Truels, or Survival of the Weakest

Abstract

We're all familiar with the concept of duels. They can be deadly, such as those fought between two suitors vying for a lady's affections or sheriffs and outlaws battling it out in the American Old West, or they can end in both parties enjoying a beer, with the loser buying a round after an intense game of darts. A duel is simply a game with two players, each having a probability of winning that game and each having an intrinsic marksmanship, or ability, associated with his or her performance. Of course, the worst player--the one with the worst marksmanship--might have an advantage such as extra bullets or extra points, but in a normal unbiased situation, common sense tells us that the higher the marksmanship, the higher the probability of winning the duel. The mathematical treatment of this game confirms our simple expectations. If more than two players participate, a series of duels might be needed to determine the absolute winner. This is the case in sports tournaments, which pair teams in all possible ways until the ultimate winner emerges. A truel is a generalization of a duel involving three players and slightly different rules.