iDB Community Forum

Viability of Intermediate Freeze-bombing. (2-player)

This is an article based on the numbers/averages side of freeze-bombing. In a world of out-of-reach 100% freezers, many items such as frost cannons and the Snowglobe staff are finding spaces in line-ups. However, if we consider the numbers, some of these options may not prove to beneficial or better than counterparts.

Equipment:
-Bombs
--Honey Potion (13.3-16, 14.63 avg - 20.482 Fierce) [HP]
--Snow Mudball (9.5-13.5, 11.5 avg - 16.1 Fierce) [SM]

-Freezers
--Green Frost Cannon (2.75-4, 3.33 avg, 10% Freeze)
--Snowglobe Staff (0-6, 1.2 avg (6 icons out of 5 chances), 20% Freeze *can break before freeze)
--Black Frost Cannon (3-5, 4 avg, 23% Freeze)

-Mid-Level Gear
--7.2-10 (common) Iconers

After some thought I began wondering if some of these freezers were actually viable in mid-level line-ups- so I thought of the system of Average Icons per Turn. This is, the amount of average icons dealt each turn leading to a freeze (using average freeze percentages) and the damage dealt with a bomb on the freeze divided by the number of turns taken to freeze (not including bomb-turn- opponent doesn’t get a turn so it doesn’t count). Hopefully, this will prove whether certain freezers are viable in line-ups- ie, give a higher damage per turn average than just regular attacking. (yes I know the strategic value, getting rid of a chunk of hp at once to possibly knock-out, but how much are you willing to sacrifice for this?)

So apart from our freezer, how many icons do we assume we get from our other weapon (slot) per turn? We may use our 9-iconers and get 3 icons blocked on average or maybe a dual duty and get 6 icons of effect, so let us assume a 6-per-turn average from our other slot. (up for debate, but for the sake of the article let us assume- if you disagree, replace 6 with whatever you think in the article)

===Green Frost Cannon===
The green frost cannon is recommended in many lower-intermediate line-ups due to its lack of breaking-before-freezing in comparison to its often compared to brother, the Snowglobe Staff. How much better/worse is it?

The green frost cannon takes an average of 10 turns to freeze (100/10). It deals an average or 3.33 icons each turn. When we add our other weapon, we get 9.33 icons each turn. If we use the HP, that’s 2.04 (20.48/10) damage each turn added, for a total of 11.37 or so icons per turn. If we had used two weapons of the same iconage, we would have gotten 12 icons per turn (give or take). This means that we get LESS icons per turn using the green frost cannon rather than just attacking etc. (for the sake of completion snow mudball is 10.9/turn)

You decide whether giving damage at once is more useful to you.

===Snowglobe Staff===
Keeping in mind that the Snowglobe staff can break before freezing, the Snowglobe staff has a 20% freeze chance and deals 1.2 icons per turn on average. Adding our other weapon, we have 7.2 icons per turn. It takes roughly 5 (100/20) turns to freeze- HP: 20.4/5 = 4.8 icons per turn and SM: 16.1/5 = 3.22 icons per turn. That makes it overall 10.42 for SM and 11.9 for HP, both less than if we had used regular weapons and a bit more than if we had used the snowball. The higher damage from the cannon makes the frost cannon more viable! (and the fact that it can't break, i'd sacrifice .4 icons per turn to not have it break)

===Black Frost Cannon===
The black frost cannon is that common stepping stone between H4000 helmet and lower/no freezers. Does it really add icons per turn? It has a 23% freeze and 4 icons per turn average, already looking much better than those already looked at. Adding our other weapon, there is 10 icons per turn. If it takes 100/23 = 4.3 turns on average to freeze, SM: 16.1/4.3 = 3.74 icons per turn and HP: 4.7 icons per turn. Here we have defiantly gotten more damage per turn- 2.74 from our SM and 3.7 from our HP- a Black Frost cannon DOES make a difference.

It is hard to say exactly how beneficial this makes freezers- by using them remember you have bypassed any sinks, burrow, downsizes and mass blockings of the sort, but you have also made yourself have to survive longer to get the pay off.

I HAVE proven however, that the green frost cannon is slightly better than the Snowglobe staff no matter what other weapons your using and despite the higher freezing chance. This will hopefully settle that argument (or fuel it, maybe).

If this test were to be used on multiple-freezers, keep in mind the number of added damage is over the whole battle and not just the first x turns.

Yours Truly, Rylix.

(Edit: Number Editing)

I like your assumptions. Your point of view about the dynamics of the freezing chances that makes sounds... like there's something wrong.

Also about the Snowglobe Staff as far as i undestood it has 1/5 to attack and the breaking chance makes the 1.2 icons per turn odd. But the main thing is how works the freezing chance. Provided that solution all could make more sense.

The Snowglobe Staff attacks in one of those 5 options, for a 6 icon attack. On average just using it, thats a 6 divided by five options, which is 1.2 icons.

I found where it was wrong- i didn't factor in fierce attack properly. The results are still near the same however- Black is better and Green/Snowglobe are still worse but not by as much. Thanks for making me think of what i did .

More figures... we will see what happen (i try this to understand better).

This is the scenario that i show: A pet of 13 boost (x1 icon multiplier just to make it easier). A battle of up to 10 turns. Use of freeze weapon until freeze (+6 icon-damage, i-dam,*). After freezing, bombing (bomb+main weap+fierce/berserk). If turns still available 12 i-dam. Bombs deals just the average (+10 from main weap).

I guess i got a counter argument... or another point of view.

Equipment:
Honey Potion=14.63 i-dam (13.3-16) [HP]
Snow Mudball=11.5 i-dam (9.5-13.5) [SM]

Freezing turns: Freezing weapon average + 6 i-dam.
Bombing turn: when HP: Berzerk=36.945; Fierce=34.482.
When SM: B=32.25; F=30.1

*Assume an average of 6-per-turn (because blocking) per weapon.

The text stressed the importante of damage, the idea beneath is that brute force (more damage per turn) will win the battle. And relies on i-dam per turn. Each turn that a freezer don't freeze damage is not dealt and, this is a key issue, damage done by the freezer is a way to lessen i-dam lost.

The damage model I made: Max damage ([{Damage from constants}*{Turns-2}]+Bombing+Freezing) - MaxMin damage ([Freezing Attempt]* Turns).

But following that path, the sooner the better about the freezer's job. An earlier freeze means less i-dam lost.

I made a couple of graphics, one shows that Black Frost Cannon freezes the soonest, then Snowglobe Staff and Green Frost Cannon is the slowest of this freezers, on terms of probability.

I used {[(1-p)^(n-1)]*p} to calculate the probability of freezing each turn.

According the graphs and intuitively; freezing soon means more total damage because less damage is lost. Is until the 6th turn that all the freezer has an almost equal probability to freeze. Before, Black FC and S-Staff has more chances of freezing; after that turn, Green FC is more likely to freeze. Therefore in the short, medium term Black FC and S-Staff are better suited (its like starting vigorously)... and in the long run Green FC is better (comparable to that last sprint to be the first who cross the finish line).


Some tables i would like to show:
Green Frost Cannon. Acording idapranum's test (100): F: 11%; Av-Dam:3.339 *assume that this damage is dealt when used.


Snowglobe Staff this one is hard to model. There're 5 posibles outcomes i focus on freezing (19.7%) and attacking (deals [6 icons*22.7%] 1.362 i-dam per turn). It can break but assume it like "doing nothing" and keeps it available.


Black Frost Cannon F: 23%; Av-Dam:4 *assume that this damage is dealt when used, data provided by rylix.


I should say that statics is not my better subject and the problem of assumptions (i like the rationality on men, there're some theories that says that rationality has nothing to do with men)...

Please correct me if needed i enjoyed doing this figures but i would love to have done a correct job.

If i understand what you said correctly, the counter-counter argument to that is that even though theres a chance on freezing earlier, theres the chance that it freezes later as well. Average is all about the long run, repeating the same process.

The reason i calculated mine in icons is that you can convert the results to any damage boost afterwoods.

Would you mind rephrasing what exactly you did? I'm finding it hard to get my head around it ^^,

However i do like the idea that you can be killed before freeze comes into effect. This makes the problem more advanced... Even though the staff is weaker if your not likely to survive the 10 rounds then green might be pointless to begin with.

Rephrasing...

If you freeze soon you:
In the short term; Is more likely to freeze and deal that massive damage needed to end the battle with you as winner.
In the long term; You lost less i-dam (i-dam just to emphasize the relation with icons displayed and the damage dealt, this way i can forget about fractionals and fake icons) and that helps you to win the battle.

Also you can see the turns when you use your freezer... so if you didn't use your freezer on the first turn your T. 1 would be the second turn of the battle. In that way Black FC and S-Staff can be use to get that last sprint.

The graphs were meant to show how a 10 turn battle would work on different scenarios. I can't merge them. Use turns but i can't figure how to make comparable i-dam and probability... it may work with a logaritmic thing or a multipliying factor but thats too far to my knowledge.

I can't see why you say that "in average weapon-name freezes on turn N". I don't think that is a derivable sentence from the probability to freeze.

I have been looking at this recently myself, using a different methodology. I`m not about to post my analysis with the war on, but suffice to say that working from different methodology and disagreeing with you assumptions, I agree in general with your conclusion:

The Black and Rainbow Frost Cannons are on average slightly better than using a 9 or 10 icon weapon in most battles. The Green Frost Cannon and Snowglobe Staff are not.

This is extremely interesting, but probability is almost never correct. I'm only taking Pre-Cal right now, but from my understanding of Exponentional, and logarithmic functions, you should by some factor be able to make a graph of your data, and have it offset by 2 similer functions, that are slighty greater, and slightly lower than the function. Doing so will give you a more complete average due to the fact that exponential equations decrease faster when the values are greater, and vice versa for when they are lower.

I'm not sure if you get the point i'm trying to make, but it will in fact give you a more complete average, although it may completely throw your data off, and will no doubt take quite a while to work out.

What uh... Exponential and Logarithmic variables are you talking about exactly? As far as i know all non-average values cancel eachother out- im sure theres not a greater chance of a for example 5 turn freeze then a 15 turn freeze on a green frost cannon-

Well technically, by probability standards there is, but I won't get into that as it is streching probability at best. I mean to distort your function by making the functions of the Green Frost Cannon for incstance...

Alright, for simplicities sake i'm gonna just say it freezes at a 10% chance...

Functions:

Y=1*10^X
Y=1*8^X
Y=1*12^X
Y=1*15^X
Y=1*5^X

Now, if you graph all those, and then find the average of them all for a set point, which should result in a freaky looking function, but would be much more accurate in my opinion as it takes a look at throwing some "luck" into the equation...

Math doesn't work that way... All right, here's why (Just because it's early doesn't mean I can't think.) Try this with 20^X and 0^X. See the problem? Further, you can't do this, because the probability is set, luck is what changes this function, luck applied to X, not the base. Power functions don't average out that way, note (3^2-2^2)/2 is 2.5, while (3-2)/2 is 1. There are Stat functions (I've done some recently) where you take a sample, and can say with any certainty (Except negatives and 100% or higher) what the population (in this case, programmed) mean is. Meaning, if I have the data, I can be 99% sure of the actual probability. This is why more data gives more certainty, n, or the sample space, is in the denominator of the Error function.