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The Duel Discounted Stochastic Game

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Abstract

A duel involves two stationary players who shoot at each other until at least one of them dies; a truel is similar but involves three players. In the past, the duel has been studied mainly as a component of the truel, which has received considerably more attention. However we believe that the duel is interesting in itself. In this paper we formulate the duel (with either simultaneous or sequential shooting) as a discounted stochastic game. We show that this game has a unique Nash equilibrium in stationary strategies; however, it also possesses cooperation-promoting Nash equilibria in nonstationary strategies. We show that these are also subgame perfect equilibria. Finally, we argue that the nature of the game and its equilibria is similar to that of the repeated Prisoner’s dilemma.

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Data Availability

We do not analyze or generate any datasets, because our work proceeds within a theoretical and mathematical approach.

Notes

  1. There is no loss of generality in assuming \(a_{1}=1\), because \(\Pi _{1}\left( \delta \right) \) is homogeneous in \(a_{1}\) and \(b_{1}\).

  2. Of course, when the game is in \(\left( 2,1,0\right) \), \(P_{2}\) is supposed to have the shot, but he cannot actually shoot, since he is already dead; similarly for \(\left( 1,0,1\right) \) and \(P_{1}\).

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  1. School of Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Athanasios Kehagias

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  1. Athanasios Kehagias
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All the work reported in this article was performed by AK.

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Correspondence to Athanasios Kehagias.

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Kehagias, A. The Duel Discounted Stochastic Game. Dyn Games Appl 14, 846–864 (2024). https://doi.org/10.1007/s13235-023-00540-9

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