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Limiting Dynamics for Q‐Learning with Memory One in Symmetric Two‐Player, Two‐Action Games

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  • J. M. Meylahn
  • L. Janssen

Abstract

We develop a method based on computer algebra systems to represent the mutual pure strategy best‐response dynamics of symmetric two‐player, two‐action repeated games played by players with a one‐period memory. We apply this method to the iterated prisoner’s dilemma, stag hunt, and hawk‐dove games and identify all possible equilibrium strategy pairs and the conditions for their existence. The only equilibrium strategy pair that is possible in all three games is the win‐stay, lose‐shift strategy. Lastly, we show that the mutual best‐response dynamics are realized by a sample batch Q‐learning algorithm in the infinite batch size limit.

Suggested Citation

  • J. M. Meylahn & L. Janssen, 2022. "Limiting Dynamics for Q‐Learning with Memory One in Symmetric Two‐Player, Two‐Action Games," Complexity, John Wiley & Sons, vol. 2022(1).
    • Handle: RePEc:wly:complx:v:2022:y:2022:i:1:n:4830491
    DOI: 10.1155/2022/4830491
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    References listed on IDEAS

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    1. Stephanie Assad & Robert Clark & Daniel Ershov & Lei Xu, 2020. "Algorithmic Pricing and Competition: Empirical Evidence from the German Retail Gasoline Market," CESifo Working Paper Series 8521, CESifo.
    2. Hopkins, Ed, 1999. "A Note on Best Response Dynamics," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 138-150, October.
    3. Timo Klein, 2021. "Autonomous algorithmic collusion: Q‐learning under sequential pricing," RAND Journal of Economics, RAND Corporation, vol. 52(3), pages 538-558, September.
    4. 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).
    5. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, December.
    6. Waltman, Ludo & Kaymak, Uzay, 2008. "Q-learning agents in a Cournot oligopoly model," Journal of Economic Dynamics and Control, Elsevier, vol. 32(10), pages 3275-3293, October.
    7. Rabin, Matthew, 1993. "Incorporating Fairness into Game Theory and Economics," American Economic Review, American Economic Association, vol. 83(5), pages 1281-1302, December.
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    Cited by:

    1. Leonidov, A. & Titov, A. & Vasilyeva, E., 2025. "Strategic exploitation in Boltzmann Q-learning in the prisoners dilemma," Chaos, Solitons & Fractals, Elsevier, vol. 201(P2).

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