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Enumeration of All the Extreme Equilibria in Game Theory: Bimatrix and Polymatrix Games

Author

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  • C. Audet

    (GERAD and École Polytechnique de Montréal)

  • S. Belhaiza

    (École Polytechnique de Montréal)

  • P. Hansen

    (GERAD and HEC Montréal)

Abstract

Bimatrix and polymatrix games are expressed as parametric linear 0–1 programs. This leads to an algorithm for the complete enumeration of their extreme equilibria, which is the first one proposed for polymatrix games. The algorithm computational experience is reported for two and three players on randomly generated games for sizes up to 14 × 14 and 13 × 13 × 13.

Suggested Citation

  • C. Audet & S. Belhaiza & P. Hansen, 2006. "Enumeration of All the Extreme Equilibria in Game Theory: Bimatrix and Polymatrix Games," Journal of Optimization Theory and Applications, Springer, vol. 129(3), pages 349-372, June.
    • Handle: RePEc:spr:joptap:v:129:y:2006:i:3:d:10.1007_s10957-006-9070-3
    DOI: 10.1007/s10957-006-9070-3
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    References listed on IDEAS

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    1. Quintas, L G, 1989. "A Note on Polymatrix Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(3), pages 261-272.
    2. C. E. Lemke, 1965. "Bimatrix Equilibrium Points and Mathematical Programming," Management Science, INFORMS, vol. 11(7), pages 681-689, May.
    3. C. Audet & P. Hansen & B. Jaumard & G. Savard, 1997. "Links Between Linear Bilevel and Mixed 0–1 Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 273-300, May.
    4. Keiding, Hans, 1997. "On the Maximal Number of Nash Equilibria in ann x nBimatrix Game," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 148-160, October.
    5. McKelvey, Richard D. & McLennan, Andrew, 1996. "Computation of equilibria in finite games," Handbook of Computational Economics, in: H. M. Amman & D. A. Kendrick & J. Rust (ed.), Handbook of Computational Economics, edition 1, volume 1, chapter 2, pages 87-142, Elsevier.
    6. Todd R. Kaplan & John Dickhaut, "undated". "A Program for Finding Nash Equilibria," Working papers _004, University of Minnesota, Department of Economics.
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    Cited by:

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    2. Fandel, G. & Trockel, J., 2013. "Avoiding non-optimal management decisions by applying a three-person inspection game," European Journal of Operational Research, Elsevier, vol. 226(1), pages 85-93.
    3. Renato Soeiro & Alberto A. Pinto, 2022. "A Note on Type-Symmetries in Finite Games," Mathematics, MDPI, vol. 10(24), pages 1-13, December.
    4. Deng, Xinyang & Jiang, Wen & Wang, Zhen, 2019. "Zero-sum polymatrix games with link uncertainty: A Dempster-Shafer theory solution," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 101-112.
    5. Gabriele Dragotto & Rosario Scatamacchia, 2023. "The Zero Regrets Algorithm: Optimizing over Pure Nash Equilibria via Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 35(5), pages 1143-1160, September.

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