WARNING: Below this paragraph there is Math. Math is scary, so you may want to look away if you're afraid of that sort of thing.

To start, there's actually 5 different damage formulas, ranging from slightly different to drastically different.

For player characters using a weapon or a power-based Persona skill, the below damage formula is used.


Where

• "Skill Power" is the base power value of the skill being used, God's Hand is 800 for example. For a weapon attack, this number if half of the weapon's "Attack" stat.
  
• "Offense" is Strength if using a weapon attack or a Slash, Strike, or Pierce Persona skill, and otherwise "Offense" is Magic.
  
• "Affinity" is 1 for enemies that are Neutral to the attack (very rarely it can be 2 instead, such as when using Fire against Priestess, and in exactly 1 fight can be 2.25), 0.5 for enemies that Resist the attack (can also be 0.25 depending on the enemy), or 1.25 for enemies that are Weak to the attack.
  
• "Level Difference" is taken from the following table, based on how much higher the attacker's level is than the defender's level.
  
  
  Level Advantage

  Damage Multiplier

  -13

  0.5

  -12

  0.51

  -11

  0.53

  -10

  0.59

  -9

  0.66

  -8

  0.75

  -7

  0.84

  -6

  0.91

  -5

  0.97

  -4

  0.99

  -3

  1.0

  -2

  1.0

  -1

  1.0

  0

  1.0

  1

  1.01

  2

  1.03

  3

  1.09

  4

  1.16

  5

  1.25

  6

  1.34

  7

  1.41

  8

  1.47

  9

  1.49

  10

  1.5
  During boss fights, the multiplier from level difference will never go below 1 for either party.
  
  However I mentioned that there are other damage formulas than that, to knock out 2 more quickly, here's how damage is calculated for enemies
  
  
  (√(Skill Power x 6 x Offense / (8 x En + Armor)) x 9 x Level Difference - 10) x Affinity Rounded toward 0
  
  Where (same as above unless stated otherwise)
  
  • "Skill Power" is not halved for basic attacks, unlike the player version
    
  • "Armor" is the armor value on the defending player's armor (in the rare case where an enemy attacks another enemy or itself, it is instead 10)
    
  • the "- 10" near the end is removed if attacking with a regular attack. (which is where the 3rd formula comes from)
  
  The 4th damage formula is used for All-Out Attacks, and is calculated as
  
  
  √(Skill Power x 15 x St / En) x 1.6 x Level Difference^2 x Affinity x Participant Count Rounded toward 0
  
  Where
  
  • "Skill Power" is half of the Attack of the main character's weapon
    
  • "Affinity" is the defending character's Almighty resistance (yes there are cases where this isn't just 1)
    
  • Participant Count is the number of party members able to participate in the All-Out Attack
  
  Finally, there's Theurgy damage, which is calculated as
  
  
  √(Skill Power x 10 x (Level + 20) x (St + Ma) / En) x 0.7 x Level Difference x Affinity Rounded toward 0
  
  Where
  
  • "Level" refers to just the attacker's level, and not the level difference used later
  
  If you're wondering about Takaya's Theurgy, that's calculated the same way an enemy using a normal skill is calculated, as would any other Theurgy if used by an enemy.

Once you have obtained your base damage value from the previous section. You can now begin applying several adjustments to that damage based off of various effects that can apply. The first, and perhaps most obvious one, are the variety of passive skills that you can use to increase damage, which I will list below (some skills, like Raging Tiger, are not listed for reasons I will explain later)

• When combining these skills, simply multiply the multipliers together (so 2 1.5x multipliers becomes 1 2.25x multiplier).
  
• If a skill is on both Persona and Equipment, it is only applied once.
  
• If the final multiplier is greater than 5.0, it is instead set to 5.0 (this is why I left out some passive skills).

There's several other multipliers that apply to damage beyond passive skills. I'll start with "game rule modifiers" (which is to say I don't know how to categorize these). This includes


Your difficulty setting introduces these multipliers


Some multipliers are specific to All-Out Attacks


Buffs (and Debuffs) use the below multipliers


A number of multipliers are applied depending on what ailments are on the attacker or defender


Equipment that says "Reduce <element> dmg" will multiply damage taken by (different tiers stack)


The final multiplier I wanted to mention is the moon cycle multiplier used by the skills Zan-ei and Getsu-ei. Those multipliers are (listed by days until full moon)



After applying all multipliers so far, clamp the value to a range from 1 to 99,999, then round towards 0.

At this point the only step left is to apply damage ranges. To do this, the game multiplies damage value by (a random whole number from 95 to 105) / 100, set the value to 1 if it would be lower, then round towards 0 one last time to get the final value seen in game.

Accuracy is calculated differently for players and enemies, when a player is attacking, it is calculated with


When an enemy is attacking, instead take the formula above and multiply it by


Round this base value down to the nearest whole number

Sukukaja and Sukunda also affect accuracy, multiplying it by


The following effects will also multiply accuracy.


Equipment that says "<element> evasion" will subtract a set value from the hit rate produced up to this point (stacks unless same description)


The new value is then clamped to be within a range of 50 to 99 and rounded towards 0, then the accuracy check is performed.
The following conditions will cause the accuracy formula to be bypassed altogether

• Skill has 100 base hit
  
• Affinity is Null, Repel, or Drain
  
• Target is Down or Dizzy
  
• Target has Distress, Freeze, Shock, or Overheat
  
• Target has the skill Firm Stance

The vast majority of skills in the game hit exactly once. Among the skills that hit more than once, a number of them hit a fixed number of times. However there are 9 skills that hit a variable number of times, and as you might expect the odds that they hit a certain number of times are needlessly complicated.

The odds actually depend on the buff stage for the attacker's accuracy/evasion (so Sukukaja = +1, Sukunda = -1). At +0 the odds of any 1 hit count is calculated as
Where max hits is the most hits the skill can give, and min hits is the least hits the skill can give. In this scenario, all possible hit counts are about equally likely.

If the attacker has -1 Accuracy and/or Evasion (so Sukunda), then the odds are instead
Where t is the number of hits expected. In this scenario, lower hit counts will be more likely to occur to a significant degree, reducing the average hit count.

If the attacker has +1 Accuracy and/or Evasion (so Sukukaja), then the odds are instead
In this scenario, higher hit counts will be more likely to occur to a significant degree, increasing the average hit count. In fact the distribution of odds for each possible hit count should be exactly mirrored from the values at -1 Accuracy and/or Evasion (so if Sukunda gives a 50% chance to hit 2 times on a 2-5 hit skill, Sukukaja gives a 50% chance to hit 5 times).

Additionally, I've included a few cheat sheets for the 9 skills this property affects in the first place.

Average Hit Count

Approximate Average Skill Power (not usable in exact calculations, but useful for comparing skills quickly)

Odds of getting X hits at
-1 ACC/EVA
+0 ACC/EVA
+1 ACC/EVA

Landing a critical hit is only possible if the skill is Slash, Strike, or Pierce skill that does power-based damage (meaning it's not a fixed number or percentage). Upon attempting to crit, the base crit rate will be calculated as

"Skill Crit" is 3 for all basic attacks that can crit.
Multiply by 0.8 if the attacker is an enemy.
Round the this base value towards 0.

The following passive skills multiply the base crit rate


Note that the above modifiers are the only ones that multiply the base crit rate at all, all others only add fixed values to it after applying multipliers.

The following passive skills add a value to crit rate after applying multipliers


There are 2 buffs (that stack with each other) that also add to crit rate


If using a basic attack with a weapon that says "+Critical rate up", then the final crit rate will be increased by


And finally, if the attack is directed at a target with one of these ailments, the final crit rate will be increased by


Round the final value towards 0.

The following additional effects also apply to crit

Whenever attempting to inflict an ailment (excluding Dizzy), the success rate is calculated as
Rounded towards zero

Where

• Affinity is the Defender's Resistance/Weakness to the element that the ailment corresponds to.
  
• Enemy Penalty is 0.8 if the attacker is an enemy, otherwise it's 1
  
• Difficulty setting is a difficulty-based modifier to ailment rates.

These are the difficulty modifiers


And these are the elements that each ailment corresponds to

To get into ailment resistances, they are substantially more complicated than they have any right to be. Some "Resist <ailment>" skills directly modify the affinity the ailment corresponds to (Resist Charm, for example), while others are multipliers applied after the fact (Resist Freeze). Additionally, if the skill is an example of the former, then the equipment that adds that skill will still be an example of the latter, and also stack with the persona skill.

The distribution of those is as follows, with "Affinity" meaning that it sets the ailment's affinity to "Resist" (just like Resist Fire, for example), while "Multiplier" means it is applied separately from resistances (much like a "Boost" skill would, but in reverse).


These multipliers also affect ailment rates

After applying all multipliers, cap at 99% and round towards 0.

Dizzy operates entirely differently from other ailments. Instead of using a formula, Dizzy just uses flat rates depending on the type of skill used to damage a Downed enemy


and then applies these multipliers (all mentioned stack with each other)

and round towards zero.

For skills that can inflict multiple ailments (such as Trickster), they actually start by randomly selecting one of the possible ailments (all possible are equally likely), then try to inflict that one. This selection process does not consider immunities.

Each ailment has a minimum and maximum turn count, as listed below


At the start of each turn, each ailment's timer will increment by 1 (or 2 if Fast Heal is equipped and the ailment is not Orgia Mode or Overheat). If the number is below the minimum turn count, then the ailment will persist. If the number is at or above the maximum turn count, the ailment will end. If the number is in-between, it will calculate a chance of recovering with

rounded towards 0

Where

• All parameters are taken from the character with the ailment, the character who afflicted it is irrelevant
  
• Affinity refers to the natural ailment resistance (which is to say it's not affected by equipment, but is affected by Persona skills)
  
• Mod is a value specific to each ailment with a variable duration

The "Mod" values for each ailment are

Start with 2 variables.

The 1st is the skill's healing power

And the 2nd is a value based on your Magic stat

then calculate your base healing power as


Multiply this value by 1.5 if Divine Grace is equipped, and round towards 0 again.
Finally, apply a damage range to the healing value, which is any whole percentage from 95% to 105% selected at random. Round towards 0 one final time for the final healing value.

The amount of experience a party member needs to level up is technically arbitrary, but it's almost always aligned with a concise formula. The total amount of EXP needed to reach a certain level can be calculated as
Where Target is the level the character should be at with the resulting amount of EXP.

The first 5 levels on Makoto, Yukari, and Junpei are an exception to this rule. Those are at the following EXP thresholds


For a Persona, you can instead use the following formula to calculate total EXP
Where

• Target is the level the Persona reaches with the resulting amount of EXP
  
• Source is the level the Persona starts at

In boss fights, turn order is fairly simple. It's determined by the order you added the party member to your own party (with Makoto always going first for obvious reasons), then enemies will take their turns, selecting their order in a similar way.

As for normal battles (including Tartarus Guardians and the like), the process used to determine turn order is needlessly convoluted (what else is new).

To start, each party (so the player's group and the enemy's group) is sorted in descending order by Agility.
Same Agility party members being sorted among themselves in descending order by Level.
Party members that are also the same level are sorted among themselves in ascending order by ID.

with the following IDs for party members

If "Advantage" occurs for either party, then the advantage party all takes their turn before the disadvantage party for the 1st turn cycle.
Otherwise, the game does a loop with the first member of each party (remember the sort order). In this loop it takes the player character's speed, subtracts the enemy character's speed from it, then uses the difference to determine the odds the player character gets added next.

However, the game also keeps track of a "streak" of consecutive player characters added. and picks the rate to add player characters at from a different table based on the size of the streak. The streak is reset to 0 upon an enemy being added to the order list. The different chances for a player character to be added next are listed below, grouped by agility difference between the player and the enemy, and by the streak counter.

Once a character gets added to the turn order list, they are removed from their respective party's list, essentially moving the next character up in line.

As for "when" turn order is decided, while I can't claim with certainty that this is the case (yet), it appears to only be up to twice. One time at the start of the fight, and another at the end of the first cycle. The second time turn order is calculated will only occur if the initial turn order was determined by "Advantage." Turn order does not appear to be recalculated in response to Makoto's agility updating. I regrettably do not (yet) know how it responds to mid-battle summons.

Let me know if I made any errors, or if there's any missing information I should include here (unless it's turn order IK I'm trying to find it I found turn order and have a basic understanding of it, feel free to ask now).

Here's a spreadsheet I put together with most of the data tables a player may find relevant
https://docs.google.com/spreadsheets/d/1f4Bhkg0gBqUSm4YR1Bba1JsQn6w6fPDrqBTaWlHla8A/edit?usp=sharing

And here's the equivalent sheet for Episode Aigis
https://docs.google.com/spreadsheets/d/1BZA_ce8oyDApezEoDW0mYeV6okerur6WPaIX9RXdeWs/edit?usp=sharing