EXAMPLE 10 A Truel

A truel is like a duel, except that there are three players. Truels are depicted in several movies, including The Good, the Bad and the Ugly (1966), Reservoir Dogs (1992), and Pulp Fiction (1994). The photo below shows three of the characters in The Good, the Bad and the Ugly, about to engage in a truel—each man is armed and planning his next move.

In a truel, each player can either fire or not fire his or her gun at either of the other two players. We assume the goal of each player is, first, to survive and, second, to survive with as few other players as possible. Each player has one bullet and is a perfect shot; no communication (e.g., to pick out a common target) leading to a binding agreement with other players is allowed, making the game noncooperative. We will discuss the answers that simultaneous choices, on the one hand, and sequential choices, on the other, give to what is optimal for the players to do in the truel.

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If choices are simultaneous, at the start of play, each player will fire at one of the other two players, killing that player.

Why will the players all fire at each other? Because their own survival does not depend an iota on what they do. Since they cannot affect what happens to themselves but can affect how many others survive (the fewer the better, according to the postulated secondary goal), they should all blaze away at each other. In fact, the players all have dominant strategies to shoot at each other because whether or not a player survives—we will discuss shortly the probabilities of doing so—he or she does at least as well shooting an opponent.

The game, and optimal strategies in it, would change if the players (1) were allowed more options, such as to fire in the air and thereby disarm themselves, or (2) did not have to choose simultaneously but, instead, a particular order of play were specified. Thus, if the order of play were , followed by and choosing simultaneously, followed by any player with a bullet remaining choosing, then would fire in the air, and and would subsequently shoot each other. ( is no threat to or , so neither of the latter will fire at and waste a bullet; on the other hand, if one of or did not fire immediately at the other, that player would not survive to get in the last shot, so both and will fire at each other.) Thus, will be the sole survivor.

In 1992, a modified version of this scenario was played out in late-night television programming among the three major TV broadcasting networks of the time, with ABC effectively going first with Nightline, its well-established news program, and CBS and NBC dueling about which host, David Letterman or Jay Leno, to choose for their entertainment shows. Regardless of their ultimate choices, ABC “won” when CBS and NBC were forced to divide the entertainment audience. In 2002, ABC, presumably to attract a younger audience than the one that watched Nightline, attempted unsuccessfully to hire Letterman from CBS. Ted Koppel retired in 2005, but Nightline continues with other hosts.

To return to the original game (all choose simultaneously), the players’ strategies of all firing have two possible consequences: Either one player survives (even if two players fire at the same person, the third must fire at one of them, leaving only one survivor), or no player survives (if each player fires at a different person). In either event, there is no guarantee of survival. In fact, if each player has an equal probability of firing at one of the two other players, the probability that any player will survive is only 25%. The reason is that if the three players are , , and , will be killed if fires at him or her, does, or both do. The only circumstance in which will survive is if and fire at each other, which gives 1 chance in 4.

If choices are sequential, no player will fire at any other, so all will survive.

At the start of the truel, all the players are alive, which satisfies their primary goal of survival, though not their secondary goal of surviving with as few others as possible. Now assume that contemplates shooting , thereby reducing the number of survivors. Looking ahead, however, knows that by firing first and killing , he or she will be defenseless and be immediately shot by , who will then be the sole survivor.

It is in ’s interest, therefore, not to shoot anybody at the start, and the same logic applies to each of the other players. Hence, everybody will survive, which is a happier outcome than when choices are simultaneous, in which case everyone’s primary goal of survival is not satisfied—or, quantitatively speaking, satisfied only 25% of the time.