On the Nash equilibria of a simple discounted duel

We formulate and study a two-player – duel – game as a nonzero-sum discounted stochastic game. Players P1, and P2 are standing in place and, in each turn, one or both may shoot at the other player. If Pn shoots at Pm (m ≠ n), either he hits and kills him (with probability pn) or he misses him and Pm is unaffected (with probability 1 − pn). The process continues until at least one player dies; if nobody ever dies, the game lasts an infinite number of turns. Each player receives a unit payoff for each turn in which he remains alive; no payoff is assigned to killing the opponent. We show that the always-shooting strategy is a NE but, in addition, the game also possesses so-called cooperative (i.e., non-shooting) Nash equilibria in both stationary and nonstationary strategies. A certain similarity to the repeated Prisoner’s Dilemma is also noted and discussed.