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Index and Robustness of Mixed Equilibria: An Algebraic Approach

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  • Lucas Pahl

Abstract

We present a new method for computation of the index of completely mixed equilibria in finite games, based on the work of Eisenbud et al.(1977). We apply this method to solving two questions about the relation of the index of equilibria and the index of fixed points, and the index of equilibria and payoff-robustness: any integer can be the index of an isolated completely mixed equilibrium of a finite game. In a particular class of isolated completely mixed equilibria, called monogenic, the index can be $0$, $+1$ or $-1$ only. In this class non-zero index is equivalent to payoff-robustness. We also discuss extensions of the method of computation to extensive-form games, and cases where the equilibria might be located on the boundary of the strategy set.

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  • Lucas Pahl, 2026. "Index and Robustness of Mixed Equilibria: An Algebraic Approach," Papers 2603.04298, arXiv.org, revised Mar 2026.
    • Handle: RePEc:arx:papers:2603.04298
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    References listed on IDEAS

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    1. Robert Wilson & Srihari Govindan, 1997. "Uniqueness of the index for Nash equilibria of two-player games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(3), pages 541-549.
    2. Andrew McLennan, 2018. "Advanced Fixed Point Theory for Economics," Springer Books, Springer, number 978-981-13-0710-2, December.
    3. Lucas Pahl, 2022. "Polytope-form games and Index/Degree Theories for Extensive-form games," Papers 2201.02098, arXiv.org, revised Jul 2023.
    4. Robert F. Brown, 2014. "A Topological Introduction to Nonlinear Analysis," Springer Books, Springer, edition 3, number 978-3-319-11794-2, December.
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    6. DeMichelis, Stefano & Germano, Fabrizio, 2000. "On the Indices of Zeros of Nash Fields," Journal of Economic Theory, Elsevier, vol. 94(2), pages 192-217, October.
    7. Pahl, Lucas, 2023. "Polytope-form games and index/degree theories for extensive-form games," Games and Economic Behavior, Elsevier, vol. 141(C), pages 444-471.
    8. Lucas Pahl & Carlos Pimienta, 2024. "Robust Equilibria in Generic Extensive form Games," Papers 2412.18449, arXiv.org, revised Mar 2025.
    9. Lucas Pahl & Carlos Pimienta, 2025. "Robust Equilibria In Generic Extensive-Form Games," Working Papers 2025001, The University of Sheffield, Department of Economics.
    10. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
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