On the Nash Equilibria of a Simple Discounted Duel

04/11/2023

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by   Athanasios Kehagias, et al.

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We formulate and study a two-player static duel game as a nonzero-sum discounted stochastic game. Players P_1,P_2 are standing in place and, in each turn, one or both may shoot at the other player. If P_n shoots at P_m (m≠ n), either he hits and kills him (with probability p_n) or he misses him and P_m is unaffected (with probability 1-p_n). The process continues until at least one player dies; if nobody ever dies, the game lasts an infinite number of turns. Each player receives unit payoff for each turn in which he remains alive; no payoff is assigned to killing the opponent. We show that the the always-shooting strategy is a NE but, in addition, the game also possesses 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.