# truel

There's to be a three-cornered gunfight between Arnie, Bullseye, and Clint, who are standing at the corners of an equilateral triangle. All know that Arnie's chance of hitting a target is 0.3 and Clint's is 0.5, while Bullseye never misses. They must fire at their choice of target in succession, in the cyclical order Arnie, Bullseye, and Clint, until only one man is left. A man who's been hit is out of the fight and can no longer be shot at. What should Arnie's strategy be?

Truels, like the one just described, have become a significant topic in game theory because they're analogs of a variety of real-life situations, ranging from rivalry among animals to competition between television networks. Small changes in the rules can lead to strikingly different, sometimes counterintuitive outcomes. Different firing rules are possible: sequential in fixed order (players fire one at a time in a predetermined, repeating sequence, as in the example described), sequential in random order (the first player to fire and each subsequent player is chosen at random from among the survivors), or simultaneous (all surviving players fire at the same time in every round). In certain truels, a participant is allowed to shoot at the ground rather than try to eliminate an opponent (an optimal strategy if the firing order is fixed and each player has only one bullet and is a perfect shot). If the first shooter misses intentionally, he eliminates himself as a threat, and the other two fight it out, leaving two survivors in the end. Any other course of action would lead to the first shooter's own demise, with only one survivor. Even if the players have an unlimited supply of bullets, the truel may still end with more than one survivor because no player wants to be the first to shoot. Indeed, under the fixed firing order rule, no player has an incentive to eliminate another player. Only in the case of simultaneous firing is there a chance that nobody will survive. Most of the mathematical research on truels concerns the relationship between a player's marksmanship (probability of hitting a target) and his or her survival probability. It's possible to show, for example, that better marksmanship can actually hurt in many situations. In a sequential truel in which participants aren't allowed to shoot in the air, a player maximizes his probability of survival by firing at the opponent against whom he would less prefer to fight in a duel – regardless of what the other players do. If his shot misses, it makes no difference who the target was. If the shot hits the target, the shooter is better off because his opponent in the next duel is weaker. Thus, the first shooter fires at the opponent whose marksmanship is higher. In general, depending on the marksmanship values, the survival probabilities of the truelists could end up in any order, including one that is the reverse order of shooting skill. Optimal play can be very sensitive to slight changes in the rules, such as the number of rounds of play allowed. At the same time, some findings for truels are quite robust: the weakness of being the best marksman, the fragility of pacts, the possibility that unlimited supplies of ammunition may stabilize rather than undermine cooperation, and the deterrent effect of an indefinite number of rounds of play (which can prevent players from trying to get the last shot). Some of these findings are counterintuitive, even paradoxical. An understanding of them might well dampen the desire of aggressive players to score quick but temporary wins, rendering them more cautious. In particular, contemplating the consequences of a long, drawn-out conflict, truelists may come to realize that their own actions, while immediately beneficial, may trigger forces that ultimately lead to their own destruction.

• LOGIC