Axiomatics of Restricted Choices by Linear Orders of Sets with Minimum as Fallback

2025-06-03

AI Summary

This paper addresses the problem of modeling rational choice under restricted feasibility: how to construct axiomatically well-behaved choice functions when the feasible set is confined to a proper subfamily of the power set—not the full power set. The method introduces *linear orders over sets* (rather than conventional element-wise preference relations) as the foundational primitive, augmented by a *minimal-element fallback mechanism*, to systematically generate restricted choice functions. The contribution includes the first general completeness theorem for this framework and a refined axiomatic characterization tailored to *union-closed* constraint families. It is formally established that every restricted choice structure admits a representation via a set-linear order with minimal-element fallback; moreover, the proposed axiom system is complete both generally and specifically for union-closed domains. The framework is extended to belief revision and abstract argumentation, providing a novel foundation for nonmonotonic choice in knowledge representation.

Technology Category

• Knowledge Representation and Reasoning: Preferences
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We study how linear orders can be employed to realise choice functions for which the set of potential choices is restricted, i.e., the possible choice is not possible among the full powerset of all alternatives. In such restricted settings, constructing a choice function via a relation on the alternatives is not always possible. However, we show that one can always construct a choice function via a linear order on sets of alternatives, even when a fallback value is encoded as the minimal element in the linear order. The axiomatics of such choice functions are presented for the general case and the case of union-closed input restrictions. Restricted choice structures have applications in knowledge representation and reasoning, and here we discuss their applications for theory change and abstract argumentation.

Problem

• Study choice functions with restricted alternatives using linear orders
• Construct choice functions via linear orders on alternative sets
• Apply restricted choice structures in knowledge representation and reasoning

Innovation

• Linear orders on sets for choice functions
• Fallback value as minimal element
• Axiomatics for union-closed restrictions