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Mathematical aspect of Gaias

Discussion in 'Guides & Database' started by Gripphon, Jan 23, 2017.

  1. Gripphon

    Gripphon

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    This thread is for those who really like to optimize their characters and like to use both math and field tests to achieve that. It is kind of math heavy with a lot of explanations and examples. Hopefully there is someone else out there who takes a more mathematical approach to a game to have fun optimizing their character.

    This thread will help you how to mathematically evaluate usefulness of some gear pieces or souls to the most benefit of what you want to achieve. Beware, however, that mathematical evaluation is NOT the same as the field test, it is just an estimation method which can to some degree of precision tell you what will benefit your character the most. This guide is for those who would like to know what kind of souls do they want for their characters or what piece of equipment might give them the most benefit without spending their time leveling souls to level 50 for the field test, though it might not be a problem with equipment.

    Mathematical evaluation is an ideal model of evaluation which is, from my experience, quite precise, but not 100%. Some things are impossible to calculate, some things are a matter of taste and so on. But then again even the field tests are not completely precise because they depend on the team, on how you actually played that round and so on. Combination of both is the best since both methods have their weaknesses. Math can give you a pretty good clue on what is actually worth testing while idea of test is to finally confirm what your math suggested.


    1. How to properly do testing


    Proper testing is extremely important because otherwise test method is useless, literally. You want your test to be either a field test in some real situation, or the most ideal test on dummy target. When you are testing between 2 souls or equipment pieces, make sure those tests are as much alike as possible for the best results. If, for example, your first dps test on a dummy target was with flawless casting, but your second test was with mistakes, then second test is to be repeated.

    Also, very important, test of 15-20 seconds is NOT a proper test! For example, some sorcerer comes, drops meteor, casts some spells and 20 seconds later he claim his dps is, don't know, 1200! If test is too short, you risk that you might be a victim of too much luck on your critical hits resulting on unrealistically high dps, or you might be a victim of unlucky streak of no-criticals resulting in unrealistically low dps. Also meteor is special because it has 120 seconds cooldown which means that it has very low influence on dps on a long run. It will mean much if you test only 20 seconds, but do a 2 minute test to drop meteor twice and you might be surprised how little it means for your dps.

    Proper test is done over a longer period of time, possibly at least a minute or two long, not shorter than that. When doing a field test on bosses, don't just check your dps after 30 seconds claiming my dps is X, you should check it in the middle of the fight or towards the end of the fight for a more precise result eliminating those lucky or unlucky streaks of criticals as much as you can.


    2. DAMAGE FORMULA

    This formula can directly tell what might be better for your character depending on his current stats and gear. Formula for average damage goes:

    DMG = AP ∙ C (1 + cc ∙ (cd-1))

    AP = either attack power or could be SP like spell power, depending what kind of damage do you calculate
    cc = critical chance. If it is 35%, cc is 0.35
    cd = critical damage. If it is 155%, cd is 1.55
    C = additional factor of damage, for example fireball has 2.5x SP and you could have talent for +10% elemental damage etc, then you would multiply SP additionally with 1.1. For a general formula you don't need this at all when comparing souls who influence your whole character, not just one skill.

    This is very basic formula for most basic comparison. All you have to do is have info on your stats between two souls/pieces of equipment you are comparing and see which one has higher expected damage aka DMG. But, this formula lacks any info on spell haste or attack speed, so use it only when both of those stats are irrelevant for comparison.


    Example 1: I wonder is extra 9 AP provided by one soul better than 7% critical strike provided by other soul if I have 200 AP, 30% critical chance and 160% critical damage.

    Note: since C is the same for both formulas for this example, it is irrelevant for comparison.

    Soul 1: 209 AP, 30% cc, 160% cd
    DMG = 209 ∙ C ∙ (1 + 0.3 ∙ (1.6-1)) = 246.62 C

    Soul 2: 200 AP, 37% cc, 160% cd
    DMG = 200 ∙ C ∙ (1 + 0.37 ∙ (1.6-1)) = 244.4 C

    Result: in this particular case AP soul wins by roughly 1%. This difference is so small it might not be seen by a tests. It is fair to say such difference is too small and could mean souls are practically of identical benefit, AP providing more constant damage while cc providing more damage spikes. This formula can effectively be used to compare various souls and gear equipment. You could compare several stats simultaneously, not one by one. See example 2 below.

    Be careful though since some skills have special talents with higher critical chance when you cast them.


    2.1. Extremities

    There is a shortcut to understand what might benefit your character most. In general the rule for dps is: the more you have, the less valuable upgrade it is. In concrete, if you in higher example assume character has 120 AP (instead of 200), you will find that extra 9 AP is a much better upgrade than adding it on top of 200 AP. Completely the same story is with critcal strike and critical damage. The more critical you have, the less valuable additional upgrade is.

    BUT, this does not mean that pursuing upgrades is not worth it because it might still be more worth it than an alternative. In general, critical strike and critical damage work hand in hand. The more critical damage you have, the more valuable critical chance upgrade is, and the more critical chance you have, the more valuable critical damage modifier becomes. Let's see.


    Example 2: you wonder is extra 9 AP provided by one soul better than 7% critical strike provided by other soul if you have 200 AP, 30% critical chance and 200% critical damage. This is similar example as before, but this time instead of 160% critical damage we have 200%. So, by discussion above we expect now for a critical chance to be more valuable than before.

    Soul 1: 209 AP, 30% cc, 200% cd
    DMG = 209 ∙ C ∙ (1 + 0.3 ∙ (2-1)) = 271.7 C

    Soul 2: 200 AP, 37% cc, 200% cd
    DMG = 200 ∙ C ∙ (1 + 0.37 ∙ (2-1)) = 274 C

    This time around mathematically soul 2 is slightly better and cc is now more valuable than before when our crit damage was only 160%.


    3. SPELL HASTE FORMULA

    Some players pursue spell haste upgrade on their characters to the maximum possible. But, is that always the best course of action? It might be or it might not, depends on what the alternative is. Don't forget that spell haste works only on SOME of your spells, not on all. Some damaging spells are instantly cast and have a longer cooldown and spell haste is irrelevant for those and is not contributing toward their dps. But, double spell haste from souls is one of the best dps modifiers in the game.

    Casting time = basic casting time / 1 + spell haste, or:

    CT = BCT / (1 + SH)

    CT = casting time after spell haste is calculated
    BCT = basic casting time, for example 2.5 seconds for lightning charge
    SH = spell haste, for example 120% we write as 1.2

    The more spell haste you have, the less valuable additional spell haste is. In general, the lower CT is, the better DPS you have. So in general to calculate upgrade of dps what additional spell haste brings us, we calculate:

    (1 + SH + bonus SH) / (1 + SH) - 1 --> percentage our dps increased by additional SH


    Example 3: My character has 180 SP and 80% spell haste. Now I have a choice either to increase my SP by 10 or spell haste by 10%. What could result in a higher DPS?

    Basically we use damage formula first to determine that increasing 180 SP to 190 SP would increase our DPS by 5.6%.

    Second step is to throw values into spell haste formula like this to get increase in dps:

    (1 + 0.8 + 0.1) / (1 + 0.8) - 1 = 0.056 = 5.6%

    Basically 10 SP and 10% spell haste seem to bring similar upgrade. BUT, remember, no character out there has complete DPS done by spells which are boosted by spell haste, so in this case SP has higher dps value than SH would have if we do a field test.


    4. DPS FORMULA


    This is basically combination of damage formula with spell haste formula. This formula will give us expected dps of some spell.

    DPS = SP ∙ C (1 + cc ∙ (cd-1)) ∙ 1/CT

    or

    DPS = SP ∙ C [1 + cc ∙ (cd-1)] ∙ (1 + SH)/BCT

    DPS = damage per second
    SP = spell power
    cc = critical chance
    cd = critical damage
    CT = cast time
    SH = spell haste
    BCT = basic cast time of a spell
    C = additional factor of spell damage, for example fireball has 2.5x SP and you could have talent for +10% elemental damage etc, then you would multiply SP additionally with 1.1


    Example 4: My necro has 180 SP, 80% haste, 35% cc and 160% cd. What is better, to take additional 9 SP and 7% critical strike or additional 10% haste and 5 SP? BCT of fireball is 2.5 seconds and necro has +10% damage talent.

    Even though in both cases BCT and 10% damage talent are the same and it is irrelevant for comparison, we will calculate everything for example sake.

    DPS1 = 189 ∙ 2.5 1.1 ∙ (1 + 0.42∙0.6) ∙ (1 + 0.8)/2.5 = 468.5
    DPS2 = 185 ∙ 2.5 1.1 ∙ (1 + 0.35∙0.6) ∙ (1 + 0.9)/2.5 = 467.8

    It seems option 1 is better. Also since necro damage is not completely dependent on spell haste and necro shield depends on SP, it would mean option 1 is even more juicy than option 2 than what numbers show. Like I said, math can't include all possible factors, but we can include them ourselves to conclude what might be better based on numbers, or better on a field test if we do that. Don't forget there is also -res aura available which does increase your damage output too.


    5. ATTACK SPEED FORMULA

    Identical to spell haste formula:

    AT = BAT / (1 + AS)

    AT = attack time
    BAT = basic attack time, this you would have to test yourself
    AS = attack speed. Don't forget that agility will give you this too based on formula.

    (1 + AS + bonus AS) / (1 + AS) - 1 --> percentage our dps increased by additional AS

    PROBLEM with AS is you really have to know what % of dps of your character actually comes from actual attacks and not spells (where AS is obviously irrelevant). For example, friend tested that nearly maxed zerker does roughly 50-55% of his DPS through attacks while spells are 45-50%. This means that whatever AS you might find for a zerker, only like 55% of it is used to increase your dps. This is determined by a test:

    TEST 1: use only spells to do damage, not a single attack
    TEST 2: use both spells AND autoattacks to do damage like you normally would
    RESULT: compare those results to determine what % of your dps comes from autoattacks. Also include the fact you do have some AS already.

    Question: why not have test with usage of only normal attacks?
    Answer: because realistically when you cast spells, you lose some amount of attacks because during that time you are not attacking at all. This is why testing only autoattacks makes no sense.

    If you realize X% of your dps comes from your autoattacks, you can easily estimate how much of an influence would additional AS bring. This X% depends on your talents and current AS obviously. The more AS you have, the more DPS will be shifted to autoattack side and the less valuable additional AS would be.

    Note: the same way you could determine for example what kind of % of total dps does make lightning charge of a sorc in comparison to his other spells. Result could be in ~80% region. For necro you could determine his fireball makes up to 65-70% of his total dps and so on. This could be useful if you want to go into more detail and more precise calculations by yourself. This might actually be necessary to do if you want to determine influence of % damage souls, like 10% fire damage vs INT soul and so on.


    Example 5: zerker does 55% of damage with autoattacks. Currently he has 40 AS (skills + talents + agility incldued) and I wonder what kind of increase of DPS could I expect with additional 10 AS?

    (1 + 0.4 + 0.1) / (1 + 0.4) - 1 = 0.071 = 7.1%
    This is additional damage if 100% of dps would come from autoattacks. But it does not, only 55% of it comes to dps on 40 AS he has, so total expected increase in dpswould be:
    0.55∙0.071 ~ 0.039 ~ 3.9% additional dps.

    Note: the higher AS you have, the more shifted dps will be toward autoattacks. This increase of DPS is estimation and it would mean your total DPS would shift more toward autoattack side meaning that after you increase your attack by 10 more AS, your test would show autoattacks do closer to 60% of total DPS done now.


    Example 6: after example 5 I would like to know does my zerker get more upgrade by increasing his AP by 9 with a soul or is 10 AS soul better? He has 200 AP total now.

    Now we use simple damage formula, but since cc,cd and everything else is the same, we simply divide upgraded AP with current AP to get increase in DPS:
    209/200 = 1.045 = 4.5%

    Since AP influences complete dps of a zerker and since we calculated in example 5 that 10 additional AS would increase his DPS by ~3.9%, this means AP is expected to give him a slightly better upgrade to his dps than 10 more AS.


    6. DPS IS NOT EVERYTHING

    Most common mistake people do in rpg games like gaias is their obsession with dps like it is the only thing in the universe that matters. Because of that they tend to end up with characters who are lacking in survivability or are lacking mana to do proper battling with their epic dps. Or, they neglect something important (skill or talent) for a team fight in a name of having 5% higher dps. Before turning your character into glass cannon or complete dps machine, ask yourself:

    a) do I have enough mana to sustain slightly longer battles without yelling "out of mana" too soon?
    b) is my character sturdy enough to not die if someone looks at him in a wrong way?
    c) do I have some specific talent or skill that can help my team big time even if it means reducing my dps by 5%?

    Ask yourself questions like that, really. Just because you can do super dps doesn't mean you are the most helpful for a team and battle in general. I still didn't play a single rpg where it was a good idea to go 100% on dps build without sacrificing something more vital to have a better functioning character overall. Obviously you can't turn casters into tanks, but you could consider taking at least 1 HP talent or mana talent if that is what your character really needs.

    Unfortunately calculation is very complicated to determine what kind of mana do you need or HP, those are determined by field test.


    7. RESISTANCES

    When it comes to defensive stats, logic is the other way around than with offensive stats: the more you have, the better upgrade it is to add even more!

    This means that stacking resistances up will greatly reduce damage you take by a spells. Let us assume you increase your resistances by 15%.
    a) you go from 20 to 35% res
    b) you go from 50 to 65% res

    in case a), damage you take is decreased from 80% to 65%, or now you would take 23% less damage from spells.
    in case b), damage you take is decreased from 50% to 35%, or now you would take a whooping 43% less damage because of that!

    In both cases you increase your resistances by 15%, but in a second case it is really a big deal. Moral of the story is, when you stack resistances, stack them up. You might become nearly invincible to elemental damage. Have you ever tried to have near 75% res? You will laugh at spells you feared previously.

    How to achieve high resistances? Bishop (22%) + dual allres soul (14%) + aura from legendary soul (5%) = 41% already, not including anything from a gear, not including potential bard in a team and possible talent to give 10% more maybe. 50% is already high enough territory though. Just be aware that increasing your resistances from 50% to 75% means you take twice less damage from spells and is the same upgrade as going from 0% to 50%. Allres mode is incredibly powerful because of that, use it wisely and make your tanks to be really tanky. Don't be afraid to stack it up, the more you stack, the more powerful it becomes.


    8. EVASION

    Identical story as with resistances. The more you have, the more influence it has. If you somehow get to 50% territory and are evaluating whether to take 10% more evasion vs 10 more armor, be aware that taking 10% more evasion in that case means you will take 25% less hits while 10 more armor might produce less desired protection. Or not, it is up to you. But, be aware that stacking evasion could make your character really tanky against physical hits. Having 75% evasion with monk means you take only every fourth attack.

    Overall, it is up to you to decide since evasion is of rather limited usage in comparison to resistances which are incredibly powerful. But, given true situation, evasion could be just as powerhouse to have. Downside is double allres mod comes with legendary soul while for double evasion mod you need blue soul.


    9. WHAT SOUL MODS TO LOOK FOR

    I'm not going to name all possibly good souls, there are too many of them. In general, you want soul that focuses the need of your character. Flat bonuses are powerhouses for early levels (bf!) and very strong for end-game. Critical strike/critical damage are strong mods too. There are plenty of those, but I'll name some I think are incredibly powerful to have.

    dual allres soul - seriously, whatever third mod is, keep it until you find an upgrade. This soul can easily make your character a supertank in end game content where spells are very dominating part of a game.

    dual haste soul - offensive mod under defensive ones makes this a supermod to get. It is hard to explain how powerful dual haste is until you try it out. Keep this soul whatever third mod is until you find an upgrade.

    dual whatever soul - when you get double mod, first think about its uses, don't just throw soul away if third mod is not really good.


    9. CONCLUSION

    While math can't fully replicate real field situation, it can still give you a very good estimation of what you can expect since above formulas are exactly that - estimation. Best combination is to use both to your advantage, first theorycrafting what kind of soul/gear do you want and then do a field test to see how it works. You can easily plan your soul vault to know what are you after. Sometimes comparing two souls can be a pain, leveling one soul for a test can be annoying thing to do. Still, if you can, do a test field for the most accurate result, but you can also use presented formulas pretty accurately to dismiss obviously inferior souls to pick up a better one.

    If you go a bit deeper into calculation you will realize in Gaias end-game content many souls easily fall in top 5% category with at most 5% difference in dps. To get strongest possible soul for a certain character, that would probably be some kinky blue soul with dual critical damage mod or something similar. But, getting top 3% soul is probably good enough I assume. Also, don't just blindly use numbers, try to realize what your character needs or what do you want to do from your character. Gaias is greatly balanced and you can basically pick 5 different souls with a similar results, each focusing character to something else.

    Hopefully someone will find this thread helpful to plan his own characters. Thank you for reading. Feel free to ask or comment anything.
     
    Last edited: Jan 23, 2017
  2. Zwiebelchen

    Zwiebelchen

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    Nice thread! It's great to see some theorycrafting on stats, because that also helps me on my stat balancing. For the soul stats I pretty much used a similar logic when planning out their modifiers as you did.

    A problem in your AP AS comparison though: armor is a problem. The more armor a mob has, the more important raw AP Is over percentage modifiers. Basicly you want slow heavy hits over smaller fast hits.

    This is why ArP is such an important stat for assassins, bards and hunters.
     
  3. Gripphon

    Gripphon

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    Thank you. :)
    Yeah. There are a lot of things to consider when doing calculations. Thank you for pointing armor thing out, I'll add it to post later.