Is there a sweet spot for Critical Chance?

[galzohar]

The amount of other stats you have that aren't crit related sure don't affect the value of crit chance vs crit damage. That is not the point. The point is that an item will have other stats that aren't related to crit which will affect both your DPS and the price of an item, and at least the price will be affected in a very unpredictable manner, so just calculating that "sweet spot" is pretty pointless.

[magicrectangle]

These other factors, x, do not change the result at all, due to the functional form of the DPS function. That is in part why I started off by writing out that functional form, it's why I explicitly noted that the solution doesn't depend on B, and it's very obvious, particularly to anyone who actually worked through the little problem.

As I said, I worked through it. You keep missing the point. As I already said, your answer is correct for the question you ask, but you're asking the wrong question. The right question is "which item upgrade increases my total dps the most, for the least money."

I've already responded to your other objections, and your tone doesn't entice me to further "discuss" anything with you at all.

Your tone is more than a little arrogant too, especially since you keep ignoring my point, and claiming that the only reason I disagree with you is that I haven't done the math (which I have, it is incredibly simple).

As I acknowledged in a previous point, your answer is correct for the question you've asked. I even said that your "sweet spot" ratio is a decent rule of thumb. What it isn't, though, is a reliable way to compare item upgrades. For that you need a holistic approach - IE spreadsheet. Obviously some items are clearly better than others, and it is obvious without a spreadsheet, but in the closer cases (where we might need such a rule of thumb) a spreadsheet is the way to go (and not much work once you have it set up).

[Dethklok]

Myth: There is no sweet spot for critical damage.
Fact: Overkill is wasted DPS.

Although you'll probably never feel like you have enough critical hit chance -- critical often trigger class-specific abilities, and your ability to one-shot monsters in a white mob is often dependent on whether you crit or not -- you don't need more critical damage than is necessary to kill the monster in question. If you spend any significant amount of time attacking white mobs, critical damage past one-shot may not improve your kill speed as much as additional critical chance or primary stat (either by allowing non-crit attacks to kill in less hits, or by making crits happen more often).

Where this sweet spot is depends on the content you are running and how many non-crit shots it takes to kill monsters you're likely to encounter; it's normally easier to get a feel for it through experimentation than crunching numbers or looking at your stat sheet. For example, if your criticals one-shot a monster while your non-critical hits do not, then more critical damage isn't going to help your run time. Or, you make two attacks against a monster, one of the two crits, and it's left with a tiny sliver of life left... you could solve this either by increasing your base damage or by increasing your critical damage.

The more attacks it takes for you to kill a monster (especially elites and bosses), the less possibility of overkill and the more accurate your stat-sheet DPS is in terms of indicating how many attacks you need to kill things. Therefore, the less you care about farming and the more you care about progressing through difficult content, the less you care about overkill; as you care less about overkill, the feeling of a sweet spot for critical damage diminishes. Therefore, this advice is primarily for farmers. Furthermore, farmers are more likely to run the same content over and over again, allowing them to tune their gear to quickly eliminating particular monster types, while non-farmers are more likely to do a more generic optimization.

Myth: Once you start going crit gear, you want to keep going crit gear.
Fact 1: Maximizing a particular stat (or set of stats) is not as effective as balancing them.
Fact 2: Every item offers something different.

The thing about crit chance and crit damage is that they have a near-exponential relationship when you are first building them up. The synergy is strong, so in the beginning there is a strong feeling of non-diminishing returns that feels very different from the diminishing returns you expect from gear bonuses. However, even critical gear eventually suffers from diminishing returns. For example, once you have 35% critical chance, getting another 5% doesn't have the impact of going from 5% to 10%; it has about 80% of the effect the first 5% would have had, assuming your bonus to critical damage was the same at both points. Similarly, getting another 50% critical damage when you already have 250% isn't the same as when you first jumped from 50% to 100% critical damage; it has only 20% the effect of the first upgrade, assuming you had the same crit chance then that you do now.

When one stat suffers diminishing returns, another stat gets an opportunity to shine. For example, consider a demon hunter with 40% critical chance, 250% critical damage (before Archery), and 300 resist all, who is already wearing 3 pieces of Nat's gear and wants to shop for Natalya's Mark. The market consensus is that crit chance is the best random mod available in slot; a 4% is available. Getting that would increase overall DPS by 5%. However, getting a Mark with, say, 60 resist all would increase survivability by 10%, and at a much cheaper gold cost. We know demon hunters love their damage, but do they value it above a 2:1 ratio? It seems they do, but perhaps they shouldn't.

This is particularly true with rings. Although they offer both crit chance and crit damage, the amount offered isn't much. Other mods, such as resist all, get relatively high values. From an item economy standpoint, this means that those mods are probably overvalued by the market currently.

[magicrectangle]

Fact: Overkill is wasted DPS.

I know you play a DH from other posts, but most classes at some gear level stop killing trash. Some DH go grenade tank and stop killing trash too, but that's less common. Pretty much all wizards, barbs, and monks are going to stop killing trash long before they reach the damage threshold of one-shotting it. DH and WD might be doing glassy builds that require them to kill trash, but if they're serious about farming speed they'll still skip as much trash as possible too.

As such, elite killing speed is the primary concern - and for that crit damage never really stops being good, as they have more than enough HP to take multiple crits to kill.

You say that overkill is primarily a worry for farmers. I say just the opposite. Farmers don't care about trash at all. We run / teleport / sprint / dashing strike / vault / whatever past/through it and spend our time killing the stuff that drops loot.

Fact 1: Maximizing a particular stat (or set of stats) is not as effective as balancing them.

True, the full functional form of your damage, which has been given partially in above posts, is:

DPS = (1+A)(1+s)(1+pD)*S*W

Where A is primary attribute / 100, s is off-weapon attack speed, p is crit chance, D is crit damage, S is the speed of your weapon, and W is given by:

W = Min + Delta/2

Where Min is your minimum damage total (from both on and off weapon sources) and Delta is your damage delta from all sources. The game is inconsistent about how it addresses max damage, it seems to actually use a Delta, as increasing min damage increases max damage, so we'll talk about it in terms of the delta, even though items show "+max damage" what they do is increase the min-max delta. Some items show +20 min, +30 max (delta), other items will write the exact same thing as +20-50 damage. Annoying.

As is obvious from the multiplicative relationship, neglecting any one of your damage stats will drag down your DPS as compared to a moderate combination of all of them. A simple explanation of this multiplicative relationship in words is: "The more of one damage stat you have, the more the other damage stats will be worth."

The same marginal cost comparison from above could be made accounting for all the variables, but the result would be unwieldy. (So just use a spreadsheet)

[Dethklok]

I know you play a DH from other posts, but most classes at some gear level stop killing trash. Some DH go grenade tank and stop killing trash too, but that's less common. Pretty much all wizards, barbs, and monks are going to stop killing trash long before they reach the damage threshold of one-shotting it. DH and WD might be doing glassy builds that require them to kill trash, but if they're serious about farming speed they'll still skip as much trash as possible too.

I was aware that some builds — Run Like the Wind barbs, Critical Mass wizards (and due to Critical Mass/Night Stalker similarities, logically tank DHs as well) — were big on this practice, but I didn't know it was as mainstream as all that. However, now that I think about it, what other classes can use high-DPS, low-survivability, low-kiting farming methods? The trick to this is skills: you need to do a lot of damage to a lot of targets in a very short amount of time.
Demon Hunter: Multishot - Fire at Will. The absolute best at what it does. I can see most DHs farming this way, once they get the memo.
Barbarian: Maybe Seismic Slam, if you go ranged with Weapon Throw, No Escape, and Into the Fray. It would be an uncommon build to say the least, but I'm pretty confident that barbarians have a diverse enough skill pool to do anything, even if that something is being a squishy DPSer. Untested; from videos the skill seems to hit a good cone, the tooltip damage% is decent, but I'm no expert on Fury cost issues. I can see the rare Barbarian farming this way.
Monks: Nothing really. Pulling in with Cyclone Strike into a Sweeping Wind doesn't count because you'd have to stand still for a bit. I can't see any Monks farming this way.
Witch Doctors: Acid Rain actually seems pretty good here. The thing with Acid Rain isn't that it's a bad skill, it's that it has such a good proc value that WDs get more value from letting tank elements (LOH etc) into their build at the cost of some DPS. However, with fetish meat shields and zombie walls for panic buttons, you could probably go for squishy DPS with this as a valid Multishot replacement. I can see some WDs farming this way.
Wizards: The closest they get is Arcane Nova. I spent a good amount of (wasted) effort trying to farm with my Wiz the way I do with my DH, and I can say that Arcane Nova has NOTHING on Multishot; the projectile doesn't travel fast enough, if it misses the damage may not center where you want it to, and its proc rate is crap which isn't synergistic with Wiz's archetypal AP on crit ability at all. I can see only a rare Wizard farming this way.

So that's most of the DHs, but only some of the WDs and the rare oddball Barbarian or Wizard... not nearly as large a sample size as I assumed earlier.

I would like to point out, however, that the capability to be squishy is generally a good thing for a farmer. It allows you to maximize DPS and time per run. Yes, Critical Mass wizards and Into the Fray barbs do get to skip trash (time saver), but take longer to kill elites much slower (time waster); I believe I'm getting it done in slightly less time and considerably more random gold drops.

True, the full functional form of your damage, which has been given partially in above posts, is:

DPS = (1+A)(1+s)(1+pD)*S*W

Although all that stuff about DPS is true, perhaps my real point is that people tend to overvalue a general area of stats, such as DPS is general. I just spent a section talking about how its nice that my DH is allowed to be squishy, but that doesn't mean I do or should ignore defense entirely. However, a lot of optimizers seem to become obsessed with getting something like DPS to its theoretical limit and completely ignore another section of the statsheet. Sometimes I'm just amazed by how the relatively small crit chance and crit damage values on rings have such a high market value (even though you can get essentially two rings of bonuses on your gloves), even when other stats are much more competitive in comparison (you can get 7/8ths of a glove-serving of resist all on a ring).

[magicrectangle]

I would like to point out, however, that the capability to be squishy is generally a good thing for a farmer. It allows you to maximize DPS and time per run. Yes, Critical Mass wizards and Into the Fray barbs do get to skip trash (time saver), but take longer to kill elites much slower (time waster);

Slower than a DH popping them with impale? Perhaps, but the travel time much more than makes up for it. Even spamming wormhole I spend more time finding elites than I do killing them, so while I may not kill them as fast as a glass DH, I'm absolutely positive that I kill more elites/hour than an equivalently geared (gold value, not DPS equivalent) DH. That gap will only get wider the more gear improves. Even if your build is "slow" at killing elites, 15 seconds for a pack instead of 5 seconds for a pack doesn't matter worth a damn if you're spending ~15-30 seconds moving from pack to pack instead of ~1-2 minutes.

Although all that stuff about DPS is true, perhaps my real point is that people tend to overvalue a general area of stats, such as DPS is general. I just spent a section talking about how its nice that my DH is allowed to be squishy, but that doesn't mean I do or should ignore defense entirely. However, a lot of optimizers seem to become obsessed with getting something like DPS to its theoretical limit and completely ignore another section of the statsheet.

That's a fair point, but a difficult one to quantify. Your goal is generally going to be to have "enough" defensive stats, and then focus on optimizing offense. "Enough" is whatever allows you to not die on most packs in the area you farm. Obviously builds that skip trash will need more defensive stats than ones that kill it, but you more than make up the time.

[Dethklok]

Slower than a DH popping them with impale? Perhaps, but the travel time much more than makes up for it. Even spamming wormhole I spend more time finding elites than I do killing them, so while I may not kill them as fast as a glass DH, I'm absolutely positive that I kill more elites/hour than an equivalently geared (gold value, not DPS equivalent) DH. That gap will only get wider the more gear improves. Even if your build is "slow" at killing elites, 15 seconds for a pack instead of 5 seconds for a pack doesn't matter worth a damn if you're spending ~15-30 seconds moving from pack to pack instead of ~1-2 minutes.

I can kill whites as I'm moving from place to place as fast as a whirlwind barb can ignore them, a little slower if I turn around to pick stuff up. (Tactical Advantage: best passive ever.) I wasn't thinking about wormhole spam though, you might actually more faster than me.

That's a fair point, but a difficult one to quantify. Your goal is generally going to be to have "enough" defensive stats, and then focus on optimizing offense. "Enough" is whatever allows you to not die on most packs in the area you farm.

"Enough" is quantifiable; however, I haven't really found where it is quantified. I think someone should do experiments on monster damage, factor out his damage mitigation, and post the results. It would be nice to know things like "how much damage does an inferno oppressor charge do?"

Obviously builds that skip trash will need more defensive stats than ones that kill it, but you more than make up the time.

I wouldn't be so sure about that. Multishot is ****ing awesome.

[magicrectangle]

I can kill whites as I'm moving from place to place as fast as a whirlwind barb can ignore them, a little slower if I turn around to pick stuff up. (Tactical Advantage: best passive ever.) I wasn't thinking about wormhole spam though, you might actually more faster than me.

Nothing is as fast as wormhole spam out in the open or on straight maps (like rakkis crossing), with decent monster density. Indoors, or crossing areas that are devoid of monsters (or just have singles or whatnot) it isn't as great.

"Enough" is quantifiable; however, I haven't really found where it is quantified. I think someone should do experiments on monster damage, factor out his damage mitigation, and post the results. It would be nice to know things like "how much damage does an inferno oppressor charge do?"

Enough not to get one-shot is easily quantifiable, but enough not to die on most packs isn't. The latter depends on build, playstyle, skill, latency...

[Dethklok]

Nothing is as fast as wormhole spam

I'm going to just concede this point, not because I am sure you're right, but because the back-and-forth doesn't seem productive.

Enough not to get one-shot is easily quantifiable, but enough not to die on most packs isn't. The latter depends on build, playstyle, skill, latency...

The math may be harder, but it's still quantifiable; the math would be determined by things like build and playstyle.

For example, I use Gloom a lot (build), and I normally panic to that instead of Smoke Screen, because Smoke Screen costs more discipline per second than Gloom (playstyle; if I had more skill, I might choose more carefully, but I almost always panic to Gloom and use SS for CC breaking almost exclusively); therefore, it's important to me that ambush attacks hit me for less than 74% of my max life (quantifiable), so that I can afford to get hit by one, pop Gloom, then still take 1 more of the same hit if necessary, while hopefully using Gloom's leech plus Multishot into a mob (combination build, playstyle, and skill) to get back to near-100% before Gloom expires; which makes me wonder how many nearby monsters I'd need to hit per Multishot to get back near max in 3 seconds, with or without getting hit another time (quantifiable knowing my DPS, attack speed, and armor/resistances). Obviously this doesn't work as well if I'm not paying attention (skill) or my connection is poor (latency), so in those sorts of situations I do something else, like level a barbarian or something.

[skedastic]

The amount of other stats you have that aren't crit related sure don't affect the value of crit chance vs crit damage. That is not the point. The point is that an item will have other stats that aren't related to crit which will affect both your DPS and the price of an item, and at least the price will be affected in a very unpredictable manner, so just calculating that "sweet spot" is pretty pointless.

Believe me, I understand that items have other characteristics, and I've addressed this point several times now. This is also a quite different objection than the claim that the optimal ratio of crit hit damage to percent depends on other characteristics which affect damage, which is the point to which I was responding.

Note we're not talking about an item, we're talking about how to think about selecting a set of items (although the result above holds for both individual items and the set of items). Spreadsheets are a useful but costly way of comparing individual items, but they are of even more limited use in coming up with a good set of gear.

If you look at previous posts in the thread, you'll see people recommending things like a 10 to 1 ratio of crit damage to probability. The objection that items have many characteristics other than crit damage and prob also holds with respect to that sort of advice, and in fact to all "theorycrafting" about gearing. Sure, at the end of the day items have to be compared on a case by case basis, but that doesn't mean we can't think about principles which will tend to give us the best gear, particularly given it isn't remotely feasible to stick every item we consider into a spreadsheet, and, again, that we should think in terms of sets of items, not individual items.

Suggesting a 10 to one ratio of damage to prob, the result above shows, makes sense if and only if you think you could get an incremental unit increase in damage for about ten times less than an incremental unit increase in probability. That's probably roughly right, somewhat rationalizing that advice, but no one fixed ratio can generally be optimal, as prices matter. If you think you can get one more percentage point of hit prob for less than 10 more percentage points of damage, that ratio should be less than ten to one, and vice versa. If your character sheet shows you have, say, 180 crit damage and 40 crit percent but you think one more point of crit percent would cost you about as much as 10 more points of damage, you should regear or select new items as you upgrade to de-emphasize percent and increase damage if you want to maximize DPS.

Aside from helping us think about total crit damage versus total crit prob, how does this result help us quickly evaluate items, without resorting to a spreadsheet? Again write DPS as B(1+pD)=BM, where B is base damage, p is crit prob, D is crit damage, and denote the crit multiplier as M. Since it follows that a one percent (as opposed to one percentage point) increase in B has the same effect on DPS as a one percent increase in M, you can quickly approximate which item will increase your damage more by thinking in terms of percentage changes instead of absolute values of DPS.

Example: one set of gloves has 5 points of crit hit prob, another has 30 points of crit hit damage, they cost the same and have the same other characteristics. Which should you buy if your goal is to maximize damage? Since other characteristics (attack speed, etc) don't affect our calculation, we can ignore them, and by the same token we can ignore all other stats on other gear besides crit hit prob and damage. But we do need to know the crit bonuses on other gear.

Suppose other gear gives you 200% crit hit damage and 25% probability. Then your damage multiplier from other gear is [1 + (0.25)2] = 1.5. If you get the gloves with 5 points of crit prob, then your multiplier will increase by (0.05)2 = 0.1 to 1.6, roughly a 0.1/1.5~=7% increase in damage. If you get the gloves with 30 points of damage, your multiplier will increase by (0.25)(.30) = 0.075, roughly a 5% increase in damage, so you should choose the gloves with probability.

What about other characteristics which affect DPS? Suppose you consider a third set of gloves which increase your primary stat by 100, and your other gear gives you 1000 of your primary stat. Since your DPS is proportional to your base damage, these gloves then increase your damage by 10%, so you would choose them over either set above. If your other gear instead gave you a primary stat bonus of 2000, then these gloves would increase your DPS by only 5%, and you'd be better off with the ones with crit prob.

Generally, thinking in terms of percentage changes, incremental prices, and sets of gear helps in quickly evaluating items on the auction house. It's not pointless, unless you think typing numbers into spreadsheets is entertaining.