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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum duels and truels

Adrian P. Flitney, Derek Abbott

Published May 16, 2003• 6 citations
Publisher

Abstract

A truel, or a three person generaliztion of a duel, is a popular model in game theory of a struggle of survival. The outcome is often sensitive to the precise rules under which the truel is performed and can be anti-intuitive. We propose a quantum scheme, along the lines of recent work in quantum game theory, for the problem of duels and truels. Interference amongst the players' strategies can arise leading to game equilibria different from the classical case.

DOI
10.1117/12.504779

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A truel, or a three person generaliztion of a duel, is a popular model in game theory of a struggle of survival. The outcome is often sensitive to the precise rules under which the truel is performed and can be anti-intuitive. We propose a quantum scheme, along the lines of recent work in quantum game theory, for the problem of duels and truels. Interference amongst the players' strategies can arise leading to game equilibria different from the classical case.
In game theory, a popular model of a struggle for survival among three competing agents is a truel, or three-person generalization of a duel. Adopting the ideas recently developed in quantum game theory, we present a quantum scheme for the problems of duels and truels. In the classical case, the outcome is sensitive to the precise rules under which the truel is performed and can be counterintuitive. These aspects carry over into our quantum scheme, but interference amongst the players' strategies can arise, leading to game equilibria different from the classical case.
We revisit the quantum two-person duel. In this problem, both Alice and Bob each possess a spin-1/2 particle which models dead and alive states for each player. We review the Abbott and Flitney result—now considering non-zero α1 and α2 in order to decide if it is better for Alice to shoot or not the second time—and we also consider a duel where players do not necessarily start alive. This simple assumption allows us to explore several interesting special cases, namely how a dead player can win the duel shooting just once, or how can Bob revive Alice after one shot, and the better strategy for Alice—being either alive or in a superposition of alive and dead states—fighting a dead opponent.
We quantize the classical game of simultaneous two-person duel. In this game, the players Alice and Bob, have a two level system, for example, a spin 1/2 particle, which models alive and dead states. Through physical considerations we write up the unitary operator which represents the classical action of firing, which quantum mechanically is represented by flipping the opponent’s spin and test its classical limits. We study three representative special cases of such duel where players start the duel: both alive; in a Bell state; and when one of them has her skills decreasing along the duel. We also present a proposal of experimental realization of this quantum game using twin photons produced by spontaneous parametric down conversion.
Recently the concept of quantum information has been introduced into game theory. Here we present the first study of quantum games with more than two players. We discover that such games can possess an alternative form of equilibrium strategy, one which has no analog either in traditional games or even in two-player quantum games. In these ``coherent'' equilibria, entanglement shared among multiple players enables different kinds of cooperative behavior: indeed it can act as a contract, in the sense that it prevents players from successfully betraying one another.