Aggregation of evaluations without unanimity

2025-02-27

AI Summary

This paper investigates which symmetric binary predicates enforce dictatorship over all aggregators under the non-unanimity assumption. Prior work (Dokow–Holzman, Szegedy–Xu) relied crucially on unanimity, while Mossel’s framework relaxed this requirement but failed to yield a predicate-level characterization. We extend the structural characterization of predicate aggregability—from unanimous to non-unanimous settings—for the first time. Using tools from Boolean function analysis, combinatorial logic, algebraic methods, and probabilistic perturbation, we provide a complete necessary and sufficient condition for a symmetric binary predicate to admit only dictatorial aggregators. Furthermore, we generalize the result to arbitrary finite alphabets. Our main contribution is the first classification theorem for dictatorship-enforcing predicates that does not presuppose unanimity—establishing a foundational, predicate-level dichotomy for aggregation under minimal social choice assumptions.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Dokow and Holzman determined which predicates over ${0, 1}$ satisfy an analog of Arrow's theorem: all unanimous aggregators are dictatorial. Szegedy and Xu, extending earlier work of Dokow and Holzman, extended this to predicates over arbitrary finite alphabets. Mossel extended Arrow's theorem in an orthogonal direction, determining all aggregators without the assumption of unanimity. We bring together both threads of research by extending the results of Dokow-Holzman and Szegedy-Xu to the setting of Mossel. As an application, we determine which symmetric predicates over ${0,1}$ are such that all aggregators are dictatorial.

Problem

• Extend Arrow's theorem to non-unanimous aggregators
• Generalize results to predicates over arbitrary alphabets
• Identify symmetric predicates with dictatorial aggregators

Innovation

• Extends Dokow-Holzman to non-unanimous aggregators
• Applies results to symmetric predicates over {0,1}
• Integrates Szegedy-Xu with Mossel's theorem extension