Solution
The Truel
Show the problem again
Three shooters duel in turns: A hits with probability 1/3, B with probability 2/3, and C with certainty. They fire in order A, B, C (skipping the dead), each choosing any target, until one survives. What should A do with his first shot?
Worked solution
A should deliberately fire into the air. If A kills C, B shoots at A hitting 2/3 of the time; if A kills B, C kills A with certainty. Either success leaves A facing a stronger opponent who shoots first. By missing intentionally, A lets B and C target each other (each rationally shoots the bigger threat), then A gets the first shot against the lone survivor: the weakest shooter maximizes survival by abstaining.
Source: The truel, a three-way duel problem analysed in the game-theory literature by D. Marc Kilgour and Steven Brams. Statement written for AxiomIQ.