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

Quantum minority game utilizing various forms of entanglement

Adrian P. Flitney, Lloyd C. L. Hollenberg

Published Dec 3, 2007• 4 citations
Publisher

Abstract

The quantum Minority game provides a means of studying the effect of multi-partite entanglement in a game theoretic setting. We study symmetric Nash equilibria and symmetric Pareto optimal strategies arising in a four-player quantum Minority game that uses an initial state that is a superposition of a GHZ state and products of EPR pairs. We find that the payoff curve for the symmetric Pareto optimal strategy is the same as that for the maximal violation of the Mermin-Ardehali-Belinskii-Klyshko inequality for the initial state, indicating a correspondence between quantum game theory and Bell inequalities. We also show that no advantage over the classical Minority game can be obtained when the initial state has only two party entanglement.

DOI
10.1117/12.774090

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The quantum Minority game provides a means of studying the effect of multi-partite entanglement in a game theoretic setting. We study symmetric Nash equilibria and symmetric Pareto optimal strategies arising in a four-player quantum Minority game that uses an initial state that is a superposition of a GHZ state and products of EPR pairs. We find that the payoff curve for the symmetric Pareto optimal strategy is the same as that for the maximal violation of the Mermin-Ardehali-Belinskii-Klyshko inequality for the initial state, indicating a correspondence between quantum game theory and Bell inequalities. We also show that no advantage over the classical Minority game can be obtained when the initial state has only two party entanglement.
A quantum version of the Minority game for an arbitrary number of agents is studied. When the number of agents is odd, quantizing the game produces no advantage to the players, however, for an even number of agents new Nash equilibria appear that have no classical analogue. The new Nash equilibria provide far preferable expected payoffs to the players compared to the equivalent classical game. The effect on the Nash equilibrium payoff of reducing the degree of initial entanglement between the players' qubits or of introducing various forms of decoherence is discussed. As the number of players increases the multipartite GHZ state becomes increasingly fragile, as indicated by the smaller error probability required to reduce the Nash equilibrium payoff to the classical level.
Game theory is central to the understanding of competitive interactions arising in many fields, from the social and physical sciences to economics. Recently, as the definition of information is generalized to include entangled quantum systems, quantum game theory has emerged as a framework for understanding the competitive flow of quantum information. Up till now, only two- and three-player quantum games have been demonstrated with restricted strategy sets. Here, we report the first experiment that implements a four-player quantum minority game over tunable four-partite entangled states encoded in the polarization of single photons. Experimental application of appropriate player strategies gives equilibrium payoff values well above those achievable in the classical game. These results are in excellent quantitative agreement with our theoretical analysis of the symmetric Pareto optimal strategies. Our results demonstrate for the first time how nontrivial equilibria can arise in a competitive situation involving quantum agents.
A quantum version of the Minority game for an arbitrary number of agents is considered. It is known that when the number of agents is odd, quantizing the game produces no advantage to the players, but for an even number of agents new Nash equilibria appear that have no classical analogue and have improved payoffs. We study the effect on the Nash equilibrium payoff of various forms of decoherence. As the number of players increases the multipartite GHZ state becomes increasingly fragile, as indicated by the smaller error probability required to reduce the Nash equilibrium payoff to the classical level.
A quantum version of the Minority game for an arbitrary number of agents is studied. When the number of agents is odd, quantizing the game produces no advantage to the players, however, for an even number of agents new Nash equilibria appear that have no classical analogue. The new Nash equilibria provide far preferable expected payoffs to the players compared to the equivalent classical game. The effect on the Nash equilibrium payoff of reducing the degree of entanglement, or of introducing decoherence into the model, is indicated.
Game theory is the mathematical framework for analyzing strategic interactions in conflict and competition situations. In recent years quantum game theory has earned the attention of physicists, and has emerged as a branch of quantum information theory [1]. With the aid of entanglement and linear superposition of strategies, quantum games are shown to yield signifcant advantage over their classical counterparts. In this paper we explore optimal and equilibrium solutions to quantum minority games. Initial states with different level of entanglement are investigated. Focus will be on 4 and 6 player games with some N-player generalizations.
Game theory is the mathematical framework for analyzing strategic interactions in conflict and competition situations.In recent years quantum game theory has earned the attention of physicists, and has emerged as a branch of quantum information theory [1].With the aid of entanglement and linear superposition of strategies, quantum games are shown to yield significant advantage over their classical counterparts.In this paper we explore optimal and equilibrium solutions to quantum minority games.Initial states with different level of entanglement are investigated.Focus will be on 4 and 6 player games with some N -player generalizations.