A Mathematical Theory of Ranking

TL;DR

This paper develops a mathematical framework for ranking systems based on pairwise comparisons, analyzing influence, factor interactions, and geometric properties to better understand and interpret ranking models.

Contribution

It introduces a pairwise margin-based analysis, proving uniqueness of influence sharing, and extends to nonlinear scoring with path-dependent decompositions and geometric diagnostics.

Findings

L1 local influence share is uniquely determined by pure factor refinement.

Global influence structure forms a convex potential gradient with a competition-graph Laplacian.

Factorwise path attribution is path-independent only in the additive case.

Abstract

Ranking systems produce ordered lists from scalar scores, yet the ranking itself depends only on pairwise comparisons. We develop a mathematical theory that takes this observation seriously, centering the analysis on pairwise margins rather than absolute scores. In the linear case, each pairwise margin decomposes exactly into factor-level contributions. We prove that the resulting L_1 local influence share is the unique budgeting rule consistent with pure factor refinement. Aggregating local shares yields a global influence structure: in log-absolute-weight coordinates, this structure is the gradient of a convex potential, its Jacobian is a competition-graph Laplacian, and Influence Exchange -- the reallocation of pairwise control across model states -- satisfies a finite energy identity with a zero-exchange rigidity law. For nonlinear scoring, the pairwise margin remains well-defined, but factor-level decomposition becomes path-dependent due to cross-factor interactions. We prove an interaction-curvature theorem: factorwise path attribution is path-independent if and only if the relevant mixed partial derivatives vanish, recovering full factorwise uniqueness exactly in the additive regime. The framework extends through local linearization and Pairwise Integrated Gradients. The geometric arc continues through permutation space, score-space hyperplane crossings, discrete exactness and triangle curl, Hodge-like diagnostics, and root-space/Weyl-chamber geometry -- organized as successive interpretive closures of the same pairwise-first analytical progression.