A truel is like a duel except there are three players involved. We will call the

Question

A truel is like a duel except there are three players involved. We will call them player A, player B, and Player C. Each player comes with his probability of hitting a target. Player A hits a target with probability pA=0.9, Player B with probability pB=0.7 and Player C with probability pC=0.2. The game is played by rounds. At each round the players shoot at each other simultaneously, hence their speed doesn’t matter. The truel is over when ONLY ONE player is still alive or if all three of them are dead. We assumed that each player has infinitely many bullets. Find the probabilities of winning for all three players including the case when nobody wins.

Question: Compute the expected number of rounds for the whole truel.

Explanation / Answer

Well we know that there is only a finite number of ways each first round can play out.
A shoots B B shoots A C shoots A
A shoots B B shoots A C shoots B
A shoots B B shoots C C shoots A
A shoots B B shoots C C shoots B
A shoots C B shoots A C shoots A
A shoots C B shoots A C shoots B
A shoots C B shoots C C shoots A
A shoots C B shoots C C shoots B

Intuitively, I know that it will always be in each players best interest (to have greatest chance of winning) to shoot the other player with the highest chance of getting a hit.
I.e. A will always try shoot B, B will always try shoot A and C will always try shoot A.

So now we know that the first round will always be:
A shoots B B shoots A C shoots A

You can now determine the possible outcomes that this round could have, based on the chance each player has of dieing, and combine the multiplicatively to determine the probability of each scenario occurring. Then using the same logic as before, determine the best options for each player for each of these following rounds, this will be rather obvious.
Continue until you reach a games end for each scenario, then combine the probabilities as per what the question asks

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