This is a simplified dice mechanic using standard d6 dice to balance 5 unique weighted distributions.
Define the following events:
A = {6,5,4,3,2}
B = {6,5,4,3}
C = {6,5,4}
D = {6,5}
E = {6}
Meaning event B is rolling a 3 or better. The probability of event B is 4:6
Weight each event as:
A(12) B(15) C(20) D(30) E(60)
The weighted events have an average value of 10. If multiple dice are thrown, the weighted average will be 10 times the number of dice:
http://anydice.com/program/35d
The dice mechanics are balanced in that each weighted event will on average produce the same value. This makes for an interesting battle mechanics. An example of a balanced battle mechanic is to assign an event to each player. A player's starting hit points are the respective event weight (ie: playerA starts with 12 hit points). Each player rolls a set amount of dice and scores a hit for each die conforming to the event:
PlayerB rolls 5d6 and gets {3,2,5,1,6}. PlayerB scores 3 hits against their opponent.
In such a battle, each player has an equal chance to win. PlayerA, having a 5:6 hit ratio, will do more damage. However playerA has less hit points. PlayerE has a poor hit ratio 1:6, but very high hit points.