In this work we present a detailed analysis using a Markov chain theory of some versions of the truel game in which a set of 3 players with different markmanships a, b and c, try to eliminate each other in a series of one-to-one competitions, using the rules of the game. The paradoxical result in this game is that under certain circumstances the player with the highest markmanship does not necessarily have the highest survival probability.
There are several versions of the truel, depending on the way the players are chosen. We can distinguish between a random firing case where each round one player is chosen randomly for shooting, and a sequential case where a fixed order for playing is defined in advance. The players, depending on the kind of truel they are playing, may adopt different strategies in order to maximize their own survival probability. We present analytical expressions for the actual distribution of winners in a truel competition when playing either the sequential or the random truel.
We also generalize the truels to N players, considering three different populations A, B, C, and then we study the dependence of the final surviving population upon the initial concentrations x_A, x_B and x_C. We will also present other variations of collective truels (N--uels), as well as a slightly modification of the original model, turning it into an opinion model.