Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization

Generalized Nash equilibrium problems

“…There are, however, a host of other problems that give rise to games with discontinuous objective functions; good entry points to the literature on the subject are (Baye et al 1993;Dasgupta and Maskin 1986a, b). There are several papers where the relaxation of continuity is pursued; the seminal one is the 1986 paper (Dasgupta and Maskin 1986a), further developments and applications are discussed in (Baye et al 1993;Dasgupta and Maskin 1986b;Reny 1999;Tian and Zhou 1995;Vives 1994) and references therein. However, with the (partial) exception of Baye et al (1993), where jointly convex GNEPs are discussed, all these papers deal only with pure NEPs.…”

Characterization of the existence of maximal elements of acyclic relations

“…Some authors have dealt with the following alternative expression for transfer lower continuity (e.g. Sonnenschein [12] and Tian and Zhou [14]):…”

Variable preference relations: Existence of maximal elements

“…To establish existence criteria for maximal elements of a variable preference relation we exploit the concept of order-completeness borrowed from vector optimization which can be interpreted as a hypothesis on aspiration points recently introduced in the theory of change (see Soubeyran 2009 [23]) or Brezis-Browder's inductivity hypothesis (Brezis and Browder 1976 [8]) when the preference relation is determined by a utility function. Similar to the case of multi-criteria optimization given in Luc [18] and recently in Flores-Bazan et al [11] the method of order-complete sets is very useful in unifying results on existence of maximal elements in various contexts, including (a) existence of maximal elements in social choice theory with non-transitive and incomplete preferences (Bergstrom 1975 [6] for acyclic relations, Tian and Zhou 1993 [25] for transfer upper continuous preferences, Zuanon 2009 [32] for weakly tc-upper semicontinuous acyclic relations, Alcantud 2002 [1] for upper continuous preference on " "-upper compact sets, Andrikopoulos and Zacharias 2009 [3] for consistent upper tc-S-semicontinuous preferences); (b) maximum ordering principles in partially ordered spaces without compactness (Altman 1982 [2], Turinici 1984 [28], Szaz 2007 [24], Zeidler 1986 [30], Zhu and Li 2007 [31] in the context of partial ordering); and (c) efficiency conditions in the framework of generalized multi-criteria optimization (Flores-Bazan et al 2008 [11]). …”

“…We say that a variable preference relation " ≤ x " is transfer upper-continuous if x < x y implies the existence of some y ∈ X and a neighborhood U (x) in X such that z < z y for all z ∈ U (x). The concept of transfer upper-continuity was introduced in Tian and Zhou 1995 [25] for a fixed binary relation on X which generalizes upper continuity of binary relations ( x < x y implies z < z y for all z in some neighborhood of x). The corollary below is a generalization of the main results of Bergstrom 1975 [6] and Tian and Zhou 1995 [25].…”