Nash’s Existence Theorem for Non-Compact Strategy Sets

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, s...

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Bibliographic Details
Published inMathematics (Basel) Vol. 12; no. 13; p. 2017
Main Authors Zhang, Xinyu, Yang, Chunyan, Han, Renjie, Zhang, Shiqing
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.07.2024
Subjects
Banach spaces
Coercivity
Equilibrium
Existence theorems
Game theory
Hypotheses
Inequality
Ky Fan inequality
Mathematicians
Minimax technique
Nash equilibrium
Strategy
two-player zero-sum game
Vector space
Zero sum games
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12132017