That’s why the coefficient on the Defense damage is 101, while on offense it’s 78. Ogives that the exponent values are also different, benefitting offense.
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Thanks for the clarification 
A big thank you to you @RedPython!
I didn’t know anything about this increased defense of the AI-led team.
How does this information affect the formula? Take a look at the first post on this topic and you’ll know it!
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Beautiful
(as only 20 characters can be)
Amazing. Thanks for updating the original (and now unified) formula.
Kudos to @RedPython for helping reveal the final key to this puzzle. 
@SolemnWolf The new formula definitely makes more sense, tho is θ an evenly distributed random between e^-0.5 and e^0.5 or is it a curve such that it’s usually around 1? I think you had something about a Guassian curve in your older version of the post? Most low damage hits seem to be near each other in my experience, otherwise even for low damage hits near 30 we’d see more 15 and 58 damage hits when θ was 0.6 or 1.6.
So maybe a full version of the formula would be?:
100 * TileD * ((θ * AttackSum / DefenseSum) ^ 1.35) * ( ColorBonus + Crit)
Where:
AttackSum = (Attack Stats of a hero * Troop buffs) for each hero * Skill% * Combos * Buffs
DefenseSum = Defense Stat of target * EnemyBonus * Buffs
TileD = 1/3 if it’s a tile, else 1
Combos = 1, 1.1, 1.2 etc depending on cascading combo multiplier for tiles, else 1
Skill% = skill stat in decimal form, else 1 if not a skill
Buffs = 1 + Skill Buffs [only 1 buff/debuff of the same type can be active at once]
EnemyBonus = 1.2 if the target is an enemy of the active player in PVP, else 1
ColorBonus = 2, .5, or 1 for tile damage depending on color strengths/weaknesses, else 1 for non tiles
Crit = 1 if it’s both a tile and rolled a crit (base 0 chance, increased with troops / skills), else 0
θ = a random number between e^-0.5 and e^0.5 with a Guassian curve distribution of σ^2=1
Note: The special effects of certain skills can change a number of these stats
Or did I mis-order something?
Edit: Fixed ColorBonus / Crit interaction via Kerridoc’s and SolemnWolf’s help. Thanks!
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The way you incorporate the crit bonus is wrong. The crit bonus is color-blind. The right hand terms of the equation should be (colorBonus + Crit) where Crit = 1 if crit, 0 otherwise.
Now I know exactly what happened here, other than my poor judgment that I can defeat this team and pressing the attack button. Thank you @SolemnWolf for this great efforts.
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You faced an all yellow team with ONE purple and to make it funnier he was a healer 
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It was Sartana ,… I believe.
But I killed her… Rawr!
I’m saying on YOUR team! 
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This is a beautiful work Lutrinae, personally I didn’t want to go into much detail because there are many exceptions with all the effects of special skills (some are powerful and without weaknesses, buffs can be cumulated if they are slightly different…).
The only (possible) error is that identified by Kerridoc: It seems that critics cancel the weakness.
Moreover, the bonus of 20% for the AI certainly concerns only the PvP fights (but we have no way to check it because the stats are only accessible in PvP …).
Finally, if you’re looking for detail for θ, the best approach I’ve found is:
θ = exp (χ)
And χ follows a Gaussian law centered at 0 with a variance of 1.07%, so in 68% of cases -10% <χ <10% → 0.90 <θ <1.11.
But this is only a statistical approach!
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Is there an error in the defense value? should it be 20% more, right?
In which case does this % def. increase apply? In the prevoius example it was not cnsidered^^ Thanks
@Oracle83, this bonus applies to AI defense for each player attack (but apparently only for PvP).
The bonus is well applied in the calculation examples (post # 13), the defense of the AI is increased by 20% from the start (Def x 120%)
Right…I didn’t consider that…my fault 
What does AI stand for?
Artificial Intelligence, in this case it refers to the algorithm that controls the defending team
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@SolemnWolf I’ve been meaning to comment for a long time. This is a beautiful piece of work and must have taken a long time to compile the data.
I have one question. I usually fight 11-12* titans and I moved down to merc on a 6* rare titan. I brought a lot of heavy hitters and I am sure that I had at least three times the summed attack power of some of the other people in the alliance. However, I only got hits of 30-40K even though my 11* titan hits were a lot higher than that, up to close to 100K. Several other mercs tried and were frustrated as well with their low scores.
I was wondering if there are cases in which the curve flattens instead of continues exponentially, maybe beyond the values you have in your graph? I am asking because long ago, SG was talking about limiting the damage of Wu Kong plus attack buffers. After a long argument things were kept the same, but we never know what has happened since then. I wish I had data but I didn’t see your post until long after this incident.
Thanks!
I think it may be due to the fact that titans defense saturates early (or at least it grows slowly), and their increasing difficulty is due more to increasing attack and hp (I’m speculating here, no real data, anyone has estimated titans stats?)
But I think that an easy way to verify that the damage formula holds for very high attack/defense ratios is playing some early provinces’ stages. Ideally with Wilbur one can reduce the enemy defense without killing them (maybe keep only one mob alive so damage sharing doesn’t apply), then buff attack, fire wu and see what happens when you drop a tile brick over the mob’s head
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Do we need count
as
DefenseSum = (Defense Stat of target * Defense Troop buffs) * EnemyBonus * Buffs
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Is this topic alive?
Just checked numbers from 1 battle. θ was always possitive.