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arXiv:quant-ph/0003036 · Physical Review Letters

Comment on “Quantum Games and Quantum Strategies''

Simon C. Benjamin, Patrick M. Hayden

Published Jul 19, 2001• 162 citations
arXivHTMLPDF

Abstract

In a recent paper, Eisert et al. presented a quantum mechanical generalization of Prisoner's Dilemma. They asserted that the maximally entangled game exhibits a unique Nash equilibrium which yields a pay-off equivalent to cooperative behaviour. In this comment we show that their observation is incorrect: there is no Nash equilibrium in the space of deterministic quantum strategies.

DOI
10.1103/physrevlett.87.069801

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In a recent paper, Eisert et al. presented a quantum mechanical generalization of Prisoner's Dilemma. They asserted that the maximally entangled game exhibits a unique Nash equilibrium which yields a pay-off equivalent to cooperative behaviour. In this comment we show that their observation is incorrect: there is no Nash equilibrium in the space of deterministic quantum strategies.
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We study a quantum game played by two players with restricted multiple strategies. It is found that in quantum game Nash equilibrium does not always exist which is totally different from that in classical game. This happens only when the initial state is in entangled state. At the same time, we find that when Nash equilibrium exists the pay off function is usually different from that in the classical counterpart except in some special cases. This presents an explicit example where quantum game and classical game differ fundamentally.