## 2.4 Mana Regen sweetspot, revisited

This is… annoying. In my last long post, I told you the sweet spot for mana regen was when your spirit was twice your int. Wider separation, close the gap for better increase of MR, and if closer widen the gap.

The rule – move toward the sweet spot – still applies. The problem is that I’ve got a conflict. Someone on elitist jerks (and I regretfully do not recall the individual) posted an elegant, SIMPLE mathematical proof that showed the sweet spot is where INT is 2/3 SPI. That’s… just a little different from what I got. I got MINE from empirical work – created a chart with a whole bunch of numbers, and crawled through looking for the best change.

I’ll be revisiting it – printing out not only a chart, but letting the autocalculations determine the ‘better’ option for a given increase. For now, though, I’m going to say that given a choice between math and bleary-eyed scrutiny of a screenfull of small numbers, trust the math. If your spirit is 600, aim for an INT of 400.

Oh, one tiny, tiny point here. Apply a braincell or two. If you’ve 600 SPI and, for example, 500 INT, don’t just strip 100 int. That’s 1500 mana and a bit of spellcrit. Worse, 600/500 generates more mana than 600/500. Given a choice between boosting more INT and more SPI, go for the SPI. Maybe – MAYBE – if you can swap some INT for SPI, you can go ahead and do so. But don’t just remove INT for “optimal mana regen”.

As a last remark — I foresee Draenic Wisdom elixirs becoming really, really popular. Add the mastery as well and most level 70 casters will see mana regen rates comparable to if not greater than what they can get from Mighty Restoration – but with extra stamina, agility… For a lot cheaper, at that.

I don’t think it was me, but I can show you a way to do the derivation. The goal of stats on gear is to provide maximum utility in a fixed budget. Basic Economics says that the way to do this is to balance all stats until the marginal utility per marginal cost of all of them are equal. Basically (du/da) / (dc/da) = (du/db) / (dc/db) [the notation is for derivatives ie calculus]. While we have a rudimentary understanding of the cost formula, the utility formula is unknown. The best and the brightest (EJ forums) don’t really have closed form functions for utility on which you can take the derivative. However, the utility derivative can be approximated (ala the shadowpriest.com coefficient simulations, or simple theorycrafting)

The ratio of stats a and b that is optimal assuming that the cost formula is (sum((c_i * x_i) ^p)) ^ (1 / p) is

x_a / x_b = (u_a / u_b) ^ (1/ (p – 1)) * (c_b / c_a) ^ (p / (p – 1))

[c_i = cost coefficient of stat i, x_i = amount of stat i, u_i = marginal utility of stat i]. Hyzenthlei’s work suggests the cost formula does have the form above and that p = 1.5. This simplifies the optimal ratio formula to x_a / x_b = (u_a / u_b) ^ 2 * (c_b / c_a) ^ 3. Furthermore his work suggests that intellect and spirit have the same cost, thus x_a / x_b = (u_a / u_b) ^ 2. In order to find the optimal gear ratio (nonsockets) of Int and Spi we just have to estimate their relative marginal utilities. To simplify things below I’m going to assume only GH7 healing and that lower ranks scale down in a way that doesn’t change the ratio significantly

Int’s marginal utility in terms of healing 15 mana * (hp/mana ave) * (percent nonoverheal) + (1/16000) * (total noncrit healing) * (percent nonoverheal of the crit bonus portion of the heal)

There are some hard numbers to estimate in the above. hp/mana average depends on 23/38 vs 1x/4x as 1x/4x is about 2% (1000 +heal) to 3% (2000 +heal) higher than 23/38 on GH7 (going to assume 7.92 as that’s the average of the builds at 2000 +heal and a 20% crit rate). percent nonoverheal is likely around 70%. percent nonoverheal of the crit bonus portion of the heal is really hard to estimate, if all heals are 70% effective 30% overheal, then it’s 0%, if 70% of heals are 100% effective and 30% are wasted it’s, 70%. Assuming that half of noncrit heals have some overhealing and on the other half, the full crit extra healing can fit in without overhealing gives a figure of 50%. Total noncrit healing is another hard factor to estimate, I’m going to leave it as a parameter for now. These assumptions reduce the Int marginal utilityin terms of additional healing to 83.16 + (total noncrit healing)/32000.

Spirit’s utility is ((1/8 mana/s * o5sr% + .3/8 mana/s * (1 – o5sr%)) * fight duration * hp/mana ave + 0.35 * 6/7 * 1.1 * empowered healing coeff * total mana spent on healing/spell cost) * nonoverheal %

I’m going to assume o5sr% of 50%, hp/mana ave as 7.92 again, empowered healing coeff as 1.16, nonoverheal % as 70%, GH7′s spell cost is 701 mana, with 6% clearcasting this reduces Spirit’s marginal utility to (0.45045 * fight duration + 0.26796/(701*.94) * mana spent on healing)

If we assume that a healer pumps 30,000 mana in heals a given fight at the 7.92 coefficent a 20% crit rate for 8/11 noncrit healing, the forumlas reduce to

Int = 88.56

Spi = 0.45045 * duration + 12.195…

duration spi util, optimal spi/1 int

60 39.22 0.20

120 66.25 0.56

180 93.28 1.11

240 120.30 1.85

300 147.33 2.77

360 174.36 3.88

420 201.38 5.17

480 228.41 6.65

540 255.44 8.32

600 282.47 10.17

Notes:

The 3 spirit to 2 int ratio occurs in fights of 214 seconds, with the assumptions above. I’d love to see what assumptions were used in computing a fixed ratio of 3 spirit to 2 int as it’s highly dependant on mana use (shadowpriest/potions/runes) and fight duration.

Longer fights make show that Int really doesn’t matter on gear as in a 10 minute fight, there should be over 10 spirit for each int.

This only applies to the stats without sockets. Since the cost formula on sockets is different (cost = sum(c_i * x_i), no exponents involved), only the best stat should generally be socketted, ie don’t socket to try and reach a ratio, socket to get the best one of all the possible stats. It also excludes innate stats from the ratio as those are not on gear.

There are a lot of assumptions in the above, the only way to get a better figure would be to plug in better numbers. ie you don’t run through 30000 mana in a 60 second fight, and you might run through significantly more in a 600 second fight, some sort of mana use/fight duration incorporating potions would be needed [10k base, 5k shadow fiend * 2 = 10k, 5 * super mana = 12k, 5 * dark runes/demonic runes = 6k, 50% o5sr at 600 spirit = 29.25k, 39 mana/5 from oil/flask = 4.68k, shadow priest at 800 dps = 38k, total = 111.93k mana to use in 10 minutes if you're going "all out", without the runes and a shadow priest, it's 61.25k, a 3 minute fight with just a shadow fiend, mana oil, flask, and 2 supers is 29,979] Remember if you’re not going “all out” (for whatever your definition of that term is), the bulk of the value of both stats disappears as you just end the fight with extra mana or consumables not used. The terms turn into Int = 5.4, Spi = 12.195…, or 5.1 Spi per Int.

For the EJ post I put up last Feb on calculating the optimal mix, see http://elitistjerks.com/f33/t9796-evaluating_itemization/

Sservis said this on February 17, 2008 at 12:35 pm |

Based on simple derivative calculus, if you use the formula (S/21.44)(I^0.5) and pick some arbitrary number for spirit + intellect (call it C) we can do this:

S + I = C

S = I-C

Y = mana regen = (S/21.44)*(I^0.5)

Y = ((I-C)/21.44)*(I^0.5)

Y = (1/C)I^1.5 – 1000/(C * I^0.5)

dY/dI = (1.5/C)I^0.5 – 1000/(2C*I^0.5)

We then take this derivative of the plot of mana regen vs. Intellect and set dY/dI = 0 (because the derivatve of a graph is is slope at that point, where the slope is zero is (in this case) the peak of the curve)

0 = (1.5/C)I^0.5 – 1000/(2C*I^0.5)

multiply both sides by C, then algebraic manipulation to:

1.5I^0.5 = 1000/2I^0.5

3.0I = 1000

I = 333.33

This means that if you have 1000 stat points to spend in spirit and intellect, from a PURELY mana regen bases standpoint, for maximal mana regen you should place 333 of those into int and 667 of those into spirit (really 1/3 and 2/3) giving a ratio of 1:2 as priestly endeavors said earlier.

HOWEVER: it should be noted that this is only for maximal mana regen from spirit and intellect. It does not take into account the mana you get from intellect; hence theoretically this is for an infinitely long fight where you initial mana pool from intellect doesn’t matter. The shorter the fight, the more your initial manna pool comprises the amount of mana you spend, and thus the more important your intellect becomes, the longer the fight is the better it is to be closer to the ideal of 1INT:2SPI

Saerax said this on February 18, 2008 at 3:41 pm |

BTW, the conclusion above is right, the math is a little off cause I messed up in copying. The C was supposed to be the 21.44 constant which is based on level (it drops out, so 1:2 is valid for all levels), the 1000 comes from the assumed 1000 points to distribute into spirit and int. I did it both ways on paper, and it typing it up mixed them together incorrectly.

Saerax said this on February 18, 2008 at 3:45 pm |

CORRECT VERSION of reply number 2

Again, this is only for your spirit based mana regen not taking into account time of fight or initial mana pool (see caveat)

Based on simple derivative calculus, if you use the formula (S/21.44)(I^0.5) and pick some arbitrary number for spirit + intellect (say 1000) we can do this:

S + I = 1000

S = I-1000

Y = mana regen = (S/21.44)*(I^0.5)

(edit: set 21.44 = C, C is a constant that is based on level, but falls out of the equation so we can not set a specific value to it)

Y = ((I-1000)/C)*(I^0.5)

Y = (1/C)I^1.5 – (1000/C) * I^0.5

dY/dI = (1.5/C)*I^0.5 – (1000/ (2C*I^0.5))

We then take this derivative of the plot of mana regen vs. Intellect and set dY/dI = 0 (because the derivatve of a graph is is slope at that point, where the slope is zero is (in this case) the peak of the curve)

0 = (1.5/C)I^0.5 – 1000/(2C*I^0.5)

multiply both sides by C, then algebraic manipulation to:

1.5I^0.5 = 1000/2I^0.5

3.0I = 1000

I = 333.33

This means that if you have 1000 stat points to spend in spirit and intellect, from a PURELY mana regen bases standpoint, for maximal mana regen you should place 333 of those into int and 667 of those into spirit (really 1/3 and 2/3) giving a ratio of 1:2 as priestly endeavors said earlier.

HOWEVER: it should be noted that this is only for maximal mana regen from spirit and intellect. It does not take into account the mana you get from intellect; hence theoretically this is for an infinitely long fight where your initial mana pool from intellect doesn’t matter. The shorter the fight, the more your initial mana pool comprises the amount of mana you spend, and thus the more important your intellect becomes; the longer the fight is the better it is to be closer to the ideal of 1INT:2SPI

Mages and locks will like int more for crit rating, and because they get less benefit from spirit based mana regen.

Druids + priests get at least 30% of spirit mana regen all the time, don’t need crit as much, and also get +healing from spirit, so obviously they might like even more spirit than the optimal.

Saerax said this on February 19, 2008 at 12:38 pm |

quoting Saerax

===========

Based on simple derivative calculus, if you use the formula (S/21.44)(I^0.5)

===========

Why should we use that formula? Do you have a reference to it’s derivation? I’m not calling it specious, but I find it odd that quadrupling intellect has the effect of making each point of spirit twice as valuable. I’d enjoy reading about where it came from to increase my understanding as I’m pretty sure that unless you’re a balance druid, intellect doesn’t affect mana regen.

quoting again

===========

and pick some arbitrary number for spirit + intellect (call it C) we can do this:

S + I = C

===========

As you can see on http://www.wowhead.com/?item=24690 it’s not a linear relationship (of intellect (+56) and of the owl (+37/37)), Hyzenthlei did the original work and found an equation that fit pretty well to stats on gear pre BC, which was the one I listed as an assumption, with p=1.5 (Hyzenthlei’s original constant, still the one wowwiki lists, although that green’s relative stats suggests it might be p=5/3 as that’s the one consistent with the different of the X)

Sservis said this on February 20, 2008 at 6:42 pm |