Abstract
AbstractWe 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.
Highlights
It is well known that fixed point theorems play an important role in game theory and mathematical economics [ – ]
To the best of our knowledge, there is no proof for the Kakutani and Brouwer fixed point theorems via the Nash equilibrium theorem, we can find in the previous literature many proofs or equivalent results for these two theorems [, ]
In Section, we show that the Kakutani fixed point theorem, Walras equilibrium theorem, and generalized variational inequality can be derived from the Nash equilibrium theorem with the aid of an inverse of the Berge maximum theorem [, ]
Summary
It is well known that fixed point theorems play an important role in game theory and mathematical economics [ – ]. In Section , we show that 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 [ , ]. (Inverse of Berge maximum theorem) Let X be a subset of the n-dimensional Euclidean space Rn, and K : X ⇒ Rm be a nonempty convex compact-valued and upper semicontinuous correspondence. (Kakutani fixed point theorem) Let X be a nonempty, convex, bounded, and closed subset of Rn, and F : X ⇒ X be a nonempty convex compact-valued and upper semicontinuous correspondence. (Brouwer fixed point theorem) Let X be a nonempty, convex, bounded, and closed subset of Rn, and φ be a continuous function from X to itself.
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