Survival of the strongest in a sequential truel

Sequential truel is a duel-like game between three players. Each players is endowed with his own marksmanship. At each turn, a player whose turn is to shoot can target any of the remaining alive opponents or abstain from shooting. The game ends when there is only one player alive or when all alive players abstained from shooting one after another. The single survivor obtains the “survivor” prize 1, while the payoff of all other players is 0. In the case the truel ends due to “inactivity”, all the players receive the payoff of 0. It is a deeply studied game with paradoxical finding that the weakest player has the highest probability of surviving in many settings, especially when the player can abstain from shooting. Here we present an explicit construction that contradicts this finding. There exists a mixed strategy subgame perfect equilibrium in which the strongest player has the highest probability of survival. This equilibrium exists for a specific order of play, in which the two stronger opponents act before the weakest one. When it exists, there are multiple subgame-perfect equilibria, including the existing stationary construction, in which two stronger opponents target each other.