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

TL;DR

<3-5 sentence high-level summary> The paper develops a discrete order-sheaf framework to localize inconsistencies in ordinal preference aggregation, introducing the Obstruction Locus and Incompatibility Index to pinpoint edge-level conflicts. It preserves the discrete structure of preferences and provides a polynomial-time pushforward (π*) under graph quotients via a constraint-DAG approach, with Mallows distributions used to study stochastic variations. Key findings show that edge-level obstructions can be localized and traced across scales, and that coarse-graining can yield empty stalks, revealing fundamental limits to aggregation under merging. This framework offers a precise, scalable diagnostic tool for social choice and judgment aggregation that complements and extends global linearization methods like HodgeRank.</paper_summary>

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.

Figures (4)

• Figure 1: Comparison between the Discrete Obstruction Locus and Linearized Cohomology for a Condorcet cycle. The discrete approach identifies all three edges as incompatible ($|\Omega_1|=3$), pinpointing where coalition formation fails. Linearized cohomology yields only the cycle rank ($\dim H^1_{\rm lin}=1$), which would be identical for any cyclic graph regardless of which edges carry conflicts.
• Figure 2: Global consistency rates and average Incompatibility Index for random preference profiles on different graph topologies. Consistency rates match theoretical expectations (e.g., $\sim 2.8\%$ for $K_3$ with uniform random preferences).
• Figure 3: Stochastic interpolation via Mallows model. (Top left) Mean $|\Omega_1|$ with standard deviation band shows smooth emergent transition. (Top right) Consistency rate increases monotonically. (Bottom) Illustrative deterministic family showing $|\Omega_1|$ transitions.
• Figure 4: Rigorous Pushforward under Quotient. Merging conflicting voters V1 and V2 results in an empty stalk at V12. The obstruction persists as a local impossibility.

Theorems & Definitions (16)

• Definition 3.1: Discrete Order Presheaf
• Lemma 3.1: Sheaf Property
• proof
• Definition 3.2: Preference Profile as a 0-Cochain
• Definition 3.3: Obstruction Locus $\Omega_1$ and Incompatibility Index
• Remark 3.1: What $|\Omega_1|$ Measures
• Proposition 3.1: Global Consistency
• Remark 3.2
• Definition 3.4: Rigorous Pushforward Sheaf $\pi_* F^\sigma$
• Remark 3.3: DAG as Intersection of Constraints