A Mathematical Theory of Ranking

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.

Let the Agent Steer: Closed-Loop Ranking Optimization via Influence Exchange

Recommendation ranking is fundamentally an influence allocation problem: a sorting formula distributes ranking influence among competing factors, and the business outcome depends on finding the optimal "exchange rates" among them. However, offline proxy metrics systematically misjudge how influence reallocation translates to online impact, with asymmetric bias across metrics that a single calibration factor cannot correct. We present Sortify, the first fully autonomous LLM-driven ranking optimization agent deployed in a large-scale production recommendation system. The agent reframes ranking optimization as continuous influence exchange, closing the full loop from diagnosis to parameter deployment without human intervention. It addresses structural problems through three mechanisms: (1) a dual-channel framework grounded in Savage's Subjective Expected Utility (SEU) that decouples offline-online transfer correction (Belief channel) from constraint penalty adjustment (Preference channel); (2) an LLM meta-controller operating on framework-level parameters rather than low-level search variables; (3) a persistent Memory DB with 7 relational tables for cross-round learning. Its core metric, Influence Share, provides a decomposable measure where all factor contributions sum to exactly 100%. Sortify has been deployed across two markets. In Country A, the agent pushed GMV from -3.6% to +9.2% within 7 rounds with peak orders reaching +12.5%. In Country B, a cold-start deployment achieved +4.15% GMV/UU and +3.58% Ads Revenue in a 7-day A/B test, leading to full production rollout.