EO3 damage Formula Analysis

Etrian Odyssey 3 is a game that many people have noticed has a weird issue with the scaling for magic attacks compared to physical attacks, Where players will find Zodiacs start to fall off in the later 4th or so of the game, while physical classes like gladiators just seem to only get better and better. I’ve always had a passing interest in things like this, so I’m putting together this doc to actually look at it more in depth than I have before, and get practice pulling apart stuff like this.

The stats we are mainly going to look at, are fairly traditional RPG stats, STRength, VITality and TEChnique(I think idk what TEC actually means lol) and ATK. These stats range from 1-99 with the exception of ATK, which ranges from 1-297.

The way damage is calculated is by finding a “base damage” through two separate formulas, depending on if the move scales on STR or TEC, and then plugging any modifiers after this. Modifiers will include passive damage increases, buffs and debuffs, weapon modifiers, and any weaknesses the monster/player will have.

The rest of this doc, until the end, is going to be a lot of math written out, if you want to see only the conclusion, feel free to skip ahead. This is a lot of self indulgent writing here.

        The STR base damage formula is as follows;

        (aATK-dDEF)* (sqrt(aSTR/dVIT)

Here, the formula diverges a little based on if the monster is the aggressor or the player is, but we will only look at the player as an aggressor here.

        aATK is only the weapons attack value
        dDEF is the monsters VIT stat.

        aSTR and dVIT are self explanatory here.

What this formula turns out to do is take the equipped weapons ATK value, lower it through the monsters defense value, and then multiply it after by a very small value. Due to double dipping on both dividing the attackers STR by the defenders vit and then taking the square root of that, we end up with a number in the range of 0.8 - 1.3, its not until we start getting large differences that we start leaving the 0.9 - 1.1 range.

Lets pull up an example of how these stats interact with each other that we can use for later examples too. Our gladiator, Sonya, wants to upgrade from the Sentence Hammer(175 ATK, 3 upgrade slots which increase his str by 1 each), to the Ukonvasara(202 ATK no upgrade slots) she is level 80, has 70 base STR and is hitting a Vampire Tree with 60 VIT.

Plugging these numbers into the STR base damage formula with the Sentencing Hammer gives us

(175-60)(sqrt(73/60)

(115)(sqrt(1.21)

115*1.103

126.848 base damage.

The ukonvasara will lower our STR by 3, but increase our aATK to 202.

(202-60)(sqrt(70/60))

(142)(sqrt(1.166)

1.42*1.080

153.377

As we can see, the weapons attack has a huge effect on the formula, while the additional STR from the first hammer only increased damage by about 3%, the ukonvasara increased our damage by about 21%, even accounting for the lost strength. This is primarily due to the enemy's VIT stat creating a sort of damage threshold and a multiplier at the end. That said, the interaction between STR and VIT is fairly nonconsequential, because in normal gameplay, the difference between STR and VIT will trend towards differences of about 25% at max. If we compared 99 str to 75 vit we end up with 1.32 which gets further lowered to 1.148, a 15% multiplier isn't that much!

        Onto the TEC base damage formula which is as follows;

        (aATK-dDEF)(sqrt(aTEC/dTEC)

        aATK is the the attackers TEC multiplied by 4(4*TEC)
        dDEF is still the monsters VIT

        The first thing to note, is that if you are doing magic attacks in EO3, you don’t use your weapons attack stat AT ALL! This means that you could potentially not even upgrade despite spending a lot of money on a new weapon, this is something that will eventually change, but it's one of the more notable parts of earlier Etrian Odyssey, mages don't need a lot of money to work, and their early upgrades matter the most.

        The next difference to note, is that the monster's Vit stat is only used once, in the first part of the formula acting as a defense threshold type stat. The monsters TEC stat is instead used as a defensive stat, while this might not change much, enemies tend to have more of their offensive stats. This also leads into a somewhat interesting interaction later in this document.

        Lets go through another example, this time looking at Hira, our zodiac. She wants to upgrade from Castor’s Records(24 ATK, +2 TEC+10%TP), to the strongest tome in the game by ATK, the Origin Tablets(148 ATK, +4TEC+10% TP). She has 77 TEC at base, and again we are going to beat up a Vampire Tree which has 60 VIT and 66 TEC.

        With Castors Records first we get +2 TEC giving us 79 TEC;

(79*4-60)(sqrt(79/66))

(256)(sqrt(1.197)

(256(1.094)

280.064

The Origin tablets will increase our TEC by 4, leading us to have 81 TEC

(81*4-60)(sqrt(81/66))

(264)(sqrt(1.227)

(264(1.107)

292.4

        We get an increase of… about 4%. Keep in mind that the Castor’s Records we got earlier is a weapon you get for free, and at the beginning stretch of the game(you get it at about level 8!). The other major thing to notice is that the base damage for Hira is double Sonya’s, even if Sonya fired an attack out of the strongest weapon in the game, the base damage would only be 178.2! So why is it that Zodiacs tend to fall off later in the game if they are working off a much better base than other classes tend to?

        The answer is… everything else AFTER we do the base damage. For the next stretch of examples, we are going to apply the rest of the modifiers to Sonya and Hira’s attempts to kill a vampire tree. Starting with the passive buffs from the skill tree you can learn. Also I will not be going into subclasses, to keep things a bit more simple for me, there will be some outside support from party members though.

        To start, lets reiterate the base damages we are working with

        Sonya with the Ukonvasara:153.7
        Hira with the Origin Tablets: 292.4

        We will be plugging these base damages into this formula;

        (B)(W)(P)(B)(C)

        Base damage, Weakness, Passives, Buffs/debuffs, Charge

        Essentially, every extra damage boost multiplies the base damage, and within themselves are also multiplicative.

        The available passive boosts to damage in gladiators skill tree are:

        Endless battle(25%)

        Club mastery(11%)

        The available boosts in the Zodiac tree are:

        ICE/FIRE/VOLT mastery(11%)

        Ether mastery(25%)

        Singularity(50% if hitting a weakness)

        These multipliers are all multiplicative with each other, so we are going to end up with

        Sonya:(153.7)(1.11)(1.25)=211.3

        Hira:(292.4)(1.11)(1.25)(1.5)= 608.5

        Now for the two major differences between the two stats, the skills which you use, and after the buffs which we can place on the two. Also we will note that the vampire tree has a 1.25x weakness to fire, and is neutral to all other elements.

        Sonya is going to use a level 10 Freezing blow, which has a damage modifier of 200%

        Hira is going to use a level 10 Fire star, which has a damage modifier of 80%

        Sonya(211.3)(2)=422.6

        Hira(608.5)(0.8)(1.25)= 608.5

        Already, Sonya has started to catch up, while Hira’s damage has not moved at all.

        Now lets look at some buffs the gladiator and zodiac can apply to their own damage.

        Sonya gets two buffs that are commonly used, berserkers vow(90% increase) damage, and charge(160% damage increase!).

Hira gets only one really usable buff, Etheric charge(200% damage increase)

Sonya(422.6)(1.9)(2.6) = 2087.6

Hira(608.5)(3)= 1825.5

The two numbers aren’t that different, but the multipliers are Sonya is getting close to a 5x multiplier, while Hira is only getting a 3x multiplier here.

Lets do one last example, this time with more buffs and debuffs in the mix

Charge tactic is a limit skill which increases all damage by 25%

Attack Order will increase physical damage by 45%

Primal drums(debuff) will increase physical damage taken by 30%
        Eagle eye(debuff) will increase physical damage taken by 35%

        Berserker vow will increase physical damage by 90%
        

        There is a buff/debuff slot limit of 3, but charge skills will not take up a buff slot so they can be added too.

        Sonya(422.6)(1.9)(1.45)(2.6)(1.25)(1.35)(1.3) = 6640.6
        Hira(608.5)(3)(1.25)=2281.8

        Almost a 3x difference between the two! Not to mention freezing blow can hit some enemies weakness which would add another multiplier that Sonya already didn’t have going into this because vampire trees only have a fire weakness! We also are using charge tactic, notably a weak buff(25% more damage), when we could be adding a bravant instead for Sonya adding a 45% damage increase instead.


        One important thing though, to finalize this, is that there is a modifier we aren’t taking advantage of for our zodiac, which is head binds. Binds in the Etrian odyssey series are a type of ailment which will reduce a stat by half, and also prevent specific moves from firing off, and head binds will reduce the TEC stat of whoever is bound. Lets redo Hiras base damage and take this into account!

(81*4-60)(sqrt(81/33))

(264)(sqrt(2.454)

(264(1.566)

413.6

        Hiras base damage went up from 292.4 to 413.6! About a 33% increase just from a head bind! Unfortunately, ailments in eo3 tend to be a bit hard to hit and also due to accumulative resistance, hitting more than 2 in a given fight is fairly hard. So you better make the head bind count.

        For the curious, Hiras new damage against the vampire tree is;

        413.6(0.8)(1.25)(1.11)(1.25)(1.5)(3)(1.25)=3228.

Now, we are missing out on some stuff, mostly for our gladiator, this is because most of the other classes we can throw on our characters are physical. Our example also compared freezing blow, a skill you may viably choose to use in some endgame encounters due to it abusing ice weaknesses, while also using physical modifiers, but also she could just use a stronger all around skill from a subclass. The Arbalest skill, Front Mortar, has a 382% modifier at max level, almost double freezing blow or gladiator skill blade rave which does 4-5 hits of 95% damage, albeit with a sword which universally have less damage(though the best sword can get atk forges which is another scaling factor actually).

         While zodiacs COULD use meteor(5-6 hits of 50% TEC bash damage), it almost requires a subclass in order to use effectively because since its not an elemental attack, it will rarely proc that extremely powerful singularity passive, and also not take advantage of the other innate passives or the extremely powerful etheric charge making a gladiator subclass almost mandatory for charge and optionally berserker vow. Important to mention too, is that the hardest postgame bosses, where this issue of scaling comes into play the most, have upwards of 50% or even 100% weakness to elemental damage!

        STR skills are generally just better to begin with, which more than makes up for the users bad base damage relative to most TEC skills, and then also get so much more support. This isn’t to say that zodiacs are bad, they are still good at contributing supporting damage(especially with the elemental limits which are, iirc, 3-4 hits of 60% TEC damage), but on a turn to turn basis physical characters will burst just as or harder, while also busting these attacks out without being rate limited by the limit gauge. Zodiacs, due to how they have less support, also shine more in situations where the player does not have access to these very powerful buff stacking strategies, most notably the earlier portions of the game.