Nash Equilibrium. Keywords Nash equilibrium noncooperative games.

Tilburg: Operations research. Tilburg : Operations research, CentER Discussion Paper.

## Approximate Fixed Point Theorems in Banach Spaces with Applications in Game Theory

AU - Morgan, J. AU - Scalzo, V. AU - Tijs, S. N1 - Pagination: 15 PY - Y1 - N2 - In this paper some new approximate fixed point theorems for multifunctions in Banach spaces are presented and a method is developed indicating how to use approximate fixed point theorems in proving the existence of approximate Nash equilibria for non-cooperative games. Access to Document From the statements of [ 8 , 9 ] we know that the fixed point theorem, minimax inequility, and Nash equilibrium theorem can be derived from the KKM principle.

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However, to the best of our knowledge, there is no proof for the Kakutani and Brouwer fixed point theorems via the Nash equilibrium theorem, although we can find in the previous literature many proofs or equivalent results for these two theorems [ 2 , 8 , 9 ]. In this paper, we fill these gaps. To obtain the Kakutani fixed point theorem from the Nash equilibrium theorem, we need an inverse of the Berge maximum theorem.

Berge maximum theorem see [ 2 , 6 ]. In , Komiya [ 10 ] considered an inverse of the Berge maximum theorem, and Zhou [ 11 ] gave a simple alternative proof.

However, in this section, by using different methods, we derive the Kakutani fixed point theorem. This completes the proof.

Walras equilibrium may be formulated as follows. From the weak Walras law we have.

In , Browder [ 13 ] first gave the generalized variational inequality, which plays a very important role in game theory and nonlinear analysis see, for example, [ 6 ] and the references therein. In this section, we apply only the Nash equilibrium theorem to conclude the Brouwer fixed point theorem and related problems, without recourse to the inverse of the Berge maximum theorem. The KKM lemma is a very basic theorem, and the Brouwer fixed point theorem can be obtained by this lemma.

The proof can be found in [ 6 , 7 ]. We still derive the KKM lemma from the Nash equilibrium theorem. The variational inequality is an important tool in the study of optimization theory and game theory [ 6 ]; we also refer to early celebrated works [ 14 ] and [ 15 ].

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- Fixed point theorems with applications to economics and game theory by Kim C. Border - PDF Drive?

Here, we deduced the variational inequality by Nash equilibrium theorem directly. Nash equilibrium is a very important notion in the game theory. In general, the Nash equilibrium theorem can be derived from the Brouwer and Kakutani fixed point theorems. However, there is no proof for the Kakutani and Brouwer fixed point theorems via the Nash equilibrium theorem.

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.

## Fixed Point Theorems with Applications to Economics and Game Theory - Kim C. Border - Google книги

We next apply the Fan extension of the Nash equilibrium theorem to give an infinite-dimensional extension of the Brouwer fixed point theorem i. Tychonoff fixed point theorem Footnote 1. Cambridge University Press, Cambridge Wiley, New York Springer, New York Nash, JF: Equilibrium points in n -person games. Debreu, G: A social equilibrium existence theorem. Science Press, Beijing Dekker, New York Nonlinear Anal.

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Nonlinear Funct. Komiya, H: Inverse of the Berge maximum theorem. Theory 9 , Rationality and Equilibrium, vol. Springer, Berlin Browder, FE: The fixed point theory of multi-valued mappings in topological vector spaces.