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Algorithm for Computing Approximate Nash Equilibrium in Continuous Games with Application to Continuous Blotto

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  • Sam Ganzfried

    (Ganzfried Research, Miami Beach, FL 33139, USA)

Abstract

Successful algorithms have been developed for computing Nash equilibrium in a variety of finite game classes. However, solving continuous games—in which the pure strategy space is (potentially uncountably) infinite—is far more challenging. Nonetheless, many real-world domains have continuous action spaces, e.g., where actions refer to an amount of time, money, or other resource that is naturally modeled as being real-valued as opposed to integral. We present a new algorithm for approximating Nash equilibrium strategies in continuous games. In addition to two-player zero-sum games, our algorithm also applies to multiplayer games and games with imperfect information. We experiment with our algorithm on a continuous imperfect-information Blotto game, in which two players distribute resources over multiple battlefields. Blotto games have frequently been used to model national security scenarios and have also been applied to electoral competition and auction theory. Experiments show that our algorithm is able to quickly compute close approximations of Nash equilibrium strategies for this game.

Suggested Citation

  • Sam Ganzfried, 2021. "Algorithm for Computing Approximate Nash Equilibrium in Continuous Games with Application to Continuous Blotto," Games, MDPI, vol. 12(2), pages 1-11, June.
    • Handle: RePEc:gam:jgames:v:12:y:2021:i:2:p:47-:d:566824
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    References listed on IDEAS

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    1. Fudenberg, Drew & Levine, David, 1998. "Learning in games," European Economic Review, Elsevier, vol. 42(3-5), pages 631-639, May.
    2. Sergiu Hart, 2008. "Discrete Colonel Blotto and General Lotto games," International Journal of Game Theory, Springer;Game Theory Society, vol. 36(3), pages 441-460, March.
    3. Brian Roberson, 2006. "The Colonel Blotto game," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 29(1), pages 1-24, September.
    4. Alexander Matros, 2007. "A Blotto Game with Incomplete Information," Working Paper 332, Department of Economics, University of Pittsburgh, revised Jul 2009.
    5. Sam Ganzfried & Conner Laughlin & Charles Morefield, 2019. "Parallel Algorithm for Approximating Nash Equilibrium in Multiplayer Stochastic Games with Application to Naval Strategic Planning," Papers 1910.00193, arXiv.org, revised Mar 2020.
    6. Noah Stein & Asuman Ozdaglar & Pablo Parrilo, 2008. "Separable and low-rank continuous games," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(4), pages 475-504, December.
    7. Kovenock, Dan & Roberson, Brian, 2011. "A Blotto game with multi-dimensional incomplete information," Economics Letters, Elsevier, vol. 113(3), pages 273-275.
    8. Adamo, Tim & Matros, Alexander, 2009. "A Blotto game with Incomplete Information," Economics Letters, Elsevier, vol. 105(1), pages 100-102, October.
    9. Sam Ganzfried, 2020. "Empirical Analysis of Fictitious Play for Nash Equilibrium Computation in Multiplayer Games," Papers 2001.11165, arXiv.org, revised Jul 2024.
    10. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, April.
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    Cited by:

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    2. Louis Abraham, 2023. "A Game of Competition for Risk," Working Papers hal-04112160, HAL.
    3. Louis Abraham, 2023. "A Game of Competition for Risk," Papers 2305.18941, arXiv.org.
    4. Le Zhang & Lijing Lyu & Shanshui Zheng & Li Ding & Lang Xu, 2022. "A Q-Learning-Based Approximate Solving Algorithm for Vehicular Route Game," Sustainability, MDPI, vol. 14(19), pages 1-14, September.

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