# Hero Grading: A mathematical analysis

By: Alice – Titan Moist alliance

The King’s Gambit was considered the strongest and most aggressive gambit in chess throughout the 1800’s, and was in fact, still considered a strong opening in the early 1900’s until Capablanca found the line beginning with 3. Bxc4 d5!. This is the evolution of all games and theory. In the beginning, before mathematics are applied to properly analyze, opinion rules the day. Specifically the opinions of those who are seen within a given community as the “experts”, those who are stronger than others. We can figuratively see flaws in any such anecdotal analysis, but we cannot accurately point them out without actually supporting such conclusions with math. Else, it is simply an “opinion fight”, which will generally be won by the existing “authority”.

Heroes have 5 effective uses:

1. Raid defense
2. War defense
3. Raid attacking
4. War attacks
5. Titan striking

To properly understand the utility of a hero within these roles, we have to understand the following about any given hero:

1. Survivability
2. Ability to effect board state, either through damage dealt, increased survivability, Decreased opponent survivability, or effective control.
3. Rate at which damage can be applied or such special ability may be used.

The following assumptions will also be used:

1. All battles are against an AI. AI abilities are used left to right, and target random targets. To make up for this handicap, the AI receives 20 percent bonus damage, autoattacks twice every 5 turns, and receives a bonus unit of mana per turn.
2. The human player receives a bonus that reduces the amount of mana gained for the opponent characters for every successive additional chain of tiles that strikes the opponent.

Fundamental damage:
Tile damage =
.82>n>1.19
33 * n * (total attack stat/total defense stat) ^ (1/ln2)

Special damage =
.94>n>1.05
100 * (special percentage damage) * n * (total attack stat/total defense stat) ^ (1/ln2)

In laymans terms:
Lianna attacking Grimm, both using troops with a 10% bonus to attack and defense respectively. Lianna is the only green in the attacking party:
Our tile expectations:
.82>n>1.19
33 * n * (802/584) ^ (1/ln2) 2 (strong attack)
.82
102.7 < actual < 1.19102.7
So between 85 and 121 damage per tile is the expected value.
To understand why stacking is so effective, let’s look at this same example using 3 lianna’s attacking.
33 * n * (2406/584) ^ (1/ln2) 2 (strong attack) = 466 damage per tile, giving us an effective range of
.82
466 < actual < 1.19
466
Tripling the attack stat does not actually triple the damage (+200%). It actually increases the damage by more than 350%

Let’s replace Grimm with Aegir (815 defense plus 10 percent defense troop), and we see the core principle of survivability:
Single Liana
33 * n * (802/897) ^ (1/ln2) *2 (strong attack) = 56 median damage
Triple Liana
33 * n * (2406/897) ^ (1/ln2) *2 (strong attack) = 266 damage per tile median.
This brings us to the concept of survivability. Aegir would survive 5 green tiles from a stack of 3 liana’s. Grimm would be dead on the 3rd.

How do we measure survivability? In truth, we would also have to examine the reality that if the third part of a chain were 3 green tiles, those tiles would do 20 percent bonus damage (each additional segment in a chain increases tile damage by 10 percent.
To even begin to analyze the state matrix of expected value, we have to understand the following:

1. Our initial board contains 35 tiles, NONE OF WHICH ARE PRESET MATCHES (time=0)
2. There are 3 tile pathways that reach the central enemy, and 1 each for the flanks (the enemies next to the tank), and 1 for the wings (the edge characters).
3. If the central enemy is removed, one pathway is clear, and the two flanks are each vulnerable to 2 pathways.
4. If the flanks are removed, the wings are also vulnerable to 2 pathways.
5. Tiles that are sent that do not hit a character reward no damage but double mana.
6. If the board contains no way to match 3 tiles, it automatically reshuffles and clears automatically for any matches created by reshuffling
7. Matching 4 tiles creates a dragon emblem in the color that was matched. Selecting this tile destroys its 4 neighboring tiles
8. Matching 5 tiles within a color creates a diamond tile. The diamond will automatically launch any tiles of that color (including destroying dragons of that color).

Assumptions:

1. Begin with a 5x7 matrix of tiles each with a 1/5 probability of being any color, you reach an n5 bound
1. replacement tiles also have a 1/5 chance for each color.
N5 bound
[ 1/5 1/5 1/5 1/5 1/5 1/5 1/5 1/5]
[ 1/5 1/5 1/5 1/5 1/5 1/5 1/5 1/5] [ 4/5 0 1/5 ]
[ 1/5 1/5 1/5 1/5 1/5 1/5 1/5 1/5] x [ 4/5 0 1/5 ]
[ 1/5 1/5 1/5 1/5 1/5 1/5 1/5 1/5] [ 4/5 0 1/5]
[ 1/5 1/5 1/5 1/5 1/5 1/5 1/5 1/5]

Repeating this same process for the possibility of 4 matches and 5 matches.
We result in the following probabilities:
There is a 71 percent chance that we can match 3 tiles at the beginning of a color of our choice), a 23 percent chance to match 4, and a 6 percent chance to match 5.

Applying a discreet time Fourier Transform (n=10 here because I’m not being paid to do this, and honestly, most battles are determined by the 10th turn.

N= (0 to 10) P(n) – d(t-n(t)) = F^-1{X(1/t)(n)
Given a randomized board, we can expect to launch a total of 62 tiles in 10 turns, with resultant cascading, with an expected value ranging from 14.6 tiles to 16.1 tiles (the color you have highest preference to launch).

In truth, in the grand picture, nowhere yet. But it does give us a sound understanding, mathematically, that the dude in your alliance who whines about bad boards all the ■■■■ time probably just doesn’t know how to play or stack their attacks.
But what we CAN do, instead, is remove opinion from grading survivability. Instead, what we can put into NUMBERS, is the attack stat that must be present in the opposing attack to reasonably expect to kill your center in 3, 4, 5, or 6 tiles.
The 5 star with the HIGHEST survivability (boss wolf), for example:
826 base defense (with 20 percent defense /7 percent health troop ~ level 15), 1545 health
990 defense, 1650 health
3 tiles requires: 550 damage per tile =
4 tiles requires 412.5 damage per tile
5 tiles requires 330 damage per tile
6 tiles requires 275 damage per tile
Now the lowest, Elena
578 base defense, 1312 health (with same troop, 594 defense, 1404 health)
3 tiles requires: 468 damage per tile =
4 tiles requires 351 damage per tile
5 tiles requires 281 damage per tile
6 tiles requires 234 damage per tile

Boss Wolf
990 defense, 1650 damage per tile divided by 66 raised to (ln2/1) multiplied by defense
550 8.333333333 4.498450501 4453.465996
412.5 6.25 3.668207656 3631.52558
330 5 3.131290838 3099.97793
275 4.166666667 2.751480855 2723.966046
Now the lowest, Elena
578 base defense, 1312 health (with same troop, 594 defense, 1404 health)
468 7.090909091 4.011757686 2318.795942
351 5.318181818 3.271339821 1890.834417
281 4.257575758 2.793923155 1614.887584
234 3.545454545 2.453794804 1418.293397

Thus, For pure survivability, The baseline numbers of Boss Wolf would be a grade of 100, Whereas Elena is a 0. It takes a combined attack stat (with troops, of nearly 4500 to kill him in 3 tiles) This does not mean one hero is better than another overall, this is JUST an analysis of pure survivability. Before you go screaming that nobody uses defense troops, yes, I understand this. But both heroes scale the exact same way. Using the raw statistics would yield the exact same percentage discrepancy.

But it also highlights on the reality that a spreadsheet that gives Boss Wolf an A on survivability and Elena a C, is mathematically useless.

Looking at one in the middle, for example, say Marjana. (712 defense, 1404 health base (856 defense, 1502 health)
501 7.590909091 4.210385265 3604.089787
375.5 5.689393939 3.431688075 2937.524992
301 4.560606061 2.933539137 2511.109501
250 3.787878788 2.571638867 2201.32287

By “spreadsheet science”, this result in a grade of 2 A’s and a B. This despite a level of disparity that borders upon the absurd in relative value.

Now that we have blown the concept of stamina grades out of the water (which is all we have done so far) let’s look at other things.

Titan utility:
A proper and maximized titan team should contain the following, usually in a 3/2, 4/1 or monochrome color stack), in order of importance

Wu/tarlak
A defense debuff for the titan
A specific color defense debuff for the titan against the strong color stack
High aggregate attack stat in the strong color against the titan
An attack or crit buffer
A healer

Nice to have utilities:

1. Shared damage
2. Cleanse for any debuffs
3. Dispeller
4. Redundant attack or crit buffs.

More to come, and help is always appreciated! I would like to start to get people to stop opining randomness and let’s actually do the math. Listening purely to opinion is going to homogenize opinion and bad judgement as opposed to actually give us an effective starting point by which we can assess actual heroes.

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Amazing work, thanks

That is a lot of words there.

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