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On the Nash equilibria of a simple discounted duel

Kehagias Athanasios

Operations Research and Decisions

2024

Vol. 34, No. 2

65--84

We formulate and study a two-player – duel – game as a nonzero-sum discounted stochastic game. Players P1, andP2 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.