On the Nash equilibria of a duel with terminal payoffs - Athanasios Kehagias (Games, 2023)
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.
Keywords: 330; duel; Nash equilibrium; stochastic games