Essay: Gamma World Artifact Use Chart
From Acaeum
In Gamma World, if a player discovers an artifact, he can try to figure out how to use it. The process is governed by the artifact use chart. Starting at the S square, the player advances through the diagram by rolling a ten-sided die until he gets to the F square, indicating success, or the skull-and-crossbones, indicating harm. The player can make 5 rolls per hour. There are modifiers if the player's character has an intelligence higher than 15 or lower than 7.
In The Dragon #25, Gary Jaquet describes the charts as pointless because they could "easily be pre-calculated and the possible pathways computed to single percentage rolls". Reducing the charts to a single roll is indeed possible if the player is allowed to make an unlimited number of rolls and always rolls until arriving at success or harm. However, most people won't find the calculation easy.
One way to do it is to convert the state diagram to a transition matrix and use matrix multiplication. In the transition matrix, the entry in row i, column j represent the chance that the player will advance to state j on the next roll if he is currently at state i.
Here is a transition matrix for chart A:
[,S] [,A] [,3] [,4] [,F] [,6] [,7] [,8] [,H]
[S,] 0.3 0.0 0.0 0.0 0 0.2 0.0 0.0 0
[A,] 0.7 0.2 0.0 0.0 0 0.2 0.0 0.4 0
[3,] 0.0 0.3 0.8 0.0 0 0.0 0.0 0.0 0
[4,] 0.0 0.5 0.2 0.4 0 0.0 0.0 0.0 0
[F,] 0.0 0.0 0.0 0.3 1 0.0 0.1 0.0 0
[6,] 0.0 0.0 0.0 0.0 0 0.6 0.4 0.3 0
[7,] 0.0 0.0 0.0 0.3 0 0.0 0.3 0.0 0
[8,] 0.0 0.0 0.0 0.0 0 0.0 0.2 0.0 0
[H,] 0.0 0.0 0.0 0.0 0 0.0 0.0 0.3 1
With software that performs matrix multiplication, one can compute powers of the matrix, and as the exponent becomes large, the matrix converges to the following:
[,S] [,A] [,3] [,4] [,F] [,6] [,7] [,8] [,H]
[S,] 0.0 0.0 0.0 0.0 0.93 0.0 0.0 0.0 0.07
[A,] 0.0 0.0 0.0 0.0 0.93 0.0 0.0 0.0 0.07
[3,] 0.0 0.0 0.0 0.0 0.93 0.0 0.0 0.0 0.07
[4,] 0.0 0.0 0.0 0.0 0.93 0.0 0.0 0.0 0.07
[F,] 0.0 0.0 0.0 0.0 1.00 0.0 0.0 0.0 0.00
[6,] 0.0 0.0 0.0 0.0 0.93 0.0 0.0 0.0 0.07
[7,] 0.0 0.0 0.0 0.0 0.86 0.0 0.0 0.0 0.14
[8,] 0.0 0.0 0.0 0.0 0.65 0.0 0.0 0.0 0.35
[H,] 0.0 0.0 0.0 0.0 0.00 0.0 0.0 0.0 1.00
The first row shows the probability that starting at S, the player will eventually arrive at F (93%) or harm (7%). Other rows show the probabilities when starting at other states.
By plotting the sum of the probabilities of the final states for increasing powers of the matrix, we can show how many rolls are likely to be needed to get to a final state. After 14 rolls, there is only a 70% chance that the player has reached a final state.
Transition Matrix for Chart B
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15] [,16] [,17]
[1,] 0.8 0.0 0.0 0.0 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[2,] 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5
[3,] 0.0 0.0 0.6 0.3 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[4,] 0.4 0.0 0.0 0.3 0.2 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[5,] 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[6,] 0.0 0.0 0.1 0.0 0.0 0.2 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[7,] 0.0 0.0 0.0 0.2 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.3 0.0 0.0 0.0 0.0
[8,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[9,] 0.0 0.0 0.0 0.0 0.4 0.0 0.0 0.0 0.4 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[10,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.1 0.0 0.0 0.0 0.5 0.0 0.0
[11,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0
[12,] 0.0 0.0 0.0 0.4 0.0 0.3 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2
[13,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.0 0.0 0.0 0.3 0.4 0.0 0.0 0.0 0.0
[14,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.1 0.0 0.0 0.6 0.0 0.0 0.0 0.0
[15,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.2 0.4 0.3 0.0
[16,] 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.0 0.0 0.3
[17,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0