Transition matrix of a paint ball game

[CTK]

Summary: Finding the transition matrix of a paint ball game where only 3 probabilities are given.

We have the following question:

Alice, Tom, and Chloe are competing in paint ball. Alice hits her target 40% of the time, Tom hits his target 25% of the time, and Chloe hits her target 30% of the time. The game is played so that all survivors shoot simultaneously, and each competitor shoots at the opponent that is most accurate. Represent the game as a Markov chain where the states of the chain are the survivors.
Hint: there is/are absorbing states.

If we want to find its transition matrix, then is it as follows:

\begin{bmatrix}
0 & 0.6 & 0.4 \\
0.25 & 0 & 0.75 \\
0.7 & 0.3 & 0
\end{bmatrix}

Note: I have no idea why it is not converting to a matrix shape, sorry, so the matrix is as follows:

P = 0 0.6 0.4

0.25 0 0.75

0.7 0.3 0

I am pretty certain that it is wrong because the hint says that there is/are some absorbing states...so any suggestions regarding how to fix my transition matrix would be appreciated, thanks.

 

[FactChecker]

There certainly are absorbing states when only one or zero shooters are left alive.
I think your set of states for the state transition matrix is not detailed enough. Remember that the states must adequately summarize the situation at any time during the game. I don't think yours does that. Suppose a state is the set of people still alive: (a,t,c), (t,c), (a,c), (a,t), (a), (t), (c), (). Can you make an 8x8 state transition matrix for that?

 

[CTK]

There certainly are absorbing states when only one or zero shooters are left alive.
I think your set of states for the state transition matrix is not detailed enough. Remember that the states must adequately summarize the situation at any time during the game. I don't think yours does that. Suppose a state is the set of people still alive: (a,t,c), (t,c), (a,c), (a,t), (a), (t), (c), (). Can you make an 8x8 state transition matrix for that?

Thanks for the reply. So if we do what you just explained above and if we denote (a,t,c) to be 1, (t,c) to be 2, (a,c) to be 3, etc. then, for example, what would be the prob. of going from (a,t,c) to (t,c)? I just want to know how to do one of them and I will try to do the rest...
Also, how come can we have a state of () where all are dead? How is it possible? Shouldn't one at least still survive?
thanks for your help, it is really appreciate it.

 

[FactChecker]

Thanks for the reply. So if we do what you just explained above and if we denote (a,t,c) to be 1, (t,c) to be 2, (a,c) to be 3, etc. then, for example, what would be the prob. of going from (a,t,c) to (t,c)? I just want to know how to do one of them and I will try to do the rest...thanks for your help, it is really appreciate it.

Starting from the state (a,t,c), the rules say that a shoots at c, t shoots at a, and c shoots at a. To arrive at (t,c) it must be true that a missed, and at least one of t or c hit a. The probability of that is (1-.4)(0.25+0.3-0.25*0.3) = 0.6*0.475 = 0.285

 

[Orodruin]

Also, how come can we have a state of () where all are dead? How is it possible? Shouldn't one at least still survive?

@FactChecker has already given you lots of useful points regarding the actual state space (which is 8-dimensional and not 3-dimensional) so let me just address this.

Consider the state (t,c). Both t and c will aim for each other and both will hit with a probability of 0.25*0.3 = 0.075. In this case the game transitions to the state (). Note: This is only one example of how you could arrive at the state ().

 

[CTK]

Starting from the state (a,t,c), the rules say that a shoots at c, t shoots at a, and c shoots at a. To arrive at (t,c) it must be true that a missed, and at least one of t or c hit a. The probability of that is (1-.4)(0.25+0.3-0.25*0.3) = 0.6*0.475 = 0.285

Thank you so much for your help, I really appreciate it.

 

[CTK]

@FactChecker has already given you lots of useful points regarding the actual state space (which is 8-dimensional and not 3-dimensional) so let me just address this.

Consider the state (t,c). Both t and c will aim for each other and both will hit with a probability of 0.25*0.3 = 0.075. In this case the game transitions to the state (). Note: This is only one example of how you could arrive at the state ().

Yes, I think I have finally figured it out. Thank you very much for your input as well, it is appreciated.