Truel

A truel is a neologism for a duel among three opponents, in which players can fire at one another in an attempt to eliminate them while surviving themselves.

A variety of forms of truels have been studied in game theory. Features that determine the nature of a truel include

There is usually a general assumption that each player in the truel wants to be the only survivor, and will behave logically in a manner that maximizes the probability of this. (If each player only wishes to survive and does not mind if the others also survive, then the rational strategy for all three players can be to miss every time.)

In the widely studied form, the three have different probabilities of hitting their target.

If a single bullet is used, the probabilities of hitting the target are equal and deliberate missing is allowed, the best strategy for the first shooter is to deliberately miss. Since he is now disarmed, the next shooter will have no reason to shoot the first one and so will shoot at the third shooter. While the second shooter might miss deliberately, there would then be the risk that the third one would shoot him. If the first shooter does not deliberately miss, he will presumably be shot by whichever shooter remained.

If an unlimited number of bullets are used, then deliberate missing may be the best strategy for a duelist with lower accuracy than both opponents. If both have better than 50% success rate, he should continue to miss until one of his opponents kills the other. Then he will get the first shot at the remaining opponent. But if the "middle" opponent is weak, it can be better to team up with him until the strongest is eliminated. The details depend on the firing order. For example, if the order is P, Q, R, with respective probabilities

and it is R's turn, R should waste his shot if:

but not do so if:

In between, R should waste his shot if:

In one of the most famous spaghetti westerns, The Good, the Bad and the Ugly, the final showdown is played out to be a climactic truel among the three main characters: Blondie ("The Good"), Angel Eyes ("The Bad"), and Tuco ("The Ugly"). The standoff remains a signature piece for director Sergio Leone and one of the best-known scenes in film history.

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