Localizing Preference Aggregation Conflicts: A Graph-Theoretic Approach Using Sheaves

TL;DR

This paper presents a novel graph-theoretic framework using sheaves to identify and localize specific conflicts in preference aggregation, preserving discrete preference structures and providing detailed insights into the sources of inconsistency.

Contribution

It introduces a sheaf-based approach that localizes preference conflicts, formalizes an Incompatibility Index, and models voter merging with a polynomial-time algorithm, advancing beyond traditional linear methods.

Findings

Identifies specific voter interactions causing aggregation failures.

Formalizes an Incompatibility Index to measure local conflicts.

Shows how graph quotients relate conflicts across scales.

Abstract

We introduce a graph-theoretic framework based on discrete sheaves to diagnose and localize inconsistencies in preference aggregation. Unlike traditional linearization methods (e.g., HodgeRank), this approach preserves the discrete structure of ordinal preferences, identifying which specific voter interactions cause aggregation failure -- information that global methods cannot provide -- via the Obstruction Locus. We formalize the Incompatibility Index to quantify these local conflicts and examine their behavior under stochastic variations using the Mallows model. Additionally, we develop a rigorous sheaf-theoretic pushforward operation to model voter merging, implemented via a polynomial-time constraint DAG algorithm. We demonstrate that graph quotients transform distributed edge conflicts into local impossibilities (empty stalks), providing a topological characterization of how aggregation paradoxes persist across scales.