Browder type fixed point theorems and Nash equilibria in generalized games
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Journal of Fixed Point Theory and Applications
Browder type fixed point theorems and Nash equilibria in generalized games Jiuqiang Liu, Mingyu Wang and Yi Yuan Abstract. In this paper, we present two generalizations of the wellknown Browder fixed point theorem, one of which is equivalent to the well-known Fan–Knaster–Kuratowski–Mazurkiewicz theorem. As applications, we apply these fixed point theorems to derive existence theorems for Nash equilibria in generalized games which generalize some existing existence theorems in the literature, including the well-known equilibrium existence theorem by Arrow and Debreu (Econometrica 22:265– 290, 1954) and the existence theorem by Cubiotti (Int J Game Theory 26:267–273, 1997). JEL Classification. D51, C72. Keywords. Browder fixed point theorem, Fan–Knaster–Kuratowski– Mazurkiewicz theorem, generalized games, Nash equilibrium.
1. Introduction Fixed point theory is of fundamental importance in mathematics and many other fields. Many problems in topology, nonlinear analysis, mathematical economics, and game theory give rise to fixed point problems for some functions or set-valued mappings. For example, in economics, a Nash equilibrium of a game is a fixed point of the game’s best response correspondence. Because of the importance of fixed point theory, there are many different forms of fixed point theorems appeared in the literature. Among hundreds of fixed-point theorems, Brouwer’s is particularly well known due to its use across numerous fields of mathematics as well as in economics. The fixed point theory of set-valued mappings began in 1937 when von Neumann [21] proved his famous minimax theorem. Then Kakutani established [16] a fixed point theorem for an upper semicontinuous set-valued mapping with nonempty, compact, and convex values defined on a compact and convex subset of a Euclidean space. Kakutani’s theorem was extended by Glicksberg [13] to locally convex topological vector spaces. This Kakutani–Glicksberg fixed point theorem is a powerful tool in showing the existence of solutions for many problems in pure and 0123456789().: V,-vol
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applied mathematics, and in mathematical economics. As for lattice structures, Tarski [24] proved a lattice-theoretical fixed point theorem, and later, Fujimoto [12] extended Tarski’s fixed point theorem to partially ordered sets. Recently, Li and Tammer [18] derived an order-clustered fixed point theorem which includes both of them. Another well-known fixed point theorem by Browder [3] is built on any Hausdorff topological vector spaces. In 1987, Tarafdar [23] provided a generalization of the Browder fixed point theorem which is equivalent to the wellknown Fan–Knaster–Kuratowski–Mazurkiewicz (FKKM) theorem proved by Fan [11]. The FKKM theorem can be used to prove many theorems such as fixed point theorems, coincidence theorems, and minimax inequalities. In this paper, we present two generalizations of the well-known Browder fixed point theorem built on any Hausdorff topological vector spaces, one of which is equivalent to
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