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Evolutionarily stable sets in quantum penny flip games

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Abstract

In game theory, an Evolutionarily Stable Set (ES set) is a set of Nash Equilibrium (NE) strategies that give the same payoffs. Similar to an Evolutionarily Stable Strategy (ES strategy), an ES set is also a strict NE. This work investigates the evolutionary stability of classical and quantum strategies in the quantum penny flip games. In particular, we developed an evolutionary game theory model to conduct a series of simulations where a population of mixed classical strategies from the ES set of the game were invaded by quantum strategies. We found that when only one of the two players’ mixed classical strategies were invaded, the results were different. In one case, due to the interference phenomenon of superposition, quantum strategies provided more payoff, hence successfully replaced the mixed classical strategies in the ES set. In the other case, the mixed classical strategies were able to sustain the invasion of quantum strategies and remained in the ES set. Moreover, when both players’ mixed classical strategies were invaded by quantum strategies, a new quantum ES set was emerged. The strategies in the quantum ES set give both players payoff 0, which is the same as the payoff of the strategies in the mixed classical ES set of this game.

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Notes

  1. Quantization here refers to “deriving a quantum version of a classical algorithm”, which is different from “the process of converting analog to digital signals” that is more popular in the wider scientific community.

  2. The \(\bar{a}\) defines complex conjugate of \(a\).

  3. The \(\dagger \) notion defines Hermitian conjugate.

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Acknowledgments

Tina Yu would like to thank Institute of Physics, Academia Sinica, Nankang, Taipei, Taiwan for their support during her visit working on the paper.

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Authors and Affiliations

  1. Department of Computer Science, Memorial University of Newfoundland, St John’s, NL, A1B 3X5, Canada

    Tina Yu

  2. Department of Software Engineering, Jerusalem College of Engineering, Jerusalem, Israel

    Radel Ben-Av

Authors
  1. Tina Yu
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  2. Radel Ben-Av
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Correspondence to Tina Yu.

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Yu, T., Ben-Av, R. Evolutionarily stable sets in quantum penny flip games. Quantum Inf Process 12, 2143–2165 (2013). https://doi.org/10.1007/s11128-012-0515-3

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  • DOI: https://doi.org/10.1007/s11128-012-0515-3

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