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

We introduce a graph-theoretic framework based on discrete sheaves to diagnose and localize inconsistencies in preference aggregation and, more broadly, in the fusion of partial rankings supplied by many overlapping sources.

Unlike linearization methods such as HodgeRank, which embed comparisons into a numerical flow, this approach stays purely ordinal and locates conflict in the interaction structure via the Obstruction Locus, identifying which voter pairs fail to cohere.

We formalize the Incompatibility Index to quantify these local conflicts and examine their behavior under stochastic variations using the Mallows model.

We further develop a sheaf-theoretic pushforward operation to model voter merging, implemented via a polynomial-time constraint digraph algorithm.

We demonstrate that graph quotients transform distributed edge conflicts into local impossibilities (empty stalks), showing topologically how aggregation paradoxes can persist across scales.