Truels and strategies for survival.

The truel is a three person competition that generalises the classic duel. In this game three players try to eliminate each other in a series of one-to-one duels until there is only one survivor. The players' marksmanship, shooting order and strategies for choosing a target play a significant role in individual's survival probability. Strategies such as shooting into the air (abstention), shooting at the strongest opponent, and shooting at the weakest opponent have been analysed in the previous literature. In this paper, for the first time, we consider suicidal and random strategies that can be chosen by the weaker player. We show that although there is no possible highest probability region for weakest player adopting suicidal strategy, the player may increase the survival probability via switching between suicidal and abstention strategies randomly. In addition, we demonstrate that there is a narrow survival area for the weakest player when the player aims randomly at two other opponents, and eventually the area fades away if the player fires randomly at himself or the other two opponents.