Colosseum Damage Formula

Note: this is the formula to use during normal Arena events, where you create a team of 5 units and fight with the aid 2 random allies. The formula to use during Battle Arena events, where you create a team of 7 units and fight without the aid of your allies, is found in a different topic: Battle Arena Damage Formula.

During Arena events, the damage units deal is determined by these factors:

• RANK - The rank of the unit. In other words, the amount of stars it has. However, RANK kan never be higher than 6! For units with 7 or more stars, use 6 as the value of RANK. For example: Orlando is a 2* unit. His rank is 2. Anton is a 4* unit. His rank is 4. Lilith is a 7* unit. Her rank is 6.
  
• HIGH BASE DMG - The highest number of the unit's 3 attack numbers: land, sea or air. Bonuses, such as skills that increase dmg by a cerain percentage, are not included. In other words, this is the number you see on the card when it's in your unit list, not when it's in battle. For example: at level 80 without any bonuses, Anton has 36528 land damage, 0 air damage and 18586 sea damage. His highest base damage is 36528. At level 100 without any bonuses, Rose has 36250 land damage, 38800 air damage and 35240 sea damage. Her highest base damage is 38800.
  
• MID BASE DMG - The second-highest number of the unit's 3 attack numbers: land, sea or air. Bonuses, such as skills that increase dmg by a cerain percentage, are not included. In other words, this is the number you see on the card when it's in your unit list, not when it's in battle. For example: continuing on from the examples of Anton and Rose, Anton's second-highest base damage is 18586 and Rose's second-highest base damage is 36250.
  
• HIGH DMG BONUS - The sum of the percentages bonus that the unit is granted on his highest base attack number. If a unit's highest base attack is his land attack, the bonus percentage is the sum of the level of your Master of Land skill and the level of your Bringer of Victory skill. For example, let's say your Master of Land skill is at level 7, your Ruler of Air skill is at level 5, your Parter of Sea skill is at level 1 and your Bringer of Victory skill is at level 0. The bonus percentage for Anton's highest attack number is 7 (for Master of land) and the bonus percentage for Rose's highest attack number is 5 (for Ruler of Air). If you have Bringer of Victory at level 10, the bonus percentage for any attack number is always 20 (for Master of Land/Ruler of Air/Parter of Sea plus Bringer of Victory).
  
• MID DMG BONUS - The sum of the percentages bonus that the unit is granted on his second-highest base attack number. If a unit's second-highest base attack is his sea attack, the bonus percentage is the sum of the level of your Parter of Sea skill and the level of your Bringer of Victory skill. For example, continuing on from the examples of Anton and Rose and the earlier described skill levels, the bonus percentage for Anton's second-highest attack number is 1 (for Parter of Sea) and the bonus percentage for Rose's second-highest attack number is 7 (for Master of Land). If you have Bringer of Victory at level 10, the bonus percentage for any attack number is always 20 (for Master of Land/Ruler of Air/Parter of Sea plus Bringer of Victory).
  

The formula to calculate the damage dealt in an Arena battle is as follows:


Arena Damage = ( 1 - RANK ^ 2 / 100 ) * ( HIGH BASE DMG * (1 + HIGH DMG BONUS / 100) + MID BASE DMG * (1 + MID DMG BONUS / 100) )

What's important to note is that "HIGH BASE DMG * (1 + HIGH DMG BONUS / 100)" and "MID BASE DMG * (1 + MID DMG BONUS / 100)" are rounded up or down to an integer number (down if the first decimal is smaller or equal to 4, up if the first decimal is greater or equal to 5) before they're added together. The final Arena Damage is rounded down to an integer number.


For example: let's calculate the Arena Damage for a level 80 Anton. Let's say your Master of Land skill is at level 7, your Ruler of Air skill is at level 5, your Parter of Sea skill is at level 1 and your Bringer of Victory skill is at level 0. Anton has rank 4. His highest base damage is 36528, which receives a 7% bonus. His second-highest damage is 18586, which receives a bonus of 1%.
Arena Damage = ( 1 - RANK ^ 2 / 100) * ...
Arena Damage = ( 1 - 4 ^ 2 / 100) * ...
Arena Damage = ( 1 - 16 / 100) * ...
Arena Damage = ( 1 - 0.16) * ...
Arena Damage = 0.84 * ...
Arena Damage = 0.84 * ( HIGH BASE DMG * (1 + HIGH DMG BONUS / 100) + MID BASE DMG * (1 + MID DMG BONUS / 100) )
Arena Damage = 0.84 * ( 36528 * (1 + 7 / 100) + 18586 * (1 + 1 / 100) )
Arena Damage = 0.84 * ( 36528 * (1 + 0.07) + 18586 * (1 + 0.01) )
Arena Damage = 0.84 * ( 36528 * 1.07 + 18586 * 1.01 )
Arena Damage = 0.84 * ( 39084.96 + 18771.86 )
Arena Damage = 0.84 * ( 39085 + 18772 )
Arena Damage = 0.84 * 57857
Arena Damage = 48599.88
Arena Damage = 48599


If you have Bringer of Victory at level 10 and you want to simplify the formula, here's how you rewrite it:

Arena Damage = ( 1 - RANK ^ 2 / 100 ) * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

Note that the base damage is the number you see on the card when it's in your unit list. Base damage times 1.2 is the number you see on the card when it's in battle (could be higher if there are any units with Atk Boost in the battle).

If you want to simplify the formula any further, you could split it into different formula's for each rank:

7* Arena Damage = 0.64 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

6* Arena Damage = 0.64 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

5* Arena Damage = 0.75 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

4* Arena Damage = 0.84 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

You might feel tempted to rewrite "( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )" as "( HIGH BASE DMG + MID BASE DMG ) * 1.2", but remember that "HIGH BASE DMG * 1.2" and "MID BASE DMG * 1.2" need to be rounded before you add them together, so you shouldn't rewrite the formula in that way.


For anyone wanting to use this formula in a Spreadsheet where you have the base damage numbers of all characters and you want to assume there are no bonuses, here's how you would rewrite the formula (you would have to replace "RANK", "LAND ATK", "AIR ATK" and "SEA ATK" with the appropiate columns):

=ROUNDDOWN((1-MIN(RANK;6)^2/100)*(SUM(LAND ATK;AIR ATK;SEA ATK)-MIN(LAND ATK;AIR ATK;SEA ATK));0)

Or if you want to assume a 20% bonus:

=ROUNDDOWN((1-MIN(RANK;6)^2/100)*(SUM(ROUND(LAND ATK*1.2;0);ROUND(AIR ATK*1.2;0);ROUND(SEA ATK*1.2;0))-MIN(ROUND(LAND ATK*1.2;0);ROUND(AIR ATK*1.2;0);ROUND(SEA ATK*1.2;0)));0)

I suspect that this is the part of above post that most people will revisit this topic for to find, so I will repeat it in this post:

Colaleon wrote:

If you have Bringer of Victory at level 10 and you want to simplify the formula, here's how you rewrite it:

7* Arena Damage = 0.64 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

6* Arena Damage = 0.64 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

5* Arena Damage = 0.75 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

4* Arena Damage = 0.84 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

Remember you need to round "HIGH BASE DMG * 1.2" and "MID BASE DMG * 1.2" before adding them together. The final Arena Damage result needs to be rounded down.



A note on the event items
During Arena events you typically have special items at your disposal. These items come in 2 sizes: small and large. Not only do they increase the amount of BT you earn with each Arena battle, but they also increase the damage your units deal. The small item adds 20% to the damage your units deal. The large item adds 50% to the damage your units deal. Only one such bonus can be activated at the same time. The bonus lasts for 15 minutes from the time the item is used.

Remember how Arena Damage is rounded down at the end of the calculation? What's important to note is that the bonus granted by items is added before this final rounding down. In other words, the final damage number is only rounded down after the item bonus is applied. The formulas are as follows:

Small item Arena Damage = 1.2 * ( 1 - RANK ^ 2 / 100 ) * ( HIGH BASE DMG * (1 + HIGH DMG BONUS / 100) + MID BASE DMG * (1 + MID DMG BONUS / 100) )

Large item Arena Damage = 1.5 * ( 1 - RANK ^ 2 / 100 ) * ( HIGH BASE DMG * (1 + HIGH DMG BONUS / 100) + MID BASE DMG * (1 + MID DMG BONUS / 100) )

Or, split into unit ranks and simplified for Bringer of Victory at level 10:

7* Small item Arena Damage = 0.768 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )
7* Large item Arena Damage = 0.96 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

6* Small item Arena Damage = 0.768 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )
6* Large item Arena Damage = 0.96 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

5* Small item Arena Damage = 0.9 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )
5* Large item Arena Damage = 1.125 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

4* Small item Arena Damage = 1.008 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )
4* Large item Arena Damage = 1.26 * ( HIGH BASE DMG * 1.2 + MID BASE DMG * 1.2 )

Remember you need to round "HIGH BASE DMG * 1.2" and "MID BASE DMG * 1.2" before adding them together. The final Arena Damage result needs to be rounded down.

oh gawd, this one is definitely one of many greatest invention I ever see in my life

grats for you Cola..

now with this, we can help mister madman completed his arena dmg data list

Great works, thank you very much. Bravo!

I added a note about event items to the second post.

HMm... very nice

Impressive

Thinking of adding a column to the wiki pages for cards. Which wiki do you guys use more? I tend to use fantasicawiki, so that's where I was thinking.

I need something to do while grinding

will try in enxt arena..not that i dont trust you..i just never acept data without proof..but thanks !!!!!

Cola:
The last two formulas you put in your OP (the ones where you take total and subtract min damage) would give slightly wrong answers, no? Seeing as you have to round each of the damages (Max and Mid) before adding them together, wouldn't taking total-min then remove that part of the formula?

Also, I remember you saying something in the original research thread that you needed to always round down the damages, but there is no such indication here. Was that a mistake in the original development, or in this one?

I ask because I'm working with Kadaan to put these numbers in the unofficial wiki and would like to get it perfect .

Sabin76 wrote:

The last two formulas you put in your OP (the ones where you take total and subtract min damage) would give slightly wrong answers, no? Seeing as you have to round each of the damages (Max and Mid) before adding them together, wouldn't taking total-min then remove that part of the formula?

Yes, you're right when you say you have to round the max and the mid damages before adding them together. There's no mistake in the formulas though. Let me try to clear up the confusion.

First, I'm going to quote the part of my original post where I think the rounding of numbers is best explained:

Colaleon wrote:

What's important to note is that "HIGH BASE DMG * (1 + HIGH DMG BONUS / 100)" and "MID BASE DMG * (1 + MID DMG BONUS / 100)" are rounded up or down to an integer number (down if the first decimal is smaller or equal to 4, up if the first decimal is greater or equal to 5) before they're added together. The final Arena Damage is rounded down to an integer number.

Let me put this in different words. A unit's attack power is increased thanks to your skills. Your skills can add 0% - 20% to the numbers. This sometimes leaves you with a decimal number. That number is rounded down if the first decimal is 0-4; it's rounded up if the first decimal is 5-9. You can verify this very easily: these numbers are also displayed in your unit's stats during quests! For instance, I have an ally with a level 140 Reve. Her ground attack (without skill bonuses) is 92,001. This is the number you see when you select him from your list of ally units during a mission. When you click the "Deploy" button, your skill bonuses are applied to the unit. In my case, I have my skills maxed so 20% is added. Theoratically, that would bring the number to 110,401.2 but that's a decimal number, so it needs to be rounded. The first decimal is 2, so the number is rounded down to 110,401. I can confirm this when I deploy Reve in a quest: her ground attack is 110,401. Now I look at a different ally who has a level 140 Paris. Her sea attack is 69,043 before bonuses are applied. Adding 20% brings the number to 82,851.6. When I deploy Paris, I can see that her modified sea attack is 82,852. This confirms that the number was rounded up where the first decimal was a 6. Feel free to run a few of these tests for yourself if you want to double-check my findings.

Now, let's look at the formulas where you suspected there might be rounding errors. If I understood you correctly, you were talking about the formulas I created for using in spreadsheets:

Colaleon wrote:

For anyone wanting to use this formula in a Spreadsheet where you have the base damage numbers of all characters and you want to assume there are no bonuses, here's how you would rewrite the formula (you would have to replace "RANK", "LAND ATK", "AIR ATK" and "SEA ATK" with the appropiate columns):

=ROUNDDOWN((1-RANK^2/100)*(SUM(LAND ATK;AIR ATK;SEA ATK)-MIN(LAND ATK;AIR ATK;SEA ATK));0)

Or if you want to assume a 20% bonus:

=ROUNDDOWN((1-RANK^2/100)*(SUM(ROUND(LAND ATK*1.2;0);ROUND(AIR ATK*1.2;0);ROUND(SEA ATK*1.2;0))-MIN(ROUND(LAND ATK*1.2;0);ROUND(AIR ATK*1.2;0);ROUND(SEA ATK*1.2;0)));0)

The first formula assumes a 0% bonus for all skills. Since the unmodified attack numbers are never decimal numbers (these are the numbers that you see when you look at the back of a card) and nothing is added to them, they stay exactly the same and will not need to be rounded! The only rounded happens at the very end, when the added attack numbers are muliplied by (1-RANK^2/100), because that multiplication can result in a decimal number. As mentioned earlier: "The final Arena Damage is rounded down to an integer number". This is a bit harder to test, because you can only test it during arena events. I came to the conclusion that the final arena damage is always rounded down when I tested this for my level 80 Ibiza. I used my formulas to calculate that her arena damage should be 43,072.68. When I tried her in the last arena event, I observed her dealing 43,072 damage. That means the number was rounded down, even though the first decimal is a 6.
The second formula assumes a 20% bonus for all skills, so in this case the modified attack numbers can be decimal numbers (unlike when we assume 0% bonus). These numbers need to be rounded. If you look at the formula closely, that's exactly what happens: there's 6 instances of the "ROUND" predicate in the formula. The answers this formula gives will be spot on, without rounding errors.

Sabin76 wrote:

Also, I remember you saying something in the original research thread that you needed to always round down the damages, but there is no such indication here. Was that a mistake in the original development, or in this one?

Yes, the final arena damage needs to be rounded down. I didn't include rounding predicates in any formulas except for the formulas for spreadsheets. Instead, I mentioned how numbers should be rounded in a note under the formulas.


This has turned into quite a wall-of-text. Sorry about that. I hope it's somewhat readable and will clear things up for you, because I think it's great that the arena attack numbers are going to be added to the wiki.

Wow.. I feel pretty dumb right now. Looking back at your formulas I have no idea how I got it in my head that you weren't rounding where you should have. Sorry you had to write all of that out.

The numbers are all added to the wiki now regardless. Thinking of doing a similar column for battle arena, but not sure if that's necessary.

I figure the main rarity pages are mostly for finding and comparing units and since the arena and battle arena damages (while not the same) are based on the same stats, the comparison can be done for battle arena using the arena damage stat. Perhaps a listing of both types of arena damage in base form and with full skills on the actual card pages would be more logical...

Just thinking out loud.

I think they're changing the damage formula for the latest Arena (Colloseum) Event for the ☆7 units (in fact the change has just been applied an hour ago XD), because the current formula is unfavorable to ☆7 units (almost all ☆7 deal less damage than a strong ☆6 like Monica using the old formula).

I'm not sure if they change it just only for the ☆7 or not, because all my ☆6 still deal the same amount of damage as before, but my ☆7 allies just started popping out 200k damage when this morning they were still dealing around the same amount of damage as my 6☆.

According to the old formula, the damage coefficient for ☆7 is
= ( 1 - RANK ^ 2 / 100 )
= ( 1 - 7 ^ 2 / 100)
= 0.51
However, from observation, it seems as if they are using the ☆6 coefficient of 0.64

I'll post more when I find out more about this.

Schdawn wrote:

I think they're changing the damage formula for the latest Arena (Colloseum) Event for the ☆7 units (in fact the change has just been applied an hour ago XD), because the current formula is unfavorable to ☆7 units (almost all ☆7 deal less damage than a strong ☆6 like Monica using the old formula).

I'm not sure if they change it just only for the ☆7 or not, because all my ☆6 still deal the same amount of damage as before, but my ☆7 allies just started popping out 200k damage when this morning they were still dealing around the same amount of damage as my 6☆.

According to the old formula, the damage coefficient for ☆7 is
= ( 1 - RANK ^ 2 / 100 )
= ( 1 - 7 ^ 2 / 100)
= 0.51
However, from observation, it seems as if they are using the ☆6 coefficient of 0.64

I'll post more when I find out more about this.

This doesn't surprise me. I swear we do more cross checking / comparisons than the Dev's themselves...

Schdawn wrote:

According to the old formula, the damage coefficient for ☆7 is 0.51
However, from observation, it seems as if they are using the ☆6 coefficient of 0.64

I'll post more when I find out more about this.

Using an ally's Lilith (non-maxed LB) and an ally's Ingrid (MLB), I've confirmed that the coefficient is 0.51. This means the formula applies to 7* in exactly the same way it does 1*-6*.

I've added lines to describe the 7* formulas where they needed to be added, making this topic up to date again.

They've altered the coefficient for 7* units. The formula for 7* units now uses the same formula as 6* units (i.e. they use the same coefficient).

The formulas in the first and second post have been altered again. Hopefully for the last time.

How come the higher the rank of the unit, the bigger the initial penalty? Is that because they have higher max level so they have a higher potential to be stronger?

Update: I renamed the topic from "Arena Damage Formula" to "Colosseum Damage Formula", because apparently people were mixing this topic up with the Battle Damage Formula topic.

I actually think we should call it a Colasseum event from now on. That would be even cooler.

seanlink wrote:

How come the higher the rank of the unit, the bigger the initial penalty?  Is that because they have higher max level so they have a higher potential to be stronger?

You should email Mobage and ask them. Let us know if you get a reply.

Spoiler:

I will no longer be updating this topic.