Is there a sweet spot for Critical Chance?

[skedastic]

Suppose you just want to maximize DPS. Write DPS as B(1+pD), where B is your DPS if your critical hit probability is zero, p is your probability of a critical hit, and D is your critical hit damage multiplier.

Holding all else equal, your problem is to choose p and D to maximize DPS, subject to a constraint representing which items you can afford. Suppose on average you must pay G( p, D ) to buy an item with characteristics { p, D }. Let G_p and G_D denote the partial derivatives of this function. G_p is the marginal price in gold of a crit hit probability and G_D is the marginal price of crit hit damage.

Write the lagrangean for this problem: max_{p, D} B(1+pD) - l [ G(p, D) - k], where k is how much gold you have to spend and l is the lagrangean multiplier. The solution is partially characterized by this condition:

p/D = G_D/G_p.

The left hand side of this expression is the optimal ratio of crit hit probability to crit hit damage. The right hand tells us that that you should set the ratio of crit hit probability to crit damage to equal the ratio of marginal prices. Note we need to know prices to determine what the optimal crit hit prob and damage are, and note the solution is independent of B, so this rule always holds no matter what your other damage-changing attributes may be.

Example: all else equal, suppose it costs you 50,000 gold to buy get an item with one unit higher crit hit chance, or 10,000 gold to buy an item with one unit higher crit hit damage. Then your ratio of crit hit prob to damage should be one to five. If prices change such that crit hit prob relatively more expensive, say it now costs 70,000 to buy one more percentage point of crit hit probability, you should get more damage and less probability, that ratio should then be one to seven.

[lanceuppercut]

^^ yikes that post blew my mind, maths was never my strong point

[magicrectangle]

Good use of calculus of variations (although it is spelled lagrangian). Unfortunately, we don't gain any useful information from it. It is impossible to get the necessary marginal price data. Not only that, but pricing not only isn't fixed, but isn't uniform. Gloves going from 4% to 6% crit don't increase the cost the same as going from 8% to 10%.

Given the lack of uniformity in prices, the only strategy that has any real world use is to compare damage gain to price on an item-by-item basis. For this reason you should simply use one of the many spreadsheets or applets that calculate your damage, and plug in potential item upgrades, then simply divide that damage boost by the cost, and go for the items giving you the highest damage/cost overall, rather than trying to pin your character on a specific ratio of crit to crit damage.

It is, however, worth noting that in terms of "itemization cost" crit and crit damage have different relative values on different items. On Gloves it is 5 crit damage to 1 crit. On amulets it is 7.65 to 1, and on rings it is 7.56 to one. Those are the only items that can get both stats on them, so what that means practically is that of the pieces that can have both stats, gloves are the most efficient place to pick up crit, and amulet is the most efficient place to pick up crit damage. This is still relatively little help though, as AH costs don't correlate well to itemization costs.

[Spankalot]

Looks good, my ingame dps shows 34411.24 and that site gives me 32944.1 --now would 5% holy and 2% cold account for the missing dps as there was nowhere to add that kind of info.

For me it missed out 6 mainstat. Also remember to tick or leave the follower passives. Overall it's within 500 damage of my actual damage now, after the 6 stat correction and checking the damage calc summary stats with my stats.

Afaik, the 5% and 2% you mention aren't involved in charsheet dps either.

[skedastic]

By "uniformity" you mean "linearity," and that's why I referred to "marginal price," not just price. The math shows what you would do given a great deal of information and calculation---it shows how one would behave optimally, which is the point of this thread. When actually confronting the auction house without the resources required to actually choose optimally, knowing the rule above is not useless, rather, it gives you a guideline to keep in mind when making choices. It shows how prices matter when thinking about a "sweet spot," and helps you quickly approximate which choices will improve your character without necessarily resorting to a spreadsheet.

Getting the items which "give you the highest damage/cost overall" is exactly the same thing as trying to fulfil the optimality condition above, it's not one or the other---if you pull out your spreadsheet and do all the calculations you suggest, you're just brute forcing an approximation to the optimal solution characterized by the math above.

When thinking about how to allocate an attribute across items, the rule that must hold if you are doing so optimally is: the marginal price of buying more of some attribute on item A should equal the marginal price buying more of that attribute on item B. For example, if it costs 10k to get one more point of crit damage on gloves and 20k on a ring, if you were to get one more point of damage on your gloves and one less on your ring, you'd wind up with the same crit damage but save 10k, so you cannot be doing things optimally. Since marginal prices at a given level of an attribute will tend to be lower on items with greater max values of that attribute (for example, going from 3.5 to 4.5 crit percent on gloves will tend to cost less than going from 3.5 to 4.5 on a ring), you'll wind up with more of a given attribute on items with higher max values of that attribute. This has nothing directly to do with the ratios of _maximum_ crit damage to crit percent you give.

[magicrectangle]

By "uniformity" you mean "linearity," and that's why I referred to "marginal price," not just price.

No, I don't mean linearity. It could be quadratic, cubic, exponential, etc and still be predictable, but it is none of those things. Aside from not being any nice neat function, the big thing you've neglected is the implicit dependence of G and its derivatives on B. Specifically (G_p)_B and (G_D)_B != 0. For this reason only a holistic approach will give an accurate assessment of potential gear upgrades (unless you find a way to magically hold B constant across gear upgrades you're considering) - and that holistic approach is exactly what the spreadsheets provide.

It shows how prices matter when thinking about a "sweet spot," and helps you quickly approximate which choices will improve your character without necessarily resorting to a spreadsheet.

I won't dispute that, as a rule of thumb it isn't bad. I overstated it when I said it was useless, I apologize.

Getting the items which "give you the highest damage/cost overall" is exactly the same thing as trying to fulfil the optimality condition above, it's not one or the other---if you pull out your spreadsheet and do all the calculations you suggest, you're just brute forcing an approximation to the optimal solution characterized by the math above.

No, you're brute forcing the exact solution (and brute force isn't a bad thing when a computer does it for you), which the above is approximating, with a non-obvious error term.

Since marginal prices at a given level of an attribute will tend to be lower on items with greater max values of that attribute (for example, going from 3.5 to 4.5 crit percent on gloves will tend to cost less than going from 3.5 to 4.5 on a ring), you'll wind up with more of a given attribute on items with higher max values of that attribute. This has nothing directly to do with the ratios of _maximum_ crit damage to crit percent you give.

I never claimed otherwise. In fact I specifically mentioned that itemization costs don't correlate well to AH costs.

[skedastic]

No, I don't mean linearity. It could be quadratic, cubic, exponential, etc and still be predictable, but it is none of those things. Aside from not being any nice neat function, the big thing you've neglected is the implicit dependence of G and its derivatives on B. Specifically (G_p)_B and (G_D)_B != 0.

What you said was "Gloves going from 4% to 6% crit don't increase the cost the same as going from 8% to 10%." That's an example of a -nonlinear- relationship. If for example C(p)=p^2, then going from p=4 to p=6 does not have the same effect on cost as going from p=8 to p=10, but the relationship is perfectly "predictable." The bit of math I gave allows for such nonlinear relationships. I don't know what you might mean by "uniform" in this context. I think you might be trying to get at the idea that G() is an abstraction: the player actually faces a finite "menu" of item choices, not continuously varying attributes. Any answer to the question posed in this thread will necessarily involve such an abstraction.

The function G() gives the expected price of an item with attributes (p, D), all else equal, on the auction house. This is a simple way of capturing the fact that buying better items tends to cost more while imposing almost no assumptions on the manner in which costs vary with attributes. B is the player's base damage. The assertion that G depends on B is incoherent: how much you pay for an item on the auction house does not depend on your base damage. You should perhaps actually work through the little problem.

The question in this thread was whether there's a "sweet spot" ratio of crit hit damage to probability. I gave a (correct) answer to that question, if we understand it to mean more formally, "characterize the ratio of crit prob to crit damage which maximizes DPS for any given amount of gold spent." The answer turns out to be simple and intuitive: the ratio of crit hit prob to crit hit damage which maximizes DPS is the inverse of the ratio of the marginal prices of those attributes, regardless of your base damage. I'm not sure what your responses add to that answer, nor what your point might be.

[magicrectangle]

What you said was "Gloves going from 4% to 6% crit don't increase the cost the same as going from 8% to 10%." That's an example of a -nonlinear- relationship. If for example C(p)=p^2, then going from p=4 to p=6 does not have the same effect on cost as going from p=8 to p=10, but the relationship is perfectly "predictable."

I'm not actually an idiot, thanks. Did you read my follow up post? it isn't "perfectly predictable." And more importantly, it isn't only a function of p and D, it is a function of all stats on the item. Prices are based in part on scarcity, in part on usefulness, and in part on psychology.

I suppose one could guess at a functional form, which probably looks something like an asymptotic growth towards the "perfect" item in all 6 stats. Of course real world prices can't be infinite (2 bil price cap on the AH?), but near enough when looking from the low end at the price climb. In any case even if we could write out a functional form, it wouldn't change the discussion we're having any.

The function G() gives the expected price of an item with attributes (p, D), all else equal, on the auction house. This is a simple way of capturing the fact that buying better items tends to cost more while imposing almost no assumptions on the manner in which costs vary with attributes.

All else will not be equal across a finite menu of available items. Hence the need for a holistic damage comparison on an item-by-item basis.

B is the player's base damage. The assertion that G depends on B is incoherent: how much you pay for an item on the auction house does not depend on your base damage. You should perhaps actually work through the little problem.

What? Unless the items you're looking at have no primary attribute, no attack speed, no +damage, they will absolutely change B, and the amount they change B will directly impact their price.

The question in this thread was whether there's a "sweet spot" ratio of crit hit damage to probability. I gave a (correct) answer to that question, if we understand it to mean more formally, "characterize the ratio of crit prob to crit damage which maximizes DPS for any given amount of gold spent." The answer turns out to be simple and intuitive: the ratio of crit hit prob to crit hit damage which maximizes DPS is the inverse of the ratio of the marginal prices of those attributes, regardless of your base damage. I'm not sure what your responses add to that answer, nor what your point might be.

You gave a correct answer to the abstract question you pose, it just doesn't provide an accurate answer to the real world question of "which item gives me the most damage for the least money?" In other words, this statement by you:

Getting the items which "give you the highest damage/cost overall" is exactly the same thing as trying to fulfil the optimality condition above

Is simply wrong, and represents the problem I'm trying to illustrate. People like the OP ask for simple rules of thumb for optimizing their damage, but these rules of thumb have non-obvious error in comparison of real items. A holistic approach (such as one of the many spreadsheets available) is the most reliable way to ensure that you're maximizing your damage.

You should perhaps actually work through the little problem.

I did, and I get the same result as you (it is only a few lines to work out anyway). But it is the problem as posed, and not the solution, that I take issue with.

[galzohar]

Given the lack of uniformity in prices, the only strategy that has any real world use is to compare damage gain to price on an item-by-item basis. For this reason you should simply use one of the many spreadsheets or applets that calculate your damage, and plug in potential item upgrades, then simply divide that damage boost by the cost, and go for the items giving you the highest damage/cost overall, rather than trying to pin your character on a specific ratio of crit to crit damage.

Yes, I believe that is the only way to shop that is anywhere near effective. Once you know what kind of % damage increase you can get for how much gold, you can tell whether a new item you see is a good deal or a bad deal.

Even that is not perfect, though, and you must be careful. If you choose to upgrade an item that gives 1% more DPS for 1mil more than your current item's value, in addition to losing the 15% fee (which can be easily factored in so not a big deal), you also increase the price of any future upgrade for said item, since 2% will probably cost you more than 2mil. But even with this downside, if you stick to significant upgrades this is the best thing you can do when looking for best "upgrade per gold spent". Also, avoid over-estimating the price of your current item. I actually often value my current item as 0 since upgrades are often 10 times more expensive than the current item making its price not too significant - Your current item price only really matters when you look at small upgrades, which you should really avoid when possible.

All this "sweet spot" talk is really useless when it comes to actually going to the AH and getting your character equipped.

[skedastic]

What? Unless the items you're looking at have no primary attribute, no attack speed, no +damage, they will absolutely change B, and the amount they change B will directly impact their price.

Let x be n-vector of all other item attributes which may affect B. The full problem is then

choose {x, p, D} to maximize B(x)(1+pD) subject to G( x, p, D ) = k.

The first order conditions with respect to p and D are

(B(x)p)/(B(x)D) = p/D = G_D/G_p,

exactly as above. 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.

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