Intellect -> Spirit Conversion on the PTR : Theorycrafting HQ - Page 11

Muphrid · 02/25/08, 10:48 PM

Originally Posted by Adoriele

Yeah, I was being dumb and assumed that the reason he pointed out that 240+340 != 600 was because it was higher, and didn't bother to actually add them. 240 + 340 is definitely not the best way to use 600 'ipoints'. Point still stands about x/2x being the best allocation, though. If no one else tackles it, I'll come up with something tonight.

[edit] Beaten by Muphrid. Since you seem to know the stat allocation formula, any idea how to account for it when determining the break-even point for int and spirit?

I would just do a straightforward optimization using Lagrange multipliers.

In doing this, however, I get a very strange result. Or perhaps not strange so much as...simpler than I anticipated.

Let R(i,s) represent regeneration and I(i,s) represent item point cost.

That is...

$\begin{align} \\ R(i,s) &= k s i^{\frac{1}{2}} \\ I(i,s) &= \left (i^n + s^n \right )^{\frac{1}{n}} \end{align}$

Now let's take some derivatives...

$\begin{align} \vec{R}^\prime(i,s) &= \left(\frac{k s}{2 i^{1}{2}}, k i^{1}{2}\right) \\ \vec{I}^\prime(i,s) &= \left(\left (i^n + s^n \right)^{\frac{1}{n} - 1} i^{n-1}, \left (i^n + s^n \right)^{\frac{1}{n} - 1} s^{n-1}\right) \end{align}$

We can considerably simplify this expression for I'...

$\vec{I}^\prime(i,s) = \left(\frac{I(i,s)}{i(1+\left(\frac{s}{i}\right)^n)}, \frac{I(i,s)}{s(1+\left(\frac{i}{s}\right)^n)}\right)$

With both gradients computed, we can perform the optimization with the lagrange multiplier...

$\begin{align} \frac{\partial{R}}{\partial{i}} &= \lambda \frac{\partial{I}}{\partial{i}} \\ k s i^{-\frac{1}{2}} &= \lambda \frac{I}{i(1+\left(\frac{s}{i}\right)^n)} \end{align}$

And also...

$\begin{align} \frac{\partial{R}}{\partial{s}} &= \lambda \frac{\partial{I}}{\partial{s}} \\ k i^{\frac{1}{2}} &= \lambda \frac{I}{s(1+\left(\frac{i}{s}\right)^n)} \\ \lambda I &= k s i^{\frac{1}{2}} \left(1+\left(\frac{i}{s}\right)^n\right) \end{align}$

This fortuitously allows us to divide the two expressions for lambda I, make a lot of stuff cancel, and get the following...

$\begin{align} 1 &= \frac{1}{2} \frac{1+\left(\frac{s}{i}\right)^n}{1+\left(\frac{i}{s}\right)^n} \\ 1 + \left(\frac{i}{s}\right)^n &= \frac{1}{2} + \frac{1}{2} \left(\frac{s}{i}\right)^n \end{align}$

Substitute $x = \left(\frac{i}{s}\right)^n$ , and we have...

$1 + x = \frac{1}{2} \left (1 + \frac{1}{x}\right)$

Which simplifies to...

$x + \frac{1}{2} - \frac{1}{2x} = 0$

Which is just a glorified quadratic with solutions x = -1, 1/2.

Flip it over for convenience.

$\begin{align} \left(\frac{s}{i}\right)^n &= 2 \\ \frac{s}{i} &= 2^\frac{1}{n} \\ s &= 2^\frac{1}{n} i = 1.5 i \end{align}$

Considerably simpler than I could have expected.

Edit: though it may be that that simplicity is due to the simplicity of the optimization; all I did was compare regen against item points, without regard to other variables like overall mana gained over a fight of X minutes or other such things.

• Adoriele · 02/26/08, 12:13 AM

Originally Posted by Muphrid

Edit: though it may be that that simplicity is due to the simplicity of the optimization; all I did was compare regen against item points, without regard to other variables like overall mana gained over a fight of X minutes or other such things.

Well, since the question asked by this thread seems to be 'How do I maximize my regen given the new regen formula', I'd say this model fits pretty well.

As to the math, though, your formula for I(i,s) includes n, which I assume to be the number of different stats allocated on the item. If that's the case, given your solution, it would seem that the more separate stats allocated on a given piece of gear, the closer Int and Spirit come to each other in weighting (i.e., given an infinite number of stats to distribute across, the best allocation of spi:int is 1:1), and that the only case for which the original assumption holds, that the optimal ratio is 2:1, is the case where each piece of gear you have is only itemized for either int or spirit.

Muphrid · 02/26/08, 12:14 AM

Originally Posted by Adoriele

Well, since the question asked by this thread seems to be 'How do I maximize my regen given the new regen formula', I'd say this model fits pretty well.

As to the math, though, your formula for I(i,s) includes n, which I assume to be the number of different stats allocated on the item. If that's the case, given your solution, it would seem that the more separate stats allocated on a given piece of gear, the closer Int and Spirit come to each other in weighting (i.e., given an infinite number of stats to distribute across, the best allocation of spi:int is 1:1), and that the only case for which the original assumption holds, that the optimal ratio is 2:1, is the case where each piece of gear you have is only itemized for either int or spirit.

Whoops, my mistake in not being clear; n is a constant believed to have value ln(2)/ln(1.5).

Frah · 02/26/08, 12:17 PM

Originally Posted by tedv

Any chance you could modify the graph to also include the partial derivative for path of steepest ascent? That line should show the optimal way to split X total spirit + int for any value of X. Just eyeballing it, it looks like a 2:1 spirit to int ratio. (Hold a piece of paper up to the graph at a 45 degree angle, making a line from the top left to bottom right corner. Then keep the slope of the line constant but move the line's offset up and down the graph. The path of steepest ascent will be the set of points this line is tangent to each of the gradient curves on the graph.)

Slim to no chance I am afraid. I have been fiddling with this for a while now and I am rather stuck. Can not really claim to be good with maple . I have graphs of the derivative of the equation with respect to int and spirit. I have no idea where to go from here.

tedv · 02/26/08, 1:20 PM

Originally Posted by Frah

Slim to no chance I am afraid. I have been fiddling with this for a while now and I am rather stuck. Can not really claim to be good with maple . I have graphs of the derivative of the equation with respect to int and spirit. I have no idea where to go from here.

Don't worry about it. The mathematical equations are much more precise, and leave us with the 3:2 ratio of spirit to int, which seems reasonable from looking at the graphs.

thedopefishlives · 02/26/08, 1:38 PM

Originally Posted by tedv

Don't worry about it. The mathematical equations are much more precise, and leave us with the 3:2 ratio of spirit to int, which seems reasonable from looking at the graphs.

Okay, that just confused me. I've seen ratios of 2:1 and 3:2 tossed about. The question is, which one is it? Or to put it in terms of the math presented above (which is over my head since I only took a bit of calculus), what is the expected value of n that we should solve with? I noted in my primitive graphs, which seem to be backed up by far more knowledgeable individuals, that the ratio changes with respect to n. Find n, we find our ratio.

Muphrid · 02/26/08, 1:48 PM

Originally Posted by thedopefishlives

Okay, that just confused me. I've seen ratios of 2:1 and 3:2 tossed about. The question is, which one is it? Or to put it in terms of the math presented above (which is over my head since I only took a bit of calculus), what is the expected value of n that we should solve with? I noted in my primitive graphs, which seem to be backed up by far more knowledgeable individuals, that the ratio changes with respect to n. Find n, we find our ratio.

As I noted, the accepted value is $n = \log_{1.5}2$ .

Smizzle · 02/26/08, 1:57 PM

Originally Posted by Muphrid

I would just do a straightforward optimization using Lagrange multipliers.

$\begin{align} \left(\frac{s}{i}\right)^n &= 2 \\ \frac{s}{i} &= 2^\frac{1}{n} \\ s &= 2^\frac{1}{n} i = 1.5 i \end{align}$

Considerably simpler than I could have expected.

Muphrid, I also indepently came up with the same answer in post #143 of this thread, confirming your results.

It turns out, if you optimize the mana return formula,

(0.001 + 0.009327 * S * sqrt(I) ) * 5 = mp5

with the constraint,

(S^x + I^x)^(1/x) = Constant

where S is spirit, I is int, and x is an ( unknown ) exponent,

the maximum return occurs when,

S = 2^(1/x) * I.

For x = ln(2)/ln(1.5), this means your total Int should equal 2/3 your total Spirit.

Frah · 02/26/08, 3:56 PM

The optimum is not a 3:2 ratio in favor of spirit. It is easy to disprove using some example data (unless i have cocked up some calculations ect)

The 2:1 result seems more likely.

CheshireCat · 02/26/08, 4:03 PM

If I'm reading this right, the conclusions are that, for maximum mana regen over a constant pool of stats, the ideal is 2:1. For maximum mana regen *per item budget point*, the ideal is 3:2. That's because stacking spirit by a higher ratio is penalized in item budget terms.

Both calculations neglect things like the initial mana given by intellect compared to fight time, which would push the figures slightly further in favor of int. (That actually makes int a good deal more valuable than my mental calculations had it.)

• Adoriele · 02/26/08, 4:05 PM

Originally Posted by Frah

The optimum is not a 3:2 ratio in favor of spirit. It is easy to disprove using some example data (unless i have cocked up some calculations ect)
[snipped picture]

The 2:1 result seems more likely.

The discrepancy you're see is that, when designing an item, you cannot simply trade one int for one spirit. So an item with 2:1 spirit:int is poorly itemized for regen, while one with 3:2 is 'perfectly' itemized for regen.

Frah · 02/26/08, 6:35 PM

Originally Posted by Adoriele

The discrepancy you're see is that, when designing an item, you cannot simply trade one int for one spirit. So an item with 2:1 spirit:int is poorly itemized for regen, while one with 3:2 is 'perfectly' itemized for regen.

Well as players we do not design items. We select items from a pool of items and try to mix our stats to get the best performance. The amount of int and spirit you have will be dictated to some extent by the items you have and they in turn will be limited by their iLevel.

Morakk · 02/26/08, 7:21 PM

3:2 is the ideal ratio if you were creating your own gear. However, we are stuck with the gear that Blizzard creates, except we aren't: the tenth spirit gem you socket doesn't give you any less regen than the first, unlike the spirit on the gear itself.

2:1, then, is the ratio that you should be concerned about when trying to optimize your own gear. Since you are in all likelihood not going to be able to get more than twice as much spirit as intellect with current itemization, spirit is almost certainly better than intellect. (Int does give you max mana, which is more valuable in a shorter fights, but spirit also gives healing, for priests especially).

If Blizzard ever lets us create our own gear, then you want to have a 3:2 ratio of spirit:intellect.

Frah · 02/26/08, 7:34 PM

I am rather confused about this 3:2 ratio really. Even in terms of gear creation.

Like i showed earlier 27 spirit and 18 int (which is 3:2) ratio is worse than 30spirit 15 int (which is a 2:1 ratio) when you plug them numbers into the regen formula. In both cases the total number of attributes given is 45 which would give them both the same iLevel.

I do not really get what this 3:2 represents.

Roywyn · 02/26/08, 7:54 PM

Originally Posted by Frah

I am rather confused about this 3:2 ratio really. Even in terms of gear creation.

Like i showed earlier 27 spirit and 18 int (which is 3:2) ratio is worse than 30spirit 15 int (which is a 2:1 ratio) when you plug them numbers into the regen formula. In both cases the total number of attributes given is 45 which would give them both the same iLevel.

I do not really get what this 3:2 represents.

"27 spi + 18 int" is cheaper than "30 spi + 15 int" which is cheaper than "45 spi".

You maximised regen when assuming "int + spi = constant".
Item value doesn't scale as "int + spi" though, but as "int^x + spi^x".

He maximised regeneration when "int^x + spi^x = constant".
That's maximisation while maintaining a fixed item value/budget.

To clarify: The issue is that "30 spi + 15 int" is not the same ilvl as "27 spi + 18 int", it's more expensive on item budget.

• Adoriele · 02/26/08, 7:59 PM

Originally Posted by Frah

I am rather confused about this 3:2 ratio really. Even in terms of gear creation.

Like i showed earlier 27 spirit and 18 int (which is 3:2) ratio is worse than 30spirit 15 int (which is a 2:1 ratio) when you plug them numbers into the regen formula. In both cases the total number of attributes given is 45 which would give them both the same iLevel.

I do not really get what this 3:2 represents.

The part I bolded is where you're wrong. Those two items would not be the same iLevel, if these were the only two stats on them. The 27 spirit, 18 int one would be a lower iLevel (34.2271 stat points allocated, compared to 35.0669). At the same iLevel, 3:2 would give 27.663 spi and 18.442 int. While these numbers aren't feasible on a set of gear, imagine filling 18 slots with this exact discrepancy. You'd lose almost 8 int, almost 12 spi.

So yes, we can only pick the gear available. However, by gearing for 3:2, you will get more mana regen, as your gear will be using its item budgets more appropriately.

Frah · 02/26/08, 8:46 PM

ok i see then. I read wowwiki for how to do item levels and got my understanding of item levels from it.

Since 2:1 ratio gives best regen and 3:2 gives best from gear. Is it then better to gem everything with spirit (to an extent) since they kind of fall out with the item level area and it would pull you closer to the 2:1?

tedv · 02/26/08, 9:14 PM

Originally Posted by Frah

ok i see then. I read wowwiki for how to do item levels and got my understanding of item levels from it.

Since 2:1 ratio gives best regen and 3:2 gives best from gear. Is it then better to gem everything with spirit (to an extent) since they kind of fall out with the item level area and it would pull you closer to the 2:1?

Yes, you should spend your gems to get closer to 2:1, which will almost certainly mean stacking spirit gems instead of int gems. Assuming you don't want +healing.

Jacinthia · 02/27/08, 12:03 PM

That was something that's confused me so far. Most of my current gear has spirit on it save for a couple items (offhand from RoS, boots from Zul'Jin, MH epic ring) but I feel most of my issue lies in regemming my current gear.

Right now I stand 500 int/395 spirit unbuffed. Most of the gems in my gear are of the 11healing/2mp5 variety.
MP5 currently stands at 410/230.

On PTR I was looking at #'s like 580/280 by just logging on with no gear changes whatsoever. My question being, is it time to socket all reds with 22healing and blues with 10spi? It looks like according to the 2:1 ratio that would be the case. However most know that the druid sets are heavy in blue sockets, and a couple red here and there. I would gain roughly 100 spirit from gems (socketing all blues with spirit) losing 90 healing from it, and the red's replacing would generate 44 of that back.

I'm aware trees don't spend much time outside FSR to make use of it, but is regen enough in 2.4 pre-gemming that you could even go the stack all 22healing gem route? Or has that completely died in 2.4? I'd love to try, but the guild bank hasn't made it to PTR yet to play with all the gems.

Thanks

Muphrid · 02/27/08, 12:06 PM

Are there not +healing/+spirit purple gems? If there are, two of those would be better than 1 +healing red and 1 +spirit blue. Gems use the itemization formulas too; they're just terribly distorted due to rounding.

Jacinthia · 02/27/08, 12:19 PM

Originally Posted by Muphrid

Are there not +healing/+spirit purple gems? If there are, two of those would be better than 1 +healing red and 1 +spirit blue. Gems use the itemization formulas too; they're just terribly distorted due to rounding.

Well that's where it's a bit iffy. Say Life-Step (R/B). You socket twin shadow pearls, giving you 18healing and 8 spirit. Compared to socketing a spinel and sapphire (22healing 10 spirit). It seems that without some epic pearl, the epic counterparts are just better.

Muphrid · 02/27/08, 12:45 PM

Originally Posted by Jacinthia

Well that's where it's a bit iffy. Say Life-Step (R/B). You socket twin shadow pearls, giving you 18healing and 8 spirit. Compared to socketing a spinel and sapphire (22healing 10 spirit). It seems that without some epic pearl, the epic counterparts are just better.

Ah, of course. Gaps in itemization strike again.

chrull · 02/29/08, 8:04 AM

This change seems so weird without giving us a 11 heal 5 spi gem. I've been eagerly awaiting it every time the PTR was updated but aparantly that's more fun than we are allowed to have.

Jacinthia · 02/29/08, 10:56 PM

Originally Posted by chrull

This change seems so weird without giving us a 11 heal 5 spi gem. I've been eagerly awaiting it every time the PTR was updated but aparantly that's more fun than we are allowed to have.

Well it would be easily implemented, except what gem do you cut it from? Cutting it from a shadowsong would be the grand idea, but no such pearls exist so far. Emeralds would make the gem too common and OP. It's a sad thing too

• constantius · 02/29/08, 11:26 PM

There's absolutely nothing wrong with cutting it from an emerald. Making it 'too common' is meaningless: our 10 spirit gem is the second-lowest demand gem, and is infinitely available. This would be taking its place.

There's basically 4 gems we use, and 2 of them are in a lot of demand: spinels (huge huge demand) and pyrestones (second-highest demand). Amethysts and Sapphires are in moderate to low demand, so it's almost never an issue to fill your gear with them.

Why should it be an issue to put it on a gem that's of low use? They put the haste gem cut onto Emeralds because it *fit* and that should help make use of the 60+ we have in the bank.