A topological characterization of the existence of non-empty choice sets (Q409719)

In several contexts arising in economic theory, it is quite typical to look for maximal elements with regard to some preference relation. In general, a preference relation could be understood as a special kind of binary relation (e.g.: a total preorder, an acyclic binary relation...) defined on a nonempty set of alternatives.NEWLINENEWLINESeveral theories have dealt with the mathematical problem of guaranteeing the existence of maximal elements with respect to different suitable kinds of preferences.NEWLINENEWLINEIn the present paper the authors consider various questions of topological nature when dealing with preferences that could have cycles.NEWLINENEWLINEThus, the framework considered is the theory of optimal choice sets, which is a solution theory with a long tradition in social choice and game theory. The authors characterize the existence of the most important solution theories of arbitrary binary relations over non-finite sets of alternative. To do so, they introduce a topological characterization of the so-called Smith sets and Schwartz sets.NEWLINENEWLINEAmong the tools used, the introduction of the notion of generalized transfer tc-upper semicontinuity which extends all the well-known continuity conditions is a key fact throughout the paper.NEWLINENEWLINEVarious well-known theorems on the existence of maximal elements of acyclic binary relations are generalized, by means of the new approach introduced.