Nash’s Existence Theorem for Non-Compact Strategy Sets

Xinyu Zhang, Chunyan Yang (), Renjie Han and Shiqing Zhang
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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

Mathematics, 2024, vol. 12, issue 13, 1-10

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.

Keywords: game theory; Nash equilibrium; Ky Fan inequality; two-player zero-sum game (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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