量子博弈—“PQ”问题

杨晓堃; 李维; 黄永畅

doi:10.7498/aps.73.20230592

摘要

研究了量子博弈中的“PQ”问题, 重点探讨了参与者策略制定、收益分配和量子效应对博弈结果的影响. 首先回顾了量子博弈理论的发展, 并对经典博弈与量子博弈进行了对比, 详细定义了参与者和策略空间, 并利用收益矩阵和量子操作符进行了建模. 为解决“PQ”问题, 提出了量子策略选择和优化算法, 并设计了针对该问题的量子算法. 在实际应用和案例研究方面, 通过评述经典双人博弈—“PQ翻硬币”问题, 表明当其中一个参与人采用量子策略时, 他可以打败他的经典对手, 获得更高的收益. 通过量化手段, 把整个“PQ”问题放在一个更为普遍和公平的条件下, 对其推广并从多个方面和角度再次对其进行研究. 同时, 给出了“不完全公平博弈”和“完全公平博弈”的定义, 对量子硬币翻转博弈(Meyer 1999 Phys. Rev. lett. 82 1052 )进行了修改, 使博弈公平, 并研究了量子硬币翻转的多轮版本. 最后对“PQ”问题的局限性与扩展性进行了讨论, 并展望了量子博弈研究的未来发展方向.

关键词:

Abstract

This paper reviews the coin flipping problem of “PQ” in a classical two-player game, and shows that when one player adopts quantum strategy, he can beat his classical opponent and gain higher returns. By using quantitative means, the whole “PQ” problem is put under a more general and fair condition, and it is generalized and studied again from many aspects and angles. Finally, we obtain some more essential and important conclusions, and explain the conclusions and its practical significance through some practical examples. At the same time, this paper gives the definition of imperfect fair game and the definition of perfect fair game, revises the quantum coin flipping game to make the game fair, and studies a multi-round version of quantum coin flipping. Some basic conclusions of fair quantum game are obtained, and the meaning of perfectly fair game is explained in practice.

Keywords:

作者及机构信息

北京工业大学理论物理研究所, 北京　100022

通信作者: 杨晓堃, xkyang1003@hotmail.com

基金项目: 国家自然科学基金(批准号: 11875081)资助的课题.

Authors and contacts

Institute of Theoretical Physics, Beijing University of Technology, Beijing 100022, China

Corresponding author: Yang Xiao-Kun, xkyang1003@hotmail.com

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11875081).

文章全文 : translate this paragraph

参考文献

[1]	Nielsen M A Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) p306
[2]	Rasmusen E 2000 An Introduction to Game Theory (New Jersey: Blackwell Publishing) p19
[3]	Cheong K H, Koh J M, Jones M C 2019 BioEssays 41 1900027 Google Scholar
[4]	Koh J M, Cheong K H 2020 Adv. Sci. 7 2001126 Google Scholar
[5]	Cheong K H, Wen T, Lai J W 2020 Adv. Sci. 7 2002324 Google Scholar
[6]	Eisert J, Wilkens M, Lewensetein M 1999 Phys. Rev. Lett. 83 3077 Google Scholar
[7]	Benjamin S C, Hayden P M 2001 Phys. Rev. A 64 030301 Google Scholar
[8]	Rajendran J, Benjamin C 2018 R. Soc. Open Sci. 5 171599 Google Scholar
[9]	Rajendran J, Benjamin C 2018 Europhys. Lett. 122 40004 Google Scholar
[10]	Lai J W, Cheong K H 2020 Nonlinear Dyn. 100 849 Google Scholar
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[12]	Lai J W, Tan J R A, Lu H, Yap Z R, Cheong K H 2020 Phys. Rev. E 102 012213 Google Scholar
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[14]	Du J F, Li H, Xu X D, Shi M J, Shi M, Wu J, Zhou X, Han R. 2002 Phys. Rev. Lett. 88 137902 Google Scholar
[15]	Meyer D A 1999 Phys. Rev. Lett. 82 1052 Google Scholar
[16]	Van Enk S J 2000 Phys. Rev. Lett. 84 789 Google Scholar
[17]	Meyer D A 2000 Phys. Rev. Lett. 84 790 Google Scholar
[18]	Huang Y C, Ma F C, Zhang N 2004 Mod. Phys. Lett. B18 1367 Google Scholar
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施引文献

[1]	Nielsen M A Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) p306
[2]	Rasmusen E 2000 An Introduction to Game Theory (New Jersey: Blackwell Publishing) p19
[3]	Cheong K H, Koh J M, Jones M C 2019 BioEssays 41 1900027 Google Scholar
[4]	Koh J M, Cheong K H 2020 Adv. Sci. 7 2001126 Google Scholar
[5]	Cheong K H, Wen T, Lai J W 2020 Adv. Sci. 7 2002324 Google Scholar
[6]	Eisert J, Wilkens M, Lewensetein M 1999 Phys. Rev. Lett. 83 3077 Google Scholar
[7]	Benjamin S C, Hayden P M 2001 Phys. Rev. A 64 030301 Google Scholar
[8]	Rajendran J, Benjamin C 2018 R. Soc. Open Sci. 5 171599 Google Scholar
[9]	Rajendran J, Benjamin C 2018 Europhys. Lett. 122 40004 Google Scholar
[10]	Lai J W, Cheong K H 2020 Nonlinear Dyn. 100 849 Google Scholar
[11]	Lai J W, Cheong K H 2020 Phys. Rev. E 101 052212 Google Scholar
[12]	Lai J W, Tan J R A, Lu H, Yap Z R, Cheong K H 2020 Phys. Rev. E 102 012213 Google Scholar
[13]	Marinatto L, Weber T 2000 Phys. Lett. A272 291 Google Scholar
[14]	Du J F, Li H, Xu X D, Shi M J, Shi M, Wu J, Zhou X, Han R. 2002 Phys. Rev. Lett. 88 137902 Google Scholar
[15]	Meyer D A 1999 Phys. Rev. Lett. 82 1052 Google Scholar
[16]	Van Enk S J 2000 Phys. Rev. Lett. 84 789 Google Scholar
[17]	Meyer D A 2000 Phys. Rev. Lett. 84 790 Google Scholar
[18]	Huang Y C, Ma F C, Zhang N 2004 Mod. Phys. Lett. B18 1367 Google Scholar
[19]	Huang Y C, Liu M, Suo M 2007 Int. J. Mod. Phys. B21 4387 Google Scholar
[20]	Chang D, Huang Y C 2008 Mod. Phys. Lett. B 22 3145 Google Scholar
[21]	Huang Y C, Lee X G, Shao M X 2006 Mod. Phys. Lett. A21 1107 Google Scholar
[22]	Huang Y C, Yu C X 2007 Phys. Rev. D 75 044011 Google Scholar
[23]	Huang Y C, Huo Q H 2008 Phys. Lett. B 662 290 Google Scholar
[24]	Liao L, Huang Y C 2007 Ann. Phys. (N. Y.) 322 2469 Google Scholar
[25]	Fei S M, Gao X H, Wang X H, Wang Z X, Wu K 2003 Phys. Rev. A 68 022315 Google Scholar

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