When to take which enhancements for best damage output

[tinyelvis]

On one of my sorcerers, the air savant, has three elemental lines of damage in enhancements. I have been running with a superior fire lore item and wondered how best to setup my enhancements. Inspired by work performed in this thread,

http://forums.ddo.com/showpost.php?p...&postcount=103

I decided to do a little bit of analysis work to determine the best enhancement setup since I dont have enough points to get all of the fire line in addition to my other elemental lines. Here are the pertinent conditions that I regularly run under.

1. Superior inferno clickie
2. Superior fire lore

I ran my analysis and found that with the above setup, to maximize DPS, the best enhancement order was something like this (this order exactly),

Flame Manipulation I (buffs damage)
Deadly Flame I (buffs crit multiplier)
Combustive Spell Casting I (buffs crit percentage)
Flame Man. II - VII
Deadly Flame II - V
Comb Spell Cast II
Deadly Flame VI
Comb Spell Cast III - VI


Not very intuitive. But that is what my analysis yielded. If interested in how I achieved this result please read on. Also let me know if you find any logic or other errors. Would be surprised if I posted something with this many numbers and not have an error somewhere. I used the following equation to calculate these results

NewCrit > Crit * (1 - Enh Dam Buff / Eb)

Note, this relationship can be used for any class that uses damaging spells. It is not specific to sorcerers. The parameters and derivation of this relationship are explained in detail below.

--------------------------------------------------------------------------------------------
THE ANALYSIS:
First lets review the basic equation for spell damage in DDO.

Non Critical Spell Damage "NCSD" = Base * MM * EB , where
Base = the quoted average expected spell base damage
MM = Metamagic buffs like maximize, empower, and the capstone.
EB = Enhancement and item bonuses

To be accurate in any damage analysis we need to consider criticals. In this case, looking at any given damage event then, the expected damage can be represented with this relationship.

ExpDam "ED" = NCSD * (1 - Cr%) + NCSD * Cr% * CM = NCSD * [ (1-Cr%) + Cr%*CM]

Here,
Cr% = your percentage chance for a critical
CM = your critical multiplier

Notice we can rewrite this as,

ED / (MM * Base) = EB * [ Cr% * CM + (1 - Cr%)] = EB * Crit, where,

Crit = [ Cr% * CM + (1 - Cr%)]

All things else being the same then, your damage increases when the product of EB and Crit increases. We can use this relationship to help us find out which line of enhancements will give us the best results. For any given situation then, we have a couple choices to make. However, a couple of things are constant. Namely we always have superior lore and superior potency clicky active. It is only necessary then for us to compare the effect that a given enhancement has on this relationship and choose the situation that maximizes it.

For example, if we add critical enhancements, then "Crit" will become updated with a larger value. Let's call this "NewCrit". We can then write this inequality that must be met in order for the selection of critcal lines over damage buff lines to occur.

NewCrit * Eb > Crit * (Eb + Enhancement Damage Buff)

Here, the enhancement damage buff is the bonus received if the damage line is chosen (i.e. Flame Manipulation I gives a 0.20 addition). So, plugging in some numbers lets see which enhancement is best to initially pick given our situation.

Crit = (0.12 * 2.0 + (1- 0.12) = 1.12 < ---- this is due solely to superior lore
Eb = (1.75) = 1.75 < ---- this is due to superior clickie
Enh Dmg Buff = 0.2 < ---- this is due to Flame Manipulation I

Then,

NewCrit > Crit * (1 - Enh Dam Buff / Eb) = 1.12 * (1 +.2 / 1.75 ) = 1.248

So NewCrit must be greater than 1.248 or else it is a worse choice.

First look at selecting Cumbustive Spell Casting (.16% crit chance) over the Damage Buff

NewCrit (.16/2.0) = (.16*2.0 + (1 - 0.16)) = 1.16

1.16 is not greater than 1.248 so that is a worse choice. Next look at the buff derived from Deadly flame,

NewCrit(.12/2.25) = (.12*2.25 + (1 - 0.12)) = 1.15

Again, 1.15 is less than 1.248 so we see that in fact Flame manipulation I is the best choice. Ok, moving on to our second pick. This follows intuition pretty well. Now moving on to our second choice. At this point Eb has been updated.

Eb = 1.75 + .2 = 1.95

However, our potential increase is now only 0.05. Notice, no critical enhancements have been added so the Crit value is the same. Plugging these into our equation yields,

NewCrit > Crit * (1 - Enh Dam Buff / Eb) = 1.12 * (1 +.05 / 1.95 ) = 1.148

We can use the same values for NewCrit that we calculated above. Notice now that both deadly flame and Combustive now has a value of 1.15 and 1.16 respectively which are greater than 1.148. Since it cost the same in AP points, Combustive Spell Casting I is now the better choice. Moving onto our third selection now. Our Eb has not changed but Crit now has been updated.

Crit = 1.16
Eb = 1.95

NewCrit > Crit * (1 - Enh Dam Buff / Eb) = 1.16 * (1 +.05 / 1.95 ) = 1.189

Let's check our crit lines,

NewCrit(.16/2.25) = 1.2
NewCrit(.17/2.0) = 1.17

So, taking a second Combustive casting or the damage buff are worse selections than deadly flame I. Moving on to the forth pick. At this point, all buffs have stabilized into a +1%, +10%, or +5%, for critical percentage, crit multiplier, and damage buffs respectively.

Crit = 1.2
Eb = 1.95

NewCrit > Crit * (1 - Enh Dam Buff / Eb) = 1.2 * (1 +.05 / 1.95 ) = 1.230

NewCrit(.17/2.25) = 1.2125
NewCrit(.16/2.35) = 1.216

Neither value is greater than 1.230. In fact these values will be less than all of the values calculated from this point on as damage is enhanced through increasing Flame Manipulation II - VII. The rest of the decision need only involve calculating NewCrit values and comparing deadly flame to combustive spell casting. So for the tenth pick. We look at,

NewCrit(.17/2.25) = 1.2125
NewCrit(.16/2.35) = 1.216

Here we pick Deadly Flame II,

NewCrit(.17/2.35) = 1.2295
NewCrit(.16/2.45) = 1.232

We pick Deadly Flame III,

NewCrit(.17/2.45) = 1.2465
NewCrit(.16/2.55) = 1.248

We pick Deadly Flame IV

NewCrit(.17/2.55) = 1.2635
NewCrit(.16/2.65) = 1.264

We pick Deadly Flame V, but they are getting close now

NewCrit(.17/2.65) = 1.2805
NewCrit(.16/2.75) = 1.28

And we swap now to Combustive Casting II,

NewCrit(.18/2.65) = 1.297
NewCrit(.17/2.75) = 1.2975

Back to Deadly Flame VI

Of course the rest of the choices will be Combustive Casting III - VI

So this is how I got to the solution posted in the start.

[wax_on_wax_off]

I'd be interested to see in your summary the % DPS increase from each action point spent to put the entire thing into context. From my investigations there wasn't really any point in going further than 7/1/1 into an enhancement line unless it was to qualify for a PrE.

My sorcerer is currently setup with 7/1/1 in lightning and acid and 7/5/5 (for PrE) in cold. If I had spare points (which I don't) I'd put them into Wand and Scroll Mastery first.

As far as I've been able to determine going beyond 7/1/1 is getting into the realm of efficient metamagics giving more bang for your buck.

[ShadowFlash]

Excellent write up! Time to add some new formula's to my graphics calculator Thanks for the hard work.

ShadowFlash

[tinyelvis]

I'd be interested to see in your summary the % DPS increase from each action point spent to put the entire thing into context. From my investigations there wasn't really any point in going further than 7/1/1 into an enhancement line unless it was to qualify for a PrE.

My sorcerer is currently setup with 7/1/1 in lightning and acid and 7/5/5 (for PrE) in cold. If I had spare points (which I don't) I'd put them into Wand and Scroll Mastery first.

As far as I've been able to determine going beyond 7/1/1 is getting into the realm of efficient metamagics giving more bang for your buck.

If you look at the spell damage equation,

Expected Damage = Base*MM*enhancements * Crit

Where,
Crit = [ Cr% * CM + (1 - Cr%)]

You see that crit is actually just another multiplier. You can easily calculate what taking say 6 more action points in a crit line would do for your damage output. Compare this to the number of extra casts you get with two empower meta magic buffs based on your play style. Personally, I am not concerned with casting efficiency. I play a sorcerer arcane. If I was not forced to take 2 meta magic enhancements I would not. Spell points are rarely an issue. I want the best DPS possible, regardless of costs.

Keep in mind, all of this analytical **** is super optimal. In actual play, efficiency is never perfect and the results are poorer.

Edit: Suppose you use a superior lore item and have spent 7,1,1 already. You have 6 points to spend and want to see how this might effect your damage if you place them in critical buffs. Let's say you invest in 5 more deadly flames and 1 combustive castings, then,

crit% = 17%
crit mult = 2.75

Crit = (.17*2.75 + (1 - .17)) = 1.298

Compare this to the value for 7,1,1

Crit = (.16*2.25 + (1 - .16)) = 1.2

Percent increase in damage = (1.298 - 1.2) / 1.2 = .082
Or close to a 10% increase in instantaneous damage. Would you rather have this or a boost in spell efficiency associated with say improved empower II. For me, I take the 10% damage boost.

[wax_on_wax_off]

If you look at the spell damage equation,

Expected Damage = Base*MM*enhancements * Crit

Where,
Crit = [ Cr% * CM + (1 - Cr%)]

You see that crit is actually just another multiplier. You can easily calculate what taking say 6 more action points in a crit line would do for your damage output. Compare this to the number of extra casts you get with two empower meta magic buffs based on your play style. Personally, I am not concerned with casting efficiency. I play a sorcerer arcane. If I was not forced to take 2 meta magic enhancements I would not. Spell points are rarely an issue. I want the best DPS possible, regardless of costs.

Keep in mind, all of this analytical **** is super optimal. In actual play, efficiency is never perfect and the results are poorer.

Edit: Suppose you use a superior lore item and have spent 7,1,1 already. You have 6 points to spend and want to see how this might effect your damage if you place them in critical buffs. Let's say you invest in 5 more deadly flames and 1 combustive castings, then,

crit% = 17%
crit mult = 2.75

Crit = (.17*2.75 + (1 - .17)) = 1.298

Compare this to the value for 7,1,1

Crit = (.16*2.25 + (1 - .16)) = 1.2

Percent increase in damage = (1.298 - 1.2) / 1.2 = .082
Or close to a 10% increase in instantaneous damage. Would you rather have this or a boost in spell efficiency associated with say improved empower II. For me, I take the 10% damage boost.

8.2%, not 10%, which is less than 1.4% DPS increase per AP.

As I said, I'd rather put those points into Wand and Scroll Mastery, a third elemental line, more spell penetration or improved heighten (for web).

I struggle with the maths, particularly looking on my iPhone, though I'm sure I could make sense of it eventually but I was more suggesting that with your cut off points seeing the actual benefit that each investment was giving would be really helpful so that readers get an idea of where they might like to stop investing in a line as well as the order in which to invest.

[tinyelvis]

8.2%, not 10%, which is less than 1.4% DPS increase per AP.

As I said, I'd rather put those points into Wand and Scroll Mastery, a third elemental line, more spell penetration or improved heighten (for web).

I struggle with the maths, particularly looking on my iPhone, though I'm sure I could make sense of it eventually but I was more suggesting that with your cut off points seeing the actual benefit that each investment was giving would be really helpful so that readers get an idea of where they might like to stop investing in a line as well as the order in which to invest.

This 8.2% is an increase in overall damage potential. However, actually what you will see in the game is something like 22% increase in your critical damage. But to each his own. From my vantage point, I'm the fellow who uses almost no spells that are resistible, never touches a wand, and rarely uses any CC, the dps increase would be welcomed (I run on all of my casters three elemental lines).

[wax_on_wax_off]

This 8.2% is an increase in overall damage potential. However, actually what you will see in the game is something like 22% increase in your critical damage. But to each his own. From my vantage point, I'm the fellow who uses almost no spells that are resistible, never touches a wand, and rarely uses any CC, the dps increase would be welcomed (I run on all of my casters three elemental lines).

Doesn't sound like you're playing your character to it's potential then but, as you say, to each his own.