Optimal First Shot in a Three-Way Duel

This brain teaser describes a three-way duel between shooters of very different accuracies, with a fixed firing order that repeats until only one survives. The candidate, taking the role of the weakest shooter, must decide whom to aim at on the first shot in order to maximize personal survival. The twist is that the other two agents are also assumed rational and self-interested, and their future choices depend on who remains alive and how threatening each survivor appears. The problem is typical of abstract game-theoretic puzzles often used in brain-teaser rounds for quant or trading interviews.

The question leans on backward induction, conditional probability, and strategic reasoning in multi-agent settings. A strong answer carefully enumerates future scenarios after each possible first action, compares survival probabilities, and correctly anticipates the incentives and responses of the stronger shooters. Interviewers look for clear structuring of the state space, disciplined probabilistic bookkeeping, and the ability to separate intuitive "shoot the strongest" heuristics from the actual optimal strategy when others are also optimizing.

What it tests

In multi-agent sequential decision problems with asymmetric strengths, the key structure is that rational agents will target the greatest threat to their own survival, not necessarily the weakest. This creates a dynamic where weaker participants can benefit by letting stronger opponents engage each other, effectively hiding behind the threat posed by others. The optimal strategy often involves minimizing your own perceived threat until the field thins, at which point direct confrontation becomes necessary. This principle holds because each agent's rational action is to maximize their own survival, which means targeting the opponent most likely to eliminate them first. The pattern is that survival is maximized by manipulating the incentives of others, not just by direct action.