I am not sure of this answer but here it is any way.
Green dice have a 1 in 6 chance of rolling a shotgun. You roll 3 dice per turn so on average you will roll one half a shotgun per turn. It takes 3 shotguns to eliminate a player with no points for the round so on average that will take 6 turns.
Green dice have a 1 in 2 chance of rolling a brain. You roll 3 dice per turn so on average you will roll one and one half brains per turn. In the 6 turns that it takes to eliminate a player on average green dice will roll 9 brains.
Yellow dice have a 1 in 3 chance of rolling a shotgun. You roll 3 dice per turn so on average you will roll 1 shotgun per turn. It takes 3 shotguns to eliminata a player with no points for the round so on average that will take 3 turns.
Yellow dice also have a 1 in 3 chance of rolling a brain. So in the 3 turns it takes to roll roll 3 shotguns on average you will also roll 3 brains.
Red dice have a 1 in 2 chance of rolling a shotgun. You roll 3 dice per turn so on average you will roll one and one half shotguns per turn. It takes 3 shot guns to eliminate a player with no points for the round so on average that will take 2 turns.
Red dice have a 1 in 6 chance of rolling a brain. You roll 3 dice per turn so on average you will roll one half a brain on each turn. In the two turns on average that it takes to eliminate you, you would roll 1 brain.
If we give one point for each green dice. 3 are rolled each turn, in 6 turns that would be 18 points plus the 9 points from the brains rolled for a total of 27 points.
For yellow to get 27 points in the 3 turns they would have to add 24 to their brains. It would take 8 points a turn. With 3 dice per turn that is 2 and 2/3 points per dice.
For red to get 27 points in 2 turns they would have to add 26 points to their one brain. It would take 13 points per turn. With 3 dice per turn that would be 4 and 1/3 points per dice.
Thus the green:yellow:red ratio should be 1:2:3.
EDIT ---
I may have miss understood part of this question. I thought points were awarded for each dice rolled and added to the total obtained from the brains rolled. If we are only changing the number of points obtained by each brain on the given color of dice it changes things.
With the ratios being points per brain rolled:
Green on average will roll 9 brains, with 1 point each.
Yellow on average will roll 3 brains, requiring 3 points per brain to equal green.
Red on average will roll 1 brain requiring 9 points per brain to equal green.
Making the green:yellow:red ration 1:3:9
I find it easier to consider the problem of not rolling a 6 with any of the three dice (the same problem with a different definition).
The chance of not rolling a 6 is 5/6. The chance of not rolling a six on any of your dice is therefor 5/6 * 5/6 * 5/6.
The chance of rolling at least one six is therefore the opposite of this:
1 - (5/6)^3 = 0.421
Best Answer
Let's look at a green die.
Let p=the odds of getting brains on a green die. What are the cases when tossing one?
So, we get that $p=\dfrac12+\dfrac13p$. Thus $p=\dfrac34$.
Similarly, you can get the probabilities for the other two colors of dice. Then multiply them out like you did with the other probabilities.