Non-binary choice functions on non-compact sets

A KKM-result and an application for binary and non-binary 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 nonbinary 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-…nite sets. …”

“…As usual, we consider that D is a family of non-empty subsets of X that contains all of the …nite subsets of X (see Moulin, 1985;Nehring, 1996;LLinares and Sánchez, 1999). Since the result obtained by LLinares and Sánchez (1999) is more general than that of Nehring (1996), we shall focus on their result for the non-binary case. The assumptions imposed to the choice function are the following, Non-emptiness: For all A 2 F(X); C(A) 6 = ;:…”

“…Some of the motivations for considering this kind of choice function are the unresolvedness of preferences (see Nehring, 1997) or the use of non-classical mechanisms (see Aizerman and Malishevski, 1981). To our best knowledge, the only results that can be found in the literature that ensure the non-emptiness of non-binary choice functions are those presented by Nehring (1996), who analyzes this problem for a class of contraction-consistent choice functions, and that of LLinares and Sánchez (1999), who solve the same problem by using weaker conditions and by enlarging the family of sets on which the non-emptiness can be ensured.…”

Mixed choice structures, with applications to binary and non-binary optimization

“…It incorporates the common background for a number of apparently disconnected results on maximality, that are derived along the section. In Subsection 3.1 we deduce that Nehring's (1996) result and its generalization by LLinares and Sánchez (1999) are particular instances of that sufficient condition for non-empty choice in mixed choice structures. The same holds for Alcantud's (2002b) model, as is shown in Subsection 3.2.…”

“…However, they seem to enable us to ensure maximization on classes wider than that of compact sets (cf. (LLinares and Sánchez, 1999), where no example is provided). The literature on this topic is virtually limited to this common approach and the aforementioned Alcantud (2002b).…”