[Paper Review] Topological Connectedness and Behavioral Assumptions on Preferences: A Two-Way Relationship

M. Ali Khan, Metin Uyanık|RePEc: Research Papers in Economics|Oct 3, 2018

Economic theories and models4 citations

TL;DR

This paper establishes a two-way relationship between topological connectedness and the consistency/completeness of preferences in binary choice models. Under continuity and connectedness, transitivity and completeness become logically entailed, resolving long-standing questions in microeconomic theory through novel characterizations of k-connectedness and its implications for preference representation.

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.

Motivation & Objective

• To resolve foundational questions about whether preferences can be consistent without being complete, or complete without being consistent, under continuity.
• To investigate the logical interplay between topological connectedness and behavioral axioms (transitivity, completeness) in preference relations.
• To generalize and unify classical results by Eilenberg, Sonnenschein, Schmeidler, and Sen using topological tools.
• To provide a comprehensive framework for understanding when continuity and connectedness jointly enforce completeness and transitivity.
• To establish novel characterizations of k-connectedness and its implications for preference representation in economic models.

Proposed method

• Introduces and analyzes k-connectedness as a topological property of the choice set, generalizing connectedness and 2-connectedness.
• Applies quotient topology techniques to reduce complex preference relations to simpler, anti-symmetric forms on quotient spaces.
• Uses continuous dual-representation and section-closure properties to analyze the topological structure of preference relations.
• Employs semi-transitivity and pseudo-transitivity as intermediate conditions to bridge completeness and transitivity.
• Applies the Eilenberg-Sonnenschein research program to derive six novel theorems on the interdependence of continuity, connectedness, and preference axioms.
• Utilizes topological separation axioms and open/closed section properties to prove anti-symmetry and continuity of induced relations on quotient spaces.

Experimental results

Research questions

• RQ1 Can a continuous binary relation be transitive without being complete, or complete without being transitive, when the choice set is topologically connected?
• RQ2 What topological conditions ensure that completeness and continuity jointly imply transitivity, and vice versa?
• RQ3 How does k-connectedness (for k ≥ 1) generalize the role of connectedness in enforcing preference consistency and decisiveness?
• RQ4 In what way does the quotient space construction preserve continuity and anti-symmetry under equivalence relations induced by indifference?
• RQ5 What are the minimal topological and behavioral assumptions under which completeness and transitivity become logically entailed?

Key findings

• Under topological connectedness and continuity, completeness implies transitivity, generalizing Eilenberg (1941) and Sonnenschein (1965).
• Transitivity and non-triviality imply completeness, extending Schmeidler (1971) to connected topological spaces.
• The paper provides four new characterizations of k-connectedness and its extensions, which are novel in both economics and topology.
• It proves that a continuous, complete, and semi-transitive relation induces an anti-symmetric, continuous relation on the quotient space, preserving key preference properties.
• The paper shows that strong separability fails under certain constructions, demonstrating the necessity of connectedness for preference representation.
• The results establish that in a connected topological space, continuity and connectedness jointly eliminate the possibility of indecisive or inconsistent preferences.