Color Stacking: The Odds

[Gryphonknight]

@Edit

Defense

This also explains why color stacking works so poorly on defense. A.I. Damage is always colorless. So the increased vulnerability is not balanced with an increased attack ( the +20% final damage bonus roughly balanced a rainbow teams attack advantage ).

A.I. Cheats

@Gryphonknight

==20% bonus damage buff==
Players can increase damage by making matches of a color strong against the defending hero. Since the A.I. Cannot, they get a flat 20% bonus ( which is roughly the bonus of a rainbow team against a single color boss due to strong matches versus a team of all neutral colors fighting the same boss ).

You can preview the raid defense team and pick the best colors to attack.

Garbled info

@KnifeWonder

5 reds for example, you get a small bonus to the overall attack too, like basically if your attack power is 100% when having a monochrome team it will be like 105/110%. Is it true or just a myth?

That sounds like a garbled version of 5 red doing double damage versus a green boss and neutral damage versus a red boss in a stage with two bosses.

Or HotM with Elemental link +5% attack +5% defense for heroes of a matching color.

Where is your happy?

@Kikyo

All I’m saying is that the current color stacking formula provides an increase in the number of offenses that can beat a given defense.

This.

I repeat it over and over and over, but attacking always has to be better than defense from a human factor point of view. Who wants to beat their head against a stone wall every day in a game.

Current research on color stacking is headed towards the conclusion that is is a very strong strategy. This is almost identical to the actually broken haste mechanic in Book of Heroes - even the Devs realized Haste was broken and should have used a linear formula. After several failed attempts to reverse course, Book of heroes just learned to live with it and move on. In fact it became a driving force for several seasonal events and seasonal event loots. Taking life’s lemons and making a lemon orchard, to mangle a common expression.

Similarly in Empires, color stacking is now a fundamental mechanic of roster building and defense planning.

Missing link

@Garanwyn

So instead of your 3-stacked tile attack being 3x as powerful as a single-hero tile attack, it’s 4.4x as powerful. That’s huge.

Fudge. Totally missed this. But it follows from the base design of MMOs. Usually in MMOs a 5* hero does exponentially increased damage versus 1* enemies to encourage updgrading heroes.

Color stacking just turns this effect up to 11. Further increasing the euphoric rush.

From a game design point of view it further reinforces the use of color specific 3*/4* ascension items making rosters of five rainbow 5* teams even more powerful and desirable than two rainbow 5* teams.

But.

Missing heroes doing a fixed 1 HP damage does balance this out especially with the 90 seconds of terror known as titans.

I will now change how I assign 3*/4* troops and class nodes for heroes I usually color stack.

This piece of info appears to be the missing link for Gryphonkit’s insanely powerful Obakon, Rigard, Tiburtus, Cyprian, Hansel attack team ( the largest factor being how their specials interact ).

Aside, Gryphonkit is really annoyed at me for being on the forums when I should be resting. Oh well, that is what weekends are for, bad choices.

MMO stat curve soft cap

@Garanwyn

But in running my own tests on PvE, this doesn’t seem to be holding up. I can’t tell yet if the problem is:

1. They split out the attacks individually at some point, like Kerridoc suggested

or

2. There really is some linearization of the X^1.35 calculation as damage increases.

It looks, after about 80 sample points, as though #2 is more likely than #1. More research is definitely needed.

I am not surprised by this result. Most MMOs have a base stat curve then a soft cap or a hard cap. This sounds like Empires implemented a soft cap where they swap out a second equation ( or a second set of constants for the core equation ) once a stat value reaches X.

In Book of Heroes the soft cap increased with the number of stars, which required us to solve the soft cap equation before we could confirm the core stat curve. Empires appears to use the same equations for 1* heroes and for 5* heroes. More research is required.

Trying to track down the value of X for the soft cap, if it exists, will dramatically simplify your research since your are trying to solve for one equation when the data results from two very different equations in an “if then” situation.

Move along, move along

You probably want to skip this if not Garanwyn”

Simplification of Complex Dynamic Systems

@Garanwyn

The simplification here is that 35!/(35-X)! can actually be cancelled down to:

35 * 34 * 33 * … * (35-X+1)

So if X = 3, for example:

35!/((35-X)! * X! = 35 * 34 / (3 * 2 * 1) = 198.333

Your a wizard Harry, Er, Garanwyn.

If I understand half your math here, you are brute forcing the total tiles for small number of X tiles per 35 replacement spaces on the assumption that 3 tile will most likely not be continguous because math is not really set up to deal with the probabilities of a colored topological problem. Simplify, simplify, simplify- may be the unofficial motto of math.

This would seem to leave a large number of replacement conditions out such as combos. Due to real world versus ideal setting the Devs have several times nerfed A.I. Mana gain from combos due to defense topological conditions - mainly centers and flanks - and that would seem to indicate perceived reward versus actual mechanical odds are an important factor in game play especially mono team play.

I am bringing this up for peer review because my past experience with these game mechanic analysis lead me to believe this exponentially compounds the feeling of euphoria and rage with a mono team without significantly effecting your base mathematical model.

But might indicate that you have simplified too much and underestimated the player impact on use and reuse of mono teams.

Edit:
Actually the exponential effect of color stacking appears to be a big factor mitigated by the RNG of boards.

Analyzing replacement conditions

I tried to simplify a colored topographical program problem by examining a small sub section of the topography problem involving normally high gain conditions- power shard formation. A condition already highly charged emotionally for a player. Further magnified by a mono team ( “Yay purple power shard” or “Boo not a purple power shard”).

@Gryphonknight

odds zero mana (and almost zero damage) from:

row on board (7)- 1 / 0.8^7 = 1 in 5
column on board (5)- 1 / 0.8^5 = 1 in 3
Power shard or initial match 5 to create power shard = 4 in 5

@Gryphonknight

For a random match 3 board, every time a shield is removed from the board, it can be replaced by any of the other colors. So removing 5 shield from the board has a 1 in 3125 chance of replacing them with 5 purple shields. If your team does not contain a purple hero, you have 5 shields that do almost zero damage to the boss and generate zero mana for your hero.

Reaching the same conclusion

There seems to be missing work to be done here but both approaches appear to be sound, if limited, analysis of boards and mono teams. Which as far as I know is a unique field limited to Empires and its clones.

The conclusion being rainbow teams and mono teams have checks and balances that make class and special skill synergy more important than color of hero.

Note: the forced exception is Elemental shield enemies where heroes can do negative damage based on color.