On the Nash Equilibria of a Duel with Terminal Payoffs

We formulate and study a two-player duel game as a terminal payoffs stochastic game. Players P1,P2 are standing in place and, in every turn, each may shoot at the other (in other words, abstention is allowed). If Pn shoots Pm (m≠n), either they hit and kill them (with probability pn) or they miss and Pm is unaffected (with probability 1−pn). The process continues until at least one player dies; if no player ever dies, the game lasts an infinite number of turns. Each player receives a positive payoff upon killing their opponent and a negative payoff upon being killed. We show that the unique stationary equilibrium is for both players to always shoot at each other. In addition, we show that the game also possesses “cooperative” (i.e., non-shooting) non-stationary equilibria. We also discuss a certain similarity that the duel has to the iterated Prisoner’s Dilemma.