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arXiv:quant-ph/0305058 · Journal of Optics B Quantum and Semiclassical Optics

Quantum two- and three-person duels

Adrian P. Flitney, Derek Abbott

Published Jul 28, 2004• 33 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1464-4266/6/8/036

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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 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.
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.
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.
Quantum games extend classical game theory into the quantum realm by incorporating quantum players and quantum strategies. Two-player quantum games have been widely studied and applied, among other areas, in optimization and decision-making. In this work, we introduce, formalize and analyze a novel class of three-player quantum games, referred to as Proxy Quantum Games (PQGs). In this setting, one player, the Controller, influences or governs the strategic operations executed by a second player, the Proxy, who, in turn, competes directly against a third player, the Opponent. The strategic influence is implemented via controlled unitary operations, whereby the Proxy’s available strategies are conditioned on the Controller’s quantum state. We further generalize the PQG framework by incorporating multiple Controllers acting on a common Proxy, thereby enabling the study of higher-order strategic interactions and multi-agent influence structures. The objective under consideration is the maximization of the Controller’s expected payoff, under various configurations of strategic control and entanglement. Comparative analyses are conducted between the PQG formulation and conventional multiplayer quantum games, highlighting key differences in strategic behavior and payoff distributions.