# What are fair point values for Zombie dice?

Described below is a modified version of Zombie dice ( http://www.sjgames.com/dice/zombiedice/ ) which the kids and I play.

As in the original game, the idea is to sequentially roll sets of 3 specialty dice (described below and in the link above) until you either voluntarily stop rolling or bust.

There are green, yellow an red dice whose faces show:

Green: 3 brains, 1 shotgun blasts, 2 footsteps

Yellow: 2 brains, 2 shogun blasts, 2 footsteps

Red: 1 brain, 3 shotgun blats, 1 footsteps

After each roll, you accumulate either brains (you ate your victim's brains) or shotgun blasts (victim shot you); footsteps are neutral (victim ran away).

If you choose to stop rolling, you get 1 point for each accumulated brain. You bust when you accumulate 3 shotgun blasts and receive no points in that game.

Instead of picking the dice blindly from a bag, as called for in the original version rules, we let the players pick which 3 dice to roll of any color available.

Obviously, nobody in his right mind would roll anything other than the green dice, since they have the most brains and least shotgun blasts.

We modified the rules by somewhat arbitrarily awarding 1 point for each green die, 2 points for a yellow die and 5 points for a red die.

So, the riskier the dice you select to roll, the more points you stand to get. This introduces a nice risk/reward element to the game.

Here is where you come in. I don't know whether our 1:2:5 point ratio is the fairest, that is, the one that balances the risk/reward of each die color.

So: What should be the fair point value of each color die?

I don't know whether one can calculate the desired values. Naturally, I could do a simulation by rolling, say, 1,000 games with each color dice and tallying the average number of brains accumulated right before the bust for each color; then, I would derive a factor by which to the yellow and red dice that equates the average outomes of the three colors.

Also, I could but now don't have the tools (e.g., a BASIC interpreter) to write a simulation program, or the time to manually do the simulation.

What say you is the right green:yellow:red ratio?

BK asked: "For one's turn must the same 3 dice be rolled each time or may one switch back and forth?"

LB says: No, you can switch dice.

After each roll (except if you bust, of course) you:

1. Keep the brains.

2. Keep the shotguns.

3. Return the footsteps to the dice cache.

4. Pick any 3 dice from the dice cache, if you want to roll again, and roll.

Note that as a game progresses and players accumulate brains and shotguns--depleting the dice cache--there will be times when a player's dice choice is limited by the dice then remaining in the cache. However, this does not happen often (we actually use 2 sets of Zombie dice and may even buy a couple more so there will always be plenty to pick from). Also, when there are fewer than 3 dice left in the cache, we replenish it by writing down on a Post-it the number of brains each player has and returning back to the cache their brain dice (could also do it for the shotguns be don't). So I think reasonable to disregard the dice depletion for our purposes.

I hope you find this problem interesting enough to give it a shot.

Thanks in advance for taking the time to read this message and for hopefully figuring out the answer to the problem.

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Is the red die supposed to have only $5$ sides? – EuYu Apr 2 '13 at 20:34
No. The red dice are six-sided, like all the other dice. The difference among the dice is the number of brains, shotgun blasts and footsteps (and color of course). – Leo Bueno Apr 2 '13 at 20:41
In the original version the red dice does have 1 brain 3 shotgun blasts and 2 footsteps, you only mentioned 1 footstep – Dominic Michaelis Apr 2 '13 at 20:41
Sorry if I'm misunderstanding something. I counted $5$ sides to the red die: $1$ brains, $1$ footsteps and $3$ shotguns. What is the last side? – EuYu Apr 2 '13 at 20:42
I wrote a simple program just to see how each die behaved under a greedy algorithm. I rolled each die until I accumulated $2$ shotgun blasts, at which time I quit the game. Under this behavior (for 1 million trials), the green die averaged $18$ points, the yellow die averaged $14$ points and the red die averaged $20.65$. – EuYu Apr 2 '13 at 21:00

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

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