We show that the Kakutani and Brouwer fixed point theorems can be obtained by directly using the Nash equilibrium theorem. The corresponding set-valued problems, such as the Kakutani fixed point theorem, Walras equilibrium theorem (set-valued excess demand function), and generalized variational inequality, can be derived from the Nash equilibrium theorem, with the aid of an inverse of the Berge maximum theorem. For the single-valued situation, we derive the Brouwer fixed point theorem, Walras equilibrium theorem (single-valued excess demand function), KKM lemma, and variational inequality from the Nash equilibrium theorem directly, without any recourse.
CITATION STYLE
Yu, J., Wang, N. F., & Yang, Z. (2016). Equivalence results between Nash equilibrium theorem and some fixed point theorems. Fixed Point Theory and Applications, 2016(1). https://doi.org/10.1186/s13663-016-0562-z
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