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Nash’s Existence Theorem for Non-Compact Strategy Sets

Author

Listed:
  • Xinyu Zhang

    (Beijing Normal University-Hong Kong Baptist University United International College, Zhuhai 519087, China)

  • Chunyan Yang

    (Division of Mathematics, Sichuan University Jinjiang College, Pengshan 620860, China)

  • Renjie Han

    (School of Economics, Chongqing Technology and Business University, Chongqing 400067, China)

  • Shiqing Zhang

    (School of Mathematics, Sichuan University, Chengdu 610065, China)

Abstract

In this paper, we apply the classical FKKM lemma to obtain the Ky Fan minimax inequality defined on nonempty non-compact convex subsets in reflexive Banach spaces, and then we apply it to game theory and obtain Nash’s existence theorem for non-compact strategy sets, which can be regarded as a new, simple but interesting application of the FKKM lemma and the Ky Fan minimax inequality, and we can also present another proof about the famous John von Neumann’s existence theorem in two-player zero-sum games. Due to the results of Li, Shi and Chang, the coerciveness in the conclusion can be replaced with the P.S. or G.P.S. conditions.

Suggested Citation

  • Xinyu Zhang & Chunyan Yang & Renjie Han & Shiqing Zhang, 2024. "Nash’s Existence Theorem for Non-Compact Strategy Sets," Mathematics, MDPI, vol. 12(13), pages 1-10, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:13:p:2017-:d:1425145
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    References listed on IDEAS

    as
    1. M. B. Lignola, 1997. "Ky Fan Inequalities and Nash Equilibrium Points without Semicontinuity and Compactness," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 137-145, July.
    2. F. Fakhar & H. R. Hajisharifi & Z. Soltani, 2023. "Noncoercive and noncontinuous equilibrium problems: existence theorem in infinite-dimensional spaces," Journal of Global Optimization, Springer, vol. 86(4), pages 989-1003, August.
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