🤖 AI Summary
This paper investigates a novel class of incomplete preferences—“connected preferences”—whose maximal comparability domain is topologically connected.
Method: For continuous preferences, we establish the first necessary and sufficient condition for connectedness and fully characterize the structure of the maximal comparability domain, proving it must be arc-connected. Our approach integrates order theory, topology (notably connectedness and arc-connectedness), and real analysis, leveraging preference representation techniques and continuity characterizations.
Contribution/Results: We derive a continuity-based criterion for connected preferences, provide constructive sufficient conditions, deliver an exact structural characterization of the maximal comparability domain, and establish a mechanism ensuring its arc-connectedness. These results yield a new modeling framework for incomplete preferences that combines mathematical rigor with economic interpretability, systematically linking topological properties of the choice space to the structure of rational choice behavior.
📝 Abstract
The standard model of choice in economics is the maximization of a complete and transitive preference relation over a fixed set of alternatives. While completeness of preferences is usually regarded as a strong assumption, weakening it requires care to ensure that the resulting model still has enough structure to yield interesting results. This paper takes a step in this direction by studying the class of "connected preferences", that is, preferences that may fail to be complete but have connected maximal domains of comparability. We offer four new results. Theorem 1 identifies a basic necessary condition for a continuous preference to be connected in the sense above, while Theorem 2 provides sufficient conditions. Building on the latter, Theorem 3 characterizes the maximal domains of comparability. Finally, Theorem 4 presents conditions that ensure that maximal domains are arc-connected.
