sailor_charon (sailor_charon) wrote in perfectrpg,

A small game for balancing characters.

A mathematically game about balance, encouraged by the topic of skill balance.

This game is called Balloons and Rocks. Each Balloon is a character. Balloons start the game with 4 rocks, and each balloon has a stick that knocks off a rock with a probability of 100%. Each balloon has a 50% chance of trying to knock off a rock, and likewise a 50% chance of not letting their opponent try and knock off a rock (zero sum for these two attributes, think of it as a 1d20 roll against another 1d20 roll in which ties are rerolled).

Rocks may be purchased for A points.
The capacity to knock of 25% more rocks on a hit may be purchased for B points.
A 5% increase in knock off chance may be purchased for C points.
A 5% increase in knock off resist may be purchased for D points.

Let's compare the cost of some various balloons now.
Base + A survives 25% longer than Base (more rocks) and thus we can say is 25% better.
Base + B does damage 25% faster than Base and thus we can say is 25% better. Base + A vs Base + B will functionally be the same battle as Base vs Base, as will Base + A vs Base + A and Base + B vs Base + B (these other options might be slower or faster, but win chances are still balanced).
Base + C will knock off oponent rocks faster and do 5% more damage per round because of it, so is 5% better than Base.

Now we assume that A = 25, B = 25, and C = 5 are the balance costs of each of these.
Base + A + C (n) and Base + B + C (m) both cost the same # of points. Each of them should be equally likely to win in a fight.

n vs m
n will knock off .1375 (.55 * .25) rocks/round off of m, m has 4 rocks so will take 29.0909... rounds to be defeated before floating away.
m will knock off .275 (.55 * .5) rocks/round off of m, m has 5 rocks so will take 18.1818... rounds to be defeated before floating away.

Obviously, m is a more powerful balloon. What went wrong? Simple, we had linear costs applied to a linear * linear benefit (defense skill * HP, offense skill * damage). We can conclude by this, that hit points and damage capacity cannot be purchasable by a linear system if their probability of use is also purchasable by a linear system. So this is yet another place where we find that a balance, that is, the disassociation of point allocation with character quality, is impossible to achieve. If you could balance this, it would a) take a computer, and b) each and every skills cost would not only need to be based on the levels of other skills, but also on the levels of the skills of the intended opponent.
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