In a truel (a three-person duel), A is weakest, C is strongest. A shoots first, then B. What is the best strategy for A, and what is the chance that A will win? To approximate this, a die is chosen, and each player must roll under a certain value to hit. Odds of hitting are given in the first line of each circle.
In the best strategy, A should deliberately miss, while the strongest trade shots. When a player is eliminated, A takes aim at the survivor. Nonintuitively, the weakest player in a truel often has the best chance of winning. The odds of winning are given in each circle. Between the circles, the duel odds are listed for the first person shooting.
Clark Kinniard's 1946 book Encyclopedia of Puzzles and Pastimes
seems to be the first appearance of this three-person game. In general, three-player games are difficult, due to the strategy of ganging up on the leader. Even in nature, when three competing animals are near each other, they will often not attack, since the unharmed third can take on the weakened survivor of the first main battle. In 1954, Martin Shibik introduced the word truel (for three-person duel) in his paper "Does the Fittest Necessarily Survive?"
For this Demonstration, there are two cases when all three players have a 1 in 3 chance of surviving. One happens at die size 84, the other at die size 120. The next equality happens at die size 280, with threshholds 35, 72, and 90.