Set up a Markov Chain (either a transition matrix or a Markov Diagram) for the f

Question

Set up a Markov Chain (either a transition matrix or a Markov Diagram) for the following problem: Three tanks fight in a three-way duel. Tank A has probability 1/2 of destroying the tank at which is fires, Tanks B has probability 1/3 of destroying the tank at which it fires and Tank C has probability 1/6 of destroying the tank at which is fires. The tanks fire together (i.e. at the same time) and fire at the strongest opponent not yet destroyed. Form a Markov chain with the states being the sets of the tanks not yet destroyed i. e. ABC, AB, AC, BC, A, B, and C or empty set. Use Mathematica to determine which tank has the highest probability of winning.

Explanation / Answer

There are 8 states. ABC, AB, AC, BC, A,B,C and none.

I will give one example of how to calculate probability.

When all three are alive, A will fire at B and B will fire at A and C will fire at A. As the strength of the tank is as follows A>B>C

When only 2 tanks are alive, they will fire at each other.

So taking the state ABC, it can go to ABC or AC or BC or C. All other states in which C is destroyed is not possible at all as noone fires at C.

P (ABC) = A fired failed * B failed * C failed = 1/2*2/3*5/6 = 5/18

P (AC) = A success * B failed * C failed = 1/2*2/3*5/6 = 5/18

P (BC) = A failed * B success * C success + A failed * B failed * C success + A failed * B success * C failed

= 1/2*1/3*1/6 + 1/2*2/3*1/6 + 1/2*1/3*5/6 = 4/18

P (C) = A success * B success * C failed + A success * B failed * C success + A success * B Success + C success

= 1/2*1/3*5/6 + 1/2*2/3*1/6 + 1/2*1/3*1/6

= 4/18

Doing it for all the other states, we will get the following markov transition matrix. Note that the row total add up to 1.

The initial state is ABC, i.e the matrix is

Eventually it gets to the state where all the tanks get destroyed.

C has the maximum probability of winning.

ABC AB AC BC A B C - ABC 5/18 0      5/18 2/9 0      0      2/9 0      AB 0      1/3 0      0      1/3 1/6 0      1/6 AC 0      0      5/12 0      5/12 0      1/12 1/12 BC 0      0      0      5/9 0      5/18 1/9 1/18 A 0      0      0      0      1      0      0      0      B 0      0      0      0      0      1      0      0      C 0      0      0      0      0      0      1      0      - 0      0      0      0      0      0      0      1     

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