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Infinitely repeated quantum games and strategic efficiency

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Abstract

Repeated quantum game theory addresses long-term relations among players who choose quantum strategies. In the conventional quantum game theory, single-round quantum games or at most finitely repeated games have been widely studied; however, less is known for infinitely repeated quantum games. Investigating infinitely repeated games is crucial since finitely repeated games do not much differ from single-round games. In this work, we establish the concept of general repeated quantum games and show the Quantum Folk Theorem, which claims that by iterating a game one can find an equilibrium strategy of the game and receive reward that is not obtained by a Nash equilibrium of the corresponding single-round quantum game. A significant difference between repeated quantum prisoner’s dilemma and repeated classical prisoner’s dilemma is that the classical Pareto optimal solution is not always an equilibrium of the repeated quantum game when entanglement is sufficiently strong. When entanglement is sufficiently strong and reward is small, mutual cooperation cannot be an equilibrium of the repeated quantum game. In addition, we present several concrete equilibrium strategies of the repeated quantum prisoner’s dilemma.

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Notes

  1. The most general definition of a single-qubit mixed strategy U is given as

    $$\begin{aligned} U=\int _{U(2)} p(g)g \mathrm{d}\mu (g), \end{aligned}$$
    (2.6)

    where \(\mathrm{d}\mu \) is a Haar measure on U(2) and p(g) is non-negative and satisfies \(\int _{U(2)} p(g)\mathrm{d}\mu (g)=1\). Throughout this article, we intend to explore mixed strategies based on Def. 2.2.

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Acknowledgements

This work was supported by PIMS Postdoctoral Fellowship Award (K. I.)

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Authors and Affiliations

  1. Department of Mathematics and Statistics and Centre for Quantum Topology and Its Applications (quanTA), University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E6, Canada

    Kazuki Ikeda

  2. Department of Physics, Osaka University, Toyonaka, Osaka, 5600043, Japan

    Shoto Aoki

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  1. Kazuki Ikeda
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  2. Shoto Aoki
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K.I. designed and performed the research, interpreted the results, and wrote the paper. S.A. helped with the calculations.

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Ikeda, K., Aoki, S. Infinitely repeated quantum games and strategic efficiency. Quantum Inf Process 20, 387 (2021). https://doi.org/10.1007/s11128-021-03295-7

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