Understanding the Mabinogi Damage System by Kakashi7

• Hei wrote on 2010-10-08 18:46
  This is a guide that I wrote for Umbrasquall's blog a pretty long time ago. I've seen it being referred to once in a while in discussions, so I thought I should put it here for better availability. Hopefully this will serve to inform some players here of how the damage system works. If nothing, I'd love some more feedback.
  
  

  Article originally written at: Understanding the Mabinogi Damage System (Article Courtesy of Kakashi7)

  
  
  //
  
  
  
  After playing Mabinogi for around two years, I noticed that many players have trouble choosing the ideal weapons and damage enchants for themselves because they are not very clear on how exactly maximum damage and minimum damage affect their overall damage outputs. Among the common questions that I hear are: does gladius’s high minimum damage make up for its lack in maximum damage in comparison to the broadsword? Or does muramasa and bastard sword egos’ max damage difference make up for their lacking in min compared to the gladius? To answer these questions, I’m going to attempt at explaining how damage works in this game.
  
  In order to understand how maximum damage and minimum damage affect the average damage output, one has to first understand the concept of balance. I’m sure that most people have already noticed that the higher the balance a player has, the closer to his maximum damage the player hits. However, we have also noticed that even with low balance the player does damage close to his max on rare occasions. The consensus is that balance stabilizes damage and increases the average output.
  
  So what is balance? Balance can be thought of as a weight determinant for the worth of maximum damage and minimum damage. The higher the balance a player has, the more his maximum damage will be worth in comparison to the minimum damage. Say if a player has 30% balance, his maximum damage would be worth 30% compared to his minimum damage which would be worth 100-30=70% in their contribution to the average damage output, a ratio of 3:7. This ratio means that at 30% balance a player can either gain 3 minimum damage or 7 maximum damage, and the additions would have the same effect on the average damage output. In other words, 3 minimum damage is worth 7 maximum damage. And with this it becomes obvious that minimum damage is of more importance than maximum damage if a player were stuck at 30% balance.
  
  For a person with damage range of 50-100 at 30% balance, his average damage output can be calculated by adding together the weighted version of the min and max damage:
  
  

  Average damage=Max*balance+min*(1-balance)
  
  or for our example:
  
  100*0.3+50*(1-0.3) = 65

  
  
  This means that on average this person will do 65 damage. The earlier equation can be simplified to: Average damage=(max-min)*balance+min, which is the formula on the wiki.
  
  For most people however, their balance is capped at 80%. This means that the min to max ratio is 8:2, or simply 4:1. Every 4 minimum damage is worth 1 maximum damage when a player manages to cap his balance.
  
  Mabinogi uses a standard normal distribution curve as the model to generate damage at random with the min~max as the range of the curve. The normal distribution curve of a person with 50~100 damage range at 50% balance would look like this (not to scale, the actual curve would be wider):
  

  [Image: http://mabiumby.files.wordpress.com/2010/05/50balance.png]

  
  
  Note that the mean of the distribution curve, or the average damage output of the player in this case is 75, half way point of the damage range.
  While the normal distribution curve of a person with the same damage range at 80% balance would look like this (notice the difference in the mean):
  
  

  [Image: http://mabiumby.files.wordpress.com/2010/05/80balance.png]

  
  
  The mean of this curve is 90, this means that usually the damage output of a normal hit will be something close to 90 (when a player is dual wielding it’s according to the damage on each hand). Notice that the range at each side of the curve is not of the same scale due to the player’s balance, this shows that half of the time damage higher than the mean will appear while the other half of the time damage lower than (though usually close to) the mean will appear. Since the range on one side is much smaller than the range on the other, the probability of the player hitting for 90~100 is much higher than the probability of the player hitting for anything significantly lower than 90. This explains the stability that comes along with higher balance.
  
  This is the basic statistics behind Mabinogi’s damage calculation system. You might ask yourself what the point of understanding all this is. Well for one, with this you would know which weapon or damage range would have the highest average damage output.
  
  

  Battle Sword upgraded to 21-74 (upgrade not available until g10)
  
  74*0.8+21*0.2 = 63.4
  
  Gladius upgraded to 41-69
  
  69*0.8+41*0.2 = 63.4
  
  Broad Sword upgraded to 0-75
  
  75*0.8+0*0.2 = 60
  
  Bastard Sword upgraded to 32-77 (upgrade not available until g10)
  
  77*0.8+32*0.2 = 68
  
  Beholder Sword upgraded to 0-89
  
  (89-0)x0.8+0 = 71.2
  
  War Sword upgraded to 0-90
  
  (90-0)x0.8+0 = 72

  
  
  This does not mean that gladius is better than broadsword though, because even though the gladius has better damage output on average, the broadsword could dish out some higher damages on rare occasions.
  
  You could also calculate the average damage output of bastard sword vs muramasa vs gladius ego at max stats to determine which is best damage wise.
  
  

  Bastard Sword ego at capped level 40 stats is 54~156,
  
  54*0.2+156*0.8 = 135.6
  
  [LEFT]Muramasa ego at capped level 40 stats is ??~149, being a muramasa owner myself though, seems like the min wont hit much higher than 55 at lvl 40 dex/int.[/LEFT]
  
  55*0.2+149*0.8 = 130.2
  
  And finally the most popular Gladius ego becomes 87~145.
  
  87*0.2+145*0.8 = 133.4

  
  
  So as demonstrated above, damage wise bastard sword wins indisputably. Gladius is not very far behind and Muramasa falls to third place, at least for now. Considering the fact that one handed sword egos are mainly used for windmill damage, bastard sword’s lack in critical % is not such a big deal for pve in my opinion. On an unrelated note, a friend of mine has asked me why I am spending so much money on a muramasa ego even though it’s the worst off damage wise. The reason is simple, after uncap the subtle difference in maximum damage between gladius, bastard sword, and muramasa is going to become larger. With 1 maximum damage being worth 4 minimum, I think you will understand why the seemingly godly minimum damage growth of gladius will not be enough to make up for its severe lacking in max. However I am not saying the gladius is going to be the worst of the 3. Because of the random nature of mabinogi’s damage system, gladius’s future ~102 minimum damage will be useful in making sure that even on rare occasions the bad damage outputs will not be horrible. This stability will definitely be useful in many occasions.
  //
• Jando wrote on 2010-10-08 19:54
  This is a great guide that opened my eyes. But he didn't add in the average damage from criticals.
• hengsheng120 wrote on 2010-10-08 20:00
  the damage balance range doesn't follow a normal distribution curve
• Hei wrote on 2010-10-08 20:18

  Quote from Jando;179374:

  This is a great guide that opened my eyes. But he didn't add in the average damage from criticals.

  
  
  Thanks.
  
  average critical damage can be calculated just from +150% of your damage range, assuming you have rank 1. So your critical range is (min+min(1.5))~(max+max(1.5)) for a normal hit.
  
  Same with additions from smash or windmill.
  
  

  Quote from hengsheng120;179377:

  the damage balance range doesn't follow a normal distribution curve

  
  
  First of all, I'd have to know how you define this term "damage balance range." I have no idea what that is.
• Jando wrote on 2010-10-08 20:25

  Quote from Hei;179388:

  Thanks.
  
  average critical damage can be calculated just from +150% of your damage range, assuming you have rank 1. So your critical range is (min+min(1.5))~(max+max(1.5)) for a normal hit.
  
  Same with additions from smash or windmill.

  
  
  What about average Critical Hit?
• Hei wrote on 2010-10-08 20:29

  Quote from Jando;179392:

  What about average Critical Hit?

  
  
  You can just apply the same equation, Average damage=Max*balance+min*(1-balance), with the range you got from adding up the critical addition.
  
  Recently there was this idea about maximum damage being solely used to calculate additions from critical hit, I tested this and didn't find any reason to believe this to be true. (so i also corrected a previous post about this)
  
  going to be away for a few days for some real life stuff, be back for discussion later.
  
  Edit: woops, sorry about all these double posts, I keep forgetting.
• Justified wrote on 2010-10-08 21:46

  Quote from Hei;179388:

  average critical damage can be calculated just from +150% of your damage range, assuming you have rank 1.

  
  
  On a crit, only your normal damage uses balance. The critical portion is always maximal, that's why many people don't care about Min.
  
  Average Crit damage would be [(Average Base damage) * Skill multiplier] + [Max damage * Skill Multiplier * Crit Multiplier (1.5 at Rank 1)]
• Phanterz wrote on 2010-10-09 01:08
  A 30% critical rate at rank 1 critical grants you an extra +45% of your max damage. (0.3*150%)
  
  e.g. 50-100 damage at 80% balance and 30% critical will be 90 damage base + 0.45(100) = 135
  
  Unless I'm mentally retarded and can't do math anymore
  
  If we factor this in to the above weapons, the broadsword will outperform the gladius, in all areas but stability.
• hengsheng120 wrote on 2010-10-09 02:43

  Quote from Hei;179388:

  First of all, I'd have to know how you define this term "damage balance range." I have no idea what that is.

  
  
  it's just the balance modified damage range. In any case, providing the the testing data and statistical validation would be nice.
• Omegatronic wrote on 2010-10-09 14:14

  Quote from Phanterz;179660:

  
  If we factor this in to the above weapons, the broadsword will outperform the gladius, in all areas but stability.

  
  
  That isn't true.
  
  First, unless we are talking about ohko's average damage is stability.
  
  second,
  Glad: 69*0.8+41*0.2 + 69 * .45 = 94.45
  Broad: 75*0.8+0*0.2 + 75 * .45 = 93.75
  
  Where the Broad shines over the Glad is when you do not have 29 or 30% crit with the Glad, with its 0% base crit rate.
  
  

  Quote from Hei;179395:

  
  Recently there was this idea about maximum damage being solely used to calculate additions from critical hit, I tested this and didn't find any reason to believe this to be true. (so i also corrected a previous post about this)

  
  
  The recent idea is that crit damage does not modify max only.
  
  I spent 30 minutes of non-stop fox killing and failed to see a single crit below modified min dmg + Max dmg*crit - def. Though, that may not have been enough of a sample to overcome my 80% balance if otherwise. Though I would expect at least one occurrence if it were otherwise, probability is a fickle mistress and my mind remains open to the possibility.
• Hei wrote on 2010-10-09 19:58
  I got back early.
  
  

  Quote from hengsheng120;179720:

  it's just the balance modified damage range. In any case, providing the the testing data and statistical validation would be nice.

  
  
  It would be nice indeed. Though I have neither the time nor the will to be collecting a large enough amount of data to prove this. I've been told that this is how it works by people who have played the foreign server, and that people have gathered a large amount of data and found this trend. I took their word because I found this explanation to be reasonable.
  
  Regarding the "damage balance range" you mentioned. Of course an actual normal curve could not have two different scales on each side. However if you think of it as a seperate correspondence this would work. The model figures out how many standard deviations away from the mean you have hit, then corresponds the damage at the same position from the balance modified damage range.
  
  Anyway my article was more about explaining how things work instead of proving it. You can disagree with it by tendency, or just go out there and test it out yourself.
  
  For magic balance where balance is 100%, I believe that the graph would look something like this for someone with range 50~100:
  

  [Image: http://img830.imageshack.us/img830/606/100balance.gif]

  
  For those players with 100% magic balance, this would explain why you still hit lower than the max damage sometimes, but still hit that same number (your max) quite often.
  
  
  
  

  Quote from Omegatronic;179919:

  
  
  The recent idea is that crit damage does not modify max only.
  
  I spent 30 minutes of non-stop fox killing and failed to see a single crit below modified min dmg + Max dmg*crit - def. Though, that may not have been enough of a sample to overcome my 80% balance if otherwise. Though I would expect at least one occurrence if it were otherwise, probability is a fickle mistress and my mind remains open to the possibility.

  
  
  
  That equation seems weird, by crit do you mean the percentage addition dependent on rank? Anyway it should be critdmg=min+modifier*max-def if it were calculated solely from max.
  
  Also I wasn't talking about what you found out recently, but what I did (because I don't know you). I believed before that additional damage from critical hit is from whatever damage occurs, and recently found people saying it's from only maximum damage.
  I did a test for criticals on defenseless rats outside of dun, and hit several times below the min+max*modifier-1 number. so unless I did anything wrong that theory goes out the door.
  
  Specifics:
  My elf has a damage range of 18-41 and rank F crit, so if maxdmg were used to calculate damage added, then it couldn’t have hit anything lower than critdmg=18+41*0.5-1=37.5. I did a 31. This would make sense if the min crit possible were: critdmg=18+18*0.5-1=26 which 31 is bigger than.
• Omegatronic wrote on 2010-10-10 01:38

  Quote from Hei;180130:

  
  That equation seems weird, by crit do you mean the percentage addition dependent on rank? Anyway it should be critdmg=min+modifier*max-def if it were calculated solely from max.

  
  
  Same formula. I was rushed and used "crit" to mean modifier from crit.
  
  

  Quote from Hei;180130:

  
  Also I wasn't talking about what you found out recently, but what I did (because I don't know you). I believed before that additional damage from critical hit is from whatever damage occurs, and recently found people saying it's from only maximum damage.

  
  
  No worries. For me it's the opposite, the idea that crit doesn't modify only max damage is the new idea. For quite a long time, the wiki stated it did. (check the discussion tab) I think we are both after the same thing, to understand the game as well as possible.
  
  

  Quote from Hei;180130:

  
  I did a test for criticals on defenseless rats outside of dun, and hit several times below the min+max*modifier-1 number. so unless I did anything wrong that theory goes out the door.
  
  Specifics:
  My elf has a damage range of 18-41 and rank F crit, so if maxdmg were used to calculate damage added, then it couldn’t have hit anything lower than critdmg=18+41*0.5-1=37.5. I did a 31. This would make sense if the min crit possible were: critdmg=18+18*0.5-1=26 which 31 is bigger than.

  
  
  Not doubting your results but to cover any differences in our tests (other than the development of the character)-- was that melee or range damage? Were you free of any other damage modifiers like elementals? Was your weapon enchanted?
  
  When I get a chance I'll spend more time trying to replicate your results. I just wont use the character with 80% balance all the time. ;-)
• Justified wrote on 2010-10-10 01:57

  Quote from Hei;180130:

  
  My elf has a damage range of 18-41 and rank F crit, so if maxdmg were used to calculate damage added, then it couldn’t have hit anything lower than critdmg=18+41*0.5-1=37.5. I did a 31. This would make sense if the min crit possible were: critdmg=18+18*0.5-1=26 which 31 is bigger than.

  
  
  Low numbers are funky to work with. It's better to draw a conclusion using higher damage along with lower balance.
  I have a setup with 88-178(?) damage with 100 crit and 23 balance I can play with if I find time.
• Kaeporo wrote on 2010-10-10 04:18
  I was under the impression that final defense was subtracted from final offense. These are the commonly accepted damage formulas.
  
  Formula: Average Damage (non-critical)
  (max-min)balance+min
  Formula: Average Damage (critical)
  (max-min)balance+min+(critical multiplier x max)
  Formula: Average Damage
  (max-min)balance+min+(critical chance x critical multiplier x max)
  
  After determining your final damage (crit/normal) subtract the opponent's defense and then multiply their protection and your damage to receive your (final damage range/average damage).
  
  If I remember correctly, damage balance is plotted on a weighted bell curve. All of those graphs were correct.
• hengsheng120 wrote on 2010-10-10 08:01
  You see, I used to believe in the theory that it does follow a standard *(that's what i meant) normal distribution curve (and yes I already knew about the min/max damage that cuts the curve) until someone refuted my idea of it too. You mentioned that the curve is actually wider (but how much), but since we don't have any test data, so we don't know the true variance of the curve.
  
  as you can see it could follow a normal distribution and still vary by a lot
  
  

  [Image: http://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/1000px-Normal_Distribution_PDF.svg.png]

  
  
  Basically, we don't know the actual probability/frequency of hitting the average damage, minimum damage nor maximum damage, which are big factors
  
  For example, we know hitting the minimum damage is not really near zero, so the frequency would probably not start at 0...
  
  also, the fact that it may not be a standard normal distribution (not totally random and governed by a formula) makes question the actual shape of the curve (that's not considered a normal distribution but some other name). Or perhaps the bell curve may be skewed towards one side (longer one side than the other and not necessarily symmetrical).
  
  sorry for not making it clear