A Truel with Better Aim

Three marksmen A, B and C have accuracies $1/2$, $2/3$ and $1$ respectively: each shot by a marksman hits its chosen target independently with that probability. They fire in the fixed order A, B, C, A, B, C, ... (skipping anyone who has died) until only one is left alive. On his turn a marksman may aim at any living opponent or deliberately miss. Each marksman maximises his own probability of being the sole survivor, and this is common knowledge. Use the equilibrium in which the strategy depends only on who is alive and whose turn it is (it is unique here). What is A's probability of surviving? Give a reduced fraction (a decimal accurate to four places is also accepted).

• graph theory
• information
• markov chains
• nash equilibrium

Source: Truel analysis in the style of Mosteller and of Kilgour and Brams, with new accuracies. Statement written for AxiomIQ.