If items were designed for maximum game impact in a specific class and role based on Blizzard's item formulas, what would they look like? Is there any way to derive how much of one attribute versus another is optimal?

It is possible to answer the second question and give ways to evaluate items quickly. Answering the first question then only requires a solid understanding of what is useful for a given class and role.

If in a specific piece of gear, trading n of attribute N for m of attribute M is a wash in terms of effectiveness, then

First some examples for those impatient and with a low math tolerance.

Example A

If a dps warrior or feral druid is willing to trade 1 Str for 2 AP then

or a dps warrior or feral druid would really like Blizzard to include Str and AP in a ratio of 1 Str for every 2 AP. As many have commented, Blizzard seems to have an aversion to Str and AP both on an item for whatever reason, but that in no way changes the result.

How much of an improvement to AP would putting both Str and AP on gear be for these classes an build? The combination of 63 Str and 126 AP has roughly the same cost as 100 Str (or 200 AP), but provides 252 AP versus just 200, for an increase in AP by 26%



Example B

What about a rogue and Str and AP?

Even a rogue who only gets 1 AP per str would like to have some Str on his gear. The optimal ratio is 1 Str for every 8 AP or 24 AP and 3 Str is preferable to 26 AP (very similar costs). Such statting on gear would give a 3.8% increase in rogue AP based AP (not counting agility based AP, it's less overall due to AP from Agi)



Example C

If a warlock that isn't afraid to life tap for mana and doesn't care about spell crits (otherwise the value of Int is hard to quantify) wants to maximize the initial mana pool, what ratio of Sta and Int should there be on the gear? Such a warlock would be willing to trade 15 Sta for 12 Int (assuming improved tap)

or the warlock described would prefer 54 Sta for every 25 Int

what if the warlock doesn't have improved tap? Then the warlock would be willing to trade 15 Sta for 10 Int, and

While these ratios may seem significantly different based on a simple talent choice (and they are), the impact picking one for gear has is minimal.

432 Sta and 200 Int is in the 54/25 ratio and 369 Sta and 246 Int is in the 3/2 ratio and both cost nearly the same amount.

For the Improved Tap warlock, the 432/200 set provides 8,184 initial mana above the race/class base, while the 369/246 set provides 8,118 above base. For the nonImproved Tap warlock, the 432/200 set provides 7320 mana over base while the 369/246 set provides 7380 mana over base. In both cases, aligning the gear with the talent build only gives about a 0.8% improvement in additional mana over being in gear aligned for the other talent build.

Note: Life Tap is not free, it costs time and all Warlocks likely get at least minimal benefits from the crit component of the Int, both suggesting that the above ratios overvalue Sta, but neither of these factors impact the example once they are assumed to not exist.


Example D

Mage Spirit v Mana/5 as a regen tool

Assumed:

The mage is continually casting, and doesn't run out of mana ever. The mage in the example chain casts Fireball every 3.2 seconds, has 5/5 Arcane Concentration, has 3/3 Arcane Meditation, uses Mage Armor, and uses Evocation every 8 minutes.

In an 8 minute cycle the mage will have the equivalent of 78.564 full regen ticks and and 221.436 partial ticks. (147.5 fireballs was used per cycle even though it's not integral) In this cycle, 1 spirit will return 44.55255 mana. 1 mana/5 would return 96 mana.

The mage in this example would like to have almost 3 Spirit for each Mana/5 on gear. A mage without Arcane Meditation derives to 1.970... Spirit/Mana/5 or about 2 Spirit for each Mana/5 (1 Spirit returns 36.248... mana in the cycle)



Example example

Once the rate that a specific build in a given set of gear would trade attribute M for attribute N, it's possible to check if the total amount of M and N in the set of gear is right or not. If it takes 5 +dam/heal to make up for the loss of 3 crit rating when fully
geared, then

or the full set of gear should include 9 +dam/heal for every 2 +crit rating. If it doesn't, then the gear set could be made "better" by shifting closer to a 9 to 2 ratio.




Where does this formula come from?

It's derived by taking the item cost formula and solving for what ratio causes the marginal power gain by a marginal change in one attribute to match the marginal power gain by a marginal change in another attribute.

Let there be a utility function of attributes a and b, u(a, b), and a cost function, c(a, b). Then the marginal value per marginal cost increase of a is

with a similar formulation for b. Since we "know" that

where k_a is the per unit cost of a and k_b is the same for b. R is the cost of the rest of the attributes on the given item.

with a similar formula for dc(a, b)/db

Taking the two marginal value per marginal cost equations and setting them equal gives

Note: I've removed the common term "2/3 * ((k_a * a)^(3/2) + (k_b * b)^(3/2) + R)^(-1/3) * 3/2" from the denominators.

Simplifying a bit more gives

If trading x of a for y of b is an even trade then the utility of those traded items must be the same. Hence du(a, b)/da = C/x and du(a, b)/db = C/y) Substituting that in with a bit more simplification gives the orginal equation.

Other comments

It's fairly easy to show that the optimal ratio of stats on a single piece of gear is the same as the optimal ratio on a kit of gear, hence every item in a kit should have the same (boring) vector of stats if it's optimally designed for in game power. No set in the game is designed this way, hence they all are suboptimally designed from a "game power within a specific item level budget" standpoint.

This assumes that set bonuses aren't compensation for it. ie wear an item that has 100 str and one that has 200 ap in this set rather than two that are 63 str and 126 ap and you can have this 2 piece set bonus cookie of 26 Str and 52 AP for the suboptimality induced because the items in the set are flavored distinctly.

Though 1str=2AP for warrior, it's not exactly true to think of it this way.
Kings benefits str but not +AP. If I have the choice between X str and 2X AP I always take the str.

Also, some stats have a usefulness cap (+def, +hit, etc) -- once you pass that threshold you could be better off choosing a different point allocation (though I guess your model says that you should rebalance the "global" allocation of +def or +hit to have the total come out to what you want...)

Too long as Horde without Paladins, I also was thinking prebuffs to minimize complicating factors in the examples as much as possible. Using 1 Str = 2.2 AP, the optimal mix in a package costing the same as 100 str is 113.77 AP and 68.83 Str.

As I commented in the Warlock example, even drastic shifts in attribute mix often has a minimal change in "power" impact. The altered blend for BoK (which buffs do you want to include, world buffs affect the final numbers also) gives 265.19... AP while the 126AP/63Str blend derived for without BoK or world buffs would give 264.6 (when BoK is up).

Once you're in the ballpark of the optimal kit, changes to the kit along the item formula generally don't push you far off optimal as you always get something semi useful for what you give up. It takes a while for the value of the allocation to diverge significantly from the value of the optimal kit.

In any case, it is easy to derive what ratio of attributes is optimal under whatever conditions you desire, once you state the conditions. Just take the rate at which you'd trade one attribute for another, the costs for the two attributes you're examining, plug and chug in the equation and see if the resulting ratio matches what your gear gives you.

Probably for the same reason that agi+AP weren't (often) allowed back when hunters got 2 RAP from agi. The philosophy seems to be:

2 stats affecting crit (int or agi, and raw crit)
1 stat affecting hit (and one more for reducing resists, for casters)
1 stat affecting ap, except allowing AP+agi since agi costs more per AP for rogue/hunter/druid

I enjoyed the OP tremendously; I didn't verify the formula but the numbers I got seemed reasonable.

That's good to know. I admit I skipped over a bit of the other examples the first time but this is a good finding to know about.

And this is also a nice result at the end.

This is true, but once you reach a "not-critted" cap in PvE, would you rather have the incremental avoidance stats or invest in other things like stamina or block value? At that point it becomes more philosophical, and tends to vary with the incoming damage profile of the encounter (duration, spiky-ness, etc), what exact distribution you will want.