Truel

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A truel is a duel involving three mutual opponents (that is, each one for themselves, not two-against-one).

This is a rare form of combat, mainly known by the final gun truel in the movie The Good, the Bad and the Ugly and the final sword truel in the movie Pirates of the Caribbean: Dead Man's Chest. Truels are also studied in game theory.

Truel should not to be confused with the trial (a grammatical number), or the Mexican standoff.

[edit] Game theory overview

The truel's mutual opponents in game theory will shoot each other until one prevails, i.e. survives. However in the widely studied form, the three have different probabilities of hitting their target. Game theorists examine these probabilities to calculate the best target to aim at (see below).

[edit] Example of a theoretical truel

Three men (let us call them A, B, and C) have decided to settle a conflict by firing at each other with pistols. A has a one-third chance of succeeding and killing his opponent, whilst B's is two-thirds and C's is three-thirds, i.e. 1. To even this score, the men will take it in turns to aim, A taking first shot, followed by B then C.

A's options are:

aim at B
aim at C
aim at neither (i.e., intentionally miss both B and C)

What should A do?

Warning: Solution details follow.

One can calculate the probabilities of A surviving the truel and find that:

if A's first shot is at B, then A's probability of surviving the truel is 50/189 = 0.26455... ;
if A's first shot is at C, then A's probability of surviving the truel is 59/189 = 0.31216... ;
if A's first shot is intentionally missed, then A's probability of surviving is 75/189=0.39682... .

Thus, A's best strategy is to intentionally miss on the first shot.

Intuitively, what this comes down to is that A stands a better chance in a duel against C, if A gets the first shot, than A does in a duel against B, if B gets the first shot.