Damage Calculations¶

General¶

Damage calculations in Summoners War is a quite long process so let us start with the total value of a single stat in combat (e.g. your ATK).

First a few definitions:

$B$ is a base stat value.
$b$ is a (de)buff value, for example for ATK it is 0.5 (50%).
$p$ is the sum of all % bonuses for a stat (this includes buildings, leaderskills, and things like Fight runes).
$f$ is the sum of all flat bonuses for a stat.
$S$ is the total in-combat value of a stat.

$S$ is calculated as

$S = (1 + b)\cdot[B \cdot (1 + p) + f]$

With the introduction of artifacts and monsters that amplify (de)buffs you have to be careful about the value of $b$ . First of all these effects are multiplicative to the base value, so for example if you have an artifact that increases the ATK buff by 10% then the ATK buff has a value of $b = 50\% \cdot (1 + 10\%) = 55\%$ , and not $60\%$ . The second thing to consider is that these amplifiers are additive with each other, so for example that 10% increased ATK buff and Miriam's passive together result in a $35\%$ stronger ATK buff, not $37.5\% (= 1 - 1.1 \cdot 1.25)$ .

Help wanted

Currently the interaction between an enhanced debuff (Deborah) and an enhanced buff (artifact and/or Miriam) is untested due to my lack of a Deborah. If you have a Deborah and are willing to help me out on this matter, please let me know.

Of course this stat is not the only thing to calculate the damage your monster will deal so let us move on. To avoid the formulas getting unwieldy (read: unreadable on mobile phones) we will proceed in small steps:

First we calculate the sum of the stats $S_i$ the skill scales on and their respective multipliers $M_i$ :

$S_{tot} = \sum_i S_i \cdot M_i$

Now most skills only scale on one single stat so most of the time it is not a sum but just $S \cdot M$ . $M_i$ can depend on a variety of external factors, for example the number of living allies, remaining HP, or the number of debuffs on the enemy.

How do I find out the skill multipliers?

The Summoners War (Reddit) community has datamined the multipliers of monsters, you can check them on swarfarm.com.

The next step is to calculate the other factors that solely depend on monster stats, the influence of critial hits $f_{crit}$ and the damage reduction factor of the enemy based on their DEF value $f_{def}$ :

$f_{def}$ is simply given by

$f_{def} = \frac{1000}{1140 + 3.5 \cdot DEF}$

Also see the section on the damage reduction formula for a few more details.

The value of $f_{crit}$ is given by

$f_{crit} = 1 + s + a^+_{cd} - a^-_{cd} + CD$

where $s$ is the value for the skillups of the skill in question, $CD$ is the monster's crit damage value (the CD you see in the monster page + the tower bonus), $a^+_{cd}$ is the sum of your current extra CD from artifacts (e.g. +4% CD on S3), and $a^-_{cd}$ is the CD penalty you get from the enemy's artifacts (e.g. if your enemy has an artifact "Received CRIT DMG -10%" then $a^-_{cd} = 10\%$ ).

Note

The value $f_{crit} - 1$ is commonly known as "total CD" since then your skills do 1 + total CD damage on a crit. With the addition of artifacts to the game this value now can contain a dynamic element (for example based on enemy HP status or if you attack on your turn) which means that technically you should consider that when calculating the optimal stat distribution (for skills scaling on a single stat you ideally want the entire bonus value of your stat (measured in % of the base stat) to be equal to your total CD) but in practice it is a far better idea to ignore non-static CD from artifacts for this purpose, e.g. always having +30% CD on S3 is fine while relying on the enemy to have full HP to gain +30% CD is not a good idea.

However note that this is only an optimal distribution, a non-optimal distribution with more stats in total will still deal more damage. Also note that you do not need to meet this 1:1 ratio exactly (which might not even be possible with your runes anyway), it is mostly a reminder that you should always try to add more of the stat you have less of, e.g. if you have 300% bonus ATK but only 100% CD then adding 10% CD is better than adding 10% more ATK.

Now we already have all the parts necessary to calculate a monster's damage in a common setup where no special effects or artifacts are in play. For this we define

$\begin{aligned} D &= S_{tot} \cdot f_{crit}\cdot f_{def}\cdot f_{fluct} \\ &= \left(\sum_i S_i \cdot M_i\right) \cdot f_{crit}\cdot f_{def}\cdot f_{fluct} \end{aligned}$

which is the damage your monster will do under these circumstances. The factor $f_{fluct}$ accounts for built-in random $(\sim \pm 2.5\%)$ fluctuations of the damage of most skills (see damage types for the skills which are not subject to this). When calculating the damage of your monster you should generally ignore this factor (read: set to 1), it is just appearing here so you are not surprised when your Lushen's S3 deals 11734, 12345, and 11996 damage on the 3 cards instead of 3 times exactly the 12005 you calculated (numbers made up).

The next big thing is additional damage from artifacts. Additional damage from artifacts deals fixed damage which means it is not influenced by the enemy's defence so basically if you have an artifact that says "Additional Damage by 1% of HP" on a monster with 10k HP then that monster will deal exactly 100 (= 1% of 10000) additional damage per hit. This means the total damage from all your additional damage artifacts $AD$ is simply the sum of your stats multiplied by the respective artifact values $a_i$

$AD = \sum_i S_i \cdot a_i$

Then finally we have all the damage amplification effects in the game, for example the brand debuff (increases your damage by 25%) and the artifacts amplifying damage against/from a specific element. Now the special thing about the damage increasing artifacts ("Damage dealt on [Fire/Water/Wind/Light/Dark] +N%") is that they do not apply to additional damage from artifacts. This means that if you have an artifact increasing your damage against some element by $f_a$ then your damage will be

$(1 + f_a) \cdot D + AD$

and not

$(1 + f_a) \cdot (D + AD)$

All other damage amplification effects $f^{amp}_{i}$ (e.g. brand, Darion's passive, etc) affect both "normal" damage and the additional damage from artifacts and are additive so for example having a Brand debuff (+25% damage) on a monster that is next to a Darion (-20% damage) will result in $25\% - 20\% = 5\%$ increased damage.

Damage Calculation Formula¶

Putting all the small things above together the final formula for the damage your monster will inflict is

$\text{Damage} = \left[\left(1 + f_a\right) \cdot D + AD\right] \cdot \left(1 + \sum_i f^{amp}_{i}\right)$

Or if anyone wants to see (most of) those smaller parts from above written out (warning, this will not be readable on mobile phones unless you hold it horizontally (and even then no guarantees)):

$\begin{aligned} \text{Damage} =& \left[\left(1 + f_a\right) \cdot \left[\left(\sum_i S_i \cdot M_i\right) \cdot f_{crit}\cdot f_{def}\cdot f_{fluct}\right] + \sum_i S_i \cdot a_i \right] \\ &\cdot \left(1 + \sum_i f^{amp}_{i}\right) \end{aligned}$

Some important notes

Skills scaling on speed are treated in a special way, they do not have a separate multiplier for SPD but instead the multiplier changes with the SPD value (with only few exception the multiplier takes the form $(SPD + A) / B$ where $A, B$ are constants based on the respective skill) and affects ATK as the stat (there is one case where the SPD scaling affects an HP scaling skill instead, but usually it is ATK).
Damage increase based on the number of debuffs, for example Kro's S3, modifies the skill multiplier and does not count in the damage amplification category. For example Kro's S3 has a base multiplier of 7.2 (720%) and gains 50% damage per debuff. Against an enemy with 3 debuffs it becomes 7.2 * (1 + 3 * 0.5) = 18 (1800%).

Damage Reduction by Defence Formula¶

The defence of a monster determines by how much the damage that monster will take is being reduced. The formula for the part of the damage that you will take is

$f_{def} = \frac{1000}{1140 + 3.5 \cdot DEF}$

Attention

Ignore DEF skills are still subject to this factor. They just work with $DEF = 0$ so you end up with $f_{def} = \frac{1000}{1140} \approx 87.72\%$ . Keep this in mind when calculating your Lushen's damage. Only fixed or pure damage ignores this factor.

Damage Types¶

Some skills behave in a special way. Those skills can be put into categories which get their own names. This is similar to how other games might have things like "magical damage" and "physical damage" and respective magical armor and physical armor stats yet not quite the same because in Summoners War there is no such differentiation (or at least not different defence values) so damage types here merely affect the damage calculation.

True Damage¶

True damage still fluctuates but is not subject to damage reduction by defence ( $f_{def} = 1$ ). The only thing currently in the game dealing true damage are bombs.

Fixed Damage¶

Fixed damage deals exactly the damage you calculate, it is neither subject to damage reduction by defence nor the random fluctuation of normal damage skills ( $f_{def} = f_{fluct} = 1$ ). Examples of fixed damage are Skogul's S2, Ramagos' S3, or Arnold's S3.

Pure Damage¶

Pure damage is pretty much like fixed damage but unlike that it does not get influenced by damage reducing passives or the brand debuff or anything else. Continuous Damage and the first part of Trinity's and Daphnis' S3 are examples of pure damage.

Hit Types¶

There are four different types of hits an attack can have: Critical Hit, Crushing Hit, Glancing Hit, and Normal Hit.

The order in which the game evaluates as what kind of hit a hit lands is as follows:

Check for glancing hit
Check for critical hit
Check for crushing hit

If none of those checks are passed, then it is a normal hit.

Critical Hit¶

Base Critical Rate is 15%.
Attribute Advantages gives +15% Critical Rate, Attribute Disadvantage gives -15%.

Crushing Hit¶

Only happens when attacking while having elemental advantage.
Deals 30% increased damage.
Chance for a crushing hit is 35%.

Glancing Hit¶

Base chance to glance is 0%.
If attacking while having Attribute Disadvantage, the chance to glance increases by 50%.
Deals 30% decreased damage.
Debuffs and ATB reduction can not happen on Glancing Hits.

Normal Hit¶

Only happens if none of the glance/crit/crush checks passed.
If your monster has skillups you need to set $f_{crit}$ to $1 + s$ with $s$ being the sum of all damage skillup values of that skill (see damage skillups).