A Short Form Essay on Linear vs Non Linear Equations and their importance in economic systems - Balancing Jutsu Cost VS Reward

After reading the new proposed costs, and seeing a somewhat limited list of newly proposed jutsu effects and their costs, I believe that there is a very important to understand issue that has been missed in the new balancing system, which I feel is extremely important to impart upon anyone who bothers to read this. It is clear, that the jutsu presented were balanced with two key idea's in mind.

1) The effects of a Jutsu are directly proportional to the cost of Jutsu in CP.

2) These ratio's were done assuming (Intentionally or unintentionally) that CP Spent is automatically guaranteed to represent Effects or Damage dealt

There are a number of very important reasons why the above concepts are flawed inherently. This is because this sort of a balance system is not a linear equation, but a nonlinear system, and the economic cost of using specific jutsu and the value of CP was not taken into account properly when potentially balancing these host of new jutsu.

Definition:

Economic Cost - The possible other uses that a currency, or in this case, CP could have gone towards to produce other effects.
Linear System - A system in which two variables directly proportional. IE: Y = mX+b
Non Linear System - A system in which the two variables are not directly proportional. IE: f(x+y) = f(x) + f(y)

If we look at two possible examples of this, let us examine both a C rank, and an A rank Ninjutsu that only do damage.


Doton: Sink Hole Manipulation
Jutsu Type: Ninjutsu
Rank: C
Chakra Cost: 9
RP Description: Causes a sinkhole big enough for one person to form underneath the target's feet suddenly dropping them into it as they are scratched and beaten by the fall into the 8 foot hole.
PVP Effect: 3d6+1d4 Damage

Doton: Rock Rain Boulder
Jutsu Type: Ninjutsu
Rank: A
Chakra Cost: 35
RP Description: The user causes Rocks and Boulders to rain down on the target, crushing them beneath the immense weight.
PVP Effect: 5d10+5d4+5 Damage

The average damage of the C rank is 13.

The average damage of the A rank is 45.

The cost of using the C rank is 9.

The cost of using the A rank is 35.

The efficiency of Chakra to Damage of the C rank is 13/9 = 1.444 Repeating.

The efficiency of Chakta to Damage of the A rank is 45/35 = 1.286 (Rounded).

The efficiency difference between the two is 0.158

These are relatively equal in terms of direct cost, VS damage dealt. However, there are two things which must be taken into account to make these numbers actually accurate. The first, is how likely one is to do the damage in question, VS the cost and the second is how much each point of Chakra is worth when being used in battle.

To answer the first question of how likely one is to do the damage in question, we will assume that in theory, two equal opponents will have a 50% chance with which to hit the opponent in question with their technique. This is rather basic and does not take into the account stuns, or other techniques which will guarantee (Or increase the odds of a hit) but this also adds onto both the total amount of chakra needed to deal this base amount of damage, as well as increases the number of rounds over which one may get this damage. I will not do those calculations here but sufficed to say it nearly quarters the efficiency assuming similar cost

Assuming a 50% cost, this actually means that one has a 50% chance to deal the damage for the chakra spent, as the chakra is spent regardless of a hit or miss. In a linear system we can do this as a probability.

((1/2 * 13)/9) = .7222 Repeating Efficiency for the C rank
((1/2 * 45)/35) = .6429 (rounded) Efficiency for the A rank.

The efficiency difference between the two is .0793.

What this shows us, is that you are in fact doing on average much less damage.

Let us look at this in another way as well.

To deal this 7.5 Real Average Damage as listed above you must spend 9 CP.

To deal the 22.5 Real Average Damage for the A rank you must spend 35 CP.

What this means is, in order to deal the same amount of damage of 22.5, you would need to use your C rank jutsu about three times. Doing so will cost you 27 CP. This is a total of 8 CP less than it takes to use the A rank jutsu. This does not take into effect passive damage which will be more effective on multiple uses at lower chakra expenditures.

The second thing we need to look at, is the inherent value of each unit of chakra. What economics states, is that as your supply of a resource falls, the more value each unit of it is. Meaning, in using a much higher rank technique at a relatively equal rate of hitting an opponent, you are expending much more chakra, meaning that the more chakra you spend at once, the more valuable the technique should be, as each additional point of chakra is that much more valuable.

There will always be a risk VS reward situation, of spending a lot of chakra to do a lot of damage, and this should be reflected accordingly. But a purely linear system does not take into account the fact that the risk of using a higher ranked technique and using more chakra is inherently not only less efficient, but also less economically viable when we talk about how much each point of chakra is worth, and how passive damage bonuses stack additionally on each jutsu spent making them even less inherently effective on higher rank jutsu.

In conclusion, the current cost for high rank jutsu VS medium and low tier jutsu for the amount of damage that they do is not efficient, and does not properly take into account the added economic cost of the jutsu's use. Meaning that you are spending more chakra, for less effect overall, that would be better spent in EVERY case mathematically, on multiple C rank jutsu's of a similar format.

Recommendations; Make the jutsu costs non-linear, or make the damage scales non-linear

E-Rank
1 - 3

D-Rank
4 - 7

C-Rank
8 - 13

B-Rank
14 - 21

A-Rank
22-35

S-Rank
36-47

S+ Rank
48-60

S++ Rank
60+

Thank you for posting your concerns. I will take this into consideration and review the systems we currently have for "inherent flaws" when I get the opportunity. I'm sure your feedback will be valuable in correcting our systems if what you say is true.