Ivars Peterson's MathTrek

January 26, 1998

# Truel in the Sun

Brilliant sunshine bakes a huddled row of ramshackle, weather-beaten buildings lining a dusty thoroughfare. Two gunfighters slam out of a decrepit saloon and stalk toward their posts at either end of the street. Facing each other, they prepare to draw their six-shooters.

Abruptly, a third gunfighter steps out. The duel has turned into a truel: the good versus the bad versus the ugly. Who shoots at whom?

Game theory typically deals with the consequences of players' actions, given the rules of the game. Researchers seek to determine the optimal outcome in a particular situation.
 In a truel, each of the participants fires bullets at the others to try to eliminate them and survive. Game theorists are interested in the nature and timing of each player's actions against his or her opponents. Truels can arise in a variety of real-life situations, ranging from fierce rivalry among animals living in close proximity to competition among the major television networks. Such three-party conflicts are also often played out in the arena of international politics. In the December 1997 Mathematics Magazine, D. Marc Kilgour of Wilfrid Laurier University in Waterloo, Ontario, and Steven J. Brams of New York University provide a detailed look at truels, highlighting how small changes in the rules can lead to startlingly different, sometimes counterintuitive outcomes. Here's a typical decision problem involving a truel. It concerns a three-cornered gunfight among Arnold, Bert, and Charlie. Everyone knows that Arnold's probability of hitting his target is .3, Charlie's is .5, and Bert never misses. The truelists must fire at their choice of target in succession, in the cyclical order Arnold, Bert, and Charlie, until only one man is left. What should Arnold's strategy be?

To study target selection, one can think of the three players as standing at the corners of an equilateral triangle. Each player shoots at an opponent at one of the other two corners. Several firing rules are possible: sequential in fixed order (players fire one at a time in a fixed, repeating sequence, as in the sample problem), 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 in the air rather than to try to eliminate an opponent. That would be the 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 terminate with more than one survivor because no player wants to be the first to fire," Kilgour and Brams note. 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 survives.

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 are not allowed to shoot in the air, a player maximizes his survival probability 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.

Kilgour and Brams examine a wide variety of scenarios, including situations that involve pacts, unlimited amounts of ammunition, different firing orders, and limits on the number of rounds.

"Optimal play can be very sensitive to slight changes in the rules, such as the number of rounds of play allowed," Kilgour and Brams conclude. "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," they continue. "An understanding of them, we believe, 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 and 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."

### References:

Brams, Steven J. 1994. Theory of Moves. Cambridge, England: Cambridge University Press.

______. 1993. Theory of moves. American Scientist 81(November/December):562-570.

Kilgour, D. Marc, and Steven J. Brams. 1997. The truel. Mathematics Magazine 70(December):315-326.

Straffin, Philip D. 1993. Game Theory and Strategy. Washington, D.C.: Mathematical Association of America.

For a sampling of decision questions (and proposed solutions), including a lengthy discussion of a pistol truel, try http://forum.swarthmore.edu/dr.math/problems/truel.html.