Equivalence results between Nash equilibrium theorem and some fixed point theorems

Fixexd point theory and applications

Jian Yu¹  Neng-Fa Wang¹  Zhe Yang² 

[1] Department of Mathematics, Guizhou University, Guiyang, P.R. China;School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China

关键词: Brouwer fixed point theorem;    Kakutani fixed point theorem;    Nash equilibrium theorem;    Walras equilibrium theorem;    KKM lemma;    variational inequality;    47H10;    54C60;    91A10;   

DOI  :  10.1186/s13663-016-0562-z

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

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.

【 授权许可】

CC BY