Topological Connectedness and Behavioral Assumptions on Preferences: A Two-Way Relationship

Abstract

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 only stem from it. The six theorems are novel to both the economic and the mathematical literature: they generalize pioneering results of Eilenberg (1941), Sonnenschein (1965), Schmeidler (1971) and Sen (1969), and are relevant to several applied contexts, as well as to ongoing theoretical work.

arXiv comment47 pages, 4 figures

References in corpus (4)

• Modularity and community structure in networks
• Completeness and Transitivity of Preferences on Mixture Sets
• Groupoids, the Phragmen-Brouwer Property and the Jordan Curve Theorem
• Corrigendum to "Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve Theorem", J. Homotopy and Related Structures 1 (2006) 175-183

Cited by in corpus (4)

• The Yannelis-Prabhakar Theorem on Upper Semi-Continuous Selections in Paracompact Spaces: Extensions and Applications
• Completeness and Transitivity of Preferences on Mixture Sets
• On an Extension of a Theorem of Eilenberg and a Characterization of Topological Connectedness
• Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot