Battle Arena Statistics

Each player has a number of statistics in the Battle Arena that determine his or her fighting abilities. Understanding these is vital to success in fighting monsters.
Current guide version: 1.828
==Experience-Related Statistics==
===Arena EXP===
By defeating monsters and completing Missions, the character gains Arena EXP. This is separate from a player's total EXP in Anti-Idle: The Game.
===Rank===
A character's Rank increases when enough Arena EXP is gained. All characters start at rank 1. The maximum rank is 500. Increasing one's rank grants the character 20 skill points and a minor increase to their major statistics. At rank 500, another 2500 more Skill Points can be looted from monsters, up to the maximum of 12500 Skill Points. Rank affects what equipment a player can use. (Note: Extra 2,500 Skill Points do NOT get reset when Rebirthing). 
Reaching a new rank provides sizeable bonuses to your stats (see table in next section) as well as a reward of pixels (100 * rank^2) and crafting materials (10 * rank^2).
Additionally, your rank and Anti-Idle level play a role in damage formulas. Your Anti-Idle level in particular determines whether you receive major penalties, which are applied against opponents higher level than your Anti-Idle level.
==Major Statistics==
Certain loot drops (red circles or Rings) provide permanent increases to these stats.
The player may increase Attack, Defence, HP, or MP by paying a number of Pixels on the Stats screen, so long as that statistic does not exceed its Soft Cap. The cost of increasing these statistics with Pixels increases as the statistic increases. After the soft cap is reached, the stats can be further increased by looting Stat drops from monsters.
The base value of each stat may not exceed the Hard Cap listed.
Name 	Effect 	Growth/Rank 	Soft Cap	Hard Cap	Modifier cap
Attack 	Higher damage dealt 	25 + Rank * 0.5	65,000 	90,000 	
Defense 	Less damage taken 	25 + Rank * 0.5	65,000 	90,000 	
HP 	Can survive stronger attacks
faster HP recharge	250 + rank * 5 	650,000 	900,000 	19,999,999
MP 	Can cast more skills
faster MP recharge	100 + rank * 2 	260,000 	360,000 	19,999,999
Accuracy	Higher chance to hit enemies 	10 + 0.2 * rank	N/A 	18,000 	
Evasion 	Higher chance to dodge an attack	10 + 0.2 * rank	N/A 	18,000 	
Modifier cap refers to the base stat multiplied by all modifiers, i.e. it is impossible by any means to exceed these values in game. Only HP and MP have such a cap.
The four basic stats can be viewed at the top of the first page of the stats window in-game. Your character's base value is given in parenthesis, while the modified values are given outside the parentheses.
; Attack
: Attack, also labeled as Attack Power, is directly proportional to the maximum damage stat, viewable at the top of the second page in the stats window. This means doubling your attack power is similar to doubling the damage you deal. Attack is not to be confused with Equip ATT, which is the attack stat on your equipment (see damage formula).
; Defense
: Defense, or Defense Power, is a base statistic that is proportional to the player's flat Damage Resist, along with other multipliers. Damage Resist is subtracted from all incoming damage you receive. It's not to be confused with Equip DEF, which is the defense stat on your equipment.
; Accuracy
: Accuracy is divided into a percentage value and a numerical value in the in-game stat window. The percentage value is the player's hit chance % and represents a roll for a guaranteed hit on an enemy. If this roll fails, the player rolls a second time on the numerical value, their accuracy. This check is given by a { (Accuracy / Division) - 0.1 }% chance to hit. Only one of these rolls needs to succeed in order for your attack to hit. There is also a third check related to your hit penalty, which is described more in the formulas below. 
: Additionally, every 1000 Accuracy increases critical chance by 1%. In exchange, the hit chance % will hit enemies regardless of their evasion, so when available, it's more useful against higher level opponents.
: Enemies only have an Accuracy stat; there is no Hit Chance for monsters.
; Evasion
: Similar to accuracy, evasion is divided into a percent value and a numerical value. The percent value is the percent chance for a guaranteed dodge, while the Evasion is divided through the enemy's accuracy followed by a minus 0.1.
: Enemies need to pass both your dodge % and evasion checks in order to land a hit. This double barrier makes evasion extremely efficient. However, after an enemy misses 9 times in a fight, the tenth hit is guaranteed to succeed, and then the pattern resets.
: Enemies have no equivalent dodge %, only an Evasion stat.
: 
== Intermediate Statistics ==
Intermediate Statistics are derived from other sources than your character's base stats, typically from equipment, or otherwise are not evolving stats that increase as your character plays through the Battle Arena or ranks up. All of these stats are on the bottom half of the first page or on the second page of the stats window. Here is a list:
Name 	Effect
Elements 	Elements provide a damage multiplier on enemies of the opposite element. A 100% element increases Max Damage by 100%. Having the same element does not carry any negative effects; the enemy will not "resist" your attacks because of being the same element.
Max. Damage 	Maximum Damage is the initial damage output based on Attack Power, Equip ATT, Rage Attack, Elements, and rank/level factors. The enemy's defense is subtracted from this value, before other modifiers like crit damage are multiplied.
Damage Resist 	The flat value here is subtracted from incoming enemy damage. Then the Damage Resist % value reduces the difference by its percent.
Mastery (Min. Damage)	Determines your minimum damage as a percentage of maximum damage, e.g., 10% mastery means your minimum damage is 10% of your maximum damage. The base value for Mastery is 5%, and it's capped at 95%.
Equip ATT 	Equip ATT is the sum of all the attack stats on your equipped items. This is multiplied by your attack power and other stats in order to derive min/max damage.
Equip DEF 	Equip DEF is the sum of all the defense stats on your equipped items. This is multiplied by your Defense Power and other stats in order to derive your flat Damage Resist, which is subtracted from the enemy's incoming attacks. Furthermore, every 100 Equip DEF increases Damage Resist % by 1 percentage.
Attack Speed 	Reduces delay between attacks/abilities. The base delay of every skill may be divided by your attack speed to determine the frame value in game (which is given when you mouse over skill tooltips). Attack Speed caps at 25, and excess Attack Speed is converted to Double Hit chance at a rate of 1 Attack Speed = 1% Double Hit.
Double Hit 	Double Hit Chance. A Double Hit means a single attack will deal its damage twice. The maximum value is 95%.
Ignore DEF 	Ignore DEF is a percentage statistic that removes the same percentage from the enemy's defense stat. If the enemy's defense is equal to your incoming damage, your damage can be reduced to 0 (or something like 5-6).
Critical Chance 	Critical Hit Chance. The chance your attack will deal extra damage (see Critical Hit Damage). In Worst Moon Mode, your attacks deal 5% of their normal damage, and you must land a critical hit in order to deal normal damage. Caps at 95%.
Critical Damage 	Critical Hit Damage. The multiplier used when you land a critical strike. This caps at 1000%.
Instant Kill 	Instant Kill is a percentage chance to instantly kill non-boss enemies below half health whenever you damage them. A high enough Instant Kill effectively cuts non-boss enemy in half.
Damage % 	Damage % can refer to boss or non-boss damage. These multiply your damage against the corresponding enemy types.
Rage Depletion 	Reduces the rate of Rage Depletion. Rage accumulates through taking damage or some skills and ranges from 0% to 100%. This percentage of Rage is the percentage of Rage Attack and Rage Speed stats being applied to your character.
Rage Attack 	Rage Attack is a multiplier onto your damage that grows as your Rage accumulates. At 100% Rage, your attacks' damage is multiplied by the Rage Attack value.
Rage Speed 	Rage Speed multiplies your Attack Speed up to a cap of 25 Attack Speed. Excess Attack Speed is converted to Double Hit chance, up to 95%, after which it's useless.
Reflect 	Reflect Damage when attacked. Regular Reflect activates automatically when an enemy bumps into you, while magic reflect activates on ranged attacks.
Magic Resist 	Magic Resist is a % stat that reduces incoming ranged magic attacks by that %. It's basically a Damage Resist % for ranged attacks.
Negate Effect 	Negate Effect diminishes the effects and duration of enemy status effects on you.
==Other Statistics==
There are yet more statistics on the 3rd page of the stats window. These are listed and discussed below. Status effects have a grace period of 0.5 seconds after expiring before they can be reapplied.
Some tertiary effects like drowning or slow, which are not listed in the stats window, are also provided here.
Name 	Effect
HP / 1 Sec 	HP Regen each second
MP / 1 Sec 	MP Regen each second
Drain 	Allows you to drain some damage to turn into HP when attacked. This stat also reduces damage taken while you are at max health.
Autosteal 	Allows you to steal Coins and Pixels based on Autosteal level. The total amount P is equal to your Autosteal level multiplied by the enemy's Pixel value. You steal a part of this value with each hit in proportion to the enemy's max health, reaching 100% of P when you kill the target.
Potion Efficiency	Every 10% to Potion Efficiency gives .1 seconds more to the duration
of a Potion (e.g 120% = 1.2 seconds instead of 1 second)
Poison Chance 	Chance of inflicting Poison when attacking. Poison deals damage every 0.5 seconds over its duration. The poison status effect is symbolized by a green water drop next to the afflicted target. Max 100%.
Poison Damage 	Damage dealt by Poison. The value is a percentage of your character's damage. Max 5000%. The default poison effect from enemies deals 20% of your health per interval, but its damage is capped at 1,000,000.
Blind Chance 	Chance of inflicting Blind when attacking. Blind reduces an enemy's Accuracy and Evasion by the -% listed under Blind Effect. Blind is portrayed by a pink question mark next to afflicted targets. Max 100%.
Blind Effect 	Accuracy and Evasion reduction by Blind and Blind Duration. Max 70%. The default effect from monsters applying poison on you is to reduce Accuracy and Evasion by 40%; Hit Chance % and Dodge % by 20% (the latter are both multiplicative).
Weaken Chance 	Chance of inflicting Weaken when attacking. Weaken lowers an enemy's Att/Def by the -% listed under Weaken Effect. Weaken is portrayed by a purple water drop next to afflicted targets. Max 100%.
Weaken Effect 	Attack and Defense reduction by Weaken and Weaken Duration. Max 70%. The default from monsters applying Weaken on you is a 40% reduction to attack, damage resist, and a 20% reduction to your Damage Resist %.
Stun Chance 	Chance of inflicting Stun when attacking. Max 100%. Stunned units have a black square with a curvy gray X inside. There is a 0.5 second grace period after a stun, during which a formerly stunned target cannot be stunned again. A permanently stunned state via repeat stunning does not exist in this game.
Slow 	While Slow is not a player stat, since you cannot afflict it yourself (except an equivalent effect with the Chrono Ring), enemy monsters will use slow. Slow reduces your attack speed significantly.
Potion Block 	The last status effect. Like Slow, this is not relevant for the player since they cannot inflict this status effect. When enemies afflict it onto the player, the player may not use any potions for its duration.
Drown 	Drown is a zone effect that damages you at set intervals, for example on the Mystic Path or in the underwater room of the Prehistoric Mission. The damage is equal to:
 0.2 * \text{OxygenPenalty}^2 - 10 * \text{OxygenPenalty} + 120
In the Endless Dungeon, this is simply
 \left ( \text{OxygenPenalty} - 30 \right ) * 900 
OxygenPenalty begins at 1 and increases by 1 every interval. You mitigate drowning damage at 1/3 your Damage Resist and 1/3 your Damage Resist %, using the formula for DamageTaken given in the formulas below.
== Stat derivations ==
This section describes how the battle arena statistics are related to one another, and what their overall contribution is to your character's performance.
== Damage Formulas ==
Damage is calculated in a series of steps.
# Total Attack Power is derived from all modifiers
# The Max Damage stat is calculated from Attack Power along with other damage modifiers
# The minimum damage is determined from Max damage
# Damage Dealt is derived, by combining min/max damage with the enemies defense and zone modifiers
# Damage penalties are multiplied in
# Remaining multipliers like boss damage and critical hits are accounted for to yield the actual damage in-game.
Numerous clauses are included within each step that modify these variables, such as Worst Moon mode, being on a hard/impossible ascension, various exceptions for certain enemies, skills, or zones, and so forth. It's not practically possible to present every detail behind the formulas here.
The aim of this section is therefore to provide a general picture of what the game is doing and how certain stats are factored in. The exceptions are usually clear when they're encountered.
=== Total Attack Power ===
Total Attack Power begins with your base attack power, which is visible in the first page of the stats window in-game, inside the parenthesed value. This is modified accordingly:
\begin{align} \text{Attack Power} = & \left ( \text{Base Attack Power} + \text{purple bonus attack power} \right ) * \\
& * \ \left ( \text{Skill 1} + \text{Skill 2} + \text{Career} + \text{Ally Attack} + \text{orange Bonus Attack} \right ) \end{align}
where Skill 1 and Skill 2 are Strength Enhancement and Invisible Weapon (the percentage part) from the Str/Dex tab in your skills. Career refers to the bonuses from your fighter career level. Ally Attack is the attack bonus from an invisible ally, should that ally provide one. Purple bonus attack power is the purple-texted flat attack power bonus on equipment, while orange bonus attack is the orange-texted attack% bonus.
The first term in the equation sums all the flat attack power values together, and the second term sums all the attack % values together. The product of the two terms is your Attack Power.
Then,
\text{Total Attack Power} = \text{Attack Power} * ( 1 + \text{Combo} )
is your final attack power, where combo is derived from the skill "Combo Power" that increases attack with combos. Your final attack power is the value in your stats window outside the parentheses.
=== Max Damage and Minimum Damage ===
Max damage is constructed by adding Attack Power to the product of the following factors:
* \left ( 7 * 10^{-6} * \sqrt{\text{Anti-Idle Level}} * \sqrt {\text{Arena Rank}} \right ) + 0.01 
* Total Attack Power
* Weapon Attack (i.e. equipment attack or Equip ATT)
* Rage Attack
* Elemental bonuses
in equation form:
 \begin{align} \text{Max Damage} = & \text{Total Attack Power} * \text{Equip ATT} * \text{Rage Attack} * \text{Elemental Bonus} * \\
& * \ \left [ \left ( 7 * 10^{-6} * \sqrt{\text{Anti-Idle Level}} * \sqrt {\text{Arena Rank}} \right ) + 0.01 \right ] + \\
& + \ \text{Total Attack Power} \end{align} 
Elemental Bonus refers to your % elemental bonus in the stat window. It's applied only if you have an elemental bonus opposite the enemy's element. If the enemy has no element, then no elemental bonus can apply. If the enemy is the same element, nothing happens; they won't "resist" your damage. Elemental bonuses always act beneficially.
If the equation is confusing, just remember the first line: Attack Power * Equip Att * Rage Attack * Elemental Bonuses. The other two lines are negligible. 
For example, notice your max damage is dependent here on level and rank. Before being scared by all the exponents and square roots, it's not a strong dependency. The difference between a rank 1, level 100 player versus a rank 500, level 9000 player—the two absolute extremes of the spectrum—is still only about a factor of 20. When you consider a more realistic scenario, such as going from level 5000 to 6000, or grinding out 50 rank-ups, the change in your max damage stat is actually going to be very small, and the other factors are much more relevant to your damage calculation. So one can safely ignore this line, as well as the lone Total Attack Power on line 3, and the equation looks much friendlier. Later, we will see where level and rank are important to you: avoiding penalties to damage and chance to hit.
If we pack all of the factors in the first two lines together into a variable X, then we have the simplified equation:
 \text{Max damage} = \text{X} + \text{Total Attack Power} 
Now, Minimum Damage is calculated, by taking Max Damage * Mastery. Then, a random number, generated by the random() function, is used to provide the initial damage dealt to the target:
\text{Initial Damage} = \text{Minimum Damage} + \text{random} \left( \text{Max Damage} - \text{Minimum Damage} \right )
This is labeled "initial damage," because it still needs to be modified, which will be the next step. 
The above formulas describe how the game generates the random damage you deal on every attack. Various contributions combine to make Max Damage, from which a Min. Damage is derived. You can view these damages in-game at the top of the second page of the stats window. Then, a random number between maximum and minimum damage is generated on every attack, and this random number is referred to here as your Initial Damage.
Now your Initial Damage needs to be modified by the enemy's defenses.
=== Damage enemy receives ===
With our hopefully impressive Initial Damage in hand, let's find out how the enemy's defense counters it. This is given by:
\begin{align} \text{Damage dealt to enemy} = & \left( \text{Initial Damage} * \text{Skill Power} - \text{Enemy Defense} * \text{Ignore Defense} \right ) * \\
& * \ \text{Area Damage} \end{align}
The term Area Damage is typically 1, but in special zones it's not. For example, in the dark pyramid, it is 0.000001, i.e., all your damage is reduced by a factor of a million. 
The Enemy Defense can be viewed in the Battle Arena menu in the bestiary, by finding the monster's entry after you've defeated it once. 
From this equation, we see that Enemy Defense is subtracted from our average damage. This means that if Initial Damage * Skill Power >>> Enemy Defense, then enemy defense is negligible. For example, if the enemy has 1 million defense, and you deal 20 million damage, then the defense only mitigates your damage from 20 to 19 million. Not a big deal. If however you dish out about 1.2 million damage, then 1 million defense is going to cut your damage by over 80%. Then you've got a problem. 
Notice that Crit Damage, Non/Boss damage, Crescendo, Mana/Spirit Power, etc. have not been factored in yet. These are multipliers that come after enemy defense has mitigated your damage. If you find your damage drop to a pathetic 100-1000 damage against certain enemies with fat defense (like CHAOS), more crit damage and boss damage won't help you. A 10x critical hit on a 100 damage attack is still only 1000 damage. For these situations, you need Ignore Defense, or you can brute force it with more Attack Power, Equip ATT, Elements, or Rage Attack and outscale the enemy's defense, so that your big crit multipliers become effective. 
PS. This makes the status effect Weaken quite powerful against high defense enemies. It reduces the enemy defense by a percentage, so your crit and boss damage multipliers can be effective. Weaken is also a powerful malady when enemies apply it to you, since it cuts your min/max damage by 40% before defense is subtracted, making the enemy's defense that much more efficient. 
=== Level Modifier (Damage Penalty) ===
Because the Battle Arena wouldn't be an Anti-Idle game mode if it didn't punish you somewhere in bewilderingly complex fashion, we need to define its hidden damage penalty on players. You cannot see this in the stats window, but you're actually losing a sizeable chunk of your damage all the time. This penalty modifier is based mostly on your level and rank relative to the enemy's level.
\text{Damage Penalty} = 0.5 * \left \lfloor \frac{\left ( \text{enemy level}\right )^{1.12}}{\text{Anti-Idle level} + \text{Arena Rank} * 100 + 999} \right \rfloor - 0.1 
This is a percentage value. A damage penalty of 40 means that your final damage dealt is reduced by 40%.
The damage penalty is modified in many ways:
* Bosses add 20 to damage penalty. You will never deal more than 80% damage against a boss monster (their names are colored purple).
* If the enemy level is greater than yours and you are under level 8999, then Damage Penalty increases by another 10 (20 on hard ascensions; 30 on impossible). Basically, if the enemy is a higher level than you, then subtract another 10% of your damage.
* Worst Moon mode adds another 5
* Impossible ascensions receive an additional contribution of ( 8,999 - your level ) / 450
* Apocalypse mode takes the final damage penalty and brings it halfway to 100%, i.e. a damage penalty of 40 increases halfway to 100, in this case to 70.
* The penalty cannot go below 0 or above 95 (except on impossible ascensions, which cap at 99).
Notice that the number 8,999 occurs a couple times. Once a player reaches 8,999, they are freed from penalties due to being lower level than the enemy, which can lead to a big boost in performance. Most super end-game content is best attempted while at level 8,999 or above.
The Damage Penalty formula looks ugly to interpret, due to the 1.12 exponent on the enemy's level. Can we instead approximate this? Kind of. At enemy level 1000, the 1.12 exponent is effectively a multiplicative factor of 2.2 ( 1000^{1.12} ~= 2.2 * 1000), while at enemy level 9000, it's a factor of 2.9 ( 9000^{1.12} ~= 2.9 * 9000). So this level exponent ranges between a small grouping of multipliers, between 2 to 3, for all realistic enemy levels, and most end-game content will be about a factor of 3. Because 2 to 3 is a small range, we can approximate it:
* End-game content, level 8000+ ~= 3 * level
* Mid-game content, level ~5000 ~= 2.6 * level
* Early-game content, level ~1000-2000 ~= 2.2 * level
Most calculations involve end-game content, so we can just stick with the 3x multiplier. With this in mind, we can make some interpretations from the damage penalty equation.
Against a level 9000 boss, you can treat the numerator as 9000 * 3 = 27000. If your level + rank * 100 are equal to 27,000, then you can expect a damage penalty of about 70%. If you're below this value, say only 18,000, then it's an 85% damage penalty and you deal half as much damage. Being underleveled is really bad! 
On the other hand, if your level + rank * 100 are at 36000 or 45000, then the penalty drops to about 60% and 50%, respectively, which are about the same penalty. There isn't much advantage to being overleveled.
So the main takeaway is to try and be either 
a) equal to enemy level or 
b) have a rank * 100 that is at least 3 times the enemy's level. (Rank 300 would cover all enemies).
c) Level 8999, in which case all the penalties except for Bosses, WM, and Apocalypse vanish. Nice!
If you don't meet one of these standards, and are struggling to complete some raid or defeat an enemy, you should experience a big boost in performance by meeting one of the above conditions. This isn't the only level handicap, however, as shown further below.
=== Final Damage Dealt ===
Now we are ready for the final damage. Recall the steps to get here:
# We calculated our total Attack Power and multiplied it by our mastery, equipment attack, Elemental bonuses, and rage attack.
# We subtracted this by the enemy's defense to find "Damage Dealt."
# We derived a damage penalty.
The final damage takes the penalty and every other remaining stat and multiplies it by Damage Dealt:
 \begin{align}\text{Final Damage Dealt} = & \text{Damage Dealt} * \left( 1 - \text{Damage Penalty} \right ) * \text{Crit} * \text{Boss Damage} * \\
&* \ \text{Mana/Spirit Power} * \text{Crescendo} * \text{Cour de Grace} * \text{etc.} \end{align}
And that's it. This is how much health the enemy loses, as well as the numbers that appear on screen when you hit the enemy.
=== Mana Power ===
Since Mana Power is impossible to understand from the in-game description, for completeness it will be explained here.
Mana Power's bonus damage is a multiplier that is derived in two steps.
# \text{Mana X} = \frac{Current MP}{50} 
# \text{Damage Bonus} = \frac{1+ \text{Mana X}}{50,000} 
At Current MP = 5 million, Mana X is 100,000, and the damage is a 2x bonus multiplier.
If Current MP is greater than 5 million, then a soft cap is implemented. Every MP above 5 million contributes at 1% effectiveness due to the soft cap. Therefore, at 10 million Current MP, Mana X is 150,000, and the damage bonus is a multiplier of 3.
If Current MP is also greater than 10 million, then we encounter a second, firmer soft cap. Every mana above 10 million contributes at 0.5% effectiveness. The hard cap for MP is 19,999,999, so including both of the soft caps, the hard cap on Mana Power's multiplier is 4, at an MP of about 20 million. 
Your Current MP is subtracted by Mana X with every attack.
=== Spirit Power ===
Spirit Power is similar but more specific. It only multiplies damage if your current Spirit is within 9 of Max Spirit. If you are missing some spirit, like after using a skill, then Spirit Power does nothing. 
The calculation steps are: 
# \text{Spirit X} = \text{Max Spirit} - \text{Current Spirit} + 10 
# \text{Damage Bonus} = \left ( 1 + \frac{\text{Spirit X}}{10}\right )
If Spirit X < 0, then Spirit X is set to 0. This is why Current Spirit needs to be within 9 of Max Spirit, or else it gets set to 0. This restriction on Current Spirit leaves available the following possible multipliers: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0.
In practice, the game only uses two of these possible multipliers: 1.2 and 1.7.
This is because your Spirit is subtracted by Spirit X with every attack; however, you also gain Spirit with every attack. It turns out that as your Spirit adds and subtracts with every attack, the game code settles on a value where its net change is 0: that means where Spirit X = Spirit regen. With Spirit Power active, you gain 2 Spirit per attack, so in most cases, Spirit X = 2, and your Current Spirit will sit at Max Spirit - 8, giving you a 1.2 multiplier. When you kill an enemy, you gain 2 spirit from the attack like usual, plus another 5 Spirit for killing the enemy. So if you kill enemy monsters in a single attack, your spirit regen is 2 + 5 = 7. This results in Spirit X = 7, and your Current Spirit will be stuck at Max Spirit - 3, giving a 1.7 multiplier.
So in order to improve your Spirit Power multiplier, you need to increase how quickly Spirit is regenerated.
=== Enemies with max defense ===
Some enemies' bestiary entries will show "max" for their defensive stat. These are an exception to the above rules, so not even the typical Ignore Defense stat will help you. The code has preset stats for such enemies, for example that gems on the mystic path take 6 damage, while gems elsewhere take 4 damage, but Secret Crystals take 2000 damage.
You can overcome max defense enemies with attacks that specifically ignore 100% def, allowing them to exceed the 95% cap of the stat, such as the spear specialized skill "Pierce" or the roundhouse kick skill. This is one of the major reasons Spears are so valuable in the end-game. Enemies in the end-game effectively have max defense all the time (their defense scales exponentially in the Endless Dungeon into the billions—hopelessly out of reach of your Initial Damage without a lot of work to overcome it).
== Defense Formulas ==
DamageTaken (the label for how much damage you receive from an attack) follows a similar process to damage dealt: Combine multipliers, factor in the level modifier, and calculate final damage.
Your defense power is aggregated just like attack power above. After doing this, your damage resist is derived. Damage resist are the numbers in the stat window in-game below max/min damage. The % number on your left is "Damage Resist %" and the flat number on the right is "Damage Resist." These are calculated as:
\text{Damage Resist \%} = \frac{\text{Equip DEF}} {100} + \text{Skill (Overpowered Tanker)} + \text{Orange Bonus} + \text{Ally} 
and
 \begin{align} \text{Damage Resist} = &\left( 1 * 10^{-6} * \sqrt{\text{Arena Rank}} * \sqrt{\text{Anti Idle Level}} + 0.002 \right) * \\ 
& * \ \text{Defense Power} * \text{Equip DEF} * \text{Elemental Def} + \\
& + \ \text{Defense Power} + \text{mana * Skill 1} + \text{health * Skill 2} \end{align} 
Skill 1 = Protection; Skill 2 = Overpowered Tanker
Like before, your arena rank and Anti-Idle level are mostly there just to make the equation look scary; their contribution is relatively small. Notice also that Equip DEF double dips: in the first formula, every 100 points of Equip DEF grants 1% damage resist, while in the second formula it's a simple multiplier for your flat damage resist.
Damage Resist % is capped at 90%, while Damage Resist is capped at 1,999,999,999. Damage Resist % can exceed the 90% cap by using Mana or Spirit Power, up to 95%.
=== Defense and Anti-Idle level ===
Next, we calculate the game's level-modifier for defense. Unlike with damage, this isn't really a penalty since it can also work in your favor.
\text{Final Damage Resist} = \text{Damage Resist} * \left \lfloor 1 + \frac{\text{Anti-Idle Level} - \text{Enemy Level}}{6000} \right \rfloor 
This bonus is capped at 2x your original defense (when you are 6000 or more levels higher). It cannot go below 0. That's right. Your flat damage resist is ZERO when the enemy is 6000+ levels higher than you. Fortunately, you've still got damage resist %, which does a better job anyways.
Additionally, if the enemy is higher level than you and your level is below 8999, then your Final Damage Resist is further reduced by 20%, which rises to a 40% reduction on impossible ascensions.
Basically, if you're up against a higher level enemy, your Damage Resist is at best 80% effective.
=== Damage Taken ===
Now we derive how much damage your character actually receives. 
Final Damage Resist is subtracted from incoming enemy damage. The resulting difference is multiplied by damage resist %. And that's how much damage your character takes. 
\text{Damage Taken} = \left ( \text{Enemy Damage} - \text{Final Damage Resist} \right ) * \left ( 1 - \text{Damage Resist \%} \right ) 
...except for a few final modifiers, such as Drain when you're at max health.
Some monsters have ranged attacks, which are also referred to as magic attacks. Ranged/Magic damage are actually calculated after the above formula for damage taken:
 \text{Ranged Damage Taken} = \left ( \text{Damage Taken} + 0.1 * \text{Ranged Attack} \right ) * \left ( 1 - \text{magic resist \%} \right ) 
Because it uses "Damage Taken," this means that ranged damage is actually calculated the same as melee damage, only it afterward adds another 10% of the enemy's ranged attack. Maybe surprisingly, this indicates that against ranged/magic attacks, having a high damage resist % is still crucial. In theory, this should combine very well with a high Magic Resist % to go beyond the limits of the Damage Resist % cap.
Weaken will reduce Damage Resist by 40% and Damage Resist % by 20%.
=== Conclusions on Defense ===
Here are some takeaways from the above formulas: 
* Higher level enemies devastate your defense. This means that defense is really only useful against weak enemies. Against strong enemies, prioritize damage resist %. 
* In the end-game, damage resist % is penalized, such as in Worst Moon mode, or the enemies deal so much damage that even its cap of 95% is not sufficient to keep you alive. As we will show below, evasion, more precisely dodge %, is the absolute best protection in the game.
=== Mana and Spirit Power ===
Mana and Spirit Power, while active, increase Magic Resist by 15%. Also, your Damage Resist % is averaged with 100%. For exampe, if you have 20% Damage Resist, then Mana/Spirit Power will give ( 20% + 100% ) / 2 = 60% damage resist.
Mana Power further boosts your flat Damage Resist by 20% of your Max MP. Spirit Power does not do the same.
While Mana Power is active, Damage Taken is applied to both your health and your MP. If MP is less than the incoming damage, then MP is set to zero and the difference is subtracted from your health, dealing you extra damage on top of the attack to your health already incurred. So you're kind of punished for having Mana Power active while at low mana.
Spirit Power in theory is also reduced by incoming attacks. However, it divides incoming damage by 50,000. This is applied to damage post-mitigation. The hard cap on modified health is 19,999,999. Even if you were at this cap and were one-shot by a 20 million Damage Taken, 20 million divided by 50,000 is only 400, so you'd only lose 400 Spirit. Your Spirit with Spirit Power is at much less risk than MP is with Mana Power.
== To-hit Formulas ==
This section describes how the game calculates a player's chance to hit. There are three steps to this: 
# The player must pass a Hit Penalty check 
# The player rolls an accuracy check 
# The player rolls a hit chance % check 
In order for an attack to not miss, i.e. to hit the opponent, the player must pass the first check; however, from the remaining two checks, the player only needs to pass one (or both) to succeed on their attack.
This can be summarized in one formula. The player's attack will hit an enemy if r(n), defined as n random numbers of r from 0 to 1, meet the following condition: 
 \begin{cases} r(1) > \text{Hit Penalty \%}\ \\
\text{AND} \\
r(2) < \text{Hit Chance \%} \ \text{OR} \ r(3) < \frac{\text{Player Accuracy}}{\text{Enemy Evasion}} - 0.1 \end{cases} 
The Hit Penalty is derived from: 
 \text{Hit Penalty \%} = \frac{\text{Enemy Level} - \text{Anti-Idle Level}}{100} 
The Hit Penalty from this equation is capped at 30. However, it can exceed the cap from other sources: 
* If the player is a lower level than the enemy and below level 8999, then it is increased by 10 (20 on hard ascensions; 30 on impossible ascensions). 
* On Worst Moon mode, it's increased by 5. 
* Hit Penalty has no minimum value. If it's negative, you will always pass the check. There should be a noticeable improvement to being within ~1000 levels of an enemy (or exceed its level). 
Because higher level enemies also have higher evasion, your Accuracy stat isn't nearly as reliable as Hit Chance %. It also helps that Hit Chance % is pretty easily acquired, especially from the Censor Sheep Invisible Ally. 
When the player is affected by blind, accuracy is reduced by 40% and Hit Chance % by 20% (multiplicatively on both). Hit Chance % is capped at 90%. Accuracy has no cap from modifiers. 
== Evasion Formulas ==
Evasion functions similar to hit chance above with a couple key differences. The enemy's hit accuracy is calculated as
\text{Enemy To-Hit} = \frac{\text{Enemy accuracy}}{\text{Player Evasion}} - 0.1 
This Hit Accuracy cannot go below 0.1.
The enemy's chance to hit you, using two random numbers from 0 to 1, r(1) and r(2), succeeds if:
 r(1) < \text{Enemy To-Hit} \ \text{AND} \ r(2) > \text{ Player Dodge \\%} 
The enemy accuracy stat needs to pass BOTH checks (against your evasion and your dodge %) in order to hit. Enemies also do not have a Hit Chance % stat to guarantee hits no matter your evasion. 
This explains why evasion is so powerful. When players strike the enemy, they have two chances to successfully hit, either by Hit Chance % or Accuracy. When the enemy strikes the player, they have only one chance to hit, via accuracy. Furthermore, dodge % is comparable to the hit penalty check that players have to pass in order to successfuly hit the enemy. For players, dodge % is pretty easily raised to its cap of 90%. This leaves enemies batting against you with a permanent 90% hit penalty.
However, the enemy cannot miss more than 9 times in a row. After 9 misses over the course of a fight, the attack following the 9th miss will always hit. Then the counter resets.
Additionally, if an attack is marked with undodgeable, then it obviously cannot be dodged. The same goes for drowning damage or the special effects like Prehistoric Mission.
When the player is affected by blind, it reduces their evasion by 40% and dodge % by 20% (multiplicatively on both). Dodge % is capped at 90%. Evasion has no cap from modifiers.
== Other Formulas ==
=== Enemy Rampage ===
Rampage is a skill on some monsters that activates after they've lost a certain percentage of their health (you can view the bestiary page to see this). Once activated, Rampage multiplies attack (melee and range), defense, accuracy, and evasion by 1.5. Movement speed is also increased by 20%.
=== Hardcore Difficulty ===
Hardcore difficulty multiplies the following:
* Level = 1.3x (max 9001)
* Attack / Ranged Damage = 1.5x
* Accuracy / Evasion = 1.5x
* Defense = 1.2x
* Rewards (EXP, Coin, Pixel) = 1.5x
* Item tier is increased by 2
* Chips drop twice as often
* Raid rewards improved and timed equipment typically permanent
* Items dropped by Collector's Pendant (lrn2craft recipe items) have a chance for improved stats.
* If an enemy does not have a ranged attack, it gains a ranged attack equal to 0.8 of its normal attack. 
* If an enemy has the Rampage ability, then the HP threshold to activate it is halved.
* If an enemy does not have the Rampage ability, then it gains the Rampage ability with 1.2x Attack and Defense multipliers instead of 1.5x. Rampage is set to activate at 50% remaining health.
* Any enemies without a skill gain a random skill
=== Worst Moon / Apocalypse ===
Some (but not all) changes from the Worst Moon and Apocalypse modes (note that Apocalypse is an extension of Worst Moon and therefore multiplies all of WM's changes implicitly). 
WM and Apocalypse can be overlapped with Hardcore, where benefits/difficulties are compounded.
Also remember that the Invisible Ally, Invisible X (IX), may be used to enter WM and is required to enter Apocalypse. IX brings a base 30% Spawn Rate before upgrades, and it drains Spirit by 1 each second. Factor these properties in under good/bad, respectively, for when you use IX.
Worst Moon
Good
* Increased by 30%: EXP
* Increased by 50%: coin, pixel and drop rate
* Item tiers + 5
* Monster Chips drop rate 3x as often.
* Raids offer higher reward ceilings.
* Raids may offer equivalent rewards at an earlier point (e.g., demon slayer armor appears on the 10th floor instead of 35th in Endless Dungeon, on 4/5 floors instead of 1/5.).
* adds an additional independent roll for stat upgrades on dropped items from Collector's Pendant. These are weapons/armor from the lrn2craft recipes.
* Increased epic encounter rate (value depends on zone, typically a 2.5x to 4x bonus)
Bad
* Reduced by 80%: Reflect, Magic Reflect, Poison Damage
* Reduced by 50%: Negate Effect, Magic Resist, Damage Resist %. Double Hit, Critical Chance, Boss/non-boss damages. Dodge %, Hit Chance %. Weaken and Blind Effect, Poison, Weaken, Blind, and Stun chances. Potion Efficiency
* Reduced by 20%: Max Spirit
* \text{Worst Moon Critical Damage} = \frac{\text{Critical Damage}}{4} + 75 (Crit Damage of 1000% is now 325%. Max is still 1000%).
* Increased by 5 percentage points: Damage Penalty and Hit Penalty (see above for definitions).
* \text{Worst Moon Max Damage} = 0.9 * \text{Max Damage}^{0.95} – this is not total damage, rather the damage before enemy defense, crit, %boss damage etc are calculated in.
* Reduced by taking x^{0.98} : Max Health, Max MP
* Reduced by taking x^{0.95} : Damage Resist, Health, and MP regen
* Rage is now lost at 2 points per interval.
* Critical hits do not grant 1 bonus combo, Robacon's level 125 ability does not function.
* Enemies move twice as fast I think (rangecount + 1)
Apocalypse
Carries all the effects of WM, MULTIPLIED BY:
* Good: EXP up by 15%; Coin, Pixel, Drop Rate by 30%; Item tiers + 1; adds a third roll with the same odds as WM for better item stats dropped by Collector's Pendant; improves some raid rewards again. 
* Bad: Reapplies the 50% reductions from WM AGAIN. This means a 75% decrease total. Crit damage also receives the same penalty again, for a total of \text{Apoc CrD} = \frac{\text{CrD}}{16} + 94 . A normal crit damage of 1000% is now 156%. Max is still 1000% (would require an impossible 9000% normally).
* Spirit is no longer reduced by 20%, instead being reduced by 40% in total.
* Defense, Max Damage, Max Health, and Max MP are now simply reduced by a further 10%.
* Apocalypse is necessary to encounter ULTRA allies
=== Some info in drop rates and Collector's Pendant ===
The Collector's Pendant allows you to loot weapons/armor that are craftable (lrn2craft recipes) or unique. Without the Collector's Pendant, you will only loot pendants, bonus rocks, epic staves and some other assorted junk from your battles that aren't in a raid.
The drop rate for an item from the Collector's Pendant is 0.01% from bosses. This percentage is further multiplied by various other rolls to determine the identity of the item. The level of the enemy is important. The possible choice of items increases for enemy levels at 1000, 2500, 4000, 6000, and 8000. If you want l2rncraft level 5 recipe items, then you would need the top threshold over 8000. The odds of selecting this class of items is about 60%, so multiplied by 0.01% chance for the item to drop, and boss monsters above level 8000 drop level 5 lrn2craft items at a rate of about 0.006%. If the boss is not a secret crystal that you mine, then they gain 100 attack and 180 defense.
Once the item has dropped, its stats can receive significant boosts, so that they are even better than crafted (unless you craft with 500/500 crystals). However, this only occurs in HC, WM, Apocalypse, or while a Gem of Good Luck is equipped. The game rolls for all stat boosts, each one being checked independently with a roll check. Hardcore always has better odds than WM and Apocalypse (ranging between 45% HC vs 35% WM/Apoc for some stats, to 90% HC versus 70% WM/Apoc for other stats). The Gem of Good Luck has a constant 50% chance to roll on each modifier. Wm and Apocalypse have identical odds. Obviously, Apocalypse overlaps with WM, so it actually rolls twice for each stat at the same percentage of success. Overlapping multiple conditions will add more rolls and should dramatically improve chances of success.
If the enemy is epic, the monster may drop one of the elemental staves or glaives. The base chance is 2.5%. This chance always exists regardless if you're on normal casual. However, the unobtanium tier is of course enhanced for HC, WM, and Apocalyse. 
Furthermore, the Gem of Good Luck has a unique roll on both the non-epic and epic dropped items for attack power (40% for epic-dropped staves).
=== Ascensions ===
Hard and Impossible ascensions apply various penalties. These may be collected here at a later date.Category:Battle Arena
|}