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Symmetric equilibrium of multi-agent reinforcement learning in repeated prisoner’s dilemma

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  • Usui, Yuki
  • Ueda, Masahiko

Abstract

We investigate the repeated prisoner’s dilemma game where both players alternately use reinforcement learning to obtain their optimal memory-one strategies. We theoretically solve the simultaneous Bellman optimality equations of reinforcement learning. We find that the Win-stay Lose-shift strategy, the Grim strategy, and the strategy which always defects can form symmetric equilibrium of the mutual reinforcement learning process amongst all deterministic memory-one strategies.

Suggested Citation

  • Usui, Yuki & Ueda, Masahiko, 2021. "Symmetric equilibrium of multi-agent reinforcement learning in repeated prisoner’s dilemma," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    • Handle: RePEc:eee:apmaco:v:409:y:2021:i:c:s0096300321004598
    DOI: 10.1016/j.amc.2021.126370
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    References listed on IDEAS

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    1. Christian Hilbe & Krishnendu Chatterjee & Martin A. Nowak, 2018. "Partners and rivals in direct reciprocity," Nature Human Behaviour, Nature, vol. 2(7), pages 469-477, July.
    2. Imhof, Lorens & Nowak, Martin & Fudenberg, Drew, 2007. "Tit-for-Tat or Win-Stay, Lose-Shift?," Scholarly Articles 3200671, Harvard University Department of Economics.
    3. Marc Harper & Vincent Knight & Martin Jones & Georgios Koutsovoulos & Nikoleta E Glynatsi & Owen Campbell, 2017. "Reinforcement learning produces dominant strategies for the Iterated Prisoner’s Dilemma," PLOS ONE, Public Library of Science, vol. 12(12), pages 1-33, December.
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    Cited by:

    1. Masahiko Ueda, 2022. "Controlling Conditional Expectations by Zero-Determinant Strategies," SN Operations Research Forum, Springer, vol. 3(3), pages 1-22, September.
    2. Ueda, Masahiko, 2023. "Memory-two strategies forming symmetric mutual reinforcement learning equilibrium in repeated prisoners’ dilemma game," Applied Mathematics and Computation, Elsevier, vol. 444(C).
    3. Wang, Xianjia & Yang, Zhipeng & Liu, Yanli & Chen, Guici, 2023. "A reinforcement learning-based strategy updating model for the cooperative evolution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 618(C).
    4. Ding, Zhen-Wei & Zheng, Guo-Zhong & Cai, Chao-Ran & Cai, Wei-Ran & Chen, Li & Zhang, Ji-Qiang & Wang, Xu-Ming, 2023. "Emergence of cooperation in two-agent repeated games with reinforcement learning," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    5. Yuan, Hairui & Meng, Xinzhu, 2022. "Replicator dynamics of the Hawk-Dove game with different stochastic noises in infinite populations," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    6. Wolfram Barfuss & Janusz Meylahn, 2022. "Intrinsic fluctuations of reinforcement learning promote cooperation," Papers 2209.01013, arXiv.org, revised Feb 2023.

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