reduced mana reservation vs. increased maximum mana (formula)

When trying to build a character, I noticed those really appealing, at first glance at least, reduced mana reservation nodes, But they were far away and I wanted to know how effective they were. so I searched the forums, and failed to find some definite answers, though many raised questions and got inconclusive and contradictory answers. Thus I decided to calculate it myself. My question was, which would give me more unreserved mana to use my skills, reduction of the reservation or increased mana percentage.

Both are acting on the base, so a 40% mana reservation aura, with a 5% mana reservation reduction, would reserve 38% of my mana. On the other hand, 8% more mana would give me 8% of my base, and 40% of that would be reserved.

This post will be divided into two parts. First, I will present my calculations for an aura that reserves 40% of mana, and my results to how 8% increased mana compares against 5% and 10% reduced mana reservation.

The second part of the post will present a general formula, usable with any aura (or several) that reserve percentage, any percent of reservation reduction, and any percent of mana increase. With that formula you could include several auras and a cost reduction gem into the calculation, check to see if you could handle it with just passives and no gem to free a gem spot, or maybe use the gem and get increased mana percentage.

I think it is a very useful and quite simple formula, and hopefully somebody else will as well. My only regret is that the post is going to be extremely long (at least in forum standards), and even considered not posting the first part. Although I feel it is essential to explain how I got there, you can just skip to part 2 right away and get the formula.

As a side note, I do not account for the increased mana regeneration that those percent increased mana would give. That would require another calculation which I might do later.



PART 1

The first thing to do is identify all the variables and name them, so that I can ignore them instead of constantly thinking about them.

X = base mana
P = (percent mana increased) / 100

P is divided by 100 so you can just input percentage. The maximum mana is easily calculated: X + P*X

those are the important variables that need to be accounted for. Now I can play with numbers. I tested for 40% mana reserved by an aura, so 5% reduced mana reservation would give me 38% mana reserved, and 10% reduced mana reservation would give me 36% mana reserved.

Step 1 - 8% increased mana
Assuming I have X base mana and P percent mana increased, what would another 8% mana increased do for me? the new mana, named M1, would be as follows

M1 = X + (P + 0.08)*X = X + P*X + 0.08*X = 1.08*X + P*X

the amount of mana reserved after the 8% increase is named R1

R1 =

40*M1
-------- =
100

40*(1.08*X + P*X)
---------------------- =
100

20*(1.08*X + P*X)
---------------------- =
50

21.6*X + 20*P*X
---------------------
50

Why I left it base 50 instead of 5 you can find out in the next steps. Now I need to know how much mana would be left, which is what I am after in the first place. for that I need to subtract R1 from M1

M1 =

50*(1.08*X + P*X)
---------------------- =
50

54*X + 50*P*X
------------------
50


M1 - R1 =

54*X + 50*P*X - 21.6*X - 20*P*X
--------------------------------------- =
50

32.4*X + 30*P*X
--------------------
50


Step 2 - 5% reduced mana reservation

The maximum mana is not changed with this passive, and will stay the same for 10% reduced mana, so it is time to name it.

M = X + P*X =

50*X + 50*P*X
------------------
50


To find out how much mana would be reserved, All I need to do is calculate how much 38% of M would be, and then to find out how much mana would be left, just as in step 1, subtract the reserved mana from the total mana. the reserved 38% mana is named R2

R2 =

38*(X + P*X)
--------------- =
100

19*(X + P*X)
--------------- =
50

19*X + 19*P*X
----------------
50


M - R2 =

50*X + 50*P*X - 19*X - 19*P*X
------------------------------------- =
50

31*X + 31*P*X
-----------------
50



Step 3 - 10% reduced mana reserved

Similar to step 2, M is the same, the reserved 36% mana are named R3

R3 =

36*(X + P*X)
--------------- =
100

18*X + 18*P*X
-----------------
50


M - R3 =

32*X + 32*P*X
----------------
50



Step 4 - two inequalities

I found out, as you will see, that the base mana, which is X, doesn't matter at all. But to show it I decided to keep it here, as well as in part 2, until it disappears naturally in the inequality. Since it is a positive number, making it disappear has no effect on the inequality.

What I will do is first is eliminate X, then try to express P using the numbers, and that would give me a number of increased maximum mana percent, above which percent reduced mana reservation is more effective than percent increased mana. To do that, I need to take the unreserved mana from step 1, the 8% mana increase, and subtract from it the unreserved mana from step 2 in the first inequality, and step 3 in the second inequality. If something is unclear, please continue reading as I will try to explain myself while calculating.

(M1 - R1) - (M - R2) > 0

remember that (M1 - R1) gives us the unreserved mana from 8% increased mana. When subtracting from it the unreserved mana from 5% reduced mana reservation, if the result is positive, then 8% increased mana is more effective as it gives more unreserved mana. Let us expand it into what we got from steps 1 and 2

32.4*X + 30*P*X
------------------- -
50

31*X + 31*P*X
----------------- > 0
50


Multiplying by 50 because it is annoying to post like this

32.4*X + 30*P*X - (31*X + 31*P*X) > 0

32.4*X + 30*P*X - 31*X - 31*P*X > 0

1.4*X - 1*P*X > 0

1.4*X > 1*P*X

Notice how both sides have X, and it is positive, so we can divide by X to remove it.

1.4 > P

1.4 is 140%, which means that in order for 8% increased mana to give you more unreserved mana your percent increased mana prior to the 8% increase must be below 140%. otherwise 5% reduced mana reservation would give more unreserved mana.

Before you start picking random base mana and percent increased mana numbers and testing, wait until you see the second inequality that compares to 10% reduced mana reservation. It would give an even lower number that as a consequence easier to test.

(M1 - R1) - (M - R3) > 0

Reminding, note what I am subtracting from what. If the inequality is true, then 8% increased mana would give more unreserved mana than 10% reduced mana reservation. Just as in the last inequality, you can scroll up and verify that I have left everything at base 50, thus I am multiplying by 50 before even writing it here. Either trust me or check yourselves.

32.4*X + 30*P*X - (32*X + 32*P*X) > 0

0.4*X - 2*P*X > 0

0.4 > 2*P

0.2 > P

This is much easier to test. Pick a random base mana, try calculating with 10% increased mana, 20% and 30%. You will see that at 20%, if you add 8% increased mana, or try with 10% reduced mana reservation, the unreserved mana would be the same.

base * (1 + ((precent_increased + 8)/100) * 60/100

compared with

base * (1 + (percent_increased / 100)) * 64/100

the 60/100 and 64/100 is the unreserved amount left from 40% and 36% mana reservation respectively.



PART 2

Now comes the juicy part. But first, to remind of our two variables and add a few new ones.

X = base mana
P = (percent mana increased) / 100
A = (added percent mana increased) / 100
R = (percent reserved by skills) / 100
D = (100 - (percent reduced mana reservation)) / 100

M = mana = X + P*X

to use the formula, input A, R and D. X is not needed as we will see again, like in part 1, when calculating the formula, while P is what we are looking for. They are all divided by 100 to transform them from percentage to decimal form. Notice that D is subtracted from 100 then divided by 100. That is because you input the percentage by which you reduce mana reservation, while I want to find out how much would still be reserved, which for 5%, for example, would be 95% of R

First, how much mana I would gain from percent increased mana. M1 is the new maximum mana.

M1 = X + (P + A)*X = X + P*X + A*X

The reserved amount is named R1

R1 = R*M1 = R*(X + P*X + A*X) = R*X + R*P*X + R*A*X

For the unreserved amount I need to subtract R1 from M1

(M1 - R1) = X + P*X + A*X - R*X - R*P*X - R*A*X

Now we need to see how much mana I would gain from percent reduced mana reservation. R2 is the reserved amount.

R2 = D*R*M = D*R*(X + P*X) = D*R*X + D*R*P*X

To get the unreserved amount, subtract the reserved from the total

(M - R2) = X + P*X - D*R*X - D*R*P*X

The last step is the inequality. If it is true, then the percent increased mana would give more unreserved mana. I will again try to isolate P, the already existing percent increased mana before the new increase A, and X will disappear again just as in part 1.

(M1 - R1) - (M - R2) > 0

Get ready for a confusing inequality as I substitute (M1 - R1) and (M - R2) for the values we have

X + P*X + A*X - R*X - R*P*X - R*A*X - (X + P*X - D*R*X - D*R*P*X) > 0

X + P*X + A*X - R*X - R*P*X - R*A*X - X - P*X + D*R*X + D*R*P*X > 0

underlines removed

A*X - R*X -R*P*X - R*A*X + D*R*X + D*R*P*X > 0

Now it is time to get rid of X. Base mana is a positive value, so the inequality remains the same if we divide by X

A - R - R*P - R*A + D*R + D*R*P > 0

Now to isolate P

A - R - R*A + D*R > R*P - D*R*P

A - R - R*A + D*R > P*(R - D*R)

Rearrange a little for convenience

A - R*A - R + D*R > P*(R - D*R)

A*(1 - R) - (R - D*R) > P*(R - D*R)

Divide by (R - D*R)

A*(1 - R) - R*(1 - D)
------------------------- > P
R - D*R


A*(1 - R) - R*(1 - D)
------------------------- > P
R*(1 - D)


A*(1 - R)
----------- - 1 > P
R*(1 - D)

As a reminder:
P is the percent increased mana you already have. no need to input it into the inequality because it is what we're looking for.

A is the percentage by which you want to increase.
R is the percentage of mana reserved by skills
D is the percentage by which you want to reduce mana reservation

All of them need to be in decimal form, so divide A, R and D, which are all percentages, by 100 before using the formula. D you can input as is, no need to subtract from 100 since in the beginning we defined D as ((100 - percent_reduced_mana_reservation) / 100). Because of this, in the final formula you can input the percentage by which you want to reduce, straightforward like A and R. Don't forget to divide them by 100!

You will get something similar to what we got in part 1. try using R=40 A=8 D=10 (divide them by 100 before using in the formula)

0.08 * (1 - 0.4)
---------------- - 1 > P
0.4 * (1 - 0.1)

0.08 * 0.6
---------- - 1 > P
0.4 * 0.9

0.048
----- - 1 > P
0.04

1.2 - 1 > P

0.2 > P

Just like we got at the end of part 1. So in this case, if your percent mana increased (before adding more) was at 20%, then adding 8% would give the same unreserved mana as reducing 10% of mana reservation. if your percent mana increased is less than 20%, then increased mana would give more, while if it is more than 20%, then reduced mana reservation would give more unreserved mana.

I hope you find this useful

Umm... wow, that's a lot of math there, I bet you could divide by zero and not have the world explode if you tried.

I don't care about math and I don't like zeros. For all I care divide by zero all you want.

What I care about is figuring out which passives would be better. And if at least one person finds it useful, maybe in a few months or years that person would search for this topic. He could have the answers right there and this post would be useful.

I'd solve the equations instead as follows:

The equation for the percentual de- or increase in your mana pool is

X = (1 + IMM)*(1 - BMR*(1-MRR))

Where IMM is the Increased Maximum Mana (1 equals 100% increased maximum mana)
BMR is the percentual Base Mana reserved (1 equals 100% mana reserved)
MRR is the Mana Reserved Reduction. (Again, 1 equals 100% reduction)

X can be differentiated with respect to IMM:

dX/dIMM = (1-BMR*(1-MRR))

and with respect to MRR:

dX/dMRR = (1+IMM)*MRR

These equations show the effectiveness of an increase in mana and a decrease in mana reserved, respectively. Rather than determining the magnitude, the two can be compared to show which one is more effective for given values of BMR, MRR and IMM.

The relative effectiveness of the stats can be shown as a ratio of the above equations:

(dX/dIMM)/(dX/dMRR) = (1/BMR + MRR - 1) / (1 + IMM)

Since increased mana nodes are usually 8% and decreased mana reserved nodes are 5%, the above equation would be most interesting when the mana increased effectiveness is only 5/8th that of the reduced reserved mana effectivess.
This is true when:

(1/BMR + MRR - 1) / (1 + IMM) < 5/8

Rewriting the equation results in:

IMM < 1.6/BMR - 2.6 + MRR

To illustrate, when your increased mana IMM and reduced reserved mana MRR are both at zero, your base reserved mana BMR needs to be larger than 61.5% in order for a 5% reduced mana reserved node to be more effective.
If you already have 40% increased mana, we can plug in a figure of 0.4 for IMM into the equation. In this case, you'd need a BMR of at least 53.3% for reduced mana cost nodes to be more effective.

As the OP noted, this doesn't take into account mana regeneration increases in terms of effectiveness.

Just as a suggestion, DAO - because I strongly support this type of post - giving us every step of your work along the way leads to overlong and virtually unreadable posts. I, as a reader, trust you to do all the intermediate algebra - what's important to me is the answer, and that pretty much got buried here. You can just stick to something like

M = (Initial state) = (Final state)

rather than taking up 5 or 10 lines with it.

Great work, and really appreciated, though!

You forget that mana regeneration depends on maximum mana, so you may often want to take maximum mana passive for example if you only take tempest shield or hatred (or other auras, including amount reserved based that would take 20-30% of you mana).

While all this is well and good, the important thing to consider is how it all effects your uptime. "unreserved mana" - well that's nice to know, but regen has way more to do with your uptime than max mana.

Uptime means: what percentage of the time can I nuke? Downtime is: what percentage of the time am I forced to be inactive because I don't have mana? Some percent of downtime is okay (even efficient) in most builds because you have to spend time walking/herding.

This tool isn't perfect, doesn't account for every possibility. In fact, I don't actually completely understand how "reduced mana reserved by %" bonuses work, so they aren't supported right now. Neither is Inner Force at the moment. But this thing will show you your uptime considering all sources of regen, the cost of the skill you spam most, attack speed, size of mana pool, mana reserved.... And then how it changes your uptime to adjust your stats in various ways.

Uptime

If you feel like using it, make a copy first. It's read only. If you add in a section for reserved mana reduction, I'll merge it :). have a look.

sorry, I ruined a lot of the sheet's calculations right before I posted the link to it. Was messing with layout and broke a number of references.

If you copied it within the last hour you got something that doesn't work too well. Should be fixed now.

"

renatobaal escribió:

I'd solve the equations instead as follows:

The equation for the percentual de- or increase in your mana pool is

X = (1 + IMM)*(1 - BMR*(1-MRR))

Where IMM is the Increased Maximum Mana (1 equals 100% increased maximum mana)
BMR is the percentual Base Mana reserved (1 equals 100% mana reserved)
MRR is the Mana Reserved Reduction. (Again, 1 equals 100% reduction)

X can be differentiated with respect to IMM:

dX/dIMM = (1-BMR*(1-MRR))

and with respect to MRR:

dX/dMRR = (1+IMM)*MRR

These equations show the effectiveness of an increase in mana and a decrease in mana reserved, respectively. Rather than determining the magnitude, the two can be compared to show which one is more effective for given values of BMR, MRR and IMM.

The relative effectiveness of the stats can be shown as a ratio of the above equations:

(dX/dIMM)/(dX/dMRR) = (1/BMR + MRR - 1) / (1 + IMM)

Since increased mana nodes are usually 8% and decreased mana reserved nodes are 5%, the above equation would be most interesting when the mana increased effectiveness is only 5/8th that of the reduced reserved mana effectivess.
This is true when:

(1/BMR + MRR - 1) / (1 + IMM) < 5/8

Rewriting the equation results in:

IMM < 1.6/BMR - 2.6 + MRR

To illustrate, when your increased mana IMM and reduced reserved mana MRR are both at zero, your base reserved mana BMR needs to be larger than 61.5% in order for a 5% reduced mana reserved node to be more effective.
If you already have 40% increased mana, we can plug in a figure of 0.4 for IMM into the equation. In this case, you'd need a BMR of at least 53.3% for reduced mana cost nodes to be more effective.

As the OP noted, this doesn't take into account mana regeneration increases in terms of effectiveness.

analysis is powerful stuff. great post! will definitely use it.

I think it has two typos though.

dX/dMRR = (1+IMM)*MRR

should be

dX/dMRR = (1+IMM)*BMR

and I think that

IMM < 1.6/BMR - 2.6 + MRR

should be

IMM > 1.6/BMR - 2.6 + 1.6*MRR

correct me if I am wrong.

"

simnel escribió:

Just as a suggestion, DAO - because I strongly support this type of post - giving us every step of your work along the way leads to overlong and virtually unreadable posts. I, as a reader, trust you to do all the intermediate algebra - what's important to me is the answer, and that pretty much got buried here. You can just stick to something like

M = (Initial state) = (Final state)

rather than taking up 5 or 10 lines with it.

Great work, and really appreciated, though!

noted. to my defense, the huge calculations with the line in the middle for division would've been much shorter, but it refused to parse my spaces, so I had to make it quite longer - each time starting a new line (which is actually 4 lines for those). but your suggestion would eliminate that problem entirely.

I already see the consequences you were talking about.

"

globbi escribió:

You forget that mana regeneration depends on maximum mana, so you may often want to take maximum mana passive for example if you only take tempest shield or hatred (or other auras, including amount reserved based that would take 20-30% of you mana).

probably didn't even read my post, where I specifically stated that I won't account for it and that I only compare unreserved mana benefits. and it wasn't even in the mess, just the last paragraph before the mess, starting with "as a side note".

This could be useful to people who need just more unreserved mana (maybe to run another aura that reserves a fixed number for example?) and assumes regeneration is fine. Or just for people who want to see if there would or wouldn't be benefits to using reduced mana reservation over increased mana for some reason or another. maybe they are trying to balance unreserved mana versus regeneration, and grabbing a couple of those would help. In either case, I have mentioned it and that the post won't deal with it.

EDIT: for all those purposes use the analysis reply which gives much more powerful tools than I did.