Generalizations of the Nash Equilibrium Theorem in the KKM Theory
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Research Article Generalizations of the Nash Equilibrium Theorem in the KKM Theory Sehie Park1, 2 1 2
The National Academy of Sciences, Seoul 137-044, Republic of Korea Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
Correspondence should be addressed to Sehie Park, [email protected] Received 5 December 2009; Accepted 2 February 2010 Academic Editor: Anthony To Ming Lau Copyright q 2010 Sehie Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. These results are compared with previously known cases for G-convex spaces. Consequently, our results unify and generalize most of previously known particular cases of the same nature. Finally, we add some detailed historical remarks on related topics.
1. Introduction In 1928, John von Neumann found his celebrated minimax theorem 1 and, in 1937, his intersection lemma 2, which was intended to establish easily his minimax theorem and his theorem on optimal balanced growth paths. In 1941, Kakutani 3 obtained a fixed point theorem for multimaps, from which von Neumann’s minimax theorem and intersection lemma were easily deduced. In 1950, John Nash 4, 5 established his celebrated equilibrium theorem by applying the Brouwer or the Kakutani fixed point theorem. In 1952, Fan 6 and Glicksberg 7 extended Kakutani’s theorem to locally convex Hausdorff topological vector spaces, and Fan generalized the von Neumann intersection lemma by applying his own fixed point theorem. In 1972, Himmelberg 8 obtained two generalizations of Fan’s fixed point theorem 6 and applied them to generalize the von Neumann minimax theorem by following Kakutani’s method in 3. In 1961, Ky Fan 9 obtained his KKM lemma and, in 1964 10, applied it to another intersection theorem for a finite family of sets having convex sections. This was applied in 1966 11 to a proof of the Nash equilibrium theorem. This is the origin of the application of the KKM theory to the Nash theorem. In 1969, Ma 12 extended Fan’s intersection theorem
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Fixed Point Theory and Applications
10 to infinite families and applied it to an analytic formulation of Fan type and to the Nash theorem for arbitrary families. Note that all of the above results are mainly concerned with convex subsets of topological vector spaces; see Granas 13. Later, many authors tried to generalize them to various types of abstract convex spaces. The present author also extended them in our previous works 14–28 in various dire
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