Showing posts with label Theorycrafting. Show all posts
Showing posts with label Theorycrafting. Show all posts

Thursday, December 18, 2008

Holy Theorycrafting: Revisiting BoK vs. BoW

So, in the PlusHeal thread where I originally posted my Blessing of Kings vs. Blessing of Wisdom theorycrafting, several questions came up about what was included (or omitted) in my calculations. Those questions were:
  1. Doesn't Blessing of Kings also have the added benefit of increasing one's chance to crit, and shouldn't that be factored in to the mana regeneration comparisons due to the Illumination talent?
  2. Due to the above, wouldn't the Divine Illumination talent (different from Illumination) further push Blessing of Kings' advantage?
  3. Blessing of Kings also increases the total mana available, since each point of Intellect also grants 15 mana. Shouldn't this be taken into account as a "variable mp5" stat which varies based on the length of the fight?
In addition, it was pointed out that Holy Guidance also converts each point of Intellect into a point of spell power at a 20% rate, and this should be factored in. Finally, some of the arguments above were counter-theorycrafted based on a 1000 Intellect model and a 5-minute fight. So, let's revisit the topic and take all of the above into account.



Alright, so let's compare Blessing of Kings and Blessing of Wisdom with 1000 Intellect and see what that gets us. I'll also try to factor in the additional mana regeneration from the added crit, although I'd like to point out that I didn't factor it in earlier because not only is it difficult to model without assuming certain playstyles or going crazy with theoretical maximums, but also because I was trying to tilt the scale in favor of Blessing of Kings as much as possible. WhileBlessing of Kings will net you 0.6% crit with 1000 Intellect (since current consensus is that it 100 Intellect grants 0.6% crit, and a fully talented Blessing of Kings with 1000 Intellect will provide exactly 100 additional Intellect), it also has the significant opportunity cost of 3 talent points in Sanctified Seals, which grants 3% crit. In other words, spending the 5 points to max out Blessing of Kings instead of 2 points to max out Blessing of Wisdom essentially results in a net loss of 2.4% crit at 1000 Intellect.

Also before we begin, I'd like to note that I'm not going to factor in the additional 1500 mana at the start of each fight as mp5, because the length of each fight is too variable to simply assume a 5 minute duration. 1500 mana is definitely a bonus, but it amounts to 5 Flash of Light casts or 1 Holy Light and 1 Flash of Light cast or 1 Beacon of Light. Also, the usefulness of 1500 mana really depends on how you view that mana added to your total mana pool. For example, someone who sees that mana added to the top of the mana pool may cast a Beacon of Light prior to the fight and think "Wow, Blessing of Kings gave me a free Beacon!", whereas I see the the mana as added to the bottom of my mana pool, so I don't see it as a real factor unless it becomes the difference of having the mana to cast a spell or not having the mana to cast the spell.

That said, let's make the assumption that maxing out Blessing of Kings doesn't force you to choose between Beacon of Light and Sanctified Seals, and you somehow manage to max out all 3 talents (though this is impossible without being level 83). Since we're pretty much riding the Hypotheticalmobile to Impossibleland anyway, we'll make a few more completely unrealistic assumptions. One such assumption is that each crit would yield the maximum mana return possible—the amount returned from a Holy Light crit. Correct me if I'm wrong, but Holy Light should cost about 1274 mana at level 80, which means each Holy Light crit restores 764.4 mana. Thus, 0.1% crit would give about 0.7644 mana per cast. We continue our voyage into Crazyville and assume we have so much haste that Holy Light is now limited by the minimum global cooldown of 1 second. So, 0.1% crit would yield 0.7644 mps. Multiply this by 6, since the additional 100 Intellect from Blessing of Kings would give 0.6% crit, not 0.1%, and then multiply by 5 because we want mp5 and not mps, resulting in a maximum of 22.932 mp5 for 1000 total pre-Kings Intellect.

As for Divine Illumination, let's also model it in a best-case scenario for Blessing of Kings. Basically, this would be to assume that when Divine Illumination is up, we chain cast our most expensive heal to maximize mana return. Our most expensive heal is Holy Light, and since we've assumed that we've capped haste to the point where you are now limited by the 1 second minimum of the global cooldown above, we'll make the same assumption now, which gives us a maximum of 15 Holy Light casts in a single Divine Illumination duration. Using the Holy Light cost of 1274 mana above, Divine Illumination would reduce this cost to 637 mana, and a Holy Light crit would restore 764.4 mana as mentioned earlier. This gives us a net gain of 127.4 mana in an additional 0.6% (or 6 out of each 1000) of Holy Light casts, averaging to a gain of about 0.7644 mana per cast. Since we assumed we can cast 15 Holy Light per Divine Illumination, we get an average of 11.466 mana each Divine Illumination. As Divine Illumination has a 3 minute (or 180 second) cooldown, this yields 36 5-second periods per cast, resulting in approximately 0.3185 mp5 (or a meager 19.11 mana over the course of the hypothetical 5-minute fight).

The equation you'd get would essentially be as follows:
BoK vs BoW
= [additional mp5 of DP + MTT + replen from BoK] + [additional mp5 of crit from BoK] + [additional mp5 from Divine Illumination + Holy Light crits] vs 109.2mp5
= [1000 * 0.057] + [22.932] + [0.3185] vs 109.2 mp5
= 80.2505 mp5 vs 109.2mp5
So, even with numbers that are completely impossible in-game (not to mention at a 1000 Intellect gear level), fully improved Blessing of Wisdom gives you about 28.9495 more mp5 with 1000 Intellect. With 1000 Intellect, Blessing of Kings gives you 1500 more mana and 20 more spell power (and 0.6% crit if you want to make that argument, although it still has an opportunity cost of 3% crit from Sanctified Seals), health equal to your pre-Kings Stamina, attack power equal to 20% of your pre-Kings Strength, a small amount of block value from Strength, a small amount of dodge and armor from Agility, a small amount of Spirit-based mana regeneration, and quite possibly a partridge in a pear tree (but only if you've been a good little boy or girl). :P

Wednesday, December 17, 2008

Holy Theorycrafting: Mana Regen of Kings vs. Wisdom

Back in the Burning Crusade era, there seemed to be a wide variety of ways one could build a Holy Paladin. Some players went Holy/Prot, others Holy/Ret, and still others Holy/Prot/Ret (I myself used a 47/7/7 build). However, since Wrath of the Lich King launched, it seems that two Holy builds have become dominant: the Holy Kings build (51/5/15) and the Holy Critadin build (51/0/18 + 2 points wherever). The primary difference between these to builds boils down to just 5 points. The Holy Kings build spends these 5 points in Blessing of Kings and Improved Blessing of Kings, and the Holy Critadin build spends 3 points in Sanctified Seals and has 2 points left over. These 2 points can be invested in a variety of different talents, but for the sake of this post, we'll invest them in Improved Blessing of Wisdom. So, ultimately, our trees will look something like this: Holy Kings (51/5/15) and Holy Critadin with Wisdom (53/0/18). (As a side note, choice between Improved Judgements and Improved Blessing of Might in tier 2 of Retribution is irrelevant, and either build can take either talent).

So, with the above 2 builds being the most dominant, the question of which is better will almost invariably arise. As the issue was brought up by a supporter of the Holy Kings build, and I myself favor the Holy Critadin with Wisdom build, I figured I would stack the deck in favor of the Holy Kings build and see when it would surpass the Holy Critadin with Wisdom build. Since Intellect is now one of the most valuable mana regeneration stats for Paladins (thanks to Divine Plea, Replenishment, and other mana restoration abilities that scale based on total mana, such as Mana Tide Totem) and Blessing of Kings added stats scale based on gear, whereas Blessing of Wisdom remains a static bonus, the former will certainly overtake the latter at some point. The question deals with where exactly that point is, and whether or not it is obtainable. Now, on with the post!



Question:
How much Intellect is needed for a fully improved Blessing of Kings to equal the mana regeneration of a fully improved Blessing of Wisdom? For the purposes of this example, I'll be stacking everything possible in favor of Blessing of Kings, with the exception of improving Blessing of Wisdom via talents.

Assumptions:

  1. Blessing of Kings would replace Blessing of Wisdom on the target.
  2. Replenishment is up on the target 100% of the time (though this is not a particularly realistic assumption).
  3. The target is a Paladin who happens to be grouped with a Restoration Shaman, and thus has access to both Divine Plea and Mana Tide Totem, so that mana regeneration due to Intellect is maximized.
  4. Blessing of Wisdom and Blessing of Kings are both fully improved by their respective talents.
  5. [Glyph of Mana Tide Totem] is used, but other glyphs are not taken into account.
  6. Relics are not taken into account.
  7. The highest rank of each spell is used.

Abbreviation Key:


Given:

Theorycrafting:
First, let's convert everything into mp5, so it's easier to compare:

mps of DP = [mana restored per DP cast] / [DP cooldown]
= [INT * 15 * 0.25] / [60]
= INT * 0.0625

mp5 of DP = [mps of DP] * 5
= [INT * 0.0625] * 5
= INT * 0.3125

mps of MTT = [mana restored per MTT cast[ / [MTT cooldown]
= [INT * 15 * (0.06 + 0.01) * 4] / 300
= INT * 0.07 * 0.2
= INT * 0.014

mp5 of MTT = [mps of MTT] * 5
= [INT * 0.014] * 5
= INT * 0.07

mp5 of replen = [mps of replen] * 5
= INT * 0.0375 * 5
= INT * 0.1875

Thus, we can say that the mp5 given by DP + MTT + replen is:

mp5 of DP + MTT + replen = [mp5 or DP] + [mp5 of MTT] + [mp5 of replen]
= [INT * 0.3125] + [INT * 0.07] + [INT * 0.1875]
= INT * 0.57

Since INT will increase by 10% with BoK, we can conclude that the mana regeneration of the above will also increase by 10% with BoK, since they scale with INT. Thus, they would yield an additional mp5 of:

additional mp5 of DP + MTT + replen from BoK = [mp5 of DP + MTT + replen] * 0.1
= INT * 0.57 * 0.1
= INT * 0.057

So, how much INT would be needed for BoK to equal BoW on the mana regeneration front?

additional mp5 of DP + MTT + replen from BoK = BoW
INT * 0.057 = 109.2
INT = 109.2 / 0.057
INT = 1915.79 (rounded up to 1916)

Now, even if you stack full INT and have access to all obtainable gear, items, item enhancements, and buffs, you'd still be hard pressed getting to 1916 INT.

As I see it, Blessing of Wisdom allows me to worry less about mana regeneration stats and instead focus on stats that give me a bit better return, such as haste and spellpower. Blessing of Kings, while nice, still functions best when you stack base stats. Don't get me wrong—it can certainly be a life-saver when the extra health, dodge, armor, crit, etc. etc. it provides means the difference between life and death, but that extra mana regeneration from Blessing of Wisdom can also prove to be the difference between a victory and a wipe.

Ultimately, I think my biggest issue with Blessing of Kings is probably that I still see it as a buff which essentially works to mitigate the effect of bad luck, whereas Blessing of Wisdom does more to allow you to play at your best. I can't speak for anyone else, but personally, I'd rather be good than lucky.

A Quick Update

I was planning on posting some theorycrafting comparing Blessing of Kings to Blessing of Wisdom yesterday, but I got caught up with follow-up theorycrafting addressing some of the issues brought up (or neglected) by the first aforementioned theorycraft. Since it's late and I'm completely exhausted from a day full of traveling, I think reformatting my original theorycraft to work on the blog will have to wait until I wake up.

If you just can't wait to see the results, feel free to check out PlusHeal.com, as I've already posted it there. Night!

Monday, December 8, 2008

Holy Theorycrafting: SoR vs. SoW+Consecration

I apologize for the lack of posts recently. Real life plus the upcoming holidays has really put a hard limit on the amount of time I have to spare, and sadly online activities tend to be the first thing on my chopping block in such situations. Anyway, to continue with the topic at hand...

In case you don't know, I am one of the moderators at the wonderful healing-oriented forums known as PlusHeal. Well, it just so happened that one of the forum posters there asked a question regarding which strategy dealt damage faster: using Seal of Righteousness for straight-up damage dealing, or using Seal of Wisdom to facilitate the mana regeneration required to be able to cast Consecration as often as possible.

As a Seal of Righteousness user myself, I found this to be a very interesting question. Since I was in the mood to do a bit of theorycrafting at the time, I went ahead and mathed it out. In addition to posting this on the PlusHeal forums, I figured I'd may as well post it here as I've been somewhat lacking in post content recently, and because I spent a good chunk of time on it.

So, let's start with the general idea. As I understand it, the entire point of using Seal of Wisdom instead of Seal of Righteousness is that the former allows you to spam Consecration on cooldown without running into mana issues, whereas the latter would result in downtime if you use Consecration too often. In other words, the whole idea comes down to the notion of Seal of Righteousness vs. Seal of Wisdom + Consecration.

Question:
Which deals more DPS, Seal of Righteousness without Consecration or Seal of Wisdom with Consecration?


Assumptions:
  1. Judgement of Wisdom and Blessing of Wisdom have 100% uptime.
  2. Divine Plea is used on cooldown.
  3. Judgement of Wisdom is used on cooldown to maximize damage output. (This does not necessarily have to be true with the Seal of Wisdom playstyle, but we'll assume it to make the math slightly easier.)
  4. One can maintain Holy Shock and Consecration spamming indefinitely with Seal of Wisdom active, but not with Seal of Righteousness active.
  5. Librams are not taken into account.
  6. There are no differences in playstyle other than which seal is used and whether or not Consecration is used on cooldown.
  7. Seal of Righteousness's Judgement and Seal of Wisdom's Judgement both have the same chance to deal a critical strike, and both deal a 50% bonus damage when they critically strike.
  8. Multiple cooldowns completing simultaneously are not taken into account.
Abbreviation Key:
Given:
Theorycrafting:
First, let's calculate the DPS of SoR and Cons:

SoR DPS
= [SoR damage per swing] * [swings / second]
= [MWS * (0.022 * AP + 0.044 * SPH)] * [1 / MWS]
= 0.022 * AP + 0.044 * SPH

Cons DPS
= [Cons damage] / [Cons cooldown]
= [8 * (Cons# + 0.04 * AP + 0.04 * SPH)] / [8]
= Cons# + 0.04 * AP + 0.04 * SPH


Next, let's figure out how much damage Judgement will contribute. Remember that unlike SoR and Cons, Judgements can deal critical strikes, so we should include that into our calculations:

Judgement bonus damage from critical strikes
= [Judgement base damage] * CritChance * 0.5

average Judgement damage
= [Judgement base damage] + [Judgement bonus damage from critical strikes]
= [Judgement base damage] + [[Judgement base damage] * CritChance * 0.5]
= [Judgement base damage] * (1 + (CritChance * 0.5))

average Judgement DPS
= [average Judgement damage] / [JCD]
= [[Judgement base damage] * (1 + (CritChance * 0.5))] / [JCD]


Now that we have the formula for the average Judgement DPS, let's calculate the numbers for both SoR Judgements and SoW Judgements:

average SoR Judgement damage
= [SoR base damage] * (1 + (CritChance * 0.5))
= [1 + 0.2 * AP + 0.32 * SPH] * (1 + (CritChance * 0.5))

average SoR Judgement DPS
= [average SoR Judgement damage] / [JCD]
= [(1 + 0.2 * AP + 0.32 * SPH) * (1 + (CritChance * 0.5))] / [JCD]

average SoW Judgement damage
= [SoW base damage] * (1 + (CritChance * 0.5))
= [1 + 0.16 * AP + 0.25 * SPH] * (1 + (CritChance * 0.5))

average SoW Judgement DPS
= [average SoW Judgement damage] / [JCD]
= [(1 + 0.16 * AP + 0.25 * SPH) * (1 + (CritChance * 0.5))] / [JCD]


We now have all the numbers we really need for our final calculations, but before we get to the really heavy math, let's go ahead and do something that will make later calculations a bit easier. Since SoR Judgements have greater AP and SPH coefficients than SoW Judgements and they both have the same CritChance and JCD, we know for a fact that SoR Judgements will deal more damage than their SoW counterparts. So, let's just go ahead and figure out exactly how much more DPS we get from SoR Judgements than SoW Judgements on average.

difference between SoR and SoW Judgement DPS
= [average SoR Judgement DPS] - [average SoW Judgement DPS]
= [((1 + 0.2 * AP + 0.32 * SPH) * (1 + (CritChance * 0.5))) / JCD] - [((1 + 0.16 * AP + 0.25 * SPH) * (1 + (CritChance * 0.5))) / JCD]
= ((1 + (CritChance * 0.5)) * ((1 + 0.2 * AP + 0.32 * SPH) - (1 + 0.16 * AP + 0.25 * SPH))) / JCD
= ((1 + (CritChance * 0.5)) * ((1 - 1) + (0.2 - 0.16) * AP + (0.32 - 0.25) * SPH)) / JCD
= ((1 + (CritChance * 0.5)) * (0.04 * AP + 0.07 * SPH)) / JCD


And now it's finally time for the really heavy math. Since we're trying to figure out which playstyle deals greater DPS, I would feel uncomfortable equating the two and then mathing it out. So, we'll just use an arbitrary symbol to separate the two sides while we figure it out. Let's go with ">?<" since we're trying to figure out which is greater than and which is less than the other.

[SoR DPS] + [average SoR Judgement DPS] >?< [Cons DPS] + [average SoW Judgement DPS]

[SoR DPS] + [average SoR Judgement DPS] - [average SoW Judgement DPS] >?< [Cons DPS]

[SoR DPS] + [difference between SoR and SoW Judgement DPS] >?< [Cons DPS]

[0.022 * AP + 0.044 * SPH] + [((1 + (CritChance * 0.5)) * (0.04 * AP + 0.07 * SPH)) / JCD] >?< [Cons# + 0.04 * AP + 0.04 * SPH]


That's pretty much as far as my rusty math skills can get me. Now there's nothing left but to plug in the numbers and see what comes out. I'm going to go ahead and input my numbers as an example.

(0.022 * AP + 0.044 * SPH) + (((1 + (CritChance * 0.5)) * (0.04 * AP + 0.07 * SPH)) / JCD) >?< (Cons# + 0.04 * AP + 0.04 * SPH)

(0.022 * 474 + 0.044 * 1071) + (((1 + (0.2201 * 0.5)) * (0.04 * 474 + 0.07 * 1071)) / 10) >?< (72 + 0.04 * 474 + 0.04 * 1071)

(10.428 + 47.124) + (((1 + 0.11005) * (18.96 + 74.97)) / 10 >?< (72 + 18.96 + 42.84)

57.552 + (1.11005 * 93.93) / 10 >?< 133.8

57.552 + 104.2669965 / 10 >?< 133.8

57.552 + 10.42669965 >?< 133.8

67.97869965 < 133.8


Alternately, you can skip the "calculate the difference between SoR and SoW Judgement DPS" step, which yields the actual DPS potential of both playstyles:

[SoR DPS] + [average SoR Judgement DPS] >?< [Cons DPS] + [average SoW Judgement DPS]

[0.022 * AP + 0.044 * SPH] + [((1 + 0.2 * AP + 0.32 * SPH) * (1 + (CritChance * 0.5))) / JCD] >?< [Cons# + 0.04 * AP + 0.04 * SPH] + [((1 + 0.16 * AP + 0.25 * SPH) * (1 + (CritChance * 0.5))) / JCD]

(0.022 * 474 + 0.044 * 1071) + (((1 + 0.2 * 474 + 0.32 * 1071) * (1 + (0.2201 * 0.5))) / 10) >?< (72 + 0.04 * 474 + 0.04 * 1071) + (((1 + 0.16 * 474 + 0.25 * 1071) * (1 + (0.2201 * 0.5))) / 10)
(10.428 + 47.124) + ((1 + 94.8 + 342.72) * (1 + 0.11005) / 10) >?< (72 + 18.96 + 42.84) + ((1 + 75.84 + 267.75) * (1 + 0.11005) / 10)

57.552 + (438.52 * 1.11005 / 10) >?< 133.8 + (344.59 * 1.11005 / 10)

57.552 + (438.52 * 0.111005) >?< 133.8 + (344.59 * 0.111005)

57.552 + 48.6779126 >?< 133.8 + 38.25121295

106.2299126 < 172.05121295


So, it seems that using Seal of Wisdomand Consecration yields more DPS than Seal of Righteousness without Consecration. Naturally, using Seal of Righteousness with Consecration would deal more than either of the alternatives, but this is not a sustainable DPS cycle (which was the goal).