Anna, Bob and Claire are doing a three-way duel (a truel). Anna has an accuracy of 1/3, Bob has 2/3, and Claire is an excellent marksman with 1 (100%) accuracy. They must take turns shooting. Since each player knows others’ skills and all are rational, what would Anna do in the competition?
If Anna starts the duel and aims at Bob, there is a 1/3 chance she will shoot Bob. In that case, Claire has the next turn and will end Anna. On the other hand, if Anna aims at Claire, there is a 1/3 chance that Bob gets a chance to aim at Anna and end her with a 2/3 chance. So, it is clear that missing shots is a better option for Anna. So, she must shoot in the air first.
In the next chance, Bob will aim at the more prominent threat, i.e., Claire. Then there is a 2/3 chance of a showdown with Anna and a 1/3 chance that the showdown will be between Anna and Claire. Let P(A) be the probability that Anna survives, P(A, B) be the chance Alice wins the duel with Bob and P(A, C) Alice wins against Claire.
P(A) = 2/3 x P(A, B) + 1/3 x P(A, C)
P(A) = 2/3 x 3/7 + 1/3 x 1/3 = 25/63
