User blog:Sylvandyr/The math behind the Battle Arena

Overview
The Battle Arena uses a point system and a game-generated random range to determine the damage done by a dragon's attack. Each factor (type, rarity, and level) contributes a certain number of points, and these points are combined to determine your chances of scoring your main objective: a critical hit to defeat your opponent.

You may increase your chances of victory by maximizing your point total, but you are not necessarily guaranteed to win even with the same pair of battling dragons.

Attack Selection
Your dragon has the Bash, Magic, and Ultimate attacks. While all damage fluctuates slightly according to a game-defined range, using the Ultimate attack guarantees a critical hit, which deals an incredible amount of damage to the enemy dragon.

If you do not wish to pay for the Ultimate attack, it does not matter whether you select Bash or Magic. While you may sometimes see differences in damage between the Bash and Magic attacks, this is actually due to the slightly random variance in damage as well as the occasional critical hit, which deals a significant amount of damage more than normal.

Point Distribution
Points are distributed differently depending upon the factors.

Type

Your dragon is assigned a type score based on how all of its colors compare to all of the enemy's colors. This is a straightforward additive process. Every type of your dragon is compared to each of the enemy dragon's types. For every strength that each of your dragon's colors has against the enemy dragon's colors, you gain one point. For every weakness, you lose one point.

Once your type score is calculated, you will receive points based on the following chart:

Rarity

Your dragon and your opponent's dragon are assigned rarity scores based upon their rarity types. However, just because a dragon is "Super Rare" in the game doesn't necessarily mean that its rarity type is the same.

The following chart lists the different rarity types, the corresponding score awarded for that type, and the dragons that fall under that category:

Once the rarity scores are calculated, the rarity index is calculated in the following manner:

Enemy rarity score - Your dragon's rarity score = Rarity index

The following chart lists the rarity indices and the corresponding points awarded towards your points from rarity:

Level

Your dragon is assigned a level score, which is calculated in the following manner:

Level of your dragon - Level of enemy dragon = Level score

Once your level score is calculated, you will receive points based on the following chart:

Determining Crit Bonus
Your point total is the sum of points from type, points from rarity, and points from level. This is mapped to your crit bonus (which increases your chance of a critical hit) as shown in the following chart:

Determining Attack Damage
Your attack roll is a random number determined by the game and is completely out of your control. The same roll affects your damage, the enemy's damage, and whether or not your dragon will land a critical attack.

Damage is calculated in the following manner:

Your dragon's regular damage = 25 + (a random number between 0 and 8)

Your dragon's critical damage = 60 + (a random number between 0 and 100)

Enemy dragon's regular damage = 34 + (a random number between 0 and 15)

Rolling a critical attack allows your dragon to deal its maximum damage for that attack. A critical max is derived from the crit bonus with the following formula:

Critical max = 1000 * (1 - (1 - Crit bonus)^(1/3))

If your attack roll falls under the critical max, then your dragon will deal a critical attack. This means that the higher your critical max, the better your chances of rolling a critical attack. You are not guaranteed to roll a critical attack unless you use your dragon's Ultimate move, which will automatically grant you a critical attack regardless of any random factors.

Conclusion
Since both dragons have 100 health and the numbers are stacked in favor of the enemy dragon, you can see that it is impossible for you to win without a critical hit (and that a single critical hit will guarantee your victory). Here is proof:

With the minimum damage of 34 per turn, the enemy dragon will always beat your dragon in 3 attacks (34 * 3 = 102).

With the maximum damage of 33 per turn, your dragon will never beat the enemy dragon with three non-crit attacks (33 * 3 = 99).

With the maximum damage of 49 per turn, the enemy dragon can never beat your dragon in 2 attacks (49 * 2 = 98).

With the minimum damage of 60 for a single crit and the minimum damage of 25 for two normal attacks, your dragon can always beat the enemy dragon in three total hits (60 + 25 + 25 = 110).

Since your dragon moves first and the enemy dragon will never score a critical hit, your dragon will always get three chances to score a critical hit. Thus, winning is all about whether or not you can score a critical hit within three turns.

Battle Arena Calculator
Select your opponent dragon and level, then press the "Find Best Matches" button. This will calculate your chances to win against the selected opponent with the different available dragons.

BattleArenaData

BattleArenaSelection