Truel
A truel is a three person expansion of a duel, in which each players can fire at one another in an attempt to eliminate them while surviving itself. [1]
Game theory overview
A variety of forms of truels have been studied. Features that determine the nature of a truel include
- the probability of each player hitting their chosen targets (often not assumed to be the same for each player)
- whether the players shoot simultaneously or sequentially, and, if sequentially, whether the shooting order is predetermined, or determined at random from among the survivors;
- the number of bullets each player has (in particular, whether this is finite or infinite);
- whether or not intentionally missing is allowed.
There is usually a general assumption that each player in the truel wants to survive, and will behave logically in a manner that maximizes individual probabilities of survival. If an unlimited number of bullets are used, the players in a truel will generally shoot at each other until only one player is left. In the widely studied form, the three have different probabilities of hitting their target.
Example of a theoretical truel
Three players (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 success probability is two-thirds and C's is three-thirds, i.e. 1. To even this score, the players will shoot sequentially, A taking the first shot, followed by B then C. The players have an infinite number of bullets, and are allowed to intentionally miss.
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?
Solution
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.