Topological connectedness and behavioral assumptions on preferences: a two-way relationship

This paper offers a comprehensive treatment of the question as to whether a binary relation can be consistent (transitive) without being decisive (complete), or decisive without being consistent, or simultaneously inconsistent or indecisive, in the presence of a continuity hypothesis that is, in principle, non-testable. It identifies topological connectedness of the (choice) set over which the continuous binary relation is defined as being crucial to this question. Referring to the two-way relationship as the Eilenberg-Sonnenschein (ES) research program, it presents four synthetic, and complete, characterizations of connectedness and its natural extensions and two consequences that stem from it. The six theorems make connections that have eluded decision theory and thereby generalize the pioneering work of Eilenberg, Sonnenschein, Schmeidler and Sen. We draw the relevance to several applied contexts, as well as to ongoing theoretical work.