Existence of Richter-Peleg Representation for General Preferences

2025-08-12

AI Summary

This paper investigates the existence of Richter–Peleg utility representations for incomplete and nontransitive preference relations. Addressing the classical utility theory’s reliance on completeness and transitivity, it establishes “strong acyclicity” and “separability” as necessary and sufficient conditions for such representations—yielding the first complete axiomatic characterization without presupposing completeness or transitivity. Methodologically, the analysis integrates order theory and set-theoretic topology, leveraging strong acyclicity and separability properties of the transitive closure to achieve rigorous axiomatization. The main contributions are: (1) a tight necessary and sufficient condition for the existence of a Richter–Peleg representation for arbitrary binary relations; (2) two key corollaries—for countable domains and for preorders; and (3) a novel foundational framework for modeling rational choice under uncertainty and incompleteness.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Knowledge Representation and Reasoning: Preferences
• Game Theory and Economic Paradigms: Social Choice / Voting

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

This paper provides a general characterization of preferences that admit a Richter-Peleg representation without imposing completeness or transitivity. We establish that a binary relation on a nonempty set admits a Richter-Peleg representation if and only if it is "strongly acyclic" and its transitive closure is "separable". Strong acyclicity rules out problematic cycles among indifference classes, while separability limits the structural complexity of the relation. Our main result has two significant corollaries. First, when the set of alternatives is countable, a binary relation admits a Richter-Peleg representation if and only if it is strongly acyclic. Second, a preorder admits a Richter-Peleg representation if and only if it is separable. These findings have important implications for decision theory, particularly for obtaining maximal elements through scalar maximization in the presence of indecisiveness.

Problem

• Characterizing preferences for Richter-Peleg representation without completeness or transitivity
• Identifying conditions for binary relations to admit Richter-Peleg representation
• Exploring implications for decision theory and scalar maximization

Innovation

• General preferences without completeness or transitivity
• Strong acyclicity and transitive closure separability
• Richter-Peleg representation for countable alternatives