Minimax inequality equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz Theorem

The purpose of this note is to give further generalizations of the Ky Fan minimax inequality by relaxing the compactness and convexity of sets and the quasi-concavity of the functional and to show that our minimax inequalities are equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem and a modified FKKM theorem given in this note

A KKM-RESULT AND AN APPLICATION FOR BINARY AND NON-BINARY CHOICE FUNCTIONS

By generalizing the classical Knaster-Kuratowski-Mazurkiewicz Theorem, we obtain a result that provides sufficient conditions to ensure the non-emptiness of several kinds of choice functions. This result generalizes well-known results on the existence of maximal elements for binary relations (Bergstrom, 1975; Walker, 1977; Tian, 1993), on the non-emptiness of non-binary choice functions (Nehring, 1996; Llinares and Sánchez, 1999) and on the non-emptiness of some classical solutions for tournaments (top cycle and uncovered set) on non-finite sets.Binary Choice Function; Non-Binary Choice Function

Coincidence theorems involving composites of acyclic mappings in contractible spaces

AbstractSome coincidence theorems involving a new class of set-valued mappings containing compact composites of acyclic mappings defined on a contractible space is proved

General Conditions for Existence of Maximal Elements via the Uncovered Set

This paper disentangles the topological assumptions of classical results (e.g., Walker (1977)) on existence of maximal elements from rationality conditions. It is known from the social choice literature that under the standard topological conditions-with no other restrictions on preferences-there is an element such that the upper section of strict preference at that element is minimal in terms of set inclusion, i.e., the uncovered set is non-empty. Adding a condition that weakens known acyclicity and convexity assumptions, each such uncovered alternative is indeed maximal. A corollary is a result that weakens the semi-convexity condition of Yannelis and Prabhakar (1983).