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Mean Field Game Approach to Bitcoin Mining

Author

Listed:
  • Charles Bertucci

    (CMAP, Ecole Polytechnique, Palaiseau, France)

  • Louis Bertucci

    (Institut Louis Bachelier, Paris, France
    Haas School of Business, UC Berkeley, Berkeley, California)

  • Jean-Michel Lasry

    (Universit\'e Paris-Dauphine, PSL Research University, CEREMADE, Paris, France)

  • Pierre-Louis Lions

    (Universit\'e Paris-Dauphine, PSL Research University, CEREMADE, Paris, France
    Coll\`ege de France, Paris, France)

Abstract

We present an analysis of the Proof-of-Work consensus algorithm, used on the Bitcoin blockchain, using a Mean Field Game framework. Using a master equation, we provide an equilibrium characterization of the total computational power devoted to mining the blockchain (hashrate). From a simple setting we show how the master equation approach allows us to enrich the model by relaxing most of the simplifying assumptions. The essential structure of the game is preserved across all the enrichments. In deterministic settings, the hashrate ultimately reaches a steady state in which it increases at the rate of technological progress. In stochastic settings, there exists a target for the hashrate for every possible random state. As a consequence, we show that in equilibrium the security of the underlying blockchain is either $i)$ constant, or $ii)$ increases with the demand for the underlying cryptocurrency.

Suggested Citation

  • Charles Bertucci & Louis Bertucci & Jean-Michel Lasry & Pierre-Louis Lions, 2020. "Mean Field Game Approach to Bitcoin Mining," Papers 2004.08167, arXiv.org.
    • Handle: RePEc:arx:papers:2004.08167
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    References listed on IDEAS

    as
    1. Schilling, Linda & Uhlig, Harald, 2019. "Some simple bitcoin economics," Journal of Monetary Economics, Elsevier, vol. 106(C), pages 16-26.
    2. Julien Prat & Benjamin Walter, 2021. "An Equilibrium Model of the Market for Bitcoin Mining," Journal of Political Economy, University of Chicago Press, vol. 129(8), pages 2415-2452.
    3. Lin William Cong & Zhiguo He & Jiasun Li & Wei Jiang, 2021. "Decentralized Mining in Centralized Pools [Concentrating on the fall of the labor share]," The Review of Financial Studies, Society for Financial Studies, vol. 34(3), pages 1191-1235.
    4. Bruno Biais & Christophe Bisière & Matthieu Bouvard & Catherine Casamatta, 2019. "The Blockchain Folk Theorem," The Review of Financial Studies, Society for Financial Studies, vol. 32(5), pages 1662-1715.
    5. Huberman, Gur & Leshno, Jacob & Moallemi, Ciamac C., 2017. "Monopoly Without a Monopolist: An Economic Analysis of the Bitcoin Payment System," CEPR Discussion Papers 12322, C.E.P.R. Discussion Papers.
    6. Zongxi Li & A. Max Reppen & Ronnie Sircar, 2019. "A Mean Field Games Model for Cryptocurrency Mining," Papers 1912.01952, arXiv.org, revised Jan 2022.
    7. repec:zbw:bofrdp:2017_027 is not listed on IDEAS
    8. Easley, David & O'Hara, Maureen & Basu, Soumya, 2019. "From mining to markets: The evolution of bitcoin transaction fees," Journal of Financial Economics, Elsevier, vol. 134(1), pages 91-109.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Julien Prat & Benjamin Walter, 2021. "An Equilibrium Model of the Market for Bitcoin Mining," Journal of Political Economy, University of Chicago Press, vol. 129(8), pages 2415-2452.
    2. Wenpin Tang & David D. Yao, 2023. "Trading under the proof‐of‐stake protocol – A continuous‐time control approach," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 979-1004, October.
    3. Wenpin Tang & David D. Yao, 2022. "Trading under the Proof-of-Stake Protocol -- a Continuous-Time Control Approach," Papers 2207.12581, arXiv.org, revised Jun 2023.
    4. Rene Carmona, 2020. "Applications of Mean Field Games in Financial Engineering and Economic Theory," Papers 2012.05237, arXiv.org.
    5. Wenpin Tang, 2023. "Trading and wealth evolution in the Proof of Stake protocol," Papers 2308.01803, arXiv.org, revised Aug 2023.

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