonstrates a general concept, so I though I'd point it out.

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 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)

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.

Quote:

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Originally Posted by 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.

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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)

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)

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Originally Posted by nand

Math is hard!

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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.